Plasma Sources Sci. Technol. 8 (1999) 457–462. Printed in the UK
PII: S0963-0252(99)03994-8
Positive ion flux from a low-pressure
electronegative discharge
T E Sheridan, P Chabert† and R W Boswell
Space Plasma and Plasma Processing Group, Plasma Research Laboratory, Research School
of Physical Sciences and Engineering, The Australian National University, Canberra,
Australian Capital Territory 0200, Australia
Received 18 November 1998, in final form 11 March 1999
Abstract. We compute the flux of positive ions exiting a low-pressure, planar,
electronegative discharge as a function of the negative ion concentration and temperature.
The positive ions are modelled as a cold, collisionless fluid, while both the electron and
negative ion densities obey Boltzmann relations. For the plasma approximation, the plasma
edge potential is double-valued when the negative ions are sufficiently cold. When strict
charge neutrality is relaxed, spatial space-charge oscillations are observed at the edge of the
plasma when the flux associated with the low (in absolute value) potential solution is less
than that of the high potential solution. However, the flux is always well defined and varies
continuously with the negative ion concentration. We demonstrate that the correct solution
for the plasma approximation is that having the greater flux.
1. Introduction
The physical and chemical processes occurring at a surface
in contact with a discharge are to a great extent determined
by the ion flux and impact energy at that surface. The ion
impact energy is largely determined by the sheath [1, 2],
since most of the potential drop occurs there, while the ion
flux is determined by the plasma [3–5], since most ions are
created there. (Conceptually, a discharge can be divided
into two parts: the plasma, a quasi-neutral region many
Debye lengths wide, and the sheath, an ion-rich boundary
layer a few Debye lengths wide separating the plasma from
the wall.) Discharges made using electronegative molecules
(e.g. SF6 or O2 ) or molecules composed of electronegative
atoms (e.g. O, F or Cl) are often encountered in plasma
processing [6]. The presence of negative ions in a discharge
can significantly reduce the positive ion flux exiting the
plasma, with implications both for plasma processing and
electrostatic probe diagnostics [7–9].
Braithwaite and Allen [10] have presented a theory for
the positive ion saturation current to a spherical probe in
an electronegative plasma. Their model is similar to that
of Schott [11], who considered a planar discharge with a
low-density, energetic electron component (responsible for
ionization) and a high-density, low-temperature, Maxwellian
component (i.e. like negative ions). In both papers [10, 11],
it was found that for the plasma approximation there are
parameter regimes for which the potential at the plasma
edge, and hence the flux, is double-valued. Braithwaite
and Allen [10] proposed that the sheath forms at the
† Permanent address: Laboratoire PRIAM, UMR 9927 CNRS-ONERA,
Fort de Palaiseau, F-91761 Palaiseau Cedex, France.
0963-0252/99/030457+06$19.50
© 1999 IOP Publishing Ltd
first location where charge neutrality is violated, and were
thereby led to the surprising conclusion that the flux is not
a continuous function of the negative ion concentration.
Although not always explicitly stated, this assumption
appears to be widely used [7, 8]. If the plasma approximation
is relaxed, Schott [11] demonstrated that these multiplesolution regimes are associated with non-neutral spatial
oscillations (quasi-periodic double layers) at the plasma
edge. Clearly, to use such theories to interpret the flux from
an electronegative discharge, one must first know how to
interpret the theories.
In this paper, we compute the flux of collisionless
positive ions exiting a planar, electronegative discharge
as a function of the negative ion concentration and
temperature. Our model is solved analytically for the
plasma approximation and numerically for the non-neutral
case. We find that when multiple solutions exist for the
plasma approximation, the correct solution is that which
gives the greater flux. Consequently, the positive ion flux
is shown to be a continuous function of the negative ion
concentration. In section 2 we present the equations that
model the discharge and non-dimensionalize them. Solutions
to the model are studied in section 3 as a function of the
negative ion concentration and temperature. Section 4 is a
brief summary.
2. Model equations
We consider a plane, symmetric discharge containing cold,
positive fluid ions of mass M+ , density n+ and velocity v+ ,
Boltzmann negative ions with a temperature T− and density
n− , and Boltzmann electrons with a temperature Te and
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T E Sheridan et al
density ne . Positive ions are created at a rate proportional
to ne and then accelerated to the walls by the self-consistent
potential φ. The centre of the discharge is at x = 0 and the
sheath–plasma boundary is at x = L. On the centre plane:
φ = 0, ne = ne0 , n+ = n+0 and n− = n−0 . Quasi-neutrality
requires n+ ≈ ne + n− in the plasma, and n+0 ≈ ne0 + n−0 at
the centre of the discharge.
The steady-state fluid equations of continuity and motion
for the positive ions in a slab geometry are [12]
d(n+ v+ )/dx = νiz ne
(1)
M+ n+ (v+ d/dx)v+ = en+ E − M+ v+ νiz ne
(2)
where νiz is the net ionization rate. The second term on
the right of equation (2) represents a decrease in ion fluid
momentum due to ions born at rest. Fully-collisional models
[6] neglect ion inertia and therefore cannot describe the lowpressure case considered here. Both the electron and negative
ion densities obey Boltzmann relations, so that
ne = ne0 exp[eφ/(kTe )]
(3)
n− = n−0 exp[eφ/(kT− )].
(4)
and
That is, the confined species are considered massless. This
assumption is valid for the negative ions so long as wall losses
dominate volume recombination [6]. Finally, φ and E are
determined self-consistently from Poisson’s equation
ε0 dE/dx = e(n+ − ne − n− )
E = −dφ/dx.
(5)
To non-dimensionalize the model equations, we
introduce the following variables [4]
ξ = x/3
ñ = n+ /ne0
η = −eφ/(kTe )
u = v+ /cse
ε = e3E/(kTe )
(6a)
where the ion acoustic speed in the absence of negative ions
is cse = (kTe /M+ )1/2 , the ionization length is
3 = cse /νiz
(6b)
and the non-neutrality parameter is
q = λe0 /3
(6c)
with the electron Debye length λe0 = [ε0 kTe /(e2 ne0 )]1/2 .
The negative ion component is characterized by its
concentration α and temperature γ ,
α = n−0 /ne0
γ = Te /T−
(7)
where we assume that γ > 1. The flux of positive ions
exiting the plasma and entering the sheath is 0s , which is
non-dimensionalized by ne0 cse , so
n+s v+s
0s
=
= ñs us
ne0 cse
ne0 cse
(8)
where the subscript ‘s’ refers to values at the sheath–plasma
boundary. Note that 0s is nearly constant in the sheath as
almost no ions are created there.
458
Using the variables given above, the dimensionless
equations describing the discharge are
(ñu)0 = e−η
(9a)
(ñuu)0 = ñε
2 0
q ε = ñ − e
−η
−αe
(9b)
−γ η
0
η =ε
(9c)
(9d)
where the prime symbol denotes differentiation with respect
to ξ . Thus, we obtain a system of four first-order, nonlinear,
ordinary differential equations in four unknowns: ñ, u, η and
ε, and three physical parameters: the plasma non-neutrality
q, the negative ion concentration α and the negative ion
temperature γ .
For the plasma approximation q = 0 (or λe0 /L = 0),
equations (9a)–(9d) describe a charge-neutral plasma with
a sheath of zero thickness. The plasma solution becomes
singular at the sheath–plasma boundary since the sheath is
demonstrably not charge neutral. If the source terms are
neglected and q > 0, then equations (9a)–(9d) represent the
sheath [2], and their solution requires assumptions about the
values of the ion velocity and electric field at the sheath edge
[12, 13]. Consequently, for q > 0 the equations describe
two distinct behaviours with disparate length scales—a quasineutral plasma and a positive space-charge sheath.
3. Results and discussion
3.1. Solutions for the plasma approximation
We are interested in investigating the dependence of the ion
flux on the negative ion parameters α and γ . We first consider
the case q = 0 (i.e. the plasma approximation) to gain some
insight into the nature of solutions to equations (9a)–(9d). In
this case, the equations reduce to
(ñu)0 = e−η
(ñuu)0 = ñη0
ñ = e−η + α e−γ η .
(10)
From the third equation, we have ñ = ñ(η). We can then
eliminate ñ from the second equation and integrate to find
u(η),
α
(e−η + α e−γ η ) (11)
u2 = 1 − e−η + (1 − e−γ η )
γ
which can then be substituted into the first equation, giving
dη
(e−η + α e−γ η )2 − (e−η + αγ e−γ η )
dξ
α
× 1 − e−η + (1 − e−γ η )
γ
1/2
α
.
= 2 e−η (e−η + α e−γ η ) 1 − e−η + (1 − e−γ η )
γ
(12)
This equation can be solved by separation of variables to
find ξ(η), although the integration would probably have to
be done numerically.
For the plasma approximation, the desire to form a sheath
is thwarted by the requirement of charge neutrality and dη/dξ
becomes infinite at the plasma edge. Using this condition, we
A low-pressure electronegative discharge
can calculate the potential at the plasma edge ηs , and then the
flux there since we know both u(η) and ñ(η). At the plasma
edge, the right-hand side of equation (12) can be neglected,
so that ηs must obey
(e−ηs + α e−γ ηs )2 − (e−ηs + αγ e−γ ηs )
α
× 1 − e−ηs + (1 − e−γ ηs ) = 0.
(13)
γ
This condition is equivalent to equating the ion velocity in
equation (11) to the Bohm speed for a plasma with two
negative Boltzmann components [10]
u2B = (e−ηs + α e−γ ηs )/(e−ηs + αγ e−γ ηs ).
For α = 0, ηs = − ln 1/2 and the positive ion density at
the sheath edge [5] is ñs = 1/2. From the more accurate
kinetic model [4], it is found that ñs = 0.4871. Thus, the
fluid and kinetic models give nearly the same results. We
can approximate ηs for the cases α 1 and α 1 from
equation (13), and then calculate the flux. We find

1
1


+
α
α1


2
2γ
0s
= ñs us =
1/γ
 1
1
1
ne0 cse


α 1.
+ α
√
γ
2
2
(14)
For γ
= 1 (i.e. the two negative species are
indistinguishable), both expressions reduce to the same
correct expression. The α-dependence of the flux when α is
small and γ is large is quite weak, making measurement of the
negative ion concentration using electrostatic probes difficult
in this regime. (Similar expressions can also be derived for
a spherical presheath [10].)
When γ is large enough, equation (13) admits two
physical solutions for the plasma edge potential, as shown in
figure 1(a). The same behaviour was also noted in [10] and
[11] for similar models. For our model, multiple solutions
exist for
√ γ > 9.90, in agreement with the analytic result
of 5 + 24 given in [10]. (This is related to the fact that
our model and that of [10] have the same Bohm velocity.)
Consequently, the flux at the plasma edge calculated using
ñ and u from equations (10) and (11) is double-valued
(figure 1(b)). (The model investigated here has also been
considered by Franklin and Snell [14], although they did
not consider the implications of multiple solutions.) It has
been proposed [10] that the sheath always forms at the first
singularity encountered (i.e. the smaller value of ηs ), leading
to the conclusion that the flux changes discontinuously at
some value of α (see figure 4 in [10]). This assumption is
reasonable in the context of the plasma approximation, as
charge neutrality cannot be twice violated. However, as we
show in the next section from a consideration of non-neutral
solutions, the physically correct solution for ηs is that which
gives the greater flux. Consequently, the flux is found to vary
continuously with the negative ion concentration.
Figure 1. (a) The potential at the sheath–plasma boundary and
(b) the positive ion flux there as a function of α = n−0 /ne0 for
γ = Te /T− = 20. In (a) we show that equation (13) admits
multiple solutions over a finite range of α. Here the line labelled
BA give the transition proposed in [10], while the line labelled
SCB gives that found in this paper. In (b) we plot the flux
calculated using the solution in (a), and compare it to numerical
solutions of the model equations with q > 0 (open diamonds and
circles). Open circles indicate values of α for which the numerical
solutions are oscillatory. The numerically computed flux agrees
with the larger value of the flux found for the plasma
approximation.
ordinary differential equations numerically to calculate the
flux at the sheath–plasma boundary as a function of negative
ion concentration and temperature.
The discharge equations (9a)–(9d) are solved as an
initial value problem rather than as a boundary value problem
(BVP). (When posed as a BVP this is an eigenvalue problem,
further complicating matters.) That is, given initial values at
the centre of the discharge, we integrate equations (9a)–(9d)
towards the wall. The solution is completely determined by
the upstream conditions, since the positive ion flow is solely
outwards. To carry out this program, we require a set of
consistent initial conditions. In particular, on the centre plane
we know u0 , η0 , ε0 = 0. For q = 0, we have ñ0 = 1 + α,
while for q > 0, ñ0 is determined by the cubic equation
ñ20 (ñ0 − 1 − α) = 2q 2 .
3.2. Non-neutral solutions
In the previous section, we considered the properties of
solutions for the plasma approximation, q = 0. In this
section we relax that assumption and integrate the governing
(15)
When q 2 1, ñ0 ≈ 1+α +2q 2 —the space charge imbalance
in the plasma is of order q 2 .
Having found the initial conditions it might seem that
we need merely integrate the governing equations using any
459
T E Sheridan et al
Figure 3. (a) Positive ion velocity and (b) plasma potential
profiles corresponding to the three cases shown in figure 2.
Figure 2. Positive ion, negative ion and electron density profiles
with q = 10−3 , γ = Te /T− = 20 and for negative ion
concentrations (a) α = n−0 /ne0 = 0.2, (b) 2.6 and (c) 5. In (b),
spatial oscillations are seen near the plasma edge.
standard numerical technique, for example, the fourth-order
Runge–Kutta method. However, within the equations lurks
the abrupt transition from the plasma to the sheath, so that
the differential equations are stiff and an implicit integration
scheme is required. We use a fully-implicit, second-order
scheme. For example, equation (9a) is written in finitedifference form as
exp(−ηi+1 ) + exp(−ηi )
ñi+1 ui+1 − ñi ui
=
1ξ
2
(16)
which is equivalent to the trapezoid rule. After writing each
equation in finite-difference form, a system of four nonlinear
equations in four unknowns (the values at the next step)
results, which is solved iteratively using Newton’s method.
The scheme is further enhanced by using a variable step size
[15] to limit the local truncation error. In particular, many
small steps are needed to accurately navigate the plasma–
sheath transition.
Density profiles are shown in figure 2 for q = 10−3 with
γ = 20 and α = 0.2, 2.6 and 5. Distances are normalized to
the plasma width L. For α = 0.2 (figure 2(a)), we find that the
discharge parameters (e.g. the Debye length and ion acoustic
speed) are determined primarily by the electron temperature.
Qualitatively, these solutions resemble those seen in fullycollisional models [6, 16]—the negative ions are confined in
460
the centre of the discharge, with the outer edge of the plasma
consisting almost entirely of positive ions and electrons. That
is, there is an ‘internal’ pre-sheath that accelerates positive
ions, decreasing their density due to continuity, and confines
negative ions, so that the negative ion density at the edge of
the plasma is negligible.
When the negative ion concentration is large (figure 2(c)),
the discharge parameters are determined mostly by the cold,
negative ion component, with the electron density remaining
nearly constant to the edge of the plasma. Here the negative ions assume the role of electrons in a two-component
plasma, with the ion-acoustic velocity and Debye length determined largely by the negative ion temperature. In this case
the Bohm speed at the sheath–plasma boundary is significantly reduced (figure 3(a)), the sheath thickness decreases
(figure 3(b)) and the potential in the plasma is quite flat since
the potential drop in the pre-sheath need only be of order
kT− /e.
Between these two cases, we observe a double-layer
(i.e. an internal sheath) followed by spatial space-charge
oscillations [11], as shown in figure 2(b). In this case, the
discharge attempts to form a sheath as in figure 2(c), but
fails. As shown in figure 1(b), these oscillatory solutions
occur when multiple solutions to equation (13) exist and the
flux corresponding to the smaller value of ηs is less than
that corresponding to the larger. As shown in figure 3(a),
at this internal sheath the positive ion flux has not attained
the required value and spatial space-charge oscillations result
as discussed in [11]. These oscillations are the fluid model
analogue of the single double-layer predicted from kinetic
A low-pressure electronegative discharge
Figure 4. Positive ion flux at the sheath–plasma boundary versus
negative ion concentration α for negative ion temperatures γ = 5,
10, 20 and 50. Data were computed with the non-neutrality
parameter q = 10−3 . The dashed lines give the approximate
expressions in equation (14). Successive curves are shifted
upwards by one unit.
theory [17–19]. After a number of oscillations, the Bohm
criterion is finally satisfied and a terminal sheath forms. The
average potential continues to increase during the oscillations
because plasma is being created. To form a sheath two
conditions must be met: charge neutrality must be violated
and the ion flux must exceed the required Bohm flux. In this
case, the former is satisfied, but not the latter. Furthermore,
as q approaches zero, the oscillations persist, although their
wavelength decreases, while the flux is nearly independent
of q. That is, in the oscillatory regime solutions with q > 0
differ qualitatively from those for which q = 0. Finally, since
the negative ion density is negligible near the wall, the flux
at the sheath–plasma boundary in the oscillatory regime is a
continuation of the small-α (i.e. electron-dominated) regime.
The α-dependence of the ion flux exiting the plasma is
shown in figure 1(b), where we compare the predictions of
the q = 0 theory to values computed numerically for γ = 20.
We find that the correct criterion for calculating 0s for the
plasma approximation is to choose the larger predicted value
of the flux. This differs from the previous proposal [10],
which was to choose the flux corresponding to the smaller
value of ηs . Using our prescription, 0s varies continuously
with α, in agreement with the numerical solutions. In fact,
the expressions in equation (14) are found to give good
approximations to the flux for all α when we choose the flux
to be the larger of the two predicted values, and the transition
from the electron-dominated to the negative ion-dominated
regime occurs very near where the two lines intersect.
The dependence of the numerically-computed flux on
negative ion concentration and negative ion temperature is
shown in figure 4. For α = 0, the flux has a value of
0.506, in good agreement with the analytic (q = 0) value
of 1/2. The (normalized) flux depends only weakly on α in
both the electron-dominated and oscillatory regimes. (The
flux normalized by the positive ion density is decreasing.)
In the negative-ion-dominated regime, the flux increases
approximately linearly with α (i.e. with the positive ion
density). In all cases, the flux is a continuous function of the
negative ion concentration. The expressions in equation (14)
are plotted for comparison and agree well with the numerical
results, particularly for the larger values of γ that one might
expect to encounter in plasma processing discharges. (Of
course, determining γ accurately might be difficult.)
The existence of the oscillatory solutions raises several
questions. First, are the oscillations ‘real’? They appear to
be a general feature of fluid models, having been found in
previous work [11], and to represent real physics within the
context of the model (i.e. they are not numerical artefacts).
However, if kinetic ions are used [17], a single double-layer
is formed [18, 19]—kinetic ions born at rest (or suffering
collisions) will fill the potential valleys that are allowed by
the fluid model. Second, do the oscillations effect the flux?
The model considered here is widely used for two-component
plasmas (i.e. positive ions and electrons) and gives results in
good agreement with kinetic theory. The model also produces
good results when negative ions are the dominant species.
The only question then is how are the electron-dominated
and negative-ion-dominated regimes to be joined when the
negative ion temperature is low? In this work, we have found
that the flux varies continuously across the transition, and the
same result has also been found for the collisionless kinetic
model [19]. That is, since the flux is an integral of the positive
ion source over the entire discharge, it is not effected by
the presence (or absence) of space-charge oscillations when
q 1. Although both the fluid and kinetic models [17]
can be solved for the collisionless limit, it may prove simpler
to extend the fluid model to moderately collisional regimes
[5].
4. Summary
We have solved a model for a low-pressure plane, symmetric
discharge that includes two negative species, each obeying a
Boltzmann relation (e.g. electrons and negative ions). Charge
neutrality was not assumed, so that the model includes the
transition from a quasi-neutral plasma to a non-neutral sheath.
Using this model, the positive ion flux exiting the plasma was
computed as a function of the negative ion concentration and
temperature. When the negative ions are sufficiently cold,
we observe spatial space-charge oscillations at the edge of
the plasma when two physical solutions for the plasma edge
potential exist, and the flux associated with the low potential
solution is less than that of the high potential solution. (There
is an analogous criterion [19] for the existence of doublelayers [18] in the kinetic model.) These oscillations occur
because quasi-neutrality is violated while the positive ions
do not yet satisfy the Bohm criterion [11]. However, it is
possible to integrate across the oscillations to recover a well
defined value for the ion flux. We find that in the oscillatory
regime the correct solution for the plasma approximation is
that which gives the larger flux. This is contrary to a previous,
and widely-used, proposal that the correct solution is that with
461
T E Sheridan et al
the smaller value of the potential at the sheath edge. Using our
new criterion, we find that the flux varies continuously with
the negative ion concentration, in agreement with numerical
solutions of the governing equations for the non-neutral case.
Simple approximate expressions (14) for the positive ion
flux were derived and shown to be in good agreement with
numerical solutions of the full set of equations for all values
of the negative ion concentration.
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