MODELING OF ELECTRON KINETICS IN MAGNETICALLY …

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.


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


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.



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



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.



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


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


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.


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.


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 .


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.



• 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


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.


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&&&


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


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


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MODELING OF ELECTRON KINETICS IN MAGNETICALLY …