Transcript Slide 1

Two problems with gas discharges
1.
Anomalous skin depth in ICPs
2.
Electron diffusion across magnetic fields
12
10
10
-3
n (10 cm )
3 mTorr, 1.9 MHz
Prf(W)
8
800
240
200
6
4
2
0
-5
0
5
10
15
r (cm)
Problem 1: Density does not peak near the antenna (B = 0)
20
Problem 2: Diffusion across B
Classical diffusion predicts slow electron diffusion across B
B
B
i
n
i
n
e
D 
D
1  (c2
/ )
2
e
ci  ce
Hence, one would expect the plasma to be
negative at the center relative to the edge.
Density profiles are almost never hollow
B
r
n
If ionization is near the boundary, the density
should peak at the edge. This is never observed.
Consider a discharge of moderate length
ION
a
ELECTRON
B
1.
Electrons are magnetized; ions are not.
2.
Neglect axial gradients.
3.
Assume Ti << Te
UCLA
Sheaths when there is no diffusion
e
HIGH DENSITY
+
+
SHEATH
LOWER DENSITY
e
B
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
e
+
+
SHEATH
APPARENT
ELECTRON FLOW
ION DIFFUSION
HIGH DENSITY
LOWER
DENSITY
e
1
2
B
(a)
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.
The Simon short-circuit effect
Step 2: 10s of msec time scale
e
+
+
SHEATH
e
+
E
ION DIFFUSION
HIGH DENSITY
B
(b)
Sheath drops change, E-field develops
Ions are driven inward fast by E-field
LOWER
DENSITY
1
2
The Simon short-circuit effect
Step 3: Steady-state equilibrium
e
+
-
e
+
+
SHEATH
+ E
e
ION
DIFFUSION
LOWEST
DENSITY
3
LOWER
DENSITY
2
HIGH
DENSITY
1
B
Density must peak in center in order for potential
to be high there to drive ions out radially.
Ions cannot move fast axially because Ez is small
due to good conductivity along B.
Hence, the Boltzmann relation holds even across B
n  n0ee / KTe  n0e 
Er
 ( KTe / en)(dn / dr )
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.
We now have a simple equilibrium problem
Ion fluid equation of motion
Mv (nv)  Mnv v  enE  Mn io v  en(v  B)  KTin  0
ionization
convection
CX collisions
Ion equation of continuity
 (nv)  nnn Pi (r )
where
neglect B
neglect Ti
Pc (r )    vi cx (r )   io / nn
Pi (r )    ve ion (r )
Result
Mv v  eE  Mnn ( Pi  Pc ) v  0
UCLA
The r-components of three equations
dv
d
 cs2
 nn ( Pc  Pi )v
dr
dr
Ion equation of
motion:
v
Ion equation of
continuity:
dv r
d (ln n) v r
 vr

 nn Pi (r )
dr
dr
r
Electron
Boltzmann relation:
(which comes from)
Er
 (KTe / en)(dn / dr)
n  n0ee / KTe  n0e
3 equations for 3 unknowns:
vr(r) (r) n(r)
Eliminate (r) and n(r) to get an equation for the ion vr
This yields an ODE for the ion radial fluid velocity:
cs2
dv
 2 2
dr cs  v
 v

v2
   nn Pi (r )  2 nn ( Pi  Pc ) 
cs
 r

Note that dv/dr   at v = cs (the Bohm condition,
giving an automatic match to the sheath
We next define dimensionless variables
u  v / cs
to obtain…
k (r )  1  Pc (r ) / Pi (r)
(v  vr )
We obtain a simple equation
du
1

dr 1  u 2
Note that the coefficient of
(1 + ku2) has the
dimensions of 1/r, so we
can define
This yields
 u nn
2 


P
(1

ku
)

i
 r cs

  (nn Pi / cs )r
du
1

d  1 u2

u
2
1

ku






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.
Reminder: Bohm sheath criterion
ne = ni = n
ns
ni
n
PRESHEATH
ne
v = cs
PLASMA
+
SHEATH
xs
x
V / Cs
Solutions for different values of k = Pc / Pi
1.0
1.0
0.8
0.8
a
0.6
a
a
0.0
-0.2
n/n0
-0.4
0.6
-0.6
0.4
0.4
-0.8
v/cs
0.2
eV/KTe
0.2
-1.0
0.0
0.0
0.5
1.0

1.5
2.0
0.0
-1.2
0.0
0.2
0.4
0.6
0.8
1.0
r/a
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”.
1.0
1
0.8
0.8
0.6
0.6
n / n0
n / n0
A universal profile for constant k
p (mTorr)
1
10
100
0.4
0.2
KTe (eV)
2
3
4
0.4
0.2
0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0
0.2
0.4
r/a
k does not vary with p
0.6
r/a
k varies with Te
These samples are for uniform p and Te
These are independent of magnetic field!
0.8
1
Ionization balance and neutral depletion
1 d
 rnv   nn Pi (Te )
nr dr
D2nn  nnnPi
du
1

dr 1  u 2
 u nn 2

u Pc  Pi   0
 
 r cs



Ionization balance
Neutral depletion
Ion motion
Three differential equations
The EQM code (Curreli) solves these three equations
simultaneously, with all quantities varying with radius.
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
Pout  Wi  We  Wr
Energy balance: the Vahedi curve
This curve for radiative losses vs. Te gives us absolute values.
1000
E c (eV)
1.61
Ec  23exp(3.68/ TeV
)
100
10
1
2
KTe (eV)
5
10
Energy balance gives us the data to calculate Te(r)
Helicon profiles before iteration
Trivelpiece-Gould
deposition at edge
Density profiles computed by
EQM
1200
1.2
Case 1
Case 2
Case 3
1000
1.0
0.8
n/n 0
2
P r ( /m )
800
600
0.6
400
0.4
200
0.2
0
0.0
0
0.5
1
1.5
2
2.5
r (cm)
These curves were for uniform
plasmas
Case1
Case 2
Case 3
0
0.5
1
1.5
2
2.5
r (cm)
We have to use these curves to
get better deposition profiles.
Sample of EQM-HELIC iteration
-3
12
8
10
5
27.12 MHz
120G, 1000W
15.0
8
4
Te (eV)
p (mTorr)
14.8
3
14.6
2
14.4
2
1
14.2
0
0
14.0
6
2
n (10
11
n
Pr
4
4
0
0.0
0.5
15.2
p (mTorr)
27.12 MHz
120G, 1000W
6
Pr (k/m )
cm )
16
12
KT e (eV)
20
1.0
1.5
r (cm)
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
r (cm)
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.
UCLA
Comparison with experiment
This is a permanentmagnet helicon source
with the plasma tube
in the external reverse
field of a ring magnet.
LANGMUIR PROBE
PERMANENT
MAGNET
HEIGHT
ADJUSTMENT
GAS FEED
It is not possible to
measure radial profiles
inside the discharge.
We can then dispense
with the probe extension and measure
downstream.
2 inches
UCLA
Probe at Port 1, 6.8 cm below tube
5
18
16
4
14
12
3
11
3
10
8
2
6
n11
KTe
Vs
Vs(Maxw)
1
Vs (V)
n (10 /cm ), KTe (eV)
65 Gauss
4
2
0
0
-25
-20
-15
-10
-5
0
5
10
15
20
25
r (cm)
1.
2.
3.
The density peaks on axis
Te shows Trivelpiece-Gould deposition at edge.
Vs(Maxw) is the space potential calc. from n(r) if Boltzmann.
UCLA
Dip at high-B shows failure of model
280 Gauss
KTe
n11
3
2
11
3
n (10 /cm ), KTe (eV)
4
1
0
-25
-20
-15
-10
-5
0
5
10
15
20
25
r (cm)
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.
UCLA
Example of absolute agreement of n(0)
Measured
Calc. L=20
Calc. L=25
Calc. L=30
10
8
Density (10
11
cm -3 )
12
6
4
2
0
0
100
200
300
400
RF power (watts)
The RF power deposition is not uniform axially, and the equivalent length
L of uniform deposition is uncertain within the error curves.
UCLA