Transcript No Slide Title

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]
http://uigelz.ece.iastate.edu
October 2007
* Work supported by Semiconductor Research Corp., Applied
Materials and NSF.
YYANG_AVS2007_01
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
YYANG_AVS2007_02
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Optical and Discharge Physics
WAVE EFFECTS IN
hf-CCP SOURCES
 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.
A. Perret et al, Appl. Phys. Lett. 83, 243(2003)
YYANG_AVS2007_03
Iowa State University
Optical and Discharge Physics
GOALS OF THE INVESTIGATION
 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.
YYANG_AVS2007_04
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HYBRID PLASMA EQUIPMENT MODEL (HPEM)
 Electron Energy Transport Module:
Electron Energy
Transport
Module
Boltzmann
equation
Te,S,μ
Es , N
Fluid
Kinetics Module
Fluid equations
(continuity,
momentum,
energy)
Maxwell
Equations
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 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
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FULL-WAVE MAXWELL SOLVER
 A full-wave Maxwell equation solver has been developed to address
finite wavelength wave effects.


 Vector potential : B   A

 Coulomb Gauge :   A  0
 Scalar potential : 
 With vector and scalar potential, Maxwell equations are:




 A

2
 2   A  (  A)  J   ()
t
t
2

A
        

t
 In 2D cylindrical coordinates, Az ,, Ar , 
solved on a staggered mesh using
sparse matrix techniques.


A
 E field : E   
t
YYANG_AVS2007_06
 i 1, j 1
 i , j 1
Ari , j 1
Azi , j
B i , j
Azi 1, j
i, j
Ari , j
 i 1, j
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NUMERICAL REPRESENTATION OF EQUATIONS
 Radial vector potential:

Art i , jt  2 Art i , j  Art i, jt
t 2
 ( 2 Ar ) ti ,. j t 
Art i , jt
 J r i , j (t ' )  
ri 2,j
1
  t   t t
rt

 Axial vector potential:

Azt i, jt  2 Azt i , j  Azt i, jt
t
2
 ( 2 Az ) ti ,jt  J z i , j (t ' )  
1
  t   t t
zt

i, j
 Scalar potential:
 (  )
t  t
i, j
t
 t t
1

 A  A
t


i, j
  m (t )   q k N k (t ) 
k
t l (t ) 
  m (t )


t  
 qe   e (t )   ql   l (t ) 

2 t 
 t

l
YYANG_AVS2007_07
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i, j
TENSOR TRANSPORT COEFFICIENTS
 With azimuthal magnetic field, the
electron flux is given by
Ari , j 1
Azi , j
B i , j
Ari , j
Ari 1, j
Azi , j 1
B i , j 1
e  qne  e  E  De  ne
Azi 1, j
Ari 1, j
Azi 1, j 1
where  e and De are the tensor
mobility and diffusivity.
Ari, j 1
2 2


Br
Bz  Br B  B  Br Bz 

A
A  2 0 2    Bz  Br B
 2  B2
Br  B Bz 
(  B ) 

2
2


B

B
B


B

B
B


B

r z
r
 z
z


 mm


q and m electron momentum transfer collision frequency.
 Fluxes of heavy particles given by momentum equations.
YYANG_AVS2007_08
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NORMALIZATION OF SPARSE MATRIX
Ar elem ents
Az elements
Ar
 elem ents 
Ar
Ari, j
E i, j
Ar
Ar
Ar elements
Ar
 elem ents 
Az elem ents
Az
Azi, j
0
0
 elem ents
Az
=
Fi, j
Az
 i, j
Gi , j


 Normalized vector and scalar potentials solved in same matrix.
YYANG_AVS2007_09
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REACTOR GEOMETRY
 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
YYANG_AVS2007_10
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Optical and Discharge Physics
 Maxwell Solution
ELECTRON DENSITY

P  j  ()
 Electrostatic Poisson Solution
 [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


A
P  j  (  )
t
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Optical and Discharge Physics
 Maxwell Solution
ELECTRON
HEATING
 Bulk ionization follows
electron density as Te is
fairly uniform.
 Electrostatic Poisson Solution
 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
YYANG_AVS2007_12
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Optical and Discharge Physics
 Maxwell Solution

 Axial E field
- 170 V/cm – 260 V/cm

 Radial E field
- 89 V/cm – 24 V/cm
CYCLE AVERAGED
ELECTRIC FIELD
 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.
 Electrostatic Poisson Solution

 Axial E field
- 130 V/cm – 250 V/cm
 Relative weak radial
electric field in the bulk
plasma region.
 Ar, 50 mTorr, 200 sccm
 160 MHz, 300 W, 48 V
YYANG_AVS2007_13
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Optical and Discharge Physics
 Maxwell Solution

 Azimuthal B
 Scalar Potential
Animation Slide
- 0.07 G – 0.07 G
- 61 V – 54 V
 Electrostatic Poisson Solution
- 65 V – 45 V
 Potential
POTENTIAL AND
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.
 Ar, 50 mTorr, 200 sccm
 160 MHz, 300 W, 48 V
YYANG_AVS2007_14
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 Maxwell Solution

 Azimuthal B
 Scalar Potential
Max = 0.09 G
CYCLE AVERAGED
MAGNETIC FIELD
- 14 V – 30 V
 Symmetric B due to out
of phase sheath motion.
 Magnitude of B is small
and not major
contributor here.
 Electrostatic Poisson Solution
- 19 V – 25 V
 Potential
 Similar scalar potential
from MS as electrostatic
potential from PS.
 Ar, 50 mTorr, 200 sccm
 160 MHz, 300 W, 48 V
YYANG_AVS2007_14b
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 Maxwell Solution
SCALING WITH
FREQUENCY
 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 .
 Ar, 50 mTorr
 200 sccm
 300 W
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 Maxwell Solution
COMPARISON WITH EXPERIMENT
 Line integrated [e]
 Ar
 50 mTorr
 200 sccm
 Poisson Solution
 [e] close to experiments from 5 to 100
MHz; Better match with MS.
 PS radial [e] is not sensitive to
frequency.
G. A. Hebner et al, Plasma Sources Sci. Technol., 15,
879(2006)
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ION FLUX
 Maxwell Solution
 Electron density
 Experiment
 Ion saturation current
 Ar
 50 mTorr
 200 sccm
 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.
G. A. Hebner et al, Plasma Sources Sci. Technol., 15,
879(2006)
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2-FREQUENCY CCP
 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, 50 mTorr, 200 sccm
YYANG_AVS2007_18
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ELECTRONEGATIVE DISCHARGE: Ar/Cl2
 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
YYANG_AVS2007_19
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 Electron density
ELECTRONEGATIVE
DISCHARGE: Ar/Cl2
 Ar/Cl2 dual frequency
 Decreasing
importance of waveeffects produce edgehigh electron
densities.
 50 mTorr, 200 sccm
 LF: 10 MHz/500 W
HF: 160 MHz/ 500 W
YYANG_AVS2007_19
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Optical and Discharge Physics
 Power deposition


A
P  j  (  )
t


A
P  j  (  )
t
 Ratio: inductive to capacitive
field

A
/ 
t
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.
 50 mTorr, 200 sccm
 LF: 10 MHz/500 W
HF: 160 MHz/ 500 W
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Optical and Discharge Physics
YYANG_AVS2007_20
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.
YYANG_AVS2007_22
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Optical and Discharge Physics