How to Control and Predict Plasma Parameters in Plasma Sources
Transcript of How to Control and Predict Plasma Parameters in Plasma Sources
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3/24/20041
How to Control and Predict Plasma Parameters in Plasma Sources
Igor Kaganovich Princeton Plasma Physics Laboratory
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Outline
Observed dependences of ne,Te on p, L
Global model for ne,Te (p,L)
Evaporating electron cooling in afterglow
Fast kinetic code for plasma sources modeling
May you have the hindsight to know where you've been, the foresight to see where you're going and the insight to know when you're going too far...
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Outline
Observed dependences of ne,Te on p, L
Global model for ne,Te (p,L)
Evaporating electron cooling in afterglow
Fast kinetic code for plasma sources modeling
May you have the hindsight to know where you've been, the foresight to see where you're going and the insight to know when you're going too far...
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19.8 cm
10.5 cm
Godyak’s Experiment is Benchmark.
The Inductively Coupled Plasmas (ICP) in a cylindrical stainless steel chamber as shown in Fig.1.
Measurements were made at
f=0.45-13.56 MHz
in argon gas pressures of 0.3-300 mTorr and
rf power dissipation in the plasma 6-400 W.
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Electron Density and Plasma Potential Versus Power
ne φpl (~6 Te) in the discharge center as a function of power (P) at f=6.78 MHz
ne scales linear with P, whereas φpl and Te ~const.
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Electron Density and Plasma Potential Versus Pressure
ne,Te in the discharge center at fixed power 50W, f=6.78MHz
More gas, more plasma and smaller Te
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Electron Density, Temperature and Potential Profiles
1mTorr 10mTorr
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Outline
Observed dependences of ne,Te on p, L
Global model for ne,Te (p,L)
Evaporating electron cooling in afterglow
Fast kinetic code for plasma sources modeling
May you have the hindsight to know where you've been, the foresight to see where you're going and the insight to know when you're going too far...
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Plasma is Partially Ionized
Plasma density ne = 109 - 1013 cm-3
Gas density ng= 3.5 1013 cm-3 p(mTorr)
– Small degree of ionization ne/ng < 10-3
– Collisions with gas atoms are dominant
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Particle Balance Determines Te
Quasi-Steady-State =>
– Rate of plasma production = rate of plasma loss, or
– Ionization frequency = loss frequency to the wall
( ) ( )iz e loss eT Tν ν=
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Calculation of Ionization Frequency
where I is ionization potential 15.8eV,VT is the electron thermal velocity.
2
3
/ 2| | ( ) ( ) ,iz e g izm I
n n f dν σ∞
>= ∫ v
v v v v
/ ,eI Tiz g T izn V eν σ −=
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Loss Frequency
Electrons are confined
( ) (0)exp( / )e e en x n e Tφ=
/s eC T M=
/loss sC Lν γ=
Potential of order Teaccelerates ions
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Particle Balance: Te (PL)
( ) ( )iz e loss eT Tν ν=
More ngL less Te
/ /eI T e
g T iz
T Mn V e
Lσ γ− =
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Experiment (left) and Global Model (right) Give Close Agreement
0.1 1 10 100 1.1030
2
4
6
8
10
gas pressure, mTorr
elec
tron
tem
pera
ture
, eV
10
0
Te
10000.1 Pgr
Te as a function of gas pressure at f=6.78 MHz
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Outline
Observed dependences of ne,Te on p, L
Global model for ne,Te (p,L)
Evaporating electron cooling in afterglow
Fast kinetic code for plasma sources modeling
May you have the hindsight to know where you've been, the foresight to see where you're going and the insight to know when you're going too far...
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Evaporative Electron Cooling in Afterglow
Experiment: Kortshagen, et al. Appl. Surf. Sci. 192, 244 (2002)Similar to Godyak’s, but ID=14cm, L=10cm
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ε
f(ε)
ε1
ε2
x
-Φ(x)
0.15610, 72ττττ,µµµµs
inelasticneTe
Inelastic Collisions (Ar Excitation) cool tail quicklyThan evaporated cooling as fast electrons leave fastτ(Te)~ τ(ne)
Evaporative Electron Cooling in Afterglow
Electron temperature can cool to 30K! Biondi (1954)
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Experimental Temperature Decay Time and Theoretical Estimate For Density Decay
Experiment Te decay time in microseconds.
Theoretical estimate for n decay in microseconds.
10 20 30 40 50 60 70
60
80
100
120
Pressure, mTorr
Loss
Tim
e, m
icro
s
120
50
τexp
701 Pexp
10 20 30 40 50 60 70
60
80
100
120
Pressure, mTorrLo
ss T
ime,
mic
ros
120
50
τloss106⋅
705 Pgr
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Outline
Observed dependences of ne,Te on p, L
Global model for ne,Te (p,l)
Evaporating electron cooling in afterglow
Fast kinetic code for plasma sources modeling
May you have the hindsight to know where you've been, the foresight to see where you're going and the insight to know when you're going too far...
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BibliographyI. D. Kaganovich et al “Anomalous skin effect for anisotropic electron
velocity distribution function”, to be published in Phys. Plasmas (2004).
I. D. Kaganovich, et al , “Landau damping and anomalous skin effect in low-pressure gas discharges: self-consistent treatment of collisionless heating”, to be published in Phys. Plasmas (2004).
I. D. Kaganovich and O. Polomarov, “Self-consistent system of equations for kinetic description of low-pressure discharges accounting for nonlocal and collisionless electron dynamics”, Phys. Rev. E 68, 026411 (2003).
B. Ramamurthi, et al " Effect of Electron Energy Distribution Function on Power Deposition and Plasma Density in an Inductively Coupled Discharge at Very Low Pressures", Plasma Sources Sci. Technol. 12, 170 and 302 (2003).
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Electron Distribution Function is not Maxwellian
Measured EDF shows departure from a Maxwellian: Cold electrons are trapped in a small rf electric field.
ε1
ε2
x
Erf(x)
-Φ(x)
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Concept of Screening Temperature
EDF is not Maxwellian,introducing Tes , so that
11/ 2
02 ( )esT n f dε ε ε
−∞ − = ∫
ln( ) /es eeE T d n dx= −
3/ 2
0
2 2 ( )3 3eT f d
nε ε ε ε
∞= < >= ∫
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In collaboration with
Oleg Polomarov, Constantine TheodosiouUniversity of Toledo, Toledo, Ohio
Badri Ramamurthi, Demetre J. EconomouUniversity of Houston, Houston, TX
Kinetic Code for Calculation of EDF and Plasma Parameters
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Overview
Calculate nonlocal conductivity innonuniform plasma
Find a nonMaxwellian electron energy distribution function driven by collisionless heating of resonant electrons
What to expect: self-consistent system for kinetic treatment of collisionless and nonlocal phenomena in inductive discharge
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Inductive Discharge
ε1
ε2
x
Erf(x)
-Φ(x)
The transverse rf electric field is given by2 2
2 2 2
4 [ ( ) ( )]yy
d E iE j x I xdx c c
ω π ω δ+ = − +
The electron energy distribution is given by*0
0( ) ,dfd D S fd dεε ε
− =
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Nonlocal Conductivity
Nonlocal conductivity G(x,x’) is a function of the EEDF f0 and the plasma potential ϕ(x).
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0
( ) ( , ') ( ') ' ( ', ) ( ') 'x L
ey y y
x
e nJ x G x x E x dx G x x E x dxm
= + ∫ ∫
•PIC code is inefficient: limited by electron time step,
•while discharge develops at ion time scale =>
•implicit description of the rf electric field•solved by spectral method
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Kinetic Equation Is Averaged over Fast Electron Bouncing in Potential Well
Dee Vee are from the electron-electron collision integral, ν* is inelastic collision frequency, upper bar denotes space averaging with constant total energy.
0
00 0
( )( ) ( ) ( ) ,k
ee ee k k k kk
udfd dD D V f u f fd d d uε
εν ε ε ε ν
ε ε ε
∗∗ ∗ ∗ ∗
+ − + − = + + −
∑
Energy diffusion De coefficient is a function of the rf electric field Ey and the plasma potential ϕϕϕϕ(x).
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Comparison with Experiment
Experimental data (symbols) and simulation (lines) (a) RF electric field and (b) the current density profiles for a argon pressure of 1 mTorr.
Te
ep
Vc δω ω
< <
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Comparison with Experiment
EEDF simulated (lines) and experimental data (symbols) for 1mTorr. Data are taken from V. A. Godyak and V. I. Kolobov, Phys. Rev. Lett., 81, 369 (1998).
ε (V)
f 0(ε)
(V-3
/2cm
-3)
10 20 30107
108
109
1010
12 W50 W200 W12 W50 W200 W
1 mTorr
R=10cm,L=10cm,antenna R=4cm
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Predictions of the Code for PPPL Plasma Source
0 1 2 3
3
4
5
6
7
Te=2/3 <ε> full model global model
T e (eV
)
Pressure (mTorr)4 5 6 7
0
2
4
6
8
400W Plasma Density 0.3 mTorr 1mTorr 3mTorr
n ex10
12 (c
m-3)
0<x<L (cm)
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Predictions Using Power Balance for PPPL Plasma Source
Power deposited into plasma:P=0.5nCs εεεε S
ε ≈ ε* +2I=43eV
N=4 1012 cm-3 at 1mTorr needed power 0.4kW vs 2kW
/s eC T M=
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Predictions of the Code for PPPL Plasma Source
0 20 40 60
10-5
10-4
10-3
10-2
10-1
EED
F (e
V-3/2)
ε (eV)
400W Normalized EEDF 0.3 mTorr 1mTorr 3mTorr
0 1 2 3 4 5 6 70
1
2
3
4
5
400W Electric field (Volt/cm) 0.3 mTorr 1mTorr 3mTorr
E (V
/cm
)0<x<L (cm)
EDF is Maxwellian due to high degree of ionizationSkin layer of 0.5 cm
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Conclusions
Te 3-5eV for steady-state, P in mTorr range
Te decay time scale is 10-50 microseconds
N=4 1012 cm-3 correspond to 0.4kW
Skin layer about 1cm
EDF is Maxwellian