INFLUENCE OF ELECTRON-ELECTRON COLLISIONS ON ELECTRON ENERGY
Transcript of INFLUENCE OF ELECTRON-ELECTRON COLLISIONS ON ELECTRON ENERGY
INFLUENCE OF ELECTRON-ELECTRON COLLISIONS ON ELECTRON ENERGY DISTRIBUTIONS AND
TRANSPORT IN INDUCTIVELY COUPLED PLASMAS*
Alex V. Vasenkov and Mark J. KushnerUniversity of Illinois
Department of Electrical and Computer EngineeringUrbana, IL 61801
E-mail: [email protected]@uiuc.edu
URL: http://uigelz.ece.uiuc.edu
May 2002
*Work supported by CFD Research Corp., NSF,Applied Materials, Inc. and the SRC
AGENDA
• Modeling of non-local electron kinetics in ICPs
• Description of the model
• Validation of the model for ICPs in Ar
• Stochastic heating in ICPs
• Modeling of electron transport for ICPs in CF4
• Summary
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ELECTRON KINETICS IN ICPs• The trend towards using high-plasma density (>1011cm-3), low gas
pressure (< 10s mTorr) sources in materials processing has resulted in renewed interest in electron kinetics in ICPs.
• Owing to the electron energy distribution (EED) is generally non-Maxwellian, a kinetic approach is required for modeling ICPs.
PUMP PORT
DOME
GAS INJECTORS
BULK PLASMA
WAFER
30 30 0 RADIUS (cm)
HEI
GH
T (c
m)
0
26
sCOILS
s
rf BIASED SUBSTRATE
SOLENOID
POWER SUPPLY
POWER SUPPLY
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ELECTRON KINETICS IN ICPs
• One of the basic problems in the kinetic simulation of ICPs is the consideration of e-e collisions, which significantly influence the thermal motion of electrons.
• There are at least three approaches to resolve e-e collisions:
(i) Particle-in-cell simulations (Nanbu et al, 1997, 2000)
(ii) Direct solution of Boltzmann’s equation (Kolobov et al, 1997)
(iii) Electron Monte Carlo simulation (Weng et al, 1990)
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HYBRID PLASMA EQUIPMENT MODEL
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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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.
• An energy of collision target is randomly selected from the EED at this location
= ∑i
iiiitrtrtrtr ∆t∆εε)f(ε∆t/∆εε)f(ε)P(ε
• Since, the number of energy bins is usually large (400-500), quick lookup techniques are used for the energy of collision targets.
• In this technique the lth cumulative probability is determined from
∑∑≤≤−
=i
iε'εε'
jl )P(ε)/P(εΠljl 1
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ALGORITHM FOR E-E COLLISIONS
• The energy of the collision partner is first randomly selected on a coarse basis, followed by a refinement within the large energy intervals.
• Assuming an isotropic velocity for the target, the probability for an e-e collision is determined using the relative speed between the electron and its target,
( ) ( ) trpreeeee vvg∆t,ggσrnP rrr−=⋅⋅⋅=
( ) 22
1
22 4)(
)(ln1)(4
gmπεqgb,
gbλgπbgσ
e
oo
o
Doee =
+=
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ALGORITHM FOR E-E COLLISIONS
)πr(r| g|'g
)πr(r|g|', g|rg|'g
y
xz
22
1
22
11
2sin1
2cos1
−=
−=±=r
rr
• If a collision occurs, then a post collision relative velocity is randomly determined as,
• Finally, the velocity of the pseudo-electron is updated with,
)vv(.v ' g.vv prtrRRfrrrrrr
+=+= 5050
• The change in velocity of the collision partner is disregarded; e-e collisions are treated as collisions between pseudo-electrons and energy resolved electron fluid.
• The consequences of e-e collisions on the targets are obtained by continuously updating the stored EEDs.
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ICP CELL FOR VALIDATION AND INVESTIGATION
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• Experiments by Godyak et al (1998) are used for validation.
• The experimental cell is an ICP reactor with a Faraday shield to minimize capacitive coupling.
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ELECTRON DENSITY FOR STANDARD CASE
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)3cm10(10 [e] −
• Electron density is largest in the middle of reactor where the electric potential is maximum.
• Ar, 10 mTorr, 100 W, 6.78 MHz
IONIZATION SOURCE FOR STANDARD CASE
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)1s3cm14(10 iS −−
• The ionization source reaches a maximum at the edge of the classical skin depth where the amplitude of the electromagnetic field decays to 1/e of its edge value.
• Ar, 10 mTorr, 100 W, 6.78 MHz
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Te FOR STANDARD CASE: Ar, 10 mTorr, 100 W, 6.78 MHz
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• The first major peak occurs in the skin depth owing to collisionless electron heating by the large electric field.
• The second minor peak corresponds to the position where the electron density reaches the maximum.
(eV) eT
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• The ion temperature reaches a maximum near the walls where ions gain energy due to acceleration in the presheath.
• In the middle of reactor the ion temperature is lower owing to thermalization with neutral species.
Ti FOR STANDARD CASE: Ar, 10 mTorr, 100 W, 6.78 MHz
(eV) iT
• The maximum of electron density is displaced from the center due to the axial location of the source function.
• The high thermal conductivity and redistribution of energy by e-e collisions produce nearly uniform electron temperatures.
100
101
102
2
3
4
5
6
7
0 2 4 6 8 10
n e (1010
cm-3
)
Height (cm)
Te (eV)
ne
Te
symbols, Exp., Godyak (1998)curves, Model
Te and ne ALONG THE CENTERLINE OF THE REACTOR
• Ar, 10 mTorr, 100 W, 6.78 MHz, r=0
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EFFECT OF E-E COLLISIONS ON THE EEDs
• Low energy electrons want to “pool” at the peak in plasma potential in the center of the reactor.
• When including e-e collisions, the fraction of low-energy electrons is smaller due to energy transfer between low-energy and high-energy electrons.
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10-2
10-1
100
101
without e-e collisions
with e-e collisions
EED
(eV-3
/2)
r=1 cmz=3 cm
10-2
10-1
100
0 2 4 6 8 10Energy (eV)
EED
(eV-3
/2) without e-e collisions
with e-e collisions
r=4 cmz=3 cm
• Ar, 10 mTorr, 100 W, 6.78 MHz
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EEDs ALONG THE CENTERLINE OF THE REACTOR
• The EEDs show a bi-Maxwellian distribution, which is typical for low-pressure inductively coupled plasmas.
• The EED in the middle of the reactor has the largest energy tail
10-4
10-3
10-2
10-1
0 5 10 15 20
Godyak (1998), z=5.0 cmModel, z=0.5 cmModel, z=5.0 cmModel, z=10.0 cm
Energy (eV)
EED
(eV-3
/2)
200 W, r=0.0 cm
• Ar, 10 mTorr, 6.78 MHz
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EEDs ALONG THE CENTERLINE OF THE REACTOR
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• In the skin layer the EED is depleted at low energies.
• EEDs near the peak plasma potential (z = 5.0 cm) have larger a low energy component.
10-4
10-3
10-2
10-1
Godyak (1998), z=5.0 cmModel, z=0.5 cmModel, z=5.0 cmModel, z=10.0 cm
0 5 10 15 20
EED
(eV-3
/2)
50 W, r=0.0 cm
Energy (eV)
• Ar, 10 mTorr, 6.78 MHz
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EEDs IN SKIN LAYER AND BULK PLASMA
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EFFECT OF PRESSURE ON THE EEDs
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10-6
10-5
10-4
10-3
10-2
10-1
z=0.5 cmz=5.0 cm
0 5 10 15 20 25 30
EED
(eV-3
/2)
1 mTorr
10-6
10-5
10-4
10-3
10-2
10-1
z=0.5 cmz=5.0 cmz=10.0 cm
0 5 10 15 20 25 30
EED
(eV-3
/2)
30 mTorr
Energy (eV)
• At 1 mTorr the EEDs are dominated by collisionless heating and by non-linearLorentz forces, which provide an axial acceleration.
• The EEDs at 30 mTorr are in a collisional regime and therefore they resemble aDruyvesteyn distribution.
• Ar, 100 W, 6.78 MHz
z=10.0 cm
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EXPERIMENTAL CELL FOR INVESTIGATION OF ICPs IN CF4
• Modeling of ICPs in CF4 involves complex surface and gas-phase chemistry.
• It is particularly challenging due to only limited data is available for electron–CFx and CFx-CFx radical collisions.
• Experiments by Singh, Coburn, and Graves [J. Vac. Sci. Technol. A v. 19, 718 (2001)] are used for a comparison.
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Te, ne for CF4, 10 mTorr, 150 W, 13.56 MHz)cm(10 [e] 310 −
• The region of largest electron density intersects with the skin layer.
• Electron temperature in CF4shows similar features to that in Ar, but it is higher in magnitude due to the dissociation.
(eV)eT
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EXPERIMENTAL AND COMPUTED EEDs
10-3
10-2
10-1
0 3 6 9 12 15
Exp., Singh et al (2001) Model
EED
(eV-3
/2)
Energy (eV)
n e= 5 1010 cm-3 n
e= 7 1010 cm-3
• Model yields the same slope as experiment.
• A peak in the computed EED occurs at about 3 eV.
• CF4, 10 mTorr, 150 W, 13.56 MHz, r=0, z=7.6 cm
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• EED is well represented by a Maxwellian distribution in the sheath
• In the bulk plasma the position of the peak shifts to higher energies approaching the skin layer.
• EED in the skin layer shows usual bi-Maxwellian features.
10-2
10-1
0 2 4 6 8 10
Model, z=10 cmModel, z=9 cmModel, z=7 cmModel, z=5 cm
EED
(eV-3
/2)
Energy (eV)
r=0.5 cm
COMPUTED EDDs IN CF4 AT DIFFERENT HEIGHTS
• CF4, 10 mTorr, 150 W, 13.56 MHz
Model, z=1 cm
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10-2
10-1
Model, r=0 cmModel, r=1 cmModel, r=2 cmModel, r=5 cmModel, r=9 cm
0 2 4 6 8 10
z=7.6 cm
EED
(eV-3
/2)
Energy (eV)
• CF4 , 10 mTorr, 150 W, 13.56 MHz
• A peak occurs in the EEDs for radii < 2 cm.
• This peak is formed due to the flux of electrons which collisionlessly propagates from the coils to the opposite wall.
• This flux is formed as a result of interaction between the skin layer and the region of largest electron density.
EDDs IN CF4 AT DIFFERENT RADII
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EXPERIMENTAL AND COMPUTED EEDs
• The energy exchange between electrons is eliminated by excluding e-e collisions.
• RF fields do not propagate beyond the normal skin layer when the cold plasma approximation is used.
• In both these cases the thermal electron motion is significantly affected.
• CF4, 3 mTorr, 150 W, 13.56 MHz
10-3
10-2
10-1
0 3 6 9 12 15
Exp., Singh et al (2001)ModelModel, without e-e collisionsModel, cold plasma
EED
(eV-3
/2)
Energy (eV)
Exp., n e
= 5 1010 cm-3
Model, n e
= 7 1010 cm-3
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SUMMARY
• We present a new method for modeling e-e collisions in which Coulomb collisions are treated using a test particle approach and, so can be readily included in Monte Carlo simulations of low-pressure and high-density plasma systems.
• The method was validated by comparing calculated and measured electron densities, temperatures, and electron distributions in inductively coupled argon plasma.
• The EEDs are significantly depleted at low energies in regimes dominated by non-collisional heating.
• A maximum in the computed EEDs in CF4 depends on the thermal motion of electrons.
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