CHAPTER 1 INTRODUCTION
1.4 Literature Survey for Modeling ICP
T T
abs plasma abs
p J Edt p t dt
T T
(1.13)The ICP source is usually operated at the radio-frequency range with
pe RF
m
, which is typical for a low-pressure RF discharge. Electrons could obtain the sufficient energy due to long mean free path in that reange frequency.
1.4 Literature Survey for Modeling ICP
The development of fluid model has been lasting for more than five decades. The early work using zero-dimensional or one-dimensional fluid models investigated the capacitively coupled discharges. Due to limitation of computer resources, the focus was on validations and refining modeling techniques by studying fundamental plasma physics. By the end of 1980s, the improvement of computer performance had improved the early work to combine the comprehensive chemistry with the Maxwell’s equations, although it is still limited as one-dimensional. Later on by the end of 1990s, several two- and three-dimensional fluid models with detailed chemistry in studying inductively coupled plasmas were developed because of the necessary knowledge in design and optimization of new plasma processes in the semiconductor fabrication industry.
Until now, some groups applied the fluid model by solving momentum equations of ions and/or electron to analyze the dynamics of thin sheath. However, most of these were applied to electropositive plasmas such as Ar, Xe, He, and C2H2 [11][12][13].
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Some other groups employed an approach, named ambipolar diffusion, to deal the presence of negative ions in plasma without considering the effect of sheath. Most of these were employed to simulate electronegative plasmas, such as Cl2,CF4, and SiH4, which are important etching processes used in semiconductor industry [11][12][13]. In addition, there has been nearly no study about parallel computing of fluid modeling for ICPs. Table 1 and Table 2 summarize the list of past studies in modeling electropositive and electronegative inductively coupled plasmas using fluid model respectively. More details of the review are described in the following.
1.4.1 Numerical Modeling of ICPs
A plasma model is a computer program that numerically solves a system of equations describing the physical and chemical processes occurring in the plasma.
Modeling and numerical simulations of plasma processing can be useful in many ways.
An improved understanding of a plasma processing system can be achieved by comparing predictions from numerical simulations with experimental observations.
The optimization of existing processes or the development of new plasma processes that offer better processing results may follow such an understanding. Modeling and simulations based on reliable physical or chemical modeling of a plasma processing system can significantly reduce the number of associated experiments that otherwise would have to be performed. Additionally, such modeling and simulations can also be used in the computer-aided design of new process system and to optimize fabricating processes within the framework of existing processing system.
Plasma simulation methods include kinetic and continuum methods [11][12][13].
The method for kinetic modeling is the Particle-In-Cell and Monte-Carlo-Collision (PIC/MCC) method, in which the plasma is modeled by a system of charged
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superparticles that move in a self-consistent electromagnetic field and collide following given collision cross sections for different processes. Fundamentally, the PIC/MCC method solves the Boltzmann equation directly using no more than collision kinetics;
however, it is generally very time-consuming and becomes impractical for a multidimensional simulation with complex plasma chemistry if the pressure is not too low. An alternative method, fluid model, is generally applied to simulate the plasma under imtermediate to high pressure conditions. It consists of the continuity equations, momentum equations and energy equations for various neutral and charged species in the plasma, which are obtained by taking velocity moments of the Boltzmann equation [14]. In addition, the Maxwell equations are often required to couple with fluid equations to obtain a self-consistent electric field. Among these equations, the momentum equation is the most difficult one to solve due to its highly nonlinear term.
Therefore, a method called the drift-diffusion approximation is used, instead of solving the momentum equation directly when the momentum transfer collisional frequency is larger than the radio frequency (RF). However, the diffusion approximation may become highly questionable when the plasma density is high and the pressure is very low in the range of mtorr for a typical ICPs (i.e., collisionless sheath). Alternatively, the assumption of ambipolar diffusion [3] that does not consider the effect of sheaths has been employed successfully in the fluid model, in which the fluid model is still valid in a plasma bulk that was demonstrated by Meyyappan [52].
1.4.2 Electropositive ICP
Table 1 summarizes the past studies we have found in the literature for simulating electropositive ICPs using fluid modeling. In 1993, Ventzek and Kushner et al. [15] were the first group to publish a 2-D hybrid model consisting of
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electromagnetic, electron Monte Carlo and hydrodynamic modules, and an offline plasma chemistry Monte Carlo simulation. They used this method to predict the distributions of the plasma density, plasma potential and ion fluxes for Ar, O2, Ar/CF4/O2 gas mixtures operated at low pressure (10-20 mTorr).
In the same year, R. Stewar and Graves et al. [16] presented a 2-D fluid model that consists of the continuity equations for electron, ions and neutrals, the momentum equations for ions, drift-diffusion approximation for electron and neutral, the energy density equation for electron and the Poisson’s equation for electrostatic potential. In this literature, they applied the assumed power profiles with solving the Maxwell’s equation. They present the effect of neutral gas pressure on the plasma uniformity for an argon discharge in the range of 1-20 mTorr and the comparison between the fluid model and the global model. The same group, Stewar et al. [17], improved their model to include the Maxwell’s equations. As a result, they had studied the power heating driven by external planar or cylindrical coils.
Lnmberoupoulos and Economou [18] used a 1-D fluid model to investigate the spatiotemporal dynamics of a pulsed-power inductively coupled argon plasma at 10 mTorr. Their model involved the continuity equations for all species, the drift-diffusion approximation for electron, the momentum equations for ions, the energy density equation for electron, the Maxwell’s equation for induced electric field and the Poisson’s equation for electrostatic electric field. Furthermore, an addition of RF bias applied in the model for modeling accelerated positive ions. The spatiotemporal dynamics and sheath thickness of argon plasma at 10 mTorr were investigated in detail by their model.
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Later, Gao and Wang et al. [20] develop a 2-D hybrid Monte Carlo/fluid model, in which the method,is similar to Stewar, et al. [17], except they have obtained the electron energy distribution function (EEDF) through Monte Carlo simulations. They have further applied this model to investigate the mode transition of ICP [21], and study the variation of plasma parameters and EEDFs due to various discharge conditions [20].
1.4.3 Electronegative ICP
Table 2 summarizes the past studies we have found in the literature for simulating electronegative ICPs using fluid modeling. Modeling electronegative ICPs is more challenging than modeling electropositive ICPs. A 2-D fluid model applied to Cl2 was first developed by Lymberopoulos and Economou et al. [22] in 1995. In their simulation, they assumed that the drift-diffusion approximation is still valid using a concept of “effective electric field”, rather than solving the full momentum equations directly. In fact, this method only works well for capacitive coupled plasmas, and it would be a failure for inductively coupled plasma if assumation of azimuthal variation is axisymmetric. We actually double that method and model they report.
In 2001 and 2002, Ramamurthi and Economous et al. [23][24] did not apply effective electric field in their model. In contrast, a method, named ambipolar diffusion for dealing with the flux of charged particles, was adopted. A self-consistent fluid model, including the Maxwell’s equation for power deposition, the electron energy density equation and the species mass balance, was proposed. The important assumptions of their model are summarized as follows: (1) The charged particle flux can be described by the drift-diffusion approximation since the pressure is larger than 10 mTorr, and (2) Electron heating is assumed to be collisional. The other assumptions
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are similar to those of most other studies in the literature. The model was used to study a pulsed-power chlorine discharge sustained in an ICP reactor with planar coil.
Hash and Meyyappan et al. [25] employed the ambipolar diffusion for solving the continuity, electrostatic electric field and momentum equations using the drift-diffusion approximation. They also solved the energy equations of neutral species and electron. The space charge induced electric fields are given directly by the Boltzmann relation assuming that the plasma is quasi-neutral. The most important result they have found was that the neutral gas in ICP reactors heats up significantly in the plasma. The model was applied to study CF4 discharge in a GEC chamber.
In 2006, Hsu and Graves et al. [19] had developed a fluid model in which they assumed the plasma is ambipolar and quasi-neutral so that the total flux of charged species outside the sheath is zero. The model equations for neutral species include overall mass continuity, momentum balance, energy balance and mass continuity for each species. In addition, the equations for charged species include the continuity equations for positive and negative ions, the energy density equation for electron, electric field from the balance of the summed flux of charged species and the Maxwell’s equations for induced electric field. They have considered plasma chemistry involving Ar, mixtures of Ar and O2 and mixtures of Ar/O2/Cl2. Moreover, they model the plasma as quasi-neutral everywhere. In their study, a comparison between the simulation and the experiment was performed.
Until very recently, Fukumoto et al. [26] present a two-dimensional fluid model, which assumes ambipolar diffusion, for etching SiO2 substrate in a CF4 ICP source, taking into account the plasma and surface chemistry of etch product species. The surface reaction model assumed Langmuir adsorption kinetics with the coverage of fluorine atoms, fluorocarbon radicals and polymers on SiO2 surface. Their numerical
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results indicated that etch products become a significant fraction of reactive ions and neutrals in the reactor, which in turn changes the plasma characteristics. While the plasma model include the surface kinetics, the density and the distribution of etch products in the reactor were changed by varying the ion bombardment energy on the substrate surface, gas pressure, mass flow rate and coil configuration, which arises in part from gas-phase reactions depending on plasma electron density and temperature.
1.4.4 Tetrafluoromethane CF
4Discharge in ICPs
There have been some reports about modeling of tetrafluoromethane (CF4) plasma in either capacitive coupled plasma or inductively coupled plasma that is used for etching purpose. Some of these reports focused on global modeling considering only gas-phase reactions [29][30][31][32], and some of them only considered feeding gas-phase reactions and highly simplified surface reactions using the concept of stick coefficients without taking any etching products into account [25][33][34][35]. It was found that etching products from the substrate significantly influence the composition of the gas phase species in a CF4 glow discharge while etching SiO2 and Si [36]. The etching of SiO2 consumes F atom to form SiF4, but the oxygen, released from the etching of SiO2, reacts with CFx radicals to form CO, CO2 and COF2 [37].
Etching products were observed as important gas-phase reactants by several experiments. For example, Coburn et al. [36] found that the escaping oxygen hinders polymerization on the SiO2 surface by forming volatile CO, CO2 and COF2, which allows etching process to continue under substrate surface without oxygen. Hikosaka et al. [38] used a quadrupole mass spectrometer (QMS) to measure CO+ and SiF3+ in high power CF4 discharge and found that they are produced mostly from quartz walls that could significantly deteriorate the etching selectivity of SiO2 in an ICP reactor.
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O’Neill and Singh [39] studied the surface effect on the concentration of gas-phase reactants in high-density etching plasmas by using ultraviolet-adsorption spectroscopy.
COFx+ and CO+ were detected in gas phase with oxygen radicals produced from a quartz substrate in CF4 and CF4/Ar ICPs [40][41][42]. The spatial concentrations of etching products such as SiF and SiF2 were resolved through laser-induced fluorescence in inductively driven discharges containing C2F6 and C4F8 by Hebner [43][44]. Later, Cruden [45] applied Fourier transform infrared spectroscopy (FTIR) to detect SiF4, CO and COF2 from window etching products in a gaseous electronics conference reference cell (GECRC). Significant amounts of etch products, SiFx+
/COFx+
(x=1~3), of quartz window were also detected by Rao et al. [48][49] and Zhou et al. [50]. To summarize, for an accurate modeling of CF4 ICP discharge, fluid model has to consider not only the gas-phase plasma chemistry but also the surface reactions.
Because etching products are important as well as gas-phase reactants resulting from feedstock gases in a high-density plasma, both the complex gas and surface chemical reactions in CF4 plasma have to be considered in fluid model to reproduce the major characteristics of etching process. Zhang et al. [51] were among the first group working on combination of plasma gas and surface model. They used a hybrid model coupled with a surface kinetics model to simulate CF4 discharge for etching Si in CF4. Fukumoto et al. [26] employed a fluid model coupled with a surface site balance model to simulate CF4 plasma etching SiO2 in a reactor with simple geometry without considering detailed gas-phase reactions of CF4. Since they solved the Maxwell’s equations using the Biot-Savart’s law analytically and fluid model without parallel computation, their method can only be applied to simple reactor geometry.
The Biot-Savart’s law is analystic solutions which do not consider the plasma induced
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electric field. In addition, the computational time may become prohibitively high for the fluid modeling combining with surface kinetic model when considering very complex gas phase and surface reactions. Thus, how to speed up the fluid modeling for a realistic inductively coupled plasma source is important, which is one of the major objectives in this thesis.
In conclusion, some groups adopted solving full momentum equations of ions instead of drift-diffusion approximation for electropositive plasma, and some groups used drift-diffusion approximation of charged species coupling with flux balance and ambipolar diffusion but for electronegative plasma. For CF4 discharge, thses reports almost used drift-diffusion approximation and flux balance to study typical cylindrical reactor and GEC chamber. The summarization of literature survey was list as Table 1 and Table 2 in detail.