12.2. ION ENERGY DISTRIBUTION 137
C
B
C
1
C
2
V
Figure 12.5: Equivalent circuit for a capacitively coupled RF plasma with the blocking condensor
C
B
and the sheath capacitors C
1
and C
2
.
Assuming an equal plasma density at both plasma boundaries, the scaling between the
sheath thickness and the voltage across the sheath is according to the Child-Langmuir law,
Eq. (6.36)
x
si
hV
pi
i
3=4
(12.29)
which results in an exponent of n D 4 in Eq. (12.27). However, it is questionable to what extent
the Child-Langmuir sheath is established during the short RF half period (see Section 6.4).
erefore, the discrepancy can be considered to be due to the simplifications involved in both
of the above models. Experiments show an exponent of n D 2; : : : 3 which is between the above
predictions, and is also consistent with a collisional sheath which would predict an exponent of
n D 2:5 (see Section 6.5).
Combining Eqs. (12.12) and (12.27), the time-averaged voltage at the powered electrode
becomes
hV
1
i D hV
p2
i hV
p1
i D hV
p1
i
1
A
1
A
2
n
: (12.30)
In the standard situation with A
1
< A
2
, this results in a negative DC “self-bias” voltage at the
powered electrode. In the limit A
1
<< A
2
; < V
p2
> becomes very small so that the DC voltage
at the powered electrode, which is of the order of the RF amplitude V
0
, becomes practically
identical to the DC potential between the powered electrode and the plasma.
12.2 ION ENERGY DISTRIBUTION
e time-varying potential between the plasma and the electrodes provides ion bombardment
of the electrode surfaces. For a simplified description of the ion energy distribution, we remain
in the picture of the ion matrix sheath. According to Eqs. (12.5), (12.6), (12.10), and (12.13),
the time-dependent sheath thickness and voltage across the sheath can be written as
s.t / D s
0
.1 C sin !t/ (12.31)
and
V .t/ D
ne
2"
0
s
2
0
.1 C sin !t/
2
D
V
0
4
.1 C sin !t/
2
(12.32)
138 12. LOW-TEMPERATURE RF PLASMAS
with the time averages < s >D s
0
and < V >D 3=8 V
0
. Solving the equation of motion for a
singly charged ion starting at the average plasma boundary position in an approximate average
electric field E D< V > =s
0
, results in an average transit time to the electrode of
ht
i
i D
r
8
3
1
!
pi
: (12.33)
In a low-temperature RF plasma, typical electron densities are around 10
10
cm
3
corresponding
to ion plasma frequencies around 10 MHz. Also, the RF frequency is in this range so that the
ion transit time is in the order of the RF period. erefore, the ion energy distribution at the
electrode results both from the oscillation of the plasma boundary and the dynamics of the ion
in the time-dependent RF field. In view of the severe simplification involved in the ion matrix
picture, it would be inadequate to solving this rather complicated problem. Instead, we will treat
two extreme cases. e distribution function of ion energies E
i
at the electrode is given by
f
e
.E
i
/ D C
dN
i
dE
i
D C
dN
i
dt
dt
dV .t /
dV .t /
dE
i
; (12.34)
where C is a normalization constant, N
i
denotes the number of ions impinging on the electrode,
and t the time at which an ion starts from the plasma boundary. For a constant plasma density
the number of ions per unit of time is constant.
First, we assume that the ion transit time is infinitely short, i.e., !
pi
>> !. en, the ion
energy distribution just reflects the distribution of the voltage across the sheath, which is given
by the middle differential in Eq. (12.34) (the last term is equal to e
1
). Calculating t explicitly
from Eq. (12.32) and differentiating with respect to V D E
i
=e yields
f
e
.E
i
/ /
1
r
1
q
4E
i
eV
0
1
2
1
p
E
i
eV
0
I !
pi
>> !: (12.35)
For !
pi
<< !, each ion is accelerated by the time-varying electric field in the sheath. For a
simple solution, we approximate the field according to (see Eqs. (12.31) and (12.32))
E.t
0
/
V .t
0
/
s.t
0
/
D
nes
0
2"
0
.1 C sin !t
0
/: (12.36)
Solving the corresponding equation of motion with the initial conditions x D s.t/ and dx=dt D
0 for t
0
D 0, calculating the transit time t
0
i
from x D 0, and evaluating for t
0
i
>> !
1
results in
a velocity at the electrode
Px.t
0
i
/ D !
pi
s
0
p
1 C sin !t (12.37)
from which the kinetic energy of the ion becomes as function of its start time t
E
i
.t/ D
eV .t/
1 C sin !t
D
e
2
p
V .t/V
0
: (12.38)
12.2. ION ENERGY DISTRIBUTION 139
From this, the last term in Eq. (12.34) is evaluated resulting in
f
E
.E
i
/ /
1
r
1
q
4E
i
eV
0
1
2
1
V
0
I !
pi
<< !: (12.39)
e ion energy distributions (IED’s) resulting from Eqs. (12.35) and (12.39) are shown in
Fig. 12.6. Basically, the distributions peak at the full applied RF voltage and zero, reflecting the
probability of residence of the voltage across the sheath. e asymmetry of the voltage function
(see Fig. 12.3) results in an asymmetry of the distribution toward low energy for transient times
being short compared to the RF period. In the inverse case, ions emitted from the boundary at
large instantaneous sheath thickness (and thereby voltage across the sheath) are more accelerated
than those emitted from the shallow sheath boundaries, which causes a shift toward higher
energies. us, higher RF frequencies result in higher average ion energies.
E
i
/eV
0
0
3
2
1
0
0.25
0.5
0.75 1
IED (arb. units)
Figure 12.6: Ion energy distributions from an RF plasma in simple matrix sheath approximation,
for RF frequencies being small (red) and large (blue) compared to the ion plasma frequency.
More realistic treatments of the RF plasma sheath without the matrix sheath approxima-
tion result in considerably smaller oscillations of the plasma boundary around the mean position.
en, for !
pi
>> !, the IED reflects the oscillation of the boundary as above, but with a much
lower amplitude. In this regime, it is independent of the ion mass. For !
pi
!, the ions per-
form a pendulum motion during the acceleration toward the electrode, the amplitude of which
depends on the ion mass. For light ions, the ion energy can even exceed the RF amplitude. An
example of a corresponding measurement is given in Fig. 12.7.
140 12. LOW-TEMPERATURE RF PLASMAS
Ion Energy/eV
0
4
3
2
1
0
200 400
600 800
Ion Current/a.u.
H
H
Ar
Ar
H
2
H
2
H
3
H
3
Figure 12.7: Experimental ion energy distribution from an ArCH
2
RF plasma, in the regime
!
pi
!. Heavier ions are more inert with respect to the oscillating electric field. Experimental
parameters: RF frequency 13.56 MHz, RF amplitude 465 V, pressure 4 10
3
mbar, Ar/H
2
D
0:4. (From D. Field et al. [14].)
For !
pi
<< !, all ions are unable to follow the RF field, so that the IED becomes rather
narrow around the mean sheath voltage, and again independent of the ion mass.
e IED becomes even more complicated when the sheath is collisional. In addition to
the RF oscillations, the mean collision frequency has a significant influence. is may lead to
highly structured IEDs even for monatomic gases as shown in Fig. 12.8. Here, the measured
IED corresponds very well to a theoretical model.
12.2. ION ENERGY DISTRIBUTION 141
E
i
/eV
0
0 0.2
0.4 0.6 0.8 1 1.2
4
3
2
1
0
1
0
f(E
i
) (arb. units)
Theory
Expt.
0.03 mbar
Ar
13.56 MHz
V
0
= 700 V
Figure 12.8: Collision-dominated ion energy distribution from an Ar RF plasma, from exper-
iment and model calculation. Due to the relatively high pressure and the relatively large RF
amplitude corresponding to a large average sheath thickness, the sheath becomes collisional.
(From C. Wild and P. Koidl [15].)
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