What are helicons?
Helicons are partially ionized RF discharges in a magnetic field.
They are basically whistler modes confined to a cylinder.
They are much different than in free space; they have E-fields.
OLD
NEW
Long cylinder Permanent magnet
Helicons pose unending problems
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• Why does the amplitude oscillate along the cylinder?
• Why is a right-helical antenna better than a left one?
• What causes the high ionization efficiency?
• Why does an endplate near the antenna increase n?
• Why is the ion temperature so high?
• Why is a half-wavelength antenna better than a full?
• Why is the density peaked at the center?
Most discharge theorists treat only collision cross sections and ion distribution functions.
The Trivelpiece-Gould mode: edge ionization
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Trivelpiece-Gould mode
Helicon mode
An electron cyclotron wave near the edge deposits most of the RF energy
0
1000
2000
3000
4000
5000
0.000 0.005 0.010 0.015 0.020 0.025r (m)
P(r
)
(arb
.)
2.5E+11
4.0E+11
6.3E+11
1.0E+12
n (cm-3)
Edge ionization should give a hollow profile
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B
n
r
0
1
2
3
4
5
-25 -20 -15 -10 -5 0 5 10 15 20 25r (cm)
n (1
01
1/c
m3),
KT
e (
eV)
0
2
4
6
8
10
12
14
16
18
Vs (V
)
n11KTeVsVs(Maxw)
65 Gauss
But density is almost always peaked at center, even in KTe
is peaked at the edge.
Previous attempt for an ICP
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Skin depth
with FL
without FL
Let’s take the simplest realistic problem
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Eliminate all unnecessary features, and not length!
L
aB
+
rLi >> a
rLe << a-
Treat a 1D problem in radius r
The problem is how to treat the ends
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HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
e
e
+
½ ½ 1/2
, ln2 2
pe pe e
e
eKT KT Mn n eM m KT m
The sheath drop is normally independent of density
Ion diffusion upsets the balance
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HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
+e
e
APPARENT ELECTRON FLOW ION DIFFUSION 1
2
(a)
The short-circuit effect “moves” electrons across B.
Sheaths change to preserve neutrality.
Electrons can now follow the Boltzmann relation.
This happens in nanoseconds.
Sheath drops interchange, creating Er
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HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
+
e
e
ION DIFFUSION+
-
E
(b)
1
2
/0
ee KTn n e
( 0)
In equilibrium, n is peaked on center
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BSHEATH
+
LOWER DENSITY 2
e
+
HIGHDENSITY 1
e
+ e LOWEST DENSITY 3
ION DIFFUSION
+
-
E
Er and diffusion must be outward if axial flow is slow.n(r) is flat in the limit of all ionization at edge.
Three equations in 3 unknowns: v, n, and
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( ) ( ) /c cx io nP r v r n
( ) ( ) 0io iM n Mn en Mn en KT n v v v Bv v E v
Ion equation of motion:
Ion equation of continuity: ( ) ( )n in nn P r v
( ) ( )i ionP r v r
Use the Boltzmann relation:
Simplify the collision terms:
/0 0
ee KTn n e n e
½, / , and ( / )e s ee KT c KT M E
Reduce to one dimension in r
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2 2
2 2 2( ) ( )s
n i n i cs s
cdv v vn P r n P P
dr rc v c
Eliminate n and to get an equation for v(r):
/ , ( ) 1 ( ) / ( )s c iu v c k r P r P r Non-dimensionalize:
22
1(1 )
1n
is
ndu uP ku
dr r cu
This is an ordinary differential equation for all the plasma
profiles.
Rescale r to see structure of the equation
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( / )n i sn P c r
( ) 1 ( ) / ( )c ik r P r P r
22
11
1
du uku
d u
22
1(1 )
1n
is
ndu uP ku
dr r cu
We had:
Rescale r:
Finally:
k contains the plasma information:
Solutions for uniform pressure and KTe
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0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0
V /
Cs
a
a
a
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0r / a
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
n/n0
eV/KTe
v/cs
Solutions for three values of k Rescale so that a 1 in each case
This profile is independent of pressure, size, and magnetic field.This profile is independent of pressure, size, and magnetic field.
It depends on It depends on KTKTee, but is always peaked at the center., but is always peaked at the center.
This profile IS modified:
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• When Te is changed or varies with r
• When nn varies with r (neutral depletion, treated later)
• When k varies with r
But the central peaking remains
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0r / a
n /
n0
110100
p (mTorr)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1r / a
n /
n0
234
KTe (eV)
Ionization balance restricts KTe for real r
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1( )n i e
drnv n P T
nr dr
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5r [cm]
V /
Cs
3.00
3.49
4.00
KTe (eV)
22
1(1 )
1n
is
ndu uP ku
dr r cu
Our previous dimensional equation
Solved simultaneously
Improved Te – p0 relation
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0
2
4
6
8
10
1 10 100p0 (mTorr)
KT
e (
eV)
2.5
5
10
Tube radius (cm)
Old, radially averaged data:M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed. (Wiley-Interscience, Hoboken, NJ, 2005).F. F. Chen and J.P. Chang, Principles of Plasma Processing (Kluwer/Plenum, New York, 2002),
The EQM program solves simultaneously:
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22
1( ) 1 (1 / )
1n
i c is
ndu uP r u P P
dr r cu
1( )n i e
drnv n P T
nr dr
2n n iD n n nP
Ion motion
Neutral depletion
Ionization balance
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5r (cm)
n (1
012
cm-3
) &
p
/ p
0
1
5
10
p0 (mTorr)
Last step: iteration with HELIC
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0
2
4
6
8
0.0 0.5 1.0 1.5 2.0 2.5r (cm)
n (1
011
cm
-3)
0
1
2
3
4
5
6
Pr (K
W/m
2)n
Pr
13.56 MHz65G, 400W
0
1
2
3
4
5
6
0.0 0.5 1.0 1.5 2.0 2.5r (cm)
KT
e (e
V)
14.4
14.6
14.8
15.0
15.2
p (mT
orr)
KTe (eV)p (mTorr)
13.56 MHz65G, 400W
Lc
a b
h
Loop antenna
Helical antenna
B0
Another layer off the onion!
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Title
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