two problems with gas discharges 1.anomalous skin depth in icps 2.electron diffusion across magnetic...
TRANSCRIPT
Two problems with gas discharges
1. Anomalous skin depth in ICPs
2. Electron diffusion across magnetic fields
0
2
4
6
8
10
12
-5 0 5 10 15 20r (cm)
n (
101
0 c
m-3
)
800
240
200
Prf(W)3 mTorr, 1.9 MHz
Problem 1: Density does not peak near the antenna (B = 0)
Problem 2: Diffusion across B
B
nn
i
ei e
B
Classical diffusion predicts slow electron diffusion across B
2 21 ( / )ci ce
c
DD
Hence, one would expect the plasma to be negative at the center relative to the edge.
Density profiles are almost never hollow
If ionization is near the boundary, the density should peak at the edge. This is never observed.
B
n
r
Consider a discharge of moderate length
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1. Electrons are magnetized; ions are not.
2. Neglect axial gradients.
3. Assume Ti << Te
B
a
ION
ELECTRON
Sheaths when there is no diffusion
HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
e
e
+
Sheath potential drop is same as floating potential on a probe.
This is independent of density, so sheath drops are the same.
The Simon short-circuit effect
Step 1: nanosecond time scale
Electrons are Maxwellian along each field line, but not across lines.
A small adjustment of the sheath drop allows electrons to “cross the field”.
This results in a Maxwellian even ACROSS field lines.
HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
+e
e
APPARENT ELECTRON FLOW ION DIFFUSION 1
2
(a)
Sheath drops change, E-field develops
Ions are driven inward fast by E-field
HIGH DENSITY
LOWER DENSITY
SHEATH
B
+
+
e
e
ION DIFFUSION+
-
E
(b)
1
2
The Simon short-circuit effect
Step 2: 10s of msec time scale
The Simon short-circuit effect
Step 3: Steady-state equilibrium
Density must peak in center in order for potentialto be high there to drive ions out radially.
Ions cannot move fast axially because Ez is smalldue to good conductivity along B.
BSHEATH
+
LOWER DENSITY 2
e
+
HIGHDENSITY 1
e
+ e LOWEST DENSITY 3
ION DIFFUSION
+
-
E
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 e
en dn dr
We now have a simple equilibrium problem
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( ) ( ) 0io iM n Mn en Mn en KT n v v v v E v v B
Ion fluid equation of motion
ionization
convection
CX collisions
neglect B
neglect Ti
Ion equation of continuity ( ) ( ) /
( ) ( )c i cx io n
i e ion
P r v r n
P r v r
( ) ( )n in nn P r v wher
e
Result
( ) 0n i cM e Mn P P v v E v
The r-components of three equations
Ion equation of motion:
Ion equation of continuity:
v v(ln )v ( )r rr n i
d d nn P r
dr dr r
3 equations for 3 unknowns: vr(r) (r) n(r)
/0 0
ee KTn n e n e
( / )( / )erE KT en dn drElectron
Boltzmann relation:
(which comes from)
2 ( )s n c idv dv c n P P vdr dr
Eliminate (r) and n(r) to get an equation for the ion vr
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
We next define dimensionless variables
to obtain…
( )rv v2 2
2 2 2( ) ( )s
n i n i cs s
cdv v vn P r n P P
dr rc v c
( ) 1 ( ) / ( )c ik r P r P r
We obtain a simple equation
Note that the coefficient of (1 + ku2) has the
dimensions of 1/r, so we can define
( / )n i sn P c r
This yields 22
11
1
du uku
d 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.
22
1(1 )
1n
is
ndu uP ku
dr r cu
Reminder: Bohm sheath criterion
n
xs x
ne = ni = n
PLASMA
SHEATH
ni
ne
+
ns
PRESHEATH
v = cs
Solutions for different values of k = Pc / Pi
We renormalize the curves, setting a in each case to r/a, where a is the discharge radius. No presheath assumption is needed.
We find that the density profile is the same for all plasmas with the same k.
Since k does not depend on pressure or discharge radius, the profile is “universal”.
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
A universal profile for constant k
These are independent of magnetic field!
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)
k does not vary with p k varies with Te
These samples are for uniform p and Te
Ionization balance and neutral depletion
1( )n i e
drnv n P T
nr dr
2n n iD n n nP
22
10
1n
c is
ndu uu P P
dr r cu
Ionization balance
Neutral depletion
Ion motion
The EQM code (Curreli) solves these three equations simultaneously, with all quantities varying with radius.
Three differential equations
Energy balance: helicon discharges
To implement energy balance requires specifying the type of discharge. The HELIC program for helicons and ICPs can calculate the power deposition Pin(r) for given n(r), Te(r) and nn(r) for various
discharge lengths, antenna types, and gases. However, B(z) and n(z) must be uniform. The power lost is given by
out i e rP W W W
Energy balance: the Vahedi curve
This curve for radiative losses vs. Te gives us absolute values.
10
100
1000
1 10KTe (eV)
Ec
(eV
)1.6123exp(3.68/ )c eVE T
2 5
Energy balance gives us the data to calculate Te(r)
Helicon profiles before iteration
0
200
400
600
800
1000
1200
0 0.5 1 1.5 2 2.5r (cm)
Pr (
/m
2)
Case 1
Case 2
Case 3
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.5 1 1.5 2 2.5r (cm)
n/n
0
Case1
Case 2
Case 3
Trivelpiece-Gould deposition at edge
Density profiles computed by EQM
These curves were for uniform plasmas
We have to use these curves to get better deposition profiles.
Sample of EQM-HELIC iteration
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0
4
8
12
16
20
0.0 0.5 1.0 1.5 2.0 2.5r (cm)
n (1
011
cm
-3)
0
2
4
6
8
10
12
Pr (k
/m2)
n
Pr
27.12 MHz120G, 1000W
0
1
2
3
4
5
6
0.0 0.5 1.0 1.5 2.0 2.5r (cm)
KT
e (e
V)
14.0
14.2
14.4
14.6
14.8
15.0
15.2
p (mT
orr)
Te (eV)p (mTorr)
27.12 MHz120G, 1000W
It takes only 5-6 iterations before convergence.
Note that the Te’s are now more reasonable.
Te’s larger than 5 eV reported by others are spurious; their RF compensation of the Langmuir probe was inadequate.
Comparison with experiment
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PERMANENT MAGNET
GAS FEED
HEIGHT ADJUSTMENT
LANGMUIR PROBE This is a permanent-magnet helicon source with the plasma tube in the external reverse field of a ring magnet.
It is not possible to measure radial profiles inside the discharge. We can then dispense with the probe ex-tension and measure downstream.
2 inches
Probe at Port 1, 6.8 cm below tube
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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
18V
s (V)
n11KTeVsVs(Maxw)
65 Gauss
1. The density peaks on axis
2. Te shows Trivelpiece-Gould deposition at edge.
3. Vs(Maxw) is the space potential calc. from n(r) if Boltzmann.
Dip at high-B shows failure of model
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0
1
2
3
4
-25 -20 -15 -10 -5 0 5 10 15 20 25r (cm)
n (1
01
1/c
m3),
KT
e (
eV)
KTe
n11
280 Gauss
With two magnets, the B-field varies from 350 to 200G inside the source.
The T-G mode is very strong at the edge, and plasma is lost axially on axis. The tube is not long enough for axial losses to
be neglected.
Example of absolute agreement of n(0)
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0
2
4
6
8
10
12
0 100 200 300 400
RF power (watts)
Den
sity
(10
11 c
m-3
)
Measured
Calc. L=20
Calc. L=25
Calc. L=30
The RF power deposition is not uniform axially, and the equivalent length L of uniform deposition is uncertain within the
error curves.