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Monte-Carlo simulation for electron-neutral collision processes in normal and abnormal discharge cathode sheath region
Vacuum/volume
Pergamon PII: 80042-207X(96)00136-4
47/number S/pages 1065 to 1072/1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0042-207X196 $15.00+.00
Monte-Carlo simulation for electron-neutral collision processes in normal and abnormal discharge cathode sheath region Wei Helin, Liu Zuli and Liu Darning, Department Wuhan 430074, P R China received
15 November
of Physics, Huazhong
1995; accepted in revised form 8 January
University
of Science and Technology,
1996
Electron-neutral collision processes in helium DC normal and abnormal discharges have been studied using Monte-Carlo simulation (MCS). Four types ofcollisions (elastic, metastable excitation, excitation and ionization collision) are considered. Comparing normal discharge with abnormal discharge, we find that the number of collisions decreases and the collision rates increase through the cathode region in abnormal discharge. The collision rates calculated from electron instantaneous energy is different than that calculated from electron mean energy. The results also show that the elastic collision is the greatest proportion and plays an important role in electron-neutral collision processes in the cathode fall region. The effects of collision on the electron mean energy is also studied. Copyright 0 1996 Elsevier Science Ltd.
Introduction Low-temperature and low-pressure discharge plasmas, especially normal and abnormal DC glow discharges, have been widely used in microelectronics manufacturing.’ The cathode fall region of a glow discharge has come under increasing scrutiny with development of experimental diagnostics for the cathode region and with applications of the glow discharge plasma. The cathode fall region is interesting because electrons in it are not in hydrodynamic equilibrium due to the proximity of the cathode boundary and the electric field that is high and rapidly varies. One of the most important aspects in analysis of a glow discharge is the description of electron-neutral collision processes in cathode fall region. The electron-neutral collisions strongly affect the charged particle transport behavior, so electron-neutral collision parameters, such as excitation and ionization rates (coefficient), must be accurately considered in the modeling of the charged particles transport in discharge. Most discharge models developed to date have been continuum models.2mh These models assume that the discharge can be simulated as a continuous fluid. Graves and Jensen* modeled discharges by dc or a combination of dc and RF electric fields using a continuum simulation. They assumed that the ionization rate is a function of the electron-energy that has a Maxwellian distribution. They neglected the elastic and excitation collision effect. Richards et ~1.~ treating argon RF discharges also used a continuum model in which the ionization rate had an electronenergy dependence calculated from the experimental fitting and the electron energy was expressed by a function of E/P (the ratio of the electric field to the gas pressure). A similar model was used
by Oh and co-workers.6 In this model, the electron energy is obtained from the experimental fitting, and is expressed by a function of E/P (the ratio of the electric field to the gas pressure). In these works, they did not consider the electron energy actual dependence of the collision rate in the discharge, and some collision processes were neglected. An alternate technique for simulating discharges is the particle modeling method. Monte-Carlo simulation (MCS) is one of these models. The MCS technique can be used to simulate problems where the effects of electron excitation collision and ionization collision are important, and one can calculate the electron-neutral collision rates if the accurate electron-neutral gas collision cross sections are known. Fortunately, these collision cross sections have been extensively measured and tabulated. The MCS has been applied by a number of workers’ ” to simulate electron transport behavior in gas discharge. Moratz’ analyzed a normal and abnormal discharge using this method, in which the position of the collision is determined by a null-collision technique. Sun et al.’ studied the normal and abnormal discharges also using a MCS technique with a null-collision method. In these papers, the electron general parameters (such as the electron density, mean drift velocity and mean energy, etc.) are studied. Some models’2~‘4 have used a hybrid simulation technique to analyze the glow discharges, in which the MCS is applied to calculate the electronneutral collision rates (coefficient) and then these collision rates are incorporated into fluid models. The results of the collision rate are not given in these hybrid models. In the present work, MCS is used to study the electron-neutral collision processes in normal and abnormal discharges, in which 1065
Wei He/in et al: Electron-neutral
collision
processes
the position of the collision is determined by the electron-neutral collision frequency. The difference discharge conditions for two discharges are considered. We calculate the number of electronneutral collisions and the collision rates in two discharges. The electron mean energies are also studied. MCS model Model condition and distribution function. Den Hartog et al.‘” have experimentally investigated the cathode fall region and negative glow region of the He glow discharge. The electric field and absolute metastable densities are given and other parameters are measured over a range of current densities from normal discharge to abnormal discharge. In this paper, the model is based on these experimental results. The cathode fall condition is listed in Table 1. The neutral atom density is N,. J,, is the ion initial current density on the cathode surface. I’, and d, are the cathode sheath potential and sheath gap, respectively. J is the total discharge current density. The secondary electron emitting coefficient 5 is equal to 0.3. There are two primary mechanisms that maintain steady state discharge: (i) secondary electron emitting, and (ii) electron-neutral ionization collision in the discharge region. In normal discharge, the radial of the discharge is smaller than the electrode radial, this discharge radial can adjust to keep the current density J at its previous value. The secondary electron is emitted only from this radial. When the discharge expands to fill the entire cathode, the discharge radial is equal to the electrode radial, and the discharge current density increases with increasing the voltage applied on the electrodes. This discharge is said to be abnormal discharge. In this case, the secondary electron may emit from the entire electrode. According to Den Hartog’s experimental result,“’ the ratio of the ion current density to the electron current density at the cathode Jlo/Jeo (or called secondary electron emitting coefficient 5) slightly varies over a range of discharge current densities from 0.19-l .50 mA/cm’, this indicates that the increasing of the discharge current density is almost independent of the 5. We assume that the increase of J only depends on the number of the secondary electron emitting unit time and area from the cathode surface (or called electron initial flux). The electron initial flux is (us CCJ,
= tJ,o
(1)
where cpZ is the electron initial flux. J,,, J,, is the electron and ion current density at the cathode (in Table 1). The electron distribution function is written (2)
Table 1. The discharge abnormal discharge
1066
parameters”
where $(z, E, p) is the total flux of the electron, $[(c, p) is the total number of the times that the ith electron has crossed the plane z, M(N) is the total number of the electrons for N electrons leaving the cathode at the same time. c, IJ and p is the electron energy, velocity and cos 8 (Q is the angle that the velocity vector makes with the z axis). Electron motion. A test electron is assumed to start from the cathode surface, and moves freely under the electric field, until it collides with a neutral gas molecule. When a collision occurs, the type of encounter is determined together with the associated energy loss by collision probability. The new direction of the electron is determined by the probability function. This process continues until the test electron exits from the cathode sheath region. We assume that the electron reaching the boundary of the cathode sheath region is absorbed by the electrodes. This entire process is repeated for a large number of electrons to generate histograms. The cathode sheath is divided into 50-80 grids. 4OOG6000 test electrons leave the cathode with an initial energy uniformly distribute in 4-l 2 eV.” The time step is 0.1 ns. This model indicates that the electron parameters do not significantly change over the range of the simulation conditions (grid number, test electron number, time step and the electron initial energy) used in our works. The electric field in cathode fall region is assumed as a function of the distance from the cathode: 2 v, E(z) = 7 1-i c (
c)
where V,, d, is the cathode sheath potential and thickness, tively. The equation of electron motion is dv,(z)ldt
= @M/m,
dv,(z)/dt
= 0
respec-
(5)
where u,(z) is the electron velocity along z-axial, q is the electron charge, m, is the electron mass, and u, (z) is the electron velocity along a radial direction. Collision process. Four types of collision are taken into account in our model: elastic, metastable excitation, excitation and ionization collision. The elastic collision cross section is taken from the calculations of LaBahn and Callaway.’ The collision cross section expression for the elastic collision is
over a range
of current
densities
from normal
J (mA/cm’)
Q, (cm)
V<>(kv)
V, (lOI cm-‘)
l
J,, (mA/cn?)
0.190 0.519 0.846 1.180 1.500
0.328 0.301 0.282 0.300 0.396
0.171 0.215 0.264 0.359 0.597
11.20 10.80 10.30 9.48 8.01
0.308 0.331 0.319 0.265 0.351
0.148 0.390 0.441 0.933 1.110
to
Wei He/in et al: Electron-neutral
collision txocesses
a,,(c) = [585.0e’-“~035”+66.0] x 10-22m2
(6)
Equation (7) is the cross section of the excitation collision, the data is taken from Ref 17. c and t,, are the electron energy and neutral excitation collision threshold (where A,, AZ, and A, are constant), respectively.
a,,(t) =
AI(~-G,) A2+ (6- d2
C
_____
+A3
1
x 10P22m2
(7)
The semi-empirical analytical expressions of the metastable excitation and ionization collision cross sections are taken from Ref. 18. A test has been made that investigates the effects of the sources of the collision cross section (for example, Refs 19, 20 and 21 for elastic, metastable excitation and ionization collision cross sections, respectively) on the results, and we find that the varying is very small. The total collision cross section is:
[i = elastic,
metastable,
then the total electron-neutral
2
0
excitation collision
and ionization] frequency
(8)
is
gas density.
When At is very small, the varying small. The next step is to compare number). For pc > R,, the electron in the step At. The electron flying
The collision
probability
in
(10) of the electron energy L is also pc with RI (a uniform random collides with the neutral atom time can be calculated
If pc < R,, the probability pc is recalculated in the next step At being replaced R, by R, -pc. When a collision occurs the type of the collision is firstly determined, this is done by assuming that the probability of p is proportional to CJ~(C),we line up the relative cross sections 0 to 1, using markers: CL,/CG, (~el+q,,e)/u,, (Q+ ome+O&CT,, (u~,+~~~+cT~~ +r~Jo~, up to 1, a random number R, is used to determine the type of collision. For an elastic collision, the electron only scatters in angle with no loss of energy due to the very large neutral atom mass. For an excitation collision the electron energy after collision is (12)
where t,, t2 is the electron energy before and after collision, and t,, is the neutral atom excitation potential. For an ionization collision, the remaining energy of the incident electron after collision is divided between the scattering electron and the new electron created in the ionization collision by a random number R,. 62
=
& new
(6, =
-dR3 L,-f,,,-62
(14)
The determination of y is more complicated since the scattering in that direction is anisotropic. In the non-ionizing, the scattering angle y is determined by RS = G
(15)
[ oj(t, 7’) sin y’ dy’
J
where oj (6) is the jth collision cross section, a, (6, y’) is the jth differential angular scattering cross section. In an ionization collision, the scattering angle is simply assumed by two uniform numbers (1
-2R6,,)
(16)
y,, y2is the (9)
PC = 1 _,I-\,,Wl
61 = <,-UC ex
$I = 2nR,
Y,,~ = cos-’
1’2
v,(t) = ~&T(~>
where N, is the neutral a time step At is:
where cZ, and t,,, are the energies of scattering electrons and new electrons created in the ionization collision, respectively. Now we have the energies of the electron after collision, the next step we need to find the scattering angle. 0 is the angle that the velocity makes with z-axis, y is the electron scattering angle, and 4 is the azimuthal angle. The new electron direction after collision is determined by 0, y and 4. Since 4 represents the scattering about the plane of incidence, it is symmetrically distributed from 0 to 27r, so it can be given by
(13)
scattering angle of the scattering electron and the new electron, respectively. The new direction of the electron motion after collision is 0, = cos-’
(cos B. cos y + sin 19~sin y cos 4)
where & and 8, are the angles respectively.
0 before
(17)
and after collision,
Results and discussion The model mentioned above has been used to simulate a helium gas discharge. The number of electron-neutral gas collisions per test electron emitting from the cathode and the transport time that an electron crosses the cathode sheath region have been calculated for two discharges. The results are summarized in Table 2, NC-ebaStlC, Nc.merasrarNC-eXCltal, Nc-lonizaland NC,,,,, are the number of elastic, metastable excitation. excitation, ionization and total collisions, respectively. It is seen, from Table 2, that with larger values of discharge current density (J = 1.5 mA/cm’), the number of collisions is smaller and the transport time of the test electrons is also correspondingly shorter due to the higher electric field. Electron transport time. The average transport time 6t of the electron crosses the near grids [z,, z,+ ,]as a function of the distance from the cathode in two discharges is calculated. The results are shown in Figure 1. It indicates that the electron takes much time to cross the near grids in normal discharge due to the weaker electric field. Near the boundary of the cathode region, the transport time is almost constant in two discharges due to the weaker electric field, the electron motion is almost isotropic in this region. Collision number. The number of metastable excitation, excitation and ionization collisions as a function of the distance from the cathode is shown in Figure 2 (A. normal discharge, B. abnormal discharge). The results in Figure 2 represent the distribution of the total number of the collisions integrating all test electrons that cross the near grids. It indicates that the number of the three 1067
Wei He/in et a/r Electron-neutral
collision
processes
Table 2. The number of the electron-neutral collisions (for four types of collisions, and total collision) per test electron emitted from the cathode and electron total transport time in the cathode sheath region J (mA/cm*)
NC.,,,,,
N‘.&,,,C
NC.,,,,,,,
NC-C,,,,,,
NC.,,,,,,,
T (ns)
0.190 1.500
143.6 18.5
135.1 13.1
0.4 0.2
4.4 1.9
3.7 3.3
200 63
types of inelastic collisions is small in the abnormal discharge. It is obvious that electron staying time 6t is small in the abnormal discharge (Figure 2), so the electron-neutral collision probability is also correspondingly small. We also find that the number of collisions increases with increasing the distance from the cathode, this is because electron transport time 6~ increases with the distance from the cathode. The number of the collision near the negative glow region decreases due to the assumption of the absorbing electrode. The number of 2rS metastable collisions per electron in different discharge current densities are shown in Figure 3 ((a) experimental results, (b) MCS results in Ref. 12, (c) MCS results in this paper). Comparing the experimental results in Ref. 12 and the MCS results in this paper, we find that the experimental results for 2’S are higher than that predicated by our MCS. Part of the reason for this is that the number of the metastable collisions in Figure 3(c) is the value only in the cathode fall region and the contribution of the negative glow region is not considered, the results in Figure 3(a) are an experimental value for the number of the 2’S metastable collisions in the cathode region and negative region. The other reason is due to the neglect of the 2’P + 2’S cascade contribution in MCS determinations of 2’S production from higher singlet levels. The MCS results in Ref. 12 are also higher than that in our model. This is because the total production for 2’S metastable level includes direct excitation and a cascade contribution of 19% of the excitation to the 3’P and higher singlet levels in Den Hargtog’s model.‘”
Electron mean energy and the electron energy distribution. The electron mean energy is often used to parametrize the swarm in nonuniform field and is therefore another quantity of interest. The evolution of the electron energy distribution is governed by the Boltzmann equation and some parametrization of the collision integral is needed to reduce the problem to a hydrodynamic one and avoid solving the specially dependent equation. This parametrization is often done by assuming a function form for the collision intergrands and transport coefficient with a parametric
-0. 0
a
0.
2
0.
4
0.
6
0.
a
f. 0
z,‘dc
i
/
o.o/ 0.
0
0.
2
0.
4
0.
6
z/de
0.
a
D 1.
0
Figure 1. The electron transport time in the cathode sheath region: (a)
normal discharge and (b) abnormal discharge. 1068
Figure 2. The electron-neutral metastable collision number: Top. normal discharge, Bottom. abnormal discharge: (a) metastable excitation collision, (b) excitation collision and (c) ionization collision.
Wei /-/e/in et al: Electron-neutral
collision
wocesses
the available energy and the mean energy can be used to estimate the electron energy loss in collisions, Figure 4 indicates that the electron energy loss increases with distance from the cathode.
1.5
i
1.
0
1
o.ot’lll”l’~‘l’l’~~ll”“ll~l”“ll’l”l’~~ 0.40
0.00
0.
current
80
1.
1.20
density(
60
mA/cm’)
Figure 3. The 2’S metastable level production per electron emitting from the cathode: (a) experimental. (b) MCS in Ref. 12 and (c) MCS in this paper.
dependence on the swarm energy and doing the integral parameters (t) (the electron mean energy). The electron mean energy is calculated by
for the
(18) Figure 4 shows that the mean energy as a function of the distance from the cathode with the available energy. It is seen, from Figure 4, that: l
l
The electron mean energy is larger in abnormal discharge than that in normal discharge throughout the cathode region. The available energy t (stands for the electron energy collisionless throughout the cathode region) is larger than the electron mean energy as zjdc > 0.02 in two discharges (J = 0.19 mA/cm’ and I .5 mA/cm2), the differences between
The former of these indicates that the electron can gain more energy from the high electric fields and that the numbers of inelastic collisions are fewer in abnormal discharge (Figure l), so that the electron energy losses also become small. The reason for the second is that the collision causes an energy loss. Since the number of the electron-neutral collisions increase (Figure 2) and the electric field becomes weaker with increasing the distance from the cathode, the electron energy loss must increase with the distance. The nonequlibrium behavior of electron swarm has been clearly demonstrated by our results. Much of the electron energy gained in the cathode fall region is lost in elastic collisions. Near the cathode, we note that the mean energy increases rapidly and follows the available energy (z/de < 0.02) in two discharges. One knows that the electron mean energy depends on the electric field and the collisions in this region. Since the electrons emitted from the cathode have lower initial energy and the electric field is higher, the electric field acceleration dominates. Some electrons cross the cathode sheath region without collision, their energy follows the available energy beamlike electron. The proportion of these electrons is very small (approximately l-3%).” The electron energy distributionf‘(c) is shown in Figure 5 (A. electron distribution with t < 80 eV, B. with 6 > 80 eV). It is clear that the ratio of the higher energy electrons to the lower energy electrons becomes larger in abnormal discharge. The higher energy electrons come from these electrons that have not experienced inelastic collision. Collision rate. The collision rate is another important parameter always used to describe the electron-neutral gas interaction in discharge plasmas, and used in the continuity equations to calculate the electron transport parameters. The electron-neutral collision rate can be obtained from Vi(Z) = Ng j”Oi(E)(c/2me)“‘f(z,
(19)
where t is the electron instantaneous energy, o,(c) (i = elastic, metastable excitation, excitation, ionization or total) are the collision cross sections, it is dependent on the electron energy,f(z, 6) is the electron distribution function. In the fluid models, assumptions have been made to relate the electron collision rate and the transport coefficient to get fluid quantities such as E/N or electron mean energy. If we examine eqn (19) to involve the electron mean energy instead of the c(z) (electron instantaneous energy), the collision rate is
;;:_:;::1._
;
c k
I
,
400~
/
/’
: Qi /
/
/’
Xl(Z) = N~~i[~(z)l~(z)
/’
;
t)dG
V(z) = [2c(z)/m,]“2
-:I_::..;:::_
< 0 ’ 0.0
, 0.2
0.4
,
,
0. 6
0. 8
,
, 1. 0
, 1. 2
z/de Figure 4. The electron mean energy and the available energy in the cathode sheath region: (a) mean energy in the abnormal discharge and (b) mean energy in normal discharge. Solid line is the mean energy and dish line is the available energy.
(20)
Some interesting results are obtained. The results are shown in Figure 6 (A. elastic collision rate, B. metastable excitation collision rate, C. excitation collision rate, D. ionization collision rate and E. total collision rate). It is seen, from Figure 6A to E, that the results from eqns (19) and (20) are different: l
The collision rates calculated from eqn (19) are higher than that calculated from eqn (20) in normal discharge, and lower than that calculated from eqn (20) in abnormal discharge. 1069
Wei He/in et al: Electron-neutral
collision
processes
L
o,o:kJ 0
20
60
e7cgy(e V)
80
excitation or ionization collisions. On the other hand, despite some electrons having higher energies than the atom threshold potential, they also do not produce excitation or ionization collisions such as the beamlike electron.22 In the latter part of the cathode region (z/de > 0.5) the electron mean energy is approximately between 5@90 eV, the excitation and ionization collision cross sections have a larger value (from eqn (7) and Refs 17 and 18) so that the differences between eqns (19) and (20) become large in this region. Comparing Figure 6A and E with B, C and D, we can see that the differences between eqns (19) and (20) for the elastic collision rate are smaller than that for the other collision rates. This is because the effects of electron energy and electron mean energy on elastic collision cross sections are smaller than that on other types of collisions (eqns (6) and (7)). It is noteworthy that the total collision rate distribution (Figure 6E) is similar to the elastic collision rate distribution (Figure 6A). The reason for this follows that the elastic collision cross section is much larger than the other types of collision cross sections over a range of the electron energy from O-250 eV, and the probability of elastic collision or the number of the elastic collision is also larger (Table 2). In other words, the elastic collision has a larger proportion in the electron-neutral collision processes. This result implies that the elastic collision plays an important role in electron-neutral collision processes. Sun et al9 indicated that the elastic collision affects the angular scattering and the electron mean energy. From Figure 6, we also find that the collision rates are larger in abnormal discharge than that in normal discharge, this is because that the electron energy (or mean energy) is higher in abnormal discharge.
Conclusion 80
250
420
energy(
590
eV)
Figure 5. The electron energy distribution. Top. electron energy t < 80 eV, Bottom. electron energy L > 80 eV: (a) abnormal discharge and (b) normal discharge.
l
As z/de 2 0.5, the differences between the results from eqns (19) and (20) become large in two discharges.
In normal discharge, the electron mean energy is lower than the neutral threshold potential of the gas atom in a large part of cathode fall region (Figure 4). In this case, the collision rates obtained from eqn (20) must be low. The fact is that some electrons have higher energy (Figure 5)22 than neutral molecular excitation or ionization potentials in this region, these electrons may be able to produce excitation and ionization collisions that do not appear in eqn (20). The differences between the results from eqns (19) and (20) become more obvious in the latter part of the cathode region (z/de > 0.45) due to very low mean energy. Near the cathode, the electrons emitted from the cathode with lower energy, their energies and mean energies are lower (Figure 4) so the differences between eqns (19) and (20) become small. In abnormal discharge, the electron mean energy is larger than the neutral atom excitation or ionization potential in a large part of the cathode region (Figure 4) so the collision rates from eqn (20) are always larger than zero. It should be noted that some electrons have a lower energy than excitation or ionization potential in the cathode region (Figure 5A). They do not produce 1070
The electron-neutral atom collision processes in normal and abnormal discharges have been simulated by using a MCS method in which the position of the collision is determined by the electron-neutral collision frequency. The electron transport time, the number of collisions and collision rates are studied. The results show that the electron transport time and the number of collisions decrease in abnormal discharge due to strong electric field. In spite of the electron transport time and the number of collisions decrease in the abnormal discharge, the collision rates increase. The nonequlibrium behavior of electron swarm has been clearly demonstrated in this paper. Much of the electron energy gained in the cathode fall region is lost during collision processes. Equally important in the cathode region is the electron-neutral collision rates, which are often used in the fluid modelling and assumed as a function of electron mean energy. We calculate the collision rates from the electron instantaneous energy v,(z) and from electron mean energy x,(z). The results show that the collision rates, v,(z)and xl(z) (especially excitation and ionization collision), are different in the two discharges, v,(z) is larger than x>(z) in normal discharge, and v,(z) is smaller than x,(z) in abnormal discharge. The differences between v,(z) and x,(z) increase with the distance from the cathode. The difference between v,(z) and x1(z) is smaller for the elastic collision and the total collision. This result indicates that the collision (excitation and ionization) rate dependence of the electron instantaneous energy should be considered during the study of the electron transport behavior in a discharge.
Wei /-/e/in et al: Electron-neutral
collision processes (B)
/
/’ -- //
/
/
. -.
/-
b 0.
oot..,,...,,l,.,...,,.‘.‘,..“..‘..’,”,..’,,,,.,,,,’ 0.00
0.40
0.20
0.60
0.80
v .... 0
I.
00
0.20
0. 00
0.
0.
0.80
40
0.
80
1.00
z /dc
2 /dc
10
CD) -v)
1
n
0.20
d s
0 ‘_
0) 1 CT W k c
\ 0.
. \
lo-
\ \
c 0
\
b
\
\
.-
.d m .d 4 0 0
0. 00
0.
20
0.40
z/de
0.80
0 0
0.80
1.
00
0.00
0.20
0.
40
z/de
0.60
0.
80
1.00
I,,,,rLLL’,,,,,,U,,,,,,,‘,,,,,,,,,,,,,,
0.00 0.00
0.20
0.
40 z
0.60
/dc
0.
80
1.00
Figure 6. Theelectron-neutral collision rate (dashed line is the results from eqn (19), solid line is the results from eqn (20)) A The elastic collision rate. B. The metastable
collision
rate. C. The excitation
collision
rate. D. The ionization
collision
rate. E. The total collision
rate.
1071
Wei He/in eta/: Electron-neutral
collision processes
Acknowledgement This work is supported dation of China.
by the National
Natural
Science Foun-
References
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7072
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Pergamon PII: 80042-207X(96)00136-4
47/number S/pages 1065 to 1072/1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0042-207X196 $15.00+.00
Monte-Carlo simulation for electron-neutral collision processes in normal and abnormal discharge cathode sheath region Wei Helin, Liu Zuli and Liu Darning, Department Wuhan 430074, P R China received
15 November
of Physics, Huazhong
1995; accepted in revised form 8 January
University
of Science and Technology,
1996
Electron-neutral collision processes in helium DC normal and abnormal discharges have been studied using Monte-Carlo simulation (MCS). Four types ofcollisions (elastic, metastable excitation, excitation and ionization collision) are considered. Comparing normal discharge with abnormal discharge, we find that the number of collisions decreases and the collision rates increase through the cathode region in abnormal discharge. The collision rates calculated from electron instantaneous energy is different than that calculated from electron mean energy. The results also show that the elastic collision is the greatest proportion and plays an important role in electron-neutral collision processes in the cathode fall region. The effects of collision on the electron mean energy is also studied. Copyright 0 1996 Elsevier Science Ltd.
Introduction Low-temperature and low-pressure discharge plasmas, especially normal and abnormal DC glow discharges, have been widely used in microelectronics manufacturing.’ The cathode fall region of a glow discharge has come under increasing scrutiny with development of experimental diagnostics for the cathode region and with applications of the glow discharge plasma. The cathode fall region is interesting because electrons in it are not in hydrodynamic equilibrium due to the proximity of the cathode boundary and the electric field that is high and rapidly varies. One of the most important aspects in analysis of a glow discharge is the description of electron-neutral collision processes in cathode fall region. The electron-neutral collisions strongly affect the charged particle transport behavior, so electron-neutral collision parameters, such as excitation and ionization rates (coefficient), must be accurately considered in the modeling of the charged particles transport in discharge. Most discharge models developed to date have been continuum models.2mh These models assume that the discharge can be simulated as a continuous fluid. Graves and Jensen* modeled discharges by dc or a combination of dc and RF electric fields using a continuum simulation. They assumed that the ionization rate is a function of the electron-energy that has a Maxwellian distribution. They neglected the elastic and excitation collision effect. Richards et ~1.~ treating argon RF discharges also used a continuum model in which the ionization rate had an electronenergy dependence calculated from the experimental fitting and the electron energy was expressed by a function of E/P (the ratio of the electric field to the gas pressure). A similar model was used
by Oh and co-workers.6 In this model, the electron energy is obtained from the experimental fitting, and is expressed by a function of E/P (the ratio of the electric field to the gas pressure). In these works, they did not consider the electron energy actual dependence of the collision rate in the discharge, and some collision processes were neglected. An alternate technique for simulating discharges is the particle modeling method. Monte-Carlo simulation (MCS) is one of these models. The MCS technique can be used to simulate problems where the effects of electron excitation collision and ionization collision are important, and one can calculate the electron-neutral collision rates if the accurate electron-neutral gas collision cross sections are known. Fortunately, these collision cross sections have been extensively measured and tabulated. The MCS has been applied by a number of workers’ ” to simulate electron transport behavior in gas discharge. Moratz’ analyzed a normal and abnormal discharge using this method, in which the position of the collision is determined by a null-collision technique. Sun et al.’ studied the normal and abnormal discharges also using a MCS technique with a null-collision method. In these papers, the electron general parameters (such as the electron density, mean drift velocity and mean energy, etc.) are studied. Some models’2~‘4 have used a hybrid simulation technique to analyze the glow discharges, in which the MCS is applied to calculate the electronneutral collision rates (coefficient) and then these collision rates are incorporated into fluid models. The results of the collision rate are not given in these hybrid models. In the present work, MCS is used to study the electron-neutral collision processes in normal and abnormal discharges, in which 1065
Wei He/in et al: Electron-neutral
collision
processes
the position of the collision is determined by the electron-neutral collision frequency. The difference discharge conditions for two discharges are considered. We calculate the number of electronneutral collisions and the collision rates in two discharges. The electron mean energies are also studied. MCS model Model condition and distribution function. Den Hartog et al.‘” have experimentally investigated the cathode fall region and negative glow region of the He glow discharge. The electric field and absolute metastable densities are given and other parameters are measured over a range of current densities from normal discharge to abnormal discharge. In this paper, the model is based on these experimental results. The cathode fall condition is listed in Table 1. The neutral atom density is N,. J,, is the ion initial current density on the cathode surface. I’, and d, are the cathode sheath potential and sheath gap, respectively. J is the total discharge current density. The secondary electron emitting coefficient 5 is equal to 0.3. There are two primary mechanisms that maintain steady state discharge: (i) secondary electron emitting, and (ii) electron-neutral ionization collision in the discharge region. In normal discharge, the radial of the discharge is smaller than the electrode radial, this discharge radial can adjust to keep the current density J at its previous value. The secondary electron is emitted only from this radial. When the discharge expands to fill the entire cathode, the discharge radial is equal to the electrode radial, and the discharge current density increases with increasing the voltage applied on the electrodes. This discharge is said to be abnormal discharge. In this case, the secondary electron may emit from the entire electrode. According to Den Hartog’s experimental result,“’ the ratio of the ion current density to the electron current density at the cathode Jlo/Jeo (or called secondary electron emitting coefficient 5) slightly varies over a range of discharge current densities from 0.19-l .50 mA/cm’, this indicates that the increasing of the discharge current density is almost independent of the 5. We assume that the increase of J only depends on the number of the secondary electron emitting unit time and area from the cathode surface (or called electron initial flux). The electron initial flux is (us CCJ,
= tJ,o
(1)
where cpZ is the electron initial flux. J,,, J,, is the electron and ion current density at the cathode (in Table 1). The electron distribution function is written (2)
Table 1. The discharge abnormal discharge
1066
parameters”
where $(z, E, p) is the total flux of the electron, $[(c, p) is the total number of the times that the ith electron has crossed the plane z, M(N) is the total number of the electrons for N electrons leaving the cathode at the same time. c, IJ and p is the electron energy, velocity and cos 8 (Q is the angle that the velocity vector makes with the z axis). Electron motion. A test electron is assumed to start from the cathode surface, and moves freely under the electric field, until it collides with a neutral gas molecule. When a collision occurs, the type of encounter is determined together with the associated energy loss by collision probability. The new direction of the electron is determined by the probability function. This process continues until the test electron exits from the cathode sheath region. We assume that the electron reaching the boundary of the cathode sheath region is absorbed by the electrodes. This entire process is repeated for a large number of electrons to generate histograms. The cathode sheath is divided into 50-80 grids. 4OOG6000 test electrons leave the cathode with an initial energy uniformly distribute in 4-l 2 eV.” The time step is 0.1 ns. This model indicates that the electron parameters do not significantly change over the range of the simulation conditions (grid number, test electron number, time step and the electron initial energy) used in our works. The electric field in cathode fall region is assumed as a function of the distance from the cathode: 2 v, E(z) = 7 1-i c (
c)
where V,, d, is the cathode sheath potential and thickness, tively. The equation of electron motion is dv,(z)ldt
= @M/m,
dv,(z)/dt
= 0
respec-
(5)
where u,(z) is the electron velocity along z-axial, q is the electron charge, m, is the electron mass, and u, (z) is the electron velocity along a radial direction. Collision process. Four types of collision are taken into account in our model: elastic, metastable excitation, excitation and ionization collision. The elastic collision cross section is taken from the calculations of LaBahn and Callaway.’ The collision cross section expression for the elastic collision is
over a range
of current
densities
from normal
J (mA/cm’)
Q, (cm)
V<>(kv)
V, (lOI cm-‘)
l
J,, (mA/cn?)
0.190 0.519 0.846 1.180 1.500
0.328 0.301 0.282 0.300 0.396
0.171 0.215 0.264 0.359 0.597
11.20 10.80 10.30 9.48 8.01
0.308 0.331 0.319 0.265 0.351
0.148 0.390 0.441 0.933 1.110
to
Wei He/in et al: Electron-neutral
collision txocesses
a,,(c) = [585.0e’-“~035”+66.0] x 10-22m2
(6)
Equation (7) is the cross section of the excitation collision, the data is taken from Ref 17. c and t,, are the electron energy and neutral excitation collision threshold (where A,, AZ, and A, are constant), respectively.
a,,(t) =
AI(~-G,) A2+ (6- d2
C
_____
+A3
1
x 10P22m2
(7)
The semi-empirical analytical expressions of the metastable excitation and ionization collision cross sections are taken from Ref. 18. A test has been made that investigates the effects of the sources of the collision cross section (for example, Refs 19, 20 and 21 for elastic, metastable excitation and ionization collision cross sections, respectively) on the results, and we find that the varying is very small. The total collision cross section is:
[i = elastic,
metastable,
then the total electron-neutral
2
0
excitation collision
and ionization] frequency
(8)
is
gas density.
When At is very small, the varying small. The next step is to compare number). For pc > R,, the electron in the step At. The electron flying
The collision
probability
in
(10) of the electron energy L is also pc with RI (a uniform random collides with the neutral atom time can be calculated
If pc < R,, the probability pc is recalculated in the next step At being replaced R, by R, -pc. When a collision occurs the type of the collision is firstly determined, this is done by assuming that the probability of p is proportional to CJ~(C),we line up the relative cross sections 0 to 1, using markers: CL,/CG, (~el+q,,e)/u,, (Q+ ome+O&CT,, (u~,+~~~+cT~~ +r~Jo~, up to 1, a random number R, is used to determine the type of collision. For an elastic collision, the electron only scatters in angle with no loss of energy due to the very large neutral atom mass. For an excitation collision the electron energy after collision is (12)
where t,, t2 is the electron energy before and after collision, and t,, is the neutral atom excitation potential. For an ionization collision, the remaining energy of the incident electron after collision is divided between the scattering electron and the new electron created in the ionization collision by a random number R,. 62
=
& new
(6, =
-dR3 L,-f,,,-62
(14)
The determination of y is more complicated since the scattering in that direction is anisotropic. In the non-ionizing, the scattering angle y is determined by RS = G
(15)
[ oj(t, 7’) sin y’ dy’
J
where oj (6) is the jth collision cross section, a, (6, y’) is the jth differential angular scattering cross section. In an ionization collision, the scattering angle is simply assumed by two uniform numbers (1
-2R6,,)
(16)
y,, y2is the (9)
PC = 1 _,I-\,,Wl
61 = <,-UC ex
$I = 2nR,
Y,,~ = cos-’
1’2
v,(t) = ~&T(~>
where N, is the neutral a time step At is:
where cZ, and t,,, are the energies of scattering electrons and new electrons created in the ionization collision, respectively. Now we have the energies of the electron after collision, the next step we need to find the scattering angle. 0 is the angle that the velocity makes with z-axis, y is the electron scattering angle, and 4 is the azimuthal angle. The new electron direction after collision is determined by 0, y and 4. Since 4 represents the scattering about the plane of incidence, it is symmetrically distributed from 0 to 27r, so it can be given by
(13)
scattering angle of the scattering electron and the new electron, respectively. The new direction of the electron motion after collision is 0, = cos-’
(cos B. cos y + sin 19~sin y cos 4)
where & and 8, are the angles respectively.
0 before
(17)
and after collision,
Results and discussion The model mentioned above has been used to simulate a helium gas discharge. The number of electron-neutral gas collisions per test electron emitting from the cathode and the transport time that an electron crosses the cathode sheath region have been calculated for two discharges. The results are summarized in Table 2, NC-ebaStlC, Nc.merasrarNC-eXCltal, Nc-lonizaland NC,,,,, are the number of elastic, metastable excitation. excitation, ionization and total collisions, respectively. It is seen, from Table 2, that with larger values of discharge current density (J = 1.5 mA/cm’), the number of collisions is smaller and the transport time of the test electrons is also correspondingly shorter due to the higher electric field. Electron transport time. The average transport time 6t of the electron crosses the near grids [z,, z,+ ,]as a function of the distance from the cathode in two discharges is calculated. The results are shown in Figure 1. It indicates that the electron takes much time to cross the near grids in normal discharge due to the weaker electric field. Near the boundary of the cathode region, the transport time is almost constant in two discharges due to the weaker electric field, the electron motion is almost isotropic in this region. Collision number. The number of metastable excitation, excitation and ionization collisions as a function of the distance from the cathode is shown in Figure 2 (A. normal discharge, B. abnormal discharge). The results in Figure 2 represent the distribution of the total number of the collisions integrating all test electrons that cross the near grids. It indicates that the number of the three 1067
Wei He/in et a/r Electron-neutral
collision
processes
Table 2. The number of the electron-neutral collisions (for four types of collisions, and total collision) per test electron emitted from the cathode and electron total transport time in the cathode sheath region J (mA/cm*)
NC.,,,,,
N‘.&,,,C
NC.,,,,,,,
NC-C,,,,,,
NC.,,,,,,,
T (ns)
0.190 1.500
143.6 18.5
135.1 13.1
0.4 0.2
4.4 1.9
3.7 3.3
200 63
types of inelastic collisions is small in the abnormal discharge. It is obvious that electron staying time 6t is small in the abnormal discharge (Figure 2), so the electron-neutral collision probability is also correspondingly small. We also find that the number of collisions increases with increasing the distance from the cathode, this is because electron transport time 6~ increases with the distance from the cathode. The number of the collision near the negative glow region decreases due to the assumption of the absorbing electrode. The number of 2rS metastable collisions per electron in different discharge current densities are shown in Figure 3 ((a) experimental results, (b) MCS results in Ref. 12, (c) MCS results in this paper). Comparing the experimental results in Ref. 12 and the MCS results in this paper, we find that the experimental results for 2’S are higher than that predicated by our MCS. Part of the reason for this is that the number of the metastable collisions in Figure 3(c) is the value only in the cathode fall region and the contribution of the negative glow region is not considered, the results in Figure 3(a) are an experimental value for the number of the 2’S metastable collisions in the cathode region and negative region. The other reason is due to the neglect of the 2’P + 2’S cascade contribution in MCS determinations of 2’S production from higher singlet levels. The MCS results in Ref. 12 are also higher than that in our model. This is because the total production for 2’S metastable level includes direct excitation and a cascade contribution of 19% of the excitation to the 3’P and higher singlet levels in Den Hargtog’s model.‘”
Electron mean energy and the electron energy distribution. The electron mean energy is often used to parametrize the swarm in nonuniform field and is therefore another quantity of interest. The evolution of the electron energy distribution is governed by the Boltzmann equation and some parametrization of the collision integral is needed to reduce the problem to a hydrodynamic one and avoid solving the specially dependent equation. This parametrization is often done by assuming a function form for the collision intergrands and transport coefficient with a parametric
-0. 0
a
0.
2
0.
4
0.
6
0.
a
f. 0
z,‘dc
i
/
o.o/ 0.
0
0.
2
0.
4
0.
6
z/de
0.
a
D 1.
0
Figure 1. The electron transport time in the cathode sheath region: (a)
normal discharge and (b) abnormal discharge. 1068
Figure 2. The electron-neutral metastable collision number: Top. normal discharge, Bottom. abnormal discharge: (a) metastable excitation collision, (b) excitation collision and (c) ionization collision.
Wei /-/e/in et al: Electron-neutral
collision
wocesses
the available energy and the mean energy can be used to estimate the electron energy loss in collisions, Figure 4 indicates that the electron energy loss increases with distance from the cathode.
1.5
i
1.
0
1
o.ot’lll”l’~‘l’l’~~ll”“ll~l”“ll’l”l’~~ 0.40
0.00
0.
current
80
1.
1.20
density(
60
mA/cm’)
Figure 3. The 2’S metastable level production per electron emitting from the cathode: (a) experimental. (b) MCS in Ref. 12 and (c) MCS in this paper.
dependence on the swarm energy and doing the integral parameters (t) (the electron mean energy). The electron mean energy is calculated by
for the
(18) Figure 4 shows that the mean energy as a function of the distance from the cathode with the available energy. It is seen, from Figure 4, that: l
l
The electron mean energy is larger in abnormal discharge than that in normal discharge throughout the cathode region. The available energy t (stands for the electron energy collisionless throughout the cathode region) is larger than the electron mean energy as zjdc > 0.02 in two discharges (J = 0.19 mA/cm’ and I .5 mA/cm2), the differences between
The former of these indicates that the electron can gain more energy from the high electric fields and that the numbers of inelastic collisions are fewer in abnormal discharge (Figure l), so that the electron energy losses also become small. The reason for the second is that the collision causes an energy loss. Since the number of the electron-neutral collisions increase (Figure 2) and the electric field becomes weaker with increasing the distance from the cathode, the electron energy loss must increase with the distance. The nonequlibrium behavior of electron swarm has been clearly demonstrated by our results. Much of the electron energy gained in the cathode fall region is lost in elastic collisions. Near the cathode, we note that the mean energy increases rapidly and follows the available energy (z/de < 0.02) in two discharges. One knows that the electron mean energy depends on the electric field and the collisions in this region. Since the electrons emitted from the cathode have lower initial energy and the electric field is higher, the electric field acceleration dominates. Some electrons cross the cathode sheath region without collision, their energy follows the available energy beamlike electron. The proportion of these electrons is very small (approximately l-3%).” The electron energy distributionf‘(c) is shown in Figure 5 (A. electron distribution with t < 80 eV, B. with 6 > 80 eV). It is clear that the ratio of the higher energy electrons to the lower energy electrons becomes larger in abnormal discharge. The higher energy electrons come from these electrons that have not experienced inelastic collision. Collision rate. The collision rate is another important parameter always used to describe the electron-neutral gas interaction in discharge plasmas, and used in the continuity equations to calculate the electron transport parameters. The electron-neutral collision rate can be obtained from Vi(Z) = Ng j”Oi(E)(c/2me)“‘f(z,
(19)
where t is the electron instantaneous energy, o,(c) (i = elastic, metastable excitation, excitation, ionization or total) are the collision cross sections, it is dependent on the electron energy,f(z, 6) is the electron distribution function. In the fluid models, assumptions have been made to relate the electron collision rate and the transport coefficient to get fluid quantities such as E/N or electron mean energy. If we examine eqn (19) to involve the electron mean energy instead of the c(z) (electron instantaneous energy), the collision rate is
;;:_:;::1._
;
c k
I
,
400~
/
/’
: Qi /
/
/’
Xl(Z) = N~~i[~(z)l~(z)
/’
;
t)dG
V(z) = [2c(z)/m,]“2
-:I_::..;:::_
< 0 ’ 0.0
, 0.2
0.4
,
,
0. 6
0. 8
,
, 1. 0
, 1. 2
z/de Figure 4. The electron mean energy and the available energy in the cathode sheath region: (a) mean energy in the abnormal discharge and (b) mean energy in normal discharge. Solid line is the mean energy and dish line is the available energy.
(20)
Some interesting results are obtained. The results are shown in Figure 6 (A. elastic collision rate, B. metastable excitation collision rate, C. excitation collision rate, D. ionization collision rate and E. total collision rate). It is seen, from Figure 6A to E, that the results from eqns (19) and (20) are different: l
The collision rates calculated from eqn (19) are higher than that calculated from eqn (20) in normal discharge, and lower than that calculated from eqn (20) in abnormal discharge. 1069
Wei He/in et al: Electron-neutral
collision
processes
L
o,o:kJ 0
20
60
e7cgy(e V)
80
excitation or ionization collisions. On the other hand, despite some electrons having higher energies than the atom threshold potential, they also do not produce excitation or ionization collisions such as the beamlike electron.22 In the latter part of the cathode region (z/de > 0.5) the electron mean energy is approximately between 5@90 eV, the excitation and ionization collision cross sections have a larger value (from eqn (7) and Refs 17 and 18) so that the differences between eqns (19) and (20) become large in this region. Comparing Figure 6A and E with B, C and D, we can see that the differences between eqns (19) and (20) for the elastic collision rate are smaller than that for the other collision rates. This is because the effects of electron energy and electron mean energy on elastic collision cross sections are smaller than that on other types of collisions (eqns (6) and (7)). It is noteworthy that the total collision rate distribution (Figure 6E) is similar to the elastic collision rate distribution (Figure 6A). The reason for this follows that the elastic collision cross section is much larger than the other types of collision cross sections over a range of the electron energy from O-250 eV, and the probability of elastic collision or the number of the elastic collision is also larger (Table 2). In other words, the elastic collision has a larger proportion in the electron-neutral collision processes. This result implies that the elastic collision plays an important role in electron-neutral collision processes. Sun et al9 indicated that the elastic collision affects the angular scattering and the electron mean energy. From Figure 6, we also find that the collision rates are larger in abnormal discharge than that in normal discharge, this is because that the electron energy (or mean energy) is higher in abnormal discharge.
Conclusion 80
250
420
energy(
590
eV)
Figure 5. The electron energy distribution. Top. electron energy t < 80 eV, Bottom. electron energy L > 80 eV: (a) abnormal discharge and (b) normal discharge.
l
As z/de 2 0.5, the differences between the results from eqns (19) and (20) become large in two discharges.
In normal discharge, the electron mean energy is lower than the neutral threshold potential of the gas atom in a large part of cathode fall region (Figure 4). In this case, the collision rates obtained from eqn (20) must be low. The fact is that some electrons have higher energy (Figure 5)22 than neutral molecular excitation or ionization potentials in this region, these electrons may be able to produce excitation and ionization collisions that do not appear in eqn (20). The differences between the results from eqns (19) and (20) become more obvious in the latter part of the cathode region (z/de > 0.45) due to very low mean energy. Near the cathode, the electrons emitted from the cathode with lower energy, their energies and mean energies are lower (Figure 4) so the differences between eqns (19) and (20) become small. In abnormal discharge, the electron mean energy is larger than the neutral atom excitation or ionization potential in a large part of the cathode region (Figure 4) so the collision rates from eqn (20) are always larger than zero. It should be noted that some electrons have a lower energy than excitation or ionization potential in the cathode region (Figure 5A). They do not produce 1070
The electron-neutral atom collision processes in normal and abnormal discharges have been simulated by using a MCS method in which the position of the collision is determined by the electron-neutral collision frequency. The electron transport time, the number of collisions and collision rates are studied. The results show that the electron transport time and the number of collisions decrease in abnormal discharge due to strong electric field. In spite of the electron transport time and the number of collisions decrease in the abnormal discharge, the collision rates increase. The nonequlibrium behavior of electron swarm has been clearly demonstrated in this paper. Much of the electron energy gained in the cathode fall region is lost during collision processes. Equally important in the cathode region is the electron-neutral collision rates, which are often used in the fluid modelling and assumed as a function of electron mean energy. We calculate the collision rates from the electron instantaneous energy v,(z) and from electron mean energy x,(z). The results show that the collision rates, v,(z)and xl(z) (especially excitation and ionization collision), are different in the two discharges, v,(z) is larger than x>(z) in normal discharge, and v,(z) is smaller than x,(z) in abnormal discharge. The differences between v,(z) and x,(z) increase with the distance from the cathode. The difference between v,(z) and x1(z) is smaller for the elastic collision and the total collision. This result indicates that the collision (excitation and ionization) rate dependence of the electron instantaneous energy should be considered during the study of the electron transport behavior in a discharge.
Wei /-/e/in et al: Electron-neutral
collision processes (B)
/
/’ -- //
/
/
. -.
/-
b 0.
oot..,,...,,l,.,...,,.‘.‘,..“..‘..’,”,..’,,,,.,,,,’ 0.00
0.40
0.20
0.60
0.80
v .... 0
I.
00
0.20
0. 00
0.
0.
0.80
40
0.
80
1.00
z /dc
2 /dc
10
CD) -v)
1
n
0.20
d s
0 ‘_
0) 1 CT W k c
\ 0.
. \
lo-
\ \
c 0
\
b
\
\
.-
.d m .d 4 0 0
0. 00
0.
20
0.40
z/de
0.80
0 0
0.80
1.
00
0.00
0.20
0.
40
z/de
0.60
0.
80
1.00
I,,,,rLLL’,,,,,,U,,,,,,,‘,,,,,,,,,,,,,,
0.00 0.00
0.20
0.
40 z
0.60
/dc
0.
80
1.00
Figure 6. Theelectron-neutral collision rate (dashed line is the results from eqn (19), solid line is the results from eqn (20)) A The elastic collision rate. B. The metastable
collision
rate. C. The excitation
collision
rate. D. The ionization
collision
rate. E. The total collision
rate.
1071
Wei He/in eta/: Electron-neutral
collision processes
Acknowledgement This work is supported dation of China.
by the National
Natural
Science Foun-
References
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