RF Discharge Modeling Through Solutions to the Moments of the Boltzmann Transport Equations
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RF DISCHARGE MODELING THROUGH SOLUTIONS TO THE MOMENTS OF THE BOLTZMANN TRANSPORT EQUATIONS M. Meyyappan, T. R. Govindan and J. P. Kreskovsky Scientific Research Associates, Glastonbury, CT 06033 ABSTRACT Simulation of if discharges through numerical solutions to the Moments of the Boltzmann Transport Equations (MBTE) is discussed. Continuity and momentum equations for the electrons and ions, and electron energy equation have been solved using an efficient finite difference scheme. Results for a 13.56 MHz argon discharge are presented. INTRODUCTION Radiofrequency discharges are increasingly used in microelectronics fabrication, particularly in plasma etching and plasma enhanced deposition. Improvements in equipment and process design require an understanding of the discharge physics and discharge chemistry. Hence, there is a need to develop accurate computational models for discharge processes. Glow discharge physics, which deals with the production, transport and loss of charged particles in the reactor chamber, is an important aspect of discharge modeling as this forms the basis for the chemical processes leading to etching or deposition. Several models have appeared addressing the discharge physics issues, and a summary can be found in ref. [1]. Of these, the continuum models have been shown to provide valuable information about the bulk of the plasma [21. With regard to the sheaths, previous works have often ignored the inertial terms in the momentum equation for ions and electrons. As the process pressure decreases, the local and convective acceleration terms assume more importance. Realizing this, we recently developed a comprehensive model for glow discharge physics which involves solution of the moments of the Boltzmann transport equations [1]. In the present work, we provide applications of this model to if discharges. DETAILS OF THE MODEL The present continuum model assumes a Maxwellian velocity distribution for the electrons. The ion temperature is assumed to be that of the background gas. The governing equations used here are the first three moments of the Boltzmann transport equations.
an 1
at
an
-
at
2
+ Ven 1 mv = gn,
(1)
+ Ven
(2)
2
v 2 = gnI
Mat. Res. Soc. Symp. Proc. Vol. 190. ©1991 Materials Research Society
150
a at
(nlmlVyl)
+ v. (nlmlylyl)
= -VP 1
+ enlVO
(3)
-nlmnlv IyI
a at
(n 2 m2 _2 ) + V. (n 2 m2 _2 yv2 ) = -VP -n
-
a
at
3
(-
- en 2VO
2
(4)
2 m22 v 2 Y 2
3
3 kTlnly1 = -nlT1 V.yI + V.KVT 1 2 2
knlTl) + V.-
2
(5)
-gnlH + nlmlvlyi.y1 1 + - nlmlg MI9yI 2
where, g = kio N exp (-Ei/kT,)
and
P = nkT
(6)
Poisson's Equation: v2
e (nI - n2)
(7)
Current relations: !1
= -nlevl,
J2 = n 2 ey 2 ,
!D =
a (VO) at
-E -
(8)
!IT = -1 + 22 + -TD In the above, subscripts 1 and 2 represent electrons and positive ions respectively. We use the following notations: n, particle density; N, neutral gas density; L,particle velocity, m, particle mass, T, particle temperature; P, particle partial pressure; ,, potential; J, current density; v, elastic collision frequ
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