2D fluid model for interactive development of ICP technological tools
The software for ICP device simulation is worked out. Discharge chamber geometry, RF power, pressure and working gas type are the input data. The results of calculation are inductor voltage, ion current density distribution on the chamber surface, steady state space distributions of the electric f...
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| Дата: | 2006 |
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| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | 2D fluid model for interactive development of ICP technological tools / A.V. Gapon, A.N. Dahov, S.V. Dudin, A.V. Zykov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 186-188. — Бібліогр.: 3 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-82306 |
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dspace |
| spelling |
irk-123456789-823062015-05-28T03:02:32Z 2D fluid model for interactive development of ICP technological tools Gapon, A.V. Dahov, A.N. Dudin, S.V. Zykov, A.V. Azarenkov, N.A. Low temperature plasma and plasma technologies The software for ICP device simulation is worked out. Discharge chamber geometry, RF power, pressure and working gas type are the input data. The results of calculation are inductor voltage, ion current density distribution on the chamber surface, steady state space distributions of the electric field, plasma density and electron temperature in the chamber. Set of 2D parameter distributions is visualized immediately after calculation. The software had been carefully verified by comparing the calculation results with real data measured experimentally. The comparison has shown that both calculated 2D plasma density and electron temperature profiles and ion current density distribution on the processed surface are quite realistic. Graphical geometry input, fast calculation and immediate result visualization makes it possible to use our software for interactive development of ICP technological tools. 2006 Article 2D fluid model for interactive development of ICP technological tools / A.V. Gapon, A.N. Dahov, S.V. Dudin, A.V. Zykov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 186-188. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 52.35.Hr http://dspace.nbuv.gov.ua/handle/123456789/82306 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| 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 Gapon, A.V. Dahov, A.N. Dudin, S.V. Zykov, A.V. Azarenkov, N.A. 2D fluid model for interactive development of ICP technological tools Вопросы атомной науки и техники |
| description |
The software for ICP device simulation is worked out. Discharge chamber geometry, RF power, pressure and working
gas type are the input data. The results of calculation are inductor voltage, ion current density distribution on the
chamber surface, steady state space distributions of the electric field, plasma density and electron temperature in the
chamber. Set of 2D parameter distributions is visualized immediately after calculation. The software had been carefully
verified by comparing the calculation results with real data measured experimentally. The comparison has shown that
both calculated 2D plasma density and electron temperature profiles and ion current density distribution on the processed
surface are quite realistic. Graphical geometry input, fast calculation and immediate result visualization makes it
possible to use our software for interactive development of ICP technological tools. |
| format |
Article |
| author |
Gapon, A.V. Dahov, A.N. Dudin, S.V. Zykov, A.V. Azarenkov, N.A. |
| author_facet |
Gapon, A.V. Dahov, A.N. Dudin, S.V. Zykov, A.V. Azarenkov, N.A. |
| author_sort |
Gapon, A.V. |
| title |
2D fluid model for interactive development of ICP technological tools |
| title_short |
2D fluid model for interactive development of ICP technological tools |
| title_full |
2D fluid model for interactive development of ICP technological tools |
| title_fullStr |
2D fluid model for interactive development of ICP technological tools |
| title_full_unstemmed |
2D fluid model for interactive development of ICP technological tools |
| title_sort |
2d fluid model for interactive development of icp technological tools |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2006 |
| topic_facet |
Low temperature plasma and plasma technologies |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/82306 |
| citation_txt |
2D fluid model for interactive development of ICP technological tools / A.V. Gapon, A.N. Dahov, S.V. Dudin, A.V. Zykov, N.A. Azarenkov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 186-188. — Бібліогр.: 3 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT gaponav 2dfluidmodelforinteractivedevelopmentoficptechnologicaltools AT dahovan 2dfluidmodelforinteractivedevelopmentoficptechnologicaltools AT dudinsv 2dfluidmodelforinteractivedevelopmentoficptechnologicaltools AT zykovav 2dfluidmodelforinteractivedevelopmentoficptechnologicaltools AT azarenkovna 2dfluidmodelforinteractivedevelopmentoficptechnologicaltools |
| first_indexed |
2025-07-06T08:48:56Z |
| last_indexed |
2025-07-06T08:48:56Z |
| _version_ |
1836886772497252352 |
| fulltext |
186 Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 186-188
2D FLUID MODEL FOR INTERACTIVE DEVELOPMENT
OF ICP TECHNOLOGICAL TOOLS
A.V. Gapon, A.N. Dahov, S.V. Dudin, A.V. Zykov, N.A. Azarenkov
V.N. Karazin Kharkov National University, 31 Kurchatov Ave., 61108, Kharkov, Ukraine,
e-mail: gapon@pht.univer.kharkov.ua
The software for ICP device simulation is worked out. Discharge chamber geometry, RF power, pressure and work-
ing gas type are the input data. The results of calculation are inductor voltage, ion current density distribution on the
chamber surface, steady state space distributions of the electric field, plasma density and electron temperature in the
chamber. Set of 2D parameter distributions is visualized immediately after calculation. The software had been carefully
verified by comparing the calculation results with real data measured experimentally. The comparison has shown that
both calculated 2D plasma density and electron temperature profiles and ion current density distribution on the proc-
essed surface are quite realistic. Graphical geometry input, fast calculation and immediate result visualization makes it
possible to use our software for interactive development of ICP technological tools.
PACS: 52.35.Hr
1. BASIC EQUATIONS
In the paper [1] 2D fluid model was built which al-
lowed to calculate basic parameters of ICP discharge in a
simple cylindrical chamber. At the same time real techno-
logical ICP devices have chambers of more complicated
shape. In this work we present software specifically
aimed to solve this problem in arbitrary chamber of cylin-
drical symmetry. It calculates the electron density, elec-
tron temperature, electric field and particle flows distribu-
tions, if the chamber geometry, gas pressure and input
power are given.
The problem is considered in cylindrical coordinates
zr ,,θ . The ambipolar diffusion and electron heat conduc-
tivity equations set was used for the transport processes
modeling [2]:
NnDa =∇−∇ , (1)
QT =∇−∇χ , (2)
)/( iaia mTD ν= , (3)
eaem
nT
ν
χ
2
5
= , (4)
nN iν= , (5)
wkQ
j
j +−= ∑2/3 (6)
where n - plasma density, aD - ambipolar diffusion co-
efficient, N - particle sources density, T - electron tem-
perature, im , em - ion and electron masses, iν , eaν , iaν
are the ionization, electron-neutral and ion-neutral colli-
sion frequencies, χ - electron heat conductivity, Q - heat
sources density, p - working gas pressure, jk -rate coeffi-
cient of j -th reaction between electron and neutral, w -
input power density. We assume that particle velocity on
the wall is equal to ion sound velocity is mTu /= .
Then, particle flow on the wall is:
sWa nunD =∇− . (7)
Heat flux on the walls is formed by the electrons with
energies more than sheath potential drop. In the assump-
tion of Maxwellian EEDF, the heat flux is:
( )eisW mmnTuT /ln2 +=∇− χ . (8)
Expressions (7,8) make up the boundary conditions for
(1,2). Rate coefficients were approximated from the data
obtained with the help of BOLSIG+ program [3].
RF electric field E
r
with the angular frequency ω re-
sults in the electric current density j
r
in plasma, that in
linear approach depends on E
r
by Ohm law:
Ej
rr
σ= ,
)(
2
eae im
nei
νω
σ
+
= , (9)
where σ is the RF conductivity of plasma. Due to axial
symmetry of the problem only angular component of the
RF electric field is not equal to zero, )0,,0( Θ= EE
r
, there-
fore, it can’t disturb the electric charge volume density
ρ , because of plasma density does not depend on angle
coordinate. Then 04 ==∇ πρE
r
, and from Maxwellian
equations and quasi-static approach we obtain that
02 =−∆ − EE
rr
δ , (10)
22
2
2- /4 c
c
i pωσ
ω
πδ =−= , (11)
where δ is the skin depth, pω is the plasma frequency.
Energy input in plasma is accounted as Joule heating and
is described by term w in right part of (6):
2
22
2
*
8
12/)Re( EEjw
ea
eap
νω
νω
π +
==
rr
(12)
Boundary conditions for (10) are
0=ΘE on the metal walls (13)
I
ac
iE
S 2
2],[ ω
τ =∇
r
on the inductor surface. (14)
In the (15) τ
r is a unit vector tangential to the inductor
surface and lied in ( r , z ) plane, a denotes wire radius of
the inductor, and I - the amplitude of the electric current
flowed through the inductor.
2. NUMERICAL SOLUTION
Transport equations set (1-2) and the equation for the
electric field strength (10) are solved iteratively to obtain
steady state self-consistent solution. The electric field,
and then heat source density w do not vary when trans-
port set is solved; in turn plasma density n , and then skin
mailto:gapon@pht.univer.kharkov.ua
187
depth δ are constant during the solution of the equation
for electric field. An initial guess is needed the iterations
to start. It is obtained from the eigenvalue problem solu-
tion ( constTe = over the discharge chamber). Total
power input in discharge is set to constant value and is
one of the input parameters. Then, input power density
w should be normalized correspondingly before the
transport equations set is treated.
The code was totally realized in MATLAB-6.0.0.8 r12
environment because of excellent partial differential equa-
tion functions library (PDE toolbox) is included in it. In
accordance with PDE toolbox paradigm, we should give
the following form to the system of equations (1)-(6),(7)-
(8):
ijij
n
j lk
j
l
ijkl
k
FuAu
x
C
x
=+
∂
∂
∂
∂
− ∑ ∑
= =1
2
1,
, (15)
The boundary conditions of Neumann type looks as:
∑ ∑
= =
=+
∂
∂n
j
i
lk
jijj
l
ijklk guqu
x
C
1
2
1,
cosα . (16)
In (15), (16) n is the number of equations, kα is the an-
gle between outward normal to the surface and direction
of kx axis. In the case of the transport equation set:
=
T
n
u ,
=
z
r
x ,
⋅
⋅
=
Qr
Nr
F , (17)
aDrCC ⋅== 11221111 , (18)
χ⋅== rCC 22222211 , (19)
suq =11 , ( )eis mmnuq /ln222 += . (20)
For the electrodynamics set (10, 13, 14) we have:
θEu = , rCC −== 11221111 , 22 / cA pω−= , (21)
0=metu , I
ac
ig ind 21
2ω
= , (22)
where met and ind subscripts denote metal surfaces and
inductor surface correspondingly. The value of inductor
current I can be obtained during the normalization of w .
All unmentioned components of C , A , q , g are equal to
zero. After applying of finite element method transport
equations take form
FKu = , (23)
where u is the solution vector and matrices K and F
depend on u . We submit (23) in the traditional form for
the nonlinear solvers applying:
0)( ==− uAFKu (24)
Equation (24) allows using the Newton iteration process
in accordance with the known formula:
nnn uuAsuu )/(1 ⋅∂∂⋅+=+ , (25)
where uA ∂∂ / is the Jacoby matrix calculated on n-th
iteration, s -parameter, which should be adjusted in limits
11.0 << s for iteration convergence. All quantities in-
cluded in A in (24) vary with u under calculation of
uA ∂∂ / , with exception of w . The last is found in nu and
assumed to be constant under Jacoby matrix calculation.
Fig. 1. Schematic diagram of the ICP reactor
3. COMPARISON WITH THE EXPERIMENT
The software had been verified by comparing the cal-
culation results with real data measured experimentally.
Fig.1. presents sketch of the working chamber of the
real setup used for the model testing and adjustment is
shown. Inductive coil is fed by RF power of 0.1…1 kW
range and 13.56 MHz frequency. Argon was used as
working gas under pressures 0.3 mTor…1 Torr. Fig.2
presents dependences of T and n on pressure.
Fig.2. T and n in the chamber center vs. argon pressure.
Bold lines – experimental results
Reliability of the model under low pressures (<10 mTorr)
is limited by the validity of the diffusive approach due to
relatively high mean free path of charged particles. So,
the discrepancies between measured and calculated values
of T and n in the low pressure region looks natural.
Fig.3 depicts the radial dependence of the ion current
density on the base of the chamber. The probe was set on
the chamber axis. Calculated curves shows similar behav-
ior with measured one. Presence of typical maximum en-
sures one in the model reliability.
Fig.4 shows normalized ion current density profiles on
substrate holder.
At the Fig. 5 the set of calculated radial distributions
of the ion current density on the substrate holder is pre-
sented. Sequence of profiles follows the behaviour of the
curve on Fig. 2.
The comparison has shown that results of calculations
at argon pressures > 0.01 Torr are in good agreement with
all obtained experimental data.
188
Fig.3. Ion current density at the chamber bottom and side
walls vs. argon pressure. Bold lines – experiment
Fig.4 Normalized j profiles on the substrate holder
Fig.5. Pressure dependence of j radial profiles
ACKNOWLEDGMENTS
This work was supported by Ministry of Industrial Policy
of Ukraine, Project 92373/60
REFERENCES
1. I. Denysenko, S. Dudin, A. Zykov, N. Azarenkov. Ion
flux uniformity in inductively coupled plasma
sourses//Physics of Plasmas. 2002, v.9, N11,p.4767-4775.
2. V.E. Golant, A.P. Zhilinskii and I.E.Sakharov. Funda-
mentals of Plasma Physics/ Moscow: “Atomizdat”, 1977.
3. G.J.M. Hagelaar and L.C. Pitchford. Solving the
Boltzmann equation to obtain electron transport coeffi-
cients and rate coefficients for fluid models//Plasma
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