Monitoring Sheath Voltages and Ion Energies
in High-Density Plasmas Using Noninvasive
Radio-Frequency Current and Voltage Measurements
Mark A. Sobolewski
Process Measurements Division, National Institute of Standards and Technology, Gaithersburg, MD 20899
Abstract. To obtain optimal results from plasma processing, the energy of ions incident on substrate wafers must be
carefully controlled. Such control has been difficult to achieve, however, because no practical method exists for
monitoring the energy distributions of ions at a wafer surface during processing. To solve this problem, we have
developed a noninvasive, model-based method for determining ion energy distributions (lEDs) that is suitable for use
during actual processing in commercial plasma reactors. The method was validated by tests performed in argon and CF4
discharges at 1.3-3.1 Pa (10-23 mTorr) in an inductively coupled, high-density plasma reactor, with radio-frequency
(rf) substrate bias at frequencies of 0.1 MHz to 20 MHz. Plasma potential waveforms and sheath voltages obtained
from the noninvasive rf technique agreed well with independent measurements made using a capacitive probe. Ion
energy distributions from the rf technique were in good agreement with distributions measured by ion energy analyzers.
INTRODUCTION
In plasma etching, substrate wafers are bombarded by
reactive neutral species and energetic positive ions.
These energetic ions are essential for etching. Etch
rates, etch profiles, and damage rates all depend on the
kinetic energy of the ions. To obtain optimal results,
ion energies must therefore be carefully optimized and
controlled. Unfortunately, no good method now exists
for monitoring energy distributions of ions at a wafer
surface during actual processing. Thus it is difficult or
impossible to know what the ion energy distributions
are, and whether they are stable or drifting with time.
Ion energy analyzers, sometimes coupled with mass
spectrometers for ion mass analysis, are used in plasma
research studies, but they are not well-suited for process
monitoring during manufacturing. When exposed to
actual processing conditions, ion energy analyzers may
fail due to deposition or etching at analyzer surfaces.
Commercial plasma reactors often cannot accommodate
an ion energy analyzer anywhere close to the wafer.
Furthermore, materials etched or sputtered from the
analyzer could possibly contaminate wafers. Ideally,
one wants to know the lEDs at the wafer itself, but to
directly measure these distributions would require
fabrication of an ion energy analyzer on each wafer,
which is impractical.
Because of these difficulties in measuring lEDs
directly, there is much interest in developing indirect
methods, which would use models to deduce the ion
energy distributions from other types of measurements.
It would be best to have a completely noninvasive
method, one that does not require anything be inserted
into the plasma reactor. This paper describes one such
noninvasive, model-based method. It relies solely on rf
measurements of current and voltage made outside the
reactor. The measurements are analyzed using electrical
models of the plasma and its sheaths, the thin regions at
the boundary of the plasma adjacent to the wafer or
other electrode surface. The method was tested by
comparisons performed in argon and CF4 discharges in
an inductively coupled, high-density plasma reactor,
equipped with rf substrate bias. Model results for the
sheath voltage and plasma potential waveforms, which
are important intermediate results needed to accurately
determine the lEDs, were validated by comparisons to
independent measurements made using an invasive,
capacitive probe. lEDs from the rf technique were
compared to distributions measured by ion energy
analyzers at grounded [1] as well as rf-biased [2]
surfaces, and found to be in good agreement.
EXPERIMENT
The plasma reactor used in this study, an inductively
coupled Gaseous Electronics Conference Reference Cell
[3], is shown in Fig. 1. Mounted on the top of the
reactor is an inductive plasma source, consisting of a
flat coil powered at 13.56 MHz and an electrostatic
shield. Wafers to be processed are placed on the
substrate electrode located below the inductive source.
Using a sinusoidal signal generator and a wideband rf
amplifier, the substrate electrode can be biased at
frequencies from 10 kHz to 100 MHz. This rfbias is
CP683, Characterization and Metrology for VLSI Technology: 2003 International Conference,
edited by D. G. Seiler, A. C. Diebold, T. J. Shaffner, R. McDonald, S. Zollner, R. P. Khosla, and E. M. Secula
2003 American Institute of Physics 0-7354-0152-7/03/$20.00
195
high-density,
inductive
plasma source
Vb(t)
-
+
VgeW
ground —o grounded
sheath
surfaces
Vps(t) powered sheath
A Vpe(t)
F>owered electrode
(a)
1
^rw
4
_Vge(t)
Rs
wall
sheath
Vps(t) 1 powered sheath
A Vpe(t)
powered electrode
(b)
FIGURE 1: Diagram of the plasma reactor and electrical
measurement system.
FIGURE 2: Circuit model of discharges in (a) argon and
(b) CF4.
designed to control ion energies at the wafer independent
of the plasma density and ion flux, which are controlled
by the inductive source. On the electrical connections
between the rf amplifier and the electrode, a current
probe and voltage probe are mounted. Signals from
these probes are digitized by an oscilloscope and
transferred to a computer. Using techniques described
previously [4-5], the computer accounts for propagation
delays and stray impedance, resulting in an accurate
determination of the current and voltage signals Ipe(f)
and Vpe(t) present at the substrate electrode or wafer
surface, and the small voltage Vge(t) induced on
grounded reactor surfaces. These signals are then further
analyzed, as described below, using an electrical model
of the discharge. Plasma potential waveforms and
sheath voltages obtained from this analysis are
compared with independent measurements made using a
capacitive probe [1] inserted into the plasma. Ion energy
distributions obtained using the model are compared
with lEDs measured either by a mass spectrometer and
ion energy analyzer system inserted from the side of the
reactor [1], as shown in Fig. 1, or an analyzer
incorporated into the rf-biased substrate electrode [2]
(not shown). Additional plasma characterization was
performed using a Langmuir probe. The Langmuir
probe provides a measurement of the dc plasma
potential which is necessary to calibrate the dc offset
inherent in the capacitive probe. It also provides
measurements of the radial variation of the dc plasma
potential and values for the electron temperature.
discharges is shown in Fig. 2(a). The model includes
two sheaths: the powered sheath—that is, the sheath
adjacent to the rf biased substrate electrode — and the
opposing sheath adjacent to grounded reactor surfaces.
There is no need to model the sheath adjacent to the
inductive source, since the electrostatic shield
minimizes any capacitive coupling between the source
and the plasma. The Ipe(t) and Vpe(t) signals, and
capacitive probe measurements of the bulk plasma
potential, V^(t), show only very small Fourier
components at the fundamental and harmonic
frequencies of the inductive source. These small
components need not be included in the analysis; they
are removed from all measured signals by Fourier
filtering. There is also no need to include in the model
any impedance associated with the bulk plasma. For
the high-density argon discharges studied here, the bulk
plasma impedance is negligible compared to the
impedance of the sheaths [6]. On the other hand,
Langmuir probe studies [3,7] have detected a radial drop
in dc potential across the bulk plasma. Because the rf
biased electrode is located near the radial center of the
reactor but grounded surfaces are located further out, this
radial dc voltage drop, Vr, should be considered to be in
series with the two sheaths, as shown in Fig. 2(a).
The powered sheath and ground sheath are modeled
using a 1-dimensional numerical sheath model described
previously [8]. An iterative procedure solves for the rf
current and voltage across each sheath, subject to the
two constraints imposed by the circuit of Fig. 2(a).
First, the sheath currents must be equal and, second, the
sum of the sheath voltages, plus Vr, must equal the
total voltage Vpe(f)—Vge(f).
To solve the model, one
METHOD
The model used to analyze the rf waveforms of argon
196
must input the Vpe(t)-Vge(t) waveform, obtained from
the rf measurements described above, as well as a
measured or estimated value for Vr. Other model input
parameters are the area of the powered electrode Ape, the
total grounded area Age, the electron temperature Te,
and the total time-averaged ion current across each
sheath, IQPS and lQgs. The model also requires the mass
mi, charge Z/, and relative flux FI of each species of
positive ion.
Several of the model input parameters can be
considered to be constants, since they vary little or not
at all over a wide range of plasma conditions. In the
argon discharges studied here, Ar+ constitutes 98% or
more of the total ion flux [9], so the model need only
consider one ionic species with m/ = 40 amu, Z/ = +e,
and JTJ = 100%. Langmuir probe measurements of argon
discharges [8] yield an electron temperature given by
kfiTe = 3 eV. This value is an effective temperature for
higher energy electrons, which are the most relevant for
the sheath model. Low energy electrons are less
important, since they are reflected back into the plasma
before reaching the electrode surface, and thus they make
no net contribution to the sheath current. Langmuir
probe measurements also yield a value for Vr = 6 V.
The dimensions of the substrate electrode and the reactor
give values of Ape = 81 cm^ and Age = 6000 cm^.
Unlike the other parameters, IQPS and lQgS vary
strongly with inductive source power, and even at fixed
source power they tend to drift. Consequently, IQPS and
lQgs are treated as fitting parameters. They are varied in
iterative fashion until the best agreement is obtained
between the measured rf current waveform Ipe(f) and the
rf current waveform output by the model, If(t).
Specifically, the fitting algorithm minimizes the sum
of the squares of the difference, Ipe(t)-It(t). Typically,
the fitting procedure yields excellent agreement between
Ipe(f) and It(f). When the fit has converged, plasma
potentials, sheath voltages, and lEDs are output by the
model. For ease of comparison, model lEDs are then
broadened by an amount (2 eV) comparable to the
energy resolution of our ion energy analyzer.
The procedure used for CF4 discharges is slightly
different. It will be discussed below. First, however,
results obtained in argon discharges are presented.
ARGON RESULTS
Figure 3 shows plasma potential and sheath voltage
results for argon discharges at three different values of
the rf bias frequency, ranging from 100 kHz in Fig. 3(a)
to 10 MHz in Fig 3(c). In the middle of each figure is
plotted Vpe(f), the voltage waveform on the surface of
the rf biased substrate electrode. Near the top of each
figure we see plots of the bulk plasma potential, Vfr(f).
The solid plot is the Vb(t) waveform measured directly
using the invasive, capacitive probe. The dotted plot is
the Vfr(t) waveform obtained from non-invasive rf
current and voltage measurements using the model-based
197
6
8
10
12
time (ps)
CD
D)
§-50
(b)
1.0
1.5
2.0
2.5
3.0
time (ps)
10MHz
-100
0.7
0.8
0.9
1.0
time ([is)
FIGURE 3: For argon discharges, waveforms of the
plasma potential V/,(0 and the powered sheath voltage
Vps(t) obtained by the noninvasive, model-based method
(dotted curves) fall nearly on top of V/,(0 and V p s ( t )
waveforms measured using an invasive, capacitive probe
(solid). The powered electrode voltage Vpe(t) is also
shown. Conditions are 1.33 Pa (10 mTorr), an inductive
source power of 200 W, a peak-to-peak Vpe(t) of 100 V and
a bias frequency of (a) 0.1 MHz, (b) 1 MHz, and (c) 10 MHz.
procedure described above. Similarly, near the bottom
of each figure we see two plots of the powered sheath
voltage, Vps(f)=Vpe(t)-Vb(t).
The solid curve is the
VpsW waveform obtained from capacitive probe
measurements of Vfr(t). The dotted curve is the Vps(t)
waveform obtained by the noninvasive, model-based
method. The agreement between the invasive and noninvasive measurements is striking. They agree within
±2 V, which is roughly the uncertainty of the capacitive
probe. Thus, one can obtain plasma potentials and
sheath voltages from the noninvasive rf technique with
an accuracy comparable to the invasive measurements.
Excellent agreement was also obtained at other bias
frequencies ranging from 10 kHz up to 20 MHz.
Our plasma reactor is not equipped to measure ion
energy distributions at the rf biased electrode, but
> 200
'(a) peak-to-peakVps(t) = 36V
CM
I 150
> 50
CD
O)
a 100
J
50
g
o
6
10
8
12
time (us)
> 50
0
10
20
30
40
50
60
ion energy (eV)
FIGURE 4: (a) Ar+ ion energy distribution measured by
Woodworth et al. [2] for argon discharges in an inductively
coupled GEC Reference Cell at 3.1 Pa (23 mTorr), with an
inductive source power of 100 W, rf bias at 13.56 MHz, and
a peak-to-peak powered sheath voltage Vps(t) of 36 V. (b)
Ar + ion energy distribution obtained from noninvasive rf
measurements obtained for the same conditions in our GEC
Reference Cell.
2.0
2.5
3.0
3.5
4.0
time (us)
50^
Woodworth et al. [2] have measured such distributions,
for argon discharges, in a GEC Reference Cell that is
ostensibly identical to ours. Figure 4(a) shows an ion
energy distribution for Ar+ reported by Woodworth et
al. Figure 4(b) shows an Ar+ distribution obtained in
our cell under the same conditions, using the
noninvasive rf method. Both distributions show the
familiar two-peaked structure. The energies and relative
amplitudes of the two peaks agree rather well. The
peaks do appear broader in Fig. 4(a) than in Fig. 4(b),
but this may simply be a result of broadening due to the
resolution of the gridded energy analyzer used by
Woodworth et al.
10MHz,
-50
0.5
0.6
0.7
0.8
time (us)
FIGURE 5: For CF4 discharges, waveforms of the plasma
potential V^t) and the powered sheath voltage Vps(t)
obtained by the noninvasive, model-based method (dotted
curves) and measured using an invasive, capacitive probe
(solid). The powered electrode voltage Vpe(t) is also
shown. Conditions are 1.33 Pa (10 mTorr), an inductive
source power of 200 W, a peak-to-peak Vpe(t) of 100 V and
a bias frequency of (a) 0.1 MHz, (b) 1 MHz, (c) 10 MHz.
split the ground sheath into two separate parts as shown
in Fig. 2(b). The orifice and other grounded surfaces
close to the radial center of the reactor are considered one
sheath (called the mass spectrometer sheath, with area
Ams and total ion current I$ms) and the more remote
grounded surfaces such as the vacuum chamber wall are
considered to be another sheath (the wall sheath, with
area Aws and total ion current IQWS). The radial drop in
dc plasma potential contributes a dc voltage in series
with each sheath, denoted Vrm and Vrw. Also, to
obtain good agreement with electrical waveforms
measured at high bias frequencies, e.g., 10 MHz, it is
necessary to include in the model a resistance Rs in
series with the wall sheath. This represents the
resistance of the diffuse plasma adjacent to the wall
sheath as well as the stochastic resistance associated
with the boundary between the plasma and the wall
CF4 RESULTS
The configuration of the plasma reactor used for CF4
studies was somewhat different than for argon. To
extend the operating range, CF4 plasmas were partly
confined by a quartz ring suspended from the inductive
source and a steel plate placed on lower electrode. [1]
The ion energy analyzer and mass spectrometer system
used in the CF4 studies entered from the side of the
reactor and protruded into the intense plasma inside the
confinement ring. The entrance orifice of the mass
spectrometer system is grounded, so the sheath in front
of the orifice is part of the ground sheath. However, the
plasma density and ion current density are much higher
close to the orifice than at the more remote grounded
surfaces. Because of this nonuniformity, it is useful to
198
sheath. It is not necessary to include any resistance in
series with the mass spectrometer sheath. Apparently,
the plasma adjacent to that sheath has a high enough
density that its resistance is negligible.
For CF4 discharges, the model of Fig. 2(b) is solved
in a similar manner to the argon model in Fig. 2(a),
except that we now solve three sheaths simultaneously.
Measured values of constants input into the CF4 model
were Ams = 13 cm2, Aws = 6000 cm2, Ape = 214 cm2,
kpTe = 3 eV, Vrm = 2 V, and Vrw = 6 V. The relative
ion fluxes were 39% CFs+, 27% CF2+, 23% CF+, and
11% F+. For simplicity, the total ion current density at
the mass spectrometer sheath and the powered electrode
sheath were assumed to be equal, that is, lQpS/ApS =
lQmsIAms. The three remaining independent variables,
/Ops, /Qw5» and RS* were treated as fitting parameters,
again adjusted until the rf current waveform output by
the model agreed with the measured Ipe(f) waveform.
Figure 5 shows plasma potential and sheath voltage
results for CF4 discharges, again at four different values
of the rf bias frequency. At each frequency, the plasma
potential Vb(t) as well as the sheath voltage VpS(t)
obtained by the noninvasive, model-based method agree
well with measurements made using the capacitive
probe. The agreement is not quite as good as that seen
for argon discharges in Fig. 3, perhaps because of the
greater number of model input parameters for CF4, or
greater uncertainty in their values.
Figure 6 shows ion energy results for CF4. lEDs
obtained from the noninvasive method are compared
against lEDs measured by the mass spectrometer, at
bias frequencies of 0.1 MHz [Fig. 6(a )and (b)] and 10
MHz [Fig. 6(c) and (d)]. Good agreement is obtained
for the energies of all peaks in the distributions, both at
the lower rf bias frequency, 0.1 MHz, where ion inertia
is negligible, and at the higher rf bias frequency, 10
MHz, where ion inertia is beginning to be dominant.
Like the mass spectrometer measurements, the model
based results show the expected narrowing of peak
separation with rf bias frequency and with increasing ion
mass at 10 MHz. The relative amplitudes of the peaks
are usually in fair agreement, but there are several
disagreements, for example CF3+ at 10 MHz. The
amplitude disagreement would appear to indicate a
failure in the sheath model, perhaps because it uses a
simplified treatment of the boundary conditions between
the plasma and the sheath. Peak amplitudes are more
sensitive than peak energies to the boundary conditions.
CONCLUSIONS
Using the model-based method described here, one
can obtain accurate plasma potential waveforms, sheath
voltages, and ion energy distributions solely from
noninvasive rf current and voltage measurements. One
potential disadvantage of the method is that it requires
values for several model input parameters. A sensitivity
analysis too lengthy to be included here identifies the
electron temperature as a particularly important
parameter. To obtain the most accurate results, the
electron temperature should be known ahead of time, or
monitored by some independent measurement technique.
mass spectrometer data
noninvasive I/V method
100kHz
(c)
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10MHz
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noninvasive I/V method
10MHz
20
40
60
80
100
ion energy (eV)
FIGURE 6: Ion energy distributions in CF4 discharges
obtained by a mass spectrometer system (a) and (c); and by
rf measurements (b) and (d). The bias frequency is 0.1 MHz
in (a) and (b) and 10 MHz in (c) and (d). Other conditions
are the same as in Fig. 5.
199