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Influence of the emission current on a beam-generated plasma
Dmitry Levko
1
and Johannes Gruenwald
2
1
CFD Research Corporation, Huntsville, Alabama 35806, USA
2
Gruenwald Laboratories, Taxberg 50, 5660 Taxenbach, Austria
(Received 1 July 2018; accepted 27 November 2018; published online 18 December 2018)
The influence of the electron emission current on the beam-generated plasma is analyzed using the
self-consistent one-dimensional Particle-in-Cell Monte Carlo collision model. It is established that
the emission current can be used to control both the ion current and the ion energy at the anode. It
is also found that for the values of the emission current of interest in the present work and the gas
pressure of 15 Pa, the plasma density is mainly controlled by the emission current, while only
1%–2% of plasma is produced by the plasma electrons being generated in the cathode sheath.
The plasma potential, which defines the ion energy at the anode, is controlled by the coupling
between the beam and plasma electrons through the excitation of electrostatic waves in the plasma
by damping the electron beam. Published by AIP Publishing. https://doi.org/10.1063/1.5046775
I. INTRODUCTION
Today, plasma etching plays a major role in the semi-
conductor industry.
1–3
There are four basic low-pressure
plasma processes commonly used to remove materials from
surfaces: sputtering, pure chemical etching, ion energy
driven etching, and ion inhibitor etching.
1
Any etching pro-
cess relies on the heavy species (either ions or neutrals or
both), and thus, having precise control of their velocity dis-
tribution functions, fluxes, and homogeneity is extremely
important.
There are several main plasma sources used for precise
etching processes. They are inductively and capacitively
coupled plasmas and electron-cyclotron resonance
plasma.
1–4
Unfortunately, these technologies exhibit several
types of radiation damages caused by the charge buildup of
positive ions and electrons
5
or radiation from ultraviolet,
vacuum ultraviolet, and x-ray photons
6
during etching.
Voltages generated by the charge buildup distort ion trajecto-
ries and lead to the breakage of thin gate oxide films, stop-
page of etching, and pattern dependence of the etching rate.
These serious problems must be overcome in the fabrication
of future nanoscale devices.
The manufacturing of semiconductor chips and solar
cells and display panels demands increasing substrate dimen-
sions.
4
Because plasma etching and deposition processes are
sensitive to radical and charged particle densities in the
plasma and ion energies at the substrate, the spatial unifor-
mity and precise control of the above parameters become
very challenging with growing plasma dimensions.
In recent years, the Naval Research Laboratory has
explored the use of electron beams generated by an external
source to generate large-area plasma.
7–10
Being a direct-
current technology, it does not suffer from electromagnetic
non-uniformities with increasing dimensions. Furthermore,
the electron energy is a readily tunable parameter for unifor-
mity control.
Another method to generate large-area plasmas was
recently proposed in Refs. 11–13. In this method, the plasma
is formed inside a hollow anode by the electrons injected
from a hot cathode. The hollow anode plasma is shielded
from the outer plasma by a transparent metal grid which
allows the generation of the plasma with a flat plasma poten-
tial profile as first described in Ref. 14. The advantage of this
discharge over the one studied in Refs. 7–10 is the absence
of the external electron beam generator. In this discharge,
the electron beam is generated in the cathode sheath. Also,
this discharge does not need an external magnetic field to
produce a homogeneous non-equilibrium plasma.
In the present paper, we analyze how the emission cur-
rent influences the parameters of plasma generated outside
the hollow anode, i.e., outside this so-called inverted fireball.
As it follows from our previous two-dimensional model-
ing,
16
this plasma can be simulated using the one-
dimensional approach. Namely, the longitudinal electrostatic
waves excited between the cathode and the grid were
obtained in Ref. 16. The plasma parameters in the transversal
direction were almost homogeneous. Therefore, in the pre-
sent study, we use the one-dimensional Particle-in-Cell
Monte Carlo Collision (1D PIC/MCC) model.
II. RESULTS OF NUMERICAL MODELING
In this section, the results of 1D PIC/MCC simulations
carried out for different emission currents (Iemm ) for helium
(He) as background gas at a pressure of 15 Pa, a cathode-
anode gap of dCA ¼10 cm, and a cathode voltage (left
boundary) of uC¼125 V are presented. The electron
reflection from the electrodes was not considered in the pre-
sent study. The model used in our studies is detailed else-
where.
17
The parameters chosen in the present work
correspond to the parameters of experiments presented in
Ref. 18 for the inverted fireball generated inside the hollow
anode by the electrons emitted from the hot cathode. Here,
we are interested only in the plasma parameters generated
outside the hollow anode, i.e., between the cathode and the
grid anode.
18
As it follows from Ref. 15, they define the
regime of the discharge operation.
The time step used in our simulations is Dt¼10
12
s,
and the space step is 10
6
m. Such a choice allows us to
1070-664X/2018/25(12)/123509/6/$30.00 Published by AIP Publishing.25, 123509-1
PHYSICS OF PLASMAS 25, 123509 (2018)
resolve the plasma frequency and the Debye length. Also,
such a time step ensures that the particle propagates only
through a small fraction of its mean free path during Dt. The
total number of macro-particles is 10
6
for both electrons
and ions. The weight of macro-particles (i.e., the number of
real particles in one computational particle) is 10
5
–10
6
depending on the emission current. The number of macro-
particles per space cell is 300–500.
The plasma density outside the hollow anode reported in
Ref. 18 is 10
15
m
3
. The value of the emission current was
not reported explicitly (although the total plasma current and
the area of the electron emitter are given) in Ref. 18, which
makes it difficult to set up the simulations. Therefore, we
varied the emission current in the range 1 lA–1 mA in order
to obtain the best agreement with the experiments.
Figure 1(a) shows the influence of the electron emission
current on the electron plasma density, and Fig. 1(b) shows
the plasma potential as functions of the distance to the cath-
ode. The latter figure does not show the cathode sheath
region. It shows only the region where u>0. Also, it is
important to note that in the present study, we distinguish
between the electrons being emitted from the cathode and
the electrons being generated in the cathode-anode gap due
to the gas ionization. The first group of electrons is called
“emitted” electrons, while the second group is called
“generated” or “plasma” electrons. The electrons of both
groups are identical except for their origin.
The dominant electron-neutral collision type in He is the
momentum transfer, rm.
19
Its cross section exceeds the cross
section of ionization (rion) for electron energies below
100 eV. Then, using the value rm310
20
m
2
, the mean
free path of emitted electrons is estimated as kem 1cm.
Figure 1(a) shows the cathode sheath thickness lsh 2cm
for Iem ¼1lA and lsh 0.25 cm for Iem ¼1 mA. Thus, the
cathode sheath is collisional for Iem 10 lA and weakly col-
lisional for higher currents. This means that in the latter
case, the emitted electrons being accelerated in the cathode
sheath enter the plasma bulk as a beam having the energy
125 eV, i.e., the beam energy is almost equal to the
cathode-anode gap voltage. For Iemm 10 lA, our simulation
results have shown several beams propagating through the
plasma [see Fig. 3(a)]. Each of these beams consists of the
emitted electrons, which experienced 1–2 collisions in the
cathode sheath.
The peak value of the plasma density as a function of
emission current is shown in Fig. 2(a). One can see that the
curve can be fitted by two linear functions having different
slopes. This means that two regimes are realized for the con-
ditions considered in the present paper. The linear function
obtained for Iem <100 lA grows faster than the one
obtained for Iem >100 lA. In order to understand this result,
one needs to examine the electron dynamics in the cathode-
anode gap (see the discussion below).
The ion current to the anode and the ion energy obtained
at the anode as the functions of the emission current are
shown in Fig. 2(b). One can see the linear growth of the ion
energy and the non-linear dependence of the ion current on
Iem. The ion energy depends on the anode sheath voltage
which increases for increasing current (see the discussion
below). The ion current to the anode is defined as
Iion ¼enivionSa, where eis the electron charge, niis the ion
FIG. 1. Influence of the electron emission current on (a) the generated elec-
tron density and (b) on the plasma potential.
FIG. 2. (a) The peak value of the plasma density and (b) the ion current den-
sity to the anode and the ion energy at the anode as the functions of emission
current.
123509-2 D. Levko and J. Gruenwald Phys. Plasmas 25, 123509 (2018)
density at the anode, vion is the ion velocity at the anode, and
Sais the anode surface area. Since vion /I
1
2
em and ni/Iem,
one has Iion /I
3
2
em, which explains the non-linear dependence
of the ion current on the emission current [see Fig. 2(b)].
The phase space of generated electrons is shown in Fig.
3(b). The density of these particles is 500 times larger than
the density of emitted electrons. In spite of this, our simula-
tion results have shown that 99% of plasma is generated by
the emitted electrons and only 1% is generated by the
plasma electrons. However, as it follows from Fig. 1(b), the
plasma potential for Iem ¼100 lA is controlled by the tem-
perature of plasma electrons rather than the emitted ones.
Indeed, Fig. 1(b) shows the plasma potential upl 0.5 V.
Our simulations have shown the plasma electron temperature
Te0.15 eV. Then, substituting 0.15 eV into the relationship
between the plasma potential and the plasma temperature
1
upl ¼Te
21þln mi
2pme
;(1)
one obtains a rather good agreement between the simulated
upl and Te. In Eq. (1),miis the He ion mass.
Figure 3(b) shows that there are a few electron macro-
particles having velocities much larger than the thermal
velocity of generated electrons. These high-velocity elec-
trons are the electrons generated in the collisional cathode
sheath by emitted electrons. As it follows from our simula-
tion, they contribute to the obtained 1% of the plasma gener-
ation by the plasma electrons. Since the density of these
energetic plasma electrons is much smaller than the density
of other plasma electrons, they do not influence the plasma
potential, i.e., the plasma density is controlled by the elec-
tron beam, while the plasma potential is controlled by the
plasma electron temperature.
Figure 4shows the typical phase space of emitted and
generated electrons obtained for Iem ¼1 mA. One can
observe a few interesting effects: (1) the electron beam
spreads in phase space when it approaches the anode, (2) a
lot of emitted electrons are thermalized, i.e., their velocities
are comparable with the average velocity of the plasma elec-
trons, and (3) some fraction of generated electrons has the
“supra-thermal” velocity, i.e., the velocity of plasma electron
is comparable to the beam velocity.
The analysis of the ionization profile for the emitted
electrons (not shown here) shows that the “supra-thermal”
electrons are the electrons generated in the cathode sheath.
Previously,
20–22
the generation of “supra-thermal” electrons
in the beam-generated plasma was explained by the accelera-
tion of plasma electrons by electrostatic waves induced by a
damping of the beam electron. However, this mechanism
was realized for a hotter plasma when the anode sheath volt-
age was 10–50 V. Then, the plasma electrons reflected by
the anode sheath were able to obtain an energy >50 eV after
2–3 circulations through the plasma. For the conditions of
our studies, the anode sheath voltage is 5 V [see Fig. 1(a)]
and the mechanism described in Refs. 20–22 is not realized.
Nevertheless, the excitation of electrostatic waves still
explains the thermalization of the emitted electrons and the
beam energy spread. Our simulations have shown the
FIG. 4. Phase space of (a) emitted and (b) generated electrons obtained for
the emission current of 1 mA.
FIG. 3. Phase space of (a) emitted and (b) generated electrons obtained for
the emission current of 100 lA.
123509-3 D. Levko and J. Gruenwald Phys. Plasmas 25, 123509 (2018)
excitation of waves having the frequency in the range
0.1–10 MHz and the electric field amplitude 210
4
V/m
(see Fig. 5). The total cross section of electron-neutral colli-
sions in He gas for 125 eV is rtot 310
21
m
2
. Then, the
mean free path of 125-eV electron in the gas at 15 Pa is
9 cm, i.e., it is comparable with the cathode-anode gap.
Therefore, the beam damping seen in Fig. 4(a) is of electro-
static nature rather than due to the energy relaxation in colli-
sions with neutrals.
In brief, the mechanism of the beam damping is
explained as follows. The propagation of the beam through
the plasma excites the two-stream instability whose growth
rate can be estimated as
22
cxpe ffiffiffi
3
p
24=3
nb
ne
1
3:(2)
Here, xpe is the electron plasma frequency, nbis the beam
density, and neis the electron plasma density. The simulation
results have shown that nb
ne10
3
. Figure 1(a) shows that the
plasma density in the vicinity of the cathode sheath edge is
10
15
–10
16
m
3
. Then, one estimates c210
8
–510
8
s
1
. The beam-neutral collision frequency is estimated as
tot ngvbrtot 0.7 10
8
s
1
, i.e., ctot and the instabil-
ity cannot be damped by the electron-neutral collisions.
The phase velocity of the electrostatic wave excited due
to the two-stream instability is comparable with the beam
velocity, while the plasma electron velocity is much smaller.
This means that only the beam interacts with this wave.
Figure 1(a) shows that the beam propagates through the
plasma having a non-homogeneous spatial profile. Then, the
approach of the beam to the center of the cathode-anode gap
leads to the increase in c. The increase in cresults in the
increase in the amplitude of the electrostatic wave, which
leads to the spread of the beam velocity distribution function.
This spread is responsible for the beam damping.
22
The damping of the electron beam excites multiple elec-
trostatic waves through the parametric instability.
22
The
interaction between these waves and the beam causes the
beam energy spreading. Also, some of these waves have
phase velocities which are comparable with the thermal
plasma electron velocity. These resonant electrons are heated
by the waves.
The interaction between these waves and the plasma
electrons causes the plasma heating.
20,21
Figure 6shows the
typical plasma electron heating profile, i.e., the power depos-
ited to the plasma electrons by the electric field per unit vol-
ume (P=V). One can see the narrow peak of P=Vin the
vicinity of the cathode and the wide peak in the region 5 cm
<x<8 cm, i.e., in the region where the electron beam
damping starts [see Fig. 4(a)]. The first peak is obtained for
the plasma electrons being generated in the cathode sheath.
Since the electric field is the highest in the cathode sheath,
even a few electrons propagating through the sheath absorb
high power.
The wide peak of P=Vseen in the plasma bulk (Fig. 6)
is obtained for the plasma electrons accelerated by the elec-
trostatic waves generated by the damped electron beam. Our
simulation results have shown that for Iemm ¼1mA98%
of plasma is generated by the emitted electrons and 2% is
generated by the plasma electrons. These 2% of plasma are
generated by the plasma electrons accelerated in the cathode
sheath. The electrons accelerated in the plasma bulk by elec-
trostatic waves cannot generate plasma since their energy
does not exceed 10 eV, i.e., it is much smaller than the ioni-
zation threshold of He.
III. ANALYTICAL MODEL
One can build the analytical model of the beam-driven
discharge for the conditions of Sec. II. Let us consider the
spatially averaged plasma parameters. Then, the density, ne,
FIG. 5. (a) Electric field obtained in the center of the cathode-anode gap as
the function of time, and (b) Fast Fourier Transform of this electric field.
Emission current is 1 mA. FIG. 6. The power deposited to the plasma electrons.
123509-4 D. Levko and J. Gruenwald Phys. Plasmas 25, 123509 (2018)
and the temperature, Te, of generated electrons can be calcu-
lated by
23
dne
dt ¼kbngnbþkengnene
s;(3)
dTe
dt ¼eionkengTe
sþP
V
S
ne
:(4)
Here, ngis the background gas density, nbis the electron
beam density, eion is the ionization threshold of He, sis the
plasma electron diffusion time to the anode, and Sis the fac-
tor which is equal to 1 if the collisionless electron heating
occurs and 0 otherwise.
23
In Eq. (3),kband keare the ioniza-
tion rate coefficients for the beam and plasma electrons,
respectively. It is important to note that Eqs. (3) and (4) do
not take into account those of emitted electrons which lost
the major fraction of its energy in inelastic collisions.
Equation (4) shows that if the collisionless heating of
plasma electrons is not realized, the temperature of plasma
electrons is controlled by the electron energy losses to
the walls (the first term on the right-hand side of Eq. (4) is
zero for such plasma). The density of plasma electrons at
t!þ1is defined as
nekbngnbs:(5)
This equation shows that the density of plasma electrons is
proportional to the beam density and the electron diffusion
time to the wall. As it follows from Sec. II, the anode sheath
voltage is almost constant for small emission currents. The
temperature of plasma electrons is almost constant as well.
Hence, at Iemm <100 lAneis controlled only by the beam
density/current which agrees with the results shown in Fig. 2.
The rate coefficient kbis calculated by kb¼rion eb
ðÞ
vb,
where vbis the beam velocity. At 125 eV, rion eb
ðÞ
1.6 10
13
m
3
/s. The electron diffusion time can be esti-
mated as sdCA=2vth , where vth 4.2 10
5
m/s is the elec-
tron thermal velocity. Then, substituting nb¼10
13
m
3
into
Eq. (5), one finds ne10
15
m
3
which agrees with the
results shown in Fig. 1(a) for small emission currents.
Now, let us assume that the collisionless plasma heating
is realized. In this case, the plasma heating rate P
Vcan be esti-
mated as
P
Vgnbeb
vb
dCA
:(6)
This equation describes the part of the energy injected with
the beam into the plasma which is absorbed by the plasma.
Here, vbis the beam velocity and gis the factor which takes
into account the efficiency of the energy conversion from the
beam to the plasma electrons. It varies from 0 to 1. For
instance, using the value of the beam density 310
14
m
3
(the value obtained for Iemm ¼1 mA) and eb¼eðjuCj
þuplÞ130 eV, one obtains nbebvb
dCA 0.5 W/cm
3
which is
in satisfactory agreement with the average value of P
Vseen in
Fig. 6. Note that Eq. (6) does not distinguish between differ-
ent energy conversion mechanisms. This information is hid-
den in parameter g.
Using the steady-state Eqs. (3) and (4), one obtains the
plasma electron density
ne1kengs
ðÞ
kbngnbs(7)
and the plasma electron temperature
eionngkeTe
sþgeb
vb
dCA
1kengs
kbngs¼0:(8)
In general, this is a transcendental equation since both keand
sdepend on the electron plasma temperature. However, it
can be simplified using the results shown in Fig. 4. The tem-
perature of plasma electrons seen in Fig. 4(b) is 1 eV. The
energy distribution function of the plasma electrons, as it fol-
lows from Fig. 4(b) (see also Fig. 7), can be presented as the
sum of Maxwellian distribution and the sum of delta-
functionals:
feee
ðÞ
fMaxw ee
ðÞ
þaXde
eet
ðÞ
e1=2
t
:(9)
Here, a¼nt=neis the ratio between the densities of the
plasma electrons in the tail (nt) and in the body of distribu-
tion function, and etis the energy of tail. The ionization rate
coefficient is calculated by
ke¼ffiffiffiffiffiffiffi
2qe
me
rð1
0
eefeee
ðÞ
riee
ðÞ
dee:(10)
The contribution of the Maxwellian function is negligibly
small for Te1 eV. Then, one obtains
ke¼aXffiffiffiffiffiffiffiffiffiffi
2qeet
me
rriet
ðÞ
:(11)
This expression should be substituted into Eqs. (7) and (8).
For a rough estimation, we assume keakeðe1Þ, i.e., we
replace the flat tail of the distribution function by the delta-
functional having the energy e1. As it follows from Fig. 7,
the energy e1can be taken equal to 30–40 eV.
Equation (7) gives the electron density
nengkbnbþkent
ðÞ
:(12)
One can see that the plasma density is comparable to the
plasma density obtained in the case of S¼0 [see Eq. (5)]
FIG. 7. Energy distribution function of plasma electrons (EEDF) for the
emission current of 1 mA.
123509-5 D. Levko and J. Gruenwald Phys. Plasmas 25, 123509 (2018)
since nt<nband kb=ke5–7. That is, the plasma density is
still the linear function of the beam density. In order to
explain the change of slope of the curve shown in Fig. 2(a),
one has to take into account the spread of the beam velocity
distribution function [see Fig. 4(a)]. This spread leads to the
decrease in the rate coefficient kbbecause now many elec-
trons have the energies for which the ionization cross section
is smaller than the ionization cross section obtained for eb.
Assuming sdCA=2vth constant, we find the plasma
electron temperature from Eq. (8)
Tegeb
vb
dCA
1akbngs
kbngeionngakbs:(13)
Using this equation, we can estimate the electron tempera-
ture for Iemm ¼1 mA. Our simulations have shown a10
3
,
vth 410
5
m/s. Then, the diffusion time is estimated as
s1.3 10
7
s and, for g¼1, the plasma temperature is
estimated as 12 eV. This value is 1 order of magnitude
larger than the value of the electron temperature obtained in
our simulations. This means that for Iemm ¼1 mA one has
g0.1, i.e., the plasma density is insufficient to absorb all
the power emitted by the electrostatic waves.
Equation (13) shows that for very small athe electron
temperature depends on the beam energy and the power
absorption factor g. One expects that gincreases for increas-
ing emission current. Therefore, one may expect that the
electron temperature increases for increasing Iemm as well.
Now, let us estimate the parameter g. The increment of
the two-stream instability is defined by Eq. (2).
22
The condi-
tion for particle trapping by the electrostatic wave is
22
eu=mec2=k2, where kis the wave number. The electric
field of the plasma wave is of the order of Eku, while the
wave number in the region of resonant wave-particle interac-
tion is kxpe=v0, where v0is the velocity of resonant par-
ticles. Then, one can obtain the energy per unit volume
absorbed by the resonant plasma electrons
W
Vmev2
0nb
nb
ne
1
3:(14)
Dividing this equation by the beam time of flight through the
cathode-anode gap, one obtains the power per unit volume
absorbed by the resonant plasma particles
P0
Vmev2
0nb
nb
ne
1
3vb
dCA
:(15)
Comparing this equation with Eq. (6), one estimates
gv0
vb
2nb
ne
1
3:(16)
The simulation results have shown that for Iemm ¼1mA
nb=ne10
3
. Figure 4(a) shows that v010
6
m/s and
vb710
6
m/s. Then, one estimates g0.02 which is
5 times smaller than the value estimated above from Fig. 6
and Eq. (13).
IV. CONCLUSIONS
The results of our one-dimensional Particle-in-Cell
Monte Carlo collision simulations of the beam-driven dis-
charge have shown that the emission current can be used to
control the ion current to the anode and the ion energy at the
anode. Both parameters increased for increasing emission
current in the range 1 lA – 1 mA. We obtained that for these
values of the emission current and gas pressure of 15 Pa, the
plasma density is mainly controlled by the emission current.
Only 1%–2% of plasma was produced by the plasma elec-
trons being generated in the cathode sheath. At the same
time, the plasma potential which defines the ion energy at
the anode was controlled by the coupling between the beam
and plasma electrons through the beam damping in the
plasma and the excitation of the electrostatic waves in the
plasma. These waves heated the plasma electrons which
results in the increase in the anode sheath potential and, as a
consequence, results in the increase in the ion energy at the
anode. More studies are needed to understand the ion extrac-
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