numerical investigation of wave effects in high-frequency capacitively coupled plasmas*

NUMERICAL INVESTIGATION OF WAVE EFFECTS IN HIGH-FREQUENCY

CAPACITIVELY COUPLED PLASMAS*

Yang Yang and Mark J. Kushner

Department of Electrical and Computer Engineering Iowa State University, Ames, IA 50011

[email protected] [email protected]://uigelz.ece.iastate.edu

October 2007

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* Work supported by Semiconductor Research Corp., Applied Materials and NSF.

Iowa State UniversityOptical and Discharge Physics

AGENDA

Wave effects in hf capacitively coupled plasma (hf-CCP) sources

Description of the model Base Case: 160 MHz, single frequency Scaling of plasma properties with frequency Scaling of dual frequency CCP (dfCCP) properties in Ar/Cl2

Concluding Remarks

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WAVE EFFECTS IN hf-CCP SOURCES

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A. Perret et al, Appl. Phys. Lett. 83, 243(2003)

Wave effects in CPPs impact plasma uniformity at high frequencies: Standing waves due to finite

wavelength tend to produce center peaked plasma.

Skin effects due to high electron density tend to produce edge peaked profile.

Electrostatic edge effects still contribute.

Relative contributions of wave and electrostatic edge effects determine plasma distribution.

Electronegative additives complicate issue by changing relationship between power and plasma density.

Plasma uniformity will be a function of frequency, power, mixture…

In this talk, results from a computational investigation will be discussed: Wave effects on plasma properties in hf-CCPs. Roles of electronegative gases on uniformity.

Iowa State UniversityOptical and Discharge PhysicsYYANG_AVS2007_04

GOALS OF THE INVESTIGATION


HYBRID PLASMA EQUIPMENT MODEL (HPEM)

Electron Energy Transport Module: Electron energy equation with

Boltzmann equation derived transport coefficients.

MCS for secondary, sheath accelerated electrons

Fluid Kinetics Module: Heavy particle and electron

continuity, momentum, energy Maxwell’s Equations in potential

form

Es, N

Fluid Kinetics ModuleFluid equations

(continuity, momentum,

energy)Maxwell

Equations

Te,S,μ

Electron Energy Transport

ModuleBoltzmann equation


FULL-WAVE MAXWELL SOLVER

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A full-wave Maxwell equation solver has been developed to address finite wavelength wave effects. Vector potential : Coulomb Gauge :

With vector and scalar potential, Maxwell equations are:

tA

ji ,

1, ji 1,1 ji

ji ,1jiAr ,

1, jiAr

jiAz , jiAz ,1

0 AAB

)()(22

2

tJAA

tA

In 2D cylindrical coordinates, , solved on a staggered mesh using sparse matrix techniques.

E field :

,, rz AA

tAE

Scalar potential :

jiB ,


NUMERICAL REPRESENTATION OF EQUATIONS

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jittt

jirji

ttjirtt

jir

ttjir

tjir

ttjir

trtJ

rA

At

AAA,,2

,

,,.

22

,,, 1)'()(2

jittt

jiztt

jiz

ttjiz

tjiz

ttjiz

tztJA

tAAA

,,,2

2,,, 1)'()(

2

)()(1)( ,, tNqtAAt k

kkmjittttt

ji

l

lllee

m

ttttqtq

tt

t)(

2)()(

)(

Radial vector potential:

Axial vector potential:

Scalar potential:


TENSOR TRANSPORT COEFFICIENTS

eeeee nDEqn

With azimuthal magnetic field, the electron flux is given by

where and are the tensor mobility and diffusivity.

e eD

and electron momentum transfer collision frequency.

22

22

22

220

)(zzrzr

zrrz

zrrzr

BBBBBBBBBBBBBBBBBBBBB

BAA

qmm

m

Fluxes of heavy particles given by momentum equations.

jiAr ,

1, jiAr

jiAz , jiAz ,1jiB ,

1, jiAz 1,1 jiAz1, jiB

jiAr ,1

1, jiAr

jiAr ,1


NORMALIZATION OF SPARSE MATRIX

Normalized vector and scalar potentials solved in same matrix.

=

elementsArArelementsAz

ArelementsAr

elementsAz

ArAr ji,

AzAz ji,

ji ,elements

ArE ji,

AzF ji ,

jiG ,00

Arelements

Azelements


REACTOR GEOMETRY

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2D, cylindrically symmetric. Ar, 50 mTorr, 200 sccm Base case: 160 MHz, 300 W (upper

electrode) Specify power, adjust voltage.

Ar for single frequency. Ar/Cl2 dual frequency

Ar, Ar*, Ar+

Cl2, Cl, Cl* Cl2

+, Cl+, Cl-

e

ELECTRON DENSITY

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[e] peaked at center with Maxwell solution (MS) due to finite wave length effect.

With Poisson solution (PS), a flat [e] profile.

Less power penetrates into bulk plasma with MS.

Ar, 50 mTorr, 200 sccm 160 MHz, 300 W, 48 V

Maxwell Solution

Electrostatic Poisson Solution

)(tAjP

)( jP

ELECTRON HEATING

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Bulk ionization follows electron density as Te is fairly uniform.

With MS, lower Te obtained in the center due to reduced ohmic heating in high electron density region .

Ar, 50 mTorr, 200 sccm 160 MHz, 300 W, 48 V

Maxwell Solution


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Maxwell Solution Axial field


E

- 170 V/cm – 260 V/cm

Radial fieldE

- 89 V/cm – 24 V/cm

Axial fieldE

- 130 V/cm – 250 V/cm

CYCLE AVERAGEDELECTRIC FIELD

Ar, 50 mTorr, 200 sccm160 MHz, 300 W, 48 V

With MS, the cycle averaged axial electric field is stronger in the center in sheath region.

As such, standing wave effect mainly enhances stochastic heating in the center.

Relative weak radial electric field in the bulk plasma region.

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Maxwell Solution Azimuthal - 0.07 G – 0.07 G

Scalar Potential - 61 V – 54 V

Electrostatic Poisson Solution Potential - 65 V – 45 V

Symmetric B due to out of phase sheath motion.

Magnitude of B is small and not major contributor here.

Similar scalar potential from MS as electrostatic potential from PS.

B Animation Slide POTENTIAL AND

MAGNETIC FIELD


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Maxwell Solution Azimuthal Max = 0.09 G

Scalar Potential - 14 V – 30 V

Electrostatic Poisson Solution Potential - 19 V – 25 V

B CYCLE AVERAGED

MAGNETIC FIELD


Symmetric B due to out of phase sheath motion.

Magnitude of B is small and not major contributor here.

Similar scalar potential from MS as electrostatic potential from PS.


SCALING WITH FREQUENCY

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Maxwell Solution

Ar, 50 mTorr200 sccm300 W

Uniform [e] at 5 MHz for MS, similar to PS.

With increasing frequency, [e] profile undergoes transition from flat at 5 MHz, to edge peaked at intermediate frequencies, to center peaked at 160 MHz.

Wider edge peak with MS at 50 and 100 MHz .


COMPARISON WITH EXPERIMENT

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[e] close to experiments from 5 to 100 MHz; Better match with MS.

PS radial [e] is not sensitive to frequency.

Ar 50 mTorr 200 sccm

Maxwell Solution

Poisson Solution

Line integrated [e]

G. A. Hebner et al, Plasma Sources Sci. Technol., 15, 879(2006)


ION FLUX

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Ar 50 mTorr 200 sccm

Maxwell Solution Electron density

Experiment Ion saturation current

G. A. Hebner et al, Plasma Sources Sci. Technol., 15, 879(2006)

MS transitions from uniform to edge peaked to center peaked from 5 MHz to at 160 MHz.

Skin effect and wave effects have different contributions with frequency.

Trends agree with experiment.


2-FREQUENCY CCP

Ar, 50 mTorr, 200 sccm

Electron density Single frequency at 160 MHz, 300 W

Dual frequency 10/160 MHz, 500/500 W

Ar has center peaked [e] for single frequency (160 MHz/300 W).

dfCCP (PLF=PHF) 10 MHz ionization source has uniform distribution.

Electrons are “seeded” where HF ionization might not occur (near edges) increasing skin effect.

Combined effects dominate over standing wave .

Edge high [e] with a small center peak is produced.


Ar/Cl2 dual frequency have similar effect of reduced importance of wave effects.

Increasing Cl2 decreases electron density and reduces axial current.

Result is weakening of standing wave effect and skin effect. 50 mTorr, 200 sccm LF: 10 MHz/500 W, HF: 160 MHz/ 500 W

ELECTRONEGATIVE DISCHARGE: Ar/Cl2


ELECTRONEGATIVE DISCHARGE: Ar/Cl2

Electron density

Ar/Cl2 dual frequency Decreasing

importance of wave-effects produce edge-high electron densities.

50 mTorr, 200 sccm LF: 10 MHz/500 W

HF: 160 MHz/ 500 W


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POWER DEPOSITION

Ar/Cl2 = 80/20, more bulk power deposition due to lower electron density.

Lower [e] produces smaller axial current, smaller Ar, Az and longer wavelength.

Ratio of inductive to capacitive field decreases.

Power deposition

Ratio: inductive to capacitive field

)(tAjP

/tA

50 mTorr, 200 sccm LF: 10 MHz/500 W

HF: 160 MHz/ 500 W

)(tAjP


CONCLUDING REMARKS A full Maxwell solver was developed and incorporated into HPEM;

to resolve wave effects. Experimental trends of transition of plasma density from flat to

edge peaked to center peaked with increasing frequency are reproduced.

At low powers, azimuthal B is not a large contributor to electromagnetic effects.

Standing wave generally increases sheath fields at center of reactor.

With dual frequency excitation, low frequency provides ionization independent of wave effect. Seeding of electrons reduces severity of high frequency wave effect.

Adding Cl2 reduces wave effects by lengthening wavelength and increasing bulk electron heating.

numerical investigation of wave effects in high-frequency capacitively coupled plasmas*

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