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Ralf Krimke, Herbert Urbassek
To cite this version:
Ralf Krimke, Herbert Urbassek. PIC/MC Modeling of an Ion Source: Case Study. Journal de Physique III, EDP Sciences, 1996, 6 (9), pp.1219-1228. �10.1051/jp3:1996181�. �jpa-00249520�
J. Phys. III France 6 (1996) 1219-1228 SEPTEMBER1996, PAGE 1219
PIC/MC Modeling of an Ion Source: Case Study
RalfKrimke (*) and Herbert M. Urbassek
Fachbereich Physik, Universitit Kaiserslautem, Erwin-Schr6dinger-Strafle, 67663 Kaiserslautern, Germany
(Received 8 January 1996, received in final form 20 May 1996, accepted 22 May 1996)
PACS.52.80.-s Electric discharges
PACS.52.75.Rx Plasma applications in manufacturing and materials processing
Abstract. Kinetic simulations on the basis of the PIC /MC code allow for the self-consistent kinetic simulation of gas discharges under conditions where other simulation methods fail. We present a case study of a low pressure asymmetric magnetized capacitively coupled RF discharge.
The one-dimensional treatment of the system can be extended to model ion extraction in the axial direction from this discharge, thus simulating an ion source. We present results on the properties
of the sheath and bulk regions, on the efficiency of the extraction, and on the dependence of the power balance of the discharge, the magnetic field and the wall material.
1. Introduction
Ion and plasma sources are important tools for a multitude of surface treatment techniques.
Since they are often based on complex gas discharge configurations, modeling of these sources is helpful for optimizing their design and performance. A large variety of modeling tools is
available, ranging from electric circuit analogues over macroscopic fluid dynamic models to
microscopic kinetic simulations ii,2]. For low pressure discharges, in which non-local efsects of the electric field on the ion and electron distribution may be strong, fluid dynamic models
usually fail [3]; in this regime, kinetic simulations offer the possibly only means to accurately model the behavior of a discharge.
As the main kinetic simulational tool, the particle-in-cell/Monte-Carlo method (PIC/MC) [4,5] has played a major role in the recent past. This method uses of the order of 10~ to 10~
pseudo-electrons and -ions to simulate the behavior of the considerably larger number of real electrons and ions in the discharge. These pseudo-particles move under the influence of the electric field which they generate by their space charge density; by collisions with neutral gas particles, they may ionize them and thus keep the discharge burning. Thus, by following the
trajectories of these pseudo-particles and the evolution of their electric fields, a modest but
hopefully representative replica of the behavior of the real particles, and thus of the entire
plasma dynamics, is obtained.
We shall show in this paper how, by using the PIC/MC method, the dynamics and the
performance of a concrete plasma source can be modeled. As an example for this plasma
source, we choose a low pressure cylindrical Ar source, which is capacitively coupled to a high
(*) Author for correspondence (e-mail: krimke@physik.uni-kl.de)
© Les kditions de Physique 1996
Ji,x .je.x
i(t) c v(t>
~
h (
~i B
Q~
Fig. I. Schematic view of the plasma source. j~,x (j~,x) are the current densities of ions (electrons).
See text.
frequency generator; an axial magnetic field helps to confine the plasma and sustains it [6].
We shall present a summary over several results which we have obtained in the recent 2 years,
concentrating after a presentation of the general discharge characteristics on the extraction process and its efficiency, and on the power balance in the discharge and its dependence on the
magnetic field and the properties of the electrode material.
2. System and Model
The ion source which we simulate is operated between two coaxial electrodes of radii ro
" 5 mm
and ri
" 35 mm in a cylindrical vessel of height h
= 45 mm, cf. Figure 1. An axial magnetic field, which will unless otherwise stated be fixed to 7 mT helps to confine the plasma.
The external circuit is simplified to a voltage source V(t)
= V~~ sin 2~un~t, with amplitude Vn~ = 500 V and frequency u~~
= 13.56 MHz; it is coupled to the inner electrode by a capacitance C
= 10 pF. The outer electrode is grounded. The discharge is operated with Ar
at temperature 300 K and pressure 1.36 mTorr [7].
The PIC/MC simulator works as follows: we use a number of N rs 24 000 pseudo-electrons and -ions to simulate the discharge behavior. The electric field generated by these particles is
calculated via Poisson's equation on an equidistant grid. The particles move under the action of their self-consistent electric field and the external magnetic field. They may collide with neutrals; here we take into account ionizing, exciting and elastic electron-Ar collisions, as well
as elastic and charge exchange ion-neutral collisions. The boundary conditions of the discharge
are calculated in consistence with the solution of the external circuit equations [5]. To enable
a realistic modeling of the collision processes, we take experimentally measured collision cross sections as input data [8-11]. Electron reflection and emission at the electrodes have been added to model particle-surface interactions [12]. The incorporation of particle extraction into the one-dimensional model is described in reference [13]. The external circuit was extended to model the discharge behaviour when connected to a matching network [14]. A schematic
presentation of the simulation algorithm is shown in Figure 2.
The statistics in the sheaths has been improved by a particle splitting scheme: on entering the sheath, each pseudo-ion is split up into a number of sub-particles with correspondingly
reduced charge and mass which are placed around the phase space position of the original particle. Furthermore, a particle subcycling scheme was implemented [15] to allow for multiple
N°9 PIC/MC MODELING OF AN ION SOURCE: CASE STUDY 1221
Calculation of forces Panicle motion
~ 4
E ~
? l
~ ~
Radlu«
~
Integration of Poisson
p~ ~j~j~ ~~~~~~~j~~ ~~~ Panicle-wall interaction
solution of external circuit (
,
j
= , Jx ,
I ~(~) ~ i~(~)
Z z
j )
J~ ~
Radlo« Ph«««In RF p«clad
Calculation of charge densiw Monte-Carlo Collisions
~
g (
(j ÷
I
j £
,
R«dlo« Radlo«
Fig. 2. Sequence of important subroutines to be performed during one time step in the PIC/MC algorithm. Each subroutine is exemplified by a graphical display of the pertinent result.
electron time steps during one time step of the slow ions in the bulk of the plasma. Both methods help to lower the computational cost of the simulation. The code was adapted to its
use on a Multiple Instructions Multiple Data (MIMD) machine. In this way the resources of parallel computers and workstation clusters could be made available. A machine independent
user interface was developed to simplify the evaluation of data produced on different computer architectures.
3. Spatial Discharge Characterization
Figure 3 shows the cycle-averaged ion density in the discharge; furthermore, the space charge
distribution is indicated. From the latter the width and the potential in the sheaths can be inferred. Due to the strong geometrical asymmetry of the discharge, the inner sheath is
considerably more pronounced than the outer one. As a consequence, the discharge is heated
mainly in the vicinity of the inner sheath edge.
In the plasma bulk, the electron energy distribution is more or less Maxwellian cf. Fig. 4).
In particular, the outer part of the bulk which we call thermal bulk is truly Maxwellian up to an electron energy of 20 eV. The inner part of the bulk is slightly non-Maxwellian as
a consequence of the sheath heating occurring in its vicinity. We note that the Maxwellian character of the plasma in the (heated and thermal) bulk makes a macroscopic description of
n,
i z ' ~ n,-ne
' '
,
i-o
' '
- '
m '
°.8
o '
$ '
O-G j ~ fl j
S
~ ~ ~
c
~ ~
~ ~
0A
~ 3 '
« ©
, « S B
j # E
° o-z
' ' '
°.§~ '
io 1.5 z-o 2.5 3.° ~.~
r (Cm)
Fig. 3. Simulation result of the cycle averaged ion and electron density in the discharge. The
spatial division of the discharge into 4 regimes is indicated.
o°
1
5
§1
r
= ,,6
o
~~t
0.5
r (cm)
Fig. 4. Spatial dependence of the electron energy distribution function.
this region in terms of fluid dynamic variables electron and ion density, temperature, current
density and energy current density possible [16]. Such an analysis allows to determine the relative importance of diffusive and drift currents, and in particular shows that thermal diffusion is of prime importance in this region for an understanding of the charged particle
currents.
The electron energy distribution function is shown in Figure 4 as a function of the radial position r. It is of interest to discuss the reasons for the strongly Maxwellian character of
N°9 PIC/MC MODELING OF AN ION SOURCE: CASE STUDY 1223
~'~
Acathodic
pan
1.0
-
~ ~
~
~j o-o
z
"
~~
>,$
~
~
0.5
~
anodic pan 1,
~'~
staff
~
(q=0)-l.5
1.5 1.0 -0.5 O-o 0.5 1.o 1.5
X (Cm)
Fig. 5. View of an arbitrary electron trajectory projected on a plane perpendicular to the cylinder
axis (see text). The inner electrode is shaded. The time average of the sheath edge is indicated.
the plasma bulk, in spite of its low density which promotes non-local effects. Here we note in
particular the magnetic field, which reduces the electron mobility to a value of less than 3% of the ion mobility, thus enhancing the effective pressure of the discharge. As a consequence, non-
local effects in the plasma bulk are suppressed. Furthermore, the electron energy distribution function does not show a decrease in the vicinity of the Ar excitation and ionization energies.
This is connected to the fact that electron collisions are rare in the bulk.
The mechanism of inner sheath heating is exemplified by showing the trajectory of an arbi- trary electron in the vicinity of the inner electrode, Figure 5. It starts in this region at a time when the phase of the RF voltage is 0. In the ensuing anodic part of this cycle, the electron
is attracted towards the inner electrode; however, it does not reach it but circles around it due to the magnetron motion. Later on, when the cathodic part of the voltage sets in, the sheath expands and with it the electron moves away from the electrode. Then the cycloidal magnetron motion caused by the E x B drift is visible. The electron energy oscillates during
its motion; it becomes sufficiently high to ionize a neutral. As a consequence the electron is scattered outwards into the bulk part of the plasma.
4. Extraction
In order to understand the operation of the discharge as a plasma source, a 2-dimensional simulation is necessary. Due to computer time restrictions, this is not readily performed. We therefore implemented the following method: in a I-dimensional simulation, ions and electrons
are continuously removed from the simulation volume. This is done by extracting a number of particles identical to twice the ion saturation current density which is given by [17]
j( = nedge eitj = 0.61nbuik eui, 11
~
O B=7mT
m B=20mT
.
nco (7mT)
m
E
C
~
5
"
d
~~9
25
5
~
N°9 PIC/MC MODELING OF AN ION SOURCE: CASE STUDY 1225
o-B
. .
O B=7mT
. m B=20mT
o.5 "
f °.~ ,
~E
$ o.3
.
E
~ o-z
o-i
o-o
5 2 5 2
~~
h (cm)
Fig. 6. (Continued.)
process which is defined by fl = jxIF as the ratio of the extraction current density jx and the dissipated power P. These results have been obtained for two different magnetic field
strengths B
= 7 mT and B
= 20 mT. For the low magnetic field case, we also indicate the results valid for an infinite cylinder; for the high magnetic field case, the pertinent simulation did not converge.
We observe that the bulk density increases with increasing plasma height. This is plausible
since the extraction process then influences the total plasma volume less. On the other hand the extraction current rises with increasing plasma height; this is plausible, since the extraction
current density is connected to the plasma density via equation (1). As a consequence it would
seem that the largest ion source would be best for the extraction process. Figure 6c shows, however, that for large sources, the extraction becomes less efficient, since more and more
discharge energy is used up in the plasma and does not serve the extraction process. For the
high magnetic field case, which is realized in experiment, hence an optimum height would be in the region of 5 cm, where the efficiency starts to drop; this coincides actually with the value
realized in experiment [6].
In Figure 6 we also see that an increased magnetic field is highly desirable for the plasma density, extraction current, and efficiency, since the magnetic field confines electrons in the
plasma bulk and lets them perform many ionizing collisions before they are attracted to a wall and lost from the plasma.
5. Power Balance: Influence of Magnetic Field and Wall Material
In Figure 7 we display the power balance in the plasma while the plasma height, the magnetic field strength and the electron reflection and secondary electron emission coefficient of the inner electrode are varied. Note that we define power as the energy loss per cycle. Energy may be lost from the plasma by ionizing and exciting collisions of electrons, and by electrons and ions
hitting an electrode. As soon as extraction is operating, this constitutes a further energy loss
exc<tat<on extract<on
In
-wall,on<zation
,~~_ ~all
/
electron -wall
~
electron -wall
<nfin<te cylinder B=7mT, h=4 scm
a) b)
extract<on exc<tat<on
<on-wall
ion-wall
excitation
ion12ation
electron -wajj
electron-wajj
~_~~~~ ~_ ~ ~~~ reflection n=0 8, secondary electron em<ss<on w0 36
c) d)
Fig. 7. Power balance of the discharge, indicating ~vhich fraction of the power input is used for ionizing or exciting collisions in the discharge, which fraction is lost by collisions of ions and electrons with electrodes, and which part is extracted. a) Infinite cylinder as a reference case; the total dissipated
power is P
= IA ~V and the ion density in the bulk is nbuik
" 1.2 X 10~°cm~~, b) Same as a), but
with finite height; P
= I.I W and nbuik
" 0.5 X 10~°cm~~. c) Same as b), but with increased magnetic
field; P
= 1.3 W and nbuik
" 2.9 X 10~°cm~~. d) Same as a), but with high electron reflection
coefficient ~ and secondary electron emission coefficient i from the inner electrode; P
= 2.0 W and
nbuik = 2.4 X 10"cm~~
process. The contribution of elastic and charge exchange ion neutral collisions is negligible.
We note that this internal, microscopic way of viewing the energy losses in the plasma may be contrasted to a macroscopic approach, in which the energy loss is viewed as the ohmic heating proportional to the product of the electric field times the current density in the plasma. Both views coincide when averaged over the discharge volume.
In Figure 7a, we plot what constitutes our reference discharge: B
= 7 mT, infinite height.
Note that most of the energy is lost in the plasma by the acceleration of ions and electrons to the wall. As soon as extraction is operating, Figure 7b, the balance is shifted to some degree
towards the bulk processes of ionizing and exciting collisions. This is due to the increased kinetic energy of the electrons. Note that not much power goes into extraction; the efficiency is still low. It can be increased if a higher magnetic field strength is chosen, Figure 7c. In this case,
N°9 PIC/MC MODELING OF AN ION SOURCE: CASE STUDY 1227
a considerable part of the power is used for ionization processes. This is obviously the regime
in which an ion source should operate. Also losses to the walls are considerably reduced.
Finally, we plot in Figure 7d the influence of the wall material on the discharge power balance.
While up to now, all particles impinging on an electrode were neutralized there, inducing no
further effects, for this simulation we used a rather high electron reflection coefficient of 0.8, and
a secondary electron emission coefficient of ions impinging on a wall of 0.36. These values are at the maximum of what is realizable [17-22]. Otherwise the discharge conditions are identical to the reference case of Figure 7a. Note that the total power of the balance is increased and so is the bulk density. Losses to the wall are still dominating, but among these now electron wall losses have increased weight, since a large number of reflected and secondary emitted electrons
are bent back to the wall by the magnetic field. Nevertheless, the absolute ionization rate has increased, as is reflected by the increased plasma density. We note that the experimental plasma source [6] operates with a high magnetic field of B = 15 mT and an inner electrode material with a high secondary electron emission coefficient in agreement with our findings.
6. Conclusions
We have presented in this paper an application of the PIC/MC method to a realistic plasma
source configuration. Since we study a low pressure system, we had to use a kinetic simulation
scheme. The results presented above exemplified that the anatomy of this discharge becomes transparent through the modeling process, and that such a modeling can be useful for discharge optimization.
The advantages of the kinetic simulation lie in the fact that a microscopic picture of the
discharge is given from which all macroscopic information can be obtained such as the
spatio-temporal ion and electron distributions, electrical fields, etc. On the other side, many important characteristics of the discharge are based on the microscopic discharge behavior, and questions may be addressed such as: to which degree is the discharge Maxwellian? Which is the discharge heating mechanism? What are the ion and the electron energy distributions at the electrodes? Such information can be gained only by a kinetic modeling. When specializing
on parts of the discharge such as the sheaths or the bulk behavior it may be possible to
perform a non-self-consistent simulation in which part of the plasma characteristics are inserted into the simulation, assuming it to be well-known. Such an approach is often impossible when the entire discharge is to be simulated and at least needs to be checked by other independent
means in particular by experiment, which is not always easily performed.
A definite disadvantage of the method used in the present investigation is its quite high computational cost. The simulation of one rf cycle needs approximately lo minutes on a usual 20 MFLOPS RISC machine. While this may not appear too lengthy, it must be borne in mind that it usually takes several hundred cycles to establish the equilibrium rf configuration of the
discharge. In comparison to this, the averaging which is needed when monitoring the plasma characteristics over something like forty cycles is negligible. Other simulation procedures
and in particular non-self-consistent Monte Carlo routines are considerably less computation time expensive.
While the simulation of plasma discharges like the one presented here may nowadays be done more or less routinely, several improvements still need to be made in order to simulate existing discharge configurations fully realistically. Usually discharges have a complicated
3-dimensional geometry, which necessitates a fully 3-dimensional spatial simulator. This is not possible nowadays; only an advance from I-dimensional to 2-dimensional schemes has been made in the last years [23-26]. Secondly, many discharges and in particular plasma
