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Transcript of PM Helicons, a Better Mouse Trap UCLA Part 1: Permanent-magnet helicon sources and arrays Part 2:...
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PM Helicons, a Better Mouse Trap
UCLA
Part 1: Permanent-magnet helicon sources and arrays
Part 2: Equilibrium theory of helicon and ICP discharges with the short-circuit effect
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Commercial helicon sources required large electromagnets
The PMT MØRI helicon etcher
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Helicon waves require a DC magnetic field
Helicon waves are excited by an RF antennaand propagate away while creating plasma
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Field lines in permanent ring magnets
The strong, uniform field is inside the rings, but the plasma cannot get out. However, the reverse external field extends to infinity.
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Place the plasma below the magnet
The B-field is almost straight, and its strength can be changed by moving the magnet.
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
-10 -8 -6 -4 -2 0 2 4 6 8 10
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Designing the magnet
It should be small and not heavy.
It is 3 in. ID and 5 in. OD, 1 in. thick.
Material: NdFeB
Internal field: 1.2 T (12 kG)
12.7 cm
7.6 cm
PLASMA
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Designing the discharge tube
A long, helical antenna has the best coupling, but plasma is lost to the walls before it is ejected.
A short stubby tube with a short antenna has more efficient ejection.
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The final tube design
1. Diameter: 2 inches
2. Height: 2 inches
3. Aluminum top
4. Material: quartz
5. “Skirt” to prevent eddy currents canceling the antenna current
5.1 cm
10 cm
5 cmANTENNA
GAS INLET (optional)
The antenna is a simple 3-turn loop, not a helix. The loop antenna must be as close to the exit aperture as possible.
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Reflections from the endplates are included.
The power absorption distribution is calculated, as well as the total plasma resistance including the
strong TG mode absorption at the boundary.
Lc
a b
h
Loop antenna
Helical antenna
B0
Designing the helicon wave: the HELIC code
Trivelpiece-Gould mode
Helicon mode
written by Don Arnush
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For small tubes, we must use the low-field peak effect
The plasma loading must overcome the circuit losses Rc:
Rp >> Rc
This is helped by adjusting the backplate position so that the reflected
backward wave interferes constructively with the forward wave.
This effect is important here because the external field of permanent
magnets is relatively weak.
This condition for good coupling sets the height of the tube
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Designing the discharge
B is uniform in r but doubles in z within tube
0
50
100
150
200
250
300
0 2 4 6 8 10 12z (in.)
Bz (
G)
0.00.50.9
radius (in.)
D
B ~ 60G at D = 6”, antenna to magnet midplane
16 cm
12.7 cm
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HELICON ARRAYS
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Uniformity with discrete sources was proved long ago
Density uniform to +/- 3%
ROTATING PROBE ARRAY
PERMANENT MAGNETS
DC MAGNET COIL
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Two kinds of arrays were tried
Staggered array
Covers large area uniformly for substrates moving in the y-direction
165 cm
53.3 cm
17.8
17.8
17.8
35.6 cm
x
y
165 cm
53.3 cm
17.8
17.8
17.8
x
yTop view 17.8 Compact array
Gives higher density, but uniformity suffers from
end effects.
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How did we choose the spacing between tubes?
10 mTorr Ar, 500W @ 13.56 MHz, Z2 @ 18 cm below tube
0
1
2
3
4
5
-20 -10 0 10 20r (cm)
n (1
011 c
m-3
)
34 cm
36 cm
D
Z1
Z2
Pyrex tube, 2-in. diam
Ceramic magnet stack
Radial profile of a single tube
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How did we choose the spacing between tubes?
The tube spacing is chosen so that the ripple is less than +/- 2%
Z-2 Z-1 Z2
y
Z
r
y
yLd
Z1
d
0
2
4
6
8
10
12
15 20 25 30L (cm)
Rip
ple
(+/-
%)
0
1
2
3
4
5
6
7
8
-30 -20 -10 0 10 20 30x (cm)
n (1
011 c
m-3
)
y = 7.5
Average over all y
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The matching circuit
The problem is that all the Z2 cables must be the same length, and they cannot all be short if the tubes are far apart. The cable lengths and
antenna inductance L are constrained by the matching circuit.
R, L
R, L
R, L
R, L
PS
N loads
Z2 - short cables
Distributor
Z1Z2
Z1 - long cable
C1C2
Matching ckt. 50W
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Design of the Medusa 2 machine
65"
21"
7"
7"
7"
7"29"
3.5"
12"7"
3/4” aluminum
Endplates: gas feed and probe port at each end.
Tubes set in deeply (½ inch)
1/2" aluminum
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Side view
165 cm
30 cm
15 cm
Probe ports
Aluminum sheetHeight can be adjusted electrically if desired
The source requires only 6” of vertical space above the process chamber
Z1Z2
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Operation with cables and wooden magnet tray
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Detail of a discharge tube for 13.56 MHz
3-turn antenna of 1/8” copper tubing
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Rectangular transmission line
50-W line with ¼” diam Cu pipe for cooled center conductor
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Operation with rectangular transmission line
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Radial profile between tubes at Z2
0
0.5
1
1.5
2
2.5
3
3.5
-25 -20 -15 -10 -5 0 5 10 15 20 25r (cm)
n (1
011 c
m-3
) nKTe
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Density profiles along the chamber
Staggered configuration, 2kW
Bottom probe array
0
1
2
3
4
5
-8 -6 -4 -2 0 2 4 6 8 10 12 14 16x (in.)
n (1
011 c
m-3
)
-3.503.5
Staggered, 2kW, D=7", 20mTorr
y (in.)
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UCLA
Density profiles along the chamber
Compact configuration, 3kW
Bottom probe array
0
2
4
6
8
10
-8 -6 -4 -2 0 2 4 6 8 10 12 14 16x (in.)
n (1
011 c
m-3
)
3.5-03.5
Compact, 3kW, D=7", 20mTorr
y (in)
Data by Humberto Torreblanca, Ph.D. thesis, UCLA, 2008.
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Part 2
Equilibrium theory of helicon and ICP
discharges with the short-circuit effect
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Motivation: Why are density profiles never hollow?
In many discharges, ionization is at the edge. Yet, the plasma density is always peaked at the center.
0
z (c
m)
PlasmaTherm ICP Source Module on top of Magnetic Bucket
Case of an ICP: B = 0
0
2
4
6
8
10
12
-5 0 5 10 15 20r (cm)
n (1
010
cm
-3)
800240200
Prf(W)3 mTorr, 1.9 MHz
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Helicon case: Trivelpiece-Gould ionization at edge
Typical radial energy deposition profiles as computed by HELIC
0
200
400
600
800
1000
1200
0 0.5 1 1.5 2 2.5r (cm)
Pr (
W/m
2 )
Case 1Case 2Case 3
First data (1991) in long, 5-cm diam tube. Since then, all n(r)’s have peaked on axis.
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Classical diffusion prevents electrons from crossing B
If ionization is near the boundary, the density profile would be hollow. This is never observed.
B
n
r
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The Simon short-circuit effect cures this
A small adjustment of the sheath drop allows electrons to “cross the field”.
HIGH DENSITY
LOWER DENSITY
SHEATHB
+
+e
e
APPARENT ELECTRON FLOW ION DIFFUSION
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Hence, the Boltzmann relation holds even across B
As long as the electrons have a mechanism that allows them to reach their most probable
distribution, they will be Maxwellian everywhere.
This is our basic assumption.
/0 0
( / )( / )e
ee KT
rE KT
n n e n een dn dr
This radial electric field pushes ions from high density to low density, filling in hollow profiles so that, in equilibrium, the density is always
peaked on axis.
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An idealized model of a high-density discharge
1. B = 0 or B 0; it doesn’t matter.
2. The discharge is cylindrical, with endplates.
3. All quantities are uniform in z and θ (a 1-D problem).
4. The ions are unmagnetized, with large Larmor radii.
5. Ion temperature is negligible: Ti / Te = 0
L
aB
+
rLi >> a
rLe << a-
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The ion equations
Motion: (Mn en v v E v B ) iKT ion Mn v
½, / , and ( / )e s ee KT c KT M E
2vv vrr s io rd dcdr dr
( ) ( ) ( )n in Q r nn P r vContinuity: ( ) ( )i ionP r v r
v v(ln )v ( )r rr n i
d d n n P rdr dr r
1-D:
Pi is the ionization probability
( )io n cx n cn v n P r
Pc is the collision probabilityUnknowns: vr(r) (r) n(r)
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Combine 2 ion equations with electron Boltzmann relationTo eliminate unknowns (r) and n(r)
2 2
2 2 2 ( ) ( ) 0sn c n i
s s
cdv v v n P r n P rdr rc v c
This yields an ODE for the ion radial fluid velocity:
Note that dv/dr at v = cs (the Bohm condition, giving an automatic match to the sheath
/ su v c / /c i cx ionk P P v v
We next define dimensionless variables
to obtain…
( )rv v
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The universal equation
Note that the coefficient of (1 + ku2) has the
dimensions of 1/r, so we can define
( / )n i sn P c r
22
1 1 01
n i
s
n Pdu ukudr c ru
This yields 22
1 11
du ukud u
Except for the nonlinear term ku2, this is a universal equation giving the n(r), Te(r), and (r) profiles for any discharge and satisfies the Bohm
condition at the sheath edge automatically.
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Solutions for different values of k = Pc / Pi
Identifying a with the discharge radius a gives the same curves in all cases.
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
Since v(r) is known, the collision rate can be calculated with the exact v(r) at each radius. No presheath assumption is necessary.
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Further developments in the theory
1. Arbitrary ionization profiles. EQM code
2. Ionization balance: Te(r) depends on p(r)
3. Neutral depletion with velocity-dependent Pc
4. Iteration with HELIC to get ionization profiles for helicon discharges which are consistent with the Te and p profiles from EQM.
5. Energy balance using the Vahedi curve to get absolute densities for given antenna power
This work was done with Davide Curelli, University of Padua
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There is absolute agreement with experiment
Single-tube apparatus with probe inside the source
PERMANENT MAGNET
GAS FEED
HEIGHT ADJUSTMENT
LANGMUIR PROBE
0
2
4
6
8
10
12
0 100 200 300 400RF power (watts)
Den
sity
(1011
cm
-3)
MeasuredCalc. L=20Calc. L=25Calc. L=30
Absolute-value agreement with theory with no adjustable paramters
15 mTorr Ar, 400W @ 13.56 MHz, 65 G
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The understanding of helicons never ends, but this is…
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ABC
ABC