Ion kinetics in an ECR plasma source
In plasma reactors it is extremely important the study for achieving greater control of critical parameters such as the flux velocities and the energy distribution of ions. These quantities are functions of the reactor physical dimensions, magnetic field profile as well as the kinetic chemical react...
Saved in:
| Date: | 2002 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
|
| Series: | Вопросы атомной науки и техники |
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/80327 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Ion kinetics in an ECR plasma source / C. Gutiérrez-Tapia // Вопросы атомной науки и техники. — 2002. — № 4. — С. 170-172. — Бібліогр.: 5 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-80327 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-803272025-02-09T09:47:29Z Ion kinetics in an ECR plasma source Gutiérrez-Tapia, C. Low temperature plasma and plasma technologies In plasma reactors it is extremely important the study for achieving greater control of critical parameters such as the flux velocities and the energy distribution of ions. These quantities are functions of the reactor physical dimensions, magnetic field profile as well as the kinetic chemical reactions occurring in the discharge. In this work, a model previously reported in [1] is improved. In that model using the drift kinetic equation approach the axial and radial ion fluxes at processing zone of an ECR plasma source are calculated. Now, the ionization rates are calculated from the energy distribution function obtained by a regularization method and using the experimental data of the electric probe measurements. Also, a more exact calculation of the external magnetic field is included. This work was patially supported by project CONACyT 33873-E 2002 Article Ion kinetics in an ECR plasma source / C. Gutiérrez-Tapia // Вопросы атомной науки и техники. — 2002. — № 4. — С. 170-172. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 52.50.-b https://nasplib.isofts.kiev.ua/handle/123456789/80327 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Low temperature plasma and plasma technologies Low temperature plasma and plasma technologies |
| spellingShingle |
Low temperature plasma and plasma technologies Low temperature plasma and plasma technologies Gutiérrez-Tapia, C. Ion kinetics in an ECR plasma source Вопросы атомной науки и техники |
| description |
In plasma reactors it is extremely important the study for achieving greater control of critical parameters such as the flux velocities and the energy distribution of ions. These quantities are functions of the reactor physical dimensions, magnetic field profile as well as the kinetic chemical reactions occurring in the discharge. In this work, a model previously reported in [1] is improved. In that model using the drift kinetic equation approach the axial and radial ion fluxes at processing zone of an ECR plasma source are calculated. Now, the ionization rates are calculated from the energy distribution function obtained by a regularization method and using the experimental data of the electric probe measurements. Also, a more exact calculation of the external magnetic field is included. |
| format |
Article |
| author |
Gutiérrez-Tapia, C. |
| author_facet |
Gutiérrez-Tapia, C. |
| author_sort |
Gutiérrez-Tapia, C. |
| title |
Ion kinetics in an ECR plasma source |
| title_short |
Ion kinetics in an ECR plasma source |
| title_full |
Ion kinetics in an ECR plasma source |
| title_fullStr |
Ion kinetics in an ECR plasma source |
| title_full_unstemmed |
Ion kinetics in an ECR plasma source |
| title_sort |
ion kinetics in an ecr plasma source |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2002 |
| topic_facet |
Low temperature plasma and plasma technologies |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/80327 |
| citation_txt |
Ion kinetics in an ECR plasma source / C. Gutiérrez-Tapia // Вопросы атомной науки и техники. — 2002. — № 4. — С. 170-172. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT gutierreztapiac ionkineticsinanecrplasmasource |
| first_indexed |
2025-11-25T14:07:32Z |
| last_indexed |
2025-11-25T14:07:32Z |
| _version_ |
1849771588324425728 |
| fulltext |
ION KINETICS IN AN ECR PLASMA SOURCE*
César Gutiérrez-Tapia,
Departamento de Física, Instituto Nacional de Investigaciones Nucleares,
Apartado Postal 18-1027, Col. Escandón, México 11801 D. F.
In plasma reactors it is extremely important the study for achieving greater control of critical parameters such as the
flux velocities and the energy distribution of ions. These quantities are functions of the reactor physical dimensions,
magnetic field profile as well as the kinetic chemical reactions occurring in the discharge. In this work, a model
previously reported in [1] is improved. In that model using the drift kinetic equation approach the axial and radial ion
fluxes at processing zone of an ECR plasma source are calculated. Now, the ionization rates are calculated from the
energy distribution function obtained by a regularization method and using the experimental data of the electric probe
measurements. Also, a more exact calculation of the external magnetic field is included.
PACS: 52.50.-b
1.BASIC EQUATIONS
In reactor region of an ECR plasma source, the geometry
of the external field ),( trBB = is not uniform. For the
case of an ECR plasma source, the magnetic field is
strong enough to satisfy
1/ < <= LLρδ , (1)
where L is the characteristic plasma length and
ZeBcvmiL /⊥=ρ is the ion Larmor radius, c is the
speed of light in vacuum and Ze is the ion charge. The
condition (1) is sufficient to apply the drift kinetic theory
[2,3] adopted along the entire work. The approximate
version of the drift kinetic equation, reported in [2,3], is
assumed appropriate in the small gyroradius limit (1).
From condition (1), one can obtain the drift kinetic
equations in coordinates ( )tU ,,,, γµr :
( )
( ) ,||
||
C
U
f
t
A
c
e
t
B
t
e
f
dt
df
t
f
i
DD
i
cg
iDD
i
=
∂
∂
∂
∂⋅++−
∂
∂+
∂
∂
+
∂
∂
+∇⋅+++
∂
∂
uvv
uvv
µφ
µ
µ
(2)
where ∫= γdff ii is the average part of the ion
distribution function with respect to the phase γ ; C is
the collision operator which includes charge-exchange
(cx), elastic collisions (el) and gas ionization by electron
impact at the rate e
vσ , φ is the electrostatic potential,
A is the vector potential, and cmZeB ic /=ω is the
cyclotron frequency. U is the total particle energy, given
by
.
22
22
|| φe
vmvm
U ii −+= ⊥ (3)
µ is the magnetic moment and DD uvv ,,|| , are the
components of the total drift velocity.
In this method the parallel and perpendicular
particle fluxes are defined as
,
,3
||||
×∇−+=
=
⊥
⊥
∫
Bm
Pnnn
vdfn
ci
DD
i
ω
BUVV
vV
(4)
where
∫
∫∫
=
==
⊥ .
; ;
3
33
vdfBP
vdfunvdfn
i
iDDiDD
µ
UvV
(5)
The charge-exchange process takes the ions
away at a rate vcxσ and returns them to the ion
distribution at zero velocity. Elastic collisions takes the
ions away at a rate velσ and return them isotropically
distributed in the center of mass reference frame. Here we
will treat the cross sections cxσ and elσ as constants
since they are insensitive to the ion energy in the range of
energies up to a few eV . In order to solve equation (1)
we will assume the following ordering [4]:
( ) ,/1~ τσσσω elcxeci nvn +> >> > (6)
where τ is the characteristic discharge time. We also
assume that
( ) .1 L
n elcx
< <
+
≡
σσ
λ (7)
These assumptions allow us to neglect, to lowest order,
the time derivatives of if in equation (1) as well as the
charge-exchange and elastic collision terms. As a result,
equation (2) reduces to [1]
( ) ,0|| ei
ci
vnnf
B
B
mB
v σδ
ω
µ vBB =∇⋅
∇×+ (8)
where 0n and n are the gas and electron densities,
respectively Also it has been used the condition of
symmetry in magnetic mirror configurations [1]
.0≅×∇ B (9)
170 Problems of Atomic Science and Technology. 2002. № 4. Series: Plasma Physics (7). P. 170-172
2.DISTRIBUTION FUNCTION
The component of equation (8) along the magnetic field
line near to the axis can be written as
( ) .1
0
||
e
i vnn
vz
f σδ v=
∂
∂
(10)
The integration of the last equation results in
( )
( ) .
,,||
0∫ ′
′
= zd
Uzv
vnn
f e
i µ
σδ v
(11)
3.CALCULATION OF ION FLUXES
From the definitions of the parallel and perpendicular ion
fluxes (4) and using the expression for the distribution
function, we obtain [1]
,
2
,
0||
0||
z
c
e
e
B
vnnv
n
B
zvnnn
eBV
BV
×=
=
⊥ ω
σ
σ
(12)
where ze is the unit vector along OZ. The corresponding
expressions for the radial and axial components of the ion
fluxes are
.
,
0
0
B
BzvnnnV
B
BzvnnnV
z
ez
r
er
σ
σ
=
=
(13)
4.MAGNETIC FIELD
The magnetic field generated by a current sheet formed
by L circular current turns is obtained using the
superposition principle, where the magnetic field for one
of the turns is used to calculate the magnetic field for the
whole current sheet
( )
( ) ,
2
,
,,
1
1
∑
∑
=
=
∂
∂
∂
∂
+=
∂
∂
∂
∂
−=
L
l
l
l
ll
z
L
l
l
l
l
r
r
k
k
A
r
A
zrB
z
k
k
A
zrB
θθ
θ
(14)
where L is the total number of current turns in the sheet,
and
( ) ( )
( )
( ) ( )
.
E
K
2
1
4,
,
4
1
2
1
2
1
12
−
−
×=
−++
=
l
l
l
l
l
l
l
k
k
k
k
k
r
R
c
JzrA
zzRr
rRk
θ (15)
Here ( )lkK and ( )lkE are the elliptic integrals of the
second and first kind respectively. ( )2/1−= ldzl is
the left-side position of the current sheet on the z axis.
Generalizing to the case of N axially aligned magnet
coils we obtain
( )
( ) ∑∑∑
∑∑∑
===
===
∂
∂
∂
∂
+=
∂
∂
∂
∂
−=
nn
nn
L
l
lmn
lmn
lmnlmn
M
m
N
n
z
L
l
lmn
lmn
lmn
M
m
N
n
r
r
k
k
A
r
A
zrB
z
k
k
A
zrB
111
111
2
,
,,
θθ
θ
(16)
where nL and nM are the number of loops in each
current sheet and the total number of nested sheets in each
of the n-th magnet coil respectively. In this case
( ) ( )
( )
( ) ( )
.
E
K
2
1
4
,
,
4
22
2
−
−
×=
−++
=
lmn
lmn
lmn
lmn
lmn
mnn
lmn
lmn
mn
lmn
k
k
k
k
k
r
R
c
J
zrA
zzRr
rR
k
θ
(17)
The particular case of three coils is illustrated in Fig.1.
The axial distributions of the magnetic field components
are plotted in Fig. 2.
Fig. 1. Schematic representation of an ECR system
showing the magnetic system with three coils.
5.DISCUSSION AND CONCLUSIONS
In order to prove agreement of the theoretical results with
experimental ones, the same parameters as in [5] are used.
These parameters are resumed in Table 1. The argon ions
are considered in calculations.
To calculate the ionization rate, it is of great
importance to know the energy distribution function. In
the case of low ion temperatures ( iT ≤ 2.5 eV), the
maxwellian distribution is a good approximation as can be
obtained from the integration of the ion part of the I-V
probe characteristic by regularization methods. Using this
result, the expression for the ionization rate is calculated.
171
From expressions (13) it is noticed that the radial
and axial ion fluxes are directly proportional to the
corresponding magnetic field components.
Fig. 2. Axial behavior of the axial (solid) and radial
(dashed) components of the magnetic field, for different
radii.
Table 1. Parameters reported in [5] and used to validate
the theoretical results of the present work.
n0=5.5E11cm3 n =4.6E12cm-3 P=10-4torr Te=3.5eV
Ii=13.8eV
e
vσ =6.24E-
10cm-3/s
Ti=0.5eV J1=180A
J2=90A J3=70A r1=r2=10.8cm r3=11.4cm
L1=L2=14 L3=46 M1=M2=12 M3=5
Z1=0.0cm Z2=24.6cm Z3=74.5 d=0.4cm
In Fig. 3, the axial distribution for both the axial
and radial ion fluxes As we can see, the axial ion flux
always increases. Otherwise, the radial flux shows a
strong irregular behavior.
Fig.3. Calculated axial ion flux (solid) and experimental
points reported in [5] (dashed) using the parameters
resumed in Table 1.
Fig. 4. Axial distribution of the radial ion flux for several
radii.
The irregular behavior of the radial ion flux a be
the source of the inhomogeneity observed in depositions
as is the case of steel nitriding.
REFERENCES
1. J. González-Damián and C. Gutiérrez-Tapia,
J.Appl.Phys., 90, 1124 (2001).
2. R.D. Hazeltine, Plasma Phys., 15, 77 (1973).
3. F. L. Hinton and R. D. Hazeltine, Rev. Mod.
Phys., 48, 239 (1976).
4. B. N. Breizman and A. V. Arefiev, Phys.
Plasmas, 9, 1015 (2002).
5. N. Sadegui, T. Nakano, D. J. Trevor, and R. A.
Gottscho, J. Appl.Phys., 70, 2552 (1991).
*This work was patially supported by project CONACyT
33873-E
172
ION KINETICS IN AN ECR PLASMA SOURCE*
REFERENCES
|