modeling of electron kinetics in magnetically …
TRANSCRIPT
MODELING OF ELECTRON KINETICS IN MAGNETICALLY ENHANCED INDUCTIVELY COUPLED PLASMAS1
Alex V. Vasenkov2, Gottlieb S. Oehrlein3 and Mark J. Kushner2
2University of IllinoisDepartment of Electrical and Computer Engineering
Urbana, IL 61801
3University of Maryland, Materials Science and Engineering & Institute for Research in Electronics and Applied Physics
College Park, MD 20742-2115
E-mail: [email protected]@[email protected]
October 2002
1Work supported by CFD Research Corp., NSF,Applied Materials, Inc. and the SRC
AGENDA
• Introduction
• Description of the model
• Ar/C4F8 reaction mechanism
• Validation of the chemistry mechanism for ICPs in Ar and C4F8
• Electron kinetics in magnetically enhanced ICPs
• The angular anisotropy of the EEDs in ICPs
• Summary
University of IllinoisOptical and Discharge PhysicsGEC-2002-02
MAGNETICALLY ENHANCED ICPs
• Studies of inductively coupled plasmas (ICPs) have focused on collisionless electron heating when the skin layer is anomalous.
• To obtain higher ionization efficiencies magnetically enhanced ICP (MEICP) sources have been being proposed to replace conventional ICP sources.
• The mechanisms for these high efficiencies are not well understood.
• This presentation reports on the computational study of non-collisional electron heating of MEICPs in C4F8/Ar are used in the semiconductor industry for selective plasma etching.
University of IllinoisOptical and Discharge PhysicsGEC-2002-03
ANGULAR DEPENDENCE OF EEDs
University of IllinoisOptical and Discharge Physics
• A traditional two-term Legendre expansion of the electron energy distributions (EEDs) in the direct solution of the Boltzmann equation is often used for modeling gas discharges.
• Is this approach valid for modeling ICP discharges when the skin layer is anomalous?
• A new technique, which involves the calculation of the Legendre expansion coefficients for the angular dependent EEDs was incorporated into the Hybrid Plasma Equipment Model (HPEM).
• The HPEM is a two-dimensional, plasma equipment model which consists of an electromagnetic module, an electron Monte Carlo simulation with e-e collisions and a fluid kinetics module.
GEC-2002-04
ANGULAR DEPENDENCE OF EEDs (CONT.)
,))(())(( 21
21 ∑ −∆±−∆±=∆Φ ++
mjkmkmjiijikj rrrw rrrδεεεδ
• As electron trajectories are integrated, the coefficients for the lthexpansion term are obtained from,
( ) ,)())((2
12, 21∑ ∑
−∆±
+∆Φ=
j nnljnnikjjkil PlwrA µµµµδε r
• where µ=cos(θ) and Pl is the Legendre polynomial,
• The angular dependences of the EEDs are reconstructed from:
( ) ( ) ,)(,,, µεµε ll
kilki PrArf ∑=rr
University of IllinoisOptical and Discharge PhysicsGEC-2002-05
Ar/C4F8 REACTION MECHANISM
• The limited electron impact cross-section data for the fluorocarbon species were collected and synthesized.
• Rate coefficients for gas phase chemistry were taken from independent studies in the literature or estimated from measurements for related species.
• The final mechanism involves 38 species and over 200 reactions.
• The mechanism was validated by comparing to measured ion saturation currents obtained with probes.
University of IllinoisOptical and Discharge PhysicsGEC-2002-06
University of IllinoisOptical and Discharge Physics
ICP CELL FOR VALIDATION AND INVESTIGATION
• An ICP reactor patterned after Oeherlein, et al. was used for validation.
• Reactor uses a metal ring with magnets to confine plasma.
GEC-2002-07
ELECTRON DENSITY FOR BASE CASE
• Electron density is largest in the middle of reactor where the electric potential is maximum.
• The static magnetic fields produce confinement in the periphery, flattening the plasma potential and [e].
• C4F8, 10 mTorr, 1.4 kW, 13.56 MHzUniversity of Illinois
Optical and Discharge PhysicsGEC-2002-08
CF2+ DENSITY FOR BASE CASE
• CF2+ is one of the dominant
ions in C4F8 plasma due to large dissociation.
• The major path for the CF2+ is:
• C4F8 + e → C2F4 + C2F4 + e
• C2F4 + e → CF2 + CF2 + e
• CF2 + e → CF2+ + e
• C4F8, 10 mTorr, 1.4 kW, 13.56 MHzUniversity of Illinois
Optical and Discharge PhysicsGEC-2002-09
ELECTRON TEMPERATURE FOR BASE CASE
• The peak in electron temperature occurs in the skin layer due to the collisionless electron heating by the large electric field.
• The electron temperature is rather uniform over the radius in the bulk plasma where electrons experience a large number of e-e collisions.
• C4F8, 10 mTorr, 1.4 kW, 13.56 MHzUniversity of Illinois
Optical and Discharge PhysicsGEC-2002-10
THE THEORY OF CYLINDRICAL PROBE• Ion saturation current in the case of low-pressure measurements is
where q = chargeni, vi = ion density, ion thermal velocityA = probe area,Vp = probe potentialη = qVp/kTi, Ti, Te = ion, electron temperaturesrs, rp = sheath thickness, probe radius.
• Ip is larger than the Langmuir current by due to the existence of the pre-sheath.
iii
ep qvAn
eTTI π2
41
∆=
−++−
−−+
+
−+−+
=∆2/1
22
22/1
22 )()(
1)exp(1/)( pps
ps
ps
p
ppsp
ps
rrrrr
erfrrr
rrrerf
rrr η
ηη
2/1)]/(2[ ie eTTπ
University of IllinoisOptical and Discharge PhysicsGEC-2002-11
0
1
2
3
4
5
6
-120 -100 -80 -60 -40 -20 0
Exp.Calc.Exp.Calc.Exp.Calc.
Vp (V)
I (m
A)
1 kW
600 W
400 W
IP VERSUS PROBE BIAS FOR ICP WITH MAGNETS in Ar
• Both the model and the experiments showed significant variations in ion saturation current with the probe collecting voltage.
• Ar, 10 mTorr, 13.56 MHz
University of IllinoisOptical and Discharge PhysicsGEC-2002-12
IP VERSUS POWER for ICP in C4F8
0
1
2
3
4
5
6
7
8
400 600 800 1000 1200 1400
Exp.Calc.Exp.Calc.
Power (W)
I (m
A)
without magnets
with magnets• The ion saturation currents are larger if a static magnetic field is used to confine the plasma.
• C4F8, 10 mTorr, 13.56 MHz, 100 V probe bias.
University of IllinoisOptical and Discharge PhysicsGEC-2002-13
POSITIVE AND NEGATIVE POWER DEPOSITION
• Ar/C4F8=20/80, 3 mTorr, 13.56 MHz, 400 W.
• The power deposition alternates between positive and negative values resulting from noncollisional transport of electrons.
• The effect of static magnetic fields on the thermal electron motion produce considerable negative power deposition.
University of IllinoisOptical and Discharge PhysicsGEC-2002-14
EEDs FOR ICP IN Ar/C4F8
r=4 cm
Coils
• The EEDs have long energy tails in the radial center of the skin layer due to stochastic heating.
• Low energy electrons “pool” at the peak in plasma potential in the center of the reactor.
• Ar/C4F8=20/80, 3 mTorr, 13.56 MHz, 400 W. University of IllinoisOptical and Discharge PhysicsGEC-2002-15
EEDs FOR ICP IN Ar/C4F8
r=1 cm
Coils
• The EEDs in the skin layer are only slightly enhanced at high energies due to the electric field being zero at r=0.
• The low-energy pooling is still evident.
• Ar/C4F8=20/80, 3 mTorr, 13.56 MHz, 400 W. University of IllinoisOptical and Discharge PhysicsGEC-2002-16
• The angular distribution of low-energy electrons is nearly isotropic and shifted to +vz due to the energy pooling.
• The anisotropy of the EEDs increases with energy owing to rf Lorentz force .
University of IllinoisOptical and Discharge Physics
ANGULAR DISTRIBUTION FOR ICP in Ar
• EEDs at r=4 cm, z=3 cm.GEC-2002-17
rfBvLrr
×~
University of IllinoisOptical and Discharge Physics• Ar, 1 mTorr, 3.39 MHz, 400 W.
GEC-2002-18
ANGULAR DISTRIBUTIONS FOR ICP in Ar
• The anisotropy of the EEDs produced by rf electric field is mostly visible at low electron energies.
ANGULAR DISTRIBUTIONS FOR ICP in Ar
University of IllinoisOptical and Discharge Physics
• EEDs at r=4 cm, z=3 cm.GEC-2002-19
University of IllinoisOptical and Discharge Physics• Ar, 1 mTorr, 3.39 MHz, 400 W.
GEC-2002-20
ANGULAR DISTRIBUTIONS FOR ICP in Ar
SUMMARY AND ACKNOWLEDGMENT• A new reaction mechanism for Ar/C4F8 was developed and validated
against measured ion saturation currents for ICPs.
• Static magnetic fields effectively confine the plasma and significantly increase the density of electrons and ions in the discharge.
• A significant region of negative power deposition result from the effect of static magnetic fields on thermal electron motion.
• The angular anisotropy of the EEDs is considerable when the skin layer is anomalous.
• Several expansion terms should be taken into account to obtain correct EEDs at large energies.
Acknowledgement is made to Dr. V. Kolobov for insight discussion of angular anisotropy of EEDs.
University of IllinoisOptical and Discharge PhysicsGEC-2002-21
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
University of IllinoisOptical and Discharge PhysicsGEC-2002-add1
ELECTROMAGNETICS MODEL
• 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
eo
m
zzrzr
zrrz
zrrzrm
o
mnq
mqiEj
BBBBBBBBBBBBBBBBBBBBB
Bqm
νσνωασ
ααααααααα
αανσσ
θθ
θθθ
θθ
2
22
22
22
22
,/
1
=+
=⋅=
++−+−+++−+−++
+
=
r&&&
v
r&&&
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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.
( )trf ,, rε
• Continuum (2D,3D):
( ) ( ) ( ) EBeeeeeee STTkTTLTStkTn +
∇⋅−Φ⋅∇−−=∂
∂ κ
25/
23
University of IllinoisOptical and Discharge PhysicsGEC-2002-add3
PLASMA CHEMISTRY, TRANSPORT AND ELECTROSTATICS
iiii SNtN
+⋅−∇= )v( r
∂∂
( ) ( ) ( ) ( ) iii
iiiiiii
i
ii BvEmNqvvNTkN
mtvN µ
∂∂
⋅∇−×++⋅∇−∇=rrrrr
r 1
( ) ijjijj
imm
j vvNNm
ji
νrr−− ∑
+
( ) 222
2
)()U(UQ Em
qNNPtN
ii
iiiiiiiii
ii
ωννε
∂ε∂
+=⋅∇+⋅∇+⋅∇+
∑∑ ±−+
++j
jBijjij
ijBijjiji
ijs
ii
ii TkRNNTTkRNNmm
mE
mqN 3)(32
2
ν
• Continuity, momentum and energy equations are solved for each species (with jump conditions at boundaries) (2D,3D).
• Implicit solution of Poisson’s equation (2D,3D):
( ) ( )
⋅∇⋅∆+=∆+Φ∇⋅∇ ∑∑
iiqt-- i
iiis Nqtt φρε
r
University of IllinoisOptical and Discharge PhysicsGEC-2002-add4