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WDS'05 Proceedings of Contributed Papers, Part II, 306–312, 2005.
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The Role of Secondary Electrons
in Low Pressure RF Glow Discharge
O. Brzobohatý and D. Trunec
Department of Physical Electronics, Faculty of Science, Masaryk University,
Kotlářská 2, 611 37 Brno, Czech Republic
Abstract. We concerned on the role of secondary electrons in low pressure rf glow
discharge. The Particle in Cell/Monte Carlo computer simulations (PIC/MC) was
applied to simulate the discharge. The influence of secondary electron emission yield
(SEY) on the plasma density was studied. Furthermore the relative density and the
electron energy probability function of secondary particles, i.e., secondary electrons
and the products of ionizing collisions of secondary electrons was determined. We
found that the secondary particles play very important role in this type of discharge.
The relative density of secondary electrons created on electrodes is 2-10%, 5-15%
for the value of SEY = 0.1, 0.2 respectively. But due to the high electric intensity in
electrode sheaths these electrons have relatively high energy and therefore they can
ionize neutral atoms. As consequence of this the relative density of the secondary
electrons (secondaries) and electrons created in ionizing collisions of the secondaries
is more than 40% and the relative density of ions created in ionizing collisions of
the secondaries is more than 20%, 40% for value of SEY = 0.1, 0.2, respectively.
Introduction
Low pressure radio frequency (rf) discharges are often used for sputtering, etching and plasma
enhanced chemical vapor deposition (PECVD). It is well-known that the secondary electron emission on
the electrodes affects the plasma density [Ruas, et al., 2002, Lieberman and Lichtenber, 1994].
In the present paper the role of the secondary electrons and the influence of the value of the secondary emission yield (SEY) on the plasma density are studied via computer simulation. Our simulation
model was based on Particle In Cell/Monte Carlo (PIC/MC) method. The PIC/MC method was used
extensively for the study of plasma and the method is well documented in [Birdsall et al.,1991] and
[Vahedi and Surendra,1991]. Modified simulation program PDP1 [Verboncoeur et al., 1993] was used.
Model description
We studied low pressure rf capacitively coupled discharge. One dimensional electrostatic model was
applied, i.e., the electric field was only in one direction perpendicular to electrodes. Our model system
was composed of two planar parallel electrodes, the diameter of the electrode was chosen 10 cm, the
distance between the electrodes was chosen 3 cm. One of the electrode was powered by rf voltage with
amplitude U0 = 150 V and frequency f = 13.56 MHz, the second one was grounded (zero voltage). Argon
(pressure 3–20 Pa) was used as working gas. For the background neutral gas we considered Maxwellian
velocity distribution with temperature 300 K. Density of neutral gas was uniform and depended only on
the gas pressure.
So we have plasma which is composed from electrons and positive ions, they move in the electric
field and collide with the neutral atoms of the working gas. Their movement can be written by the
Lorentz equation. Only the binary collisions were considered, as can be seen in the Table 1.
Table 1. Collision processes used in the
15.7 eV.
electrons + Ar
elastic collision
excitation
ionization
Ar+ + Ar
scattering
charge transfer
simulation. The energy necessary to the ionization is about
e + Ar → e + Ar
e + Ar → e + Ar∗ (11.55 eV)
e + Ar → 2e + Ar+ , εi = 15.7 eV
Ar+ + Ar → Ar+ + Ar
Ar+ + Ar → Ar + Ar+
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BRZOBOHATÝ AND TRUNEC: THE ROLE OF SECONDARY ELECTRONS . . .
Interactions of charged particles with the electrodes were considered, e.g., absorption, secondary
emission. The secondary emission was included through the secondary emission yield (SEY). Generally
the SEY depends on the energy and impact angle of impacting electron or ion. In our case we studied
mainly the role of secondary electrons in the discharge. The dependence of the SEY on impact energy
and angle was not considered in our calculations. We considered the same value of the SEY for both
electrons and ions.
Our computer model is based on two methods, i.e., Particle In Cell method and Monte Carlo method,
so a particle computer simulation was applied. The PIC method was used for calculation of motion of
charged particles and the MC method was used for simulation random processes, e.g., collisions of charged
particles. Both used methods are well-known in the field of computer simulation. In next two sections
we briefly describe both methods.
Particle In Cell method
If we want to calculate the motion of N charged particles in electromagnetic field (in our case only
electric field) then we have to solve self-consistently equations of motion
mi r̈i = qi E = −qi ∇φ, i = 1, . . . , N,
with Poisson equation
∆φ = −ρ(r1 , . . . , rN )/ε0 .
Both equations were solved by finite-difference method. The equations of motion were solved by
well-known leap-frog method :
n+1/2
mi
vi
n−1/2
− vi
∆t
= qi E n ,
rin+1 − rin
n+1/2
= vi
, i = 1, . . . , N,
∆t
n+1/2
where vi
is velocity, r n is position of ith particle, n denotes time step (tn+1 = tn + ∆t).
The Poisson equation has form:
φj−1 − 2φj + φj+1
ρj
= − , j = 1, . . . , M − 1
2
(∆x)
ε0
where φj is potential and ρj is density on the jth grid point of mathematical mesh. Boundary conditions
φ0 = U0 cos(2πf t), φM = 0 were used. We need to know charge density ρj at the grid points (used in
our finite-difference schema) for the solution of Poisson equation. This charge density was obtained from
the positions of charged particles (r1 , . . . , rN ). The method of determination of electric charge density
from the positions of particles gives the name of the whole method – Particle In Cell. There are many
ways how to do it. We used so called Cloud In Cell–Particle In Cell method that linearly assigns the
charge of the particles to the grid points.
Note that for the spatial and the time step exist the conditions of precision. The spatial step is then
given by the condition l∆x
< 1, where lrfsh is diameter of plasma sheath. The spatial step ∆x is about
rfsh
4 × 10−4 m. The time step is given by the condition ωpl ∆t < 0.2, where ωpl is the plasma frequency.
This condition is in our case fullfiled for ∆t < 10−11 s. Note that the plasma frequency depends on the
plasma density and its value was then about 109 s−1 for our plasma density (≈ 101 5 m−3 . More detailed
description of the PIC method can be found in [Birdsall et al., 1991].
n
n
The computation cycle was composed from four steps, i.e., 1. (r1n , . . . , rN
) → ρ(r1n , . . . , rN
), 2.
n
n
n
n
~ , 3. collissions of charged particles using Monte Carlo method and 4. E
~n →
ρ(r1 , . . . , rN ) → φj → E
i
i
n+1
(r1n+1 , . . . , rN
), tn+1 = tn + ∆t, jump to the step 1.
Monte Carlo method
The MC method is method which make use of (pseudo)random numbers with a given probability
function. The MC method is very often used for simulation of random processes, e.g., collisions of
charged particles with neutrals, secondary emission on the electrodes and generation of initial velocities.
The MC method – used in our model for simulation of collisions – is based on the calculation of
probability of collision in time (t, t+∆t). Time element ∆t is used in finite difference schema of equations
of motion. In this method the probability of collision in (t, t + ∆t) is given by
P (∆t) = 1 − exp(−ν∆t).
307
(1)
BRZOBOHATÝ AND TRUNEC: THE ROLE OF SECONDARY ELECTRONS . . .
The collision frequency of ith type of collision is given generally by νi (r, ε) = nσ(ε)i vr (ε), where
n is the density of background neutral gas, σ is collision cross section and vr is the relative velocity
of colliding particles, ε is the energy of thePcolliding particle. The null collision method was used and
therefore the total collision frequency νt = i νi is constant. From this probability a number of particles
that will collide in time (t, t + ∆t) is determined by following equation Ncoll = P (∆t)Ntot , where Ntot
is total number of particles – electrons or ions. This method is more detailed discussed in [Nanbu et al.,
2000].
This method determines when a collision occurs but says nothing about its type. The type of
collision is given by
ic
ic
X
X
νi−1
νi
≤r<
,
(2)
νt
ν
i=1
i=1 t
where ic is the type of collision that will occur and r is the random number uniformly distributed in
(0, 1). The knowledge of differential cross section and total cross section is very important for the MC
method because these quantities determine the probability (2) of the given type of collision and the solid
angle for scattering. The collision data for argon can be found in [Lieberman and Lichtenberg, 1994] or
[Vahedi and Surendra, 1994].
The SEY gives the probability of secondary emission after impact of electrons, ions, respectively.
The condition for realization of the secondary emission is then given by r ≤ SEY, where r is again the
random number uniformly distributed in (0, 1).
Results of the simulations
In our simulations we monitored the secondary electrons and products from the ionizing collisions of
secondary electrons with neutrals (their trajectories and collisions) in accordance with model assumptions
in the some way as other plasma charged particles. The influence of the value of the SEY on the plasma
density was studied. The plasma density and the electron energy probability function were measured in
the middle of the system. We chose two values of the SEY, i.e., 0.1 and 0.2 and we studied the influence
of change of the value of the SEY on the plasma density.
We studied three groups of electrons. First studied group was composed of secondary electrons
created on the electrode – secondaries. Second studied group was composed of secondaries and electrons
created in ionizing collisions of secondaries – SECI and third group of electrons was all electrons in
system. Note that the SECI included electrons created in the ionizing collisions of SECI.
It is known that in the sheaths (electrode regions) is the highest electric field in the system and
therefore the secondary electrons and electrons created in sheaths gain a lot of energy. The energy that
secondaries gained in the plasma sheath is proportional to the plasma potential (in our case about 70 V)
and to phase of the rf field, in which the secondaries were created on the electrode. We calculated the
electron energy probability function (EEPF) for secondaries to show the relatively high temperature of
secondaries, further we calculated the EEPF for SECI and finally we calculated the EEPF for all electrons
in our system. The EEPFs are shown in Fig. 1 (pressure 4 Pa). As can be seen the secondaries have
Maxwellian-like distribution
with relatively high temperature about 3 eV. Note that the EEPF is defined
√
by f (E) = F (E)/ E, where F (E) is electron energy distribution function (EEDF) √
and E is energy
of
E
electrons. The EEDF of the Maxwellian distribution has this form F (E) = √2π (kT1)3/2 E exp − kT
. So
the EEPF of Maxwellian distribution is represented by a line in the semi-log plot. The temperature of
electrons is then simply given by tangent of this line. The SECI have bi-Maxwellian-like distribution as
well as all electrons in our system, the lower temperature is about 0.5 eV and the higher is about 3 eV.
This bi-Maxwellian distribution was measured by Godyak et al., [1990] too, here the lower temperature
was 0.34 eV and the higher temperature was 3.1 eV (the distance between the electrodes was 2 cm, the
linear diameter was 14.2 cm).
The high-temperature group of electrons is created in sheaths and the low-temperature group of
electrons is created in ionizing collisions. Furthermore this bi-Maxwellian effect is forced by Ramsauer
effect in argon. The low-temperature group of electrons has mean energy close to the energy of the
Ramsauer minimum (in the cross section of the elastic collision of electrons with neutrals) therefore they
move almost collisionlessly. Due to their low energy they cannot overcome ambipolar potential barrier in
the plasma body, so they cannot reach the oscillating plasma sheath interface, where stochastic heating
take place, thence they cannot gain a lot of energy. Number of low-temperature electrons is 80 % for
low pressure and the number decreases with pressure.
On the other hand the high-temperature group of electrons effectively interacts with argon atoms (in
308
BRZOBOHATÝ AND TRUNEC: THE ROLE OF SECONDARY ELECTRONS . . .
1016
secondaries
SECI
all electrons
f(E) [eV-3/2 m-3]
1015
1014
1013
1012
1011
1010
2
0
4
6
10
8
E [eV]
12
14
Figure 1. The electron energy probability functions of studied groups of electrons.
4
secondaries
SECI
all electrons
kTH
3.5
kT [eV]
3
2.5
2
1.5
1
kTL
0.5
0
2
4
6
8
10 12
p [Pa]
14
16
18
20
Figure 2. The pressure dependence of temperatures of studied groups of electrons in the system.
collisions) and boundaries (electrodes). They compensates their energy losses through stochastic heating
on the oscillating plasma sheath boundaries. Unlike the low-temperature electrons the high-temperature
electrons easily overcome the ambipolar potential barrier and collide more frequently with the axial
plasma boundaries [Godyak et al., 1990].
The bi-Maxwellian-like distribution is observed only at low pressures (up to 12 Pa). The Ohmic
heating is more important for higher pressures and therefore the bi-Maxwellian like distribution came
to Mawellian like distribution, i.e., the low-temperature group of electrons are thermalized by Ohmic
heating. The pressure dependence of temperatures of all groups of electrons in the system is shown in
the Fig. 2. It can be seen that both temperatures merged with growing pressure.
The influence of the value of the SEY on the plasma density was determined. The pressure dependence of the plasma density, calculated in the middle of the system, is shown in Fig. 3. If we increase
the value of the SEY from 0.1 to 0.2 the calculated plasma density increases of about 30%, see Fig. 4.
The relatively high errorbars are mainly given by number of particles in the computer cell in mathematic
mesh. The experimental determination of the influence of the material of the electrodes (different values
of the SEY) can be found in [Ruas et al., 2002]. But the values of the SEYs of electrodes material were
309
BRZOBOHATÝ AND TRUNEC: THE ROLE OF SECONDARY ELECTRONS . . .
6
5.5
5
n [1015 m-3]
4.5
4
3.5
3
2.5
2
SEY = 0.1
SEY = 0.2
1.5
1
2
4
6
8
10 12
p [Pa]
14
16
18
20
18
20
Figure 3. The influence of the value of SEY on the plasma density.
40
nrel
38
nrel [%]
36
34
32
30
28
26
2
4
6
8
10 12
p [Pa]
14
16
Figure 4. The influence of the SEC on the plasma density in dependence on pressure. On the ordinate
there is plotted the relative increase of plasma density when increasing the SEC from 0.1 to 0.2. The
relatively increase of plasma density nrel = n0.2 /n0.1 − 1, where n0.1 , n0.2 is plasma density in the middle
of the system for the SEY = 0.1, 0.2, respectively.
not determined. So the comparison with experiment could not be done.
The density (in the middle of the system) of secondaries was relatively low, i.e., nsecele/ntotal =
3 − 10 %, 5 − 16, for the value of the SEY = 0.1, 0.2 respectively. Newertheless they play, due to their
relatively high temperature, very important role. They have enough energy to ionize neutral atoms and
therefore the relative concentration of the SECI is more than 40%, as you can see in Fig. 5. Ions created
in ionizing collisions of secondary electrons comprise more than 20%, 40% for SEY =0.1, 0.2 respectively,
as can be seen in Fig. 6. So the change of the value of the SEY from 0.1 to 0.2 caused that the relative
density (nsec ion /ntot ) of ions created in ionizing collisions of secondaries increase by about 20 %. This
is caused by increase of secondaries by about 2 − 6 %.
310
BRZOBOHATÝ AND TRUNEC: THE ROLE OF SECONDARY ELECTRONS . . .
35
SEY = 0.1
SEY = 0.2
30
nsec ele/n [%]
25
20
secondaries
15
SECI
10
5
0
2
4
6
8
10 12
p [Pa]
14
16
18
20
Figure 5. The relative density (nrel = nsec ele /n, where nrel is relative density in the middle of the
system, nsec ele is density of secondaries or SECI and n is total plasma density) versus pressure.
60
55
nsec ion/n [%]
50
45
40
35
30
25
SEY = 0.1
SEY = 0.2
20
2
4
6
8
10 12
p [Pa]
14
16
18
20
Figure 6. The relative concentration of ions created in the ionizing collisions of the SECI versus pressure
(nsec. ion /n, where nsec. ion is density of ions created in the ionizing collisions of secondaries, n is total
plasma density).
Conclusion
In the present contribution the role of secondary electrons was studied via computer simulation.
The computer model was based on the Particle In Cell and Monte Carlo method. We shown that the
secondary electrons are very important due to their relatively high temperature. More than 40% of all
electrons in the system was created on electrodes and in ionizing collisions of secondaries. More than
25% of all ions are ions created in ionizing collisions of secondaries. The EEPFs for different groups of
electrons, i.e., for secondaries, for secondaries and electrons created in ionizing collisions of secondaries
and for all electrons in the system, were calculated. We shown that the secondaries have Maxwellian-like
EEPF. The secondaries and their ionizing products have bi-Maxwellian-like distribution (for pressure
less than 10 Pa) as all electrons in the system. These interesting results could be important in plasma
processing like plasma deposition or sputtering.
311
BRZOBOHATÝ AND TRUNEC: THE ROLE OF SECONDARY ELECTRONS . . .
Acknowledgments.
This research has been supported by the Czech Science Foundation under project
No. 202/03/0827 and grant FRVŠ No. 2910/2005.
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