Download On the possibility of negative electron mobility in a decaying plasma

J. Phys. D: Appl. Phys. 33 (2000) 375–380. Printed in the UK
PII: S0022-3727(00)05647-3
On the possibility of negative electron
mobility in a decaying plasma
N A Dyatko†, A P Napartovich†, S Sakadžić‡, Z Petrović‡ and
Z Raspopović‡
† Troitsk Institute for Innovation and Fusion Research, Troitsk, Moscow Region,
142092 Russia
‡ Institute of Physics, University of Belgrade, PO Box 57, 11001 Belgrade, Yugoslavia
Received 1 July 1999, in final form 3 November 1999
Abstract. The electron energy distribution function and the electron drift velocity are
studied numerically in a decaying plasma in the external electric field in the mixture
Ar:F2 = 1:0.005 at atmospheric pressure. For a reduced electric field strength over the range
0.055–0.55 Td the negative electron mobility was predicted earlier by the solution of the
Boltzmann equation using the two-term approximation for the velocity distribution function.
The applicability of this approximation for calculations of negative mobility is discussed.
The Monte Carlo simulation method is used to verify results obtained from Boltzmann
equation analysis. It is shown that there is a reasonable agreement between the Boltzmann
equation and Monte Carlo results.
1. Introduction
In the last few years a number of papers on the
negative mobility of electrons in low-temperature plasma
have appeared [1–20]. This phenomenon is sometimes
called absolute negative conductivity, because the plasma
conductivity is mainly governed by the electrons. Several
different physical situations are presently known in which
negative electron mobility has been studied.
At first, negative electron mobility was predicted to occur
in weakly ionized relaxing plasma in heavy rare gases on the
time scales of the order of the plasma electron thermalization
time [1, 2]. The results of the theoretical work [2] initiated
the publication of previously unpublished experiments [3]
in which transient negative conductivity on a nanosecond
time scale was detected in relaxing Xe plasma, ionized by a
hard x-ray pulse. In [4, 5] a modification of the experimental
conditions was proposed by adding a small amount of Cs to
Xe in order to use laser radiation instead of x-ray radiation,
since Cs atoms have low ionization energy.
Another physical situation was considered in theoretical
works [6–10], where the possibility of negative conductivity
under steady-state conditions in externally ionized (electron
beam discharge) gas mixtures was investigated. In this case
gas mixtures with a small amount of electronegative gas were
studied: Ar:CCl4 [6], Ar:F2 [7, 10], Xe:F2 [8, 9]. So far
no experiments were made to detect steady-state negative
electron mobility in these mixtures. We can mention the
experimental paper [11] in which the Q-factor of a resonator
filled with a Xe–Hg–Cs–CO gas mixture was measured.
The Q-factor was found to increase when the gas mixture
was irradiated by ultraviolet radiation in order to produce a
steady weakly ionized plasma. The authors of [11] explained
0022-3727/00/040375+06$30.00
© 2000 IOP Publishing Ltd
this experimental result by the fact that the mobility of the
steady plasma was negative, but in the theoretical work [12]
this statement was disproved. The possibility of negative
electron mobility in Ar:Na and Ar:Na:N2 photoplasma was
also theoretically studied in [13, 14].
Finally, negative electron mobility was predicted in
Ar:NF3 plasma under steady-state Townsend conditions [15]
and in a decaying Ar:NF3 [16–18] and Ar:F2 [19, 20] plasma
after a short pulse of ionization. In the latter case the quasisteady-state negative electron mobility exists along with a
decreasing of the electron concentration due to attachment
processes.
All of the theoretical studies mentioned above were
made by solving an appropriate Boltzmann equation using
the so-called two-term approximation for the electron energy
distribution function (EEDF). In fact, the applicability of this
approximation for the electron mobility calculations under
conditions when the mobility is negative is not clear enough.
In particular, we are concerned with the calculations of
negative electron mobility in decaying plasma. The purpose
of the present work is to clarify the situation by calculation of
the electron drift velocity in a decaying Ar:F2 plasma using
two different methods: a Monte Carlo (MC) simulation and
the solution of the Boltzmann equation (BE) with the twoterm approximation for the distribution function.
2. Statement of the problem
The solution of the BE for the electrons in a homogeneous
plasma acted upon by a steady electric field is usually
based on the expansion of the distribution function f (E
v ) in
375
N A Dyatko et al
Legendre polynomials Pn (cos θ):
f (E
v) =
∞
X
fn (v)Pn (cos θ)
(1)
0
where vE is the electron velocity, θ is the angle between
the electron velocity and the electric field directions and
the functions fn depend only on the absolute value of the
electron velocity. In many cases, when the anisotropy of
the distribution function is low, the expansion (1) can be
restricted to two terms:
f (E
v ) = f0 (v) + cos θ f1 (v)
(2)
where f0 (v) is the symmetrical part of the distribution
function and f1 (v) describes the directed motion of the
electrons along the electric field. The common condition for
the applicability of the two-term approximation is that the
momentum transfer frequency νm (u) should be considerably
larger than the inelastic collision frequency νin (u) at each
energy u from the energy interval under consideration.
The relation νm (u) νin (u) leads to an almost perfect
description of the EEDF with approximation (2)†.
In the case of the two-term approximation, the function
f1 (u) is defined as (in the following we will use variable
u = 0.5mv 2 instead of v)
f1 (u) =
eE df0 (u)
NQm (u) du
3. Solution methods
(3)
and the electron drift velocity W is calculated from the
formula [22]
r Z
1 2 ∞
uf1 du
W =−
3 m 0
r Z ∞
eE 2
u df0 (u)
= −
du
(4)
3 N m 0 Qm (u) du
where e and m are the charge and mass of the electron, E is
the electric field strength, Qm (u) is the momentum transfer
cross section and N is the number of atoms and molecules per
unit volume. The normalization condition for the function f0
may be taken in the form
Z ∞
√
uf0 (u) du = 1
0
Expression (4) shows that, in order for the electron drift
velocity to be negative, the derivative of f0 (u) should be
positive (df0 /du > 0) in a certain energy range. Such
a function has a local maximum at some energy u = u∗
and is usually called ‘the inverse function’. According to
the expression (3), f1 (u) is equal to zero at u = u∗ and
is small at energies near u = u∗ . Our estimations also
showed that, if the calculated drift velocity is negative, its
value is small with respect to two parts (positive, where
df0 /du < 0, and negative, where df0 /du > 0) of
integral (4). For these reasons it is not obvious that the
contribution of the higher-order terms of expansion (1) to
the electron drift velocity is negligible even if the relation
† Even under these conditions, the coefficient of transverse diffusion was
found to be inadequately calculated by the two-term approximation in the
case of argon due to a rapidly increasing cross section [21].
376
νm (u) νin (u) is satisfied. In order to clarify the situation,
it is necessary to compare results of the BE (using the
two-term approximation) calculations with results of a more
accurate method. For this purpose, a MC simulation method
was chosen. Calculations were made for decaying plasma
conditions similar to those described in [19, 20].
We considered a uniform weakly ionized Ar:F2 =
1:0.005 plasma in an external electric field, assuming that
the initial densities of electrons and positive ions are equal.
A plasma with such initial conditions can be created by a short
ionizing pulse. If the applied electric field is not too strong,
then the electric loss rate caused by electron attachment to
fluorine molecules will be higher than the ionization rate, so
plasma will start to decay and, simultaneously, the EEDF will
begin to deviate from its initial shape. The shape of the EEDF,
the electron drift velocity and the mean electron energy were
calculated as functions of time in a decaying Ar:F2 plasma.
All calculations were carried out for a gas temperature
T = 300 K and at atmospheric pressure. The initial
EEDF was considered to be Maxwellian with a prescribed
temperature Te0 . The degree of the initial plasma ionization
was assumed to be fairly low, so we did not incorporate
electron–electron and electron–ion collisions and electron
recombination processes in calculations.
3.1. Boltzmann equation
We numerically solved the time-dependent BE for the spherically symmetric part of the EEDF, F (u, t) = n(t)f0 (u, t),
with the initial condition F (u, t) = n(0)f0 (u, 0), where n(t)
is the electron concentration and t is the time. The corresponding BE has the form
√ dF
= IE + St (F )
u
dt
(5)
where the quantity IE describes electron heating in an
external electric field applied to the plasma and St (F ) is
the collision integral incorporating both elastic and inelastic
collisions.
In the computations, we took into account the following
processes: the elastic scattering of electrons by Ar atoms
and F2 molecules; the excitation of the vibrational levels
and electronic states of F2 molecules; and the electron
attachment to F2 molecules. The set of cross sections for the
interaction between electrons and F2 molecules was chosen
in accordance with [23]. The transport cross section for
electron scattering by Ar atoms was taken from [24]. The
time-dependent equation (5) was solved by the method that
was used in [18].
Some of the cross sections used in the calculations are
shown in figure 1. It is easy to estimate that the relation
νm (u) νin (u) is satisfied, taking into account the fact that
we considered a low concentration of the F2 admixture.
3.2. Monte Carlo simulation
The Monte Carlo code used in this paper is essentially the
same as the one used in our previous studies of electron
Negative electron mobility in a decaying plasma
1E+2
2
1E+1
1E+1
1
1E +0
f 0 (u ), eV
Q (u ), 1 0
-1 6
-3 /2
cm
2
3
1E+0
1 E -1
1 E -2
5
1 E -3
3
4
1E -4
4
2
1
1 E -1
1E -5
1 E -2
1 E -1
1E+0
1E -6
1E+1
u , eV
1E -7
Figure 1. Cross sections used in our study: 1, the transport cross
sections for Ar; 2, the transport cross sections for F2 ; 3, the cross
sections for electron attachment to F2 molecules; 4, excitation of
the vibrational levels of F2 molecules.
transport in time varying electric fields [25, 26]. In order to
determine the moment of the collision, small time steps are
followed with the length of a step being equal to a small part
of the mean time between collisions. The number of steps
is chosen in such a way that further reduction in the length
of the time steps does not affect the results. We calculated
all the properties of electron swarms including the EEDF,
mean electron energy and electron drift velocity. Two drift
velocities were determined [27–30]:
Wb =
and
Wf =
d
hri i
dt
dxi
= hvi i
dt
bulk drift velocity
flux drift velocity
(6)
(7)
where xi and vi are the position and velocity of the ith
electron.
It follows from equation (7) that the flux drift velocity
is the mean velocity of electrons. If we are considering the
region of a uniform plasma, this value should be compared
with the drift velocity obtained from BE analysis (4). The
bulk drift velocity (6) characterizes the motion of the total
ensemble of electrons and is the velocity of displacement
of the mean position of the electron swarm. The bulk
and the flux drift velocities were shown to be identical for
‘conservative’ (number conserving) collisions while they
differ in the case of ‘non-conservative’ (number changing
collisions such as attachment or ionization) transport.
As non-conservative collisions dominate the transport in
the present case, it is important to note that our code includes
both attachment and ionization correctly, as was confirmed by
benchmark calculations and comparisons with other available
results [31, 32]. The code was designed in such a way
to maintain a constant number of electrons throughout the
simulation (106 ) by replacing each electron lost by another
0
1
2
3
4
5
6
7
u , eV
Figure 2. The established distribution function f0 (u) in a
decaying Ar:F2 = 1:0.005 plasma. 1, E/N = 0.04 Td;
2, E/N = 0.07 Td; 3, E/N = 0.1 Td; 4, E/N = 0.5 Td;
5, E/N = 1 Td.
electron from the remaining electrons. The technique was
shown to be in excellent agreement with the results obtained
without such replacement [32] for general benchmarks. The
code employs the technique of keeping the number of test
electrons fixed by effectively introducing a constant collision
frequency ionization without energy loss, which is a standard
procedure in Monte Carlo simulations [33, 34]. Such a
process allows us to keep the number of test particles fixed
and to maintain an acceptable statistical uncertainty. In
addition this procedure implies that each electron represents
a different number of real electrons at different times.
Since we do not calculate the field distribution or since,
under swarm conditions, there is no interaction between
the swarm particles, we do not need to follow explicitly
the changing number of real particles (which we do but it
has no consequence on the conditions of the simulation).
The decaying swarm is thus described only by the changing
properties of the electrons starting with the initial conditions.
In our case the electron number, for the entire period of
decay, drops by eight orders of magnitude so the technique
without compensation could not be easily applied. In the
region where both techniques were applied, results were in
excellent agreement. Generally, the calculations were quite
lengthy in order to achieve reasonably good statistics over the
entire period of relaxation. Simulations were carried out on
a Origin 2000 Silicon Graphics computer and usually lasted
between 24 hours and several days.
4. Results and discussion
Let us consider first the results of BE calculations. Our
computations show (see also [17–20]) that in a decaying
plasma the EEDF approaches its steady-state profile f0 (u)
377
N A Dyatko et al
3E+5
W , cm s
-1
1
2E+5
5
1E+5
0E+0
4
2
3
-1 E + 5
1 E -3
1 E -2
1 E -1
1E+0
E /N , T d
Figure 3. The established electron drift velocity as a function of
E/N in a decaying Ar:F2 = 1:0.005 plasma. The numbered circles
on the curve correspond to the numbers of the curves in figure 2.
and the electron density becomes an exponentially decreasing
function. In other words, after a certain time interval, the
function F (u, t) takes the form
F (u, t) = n(t)f (u)
where n(t) ∼ exp(−νt) and ν corresponds to the attachment
rate constant rate calculated from the established f0 (u). For
a given E/N value, f0 (u) and ν do not depend on the initial
conditions (Te0 ).
Figures 2 and 3 show the established f0 (u) and drift
velocity as functions of the electric field strength. We note
an increase of the drift velocity in the range E/N = 0.001–
0.053 Td, followed by an abrupt transition from a positive
value at 0.053 Td to a negative value at 0.055 Td. In the range
0.055–0.55 Td the drift velocity is negative and its minimum
value is about −5×104 cm s−1 . For E/N > 0.55 Td the drift
velocity again becomes positive and increases with E/N .
Such an unusual dependence of the electric drift velocity
on the electric field is due to the shape of the attachment cross
section. An explanation of this phenomenon was given in
previous papers [18, 20] in terms of two groups of electrons.
Here we briefly repeat this explanation. The plasma electrons
can be divided into two groups concerning: (i) electrons with
energies u < um , where um = 0.08 eV is the energy at which
the cross section for electron attachment is maximum (see
figure 1), and (ii) electrons with energies u > um . Particle
exchange between these groups is hindered, because they
are separated by a barrier whose role is played by electron
attachment. As the plasma decays, the electron density
of both groups decreases, and the shape of the established
distribution function f0 (u) is governed by the ratio between
the decay rates of these electron populations. If the electric
field is weak, the rate of electron loss in the first group is
low, because, in the low-energy range, the attachment cross
section is small. Due to elastic and inelastic collisions,
the energy of electrons from the second group falls in the
range u < 0.4 eV, in which the attachment cross section
378
is large and, hence, the electron loss rate is very high. As a
result, the established f0 (u) is governed by electrons from the
first group and is a monotonously decreasing function (see
curve 1 in figure 2). The corresponding value of the electron
drift velocity is positive. As the electric field increases,
the mean electron energy grows and the attachment for the
electrons from the first group rises, since in the range u < um
the attachment cross section increases with energy. For
E/N > 0.55 Td the rate of electron loss for the first group
is higher than that for the second group. In this case the
established f0 (u) is governed primarily by the electrons from
the second group and has inverse form (see curves 2 and 3 in
figure 2). The corresponding drift velocity becomes negative.
As the electric field increases further, the electrons from the
first group acquire sufficient energy (during their lifetime) to
overcome the attachment barrier in the energy space. In this
case, a weakly inverse or even monotonous function f0 (u)
forms (see curves 4 and 5 in figure 2) and the drift velocity
becomes positive.
A physical explanation of the negative drift velocity may
be given in the following manner. The shape of the EEDF
(see curves 2 and 3 in figure 2) is such that the majority of the
electrons have an energy above the Ramsauer minimum. The
momentum transfer cross section for Ar grows rapidly with
increasing energy in this energy range. The group of electrons
that are moving in such a way to gain energy from the field
has an increased probability of elastic collisions (which lead
to randomization of the velocity direction). The electrons
that are moving against the field lose their energy and the
probability of elastic collisions decreases. As a result, a flux
of electrons against the field is generated. This effect would
be lost if a large group of low-energy electrons was formed
below the Ramsauer minimum. In the case studied here,
the attachment at low energies prevents the formation of the
low-energy group. Thus the effect, which would otherwise
be transient [2], becomes stationary.
The comparison of MC and BE results was carried out
for E/N = 0.1 Td and Te0 = 0.666 eV. Figure 4 shows the
EEDFs calculated at different times. It follows from figure 4
that both methods of calculation give the same scenario of
EEDF variations with time: a fast decrease of the low-energy
part and formation of an inverse distribution function with a
maximum in the energy range 1–1.5 eV. At a time of about
75 ns, the EEDF is nearly established. Such a scenario is
in agreement with the qualitative explanation of the EEDF
formation presented above. From figure 4, one can also see
that at early times the MC and BE distribution functions are
graphically indistinguishable, but their established forms are
slightly different.
Figure 5 shows the calculated electron drift velocity as
a function of time. It follows from figure 5 that the time
evolution of the drift velocity is the same in both cases and
the established values are negative. In the case of the MC
simulation the drift velocity reveals appreciable statistical
fluctuations in spite of the large number of electrons involved
in the calculations. This is, obviously, because of the small
value of the drift velocity with respect to the thermal electron
velocity. The averaged MC drift velocity has an established
value of about −4.26 × 104 cm s−1 . The BE calculations
give the value −4.74 × 104 cm s−1 , so the discrepancy
Negative electron mobility in a decaying plasma
1 .8
1
1 E -1
1 .6
2
U m e an , e V
f 0 (u ), eV
-3 /2
1E+0
1
1 E -2
2
1 E -3
1 E -4
1 .4
1 .2
1 .0
0
1
2
3
4
5
u, eV
0 .8
1E+0
0
f 0 (u ), eV
-3 /2
4
1 E -1
3
1 E -3
1 E -4
1
2
3
u, eV
Figure 4. Comparison of BE (full curves) and MC (full circles)
distribution functions f0 (u) in a decaying Ar:F2 = 1:0.005 plasma
at different times for E/N = 0.1 Td. 1, t = 2 ns; 2, t = 5 ns;
3, t = 25 ns; 4, t = 75 ns.
W , W f , cm /s
4E+5
3E+5
2E+5
1E+5
0E+0
-1 E + 5
0
20
40
60
t, n s
80
100
Figure 6. Comparison of the time evolution of BE (full curve) and
MC (full circles) electron mean energies in a decaying
Ar:F2 = 1:0.005 plasma for E/N = 0.1 Td.
1 E -2
0
20
40
60
t, n s
80
100
Figure 5. Comparison of the time evolution of the BE drift
velocity (full curve) and MC flux drift velocity (full circles) in a
decaying Ar:F2 = 1:0.005 plasma for E/N = 0.1 Td.
is about 10%. The time variations of the mean electron
energy, Umean , in a decaying plasma are sufficiently close
in both cases (see figure 6), the discrepancy of established
values being about 6%. The non-monotonous behaviour of
curves seen in figure 6 is explained by rapid attachment of
slow electrons at an earlier stage of decay, followed by a
slower deceleration of electrons in collisions with atoms and
molecules.
It is important to note that we have found that the bulk
drift velocity differs considerably from the flux drift velocity
and that it is always positive. This supports the claim that,
overall, electrons move in a direction determined by the
electric field. The positive drift is brought about by the fast
moving, but short-lived, electrons so on average we have
a negative drift velocity and positive spatial motion of the
swarm itself. As a result, the fast and slow electrons may
be resolved spatially and it may also be possible to form an
explanation based on the diffusion of two different types of
spatially-resolved particles.
In an attempt to verify the MC procedure of keeping
the number of particles constant we have performed several
simulations with a changing number of particles and could
follow it only to 0.07 µs where number of electrons drops
to one or less for initial number of 1 000 000. Even with 10
times more initial electrons we could not extend the results
much further but still at 0.05 µs the number of particles is
sufficient to confirm the negative drift velocity and at that
time the drift velocity reaches the constant value. These
results were not shown here as the uncertainty increases
rapidly with decaying number of electrons but they may be
used to confirm our procedure. Further improvement may be
achieved if we average the drift velocity over times after the
swarm has decayed to its quasi-steady state form, and these
results again confirm our procedure. Thus we may conclude
that the Monte Carlo procedure was sufficiently accurate to
confirm the observation of the phenomenon of the negative
quasi-steady state drift velocity and that possible inadequacy
of the two-term approximation is not the cause of the observed
effect.
5. Conclusion
The main conclusion is that the Monte Carlo calculations
confirmed the predictions of the existence of negative electron
mobility in a decaying Ar:F2 plasma, which were made
from solutions of the Boltzmann equation in a two-term
approximation for the EEDF. The small difference in the
379
N A Dyatko et al
obtained results is probably due to the inaccuracy of twoterm approximation.
The observation of the quasi-steady state negative
drift velocity may be associated with the observation of
the transient negative conductivity [1–3] which was also
experimentally confirmed [3]. The decaying swarm is
a good model for the decaying plasma which may be
used for a possible experimental verification of our result.
Our result may also be of interest for modeling of rf
plasmas.
Acknowledgments
Three of the authors (ZR, SS and ZP) are grateful to MNTRS
project 01E03 and to the Institute of Physics Computer
Facility for support.
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