J. Phys. D: Appl. Phys. 29 (1996) 2098–2103. Printed in the UK
Electron energy dynamics in a RF
discharge with trapezoidal driving
voltage: comparison of experiment
and particle-in-cell Monte Carlo
collision simulation
E Quandt, H-M Katsch and P Mark
Institut für Laser- und Plasmaphysik, Universität Essen, D-45117 Essen, Germany
Received 13 February 1996, in final form 29 April 1996
Abstract. We report on the properties of a special RF discharge characterized by
a trapezoidal shape of the driving voltage pulse. This driving mode offers an
advantage in comparison to the common sinusoidal voltage—in allowing the
separation in time of effects which are connected with the time derivative of the
driving voltage (displacement current, α-mechanism) from effects connected with
the amplitude of the driving voltage (production of electrons at the electrodes,
γ -mechanism).
The energy gain and loss of electrons in this RF discharge are studied in a
collision-dominated helium discharge with special emphasis on the α-mechanism.
Experimental results of space- and time-resolved plasma-induced emission
spectroscopy are compared and combined with the results of a particle-in-cell
Monte Carlo collision simulation. It is demonstrated that the electrons gain energy
during the temporal change of the discharge voltage at high neutral particle
pressure (400 Pa) in the vicinity of both sheaths and that they dissipate energy in
these regions by inelastic collisions. At a lower neutral particle pressure (100 Pa)
the electrons gain energy in the vicinity of the expanding sheath and lose energy at
the contracting sheath. In this case the energy loss by inelastic collisions takes
place in the bulk plasma.
1. Introduction
Low-temperature RF plasmas are becoming increasingly
interesting because of the technical application potential for
materials treatment [1]. The parallel plate configuration is
applied on a large scale in the semiconductor industry, e.g.
for etching and layer deposition. A basic understanding of
the mechanisms of radio frequency discharges is required
to overcome the widespread method of ‘trial and error’
in process control and to arrive at an optimum choice of
the most important plasma parameters. These discharges
have therefore been investigated experimentally [2, 3], by
analytical modelling [4, 5] and by computer simulations
[6–8] over recent years.
The experimental and theoretical studies refer so far
to sinusoidal excitation of the discharges whereby the
heating mechanisms are investigated mostly by spaceand time-resolved plasma-induced spectroscopy. With the
exception of RF discharges in the upper kilohertz domain,
interpretation of time-resolved plasma-induced emission
leads to the following problem. The largest contribution
to electron heating on the basis of the displacement current
c 1996 IOP Publishing Ltd
0022-3727/96/082098+06$19.50 (α-mechanism) is expected with only a short temporal delay
in connection with the highest rate of change of the voltage,
e.g. at the zero-crossing of the driving voltage. A quarter
period later, the highest voltage is reached at the electrodes
so that at the respective cathode a contribution to the
electron gain is expected by secondary electron emission
(γ -mechanism). Both mechanisms lead to excitation and
ionization with a continuous transition between the two
mechanisms. Furthermore, with the common 13.56 MHz
discharge the time delay between the zero-crossing and the
maximum of the discharge voltage is of the same order of
magnitude as the typical lifetime of excited states of atoms,
so that identification of the heating mechanisms on the basis
of the time-resolved light emission is often ambiguous.
For this reason a special type of discharge was
developed whereby the driving voltage exhibits a
trapezoidal profile in time. This allows one to distinguish
between effects which are present during the steep voltage
edges which are connected to the α-mechanism and
between effects which are present during the voltage plateau
and which can be connected to the γ -mechanism. With
RF discharge with trapezoidal driving voltage
this mode of excitation the question of energy gain and
loss of electrons based on the α-mechanism in a collisiondominated helium discharge is investigated. In order to
arrive at this goal the results of space- and time-resolved
plasma-induced emission spectroscopy are compared and
combined with results of particle-in-cell (PIC) Monte Carlo
collision (MCC) simulations with special emphasis on the
space- and time-resolved total excitation of neutrals and the
electron energy gain (j · E). Variation of the amplitude of
the driving voltage allows one to demonstrate the transition
of a discharge driven mainly by the displacement current
and predominantly by secondary electron emission (α-mode
to γ -mode transition).
2. Experimental set-up
The discharge is operated in helium between two circular,
parallel, stainless steel electrodes (diameter 12 cm, distance
4 cm) in the pressure range 100 Pa to 400 Pa.
The arrangement is similar to the GEC reference cell.
The background pressure in the vacuum chamber is
approximately 10−6 Pa, the average gas flow amounts to
30 sccm.
The discharge is radially confined by a floating
cylindrical electrode to provide a defined loss and defined
pressure independent of the discharge volume so that
better comparability with the one-dimensional model can
be expected. Openings within this electrode, covered by
fine meshes or provided with long tubes, allow both gas
supply and optical diagnostics.
The discharge voltage has a trapezoid-like profile
with exponential wings (τwing ' 30 ns); the frequency
is 300 kHz. Alternating between the electrodes, the
instantaneous anode is grounded, and the cathode is driven
with the trapezoid voltage pulse (voltage of the plateau
160 V < Utrapez < 300 V, negative with respect to ground).
The input power is less than 5 W. A schematic plot of the
power circuit is shown in figure 1.
3. Diagnostics
A schematic overview of the diagnostics is given in figure 2.
The plasma density is measured during the voltage
plateau, resolved in time and space with a Langmuir
probe. The temporal resolution of the probe measurement is
essential, because strong fluctuations of the probe current
during the voltage change and in the first quarter of the
voltage plateau are observed. These fluctuations would
lead to an error in time-integrated measurements. A
temporal dependence of the probe characteristic is not
observed during the rest of the voltage plateau. A technique
developed for pulsed plasmas [9] has been used; it is based
on the measurement of the probe current as a function of
time for a specific setting of the probe voltage. By changing
the voltage setting in a sequence of steps and combining
the relevant probe currents the probe characteristic can be
constructed for each temporal point within the period. The
temporal resolution of this method is 100 ns, the spatial
resolution is about 1 mm.
For spatiotemporally resolved optical emission spectroscopy a small section of the plasma on the symmetry
axis is imaged onto the entrance slit of a monochromator
with a movable mirror and lens system. Several lines (e.g.
31 P–21 S, 33 P–23 S, 41 S–21 P, 43 S–23 P) were observed but
did not show significantly different results. The emission
profiles discussed in addition refer to the atomic transition 31 D–21 P (667.8 nm). This line was chosen for three
reasons. First, excitation in the upper 31 D state from the
metastable 21(3) S state (excitation energy 2.5(3.3) eV) can
be neglected. An estimation shows that the rate for excitation in the upper state is much larger for excitation from
the ground level than from the metastable level:
Z
f (E)σexc(11 S−31 D) n(11 S) dE
Z
f (E)σexc(21(3) S−31 D) n(21(3) S) dE.
Second, the radiative lifetime of the upper level is very
short (τlif e ' 15 ns). The application of a commonly
used numeric deconvolution algorithm to reconstruct the
temporal evolution of excited levels [10] (which is often
questionable [11, 12]) is not necessary. The lifetime of
the upper level is much shorter than the time constant
of the voltage change (τwing ' 30 ns) and the duration
of the voltage plateau (tplateau ' 1.7 µs), thus a
temporal superposition of the emission initiated by the
voltage change (α-mechanism) and by the voltage plateau
(γ -mechanism) should not be expected. Due to the
small velocity of the excited atoms (gas temperature
approximately 300 K) the effect of the lifetime of the
upper level on the spatial structure of the emission can
be neglected. Third, as estimated, radiation trapping of
this line can be neglected (optical thickness approximately
0.001). Consequently, the emitted light reflects directly the
excitation of the upper level of the observed transition from
the ground state (excitation energy 23.1 eV) and therefore
the electron energy loss and the ionization (ionization
potential 24.6 eV).
The spatial profile of the density of metastables is
obtained by exciting the 31 P level from the metastable 21 S
state with a dye laser (501.5 nm) and detecting the emission
of the same transition.
4. Simulation set-up
The PIC MCC simulation codes PDP1 (PC version) and
XPDP1 (workstation version) developed by Birdsall [8]
are used. The PDP1 simulation runs with approximately
9000 computer particles, a spatial cell width of 0.2 mm
and a mean time interval of 5 × 10−11 s. These values are
chosen under the conditions ωpe 1t < 2 and λD /1x > 2
[13, 14]. The number of computer particles is chosen with
regard to stability and minimization of computation time
and is in agreement with the literature [e.g. 15] taking
into account the lower plasma density in the trapezoid-like
driven discharge. The cross sections for particle–particle
collisions are taken from experimental and theoretical data
(electron–atom: elastic collisions [16, 17], excitation [18],
ionization [19] ; ion–atom: elastic collisions [19], charge
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E Quandt et al
Figure 1. Schematic plot of the power circuit: 1, electrodes; 2, power resistors; 3, MOSFET transistors; 4, amplifier;
5, inverting amplifier; 6, opto coupler; 7, pulse generator.
Figure 2. Schematic of experiment and diagnostics.
transfer collisions [20]) and are fitted to the general profiles
(cross section versus incident particle energy) for electron–
atom and ion–atom collisions given by Birdsall [8]. A
value of γ = 0.18 [18] is adopted for the coefficient of
secondary electron emission due to the incidence of ions on
the electrodes. For more details on the simulation code the
reader is referred to the literature [8, 13]. At first, simulated
discharges with parameters of the experiment always ‘died’.
On the basis of this observation we decided to consider the
role of metastables neglected so far in the simulation. In
order to take account of their influence, a sinusoidal spatial
profile was adopted (this relative spatial variation of the
metastables was inferred from laser-induced fluorescence
measurements). Ionization of these metastables with a
cross section σmax = 1.5 × 10−19 m−2 [18] and secondary
electron emission due to the incidence of metastable atoms
(γ = 0.7, see [21]) was taken into account. The
maximum value of the metastable concentration was then
varied in the simulation until the resulting spatial-resolved
2100
plasma density was in agreement with the experimentally
determined value (e.g. np = 3 × 1014 m−3 at p = 100 Pa,
Utrapez = 160 V). In this way, a density of metastables of
1.5 × 1016 m−3 was determined for the trapezoid discharge
at 100 Pa and 160 V. Riley et al [22] measured in a
13.6 MHz helium discharge (np = 2 × 1015 m−3 , p =
0.5 Torr, 50 V zero-to-peak voltage) a total density of
metastables of approximately 1.5 × 1017 m−3 . Taking
into account that in the 300 kHz trapezoid discharge the
plasma density is lower by nearly one order of magnitude
than in the 13.6 MHz sinusoidal discharge, a density of
metastables of 1.5 × 1016 m−3 seems realistic for this
discharge. Comparing ionization rates of metastables and
of ground level atoms (in the simulation) leads to the result
that, in the investigated discharge, the dominant role of
metastables is the emission of secondary electrons. At
densities of metastables higher than around 1 × 1017 m−3
Penning ionization has to be taken into account as an
additional process [23].
RF discharge with trapezoidal driving voltage
Figure 3. Emission on the 667.8 nm helium line.
Utrapez = 300 V, T = 3.33 µs, electrodes at x = 0 m and
x = 0.04 m. (a ) p = 100 Pa, (b ) p = 400 Pa.
5. Results
Figure 3 shows the spatio–temporal emission profiles of a
discharge running at high voltage (300 V) and different
pressures (100 Pa, 400 Pa). The emission signal is
maximum during the trapezoid plateau and drops from the
cathode to the anode. This light is obviously caused by
secondary electrons. Assuming that excitation out of the
ground level and ionization are caused by collisions of
neutrals with electrons of similar energy, this discharge is
driven by secondary electrons and can be described as γ type. Comparing the emission at low (figure 3(a)) and high
pressure (figure 3(b)), the dependence of the penetration
depth of secondary electrons in the plasma on the pressure
can be noted.
Figure 4(a) shows in contrast the spatio–temporal
emission profile of a discharge running at low voltage
(160 V) and high pressure (400 Pa). Here, emission during
the plateau is reduced, and a signal during the voltage
change appears with a peak in the vicinity of the expanding
sheath (switching the neighbouring electrode from anodic
to cathodic). This emission is caused by electrons gaining
energy by mechanisms due the change of discharge voltage.
Therefore, this discharge can be described as α-type.
Figure 4. Trapezoidally driven helium discharge at 400 Pa
pressure. Utrapez = 160 V, T = 3.33 µs, electrodes at
x = 0 m and x = 0.04 m. (a ) Experiment: emission on the
667.8 nm helium line (corresponds to electron energy loss).
(b ) Simulation: total excitation (corresponds to electron
energy loss). (c ) Simulation: j · E (corresponds to electron
energy gain).
Figure 4(b) shows the spatio–temporal excitation of a
simulated discharge with the same set of parameters as
figure 4(a) (160 V, 400 Pa). It is also maximum during
the voltage change in the vicinity of the expanding sheath.
The agreement of measured emission (figure 4(a)) and
simulated excitation (figure 4(b)) is obvious.
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E Quandt et al
Figure 5. Trapezoidally driven helium discharge at 100 Pa
pressure Utrapez = 160 V, T = 3.33 µs, electrodes at
x = 0 m and x = 0.04 m. (a ) Experiment: emission on the
667.8 nm helium line (corresponds to electron energy loss).
(b ) Simulation: total excitation (corresponds to electron
energy loss). (c ) Simulation: j · E (corresponds to electron
energy gain).
The fate of an electron from gaining to losing energy
is marked by two kinds of collisions—the directional
energy gained in the electric field is randomized by elastic
collisions with atoms and is lost by inelastic exciting or
ionizing collisions with atoms. The mean free path for
these collisions leads to an estimate of the spatial distance
2102
of energy gain and loss. At a pressure of 400 Pa λel is
approximately 0.2 mm and λinel approximately 4.5 mm.
This shows that, compared with the dimensions of the
discharge (40 mm), energy gain and loss occur in close
proximity. This is confirmed by the appropriate simulated
energy gain (j ·E, figure 4(c)), which exhibits a peak in the
vicinity of the expanding sheath also. An additional smaller
energy gain in the plasma bulk and in the vicinity of the
contracting sheath (switching the neighbouring electrode
from cathodic to anodic) is obviously related to electrons
with an energy below the excitation energy. According to
a previously published analytical model [24], the energy
gain in the different discharge regions can be explained
as follows. Close to the expanding sheath energy gain
occurs due to collision-dominated sheath oscillation heating
(reflection and acceleration of electrons at a sheath edge
moving towards them) and ohmic heating due to elastic
collisions of the plasma electrons carrying the displacement
current. In the plasma bulk, energy gain can be attributed
only to ohmic heating. In the vicinity of the contracting
sheath, energy gain is again the sum of sheath oscillation
heating and ohmic heating. In this case, however, the
effect of the sheath edge moving away from the electrons
results in a negative sheath oscillation heating (diffusion
cooling). Because the amount of (negative) oscillation
heating is smaller than the amount of (positive) ohmic
heating, a positive energy gain results. This corresponds
to the field reversal heating observed in a sinusoidal RF
discharge [25, 26] .
At low pressure, the assumption that energy loss and
gain occur close together never holds. The mean free paths
for elastic and inelastic collisions for a pressure of 100 Pa
are approximately λel = 0.7 mm and λinel = 18 mm
respectively, figure 5(a) shows the measured emission
profile at 100 Pa and a voltage of 160 V. The emission is
again maximum during the voltage change, but is broadened
in space. In the simulation, the total excitation (figure 5(b))
at a pressure of 100 Pa shows similar behaviour. The
energy gain of the electrons in the simulation for a pressure
of 100 Pa (figure 5(c)) shows a positive peak during the
voltage change in the vicinity of the expanding sheath
and a negative peak (electron cooling) in the vicinity of
the contracting sheath. The energy gain in the different
discharge regions occurs on the basis of the different
heating mechanisms in an analogous way to the highpressure (400 Pa) case. However, there is now a weaker
influence of the ohmic heating as compared to sheath
oscillation heating in the vicinity of the sheaths due to a
reduction of elastic electron–atom collisions. In the vicinity
of the contracting sheath the addition of positive ohmic
heating and negative sheath oscillation heating (slowing
down of the electrons) therefore results in negative electron
heating, equivalent to electron cooling. The influence
of stochastic sheath oscillation heating [27, 28] can still
be neglected, but has to be taken into account at lower
pressures.
6. Conclusion
Comparing and combining experimental measurements and
particle-in-cell Monte Carlo collision simulations allowed
RF discharge with trapezoidal driving voltage
us to identify the spatial and temporal characteristics
of energy gain and loss of electrons in a special radio
frequency discharge driven by a trapezoidal voltage profile.
In this collision-dominated discharge, the energy gain
of the electrons occurs predominantly in the vicinity of
the sheaths; at low pressure, this gain becomes negative
in front of the contracting sheath.
Energy loss of
the electrons occurs at high pressure similarly in the
vicinity of the sheaths; at low pressure, however, it
occurs in the plasma bulk. The applicability of the
results to a sinusoidal driven discharge was verified by
an analogous investigation of a 3.2 MHz symmetrically
coupled sinusoidal discharge. Good agreement between
experiment and simulation was also obtained in this way.
Due to the nature of the sinusoidal driven discharge, there
is a complex superposition of α- and γ -mechanisms over
most of the cycle—the related emission profiles are thus
difficult to interpret.
Acknowledgments
We are grateful to W Saure for the development of the
trapezoid pulse generator. This work is supported by
the ‘Deutsche Forschungsgemeinschaft’ in the frame of
Sonderforschungsbereich 191.
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