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31st ICPIG, July 14-19, 2013, Granada, Spain
8
Diffusion of electronegative low-pressure plasma through a localized
magnetic filter
D. Levko1, L. Garrigues1,2, G. J. M. Hagelaar1,2, J. Bredin3, D. Rafalskyi3 and A. Aanesland3
1
LAPLACE (Laboratoire Plasma et Conversion d’Energie), Universite de Toulouse, UPS, INPT
Toulouse, 118 route de Narbonne, F-31062 Toulouse cedex 9, France
2
CNRS, LAPLACE, F-31062 Toulouse, France
3
Laboratoire de Physique des Plasmas, CNRS – Ecole Polytechnique, 91128 Palaiseau Cedex, France
Transport of electronegative SF6 low-pressure plasma through the magnetic filter is studied using a
one-dimensional fluid model.The influence of heating power and amplitude of magnetic field on
plasma parameters is studied. The analysis of fluxes in drift-diffusion approximation shows that
the discharge chamber can be divided into three main parts. The first region is the positive ion –
electron sheaths situated near both walls. The second region is the region where the diffusion of
plasma species is ambipolar. The third region corresponds to the center of the magnetic filter
where the electron and ion diffusion becomes independent of each other diffusion.
1. Introduction
Electronegative plasmas are plasmas composed
of positive and negative ions and only a small
fraction of electrons. Nowadays these plasmas are
mainly applied in plasma etching technologies (see
[1] and references therein). Recently it was proposed
to use electronegative plasma in a new type of
thrusters for electric propulsion (PEGASES thruster)
[2]. The advantage of such thrusters is the possibility
to use both positive and negative charges for thrust,
which eliminates the need for additional downstream
current and space charge neutralization.
There are two main ways to produce the
electronegative plasma [1-6]. The first way is
generation during the afterglow of electronegative
discharges when the electron temperature decreases
and the rate of electron attachment to electronegative
molecules increases. The second way is the use of
magnetic filters, which decreases both the electron
mobility and diffusion through the plasma and also
increases the rate of electron attachment due to the
decrease in the electron temperature across the filter.
This work presents the results of a onedimensional (1D) fluid model of transport of
electronegative SF6 low-pressure plasma through a
localized magnetic filter. The influence of the
magnetic field strength and heating power is studied.
The analytical model will be compared with
experiments carried out in the PEGASES II
prototype described in [7].
2. Numerical model
In order to study, in 1D, the diffusion of lowpressure electronegative SF6 plasma through the
magnetic filter a fluid model [8] is used. The three
component plasma, consisting of electrons, SF6+ and
SF6- ions is considered. At low pressures the main
ion species in SF6 electronegative plasma are
SF5+, SF3+ and SF6 - (see, for instance, [6]). For
instance, at 1.3 Pa and with unmagnetized
species the mobility of each ion are
 (SF5 )  4.49 m 2 V-1s-1 ,  (SF3 )  4.95 m 2 V-1s -1
and  (SF6 )  4.1 m 2 V -1s-1 , while the electron
mobility is  e  1.16 103 m 2 V -1s -1 [6]. One can
see that the mobility of SF5+ and SF3+ are
comparable. Therefore, we suppose in our
model that the plasma consists only of one
positive and one negative type of ions, and
electrons.
The system of continuity, momentum and energy
conservation equations is solved for these
components. The mobility and diffusion coefficient
of the two types of ions are assumed equal.
In our model we consider the plasma located
between two parallel grounded walls separated by
the distance of 0.12 m. The heating power is
introduced into the right half of the system, and the
magnetic field has a Gaussian profile which
maximum is in the left half of the system [see Fig.
1(a)]. Mass of SF6 ions is large enough and they are
unmagnetized in the considered magnetic fields. The
temperature, mobility and diffusion coefficients of
negative and positive ions are assumed equal. Rate
constants of ionization, attachment and excitation
processes were taken from [9]. Rate constants of
attachment and ionization processes are the sum of
all considered in [9] attachment and ionization
reactions. Excitation reactions were taken into
account only in the electron energy balance
equation.
3. Results and discussion
31st ICPIG, July 14-19, 2013, Granada, Spain
Fig. 1(b)-(d) show the dependence of plasma
parameters obtained at a power of 50 W, gas
pressure of 1.3 Pa and a maximum magnetic field
strength of 0.01 T. Fig. 1(b) shows that two distinct
regions exist, namely, the positive ion sheaths
existing near both walls and the region consisting of
positive and negative ions and small impurity of
electrons.
Fig. 1(b) and Fig. 1(c) show that both the electron
density ne and the electron temperature Te decreases
in the magnetic filter. Te drops to a value Te< 4.9 eV
where the attachment dominates over the ionization.
It leads to a decrease of ne and a slight increase of nn,
hence the electronegativity α = nn/ne increases.
Fig. 1(d) shows that the existence of the positive
ion sheath near the walls leads to zero flux of
negative ions towards the walls. Therefore, in order
to extract these ions from the plasma one needs to
apply the bias voltage.
Fig. 1. (a) Scheme of the discharge chamber. (b) Profiles
of plasma components densities and α. (c) Spatial
dependence of Te and φ. (d) Spatial dependence of fluxes
of plasma components. Power is 50 W, amplitude of
magnetic field is 0.01 T, gas pressure is 1.3 Pa.
In
the
drift-diffusion
approximation
quasineutrality and balance of fluxes give the
ambipolar electric field:
D p ne   ( De ne )  C
E
.
(1)
( i   e ) ne  2i nn
In eq. (1) Dq and µq are the diffusion coefficients and
mobility of plasma species, C  e  n  p is the
constant current passing from the left to the right
wall due to inhomogeneous heating of electrons.
Indexes e, p and n denote the electrons, positive and
negative ions, respectively. Further the Einstein
relation Dq  Tq  q is supposed.
The analysis of the various ion and electron
fluxes shows that the volume between walls can be
separated into three main parts. The first region [see
Fig. 1(a)] is the positive ion sheaths located near
both walls. The second region is the region of
electronegative plasma with the small impurity of
electrons. It is in the heating region and in the
magnetic filter where B is small [see Fig. 1(a)]. The
diffusion of plasma species in the driver is
ambipolar and electron and negative ion fluxes are
e  2 iTene  C ,
(2)


n  2Di i nn  iTene  C i . (3)
e
e
A homogeneous Te [see Fig. 1(c)] was imposed in
order to obtain these equations [see Fig. 1(b)].
Finally, the third region is also the region of
electronegative plasma with small impurity of
electrons. However, here the electron diffusion is
independent on the ion diffusion and ion diffusion
depends slightly on the electron diffusion. The
fluxes here are defined by:
e  ( De ne ) ,
(4)
n   Dinn  0.5( De ne )  0.5C .
(5)
Inside this region the magnetic field has to satisfy
the condition
m
B  e e  e* / 2 i  1 ,
(6)
qe
where me and qe are the electron mass and charge,
respectively, νe is the electron momentum transfer
frequency, and  e* is the electron mobility without
magnetic field.
The comparison between fluxes (2) and (4)
shows that the diffusion coefficient of electrons in
the driver is larger than the one in the magnetic
filter, since there µe~µi, but (2) also depends on
electronegativity. Eq. (3) and (5) show that the
diffusion of negative ions is defined by the terms in
31st ICPIG, July 14-19, 2013, Granada, Spain
front of nn and ne , as well as the term
containing C. In the driver the electron mobility does
not depend on magnetic field and 2 i / e  1 .
Therefore, both the first and third terms in (5) are
smaller than the ones in (3). On the other hand, in
the magnetic filter µe~µi and the second term of (5)
exceeds the second term of (3). Thus, the diffusion
of negative ions in the magnetic filter is defined by
Di and C and could be considered as free, while the
diffusion in the driver is ambipolar and is defined by
the second term in (3). Fig. 1(d) shows that the sign
of n is not changed in the right side of the
discharge chamber. Since both ne and nn are
decreasing functions in the magnetic filter, the first
and second terms in (3) and (5) have different signs.
Therefore, one can conclude that the terms
containing C, and hence the current flowing in the
system, play a significant role in n .
The dependence of the plasma parameters on the
amplitude of magnetic field is shown in Fig. 2. It
was obtained that the decrease of B leads to decrease
of α in the magnetic filter and to increase of α in the
driver. The model showed that α in the magnetic
filter increases with the increase of B due to the
decrease of ne since nn remains almost constant
beginning from some value of B.
Fig. 2(c) shows that φ increases with the increase
of the amplitude of magnetic field despite of the
growth of α. If one would neglect the constant C the
electric field (1) and plasma potential tends to zero
when    . However, Fig. 1(c) shows that it is not
the case. Therefore, the behavior of E is influenced
by C. In the limit of high α one can neglect ne and
ne and write in the whole discharge chamber
C
E
.
(8)
2 i nn
Therefore, when electronegativity is large, the
plasma potential is not zero and is defined by C.
The fluid model shows that the flux of negative
ions depends significantly on the magnetic field.
When B is small the flux of negative ions is defined
by (3) in the whole discharge chamber. The first
negative term is negligibly small in comparison with
positive second and third terms. Therefore, the flux
n is directed opposite to ne , i.e. the negative ions
move from the walls to the center of the chamber.
When B increases the role of the first term in (3)
increases and the profile of n changes. If magnetic
field exceeds 0.01 T the flux is defined by (5) with
the dominant role of the first term, i.e. n follows
nn and negative ions move towards the walls.
The study of the influence of power on plasma
parameters showed that the increase of P leads to
decrease of both α and Te in the driver, while the
plasma potential and Te in the magnetic filter
increases. The fluid model shows that the growth of
P leads to increase of both electron and ions
densities. However, ne grows as the linear function
while nn grows according to non-linear law. The
latter can be understood from the negative ion
balance equation:
k a n g ne  k r nn2 .
Here ka and kr are, respectively, attachment and
recombination rate coefficients, and ng is the gas
density. One can see that nn  ne . Therefore, α is
Fig. 2. Dependence of α(a), Te (b), andφ(c) on amplitude
of magnetic field. Power is 50 W and gas pressure is 1.3
Pa.
the non-linear function of P, namely   1 / P .
According to (1) the decrease of the
electronegativity increases the role of electrons in
determining the value of the plasma potential. It was
obtained that the increase of P leads to decrease of α
and, as a consequence, leads to increase of the
plasma potential.
Experiments were carried out at similar
conditions in SF6 gas: the distance between walls
was 11 cm, gas pressure was 0.13 Pa, and power
dissipated in plasma was 100 W. It was obtained that
the plasma potential was 0.68 V, the positive and
31st ICPIG, July 14-19, 2013, Granada, Spain
negative ion fluxes were, respectively, 0.93 mA/cm2
and 1.0 mA/cm2. The similarities and discrepancies
between the analytical model and experiments will
be discussed in detail
4. Summary
Transport of electronegative SF6 low-pressure
plasma through a localized magnetic filter is studied
using the one-dimensional fluid model.
It is shown that the plasma segregates into
several regions. First, the positive ion sheaths exist
near both walls. Second, the region of
electronegative plasma with a small fraction of
electrons is located in the driver. This region is
characterized by ambipolar diffusion of all plasma
species. Finally, the region located in the middle of
magnetic filter. This is a region of high
electronegativity, where the electron diffusion is
independent of ion diffusion and ion diffusion
depends slightly on the electron diffusion.
The model shows that the presence of the
magnetic filter influences significantly on the
plasma parameters. In the magnetic filter the
electron temperature drops to a value where
attachment dominates over ionization. The latter
causes the decrease of electron density. Also, the
fluxes of negative ions and electrons decrease
significantly in the magnetic filter compared to the
driver region.
It is shown that the increase of the maximum
magnetic field strength leads to a decrease of the
electron temperature in the magnetic filter what
causes the decrease of the electron density and the
growth of the plasma electronegativity. However,
magnetic field strength does not influence
significantly the Te and α in the driver. We also find
that the increase of B leads to increase of the plasma
potential despite of increase of plasma
electronegativity.
The increase of heating power leads to increase
of both the electron and negative ion density
although electronegativity decreases. It is obtained
that the growth of P causes the growth of both
plasma potential and negative ion flux.
5. Acknowledgements
This work is supported by the EPIC “Strongly
Electro-negative Plasmas for Innovative Ion
Acceleration” project funded by ANR (Agence
Nationale de la Recherche) under Contract No.
ANR-2011-BS09-40.
6. References
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