# Numerical and experimental investigation of nonequilibrium microwave argon plasmas

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Numerical and Experimental Investigation of Nonequilibrium Microwave Argon Plasm as A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy By Yunlong Li, B.S.M.E., M.S.M.E. Xian Jiaotong University, Xian, China, 1992, 1995 May 2000 University of Arkansas, Fayetteville Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number 9987250 __ ® UMI UMI Microform9987250 Copyright 2000 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This dissertation is approved for recommendation to the Graduate Council DISSERTATION DIRECTORS DrTLafry A. Rde Dr. Matthew H. Gordon DISSERTATION COMMITTEE "D r. KnaledTTassouni - Dr. Ajay Malshe Dr. R i^ C o u v illio n Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A C K N O W LED G M EN TS I would like to thank Dr. Larry Roe and Dr. M att Gordon for their support throughout my Ph.D. program. Also, thanks to my committee members: Dr. Khaled Hassouni, Dr. Ajay Malshe, Dr. Min Xiao, and Dr. Rick Couvillion. Special thanks go to my wife, Yuhong Cai, who has made these years the best ever. The Mechanical Engineering Department supported my Ph.D. study all through the past four and half years. I want to thank Dr. Schmidt, the department head, for his consideration and kindness. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CO NTENTS Page ACKNOWLEDGEMENTS.................................................................................. iii TABLE OF CONTENTS.......................................................................................iv LIST OF SYMBOLS.............................................................................................vi LIST OF TABLES..................................................................................................x LIST OF FIGURES............................................................................................. xi CHAPTER 1 INTRODUCTION............................................................................. 1 1.0 Motivation.................................................................................................................................1 1.1 Introduction.............................................................................................................................. 2 1.2 Diamond CVD Processes...................................................................................................... 4 1.3 Fundamental Plasma Processes and Forms of Nonequilibrium..................................... 9 1.4 Plasma Diagnostics.............................................................................................................. 12 1.5 Plasma Modeling..................................................................................................................13 1.6 Objective............................................................................................................................... 22 CHAPTER 2 EXPERIMENTAL FACILITY......................................................... 24 2.1 Microwave Plasma Reactor................................................................................................ 24 2.2 Emission/Absorption Systems............................................................................................ 26 CHAPTER 3 EMISSION/ABSORPTION DIAGNOSTIC TECHNIQUES............. 30 3.1 Emission Spectroscopy....................................................................................................... 30 3.2 Absorption Spectroscopy.................................................................................................... 37 3.3 Spectral Line Broadening Theories.................................................................................... 40 CHAPTER 4 MODELING................................................................................... 48 4.1 Pseudo-1-D Plasma Model ............................................................................................... 48 4.2 Electromagnetic Model........................................................................................................ 56 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Two Dimensional Plasma Fluid Model...............................................................................60 4.4 Two Dimensional Argon Plasma Model.............................................................................72 4.5 Two Dimensional Collisional Radiative Model.................................................................. 85 CHAPTER 5 ENERGY BALANCE STUDY........................................................ 89 5.1 Pseudo-1-D Argon Model Results......................................................................................90 5.2 Experimental Global Reactor Energy Balance................................................................. 96 5.3 Control Volume Heat Transfer Analysis.............................................................................99 5.4 2-D Argon Fluid Model Results......................................................................................... 104 CHAPTER 6 EXCITED STATES OF THE ARGON PLASMAS....................... 114 6.1 The Experimental OES D ata.............................................................................................115 6.2 Absorption Measurement of the Metastable S tate.........................................................116 6.3 Characterization of the Argon Plasma with the 2-D Fluid Model................................. 121 6.4 Characterization of the Argon Plasma with the 2-D C R M ............................................. 140 6.5 Non-uniformity of the Argon Plasmas.............................................................................. 159 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS............................162 7.1 Conclusions.........................................................................................................................162 7.2 Recommendations..............................................................................................................165 REFERENCES..................................................................................................167 APPENDIX A .....................................................................................................173 APPENDIX B .....................................................................................................178 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF SYMBOLS The following is a list of symbols used frequently in this dissertation. A number of symbols have been used for several different purposes. a a, b A x Voigt parameter Coefficients in the curve-fitted equation for reaction rates Area Einstein coefficient, transition probability from energy level I to j, or at wavelength X B Magnetic flux density B,| Stimulated absorption or emission probability c Speed of light, = 3e8 m/s c Specific heat C Constant d Stark shift D Diameter D Diffusion coefficient Dx The absolute calibration factor E Electric field strength E Electric field f Electron energy/velocity distribution function f Gas flow rate F, Fx, Fy F, Flux density vector and its component in x and y directions External force g Relative velocity g Degeneracy h Planck’s constant, = 6.6262e-34 Js h Heat transfer coefficient H Magnetic field I |(v) Signal intensity Lineshape function J Current density k Boltzmann’s constant, = 1.3806e-23 J/K k Spectral absorption coefficient k Reaction rate vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. K Total production or loss I Angular momentum quantum number L Chord length m Mass m Mass flow rate n, N Number density Nu Nusselt number P Pressure P Power Pr Prandtl number q Electron charge Q Heat rate Q Cross-section Q Electron partition function r R Re s, p, d, f S, P, D, F t T u. v, U, V Radial coordinate ratio Reynolds number Electronic state Atomic state Time Temperature Velocity V Signal strength V Volume w Electron impact parameter W x x,y,z Chemical production or consumption rate Mole fraction Cartesian coordinates Z Effective nuclear charge a Ion-broadening parameter 8 Skin depth y Reduced velocity e Continuum emission e Energy level e Emissivity VII Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e permittivity 4> Lineshape factor X Wavelength X Energy relaxation length, or mean free path X Thermal conductivity n Dynamic viscosity po Magnetic permeability I* Ion mobility v Collision frequency v Frequency p Density, or charge density po Debye radius a Collision cross-section a Plasma conductivity a Ratio of electron to ion collision frequency t Shear stress t Radiative lifetime cd Angular frequency cDpe ^ Electron plasma frequency The free-bound Biberman factor The free-free Biberman factor (p Conserved variable A Characteristic diffusion length ¥ Conserved variable per unit volume n Collision integral Subscripts 0 1 ,2 aa Reference value, or at center frequency At energy le v e P I” or“2" Atom-atom abs Absorbed amb Ambipolar D Doppler, or based on diameter e Electron ea Electron-atom viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ei g, a Electron-ion Gas i Ion i, j At energy level ‘ I* or “j’ m Medium value, or mean (bulk) value p Plasma, or perturber R Reaction “R’ s Species ‘ s' sub t th Substrate Total Thermal x, y, z, r In a given direction X At a given wavelength v At a given frequency <P Conserved variable ¥ Conserved variable per unit volume Superscripts ea ei • Electron-atom Electron-ion Excited state, or the metastable state (4s) for argon, or the mean value Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Chapter 3 Table 3.1 Physical classifications, mathematical approximation, and lineshapes of various line-broadening mechanisms Chapter 4 Table 4.1 The Conservation Equations Table 4.2 Species considered in 2-D argon plasma chemistry Table 4.3 2-D Argon Plasma Chemistry Model Chapter 5 Table 5.1 The Ranges of Studied Parameters Used for Argon Plasmas at Pressure of 5 Torr Table 5.2 The Experimental Data for Microwave Argon Plasma Energy Balance Table 5.3 Calculated Enthalpy Data for the Argon Plasma in Each Cooling Line with an Argon Flow Rate of 250 seem and an assumed gas exit temperature of 350 K Table 5.4 Heat Transfer Analysis Data for the Control Volume Encompassing the Argon Plasma (250 seem argon flow, 5 Torr) Table 5.5 The Total Heat Predicted from 2-D Model Comparing with the Global Energy Balance Study and Control-Volume Analysis Results Chapter 6 Table 6.1 The Excited State Number Densities Calculated from the OES Data Comparing with the Numerical Predictions At Ne = 5e17 m-3, Tg = 350K Table 6.2 The calculated argon linewidths (FWHM) with different gas temperatures at Ne=1e18 rn3, Te=10.000K Table 6.3 The experimental results of the excited state number densities comparing with the 2-D CRM predictions for an argon plasma at 5 Torr, 250 seem (Best matched case at microwave power of 30W) Table 6.4 Error between peak values comparing the reaction rates calculated from 2-D CRM and the curve-fitted equations APPENDIX A Table A.1 Spectroscopic Notation of Selected Argon Energy Levels Table A.2 Transition Lines Spectroscopic Data (from NIST Atomic Spectra Data) x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Chapter 1 Figure 1.1 Hot Filament CVD Diamond Reactor Figure 1.2 Combustion Flame Assisted CVD Process Figure 1.3 RF Plasma Enhanced CVD Process Figure 1.4 Fundamental Plasma Processes Chapter 2 Figure 2.1 The WAVEMAT microwave plasma reactor Figure 2.2 Schematic of the microwave reactor system Figure 2.3 Schematic of the optical emission spectroscopic measurement system Figure 2.4 Schematic of the absorption measurement experimental setup Figure 2.5 Some pictures of the absorption measurement setup Chapter 3 Figure 3.1 Calibration of the tungsten lamp passing through the belljar Figure 3.2 Calibration of the tungsten lamp without passing through the belljar Chapter 4 Figure 4.1 The iterative scheme to obtain a self-consistent solution Figure 4.2 Schematic of a control volume AxAy for <1>conservation Figure 4.3 Non-Maxwellian EEDF plots with averaged Te = 1.07e4 K and 300K <Tg<600K comparing with the Maxwellian EEDF Figure 4.4 Comparison of the curve-fitted reaction rates (Kea(0,1)) and those calculated from the CRM as a function of electron temperature Figure 4.5 Comparison of the curve-fitted reaction rates (Klea(O)) and those calculated from the CRM as a function of electron temperature Figure 4.6 Comparison of the curve-fitted reaction rates (Kiea(1)) and those calculated from the CRM as a function of electron temperature Figure 4.7 The iterative scheme of the coupled 2-D plasma model and the EM model Figure 4.8 The simulation domain of the microwave plasma discharge with the WAVEMAT reactor XI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.9 The Flow Chart of the Iterative Scheme for the 2-D Collisional-Radiative Model (CRM) Chapter 5 Figure 5.1 The comparison of non-Maxwellian and Maxwellian EEDFs At Ne = 7e17 nrf3, Tg =350 K, and P= 5 Torr Figure 5.2 The electron energy loss terms with non-Maxwellian EEDF changing with the averaged electron temperature at Ne = 7e17 m'3, Tg = 350K and P = 5 Torr Figure 5.3 Eiectron production and loss rates changing with the averaged temperature at ne of 1e18 m"3 and T0 of 350 K Figure 5.4 The self-consistent solutions (Ne-Te pairs) with non-Maxwellian EEDF at Tg = 350K, P = 5 Torr Figure 5.5 Microwave power deposition with non-Maxwellian EEDF changing with the electron number density while electron number density and energy conservation are imposed Figure 5.6 The experimentally measured and theoretically calculated total continuum emission changing with the electron number density Figure 5.7 The schematic used for the control-volume heat transfer analysis Figure 5.8 Electron temperature distributions at the centerline changing with the microwave power (grid 20 is at substrate) Figure 5.9 Gas temperature distributions at the centerline changing with the microwave power Figure 5.10 The spatial distributions for microwave argon plasma at 5 Torr and 680W Figure 5.11 A vertical stacked column graph that shows the total heat along the belljar walls and the substrate surface changing with the microwave power Figure 5.12 Heat fluxes along the belljar walls changing with the microwave power (5 Torr and 250 seem) Figure 5.13 Heat fluxes along the substrate surface changing with the microwave power (5 Torr and 250 seem) electron Chapter 6 Figure 6.1 The Boltzmann plots showing that the experimental data can only be matched by the model predictions with the changed rates Figure 6.2 The Voigt parameter a changes with gas temperature at an electron number density of 1e18 m'3, and an electron temperature of 10000K Figure 6.3 The Boltzmann plot of the numerical predictions with the experimentally measured 4s state number density and the OES data Figure 6.4 Spatial distributions of the argon plasma parameters at 5 Torr, 250 seem and xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10W Figure 6.5 Spatial distributions of the argon plasma parameters at 5 Torr, 250 seem and 30W Figure 6.6 Mol fraction of 4s state at the centerline changing with the power Figure 6.7 Mol fraction of electron number density at the centerline changing with the power Figure 6.8 The comparison of gas temperature distributions at the centerline with different plasma pressures Figure 6.9 The comparison of electron temperature distributions at the centerline with different plasma pressures Figure 6.10 The comparison of the electron number density distributions at the centerline with different plasma pressures Figure 6.11 The comparison of 4s state number density distributions at the centerline with different plasma pressures Figure 6.12 The spatial distribution of the 4s state number density at 5 Torr and 10W Figure 6.13 The spatial distribution of the 4p state number density at 5 Torr and 10W Figure 6.14 The spatial distribution of the 5p state number density at 5 Torr and 10W Figure 6.15 The spatial distribution of the 5d state number density at 5 Torr and 10W Figure 6.16 The averaged 4s-state number density distribution along the z-direction from the numerical predictions comparing with the results from the absorption measurement Figure 6.17 The averaged 4p-state number density distribution along z-direction from the numerical predictions comparing with the experimental results Figure 6.18 The averaged 5p-state number density distribution along z-direction from the numerical predictions comparing with the experimental results Figure 6.19 The averaged 5d-state number density distribution along z-direction from the numerical predictions comparing with the experimental results Figure 6.20 The wire frame plot of the comparison between MW PD profiles used in 2-D fluid model and predicted from 2-D CRM at 5 Torr and 10W Figure 6.21 The wire frame plot of the comparison between the MW PD used in 2-D fluid model and predicted from 2-D CRM Figure 6.22 The comparison of the spatial distributions of MWPD used in the fluid model and predicted from the 2-D CRM at 5 Torr and 10W Figure 6.23 The comparison of spatial distributions of MWPD used in fluid model and predicted from the 2-D CRM at 5 Torr and 30W xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.24 The comparison of Kea[0,1] rates calculated from 2-D CRM and from curve-fitted equation Figure 6.25 The comparison of Kea[1,0] rates calculated from 2-D CRM and from curve-fitted equation Figure 6.26 The comparison of Klea[0] rates calculated from 2-D CRM and from curve-fitted equation Figure 6.27 The comparison of Klea[1] rates calculated from 2-D CRM and from curve-fitted equation Figure 6.28 4s State number density comparison between the results from the 2-D CRM and the 2-D fluid model at 5 Torr, 10W Appendix A Figure A.1 The Optical Emission Scan of Microwave Argon Plasma at 680W, 5 Torr, and 250 seem flow rate (Data taken on 04/16/98) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER ONE INTRODUCTION 1.0 M o tivatio n Chemical Vapor Deposition (C V D ) plays an important role in the semiconductor industry. It offers an easy and cost-effective w ay to deposit thin films on certain substrates by varying input gases. Diamond is an attractive material that offers many potential uses despite its high cost. The High Density Electronics C enter (H iD EC ) of University of Arkansas, which was established in 1991, expanded its Thermal Management Program to include synthetic diamond substrates for Multi-Chip Modules (M CM s) to take advantage of diamond's high thermal conductivity. In 1992, HiDEC purchased W A V E M A T s (M odel M P D R 3135) Microwave Diamond Deposition System for its potential of high efficiencies, scaling and minimal contamination. Although high quality thin films are achievable, the understanding of the gas phase reactions and surface chemistry is fa r from complete. To characterize the microwave plasma, optical emission data w ere used in earlier studies [Gordon, 1996, Kelkar, 1996 and 1997] to verify the predicted excited state num ber densities from an in-house Collisional Radiative Model (CRM ). However, it was very difficult to match the measured excited state number densities with the predicted ones. Further study [Kelkar, 1 99 7 and Li, 1997] showed a discrepancy between the m etered input power and the numerically predicted power. It was thus concluded that the metered pow er could 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. not be used as input in the model. Therefore, a 2-D model, which couples the electromagnetic fields and plasma discharge model, is employed to achieve selfconsistency. For the microwave argon plasma, this is the first self-consistent 2 -D model. 1.1 Introduction Plasma is the fourth state of matter in the universe. From a phenomenological point of view, an ionized gas (or plasma) is distinguished from a room-temperature gas primarily by its ability to conduct electricity. At temperatures of about 100 ,00 0,000 K, gases are fully ionized. T h e description of this kind of gas is much sim pler as compared to the general situation. For most other engineering applications, the gas temperatures are lower than 20,000 K, and thus such gases are partially ionized [Mitchner and Kruger 1973]. Comprised of electrons, ions, and neutral species, partially ionized gases open a door of vast wealth of new physical phenomena and add a lot of complexity to their characterization. Natural diamond was first characterized to be of organic origin by Sir Isaac Newton. In 1955, General Electric first documented synthetic diamond by using a High-Pressure, High-Temperature (H P H T) process. However, a later study of the notes of W . G. Eversole of Union Carbide Corporation would determine that diamond had actually been deposited using a Chemical Vapor Deposition (C V D ) process in 1952 [Lettington, 1994]. This evidence shows that the first 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. reproducible synthetic diamond process was CVD instead of HPHT. Diamond, with its attractive optical characteristics and unbeatable mechanical hardness, is a perfect material for use in windows, lenses, and mirrors. Except the electron mobility, diamond’s electrical properties also exceed virtually those of all other semiconductor materials currently in use [Yodor 1990]. Therefore, the potential use of diamond as a semiconductor material has attracted a lot of interest. Among the various diamond CVD techniques, microwave plasma C VD has gained considerable importance because of its capability to produce high quality diamond films with reasonable growth rates and areas. However, the optimization and development of microwave plasma reactors are very difficult since it is hard to predict the performance of the reactors. The difficulty comes mainly from the fact that the microwave field and the plasma are highly interactive. A development based on trial and error is very time-consuming and costly; thus the numerical simulation of microwave plasma reactors has gained considerable interests. Moreover, the physics and chemistry involved in the microwave plasma are far from fully understood. Numerical modeling can thus add a lot of basic understanding. Before we discuss the microwave plasma C V D processes in detail, a review of diamond CVD processes is presented for better understanding of the CVD mechanisms. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 Diamond CVD Processes The diamond C V D processes, according to their different activation techniques, are well summarized in Celii and Butler, 1991 and Palmer, 1992. And they are discussed below. 1.2.1 Filament-Assisted CVD The first diamond nucleation on a non-diamond substrate was achieved by a Hot Filament-assisted CVD process (H FC V D ) [Matsumoto, 1982]. HFCVD is considered as the simplest of all methods except the combustion flame CVD. In this method, diamond particles or thin films are deposited on a heated substrate from a mixture of m ethane and hydrogen activated by a hot tungsten filament. An experimental laboratory setup can be constructed with a capital investment of $15,000. Thus it makes HFCVD among the least expensive techniques [Taher 1999]. A typical H FC VD reactor is shown as in Figure 1.1. Filament Methane Substrate Hydrogen Pump Figure 1.1 Hot Filament CVD Diamond Reactor Diamond deposition using the hot filament method is achievable under a 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. wide range of deposition conditions. The deposition param eters should be taken into consideration for good quality film, such as the filament tem perature, the filament-substrate distance, the substrate temperature, the reactor pressure and the m ethane concentration. 1.2.2 Combustion Flames The growth of diamond at atmospheric pressures using combustion flames offers the simplicity of the process method and low cost of the experimental apparatus, as well as the high growth rates that is already observed [Matsui 1990]. A schematic diagram of the CVD process using a premixed oxyacetylene flame [Capelli, 1990] is shown in Figure 1.2. The diamond thin film is deposited at the second flame boundary, and no diamond deposition occurred at the central area. This suggests that OH and 0 radicals, which are present in the second flame layer, are critical to the diamond growth. Silicon Substrate Diamond Thin Film dnmary Flame Diffusion Nozzle Figure 1.2 Combustion Flame Assisted CVD Process 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Using optical emission, Hirose et al. [1990] described the conditions for production of optically transparent diamond crystals. Spatial and parametric variation of C 2 and O H concentration were obtained and indicate equilibrium concentrations. By correlation with the deposition rate, CH, C 2 , or C H 3 were considered candidates for growth species. 1.2.3 Low Pressure Plasmas Unlike the filament CVD process, plasmas offer a ready source for self diagnostic information in the form of visible optical emission. Active species in the plasma can be identified by Optical Emission Spectroscopy (O E S), which could potentially be used for process control. 1.2.3.1 DC Plasma Enhanced CVD In the DC plasma enhanced CVD processes, a plasma of m ethane- hydrogen mixture is excited by applying a DC bias across two parallel plates, one of which is the substrate. The substrate can be attached to either the anode or cathode. Another configuration of this method uses a hollow cathode, which is m ade of refractory metal. This method allows a stable discharge to be maintained at lower voltages than the planar cathode. However, there are problems encountered with diamond films by DC plasmas, including high stress and high concentration of hydrogen. Moreover, the diamond films may be contaminated by the erosion of the electrodes [Spear, 1993]. The high-pressure DC arcjet plasma is also a DC plasma process but its 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. operating conditions are closer to combustion flame techniques. This process offers an extremely high deposition rate, but this deposition can only occur over a very small area and may be contaminated by graphite and plasma-generated materials. RF Generator Match Electrost. Screen / 6 000 OOOO Probe Metal Figure 1.3 Experimental Setup of the Inductively Coupled RF Plasma 1.2.3.2 RF Plasma Enhanced CVD Plasma etching and deposition of thin films using low-pressure radio frequency (R F) glow discharges is currently in widespread use in the microelectronics industry. A lot of experimental and numerical research work has been devoted to this method. A purely inductively coupled rf plasma experimental setup [Kortshagen, 1995] is shown in Figure 1.3. In this figure, no substrate is shown, and this structure was used for research work only. A pulsed 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. probe technique was used to measure the electron energy distribution function (E E D F) for the RF plasma discharges. For RF plasma enhanced CVD processes, the substrate is generally raised to above 800° C by inductive and plasma heating. High power in the discharge was reported to be necessary for efficient diamond growth. This is due to the fact that the average electron energy in a RF discharge at 13.56 M H z at 1 Torr is about 4 eV, which is insufficient to dissociate hydrogen. However, the high power will result in physical and chemical sputtering from the walls of the silica tubes, and the diamond films thus were frequently contaminated with silicon carbide [Spear, 1993], The driving force to use this method is the availability of the fully developed equipment capable of depositing large areas. However, uniformity is still a big concern. It is hard to synthesize high quality coatings at a reasonable rate using this technique. 1.2.3.3 M icrow ave P lasm a Enhanced CVD Microwave plasma enhanced CVD processes have been used more extensively than any other methods for diamond film growth. Microwave deposition has two distinctive advantages over other techniques: • Contamination can be avoided since it is an electrodeless process, • The microwave discharge has a higher plasma density with higher energy electrons than RF discharges due to higher frequency electric field. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, as shown in the later sections, it is very hard to characterize microwave plasmas both experimentally and numerically. Due to the strong microwave field inside the cavity, the results of direct-probing measurements are not reliable. Therefore, the non-intrusive measurements, such as O ES and Laser Induced Fluorescence (LIF) are desirable. On the modeling side, the composition and temperatures of the microwave plasmas strongly depend on the electromagnetic field, which is in turn affected by the discharge itself. Therefore, a model of such a microwave-plasma-enhanced CVD system must self- consistently solve for both the electromagnetic fields and the plasma discharge [Hassouni, 1998]. 1.3 Fundamental Plasma Processes and Forms of Nonequilibrium In a partially ionized gas, there are generally six "basic" kinds of particles [Mitchner and Kruger, 1973]: photon, electron, ground-level atom/molecule, excited atom/molecule, positive ion and negative ion. Figure 1.4 shows the various energy transfer processes in partially ionized plasma. The input power accelerates the charged particles under the influence of the applied electric field. Most of the energy is absorbed by electrons because the energy imparted to the charged particles between collisions is inversely proportional to the particle mass. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Heat Conduction and Convectior Radiation Losses Electric Energy Source ^ t , Joule Heating I Ion Gas Electron Gas Reactor Wall Recombina tion ofIons and Electrons Free Radical Recombina tion Elastic and Inelastic Collisions Neutral Gas Radiatio n Losses Heat Conduction and Convection Figure 1.4 Fundamental Plasma Processes The energy is transferred from electrons to neutral particles and ions by collisions, which can be classified as either elastic or inelastic collisions. The collision is elastic if the internal energy of the neutral particle/ion remains the sam e after the collision. Otherwise, it is called inelastic, in which the electrons may induce dissociation, excitation and/or ionization. The ionization reactions will produce electrons as well as positive or negative ions. Due to recombination in the gas phase and the convection and diffusion loss of electrons, a balance is established according to the input energy to maintain the charge neutrality of the plasma. The electron induced dissociation reactions will produce active atoms and/or radicals. These particles will also reach a balance by recombination and diffusion losses. High-energy electrons will also populate the number densities of 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the excited states of either atoms or molecules. At steady states, the number densities of these excited states are balanced by the radiation losses, in which the energy is released as photons with certain frequency. To understand the plasma processes, it is essential to have the knowledge of the thermodynamic states of the plasmas. The principle of detailed balancing states that the different reaction rates for each microscopic process and for the corresponding inverse process are thermodynamic equilibrium [Mitchner and equal under conditions of Kruger, 1973]. W hen all the microscopic processes in the plasma are in detailed balance, including the radiative fields, the plasma is said to be in complete thermodynamic equilibrium (CTE). For plasma in CTE, all the particles in the plasma can be described by Maxwellian velocity distribution functions at the sam e temperature. The plasma is homogeneous and optically thick, and the radiation field follows a Planck distribution at the same temperature. The internal energy states (rotational, vibrational, and electronic) of each particle will follow Boltzmann statistics at the same tem perature. Moreover, the neutral and charged particles will satisfy Saha equilibrium, and the plasma will be in chemical equilibrium at the same temperature. Few plasmas satisfy all the conditions for C TE . Actually, many plasmas can be described as being in local thermodynamic equilibrium (LTE), where the collisions are sufficiently numerous to maintain the Maxwellian, Boltzmann, Saha, and chemical equilibrium although a black body radiation field may not be 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. observed. In terms of nonequilibrium, which implies any deviation from the state of CTE, it can be classified as kinetic nonequilibrium, ionization nonequilibrium, chemical nonequilibrium, and Maxwellian nonequilibrium. The collisions among plasma constituents will redistribute the energy and momentum. Thus the plasma is reaching its equilibrium by collisions. However, the energy transfer efficiency of the electron-neutral collisions m ay be too low to reach a steady state temperature due to the much lighter mass of electrons. On the other hand, the energy coupling efficiency of the ion-neutral collisions is very high because of their comparable masses. Therefore, the ionic and neutral species can generally be described with the same tem perature, while the electron gas has to be described by a much higher electron temperature. This kind of plasma is referred to as two-temperature plasma. However, even the assumption of a two-temperature plasma may not be acceptable in some cases. The electron energy distribution function (EEDF), which is non-Maxwellian, may significantly affect the plasma transport properties, excited state distributions, and degree of ionization, etc. In the presence of cold surfaces, such as the water-cooled reactor walls and substrate, the kinetic rates of some reactions may not be able to achieve chemical equilibrium across the plasma. There may exist large gradients of the gas and electron temperatures as well as the concentrations of certain species. Therefore, a complicated two-dimensional or three-dimensional model is needed to treat all these non-equilibrium effects that may significantly affect the plasma 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. discharges. 1.4 Plasma Diagnostics The diagnostic techniques are critical for the understanding of diamond CVD processes, especially the low-pressure plasma enhanced CVD processes. Quantitative determination of the composition and tem peratures of various species present in the gas phase will provide the necessary data for plasma modeling. In-situ diagnostics may also be used for the plasma process control. The diagnostic techniques used for diamond CVD processes can be divided into three categories: sample extraction, physical probe, and optical methods. Sam ple extraction methods, such as mass spectroscopy, perform an ex-situ analysis of the extracted portion of the reactive gas. Physical probe methods, such as those using a Langmuir probe, introduce a probe into the reactor cham ber to measure the electric potential, electron density and temperature, etc. However, these two methods may not be available for some plasma processes. For microwave plasmas, the probe method is not appropriate since the probe will disturb the microwave electromagnetic field significantly. Since the microwave plasma is highly nonequilibrium, the ex-situ measurements cannot provide accurate information. Therefore, optical methods are preferred because of their non-intrusive nature. The optical methods used in this research will be discussed in detail in Chapter 2 and 3. 13 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 1.5 Plasma Modeling The power of computers (in both speed and memory) has increased by approximately 107 over the last 40 years. As a result, computer modeling has become a powerful tool for exploring the physics and chemistry of plasmas. Although most substance in the universe is in the form of plasmas, they present us with a wealth of complexity not found in ordinary fluids. W hat m akes it even harder to characterize the plasmas is that plasmas are difficult to experimentally probe. Thus, to m eet the great challenges associated with plasmas, computer modeling offers an affordable and effective way to achieve more understanding. 1.5.1 Non-local Approximation In recent years, the modeling of nonequilibrium plasmas has gained in importance for the development and understanding of plasma sources. One of the major challenges is the development of a simple but realistic description of the spatial dependence of the electron energy distribution function (E E D F ) in spatially inhomogeneous nonequilibrium plasmas. A number of methods have been developed to treat this problem, such as the Monte Carlo method, the particle in cell method (PIC), the converted scheme method, or the direct numerical integration of the electron Boltzmann equation. However, these methods are often very slow if multidimensional systems are considered. Thus to develop spatially two- or three-dimensional discharge models, some approximation methods are considered. For low-pressure, w eakly collisional 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. plasmas, the “ nonlocal approximation" is particularly suited [ Kortshagen, 1996]. The nonlocal approximation addresses weakly collisional plasmas, w here the energy relaxation length (Xc) exceeds the discharge dimensions in the range of the kinetic energies of interest. A ( u ) = — -— Na„(u) ,, (1.1) X («) = -----------------------N ( a a (u ) + cTXu )) where Xm is the m ean free path for elastic momentum transfer of the electrons, and X is the m ean free path for all kinds of inelastic collision processes; u is the electron kinetic energy; a is the cross section area. The Boltzmann solver we used in this research, ELEND IF, relies on the nonlocal approximation [Morgan, 1994]. Using the assumption that the electron velocity distribution function (E V D F) is nearly isotropic, it can be replaced by the first two terms of a spherical harmonic expansion as will be shown in Equation 4.1. 1.5.2 Argon Models Argon is often used in plasma studies for two reasons. O ne is that argon is a frequently used rare gas in laboratory studies of plasmas and industrial applications. Argon can either be a earner gas or the gas introduced before other gases for safety and better tuning. Another reason is that extensive studies of 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. argon collisional radiative models have been conducted, especially the argon plasmas with the Maxwellian EEDF [Vlcek, 1985]. For simplicity, zero-dimensional models are generally developed first, which may be modified with diffusion to account for the effects of the geometry. Vlcek has developed a numerical method for the Boltzmann equation to obtain the EE D F in a nonequilibrium argon plasma characterized by a set of parameters (Te, T a, T it ne, etc.) [Vlcek, 1985], Vlcek then developed an argon C R M with 65 discrete effective levels. This model provided information about mechanisms populating the excited levels under various conditions in non-isothermal plasmas and about the effects caused by the departure of the actual E E D F from the corresponding Maxwellian distribution. This model proved that in some region the assumption of the Maxwellian EEDF was not justified [Vlcek, 1986 and 1989]. Braun and Kune developed a three-level atomic model that was used to determine the steady-state collisional-radiative coefficients in nonequilibrium partially ionized argon plasmas. Rate equations for populations of the atomic levels then were coupled to an electron Boltzmann equation that included inelastic processes. The solution of these coupled equations yielded an analytical form of non-Maxwellian EEDF [Braun, 1987]. Ferreira [1985] investigated the contribution of the ionization from the two metastable and the two resonance levels of argon to the total ionization rate in a low-pressure argon positive column. The results showed that the values of the predicted m aintenance field w ere considerably lower than those with only ground state. A kinetic model that 16 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. included 25 levels was developed by Repetti et al. [1991] to the study of relaxation of an atmospheric thermal argon plasma from equilibrium condition near 1 eV. It was shown that resonance radiation trapping was an important process in such plasmas. This paper also suggested that future work should include spatially dependent processes such as particle diffusion and plasma expansion, as well as photoionization modeled as trapping of recombination radiation. Benoy et al. [1993] discussed the radiative energy loss in a non equilibrium argon plasma which allowed for deviation from local Saha equilibrium. Kelkar et al. [1997] developed a self-consistent 25-level CRM that could solve for the excited state number densities as well as the power deposited into the argon plasma. This model, as outlined in Chapter 4, can be considered as a pseudo-1 -dimensional model by considering the electron diffusion loss term. Lymberopoulos et al. [1993 and 1998] developed one-dimensional fluid models for a 13.56 MHz argon glow discharge and for a pulsed-power inductively coupled argon plasma. For a pressure of 1 Torr, metastables w ere found to play a major role in the discharge despite the fact that their mole fraction was less than 10"5. The studies suggested that neutral transport and reaction must be considered in a self-consistent manner in glow discharge simulations. 1.5.3 Hydrogen/Hydrocarbon Models for Diamond Deposition Although the main purpose of this dissertation is to study microwave argon 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. plasmas, microwave hydrogen plasmas were also studied in comparison with the argon plasmas. Hydrogen and hydrocarbon models are much more complicated than argon models due to the fact the plasma chemistry will affect the discharge significantly. Since the hydrocarbon gas only consists 0 .5 -2 % of the gas mixture in the diamond CVD processes, some studies have devoted to the pure hydrogen modeling. However, to understand the mechanisms of the diamond CVD processes, hydrocarbon chemistry has to be included in the model. Kune and Gundersen [1983a and 1983b] developed a pure hydrogen model intended to have a detailed understanding of the physics of hydrogen thyratrons. Their model included collisional and radiative processes involving excited states of atomic hydrogen. A Maxwellian E E D F was assumed for their rate calculations. They concluded that the step-wise ionization from atomic hydrogen excited states played a dominant role and the inverse three-body recombination in the gas phase was the major electron loss mechanism. In Kune’s later work [ Kune 1987], the role of atom-atom inelastic collisions in twotemperature nonequilibrium plasmas was investigated. Their results can be applied to plasmas that have atomic hydrogen as one component with large enough amounts so that e-H and H-H inelastic collisions and interaction of these atoms with radiation dominate the production of electrons and excited hydrogen atoms. Burshtein et al. [1987] investigated the effects of quenching of excited hydrogen atoms by H2 molecules. The delayed coincidence method was used to 18 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. measure the quenching of the intensities of the Ha, Hp, and HY Balmer lines in the afterglow. St-Onge and Moisan [1994] investigated the hydrogen atom yield in RF and microwave hydrogen discharges. A particle balance model was developed to solve for the H, H2, H+, H2\ and H3+ species. It was observed that the H-atom concentration decreased when the wall temperature increased, due to the increased efficiency of atomic recombination on the wall. Several researchers have investigated the effects of various electronneutral interactions with the EE D F through the solution of the Boltzmann solver. Loureiro and Ferreira [1989] reported that superelastic vibrational collisions strongly enhanced the tail of the E E D F that significantly increased the electron excitation rates. Colonna et al. [1993] found that the electron-electron collisions must be included in the E E D F solver for post discharge conditions. Capittelli et al. [1994] showed the dependence of EE D F on superelastic electronic collisions by specifying parametrically the concentrations of the electronically excited states both for H2 and atomic hydrogen. They concluded that the EEDF and related properties depended on electronic superelastic collisions at low electric field (<30 Td) when the average electron energy was less than 1.5 eV. Kelkar et al. [1997] developed a comprehensive hydrogen model, which coupled the non-Maxwellian EEDF with the zero-dimensional CRM modified with the ambipolar diffusion. The predicted excited states number densities were compared with the optical emission data. T h e microwave power absorbed by the plasma was also calculated and compared with the experimental data. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For the modeling of the diamond CVD processes, different approaches have been adopted for different deposition systems. Surendra et al. [1992] developed the self-consistent particle-fluid hybrid simulations to study the structure of hydrogen dc discharges between parallel plates. Dissociation of H 2 in the anode region contributed significantly to the flux of atomic hydrogen to the anode, where diamond is typically grown. In this model, a M onte Carlo simulation was used to describe individual energetic electrons in the cathode sheath, while the electrons and ions in the low-field region of the discharge were modeled as fluid. Koemtzopoulos et al. [1992] developed a model to predict the degree of gas dissociation in an H 2 microwave discharge in a tubular reactor. Ambipolar diffusion rates were calculated for the electron number density balance. The E E D F solved from the Boltzmann equation was not Maxwellian. It was found that lowering the total gas density and/or increasing the pow er would enhance gas dissociation as well as atom density. Flow rate had a minor effect on atom density, and the addition of 1% CH4 to the hydrogen discharge did not appreciably affect the EEDF. Bou et al. [1992] adopted a model that took into account 106 reactions for kinetic calculations in a microwave plasma of 1% CH4 gas mixture with H * This model did not consider the energy conservation, and only electro-neutrality was imposed in the plasma volume. The increase in the electron temperature, electron density, or in the number of effective electrons raised the amount of H 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. atoms and consequently induced an increase in C atom concentration. T ah ara et al. [1995] used a kinetic model to analyze the particle-species compositions in the C H 4/H 2 microwave plasmas, in which 14 neutral species and 16 ionic species w ere considered. Several assumptions w ere m ade in particular, such as the Maxwellian EEDF, chemical equilibrium, and uniform tem perature and pressure across the cylindrical plasma volume, etc. The diamond-like-carbon synthesis, plasma diagnostic measurements and kinetic model analyses were conducted to correlate between plasma properties and the film features. M icrowave enhanced plasmas are of current interests and have already demonstrated good potential in moderate pressure (1 -1 5 0 Torr) plasma processing applications including diamond film deposition. Tan and Grotjohn [1994 and 1995] have developed a self-consistent model of the electromagnetic field and plasma discharge in a microwave plasma diamond deposition reactor. Their modeling is well described in Section 5.2. The understanding obtained from the electrom agnetic solutions allowed the reactor design to be analyzed for improvement and optimization of such quantities as the uniformity of microwave power absorption. Hassouni [1994, 1996, 1997 and 1998], Scott [1996] and Gicquel [1994, 1996 and 1998] developed a series of microwave hydrogen models from zero-D to two-D. A lot of experimental data w ere obtained to facilitate the plasma modeling, such as the ground state and excited state Hatom densities. These data were also used to validate the microwave plasma model. For the detailed two-dimensional plasma modeling, one can refer to 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Section 4.3. A self-consistent model of the microwave fields and the plasma discharge in a moderate pressure (2 5 0 0 -1 1 0 0 0 Pa) resonant cavity type reactor was recently developed by Hassouni et al. [1999]. The self-consistent model presented by Hassouni et al. differs from those reported in other papers. Generally, the models we addressed before require experimental data, such as the power needed to sustain the plasma and the tem peratures for different modes (electron, gas or ion temperatures). Moreover, this self-consistent model addresses the coupled phenomena of chemistry, energy transfer, species and energy transport. 1.6 Objectives Based upon the above literature review, one can conclude that more work is needed to better understand plasma discharges as well as the diamond deposition processes. In this thesis, the major purpose is to add more understanding to the nonequilibrium microwave argon and hydrogen plasmas. Several problems that have occurred in our zero-dimensional argon model cannot be solved by this model alone, such as the big discrepancy between the predicted power absorption and the experimental measurement. Developing a two-dimensional self-consistent argon model is promising. M ore experimental data can also add more understanding to the study of the microwave plasmas. The objectives of this dissertation are to: 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Develop a self-consistent pseudo-1-D plasm a model, which couples the Boltzman solver (E LE N D IF) with our CRM; • Perform the energy balance study on the microwave reactor to gain a better understanding of the energy transfer mechanisms in the non equilibrium plasmas; • Demonstrate the potential of OES as a quantitative diagnostic method for microwave plasmas when used with the numerical model; • Perform the absorption measurement for the metastable excited state of argon plasma for additional experimental data to verify the CRM; • Develop a self-consistent two-dimensional model of the electromagnetic field and the plasma, which will provide more detailed information about the plasmas and the microwave fields. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH A PTER TWO EXPERIM ENTAL FA CILITY 2.1 Microwave Plasma Reactor Th e experimental research is conducted on the W A V E M A T microwave plasma reactor (Model M P D R -3135). A picture of the W A V EM A T reactor is shown in Figure 2.1. This 7-inch diameter cavity microwave plasma reactor is designed for large-area uniform diamond film deposition. The microwave probe assembly and sliding short form the top portion of the cavity. The lower section of the reactor consists of the bottom surface, the base plate and a metal plate. Figure 2.1 The W A V E M A T microwave plasma reactor T h e microwave is produced by Sairem microwave power supply with a frequency of 2.45 G H z and maximum power of 6 kW. The microwave energy is coupled into the cavity through the waveguide and probe. The microwave power is measured by two power meters, which indicate the incident power and 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. reflected power respectively. The microwave discharge can be created when excited in a single cavity electromagnetic mode. A ball shaped plasma is formed inside the quartz bell jar, and the substrate is placed near the plasma. T h e typical operation conditions for diamond deposition are 30-80 Torr pressure, 300 seem hydrogen, 3-5 seem m ethane flow rates and 1.6 kW input power. However, for pure argon plasmas, the chamber pressure is reduced to about 5 Torr to stabilize the plasma. The pressure in the chamber is controlled by a throttle valve and a mechanical pump (Franklin Electric). All the gas flows are controlled by MKS mass flow controllers. 145GHz j Microwave i Probe Cooling Short Cooling Short Optical Pyrometer Probe Bell Jar Cooling Argon plasma [_ Applicator Cooling Substrate Chamber Cooling Substrate Cooling Figure 2.2 Schematic of the microwave reactor system 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.2 shows a schematic of the microwave reactor configured for the energy balance study. The base plate, chamber, probe, short and substrate are cooled by water, and the belljar is cooled by air. An optical pyrom eter is installed on the reactor to monitor the substrate surface temperature, which is a critical param eter for diamond deposition. The belljar temperatures are m easured by the infrared thermocouples by E X ER G E N . For the purpose of energy balance study, several K-type thermocouples are installed on the cooling-water and cooling-air lines to monitor the inlet and outlet temperatures. 2.2 Emission/Absorption Systems The primary diagnostic system is composed of a 0.5 m S P E X 500M monochromator, Hamamatsu R -928 PMT, the monochromator driver (M SD), and fiber optic assembly (Figure 2.3). 2.2.1 Emission system The emission equipment involves a dedicated com puter for data acquisition, a tungsten lamp for calibration, an optical pyrometer, and a set of collecting optics. A 1-m W H e-N e laser is used for assuring alignment and for checking spectral response of the system. The accuracy of this system agrees with the reported value of ± 5 A, and the results are repeatable within ±1 A. 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Aperture Stop Optical Fiber Microwave Plasma Monochromator D ata Acquisition Computer Figure 2.3 Schematic of the optical emission spectroscopic measurement system Figure 2.3 shows the optical setup used for collecting and focusing the emitted light onto the entrance slit of the monochromator. A 40-cm focal length (2” diameter) lens is used to collect the light from the plasma emission. T h e light then is passed through a typical 0.5” aperture stop. A 30-cm lens focuses the image of the plasma onto the fiber optic cable. To determine the absolute intensity of emission, it is necessary to calibrate the emission system. The tungsten lamp is used as a known source of emission (3 0 0 0 -8 0 0 0 A). T h e temperature of the lamp stripe is measured by a Pyro-micro® vanishing filam ent optical pyrometer. 27 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 2.2.2 Absorption system Figure 2 .4 and Figure 2.5 show the absorption experimental setup. The emission from the CENC O argon spectrum lamp (Model # 87210) was collimated with a 10-cm focal length lens through a typical 1-cm aperture stop. Then another Argon Lamp r Microwave Argon Plasma chopper Oscilloscope MSD Lock-in Amplifier Monochromator Figure 2.4 Schematic of the absorption measurement experimental setup 10-cm focal length lens imaged the lamp onto the optical chopper. The beam was collimated once more by another 10-cm focal length lens. A 40-cm lens focused the beam through the holes on the reactor cavity window and through the plasma. Another 40-cm focal length lens collimated the beam again. The collecting optics was the same as the emission system. The analog signal from P M T was the input to the MSD controller. W e then connected the analog signal 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to the lock-in amplifier in order to m easure small signals. The lock-in amplifier is synchronized with the chopper frequency to maximize the probed signal. Thus the absorption m easurem ent system can read very w eak signals, especially useful for the signal after absorption through the plasma volume. For the wavelength used for absorption measurement, the monochromator is preset to its position. However, due to the fact this wavelength may be shifted, it is safe to scan the whole emission spectrum before presetting the monochromator to this specific wavelength. (a) (b) (c) Figure 2.5 Some pictures of the absorption measurement setup: (a) the argon lamp with the focusing lenses; (b) the Wavemat reactor with the collecting optics; (c) the monochromator and P M T 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER THREE EM ISSIO N /A BSO RPTIO N DIA G N O STIC TEC H N IQ U ES 3.1 Emission Spectroscopy Emission diagnostics is the oldest and most commonly used non-intrusive plasma diagnostic technique. The equipment for emission spectroscopy is relatively inexpensive and commercially available from a variety of vendors. Therefore, despite the limitation on the spatial and spectral resolution, emission diagnostics will continue to be a useful and reliable method for plasma diagnostics. 3.1.1 Absolute Line Intensity Absolute atomic line emission leads directly to the population (nO of an excited state through the following relation: A id k ", = - t JT L hcAtJ ( 3 -1 > w here ly is the emission intensity, Xq is the transition’s wavelength, h is the Planck’s constant, c is the speed of light, and A,- is the Einstein transition probability. This measurement is independent of lineshape if the entire line’s emission is collected. This measurement is also independent of LTE or PLTE if the emission lines investigated are optically thin over all plasma operating 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. conditions. T h e optically thin condition may be expressed as: f exp i hv \ « 1 (3.2) where T ex is the characteristic temperature associated with the excited states, and v is the frequency of the measured atomic line. Two measurements are made to obtain n,-. One measurement is m ade at the transition’s center wavelength and collects both line and background continuum emission. Another one is performed 10A away from the center wavelength and collects only the continuum emission. It is assumed that the continuum emission does not change appreciably over 10 A. Then the subtraction of the latter m easurem ent from the former one will yield the desired emission intensity. After the population of a given excited state n\ is determined, one can define a temperature T lte based on the following Maxwell-Boltzmann relation: n. n„ g, Qa — = — e. exp \ (3.3) kTLTEi J In the above equation, the total number density na can be replaced by the ground number density ni, p/kTg, since for all experimental condition reported here, the fractional ionization is less than 0.5% . Qa is the electronic partition function. It can also be replaced by its ground state degeneracy (gi) for the same reason. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.2 Relative Line Intensities By using two or more excited state number densities, the ground state number density in equation (3.3) can be eliminated and we have the following Boltzmann relation: n s (e -e nj Sj \ — = — exp —-----^ kkT 1b (3.4) y The slope inferred from a plot of the natural logarithm of n/gj versus upper state energy 8j is then inversely proportional to the temperature. This relation does not require thermodynamic equilibrium between the ground and excited states. This method is valid for PLTE and LTE. 3.1.3 Continuum Radiation Continuum emission results from interactions between free electrons and ionized or neutral particles in the plasma. Absolute emission measurement of continuum radiation can provide information about both absolute electron number densities and electron temperatures. For singly ionized plasmas, the total continuum emission represented as follows: 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. can be 8 — 8 J jj + 8 g- + 6 g (3.5) C x = 1.63 x 10-43 Wm*Kl/2sr~l C 2 = 1.026 x 1(T34 WrrrK-V2sr-1 where na, ni, ne and X denote the neutral particle, ion, electron number densities and the wavelength. The needed cross-section information describing the freebound recombination is contained in the so-called free-bound Biberman factor, £fb. Similarly, the free-free electron-ion information in the free-free Biberman factor term contains the cross-section The free-free electron-atom term also requires an appropriate cross-section, Q (Te), which is taken from Deveto [1973]. 3 .1 .4 C alib ratio n Calibration is a critical step for the optical emission spectroscopy measurements. A tungsten lamp was used here acting as the calibration light source, and the temperature of the tungsten filament was measured by a pyrometer. T h e absolute calibration factor is determined as: v lam p (3-6) 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w here Viamp is the signal measured, e is the emissivity of the tungsten filament with a value of approximately 0.45 [Larrabee, 1959], Ci and C 2 are constants with C 1 = hco2, and C 2 = hco/k, dX/dx is the reciprocal iinear dispersion, and Wex# is the width of the exit slit of the monochromator. The emission intensity can be calculated using the following expression: j _ ^ngnal (3.7) "~ ~ d J w here L is the chord length, and Vsignai is the measured signal strength. Since the OES signal passes through not only the collecting optics, but also the belljar, it is interesting to compare the calibration of the tungsten lamp signals with and without the belljar. As shown in Figure 3.1 and 3.2, the two calibration curves are different. Thus it will introduce large uncertainty to the excited state number density measurements if a wrong calibration curve is used. For the calibration curve of the light passing through the belljar, it is observed that the emission light with wavelength smaller than 4000 A was not able to pass through the belljar. Comparing Figure 3.1 with 3.2, not only the m easured signal strength is lower in Figure 3.1, but also the shape of the calibration curves is slightly changed. Argon plasma should not have any contamination on the belljar surface, thus the condition of the belljar remain the same during the calibration procedure and the O ES measurements. The O E S signal also passes through the optical windows on the reactor. The windows were tested to show that their effects on the OES signal were negligible. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Signal Strength ( a.u.) 3500 4000 4500 5000 5500 6000 6500 7000 7500 8000 wavelength (A) Figure 3.1 Calibration of the tungsten lamp signal passing through the belljar Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Signal Strength (a. u.) 1.0000E+07 1.00006+04 3500 4000 4500 5000 5500 6000 7000 7500 8000 wavelength (A) Figure 3.2 Calibration of the tungsten lamp signal without passing through the belljar 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 Absorption Spectroscopy T he transmission of light through a linearly absorbing medium can be described by the Beer-Lam bert relationship: (3.8) W here T(v) is defined as the spectral transmittance of the medium, l(v) is the intensity at frequency v observed after propagation through the absorbing medium, lo is the incident intensity of the probe beam, and k(v) is the spectral absorption coefficient. The two strongly absorbing transitions that can be used to measure 4s state number density are at 8115 A and 763 5 A. The latter transition was chosen because the system’s spectral response decreases quickly towards the infrared. An argon low-pressure discharge lamp is used as our source for this 7635 A signal. This lamp operates at low pressure, and the gas temperature is close to room temperature. These conditions are similar to the lamp used in the measurements m ade by Baer et al [1993]. His m easurements showed that the FW H M (full width at half of the maximum value) of the lamp’s 8115 A line is 0 .022 A. Both this transition line and the 763 5 A one have the same metastable lower state and are predominantly Doppler broadening. W ith respect to line shift, Griem’s [1964 and 1974] theoretical work shows that this shift is negligible and is also supported by Baer’s experimental work. Therefore, this lamp is used as the 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. source for a centerline absorption measurement. The spectral absorption coefficient near a spectral line may be expressed as [Com ey 1977]: C I g ^ \J; g2” (3.9) g A z = SzB 21 B2l = c 3A 21/ ( S t B i v 3 ) Substituting Equation 3.7 into Equation 3.6, w e get: T(v) s I ( v ) I I 0 = exp|(n2B2]- , hBX2)<Kv)— c (3.10) w here B 12 and B 21 are the stimulated absorption and emission probabilities, respectively, and 4>(v) is the associated line shape. Here, ni and n2 denote the lower and upper state number densities. The num ber density of the upper state, n2, can be obtained through emission. As shown in Equation 3.9, Bi2 and B2i are converted from the Einstein coefficient A2i [W iese et al., 1966], If w e take the natural logarithm of Beer's law, w e obtain the following equation: In —— T h ree measurements = (n ,5 21 - n xBn )(j){v) are required to hvx c (3.11) experimentally determine the transmission of a probe beam. The first one is taken with just the lamp in operation (L). This gives the reference signal lo. A second measurement is taken 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with just the plasma in operation (P). As shown in Figure 3.4, the optical chopper is located between the discharge lamp and the plasma, allowing filtering out the plasma’s emission. However, the chopping wheel alternates between equally spaced open and solid segments so that a fraction of the plasma emission can be reflected off of the solid segments and thus a signal will be generated also at the chopping frequency. This extraneous signal is phase shifted exactly 180’ relative to the lamp’s signal. T h e phase sensitive detector will display the difference of the lamp’s signal and the plasma signal (the lamp’s signal is zero in this particular measurement). W e have set the detector's phase such that the lamp produces a positive signal. The plasma alone thus will produce a negative signal which represents an offset. Finally, the last m easurem ent is taken with both the lamp and plasma in operation (LP). The desired transmission is then: n v ) = l ( , v ) / I a = ( L P ~ P) 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.12) 3.3 Spectral Line Broadening Theories A spectral line m ay be broadened and shifted by a variety of processes that depend on certain environmental parameters. Therefore, accurate measurements of a spectral lineshape should be combined with an adequate theory for the determination of important plasma parameters. For a more detailed description of broadening theory, one should refer to Griem, Breene, Shore, or Sobelman [Griem, 1964 and 1974][Breene, 1961 [[Shore, 1968][Sobelman, 1981] 3.3.1 Natural Broadening Natural broadening is due to the finite lifetimes of the states involved in the probed transitions. A Lorentzian lineshape is described as: r, . A v v /2;r K v ) = - ----------------------(v '-v 'o ) tttt + (A v V 2 ) (3.13) where vo is the linecenter frequency and Avn (sec*1) is the full width of the line at the half-maximum intensity values (FW HM ). The integrated area under the lineshape is normalized to unity. The width is proportional to the sum of the inverse radiative lifetimes of the two states and m ay be expressed as: a ,.--!-' ' where ti and 12 1 — 1 i- — In vr> (3.14) Tu are the total radiative lifetimes of states 1 and 2, respectively. The total radiative lifetime may be determined from the sum of all radiative decay 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rates from that level. (3.15) where A] is the spontaneous emission coefficient for the transition i-> j w here the index j runs all over the possible lower states. 3.3.2 Doppler Broadening Doppler broadening is caused by the Doppler effect and is described as an inhomogeneous broadening mechanism since it does not affect all particles equally. Specifically, for an emitter (absorber) approaching the observer (light source) with a relative velocity Vo, the effective emission (absorption) frequency is shifted as shown in the following equations: (3.16) or equivalently, w here Av and AX represent the Doppler shift in frequency and wavelength, vo and Xo represent the frequency and wavelength in the stationary reference frame. For a system in thermal equilibrium, the number density of particles in state j moving with a velocity between Vo and V q + dVo m ay be described by 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Maxwellian distribution: N, n tfo V n j = 7tU1V. -exp V VD V dVn V r,j f (3.18) where N* is the total number density of particles at state j. V p= (2kT/m )1/2 is the most probable veiocity. Substituting Equation 3.15 into Equation 3.16, we obtain an expression that describes the number of particles with linecenter frequencies shifted from vo to vo + dv: NjCc/vo) nj {v)dv = exp c v-v n V dv (3.19) Since the emitted or absorber power from an optically thin spectral line is proportional to the number density 0 j(v)dv, the resulting Doppler broadened profile (Gaussian shaped) can be expressed as: f I(v) = me 2 f 1/2 exp 2nkTv o v-vn me 2kT (3.20) The Doppler FW HM is given by: f8ln2 fcT A vD = v 0 n I/2 (3.21) me' In terms of the Doppler width, the profile can be rewritten in the following expression that is more convenient: f f 41n2y /(v ) = V «■ J f M—1/ \ 2> v -v 0 1 - — exp -41n 2 Avd I VD J y V. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.22) 3.3.3 Pressure Broadening In addition to natural and Doppler broadening, a spectral line may be affected by interactions with neighboring particles (i.e., molecules, atoms, ions, and/or electrons). Pressure broadening effects can be classified either by the mathematical approximations m ade in treating the perturbation or by the types of the perturber. The following sections will review the types of interactions classified by the type of perturber. 3.3.3.1 van d erW aals Broadening Interactions broadening, result with neutral from the particles, also dipole-induced known dipole as van forces. der W aals The impact approximation is used and the width (Avc) and shift (Avs) in the ideal case are given by Breene [1961]: Avc = 2.71N C l ' 5V V5 (3.23) A v, =0.98 N pC ; ,5V where Np is the perturber num ber density and V is the mean relative velocity between the atom and the perturber. 3.3.3.2 Resonance Broadening Resonance interactions occur between identical particles and are often significant for transitions that involve the ground state through an allowed 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. transition. Classically, the broadening may be caused by the efficient energy exchange between the two identical oscillators and effectively results in a reduction of the level’s lifetime [Breene, 1961]. Due to symmetry, resonance broadening of the transition (the transition involving the ground and first-excited states, denoted 1 and 2, respectively) will result in an unshifted Lorentzian profile with a width (FW H M ), Avres, which is given by [Griem, 1964]: f _ V '2 = - gx " 2\g i j (3.24) e0m,arl2 where so is the permittivity of free space, Ni (rrf3) is the population number density in state 1, g is the degeneracy of a given state, and fo. coi2 are the oscillator strength and angular frequency of the resonance transition, respectively. Avres reflects the energy uncertainty of state 2 as a result of the lifetime reduction due to the resonance interaction since the lifetime of state 1 (the ground state) is effectively infinite. 3.3.3.3 Stark Broadening Stark broadening, or pressure broadening caused by charged particles, can be treated with a classical electrodynamics approach that considers the effect of an external electric field on the energy levels of an atomic system and the resulting influence on a spectral line. An external electric field F applied to an atomic system effectively distorts the electron distribution and induces an electric dipole aF , where a is the atomic polarizability. The interaction between the 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. induced-dipole moment and the electric field will shift the atomic energy levels by an amount proportional to the interaction energy given by a F 2. Therefore, the Stark shift varies quadratically with the electric field strength. The motion of charged species within a Debye sphere gives rise to varying electric fields and results in the broadening of a spectral line. The lineshape may be determined from a combination of both the ion and electron broadening effects. Thus the Stark-broadened profile of an isolated neutral non-hydrogenic atomic line can be considered as a convolution of the electron-impact profile with the quasi-static profile for ion broadening. Tabulated theoretical Stark parameters relate the broadening and shift of spectral lines to electron number density and temperature. For predominantly singly-ionized plasmas, the total theoretical width (F W H M ) due to the quadratic Stark effect, wth (A), and the corresponding total theoretical Stark shift, dm (A), are given by the relations [Griem, 1964 and Baer, 1992]: ajA * 2[1 +1.75 x 1O'4ntx'4a (l - 0.068/;,1'6Tt' x' 1)]1 (T16wnt dth * 2[c//w ± 2.00 x 10'J«r,/4a (l - 0.068«t l/<T e~l/: )]10"16wnt w here w(A) is the electron impact parameter, a is the ion-broadening parameter, and (d/w) is the relative electron-impact shift. The Stark shift is towards longer wavelengths except for the negative values of (d/w). Equation (3.24) includes the param etric dependencies on ne and T e so that the state-specific parameters may be used directly from the table. The restrictions on the applicability are given by the following relations (in cgs units) [Griem, 1974]: 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.05 < a r n. \' 1/4 V1016 j < 0 .5 ^3 2 6 ) R = ^ - = 6 u l K V 6 { e 2 / k T ) xn- n \ ' 6 < 0 .8 a 3x ft.tin *2Y = ------------- v.v, m.Z "1^ >1 /3 n. where a is a measure of the relative importance of ion broadening; R is the ratio of the mean distance between electrons, pm, where 47tpm3ne/3=1, and the Debye radius, po, where pD2 = kT/47cnee2; ve,vi are the mean relative electron and ion velocities, respectively; Z is the effective nuclear charge; n’ is the effective principal quantum number of the upper state; and a is the ratio of electron to ion collision frequency and indicates the relative significance of the time-varying fields generated by the respective charged species. For parameters outside the ranges specified by Equation 3.26, the Stark broadening and shift equations should be appropriately modified or the entire theoretical profiles should be used [Griem, 1974]. Table 3.1 mathematical presents a brief discussion of the physical classifications, approximation, and lineshapes of various line-broadening mechanisms [Baer, 1993]: 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.1 Physical classifications, mathematical approximation, and lineshapes of various line-broadening mechanisms Hom ogeneous Broadening Inhom ogeneous Broadening Stark (charged perturbers) Resonance (like perturbers) VanderW aals (neutral perturbers) Physical Classification Other Doppler (relative velocity) Mathematical Approximation Maxwellian Velocity Distribution Impact Quasi-static Other (Avc tp^ « 1) (Avc tp >> 1) Lineshape Other Gaulsian Lorentzian Modified Lorentzian 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER FOUR MODELING 4.1 Pseudo-1-D Plasma Model This section describes the Collisional-Radiative-Model (C R M ) used in our research to predict the excited state number densities as well as the electron number density, electron temperature, the deposited microwave power, etc. The reason to label the CRM as a pseudo-1-D plasma model is that CR M itself is a P, Tg, ne, R, Lj y, etc. / E/N / A EEDF No D n e /d t= 0 Yes Figure 4.1 The iterative scheme to obtain a self-consistent solution 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. zero-D model. However, the electron diffusion loss is taken into consideration so that one can estimate the effects of the electron diffusion and the cavity geometry. Figure 4.1 shows the iterative scheme to obtain a self-consistent solution. First, the program reads in the geometric data as well as the operation conditions, such as the chamber pressure (P). Some estimated values are also input, such as the excited state number densities, electron number density, the electron and gas temperatures, and the electric field strength (E/N is used where E stands for the electric field strength and N stands for total density). Both argon and hydrogen models have been developed. One can refer to Kelkaris thesis for more detailed information [Kelkar, 1997]. Here only the argon model is discussed for the interests of this research. 4.1.1 E lectro n E nergy D istribu tion Function T h e general form of the Boltzmann equation can be found in any statistical mechanics text, such as [Mitchner and Kruger, 1973]: at V n .J . mt = 2 X (4 ' 1) W here fe is the electron velocity distribution function (EVD F). c is the electron velocity. Fe is the external force that can be simplified as qE. ne is the electron number density. me is the mass of electron. Rer represents collisional rates between electrons in a particular velocity class and species r. Further simplification is needed in the general case of non-equilibrium. W e 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. assume that the EVDF is nearly isotropic so that it can be replaced by the first two terms of a spherical harmonic expansion: /( W .( n + /,( i') ~ (4'2) Often it is desirable to work with Electron Energy Distribution Function (E E D F) rather than EVDF. Using the relation E=mV2/2, one can convert between the EV D F and EEDF: / . w = ^ 8 7T / , w < 4 ' 3 ) The distribution function is normalized such that: \af ( s ) s V2d s = 1 0 (44) A commercially available software, ELENDIF, is used as the Boltzmann solver. It includes the terms for ionization, attachment and recombination, photon-electron processes, as well as an external source of electrons such as an electron beam. The diffusion effect can also be taken into consideration by ELENDIF. 4.1.2 25-Level Argon Model An n-level atomic collisional-radiative model was developed to predict the excited state number densities. The ground state and electron num ber densities are held constant during calculation. The following electron and heavy particle collisional reactions and radiative reactions w ere used to develop the non-linear 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. system of equations: A m+ e An + e n> m A„m + M <-> A„ft + M n> m A m + e <-» A ' + e + e (4.5) Am + M <-> A~ + e + M Iff A„ —> A m + h v n> m A * + e - > Am m+ h v For each excited state number density, a rate equation was generated based on the above reactions as follows: + «,*r')+ (4.6) j<m j>m where nm represents the number density of an atomic excited state (m>1); ne, nj are the electron and ion number densities, respectively; k is the appropriate reaction rate coefficient; nt is the total number density, which can be calculated as (P/kTg); the symbol M represents the major species present in the plasma. In Equation 4.6, the first two terms account for the electron-impact and heavy particle-impact excitation and de-excitation. The third term is for the electron and heavy particle induced ionization. The fourth term represents the respective recombination reactions. The fifth and sixth terms account for radiative excited state transitions. And finally, the last term is for radiative recombination. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1.3 Electron Conservation Equation Electrons are produced inside the bulk plasma once the discharge is formed. They are lost either by recombination in the bulk plasma or by diffusion and convection to the cavity wall and substrate, or to the after-glow region. The electron number density balance can be expressed in the following equation: dne _ kr — — — A , „ A at r TS r^diff _ kr cairv (4.7) w here Kj, Kr, ( W , Kconv represent the total production of electrons by ionization, the total loss of electrons by recombination, the electron loss rate due to diffusion and the electron loss rate due to convection, respectively. For pure argon plasmas, the electron number density balance can be written as the follows, according to the set of reactions present in Equation 4.5: tin 24 - / \ £ »„ ( "AT"+ - 24 I «.», ( » . * - " + ) - m=l ^ > ix n,nk‘ Da * e / x 4 x l 0 17 ~ n , ------------ --- ------------------------------------------A nt Vp ( ‘ ) In the above equation, the terms of diffusion (neD a/A 2) and convection (the last term) need more discussion since they can affect the solution of CRM significantly. Diffusion is due to the existence of the gradient of concentration of the spatially distributed species. For charged particles in plasmas, such as electrons and ions, one must account for ambipolar diffusion, which occurs in moderate 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. pressure plasmas and ensure the electrical neutrality for a length scale greater than the Debye length [Liebermann, 1994]. The electron free diffusion rates are much higher than those of ions because of their lighter masses. Therefore, the electrons diffuse faster than the ions, disturbing the quasi-neutrality of the plasmas. However, since more electrons diffuse outside the bulk plasma and accumulate at the boundary, the so-called sheath region will be formed. A significant voltage drop will exist across the sheath, with the voltage drop varying from several volts to hundreds of volts. This resulting space charge field will retard the electron diffusion and increase the ion diffusion so that the quasineutrality can be restored at all points inside the bulk plasma. For argon, only one ion (Ar+) is considered here, the ambipolar diffusion coefficient is calculated as follows [Cherrington, 1973]: (4.9) 11 J v w here Dj represents the ionic diffusion coefficient; T e, Tj represent electron and ion temperatures, respectively. To calculate the ionic diffusion rate, Einstein’s relation is applied here [McDaniel, 1973]: (4.10) where p* is the ionic mobility, which is given by the following relation: Mi = Mio 53 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (4.11) where 1*0 is the reduced ionic mobility, NL is the Loschmidt number (2.689x1025 nrf3), and nt = P/kTg represents the total gas density. The reduced ion mobility for Ar+ (in argon) is 1.535±0.007 cm2/Vs [M cDaniel, 1973]. The characteristic diffusion length (A) for a cylindrical geometry is given by [Chem'ngton, 1979]: where L and R represent the length of radius of the plasma volume, respectively. For a plasma in contact with a cold surface (reactor walls, substrate, etc.), the electrons and ions diffusing out are lost by recombination. Thus the surface acts as a third body and absorbs the energy released in the recombination reactions. For plasmas not in contact with a surface, the diffusing electrons and ions may either combine in the surrounding gas or be pumped out. The convection rate coefficient for the charged and neutral particles is given by: f 4 x l 0 17 (4 .1 3 ) conv where f represents the gas flow rate in seem (4x10 17 is a conversion factor from seem to particles/s). 4.1.4 Pow er Balance Equation The input microwave energy accelerates the charged particles present in 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the plasma. As a result of ions’ heavier mass, the energy absorbed by the ions in the bulk plasma is almost negligible. However, ions can be accelerated in the DC sheath at the plasma boundaries and absorb a small part of microwave energy. The total power balance can be written as: (4.14) The electron energy (Pabs.e) balance can be written as: (4.15) Pabv 'E -Q la s + Q in c la s + Q ra d + Q c o n v + Q d ijr The above equation shows that the electron energy is gained by the electrical (ohmic) heating. Most of the electron energy is lost by the elastic and inelastic energy exchange in electron-neutral collisions, and the radiative losses that comprise free-free and free-bound collisions between electrons and ions or neutrals. The third and fourth terms in equation 4.15 represent the kinetic energy loss by electron diffusion and convection, respectively. The free-free and free-bound radiative losses were calculated by integrating the continuum emission over a wavelength range of 10 nm to 100 pm. The expressions for other terms in equation 4.15 can be written as follows: n (4.16) = n. ' Z kj£1nj j 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The kinetic energy loss due to ionic diffusion and convection is given as [Kelkar, 1997]: Pa b s j = 4.2 ( n,D. w. y x 4 x ! 0 ITY », V, J (4.17) ' Electrom agnetic Field Model The pseudo-1-D model discussed in the above section is not self- consistent. One should specify several important parameters such as the electron number density, gas temperature, and the electrical field to get a satisfactory solution. Another important parameter, the microwave input power, cannot be used in the model. As shown in the later sections, w e found a big discrepancy between the reading from the pow er meters and the prediction from the CR M . Therefore, a 2 -D plasma model that couples the electrom agnetic (EM ) model is desirable for further investigation. 4.2.1 Governing equations of the EM m odel The behavior of the electromagnetic energy and how it couples to and excites the plasma discharge depend on the geometry of the structure, the input power coupling structure, and the parameters of the plasma discharge (size, density, pressure, composition, etc.). The finite difference time-domain method (F D T D ) is used to solve the microwave electromagnetic fields in the plasma reactor [Tan, 1994 & 1995]. Since the microwave wavelength (X=c/v=0.122m ) is sm aller than the dimension of the electromagnetic field confinement region, a 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. full w ave solution of the Maxwell equation is necessary. The Maxwell equations are given by: V x E = - m0 — V x / 7 = e— + J (4.18) dt V-£ = — e V-B = 0 where E is the electric field, H is the magnetic field, B is the magnetic flux density, J is the current density, s is the permittivity, po is the permeability, and p is the charge density. The plasma discharge contributes to these four equations through the current density and the charge density terms. T h e current density is determined by solving the momentum transport equation for electrons in the microwave frequency range simultaneously with the Maxwell equations 4.18. The momentum transport equation can be written as: m. ^ — = -q E -m ,vtjrv (4 -1 9 ) dt And the current density is: 7 J = -qnev where v isthe - (4.20) average electron velocity,qis the electron electron mass, veff is the effective collisionfrequency, charge, me is the and ne is the electron density. The expression for the effective collision frequency can be written as: ,V e f f-^ a v th nt ° 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <4 -2 1 > w here a is the momentum transfer cross-section for electron-heavy particle collisions, vtt, is the electron thermal velocity, and nt is the total neutral gas density. The microwave power density absorbed by the plasma, M W P D , is calculated for each grid point with the following expression: (4.22) MWPD - J ■E 4.2.2 Electrom agnetic Properties of Plasma Discharges For the collisional, non-magnetized plasma discharges, such as the one used for diamond CVD deposition, the frequency of excitation is generally less than the plasma frequency. Therefore, the microwave electromagnetic fields do not propagate through the plasma discharge. In this case, the electromagnetic w ave will penetrate into the plasma discharge a distance on the order of the skin depth (defined in Equation 4.23). Then for the region of the discharge within approximately a skin depth of the plasma surface, the electron gas absorbs the energy from the microwave fields. The general expression for the skin depth 5 of a transverse electromagnetic wave penetrating into plasma can be written as [Bittencourt 1986]: r (4.23) — = -Lm ag — <a 8 c 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where c is the speed of light, v is the electron collision frequency, o is the microwave frequency, and £ope is the electron plasm a frequency, which is given by [Mitchner and Kruger 1973]: r cop* = t \ 1/2 ne2 ] (4.24) For example, if the electron density is given as n=5e17 nrf3, the electron plasma frequency cope will be equal to about 40 G Hz. For a typical argon plasma discharge, with the collision frequency of v= 5 x 1 0 10 H z and a microwave frequency of 2 .45 GHz, the skin depth is approximately 5 cm. If the collision frequency is reduced to 5 GHz, the skin depth will be about 1.2 cm. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Two Dimensional Plasm a Fluid Model In the moderate pressure plasma flows, the assumption of a continuum description of the flow is valid and the plasma species and energy transport are governed by the fluid dynamic equations, which express the conservation of mass, momentum and energy in the plasma flow [Prelas et al., 1998]. In this section, details of the modeling of the nonequilibrium plasma will be discussed. For argon plasmas, a two-temperature flow assumption is m ade for the expression of the transport equations. For previous hydrogen plasmas studies, a three-temperature flow assumption was m ade due to the existence of the vibrational modes of the molecules [Hassouni, 1997], 4.3.1 Plasma Flow Descn'ption For thermo-chemically nonequilibrium plasma flow, one has to write conservation equations for each species, each energy mode and each momentum component. G enerally we use mass density (ps), the momentum per unit volume (pU) and energy per unit volume (Emode) to denote the conserved variables. 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. COcD 7ET Figure 4.2 Schematic of a control volume AxAy for O conservation Considering a control volume element, as shown in Figure 4.2, the conservation equation for each conserved variable 0 (mass, momentum, energy, etc) can be written as: = Fxmx (AyAz) + Fymx (AxAr) A/ (4.25) - Fxoul x (AyAx) - FyMt x (AxAz) + where AV=Ax*Ay with Az=1 for a control volume element AxAy and the sourceterm can be denoted as coa>. T h e conservation equation can be written in the following differential format: ^ _ = dFx_Sfy_+ w dt dx Ay * 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.26) where ¥ = — dV , which is the conserved variable per unit volume. The following table shows the formulation for each conserved variable in Cartesian coordinate systems. For cylindrical system, the equations should be changed for axial and radial directions. Table 4.1 The Conservation Equations Continuity Equation ps for species "s" Momentum Component Equation pu (p the plasma total mass density) Electron Energy Equation Ee Total Energy Equation Equation Conservation Equation Conserved Variables # v (p su + p% u ,) = W, where u is the plasma averaged velocity, and us is the diffusion velocity of species "s" relative to the averaged velocity x-direction: v((/3«)u + P - r*)= pgx y-direction: v((pv)u + P - r y)= pgy (4.27) (4.28) V f t . v r , - J , h , ) + M W PD -Q ,_ , - 0 , _ , = 0 where Xe, Je, he denote the thermal conductivity, the mass diffusion flux and electron enthalpy, respectively (4.29) v T a ,V 7 ; + x y T ' - Y , J 1h y + M W P D - Q raJ = 0 E * J where X( is the thermal conductivity of heavy species translational mode, Qrad is the power lost from the plasma by radiation V (4.30) The continuity equation can also be written as the following equation for simplicity: 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Ms, DSl Xs, and W s denote the molar mass, diffusion coefficient, mole fraction, and chemical production rate of species "s", respectively. M and p are the averaged molar mass and total mass density of the plasma. 4.3.2 • Flux and Source Term Expressions for the Transport Equations Species Diffusion Velocity To write expressions for species diffusion velocities in a multi-component flow is very difficult. Thus an approximation similar to Fick’s Law is used, which uses an effective species diffusion coefficient in the plasma mixture, [Curtiss et ai., 1949 and Lee, 1984]: (4.32) P, • Shear Stress Tensor Components For the Newtonian fluids, the shear stress tensor components are expressed as following [Bird 1960]: at 2 at + av (4.33) (4.34) where p is the averaged plasma viscosity. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • E lectron-Translatio nal Energy T ra n s fe r S ource T erm This term can be derived from classical kinetic theory. The expression is as following [Vincenti and Kruger, 1967]: (4.35) w here M is the species molar mass. Subscript e and s denote electron and species "s" respectively. Oe-s is the electron-heavy particle momentum transfer collision cross-section, and n is the total plasma number density. • Electron Energy Loss by C h em ical A ctivatio n This term is determined by the reaction rate (A E r ). (V r ) and activation energy The expression is as following: Q .-c (4.36) = Re actions W here 4 .3.3 cxr* is the electron stochiometric coefficient in reaction "R". T ra n s p o rt C oefficients Yos [1963] derived the rigorous expressions for the transport coefficients of multi-temperature gas mixture flow in thermal equilibrium (all distribution functions are close to Maxwellian about their inherent temperatures). Lee adapted these expressions for thermally nonequilibrium flows [Lee 1984]. • V is c o s ity The viscosity of a nonequilibrium multi-component flow depends on the second order collision integrals (H s/2,2*). The expression of the calculation is as 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. following [Lee, 1984]: mexe msxs (4.37) ™<Tm) £ x X ^ ) + * eA < 2(7;) \ r where xs is the mole fraction of species "s", and Asr(2)( T ) is defined as: 16 / \ 1/2 2 msmr n£l Krd<T{ms + m r ) ( 2. 2 ) (4.38) In the above equations, ms denotes the mass of species "s". For a plasma only having one major species, the viscosity calculation can be simplified from the Wilke Equation as following [Wilke, 1950]: r \ (4.39) * - Z k *s where xs is the mole fraction of species "s", (is is the viscosity of the pure gas "s", and cbsk is given as: - 1/ 2 ( 1+ E l V ' ' i+^ Mk Y jy ' M kV '4' (4.40) \ Mkj The calculation of ps is given by Bird [1960] and W ilke [1950]: (4.41) M s • = 1 Therm al Conductivity The thermal conductivity consists of two terms, one for the translational thermal conductivity and the other for the rotational energy mode if molecules 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are present in the gas mixture. The translational thermal conductivity of heavy particles is given as: (4.42) 4 r = - * £ 4 r £ a „ * , A ™ ( r f ) + 3 .5 4 * .A ™ ( r ,) V r where asr is defined as: \ 1 a„= I + \ -0 1 l 0 .4 5 -2 .5 4 — mr mrJ f (4.43) \2 1+ ^ V mrJ If all the rotational energy modes are assumed to fully excited, the overall rotational thermal conductivity of a mixture is given as: x. s -M o le c u le (4.44) Xx,Al"(rs)+xXUrj \ r where Asr(1)(T) is defined as: 1/2 2 msmr A“ ’ ( n = | (4.45) nQ!(i.D jdcT{ms + m r ) If vibration energy mode is also involved, the vibrational thermal conductivity should also be calculated. The equation is the same as Equation 4.45 with the vibrational temperature. The electron thermal conductivity is given as follows: 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As in the case for viscosity, it is time-consuming to calculate the thermal conductivity according to the above equations. For the plasmas with only one major species, the thermal conductivity may be well estimated from the one of the pure gas component. The conductivity can be calculated as following by Eucken relations [Bird, 1960]: (4.47) Ira n i (4.48) where Ctrans, Crot, and Cv are the specific heat for the translational, rotational and vibrational modes. The electron thermal conductivity can also be simplified as follows [Jaffrin, 1965]: K 15k nkTe 6 4 (1 + V 2 )OeJ me (4.49) where Q ee is the electron-electron collision cross section. • Diffusion Coefficients The binary diffusion coefficient Dsr of a species "s" in another species "r" is given by [Yos, 1963]: (4.50) Dsr PA{H (T ) 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e diffusion coefficient of a neutral species "s" in a gas mixture can be expressed as follows (Curtiss, 1949]: _ M, M For charged species, 2 > r/D J ambipolar diffusion has to be taken into consideration to ensure the electrical neutrality for the length scale that is larger than the Debye length. The calculation is the same as discussed in the pseudo1-D model: f T \ (4.52) 1+ ^ Damb - A T \ &J T h e electron diffusion coefficient is derived from the assumption of electrical neutrality and zero electric current. It is given as [Delcroix, 1958]: ' Z ( zsM sxsD s.amb) —_£________________ n (4-53) e.am b 1 - - ^ X, s where zs is the electrical charge of the ion species "s". • Comments on the Calculation of Transport Coefficients In the calculations for most transport coefficients, the collision integral jS d e fin e as the weighted average of a collision cross section in the form as [Lee, 1984]: =f r smx d x d r e_rV 2m+3(l - c°smx) sinxdxdy 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.54) where csr = asr(x,g) is the collision cross section for the collision pair s-r, x is the scattering angle in the center of mass system, g is the relative velocity of the colliding particles, and y = [msmr/2(ms+mr)kT]1/2g is the reduced velocity. From the above equation, one can see that the accuracy of the calculated transport properties is greatly dependent on the collision integrals. The collision cross sections for ion-neutral and electron-neutral interactions still need more investigation. 4.3.4 Boundary Conditions The boundary conditions are critical to the numerical modeling. Different boundary conditions for the computation domain will generate different results. The boundary conditions may be provided by experimental measurements, or extracted from the conservation laws. For the modeling of the cylindrically symmetric deposition reactor, the computation domain consists of the belljar walls, the substrate surface, and the symmetric axes. • At the Sym metric Axis Considering the axi-symmetric computation domain, at the symmetric axis, zero gradient boundary conditions are imposed to all the parameters: *®-0 dr " ° = xs,u,v,Ts^andTt 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4 -55> • At the Belljar W all At the belljar wall and the substrate surfaces, the velocity components should vanish. The gradient of pressure will approach zero. Without a detailed sheath model, one can assume that the gas temperature is in equilibrium with the wall temperature and that the gradient of electron tem perature is zero due to the fact that the accommodation of electron energy at the wall is very weak [Scott. 1993]. The resulting boundary conditions are as follows: 11 - =v 0 = dP ------- =0 (4.56) (4.57) d r wall T g.wall - T wall dT. =0 (4.58) (4.59) d r wall • At the Substrate Surface The resulting boundary conditions at substrate surfaces are as following: W= V = 0 dP 0 = (4.60) (4.61) dz sub Tg.sub = Twall 57; = 0 dz sub 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.62) (4.63) • Surface Chemical Kinetics The boundary conditions for the chemical species concentrations at the substrate surfaces and walls can be derived from the modeling of the surface chemistry kinetics. Some models are available for diamond deposition plasmas. The resulting boundary conditions can be expressed as [Scott, 1993 and McMaster, 1994]: -D w here s% rrux W Si surface dx = W (T k c r = 1 N species J\ VY s.surfaceK1 s ->K ->C r ’ r (4.64) surface is the production or consumption rate of the species "s" by surface reactions. As shown in the above equation, it depends on the surface temperature, the rate constants of the surface reactions, and the species mole fractions at the surface. 71 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 4.4 Two Dimensional Argon Plasma Model The 2-D plasma model was originally designed for hydrogen plasma discharges in France. Characteristics of the hydrogen plasmas have been studied extensively by applying this model [Hassouni, 1999]. However, no study of the 2-D collisional-radiative-model was done according to the literature review. Moreover, no such 2-D model for the microwave argon plasma is available. Some studies of 2-D EED F have been reported [Kortshagan, 1999], but their models are very crude thus cannot provide useful information such as excited state number densities, power densities, and gas/electron temperatures. Therefore, we constructed a 2 -D argon plasma model. This 2-D model adds more understanding to the microwave non-equilibrium plasmas since it provides more detailed spatially resolved and self-consistent data. 4.4.1 Argon Plasma Chem istry Model The 2-D model requires more calculation time than the 0-D models. Although the time issue may not be critical, constructing a model that includes all the excited states is not an efficient way for research and development. Therefore, four species are considered for the argon plasma chemistry. This approach has been widely adopted by other researchers [Lymberopolous, 1998], Tab le 4.2 and 4.3 show the species and reactions considered in the argon plasma chemistry. 72 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Table 4.2 Species considered in 2-D argon plasma chemistry Species Argon ground level Symbol Ar Argon 4s excited state (metastable) Ar* Argon ions Ar* Electrons e Table 4.3 2-D Argon Plasma Chemistry Model Number Reaction Description 1 Ar + e -> Ar* + e Electron-atom excitation reaction 2 Ar* + e -» Ar + e Electron-atom de-excitation reaction 3 M + A r-> M + Ar* Atom-atom excitation reaction 4 M + Ar* -> M + Ar Atom-atom de-excitation reaction 5 Ar + e -> Ar* + 2e Electron-atom ionization reaction 6 Ar* + 2e -» A r+ e Three body electron recombination Atom-atom ionization reaction 7 M + Ar M + Ar* + e 8 M + Ar* + e -> M + Ar Three body atom recombination 9 Ar* + e -» Ar* + 2e Electron-atom ionization reaction 10 Ar* + 2e -» Ar* + e Three body electron recombination 11 M + Ar* -> M + Ar* + e Atom-atom ionization reaction 12 M + Ar* + e -> M + Ar* Three body atom recombination 13 Ar* -> Ar + hv Radiative de-excitation reaction 14 Ar* + e -> Ar + hv Radiative electron recombination 15 Ar* + e -» A r + hv Radiative electron recombination In Table 4.2, the m etastable excited state (4s) is considered along with the argon ground state, ion and electron as the four species in the argon plasma chemistry. Although only the 4s excited state is included, good results have been 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. obtained because excited state ionization is dominant from the 4s state, and 2 step ionization will be discussed in Chapter 5. Neglecting the excited state ionization will cause large error in characterizing the argon plasmas. In Table 4.3, the third body atoms, denoted as "M", are simply argon atoms in the pure argon plasma discharges. All the reaction rates can be obtained in the CRM by specifying the values of E/N, electron number density, pressure, and gas temperature. The curve-fitted rate expressions w ere obtained by curve-fitting the rates with the averaged electron temperature and electron number density. Surface chemistry for the 2-D model is necessary to simulate the effects of the existence of the physical walls. In this argon model, it was assumed that 100% of the argon excited state particles and ions were de-excited or recombined to the ground state at the wall. In the argon plasmas, since no molecules are available, only the atom translational and electron energy modes are considered. As discussed in the Section 4.3.1, for each energy mode, an energy conservation equation is needed. Due to the fact that the EEDFs of the argon plasmas w e investigated are highly non-equilibrium, the energy transfer terms should also be curve-fitted. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.2 Curve-Fitted Reaction Rates In this section, some of the reaction rates are discussed. T h e pseudo-1-D model was used to generate the rates at first. Then they w ere curve-fitted to some function of electron number density, electron temperature, or gas temperature if needed. Since the Boltzmann solver is no longer coupled inside the 2-D model, these curve-fitted reaction rates are used instead of the EEDF to calculate them directly. 10' 1 Li. N, = 5e16 m'1 Nt = 5e17 m'1 N, = 5618 m'1 Maxwellian Distribution 0 5 10 15 20 Energy [eV] Figure 4.3 Non-Maxwellian EEDF plots with averaged T e — 1.07e4 K and 300K < T g < 600K comparing with the Mawellian EEDF Figure 4 .3 shows three EEDF plots with the same averaged electron 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature at 1.07e4 K. From this figure, the effect of electron number density to the EE D F shape is very clear. At a high electron number density (5 e 1 8 nrT3), the Boltzmann plot is close to the Maxwellian distribution, which is a straight line. With lower electron number densities, the high-energy tail of EEDF will deviate from the Maxwellian distribution much further. T h e parameters changed in the pseudo-1-D model are electron number density (Ne), gas temperature (Tg), and E/N, which corresponds to an averaged electron temperature (Te). To curve-fit the reaction rates, w e use: K r^ , = c { \ . 0 - e - ^ ) ' T e ‘’ (4 '65) Microsoft Excel’s Solver was used to optimize the parameters for the curve-fitted equations. The exponent “a” is introduced to account for the degree of the non-Maxwellian effects of EEDF. Figures 4 .4 -4 .6 show the comparison of the original calculated reaction rates with the curve-fitted ones. The curve-fitting range for electron temperature is from 5000K to 30000K . This is the sam e range that is specified inside the 2-D model. Larger error is present when both the electron number density and electron temperature are low (ne < 1e17 nrf3 and T e < 10000 K). Fortunately, most of the ne- Te pairs are not in this region. Thus the numerical uncertainty introduced by the curve-fitted rates is still reasonable (<30% ) as discussed in Chapter 6. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. .-16 0 0 c ,-17 o o CO K e a(0 ,1 ) 0 c o eg o X 0 E o i-2 0 Ne=5e17 m Ne=1e18 m-3 ,-21 Ne=5e18 m"3 0I c o Curvefitted Rate cS 0 0 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 T e (K) Figure 4.4 Comparison of the curve-fitted reaction rates (Kea(0,l)) and those calculated from the C R M as a function of electron temperature. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rate ionization Klea (0) reaction 10 .-20 • •• 10" electron-atom Ne=5e17 m ,-22 N9 =1e18 m 3 N9 =5e18 m 3 ,-23 Curve-fitted Rate .‘2* 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 T e (K) Figure 4.5 Comparison of the curve-fitted reaction rates (Klea(0)) and those calculated from the C R M as a function of electron temperature. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 2 c o K lea(1) o 0 0 ,-14 c o ra N 'c o E 0 <1 3 c Ne=5e17 m o N,=5e17 m'3 o Curve-fitted Rate Ne=5e17 m'3 0 0 .-16 8000 10000 12000 14000 16000 18000 20000 22000 24000 26000 T.(K) Figure 4.6 Comparison of the curve-fitted reaction rates (K lea(l)) and those calculated from the C R M as a function of electron temperature. 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.3 Iterative Scheme between the Plasm a Model and the EM Model The 2-D argon plasma model is fully self-consistent. Once the plasma chemistry and boundary conditions are specified, the only information necessary to start the calculation is the plasma pressure and microwave power. Unfortunately, as we will discuss in the energy balance study, the microwave power measured from the power meters on the reactor could not be used. Thus a parametric study of the effects of microwave power on the argon plasma discharges is necessary. In this section, the iterative scheme between the plasma model and the EM model is discussed as following. Figure 4.7 is an iterative scheme showing the coupling between the 2-D plasma fluid model and the EM model. The self-consistent solution is obtained by iterating between these two models until a converged solution is reached. Due to the different grid structures in the two models, grid interpolation is needed for the exchange of information between these two models. The microwave power density (M W PD ) is obtained from the EM model and serves as the input to the plasma fluid model. On the other hand, the fluid model provides the information about the plasma discharge, such as the electron number density, electron temperature, gas temperature, etc. The major difference between the two grid structures is that w e have to consider the whole cavity volume for the EM model, while w e only consider the region inside the belljar for the plasma fluid model. The two grid structures will be 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shown in the next section. Te;>Tg, lie a —- ■ ^1 V Grid Interpolation | Argon Plasma 2-Temperature Model (300-400 Iterations) I 1 Electromagnetic Model E, H , J, M W P D (15-25 Microwave periods) Grid Interpolation MW PD Figure 4.7 The Iterative Scheme of the Coupled 2-D Plasma Model and the E M Model W e will show that the argon plasma is highly non-uniform. Therefore, we do need a 2-D model instead of the 0-D model that assumes a uniform plasma discharge. Also due to the high non-uniformity, more than 15 iterations will be needed to achieve a converged solution for the microwave argon plasma. 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.4 Grid Structures and Boundary Conditions As discussed in the above section, the grid structures for the plasma fluid model and the EM model are different. Therefore, for a specific reactor design, two sets of grids are generated from the reactor geometry. Figure 4.8 shows the reactor geometry considered for the models. One can refer to the schematic of the reactor in Figure 5.8 to have a better idea about the reactor structure. Due to axi-symmetry, we only need to consider the simulation domains as shown in the Figure 4.9. The grid structure is half of the cross-section of the reactor cavity in the r and z directions. For the FDTD EM model, the maximum number of grids used is 100x130 in r and z directions, respectively. The grid spacing in the plasma discharge region is set to be denser than in other regions, both in r and z directions. For the plasma model, the num ber of grids is 40x72 in r and z directions, respectively. The plasma grid is defined only inside the belljar where the plasma discharge is actually confined. The grid spacing is varied to be denser in the region of the plasma discharge ball. The base plates, substrate holder, probe, short, and the cavity walls form the boundaries for the EM model simulation region. By assuming these boundaries are made of perfect conductors, the boundary conditions for the electric fields on these surface are that only the normal components of the 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Simulation Boundary Plasma Simulation Boundary Belljar | Substrate i Holder ^ X w x w w w w W iw w w w w w w ^ 777777, I Figure 4.8 The Simulation Domain of the Microwave Plasma Discharge with the W A V E M A T Reactor electric fields exist, while the tangential components on these surfaces are set to zero. The probe is actually a coaxial antenna. At its input end is an open boundary where the electric field computed is unbounded. At this boundary, a truncation method was used to prevent any artificial reflection of outgoing waves [Tan and Grotjohn, 1994, 1995], The coordinate system sets z= 0 at the top of the substrate holder. The technique used to excite the electromagnetic field is to select the grid 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. points on the cross section plane of the coaxial probe as the source points. The time-varying electric field components are assigned to these points based on the theoretical transverse electromagnetic (TE M ) w ave solutions in a coaxial structure. The electromagnetic w ave then propagates downward into the cavity where the microwave power is absorbed by the discharge. The reflected electromagnetic w ave will propagate to the end of the open boundary of the input power antenna and be terminated by the non-reflecting boundary. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Tw o Dimensional Collisional-Radiative Model For the highly non-equilibrium plasmas we investigate, one should consider the non-Maxwellian effects of the EEDF. This requires the coupling of a Boltzmann solver for the derivation of the EEDF. Moreover, for our research interests, the excited state num ber densities are used for comparison with the experimental data. Therefore, a 25-level CRM is necessary to gain the information of all the excited states. If we couple this C R M with the 2-D fluid model, the computation load will make this approach infeasible. Another problem is that our Boltzmann solver is not a 2-D code, thus the E/N data cannot be imported from the electromagnetic model directly. Therefore, the 2-D fluid model for pure argon only has four species: Ar, Ar' (4s), A r+, and electron, while the 2-D CRM still keeps 25 levels. Inside the 2 D CRM, all input parameters come from 2-D fluid model. Recalling the numerical schematic for the pseudo-1-D model (Figure 4.1), electron number density balance is achieved by changing the E/N value. In the 2-D CRM , the electron number density balance equation is no longer implemented because in the 2-D fluid model, the electron number density balance has been realized by the continuity equation for electrons. However, the E/N value is again changed inside the 2-D CR M to ensure the resulting averaged electron temperature from the Boltzmann solver is equal to the one from 2-D fluid model. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Gas Temperature (Tg) Electron Temperature (Te) Electron Number density (Ne) Ion Number Density (N i) Electrical field => E/N SaSSSliEJfi SdSSSaSEfii ’ W r- ' < Excited State Number Densities (File: excite.dat) Microwave Power Density, E/N (File: Pabs2d.dat) Reaction Rates (File: rate.dat) Other data if needed (EEDF.etc.) 2-D CRM Figure 4.9 The Flow Chart of the Iterative Scheme for the 2-D Collisional-Radiative Model (C R M ) 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4 .9 shows the iterative scheme of the 2-D CRM . T h e outputs from 2-D fluid model are used as inputs to the 2-D CRM. The excited state number densities from the outputs of the 2-D CR M are of primary interest since w e can compare the results with our experimental measurements. Other outputs include the reaction rates and the microwave power density. The gas temperature remains unchanged in the 2-D CRM, while the value of E/N is changed inside the 2-D CR M to match the averaged electron temperature with the one from the 2-D fluid model. Another iteration between the 2-D CRM and 2-D fluid model can be formed to achieve higher accuracy. In this case, the reaction rates can be first written into a data file inside the 2-D CRM . Then in the 2-D fluid model, all these reaction can be read directly from this data file instead of calculating the rates from the curve-fitted equations, which could produce some uncertainty. Thus there will be two major iterations to get a converged solution (Fluid M odel/EM Model, and Fluid M odel/CRM ). The tradeoff between the 2-D fluid model and the 2-D C R M is that the 2 D fluid model sacrifices the accuracy of the plasma chemistry model to gain efficiency in numerical calculations. And the 2-D CRM can provide more detailed data of excited states, but is not able to consider the 2-D effects within the model itself. More detailed structures of the plasma fluid model, the EM models and the 2-D C R M can be found in the Appendix B. The codes for the fluid model and the 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. EM model were written in the W A T C O M FO R TR A N format. The 2 -D C R M codes were written in Fortran 90 and C languages. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER FIVE E N ER G Y BALANCE STUDY For the study of the nonequilibrium microwave argon plasmas, both numerical and experimental approaches have been used to gain a better understanding. In this chapter, numerical results will be presented to show that the argon plasmas absorbed only a small portion (2-5% ) of the metered microwave energy (800W forwarded, and 120W reflected) under the studied conditions (5 Torr and 2 5 0 seem argon flow). An energy balance study was then conducted to finalize this issue. A global reactor energy balance was performed on the microwave CVD reactor to observe how the microwave energy is dissipated into the cooling lines. After we understood the global reactor energy balance, a control-volume heat transfer model was constructed to perform the heat transfer analysis with a control volume encompassing the plasma discharge region directly. The 2-D argon model provides more detailed data about the heat fluxes around the plasma simulation boundaries. These data were used to check the results of the energy balance study. Good agreem ent was achieved between the argon 2-D model and the energy balance study results. To begin the discussion with the energy balance study, the pseudo-1-D model results should be introduced first. 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 P seudo-1-D A rgon M odel R esults For the non-Maxwellian EEDF, the Boltzmann solver was coupled with the argon CR M to solve for the excited state number densities and the predicted power deposition. In this case, the input parameter E/N, which represents the ratio of the electric field strength and the total number density, was varied instead of electron temperature. In this coupled CRM, the electron number density balance was imposed in the model. The ranges of the parameters studied are as follows: T ab le 5.1 T h e Ranges o f S tudied P aram eters Used fo r A rg o n P lasm as at Pressure o f 5 T o rr E/N (Td) 0.005 - 2 . 0 T g (K ) 300 - 600 ne (m*3) 8e16 - 3e19 As shown in Figure 5.1, the non-Maxwellian EEDF deviates from the Maxwellian distribution dramatically at high energies. This phenomenon could impact the plasma chemistry significantly. Therefore, non-Maxwellian E E D F was used for all the models used in this research work. Figure 5.2 shows the relative electron-energy loss terms changing with the averaged electron temperature. For the conditions studied, at lower electron temperature, the elastic energy loss term is dominant. However, at higher electron temperature, the inelastic energy loss term becomes dominant. For 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. argon plasmas, the elastic energy transfer between the electrons and the neutral particles is the major source for gas heating at low electron temperatures (Te < 14000 K). However, at higher electron temperatures, the reaction rates of the inelastic collisions become much higher. Thus the inelastic energy loss term becomes dominant at higher electron temperatures. The electron production and loss mechanisms are demonstrated in Figure 5.3 as a function of the averaged electron temperature for a particular ne and T g (1e18 nrf3 and 350 K, respectively). Under the studied conditions, the dominant production and loss mechanisms are excited state ionization and ambipolar diffusion terms. Therefore, the electron number density balance w as mainly determined by these two terms. In Figure 5.3, the point where the excited state ionization curve crosses the ambipolar diffusion curve actually means that the electron number density balance is satisfied. Thus for certain ne and T g, the electron temperature can be determined. In Figure 5.3, the electron temperature that satisfies the electron conservation is determined to be around 9000 K. Figure 5.4 shows the solutions (N e-T e pairs) generated by varying E/N at different electron number densities. As discussed in Figure 5.3, the dominant production mechanism was the excited state ionization. Therefore, the solutions change dramatically if one neglects two-step ionization. This is critical in simplifying the 2-D fluid model since one had to include two-step ionization. Figure 5.5 shows the microwave power predicted from the C R M changing with the electron number density while the electron number density and energy balances were imposed. From this microwave power deposition curve, it is 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. clear that if 6 8 0 W is absorbed by the argon plasma, the electron number density has to be higher than 2e19m '3. For the argon plasma at the conditions as studied, such a high electron number density is not feasible for the microwave to penetrate the argon plasma, although it is desirable. From the 2-D model results discussed in Chapter 6, it will show high electron number density may be possible on the edge of discharge region. However, the microwave can only penetrate within the skin depth. Thus a uniform argon discharge with electron number density higher than 2e19m '3 is impossible. Such a high electron number density also exceeds the reported experimental measurements performed in the downstream region of microwave plasma by 1-2 orders of magnitude [M ak 1996]. Moreover, for the electron number higher than 1x1019 m'3, it is impossible to numerically match the measured excited state number densities within a factor of 10. In Chapter 6, the excited states will be studied to characterize the argon plasmas. Such a large electron number density (> 1e19 m*3) also conflicts with the continuum emission data. Since large uncertainty is associated with the continuum emission measurements, the continuum emission data was only used to determine the upper limit for electron number density. Figure 5.6 shows the calculated continuum emission curve with the experimental measurement data. The upper limit of electron number density determined from this plot is about 2 e 1 8 rn3, which corresponds to a much lower microwave power (< 100W ) as determined in Figure 5.5. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P = 5 Torr f = 250 seem 10-‘ n0 = 7e17 m T„ = 350 K % > <D • Non_Maxwsliian — Maxwellian 1 -1 0 10 ‘ 1-13 IQ- 0 5 10 20 15 e [eV] Figure 5.1 The comparison of non-Maxwellian and Maxwellian EE D F At Ne = 7el7 m'3, Tg =350 K , and P= 5 T o rr 100 P = 5 Torr f = 250 seem nfl = 7e17 m T„ = 350 K S o _l >. E> Inelasbc Elastic Diffusion and Convection ® c UJ 40 u UJ 0 10000 5000 15000 20000 Te [K] Figure 5.2 The electron energy loss terms w ith non-Maxwellian EE D F changing with the averaged electron temperature at Ne = 7el7 m"3, Tg = I S n K an ri P = 5 T n r r 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (0 a: 10 "I Argon Plasma 5 Torr 10 " 10 * ■* ■Ground State Ionization - Excited State Ionization ■Total Recombination - Ambipolar Diffusion nt = 1e18 m'3 T0= 350 K 1o’*! 10"! 10" — 6000 8000 — i— 1---- i— 12000 10000 14000 ----1 16000 T [K] Figure 5.3 Electron production and loss rates changing with the averaged electron temperature at ne of le!8 nT5 and T t of 350 K 30000 28000 P = 5 Torr Argon Flow 250 seem T = 350 K 26000 24000 22000 with 2-step ionization without 2-step ionization 20000 £ 18000 l - “ 16000 14000 12000 10000 8000 6000 1E18 1E17 1E19 ne [ m l Figure 5.4 The self-consistent solutions (Ne-Te pairs) with nonMaxwellian EEDF at T s =350K and P =5 To rr 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 250 seem P = 5 Torr T a = 350 K 10 ' 10' 10 n. [m3] Figure 5.5 Microwave power deposition with non-Maxwellian EEDF changing with the electron number density while electron number density and energy conservation are imoosed 250 seem Argon P = 5 Torr T9 = 350 K Upper Limit Determined from Experiment Total Continuum Emission ” 10 ' - 10* ■ T " I I 1 T I T "i I | 1E18 1E17 i r i t i p1 ‘i 111 1E19 ne[nr3] Figure 5.6 The experimentally measured and theoretically calculated total continuum emission changing with the electron number density 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.2 Experim ental Global Reactor Energy Balance A global reactor energy balance was performed to check the accuracy of the power meters. The schematic of the reactor with the cooling lines installed for the global energy balance study is shown in Figure 2.2. The temperatures of the cooling lines w ere measured by the K-type thermocouples installed specifically for this study. The cooiing w ater and air were pre-cooled (13.3°C for water and 14.0°C for air respectively), then flowed through the base-plate/applicator walls, chamber, substrate, probe, short, and belljar cooling lines separately. The inlet and outlet temperatures were measured for each cooling line. Thus the actual absorbed energy by the cooling lines can be obtained by calculating the enthalpy increase. Table 5.2 and 5.3 show the experimental measurements and the calculated enthalpy data for the argon plasma: Table 5.2 The Experimental Data for Microwave Argon Plasma Energy Balance Cooling Lines Inlet Temperature (°C) Flow Rate (LPM) Outlet Temperature (°C) m l (prob) 1.55 14.0 m2 (short) 2.88 14.0 m3 (base plate/applicator) 5.73 14.6 m4 (not used) N/A N/A m5 (MW generator) 5.77 N/A m6 (chamber) 1.02 14.4 m7 (substrate) 0.72 14.0 13.8 Belljar air cooling 10 scft3/min 14.0 21.5 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.3 Calculated Heat Transfer Rates for the Argon Plasma in Each Cooling Line with an Argon Flow Rate of 250 seem and an assumed gas exit temperature of 350 K Flow Rate (kg/s) Cp (J/kg K) A T(K ) P,bs(W ) Experimental Uncertainty (W) prob 0.0258 4182 0.2 21.6 21.6 short 0.048 4182 0.2 40.1 40.1 Base-plate/ Applicator 0.0955 4182 0.8 320 80 chamber 0.017 4182 0.6 42.7 14.2 Substrate 0.012 4182 0.2 10.0 10.0 Belljar air cooling 0.00548 (air) 1003.5 7.5 42.6 1.1 6.81 e-6 (argon) 520.3 77 0.1842 0.6 477 168 Cooling Lines Argon gas convection Total In Table 5.3, a total of 477±168 W microwave-energy can be accounted for in the cooling lines. The bulk of the microwave energy (320±80 W ) is dissipated into the base-plate/applicator cooling line. The energy carried away by the argon gas convection is negligible (0.2W ). Therefore, although the argon exit temperature was not measured in the experiments, the assumption on this temperature will not affect the final results significantly. The large uncertainty associated with the enthalpy data was mainly due to the poor resolution of the K-type thermocouples and the low inlet-outlet temperature differences. The uncertainty caused by the thermocouple reading 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. alone can be added up to 167.6W , about 25% of the metered microwave energy (680±40 W ). Comparing the total energy absorbed in the cooling-lines (477±168 W ) with the metered microwave energy, a 30% difference is calculated. That indicates the enthalpy data calculated from the cooling lines can match the metered microwave energy within a conservatively estim ated experimental uncertainty (about 30% ). M ak et al. [1996 and 1997] calculated the power absorbed at the reactor walls by probe measurement of electric fields in a similar microwave reactor. Their measurements were performed in a different operating regime (5 mTorr, 277 W input power) and in the absence of the substrate. They concluded that the wall losses amount to only 1% of the forward power and assumed that 99% of the input power is absorbed by the plasma. However, they did not measure the power absorbed by the base-plate and the short. Also, their Langmuir probe measurements of electron number density in the bulk plasm a are a factor of 4 lower than those predicted from a simple global reactor m odel. Since electron density and power follow almost a linear relationship, a factor of 4 lower electron density reflects a factor of 4 lower absorbed power. Thus, our analysis of their reported data suggests only about 65 W is absorbed by their plasma. Som e discussion is necessary on the validity o f assumptions and uncertainties involved in the numerical predictions and experim ental data. O ur pseudo-1-D plasma model with ambipolar diffusion correction assumes uniform bulk plasma and neglects the multidimensional effects. model of a 13.56 M H z argon glow discharge by A one-dimensional Lymberopoulos 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and Economou [1993 and 1998] showed that a greater fraction of power was dissipated into the sheaths near the plasma-surface interface. Therefore, a control-volume heat transfer model was constructed, which was independent of the numerical modeling. As discussed in the next section, it will demonstrate that it is impossible to remove 60 W microwave energy out of the control volume that encompasses the plasma discharge region. 5.3 Control Volume Heat Transfer Analysis Figure 5.7 shows a control volume encompassing the plasma, belljar, base-plate, and substrate-holder assembly. Microwave energy absorbed by the Microwave Power -r Q nui Qbj by belljar Cooling air Argon Plasma Base-plate Substrate Holder Y Qcj by substrate Figure 5.7 The schematic used for the control-volume heat transfer analysis 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. plasma represents energy flowing into the control volume. conduction through the substrate, gas convection, Plasm a radiation, and forced air-cooling comprise the energy loss terms for the control volume. Plasma radiation consists of line and continuum emission and was calculated based on the radiative rates in the CR M and amounts to approximately 0.65W , mainly due to the line radiation. The free convection heat transfer between the plasma and the substrate is calculated as follows: Q ,.,= h „A ,{T l - T , ) <5 ' 1> where hps represents the average heat transfer coefficient between the plasma and the substrate and is calculated as follows [Incropera and Dewitt, 1990]: _ NiipA = 0.664Rep2 Pr1' 3 A ps~ D ~ (5.2) D where A is the thermal conductivity of the argon plasma, and D is the diameter of the substrate. A value of 20 W /m 2 K was calculated for hps. The energy transfer between the plasma and the substrate is found be less than 6 W based on the gas and substrate temperatures of 350 K and 287 K, respectively. The gas temperature was assumed to be 350K since the measured belljar temperatures were close to room temperature (about 296K). The substrate temperature was actually measured by the reactor. If the gas temperature is assumed to be at 500K, the heat transfer rate is found to be 19.5W . This result is consistent with the global energy balance study that found only a small amount of energy (about 15W ) was dissipated into the substrate cooling line. The energy carried by the gases leaving the control volume is calculated 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as follows: Q.=«.Cn<-T. - V (5 '3) where mg and Cpg represent mass flow rate and specific heat of the gas, respectively. The inlet and outlet gas temperatures are represented by T g,i and T g, respectively. The gas transport heat loss was found to be negligible (about 0.2W ). The heat transfer between the belljar and cooling air is calculated as follows: a, where Abj, T bj, temperature, and and Ta mean (54) represent the belljar surface area, measured belljar air temperature, respectively. The heat transfer coefficient for the forced air cooling of belljar is represented by hb| (150 W / m2 K). It is also calculated according to Equation 5.2 since the gas flow is laminar, but with the thermal conductivity and Prandtl number of air. The belljar and cooling air mean temperatures were 294K and 2 9 1 K, respectively. If the belljar surface area above the substrate is used, the amount of heat dissipated in the cooling air was found to be 13.7 W . If the whole surface area of the belljar is used, and also accounting for the uncertainties associated with the temperature and flow measurements, a maximum of 35 W heat transfer rate is obtained. Based on an estimated contact area between the base-plate and the belljar and experimentally measured base-plate temperature (287K), the conduction through the base-plate was found to be insignificant. As shown in Figure 5.8, the base-plate barely touches the outside of the belljar while 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the applicator walls are not encompassed in the control-volume at all. To maintain vacuum conditions inside the belljar during operation, an O-ring is designed to be between the belljar and the base-plate as a sealing device. Since the O-ring is made of rubber, it should insulate the belljar and base-plate. The theoretical calculations in the control-volume heat transfer analysis are very straightforward. Recalling the results in the global energy balance study, the bulk of microwave energy (320W ) is absorbed by the base-plate/applicator w ater cooling line. If this amount of energy comes from the argon plasma, it has to go through the belljar and dissipate into the cooling air, and then dissipate into the applicator walls. Therefore, a total of 3 6 2 .6 W microwave energy has to be transferred through the belljar (320W into the applicator wall and 4 2 .6 W into the belljar air cooling line). For this heat transfer rate, a belljar tem perature of 370K is needed. That is 76K higher than the experimentally measured belljar temperature, far exceeding the claimed uncertainty of the infrared thermocouples (about 2% accuracy). Therefore, w e can conclude that the bulk of the microwave energy actually bypassed the argon plasma and dissipated into the water cooling lines. Table 5.4 is a review of the results obtained in this section. The maximum heat transfer rates are obtained by accounting for the uncertainties in the tem perature and flow measurements. Also considering the possible minimum value for the heat transfer rates, we conclude that only 10-60 W can be removed from the control volume through thermal processes. These results support the conclusion in the pseudo-1-D model. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.4 Heat Transfer Analysis Data for the Control Volume Encompassing the Argon Plasma (250 seem argon flow, 5 Torr) AT(K) Heat Transfer Rate (W) Maximum Heat Transfer Rate (W) / / 680±40 720 / / / 0.65 1.0 Convection (plasma-substrate) 20 4.56e-3 63 5.76 19.5 Convection (belljarcooling air) 150 3.04e-2 3 13.7 35 / / / 0.2 0.8 20.3 56.3 Heat Transfer Terms Heat Transfer Coefficient (W/m2 K) Area (m2) Metered Microwave Energy / Plasma Radiation Convective Gas Flow Total Loss 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4 2-D Argon Fluid Model Results The control-volume heat transfer analysis and the pseudo-1-D model results suggest that only 10-60W microwave power could be deposited in the argon plasmas under the studied conditions. T h e 2-D argon fluid model will provide more detailed information on the spatial temperature distributions, microwave power density in the discharge region, and heat fluxes along the boundaries. Except for the boundary conditions and the plasma chemistry, the only two parameters varied in the 2-D fluid model are pressure and microwave power. Since the metered microwave power cannot be directly used, a parametric study by varying the microwave power is helpful. The heat fluxes along the substrate surface and the belljar walls will be a good check with the experimental data obtained in the global energy balance. 5.4.1 Characteristics of Plasma Discharge by Varying Microwave Power It is clear in the above discussions that the microwave power is significant in characterizing the microwave argon plasmas. Therefore, a parametric study by varying the microwave power was conducted to see the characteristics of the plasm a discharges. Although w e have concluded that it was impossible to deposit 6 8 0 W into the argon plasmas at our running condition, one case with input microwave power at 6 8 0 W was also tried for comparison. T h e microwave powers used in this parametric study are 2W , 5W, 10W, 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30W , 50W , 10OW, and 680W . No self-consistent solution was available for both the 2W and 5 W cases due to the curve-fitted rates. Figures 5.8 and 5 .9 show the electron and gas temperature distributions. Four power levels (10W , 30W , 100W, and 680W ) w ere used for the simulation. For simplicity, only the distributions at the centerline w ere shown in the figures. However, the peak values of the plasma parameters may not be on the centerline at all. Thus, these figures can only show the general trends of the variation of these parameters changing with the microwave plasma power. The gas temperature distributions can be affected by the microwave power significantly, as shown in Figure 5.9. From the energy balance study, we stated that all the microwave power absorbed by the plasma has to be dissipated either through the belljar and substrate, or by radiation. To dissipate more power from the plasma, a higher gas temperature is expected. 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18000 * — 10W -•— 30W ^ -- 1 0 0 W ■r— 680W 16000 14000 12000 t- 10000 8000 6000 4000 20 30 40 60 50 70 z direction grid Figure 5.8 Electron temperature distributions at the centerline changing with the microwave power (grid 20 is at substrate) 1400 1200 — 10W —• — 30W 1100 —A— 100w 1300 —▼— 680W 1000 900 * 800 05 I- 700 600 500 400 300- 200 20 30 40 50 60 70 80 z direction grid Figure 5.9 Gas temperature distributions at the centerline changing with the microwave power 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is interesting to see the spatial distributions with microwave pow er of 680W . Figure 5 .10 shows the spatial distributions of the gas and electron temperatures, the microwave power density, and the electron number density. Although w e can see much higher microwave power density and gas temperature distributions, there is almost no change with the electron temperatures. This is consistent with our previous discussion. From Figure 5.10(d), it is clear that the lowest electron number density is about 1e19 m"3, while the highest electron number density surpasses 7e19 m"3. Previous investigation with the continuum emissions showed that the averaged electron number density could not go above 1e19 m*3. Therefore, the 6 8 0 W case is impossible for our discharge conditions. However, the continuum emission has a large uncertainty and it does not account for the non-Maxwellian effects. Thus it may not be able to rule out all the high-energy cases. Thus we should find another method to determine it quantitatively. Heat fluxes are obtained in the 2-D fluid model. Combining this information with the experimental global energy balance study and the control-volume heat transfer analysis, it offers a better way to determine the microwave power as discussed in the following section. 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ N 1.1E4 ~N)1.2E4 9.6E3 9.6E3 2E4-1E4- 'sV \ 10- .3E4 I 8.3E3 E o 1.1 E4 \ \ ■1,2E4\ N 4- 7E3 -1.3E4 2- -6 (b) -4 0 -2 2 4 6 r(cm) Figure 5.10 The spatial distributions for microwave argon plasma at 5 Torr and 680W: (a) gas temperature distribution (K), (b) the electron temperature distribution (K). 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -6 -4 -2 (c) 0 2 4 6 r(cm) 1.9E19 1.9E19 10H E19 E u_ nT 8- 1E19 1.9E19 1.9E19 :.8E19 I 1E19. S II1 ^ 4 .6 E 1 9 -6 (d) -4 0 ■2 2 4 6 r(cm) Figure 5.10 (cont.) (c) the absorbed microwave power density (W/m3), (d) the electron number density distribution (in'3) 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.4.2 Heat Fluxes in the 2-D Fluid Model The control-volume heat transfer analysis showed that about 10-60 W microwave power could be absorbed by the argon plasma. Moreover, the heat transfer rate from the plasma to substrate has already been calculated in the global energy balance study. The control-volume heat transfer model also provides detailed information on the heat transfer rates both from the belljar to cooling air and from plasma to substrate surface. Since the 2-D fluid model has detailed information about the temperature distributions, it is interesting to compare the heat flux calculated from the 2-D model with the results from the control-volume heat transfer model. Table 5.5 lists the heat transfer rates calculated from the 2-D model by integrating the heat fluxes along the belljar walls or substrate surface, from the control-volume heat transfer analysis, and from the global energy balance study. It should be noted that although the heat flux along the top of the belljar was not calculated, the heat transfer rates along the belljar walls and substrate surface have almost reached the specified microwave power for 10W , 30W , 5 0 W and even 10 0 W case. For the 6 8 0 W case, more heat is dissipated through the top of the belljar. For the 10W case, the total heat along the belljar walls and substrate surface (10.6W ) has exceeded the specified 10W. This error could be introduced during the grid interpolation and the iteration between the EM mode! and fluid model. Combining the results from both the global energy balance study and control-volume heat transfer analysis, the total heat along the belljar walls 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. should be lower than 35W , and the total heat along the substrate surface should be lower than 19.5W . As shown in Table 5.5, the cases with microwave power higher than or equal to 5 0 W cannot m eet these criteria. Then it is concluded that the possible microwave pow er absorbed in the plasma should be lower than 50W . For better determination of the microwave power, the uncertainty of the control-volume heat transfer analysis and the global energy balance study should be reduced. Since most of the uncertainty comes from the temperature measurements, better instrumentation of the temperature measurements is needed. Table 5.5 The Total Heat Predicted from 2-D Model Comparing with the Global Energy Balance Study and Control-Volume Analysis Results Total Heat along Belljar Walls (W) Total Heat along Substrate Surface (W) 10 W 9.72 0.87 30 W 23.2 1.91 50 W 39 5.36 100 W 63.7 18.3 680 W 337 35.6 Nominal 13.7 5.76 Maximum 35 19.5 Nominal / 10 Maximum / 20 Items Microwave Power Specified in the 2-D Model Heat Transfer Rate from Control-Volume Analysis Heat Transfer Rate from Global Energy Balance 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5.11 shows the plot of the total heat changing with the microwave power, which is the same result as in Table 5.5. Figure 5.12 and 5.13 show the heat fluxes along the belljar walls and the substrate surface changing with the microwave power, which are calculated from the 2-D model results. T h e total heat transfer rates are obtained by integrating the heat fluxes with the corresponding surface area. In Figure 5.15, the heat fluxes increase with the microwave power due to the increase in the gas temperature. However, Figure 5.16 shows no significant change between the heat fluxes of the 1 0 0 W and 680 W cases. This is caused by the high non-uniformity of the plasma discharge, as shown in Figure 5.9, 5.10 and 5.13) 1000 • l = j Total Heat along Belljar Walls llllllll Total Heat along Substrate Surface 100 - (0 0) X 0067 10W 30W 50W 100W 680W Microwave Power Figure 5.11 A vertical stacked column graph that shows the total heat along the belljar walls and the substrate surface changing with the microwave power 112 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. (0 ra a3 m <u B "g § I (0 X _2 - A Li • — 30W — 5 0W —T— 1oow ra • — 680W 0) X 0.12 Figure 5.12 Heat fluxes along the belljar walls changing with the microwave power (5 Torr and 250 seem) 03 0 ■C 10s i 3 CO 10W — a — 50W 03 — 0— 680W 1« — • — 30W —T — 100W 10*1 C=O ~E | I 101 X 3 ra <D X 102 1 — 0.00 I— 0.02 0.01 0.03 0.04 0.05 r (m) Figure 5.13 Heat fluxes along the substrate surface changing with the microwave power (5 Torr and 250 seem) 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER SIX EXCITED STATES OF THE AR G O N PLASM AS Optical emission spectroscopy provides an inexpensive and non-intrusive diagnostic method to probe the plasma parameters, especially for the microwave plasmas we studied where the Langmuir probe cannot be used in the reactor cavity. Excited state number densities can be calculated directly from the O ES data. Then how sensitive the measured excited state number densities are to the plasma parameters, such as electron number densities and temperature, becomes an interesting topic. W e used the OES measurements in tandem with the pseudo-1-D model to investigate the microwave argon plasmas [Kelkar 1999]. However, the model predictions could not match the experimental results within the experimental uncertainties. As discussed in Chapter 5, w e also could not match the predicted microwave power from pseudo-1-D model with the metered power (680W ). These two topics motivated us to have further investigation of microwave argon plasmas both experimentally and numerically. The absorption measurement was conducted to obtain 4s state number density, which is the metastable state of the argon plasma. These data will be helpful in characterizing the argon plasma. Both the 2-D fluid model and 2-D C R M were constructed to model the 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. plasma more accurately. The 2-D results show high non-uniformity of the argon plasma discharges, in contrast to the uniform assumption implicit in the pseudo1 -D model. 6.1 The Experimental OES Data For the microwave argon plasma with param eters of 6 8 0 W input power (8 0 0 W forward power and 120W reflected power), 5 Torr pressure and 250 seem argon flow, the measured excited state number densities from O ES are listed in Table 6.1. The lines used in the O ES measurements are listed in Appendix A. O ur CRM results indicate that some of the 4p-4s and 5p-4s transitions are selfabsorbed. Therefore, the experiments were m ade by choosing optically thin transitions. Table 6.1 The Excited State Number Densities Calculated from the OES Data Comparing with the Numerical Predictions At Ne = 5el7 nT3, Tg = 350K Numerical Prediction with Changed Coefficients (n/gj) Experimental Uncertainty Level Data (n/gj) A4p-4s = Alp-41 “ 3.34x107 s'1 1.0x10® s‘1 QjjM* X 10 4p 2.29e13 40% 1.1 x1013 2.48 x1013 2.22 x1013 Sp 6.97e11 56% 3.15x10” 5.53 X1011 5.74x1011 Sd 1.60e11 43% 9.67 x101° 1.24 x1011 1.22x10” 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 6.1 excited state also lists the experimental uncertainty associated with the number densities. The uncertainties are mainly due to the calibration factor and the transition coefficients. The 5p state has the highest experimental uncertainty (56% ) because of its wavelength. In T ab le 6.1, the numerical predictions with changed coefficients are listed, which are the results of trying to match the experimental data within the experimental uncertainties. Kimura et al. [1985] claimed that some of the transition probabilities had a large uncertainty of one magnitude. Thus some of coefficients w ere changed to check the sensitivity of the excited state number densities to these coefficients. For detailed information, one should refer to the discussion in Chapter 6 of [Kelkar, 1999]. 6.2 A bsorption Measurement o f the Metastable State The motivation for the absorption measurements came from the Boltzmann plots for the predictions from our pseudo-1-D model to match the measured excited state number densities from O ES measurements. Figure 6.1 was generated to show that the measured excited state number densities could be matched by the model predictions only with changed rates. The experimental data points w ere drawn with the experimental uncertainties. However, it was observed that the 4s state (metastable state) number density varied with different electron number density and temperature pairs, which cam e from the pseudo-1-D model. The measured 4s state number density 116 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. will then provide another method to verify our models. 250 seem Argon P = 5 Torr T „ = 350 K ■ 0 n8=1x1017 n r3 ; T b=1 6855 K V n9=1x1018 m-3 ; T 9=9160 K n9=1x1019 m"3 ; T 9=8118 K n9=1x1018 m"3 with changed rates Experimental 5 10 15 Energy [eV] Figure 6.1 The Boltzmann plots showing that the experimental data can only be matched by the model predictions with the changed rates W e m easured the 4s population through absorption measurement at 7635 A. A typical optical emission scan of the argon plasma running at 5 Torr and 6 8 0 W condition can be found in the Appendix A. The line we used for absorption measurement, 7635 A, actually is the strongest line. The calculation of the 4s state number density needs the lineshape factor. The lineshape can be calculated from the line-broadening theories as described in Chapter 3. It turned out that the Doppler broadening was the dominant mechanism [Li, 1999], This actually eased the necessity for us to get a measured lineshape that requires a tunable laser system. The calculated argon linewidths 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (F W H M ) are listed in Table 6.2. The data in this table w ere generated at the following conditions: electron number density, N e = 1 e18 rrf3, T e= 10 ,00 0 K. Table 6.2 The calculated argon linewidths (F W H M ) with different gas temperatures at N e=lel8 m'3, Te=l0,000K M ech an ism S ym b ol Linew idth (FW H M , unit in Hz) Tg = 350k 450k Center Frequency Natural Broadening V a n der W aals Vo 3.93E + 1 4 Avn 5.33E + 0 6 Resonance Avr 0.00E +0 0 Avs 4 .95E + 0 5 Stark Broadening Doppler Broadening A vc A vq 2.87E +05 8.29E+08 2.41 E +05 9.40E +08 550k 2.10E +05 1.04E +09 To investigate the effects of gas temperature, the Voigt param eter “a”, defined as (ln 2 )1/2AvH/Avo, was used. For N e=1e18 nV3, and Te=10,000K , Figure 6 .2 is generated by the theoretical calculations. The Voigt param eter "a" actually indicates the significance of either the homogeneous or inhomogeneous broadening mechanisms. Figure 6.2 shows that the Voigt param eter a « 1 with the gas tem perature changing from 3 5 0 K to 550K . As the gas temperature increases, the Voigt param eter “a” decreases. Therefore, the Doppler broadening mechanism is dominant for the argon plasm as at the conditions w e studied. 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 . 506-03 6006-03 5. 506-03 4506-03 4. 006-03 ' 300 350 400 450 500 550 600 Gas Temperature (K) Figure 6.2 The Voigt parameter “a” as a function of gas temperature at an electron number density of le !8 m'3, and an electron temperature of 10000K For a Doppler broadening dominant lineshape, the lineshape function can be expressed as: 7(v) = ( 41n2 V nW2 1 -exp -41n2 (6-1) / With this lineshape, the 4s-state number density was calculated from the experimental results, which is 6.50e1 6 nrf3. The experimental uncertainty for the 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4s state measurement is about 42% , which was estimated conservatively from the uncertainties in the absorption measurements and the uncertainty associated with the transition line. Figure 6.3 is the Boltzmann plot of the numerical predictions with the experimentally measured 4s state number density and the OES data. The 4s state is drawn with an error bar for its 42% uncertainty. It shows that although the numerical prediction with the changed coefficients can match the experimental OES data, it cannot match the measured 4s state number density within the experimental uncertainty. Therefore, more modeling work is needed in this study. n#=1e17 m'3, Te=16855K n#=1e18 m’3, T e=9160K — na= 1e19m '3, T #=8118K — ne=1e18 m‘3 with changed rates • Experimental OES/absorption Data s(eV) Figure 6.3 The Boltzmann plot of the numerical predictions with the experimentally measured 4s state number density and the OES data 120 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 6.3 Characterization of the Argon Plasma with the 2-D Fluid Model From the energy balance study, it was concluded that not all the microwave power could be absorbed by the argon plasmas at our studied conditions. To investigate the role of excited states in the argon plasmas, the study with the base set in the 2-D fluid model will be discussed first. The microwave power and plasma pressure were varied to study the sensitivity of the excited states to these two parameters. Then the 2 -D CR M was used to generate the detailed excited state number densities. Since the higher microwave power cases (> 50W ) have been ruled out in the energy balance study, the experimental results were compared to the 2 -D predictions only from the lower power cases (10W , 30E, and 50W ). Good agreem ent was found between the experimental and numerical results. The 2-D results show high non-uniformity with the argon plasmas at the studied conditions. And the excited state number densities can be used as the indicator of the non-uniformity. 6.3.1 A rgo n P lasm a a t 5 T o rr and 10W /30W After much study with the 2-D model, the base sets we chose are at the * pressure of 5 Torr and the microwave power of 10W or 3 0 W at a frequency of 2.45 GHz. 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e results of the simulation data for the plasma discharge at 5 T o rr and 10 W are shown in Figure 6.4, which indicates high non-uniformity in the plasma discharge region. Figure 6.4 (a) shows the gas temperature distribution in the simulation region. The shadowed area represents the substrate holder. The quartz belljar forms most of the walls. The dimensions for the full plasma simulation region are from -7cm to 7cm and from Ocm to 12 cm, in the r and z directions, respectively. The spatial distribution shows a peak value of the gas temperature of 472 K at the centerline about 4 cm above the substrate. Figure 6.4 (b) shows the spatial distribution of the electron temperature. Not like the gas temperature distribution, the electron temperature has peak values at different places. This is caused by the non-uniformity of the microwave power density that in turn is affected by the electric field. Figure 6.4 (c) and (d) are the distributions of the mol fractions of the 4s excited state number density and the electron/ion number density. In the pure argon plasma discharge, the electron and ion number densities are the sam e if we neglect the second ionization from the Ar+ ions. It is clear that the excited state num ber density and the electron number density are highly non-uniform. Figure 6 .4 (e) and (f) show the spatial distributions of the microwave power density and the electric field. The microwave power density distribution varies a little from the electrical field strength distribution. This could be caused by the fact that both the electric field strength and the electron flux affect the actual pow er deposition. Due to the high non-uniformity of the 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. electron distribution, the electron flux should be expected to be highly non-uniform spatially. Figure 6.4 (g) is a gray scale map for the electric field strength. In this figure, it is much easier to see the non-uniformity of the field strength. Figure 6.5 shows the distributions for the 3 0 W case. Figure 6.5 (a) and (b) show the gas temperature and electron temperature distributions. T h e gas temperature in the 30W case is higher than in the 10 W case, which is caused by the increase in the microwave power density. However, the electron temperature is lower than in the 10W case. To understand this phenomenon, one can refer back to Figure 5.4, where the electron number densities and temperatures were in pairs for a self-consistent solution. Actually at higher electron number densities, lower electron temperature is needed to sustain the plasma. From Figure 6.5 (d), it can be shown that in the 3 0 W case, the electrons/ions number densities are much higher than in the 10W case. The highest electron mol fraction in the 3 0 W case is about 4.3e-5, while in the 10W case, the highest electron mol fraction is about 4.0e-6. Although the total particle number density should be lower in the 3 0 W case due to higher gas temperature, there still is a difference about one magnitude between the 10 W and 3 0 W cases, which is significant for an increase in the microwave power with only a factor o f 3. As shown in Equation 4.15, the absorbed microwave power should have roughly a linear relation with the electron number density. Therefore, it is concluded that the non-uniformity is even worse in the 30 W case than in the 10 W case. 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, the 4s number densities in the 10W and 30W cases do not have as large difference. Comparing Figure 6 .4 (c) and 6.5 (c), roughly a factor of 3 increase in the p eak values of the 4s state number density is observed. This is caused by lower electron temperatures in the 30W case. As discussed in Chapter 5, higher microwave power will generate higher electron num ber density. However, lower electron temperature will go with the higher electron number densities, then reduce the sensitivity of the excited state number densities to the plasma parameters. Figure 6 .5 (e) and (f) show the absorbed microwave power density and the electric field distributions. Unlike the 10W case, here the higher microwave power densities and the higher electric fields are much closer to either the belljar walls or the substrate surface. Therefore, the non-uniformity is even worse in this case. As shown in Figure 6.5 (e), the electric field is higher close to either the belljar walls or the substrate surface. Since the surfaces are acting as catalysts to the recombination or de-excitation reactions, it is possible that the plasma absorbs more energy in the regions closer to the walls and then maintains the electron number density balance in these regions by recombination at the surfaces. Figure 6 .5 (g) is a wire frame plot of the electric field inside the EM simulation region. This plot shows that the microwave is unable to penetrate the whole plasma discharge. High electric field exists along the belljar walls and substrate surface, while the bulk of the plasma discharge has relatively low electric field. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 6 - 4 -2 0 2 4 6 4 6 r(cm) '1.1 E4 5E4 8- .3E4 _ \ 9.7E3 1.1E4' ■1.3E4 E «N* 4- 9.7E3' 2- -6 -4 0 •2 2 r(cm) Figure 6.4 Spatial distributions of the argon plasma discharge characteristics at 5 Torr, 10W: (a) Gas Temperature (K ), (b) Electron Temperature (K). (Note: the shadowed area is the substrate holder.) 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. s 2.5E4E-6 10- 4E-6 E N o_ IE-6 ;.5E-7 I.3E-7- 4- 2 - -6 -4 0 •2 2 4 6 r(cm) Figure 6.4 (cont.) (c) the mol fraction of 4s state number density, (d) the mol fraction of electron/ion number density 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12- 10- 8 - E* 6 * 'Zt 4- 2- 0 -6 -4 -2 0 2 4 6 2 4 6 r(cm) 12 10 8 "g 6 4 2 0 -6 -4 -2 0 r(cm) Figure 6.4 (cont.) (e) the absorbed microwave power density (W /m3), (f) the electrical field strength (V/m) 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -6 -4 -2 0 2 4 6 r(cm) Figure 6.4 (cont.) (g) the gray scale map o f the electric field strength (V/m ) 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -6 -4 -2 0 2 4 6 r(cm) Figure 6.5 Spatial distributions of the argon plasma discharge characteristics at 5 Torr, 30W: (a) Gas Temperature (K), (b) Electron Temperature (K). (Note: the shadowed area is the substrate holder.) 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -6 -4 -2 0 2 4 6 r(cm) r(cm) Figure 6.5 (cont.) (c) the moi fraction of 4s state number density, (d) the mol fraction of electron/ion number density 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.4E5 5.2E4 5.2E4 7 2E3 7.2E3 12- •1,6E: 10- 8 - 6.3E2 Figure 6.5 (cont.) (e) the absorbed microwave power density (W/m3), (0 the electrical field strength (V/m) 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. probe position 1.4x10 belliar walls o 1 .2 x 1 0 © substrate holder ® 1 .0 x 1 0 s position o 8 .0 x10 * •c £ 6 .0 x 1 0 * 4 .0 x 1 0 2 .0 x 1 0 (g) in ner m etal chuck position Figure 6.5 (cont.) (g) the wire frame plot of the distribution for the electric field inside the EM simulation region (unit in W/m) 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From the above discussion, excited states are not very sensitive to plasma parameters due to the fact that when the electron number density increases, the electron temperature decreases in a self-consistent solution. To explain this phenomenon, the electron conservation equation should be revisited. The electron conservation equations in the pseudo-1-D and 2-D models should be sufficient for the purpose here. For simplicity, Equation 4.8 in the pseudo-1-D model is used. This equation is copied here for convenience: (6 .2 ) Also as shown in Figure 5.3, the dominant electron production and loss terms are excited state ionization and ambipolar diffusion loss, which correspond to the first and fourth terms in the right-hand-side of Equation 6.2, respectively. The electron production term actually is a function of electron number density and temperature since the reaction rates are mainly functions of the electron temperature as shown in Equation 4.65, which is also copied here for convenience: K re a c tio n = wc(l\ *w .0 - e~Ne/Ne" yJ Teb (6.3) The parameter “a” reflects the EEDF’s deviation from Maxwellian. For higher electron number densities, the EEDF will be Maxwellian, and “a" will approach zero. Therefore, the reaction rate is a function of electron temperature 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. at higher electron number density. Thus at higher electron number densities, Equation 6.2 can be simplified to the following if only the dominant production and loss terms are included and the steady state solution is considered: For a pure argon plasma, electron and ion number densities are the same. Then further simplification of Equation 6 .4 will lead to an equation with only one unknown variable, that is, the electron temperature. Therefore, for high electron number densities, the electron temperature is actually a constant determined by the pressure, cavity geometry and ambipolar diffusion rate. It is no longer affected by the electron number density. However, for low electron number densities, the non-Maxwellian effect will reduce the reaction rate. Thus a higher electron temperature is needed to maintain electron conservation. 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3.2 Effects of the Microwave Power In Chapter 5, the effects of the microwave power have been discussed related to the energy balance. In this section, the effects of microwave power on the electron and excited state number densities are discussed. Figure 6.6 and 6.7 show the 4s state and electron mol fractions changing with the microwave power. Figure 6.7 shows that the electron number density can be affected by the microwave power significantly. From the review of the plasma processes, the electrons play a very important role here. Almost all the microwave power is absorbed by electrons first, then the energy is transferred to other particles in the plasma discharges through electron-heavy particle reactions. In Figure 5.11, it is observed that the electron temperature distributions did not change significantly with different power levels, which is consistent with our pseudo-1-D model solutions. Figure 6 .6 shows that the 4s state number density will change with microwave power. However, the change is not so significant. This leads to a fact that the excited state number densities are not very sensitive to the microwave power. 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1E-4 as ra 55 T3 a> o iS <0 IE-5 1E-6 *♦— O — ■— — *— — a— — c o o as Li as o 10W 301/7 100W 680W 5 1E-9 20 30 40 50 60 70 z direction grid Figure 6.6 Mol fraction of 4s state at the centerline changing with the power w c o w o as lii o c o — — 10W 30W — a — 100W 1E-7-; — ▼— 680W IE-8 20 30 40 50 60 70 z direction grid Figure 6.7 Mol fraction of electron number density at the centerline changing with the power 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3.3 Effects of the Plasma Pressure It was very hard to get a stable argon discharge experimentally. Our W A V E M A T reactor was designed for diamond CVD processes. Therefore, it was optimized for the hydrogen plasmas at pressure around 4 0 Torn However, at such pressures, the argon discharge was not stable. Thus the pressure was reduced to 5 Torn. Argon discharges at lower pressure (at the magnitude of mTorr) were studied previously. However, most of the reactors were parallel plates or other simpler structures. Three pressure conditions (3, 5, and 8 Torr) were simulated for the discharge with microwave power of 10W . Figures 6.8 - 6.11 compare the distributions of the gas, electron temperatures, and the electron, 4s state number densities. No significant changes can be observed in these figures. Comparing the results of 3 Torr and 8 Torr, the pressure is increased roughly three times. However, the peak values of gas temperature, electron temperature, electron number density and 4s state only changes about 30K (<10% ), 1000K (<10% ), 6e17 rrf3 (within the same order), and 5e17 m*3 (within the same order), respectively. Thus the plasma parameters are not very sensitive to pressure over the range investigated. 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — ■— 3 Torr XL 480 <u c TT ffl C a> a 460- ® 400- <U (U W 3 W 0) 380- E a) 320 5 Torr — a — 8 Torr 440 420- 360 340- Q. I</> (0 O 300 280 ~r~ 20 T - ~r~ —r~ —r~ —r~ 30 40 50 60 70 -l 80 z direction grid (20 at the substrate) Figure 6.8 The comparison of gas temperature distributions at the centerline with different plasma pressures 18000 17000- < D u. 3 2 <5 Q. E <D H C o L. o 0) UJ — ■— 3 To rr m• 16000150001400013000- — • — 5 Torr — *■— 8 To rr 1 2 0 00 - A I 1 1 0 00 10000- ' i 9000 8000- A J 7000- A £ 6000 5000 4000 3000 • A A —r~ 20 - 1- l 30 40 I50 - T" - r- 60 70 ~1 80 z direction grid Figure 6.9 The comparison of electron temperature distributions at the centerline with different plasma pressures 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0) •*r E 3 II J3 Z C 10 " 3 Torr — • — 5 Torr - — a — 8 Torr s o _© UJ 10 i T" 20 30 — r~ 40 —T~ ~r~ 50 60 T “ “I 80 70 z direction grid (20 at the substrate) Figure 6.10 The comparison of the electron number density distributions at the centerline with different plasma pressures 10 '* 1017 '£ 10 " </> c <D Q k. 0) E 3 14 10 — ■— 3 Torr Z — • — 5 Torr © — a — 8 Torr 13 co in rr 10 " 20 30 40 50 60 70 80 z direction grid Figure 6.11 The comparison of 4s state number density distributions at the centerline with different plasma pressures 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4 Characterization of the Argon Plasma with the 2-D CRM The 2-D fluid model provided the spatial distributions of gas and electron temperatures, as well as other parameters. The 2 -D CR M will then use these parameters to calculate all the excited state num ber densities, which will enable us to compare them with the experimental data. Since w e follow the similar plasma chemistry for argon discharges in both the 2-D fluid model and the 2-D CRM, similar results of the 4s-state number densities, microwave power densities, and other parameters are expected. 6.4.1 Predicted Excited-State Number Densities Distribution Figures 6 . 1 2 - 6 . 1 5 show the spatial distributions of the 4s, 4p, 5p, and 5d state number densities at a pressure of 5 Torr and the microwave power of 10W . The high non-uniformity is clearly shown in these figures. The spatial distributions of the excited state number densities follow the sam e distribution profile, which follows the profile of the electron number density distribution. Due to the fact that there is not a large change with the electron temperature in the discharge region, the electron number density affects the excite-state number densities more significantly. For the microwave power level at 30W , similar results were obtained. It is of interest to compare the experimental results with the predictions from the models, recalling that w e could not match all the three excited states (4p, 5p, and 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5d) simultaneously without changing any rates in the pseudo-1 -D model. One assumption of our pseudo-1-D model is the uniformity of the discharge. However, as shown in Figures 6.12-6.15, this assumption can no longer be held after the study of 2-D modeling. To compare the numerical data with the experimental data, the predicted excited state number densities must be reduced to the O ES signal along the probing path. If axi-symmetric distribution is assumed, the general expression for the averaged optical emission signal along the r-direction is: I,iavs= 2 \J l , Iij .rr d r / D = — — All// J\ n i . r dr I D y.tfvs ^ (6 -5 ) where Xy is the wavelength of the used transition line, D is the diameter of the belljar, r is the radius from the centerline of the belljar, ni,r is the excited state number density at radius r, Aj is the transition probability. For the absorption measurements, the Beer-Lambert relationship (Equation 3.7 and 3.8) also relate the spatially distributed excited state number density to the measured transmission of a probe beam of light. For the uniform mesh spacing, the above expression can be simplified further. Although the non-uniformity along the radius direction cannot be shown after this averaging procedure, it still can be clearly shown by the non-uniform distribution along the z-axis. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 .4 x10 1.2x10 £ • 1.0x10 ° o 8 .0 x 1 0 " 6 .0 x 1 0 m 4 .0 x 1 0 2.0 x10 0 -7 Figure 6.12 The spatial distribution of the 4s state number density at 5 Torr and 10W 5x 10 £ 4X10 X "tn Q 3x10*' s a E 3 2 © 2x 10 1x10 Figure 6.13 The spatial distribution of the 4p state number density at 5 Torr and 10W 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 .5x10 0 -7 Figure 6.14 The spatial distribution of the 5p state number density at 5 T o rr and 10W 1.4x10 1.2x10 £ 10 c ffl Q o •O 1.0x10 8.0 x10 E a Z 6 .0 x 1 0 3 a in 4.0x10 2.0x10 0 -7 Figure 6.15 The spatial distribution of 5d state number density at 5 T o rr and 10W 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. During the study of the 2-D CRM, the plasma pressure is set at 5 Torr while the microwave power is changed from 10W to 50W . As a result of the previous sections, microwave powers higher than 100W are excluded. The 5 0 W case was added later on trying to get a prefect match between the experimental and numerical results, which turned out to be not successful. Figures 6.16-6.19 show the excited-state number density distributions along the z-direction compared with the experimental results. The experimental uncertainty was shown in the figures for each excited state. It is interesting to observe that the 50W case does not increase the excited-state number densities at all. Therefore, excited-state number densities have only a w eak dependence on the microwave power. Figure 6 .1 7 shows that none of the cases studied provides a good match between the numerical and experimental uncertainty (40% ). experimental results for 4p-state within the However, all other figures show that the 3 0 W case can match the numerically predicted 4s, 5p and 5d state number densities with the experimental ones within the experimental uncertainty. Even for the 4p state, the error between the numerical and experimental results is about 64% , which is much better than the results we obtained from the pseudo-1-D model. For the larger error associated with 4p state, a lot of factors may play their roles here. First of all, the top of the rector (short) actually is adjustable for better tuning during the experiments. A few millimeters adjustment was always 144 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. reasonable to obtain a stable plasma discharge. Second, the excited state number density distribution is significantly non-uniform. The values on the grids next to the position of the probe beam may introduce an error larger than 15% . The simulation region for the reactor was based on the design values. Thus a small deviation of the actual short position may introduce much larger error than expected. The uncertainty caused by the model itself should also be considered. In the next two sections, the microwave power density and the curve-fitted rates will be studied to estimate the uncertainty caused by these parameters. It needs to be pointed out that although the excited-state number densities are not good indicators for the plasma parameters (power, electron number density, electron temperature, etc.), they do provide valuable information about the non-uniformity of the discharge. With the scale-up of the plasma reactors, the uniformity becomes a critical problem for the designer. Therefore, the optical emission spectroscopic data may be an easy way to monitor the uniformity inside the plasma discharge region. In summary, the experimental and numerical data are listed in Table 6.3 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 6.3 The experimental results of the excited state number densities comparing with the 2-D C R M predictions for an argon plasma at 5 Torr, 250 seem (Best matched case at microwave power of 30W) Best Matched 2-D CRM Data at 30W Experimental Data Excited States nj/gj (m-3) nj (m*3) Uncertainty nj (mJ ) Uncertainty 4s 5.42e15 6.5e16 42% 8.09e16 +24.5% 4p 2.29e13 8.24e14 40% 2.99e14 -63.7% 5p 6.97e11 2.51e13 56% 1.31e13 -47.8% 5d 1.600e11 9.60e12 43% 6.59e12 -31.4% 146 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 3.0x10" - 2.5 x 1 0 "- U) (D Q 2.0x10 17- xi 1.5x10"- c experimental E 3 Z ffl re O ) ■ in 1.0X10” - 1 5.0 x 1 0 "0.0 - I 10 20 30 —I— T - I 40 50 60 T ” 70 -1 80 z-Direction Grid (0 at the bottom) Figure 6.16 The averaged 4s-state number density distribution along the z-direction from the numerical predictions comparing with the results from the absorption measurement 1.4x10'* ” 1.2x10'* ” % >» expenmental 1 .0 x 1 0 "- c 0) Q l. 0) 8 .0 x10'*- X) 6.0x1014 E 3 z 4 .0 x10'*0) 2 0) i Q. 2 .0 x10'*0 .0 -2.0x10" i - I- I - i— 10 20 30 40 i— |— 50 i— |— i— 60 r - 70 I 80 z-Direction Grid (0 at the bottom) Figure 6.17 The averaged 4p-state number density distribution along zdirection from the numerical predictions comparing with the experimental results 147 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10W 8x10 30W 7x10' — — 50W >» 6x10' experimental a) 5x10’ <D 4x10 5 3x1013" 2x10 A 1x10 m -1x10 0 30 20 10 40 50 60 80 70 z-Direction Grid (0 at the bottom) Figure 6.18 The averaged 5p-state number density distribution along z-direction from the numerical predictions comparing with the experimental results 'E >. '55 c a> Q w (U XI e b 3 Z © ra 4.5x10 13- 10W 4.0x1 O'3- 30W 50W 3.5x1 O'3- - — experimental 3.0x10132.5x10132.0x10,J - /VV 1.5x10’31.0x10 11" 35 5.0x10tJ- •6 in 0 .0 -5.0x101J- - 1- ~I- l I ~ r- i I - 1 10 20 30 40 50 60 70 80 z-Direction Grid (0 at the bottom) Figure 6.19 The averaged 5d-state number density distribution along zdirection from the numerical predictions comparing with the experimental results 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4.2 Effects of the Microwave Power Density Microwave power density (M W PD ) distribution is one of the most important parameters used in our 2-D fluid model. The total microwave power is specified in the EM model, then the M W P D distribution is calculated and serves as the input to the 2-D fluid model for the next iteration. During the iterations, the M W PD distributions may introduce some error due to the interpolation. The highly non-uniform argon plasma discharge also introduces more uncertainty with M W PD. T h e 2-D CRM generates the M W P D results by calculating the electron energy balance. Therefore, comparing the M W PD distributions used in the fluid model and the CRM will be a quantitative method to evaluate the uncertainty with these models, which could attribute to the uncertainties of microwave power and the excited state number densities. Figure 6 .2 0 shows the comparison of two spatial distributions of the microwave power densities at 5 Torr and 10W; one was used for the 2-D fluid model and the other was predicted from the 2-D CRM . To show the difference graphically, the 2-D fluid model results are shown in the right half of the base plane, with the 2-D CRM results shown in the left half. T h e two distributions have similar profiles as shown in Figure 6.20. However, the predicted peak value of the M W P D in the 2-D CRM (about 1.2e5 W /m 3) is higher than the one used for 2 D fluid model (about 6e4 W /m 3). Therefore, in terms of the peak values, roughly a 149 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 50% uncertainty is introduced with the M W P D distributions. In our 2-D CRM , the electron tem perature distribution was taken from the 2-D fluid model. Then the E/N value was iterated inside the C R M to get the predicted averaged electron temperature converged to the value that is provided by the 2-D fluid model. The microwave power absorbed by the plasma was calculated from the electron energy balance inside the 2-D C R M . Since we followed much smoother profiles of electron and gas temperature distributions, the predicted absorbed M W PD was also much smoother than the one generated from the 2-D electromagnetic model. Figure 6.21 compares the M W PD distributions used in the 2-D fluid model to those predicted from the 2-D CR M at 5 Torr and 30W . In this figure, the difference between these two distributions is much smaller. Comparing the peak values, only an 18% uncertainty is calculated for this case. Figure 6 .2 2 and 6.23 are the contour plots for the same distributions in Figure 6.20 and 6.21, respectively. These plots provide a quantitative method to compare the two profiles. Also the discrepancy between the two distributions is shown more clearly in these two figures. The large uncertainty with the M W P D distribution in the 10W case could be reduced if an iteration scheme between the 2-D CR M and 2-D fluid model is employed. However, for the 3 0 W case, the 18% uncertainty is acceptable. 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.4x10* ■" 1.2x10* ' M W P D Used in the 2-D Fluid Model 1.0x10* " E 8.0x104 ‘ 6.0x104 ‘ 4.0x104 2.0x104 0 ft MW PD Predicted from 2-D CRM Figure 6.20 The wire frame plot of the comparison between M W PD profiles used in 2-D fluid model and predicted from 2-D C R M at 5 Torr and 10W 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6.21 The wire frame plot of the comparison between the M W PD used in 2-D fluid model and predicted from 2-D C R M 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ,4E' 4- MWPD used in Fluid Model 2 1.9E4 - 2 E 2. E o - 2- MVyPD P re d ic te d from 2 - D C R M I.7E4 1.9E4 -4 - -6- z(cm) Figure 6.22 The comparison of the spatial distributions of M W PD used in the fluid model and predicted from the 2-D C RM at 5 Torr and 10W 1E4 MWPD Predicted I from 2-D CRM ,6E! 1E4 \ _L5T864rJ._. E u - 22E5 -4 - .6E5 hi u 1 — -------- ^1.1E5 -6 - 8E4' 12 MWPD Used in Fluid Model S.8E4 5.8E4. 10 1.1E5 m 5.8E4 5.8E4^—^ 6 8 4 2 0 z(cm) Figure 6.23 The comparison of spatial distributions of M W P D used in fluid model and predicted from the 2-D C R M at 5 To rr and 30W 15 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4.3 Effects of the Reaction Rates As discussed in the modeling part, the reaction rates w ere curve-fitted to avoid the troubles in the calculation time and programming techniques. Large errors between the curve-fitted rates and the exact ones w ere observed, especially when the electron number densities and the electron temperatures were both low. Therefore, it would be better if this can be avoided in the actual cases. Figure 6.24 - 6.27 show the comparisons between the rates calculated frum the 2-D CRM and the curve-fitted rates for the 5 Tonr and 1 0 W input power case. Other curve-fitted rates are not presented here because they are not as sensitive to the non-Maxwellian EEDF as are the four rates shown in these plots. The rates for Kea[0,1] and Kea[1,0] have larger error than the other two rates (Kiea[0] and Kiea[1]) between the 2-D CRM calculations and the curvefitted calculations. Comparing the peak values of the two profiles shown in the same plot, the errors caused by curve-fitting are listed in Table 6.4. The problem associated with the low electron number density and/or low electron temperature did not show up here. This actually is good for our analysis. Even in Figure 5.53 where the biggest discrepancy occurs, the peak value calculated from the 2-D CRM is about 2 times higher than the peak value in the curve-fitted calculation. Checking the electron number density distribution for 5 Torr and 10W case, the range is from 1e17 to 1e18 m*3, which is a much sm aller range than the one used in the curve-fitting process. Therefore, the low 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. electron number density region actually has been avoided. To have an accurate curve-fitted equation, a smaller electron number density range should be used that will not introduce too much error. Table 6.4 E rror between peak values comparing the reaction rates calculated from 2-D C R M and the curve-fitted equations Error between Peak Values Rates Kea[0,1], electron-atom excitation 48.8% Kea[1,0], electron atom de-excitation 44.4% Kiea[0], electron-atom ground state ionization < 5% Kiea[1], electron-atom excited state ionization < 5% In the above discussion, Kea[0,1] and Kea[1,0] rates have the largest errors. It is interesting to see how much uncertainty this error will introduce between the results from both the 2-D CRM and 2-D fluid models. Figure 6.28 shows the 4s-state number densities obtained from both the 2-D C R M and fluid models. From this figure, a 30% uncertainty has been observed between the two distributions if peak values are again used to evaluate the uncertainty. That means the error from the reaction rates is actually lessened in the final results. Comparing Figures 6.2 4 and 6.25, Kea[0,1] and Kea[1,0] rates both have larger values in the 2-D CRM. During the calculation, the combination of the uncertainties from these two rates may generate a lower uncertainty for 4s state 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. number density. For accurate results, more accurate curve-fitted reaction rates are desired. However, there is always a tradeoff between a more efficient working model and more accurate results. :4.1E-20 1.1E-2I Kea[0,1] calculated 4 - from 2-D CRM 'E - o - 2- Kea[0,1] calculated from 1E-21 curve-fitted equation 12 10 6 8 4 2 0 z(cm) Figure 6.24 The comparison of Kea[0,l] rates calculated from 2-D CRM and from curve-fitted equation 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.9E-176.3E-17 4- Kea[1,0] calculated from the 2-D CRM 5.6E-17- 3.5E-17 4.2E-17. 2.1 E-1 E-TS' 1.4E-17 Kea[1,0] calculated from the curve-fitted equation 2.1 E-17 3.5E-17 - 6 1.8E-17- 12 10 6 8 4 2 0 z(cm) Figure 6.25 The comparison of Kea[l,0| rates calculated from 2-D C R M and from curve-fitted equation 6- ’6.7E-1 E-21 4- Klea[0] calculated from the 2-D CRM 2- E o - 21E-23. Klea[0] calculated from the curve-fitted equation -4 - - 6- 12 10 8 6 4 2 0 z(cm ) Figure 6.26 The comparison of Klea[0] rates calculated from 2-D C R M and from curve-fitted equation 157 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. .6E-1 L7E-1 4- Klea[1] calculated from the 2-D CRM 2E-1& ‘3.3E-15 - 2 - Klea[1] calculated from -4 - - the curve-fitted equation 612 10 6 8 4 2 0 z(cm) Figure 6.27 The comparison of K Ien[l] rates calculated from 2-D C RM and from curve-fitted equation 6 - / 6.4El7r I.7E17- 4s number density 4- in 2-D CRM 2- - 2- 4s number density 1E17 -4 - in 2-D fluid model 3.7E17 - 6 V6.4E17 - 12 10 6 8 4 2 0 z(cm) Figure 6.28 4s State number density comparison between the results from the 2-D C R M and the 2-D fluid model at 5 Torr, 10W 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.5 Non-uniformity of the Argon Plasmas The 2-D model results show that high non-uniformity exists in the microwave argon plasmas under the studied conditions. It should be noted that the W A V E M A T reactor studied was originally designed for diamond CVD process. That could explain why major problems with the hydrogen modeling were not previously encountered. For the extensive microwave argon plasma study conducted in this research, a lot of problems have been solved, and the conclusions drawn from previous studies are also confirmed by this study. The applications of the 2-D fluid model and the 2-D C R M could be far beyond the areas shown in this study. In this study, although w e failed to relate the OES data to the plasma parameters directly, it is clear that the O E S data can be used to indicate the uniformity inside the discharge region. This should be an inexpensive and easy technique to implement for most reactors if spatial OES measurements are possible. The spatial distributions for the electron number density (Figure 6.4(d) and 6.5(d)) show that electrons are swarming towards either the belljar walls or the substrate surface. This is unique for the argon discharge. Hydrogen plasmas will form a perfect ball above the substrate if finely tuned. During the experiments, it was observed that the argon plasma filled the whole belljar volume. To understand this phenomenon, the introduction of the plasma conductivity should be helpful. T h e plasma conductivity is defined as [Lieberman 1994]: 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where cope is the plasma frequency, a is the microwave frequency, and vm is the electron-neutral collision frequency. For a « vm, the plasma conductivity can be simplified as the dc plasma conductivity, which is defined as: e \ G dc — (6.5) ~ V * ™ 'V m In the above equation, the electron-neutral collision frequency is related to the neutral particle number density. A plasma with lower pressure will have lower neutral particle number density. Then the electron-neutral collision frequency will be lower, and the plasma conductivity will become higher. Since the belljar walls or the substrate surface provide more efficient electron-ion recombination reactions, electrons will always try to escape the discharge volume. Thus a sheath is formed to confine the electrons inside the discharge region, which we have not put into the models yet. Another factor to consider for the non-uniformity inside the plasma discharge region is that the simulation region is axi-symmetric. That means, although the grid points on the r-direction may be uniform, the actual volume each grid represents is totally different (V=2nrZ). The grids that are closer to the wall will have larger volume. At the start point of the 2-D model calculations, a ball-like microwave power density is assumed. However, if the electrons in this assumed discharge region cannot absorb all the input power, the discharge 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. region has to be expanded to reduce the power density. On the other hand, even if more electrons are generated in the discharge region to absorb the input power, when the calculation of the electromagnetic field begins, the electrons will escape the original discharge region. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH APTER SEVEN C O NC LU SIO NS AND REC O M M EN DATIO N S 7.1 Conclusions The major focus of this dissertation is to investigate the nonequilibrium microwave argon plasmas. The motivation for the argon study originated from our previous research work [Kelkar 1999 and Li 1997]. Two major objectives of this dissertation are to solve the puzzle with the microwave power absorbed in the microwave argon plasmas, and to match the numerically predicted excited state number densities with the experimentally measured results within the experimental uncertainties. A better understanding of the microwave argon plasmas has been achieved. The major achievements and conclusions from this study are listed as follows: ❖ Microwave power is the most important parameter used in the plasma modeling, especially for the 2-D models. It affects discharges under the studied conditions significantly. the argon plasma However, a large discrepancy exists between the predicted microwave power from the pseudo1-D model (<1 00 W ) and the metered power during the experiment (8 0 0 W forward power and 12 0 W reflected). ❖ An energy balance study was conducted by performing a global energy balance study on the W A V E M A T reactor, constructing a control-volume heat 162 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. transfer model, and a self-consistent argon 2-D fluid model with the EM model. T h e global energy balance study and the control-volume analysis provided solid evidence that only a small amount of microwave energy (106 0W ) is absorbed by the argon plasma. The 2-D fluid model generates the heat fluxes along the belljar walls and the substrate surface, which confirm that upper limit. By combining all the conclusions from experimental and numerical results, a microwave power less than 50W w as confirmed for the argon plasmas under the studied conditions (5 Torr, 250 seem argon flow). Optical emission spectroscopic (O E S ) measurements w ere conducted to m easure the excited-state number densities. The three excited states used in this study w ere 4p, 5p, and 5d levels. The experimental uncertainties associated with the excited-state number densities are 40% , 56% and 43% , respectively. T h e absorption measurements w ere conducted to measure the 4s state number density. These data are independent of the OES m easurem ent and provide additional experimental results to verify our models. T he numerical models used to predict the excited states are the pseudo-1-D model, the 2-D fluid model and the 2-D Collisional-Radiative Model. The pseudo-1 -D model assumes a uniform microwave power density. However, it was impossible to match the excited state number densities within experimental uncertainties. The 2-D model, which is the first 2-D microwave argon model, provides the detailed spatial distributions for the excited state 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. number densities. It is observed from the 2-D results that the argon plasmas under studied conditions are highly non-uniform. The excited state number densities predicted from the 2-D models are more accurate than the psedo-1 D model. T h ree of the four predicted excited states number densities (4s, 5p, and 5d) can match uncertainties. The the experimental results within the experimental prediction of 4p state number density matches the experimental results with an uncertainty of 64% . The best matching case from the 2-D model has a microwave power of 30W , which is consistent with the energy balance study. Large uncertainty with the numerical predictions could also be introduced from the uncertainties of the microwave power densities caused by the iterations between the 2-D fluid model and 2-D CRM and from the curve-fitted reaction rates used in the 2-D fluid model. The high non-uniformity of the argon plasmas is rather unique and should be avoided if this kind of plasma is going to be used. At the pressure studied (5 Torr), the electron-neutral collision frequency is not high enough to reduce the plasma conductivity. Thus the recombination reactions are not able to keep electrons from swarming towards the walls, where the surface recombination reactions are more efficient. With the extensive experimental and numerical investigation of the nonequilibrium argon plasmas, a better understanding of the argon plasma has been achieved. It was confirmed that only about 3 0 W microwave-power 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. was absorbed in the argon plasma. The OES and absorption data were used to verify our pseudo-1-D and 2-D models. However, it has shown that no simple relations could be found to relate the spectroscopic data to plasma parameters such as electron number density or tem perature directly. They have to be used carefully in the models. 6.2 Recommendations This study was a continuation of previous research at the University of Arkansas. After this study, there are still some investigation. topics that need further Due to the limitation of the research facilities and funding resources, it is highly recommended that more cooperative work should be conducted in the future that will enable us to attack more sophisticated problems. ❖ First of all, spatial measurements of the OES and absorption data are highly recommended. Although the techniques are straightforward, it does require a reactor that allows spatial measurements. ❖ Although absorption measurements can generate the 4s number density using a theoretically calculated lineshape factor, an accurate lineshape is still desirable by using a tunable diode laser. Since the line-broadening mechanisms are dominated by the Doppler broadening, the actual measured lineshape could help determine the gas temperatures in the plasma discharge region. ❖ The 2-D model showed that the plasma discharge could be very sensitive to 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the geometry. The study of the 2-D model with changing geometry should be very helpful to the reactor designers. If a certain reactor needs to be scaledup to larger w afer size, the 2 -D model should also be good at checking the effects of the changed geometry. The curve-fitted reaction rates introduce numerical uncertainty to the 2-D models. For better curve-fitting results, the gas temperature should be considered in the curve-fitting process. Dividing the electron number density range into some small sub-ranges will also reduce the error. For the experimental study of the argon plasmas, instability is one issue that we could not tackle at this time. Some researchers have begun to consider the instabilities in low-pressure discharges [Lieberman 1999], This could be an interesting topic for the plasm a processing industry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES: Baer, D. S., 1993, "Plasma diagnostics with semiconductor lasers using fluorescence and absorption spectroscopy," Ph.D. Dissertation, Stanford University. 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Vlcek, J., and Pelikan.V., 1986, “Excited level populations of argon atoms in a non-isothermal plasma," J. P h y s . D . 'A p p l. P h y s ., Vol. 19, pp. 1879-1888. Vlcek, J., 1989, "A collisional-radiative model applicable to argon discharges over a wide range of conditions. I: Formulation and basic data," J . P h y s ic s . D , A p p lie d P h y s ic s , Vol. 22, pp. 623-631. White. H.E., 1934, In t r o d u c t io n to A t o m ic S p e c t r a , McGraw-Hill, New York. Wiese, W.L., Smith, M.W., and Glenn, B.M., 1966, A t o m ic T r a n s itio n P r o b a b ilit ie s , U.S. National Bureau of Standards, National Reference Series 4, U.S. Government Printing Office, Washington D.C., Vol. 1. Wilke, C.R., 1950, "A viscosity equation for gas mixtures," J . C h e m . P h y s ., Vol. 18, No. 4. Yodor, M.N., 1990, ‘The vision of diamond as an engineered material," S y n t h e t ic D i a m o n d E m e r g in g C V D S c ie n c e a n d T e c h n o lo g y , Spear, K.E. and Dismukes, J.P. Eds., John Wiley & Sons Inc., New York, N.Y. Yos, J.M., 1963, "Transport properties of Nitrogen, Hydrogen, Oxygen and air to 30000 K," T e c h n ic a l M e m o r a n d u m R A D T M - 6 3 - 7 , AVCO-RAD, Wilmington, Mass. 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A PPE N D IX A ARG O N SPEC TR O SC O PY Following is a simple introduction to the notation of the atomic spectroscopy. For electrons, when the angular momentum quantum number, £=0,1,2,3, the electronic state is denoted as s, p, d, f, etc. If w e put the main quantum number before these letters, w e get the symbol for the electronic states. The capital letters S, P, D, F are used to denote the atomic states. At the upper left corner, a number such as 2 denotes that it has a double structure. And at the lower right comer, the j quantum number is put there. Although S states only have single energy level structure, we still use 2S symbol. However, the calculation of argon transition strengths between the excited states requires the use of jl-coupling schemes due to the complex interactions between the atomic-core electrons and the valence electrons [White, 1934; Katsonis & Drawin, 1980]. Therefore, the notation describing the electronic states with the conventional LS-coupling may not be appropriate. Argon is the fortieth elem ent in the periodic table and has a nominal atomic weight of 39.95. In the ground state, argon has a closed 3p shell with a total spin S=0. Singly excited atomic states have the configurations as 3s23p5nls, where n, I, s are the principle, angular and spin quantum numbers of the excited electron, respectively. 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e energy level structure of argon is typical of all the noble gases. It is clear to see that there is a large energy gap separating the first excited state from the ground state. As a result, the electrons in the atomic core will effectively couple their respective angular momentum and spin vectors before interacting with the outer electron. For example, the electrons in the atomic core first form the angular and spin vectors, denoted lc and sc, which then couple together to yield the total core angular momentum vector j c, where j c = lc + sc. Since sc = 1/2, the two lowest energy level of the argon ion Ar(ll) will have the two possible quantum number jc = 1/2 or 3/2. T h e two states are denoted as 2P i /2 and 2Pzt2, where the 2P ^ is 1431 cm*1 above the 2Pi /2 ground state. The angular momentum vector I of the excited (outer) electron then couples with the total core angular momentum vector j c to form the intermediate vector K, w here K = jc + I. The intermediate vector K then couples with the electron spin vector s to form the total angular momentum vector for the entire atom, J, where J = K + s. The splitting of the atomic core forms two term systems. O ne is referred to as the "primed system" with jc = 1/2, and the other is called as the "unprimed system" with jc = 3/2. These two term systems may be described as the following nomenclature [Katsonis and Drawin, 1980]: [ 2P i /2 ] nl [ K]j for jc = 1/2 [ 2P 3/2 ] n l [ K ] j for jc = 3/2 Table A.1 shows the spectroscopic notation for selected argon energy levels [Li, 1988], 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. jl LS Paschen E, (c m 1) 3p6 'So ‘So Ipo 0 3p5 (2P3/2°)4s [3/2]2° 3p5 4s 3P2° Is5 93143.800 3p5 (2P3n 4s [3/2],° 3p5 4s 3P,° Is4 93750.639 3p5 (2P i /2°) 4 s ' [l/2 ]0 3p5 4s 3P0° is3 94553.707 3p5 (2P i /2°) 4 s' [l/2 ]0 3p5 4s ‘P,0 Is2 95399.870 3p5 (2P3/2°) 4p [5/2]3 3pJ 4 p 3D 3 2p9 105462.804 3p5 (2P3/2°) 4p [5/2]2 3p5 4p 3D 2 •a00 to Table A .l Spectroscopic Notation of Selected Argon Energy Levels 105617.315 3p5 (2P3/2°)4p [3/2], 3p5 4p 3D , 2p7 106087.305 3p5 (2P./2°) 4p' [3/2], 3p5 4p *P, 2p4 107131.755 3p5 (2Pi/2°) 4p‘ [3/2]2 3p5 4p 3P2 2p3 107289.747 3p5 (2P i/2°) 4p' [1/2], 3p5 4 p 3P, 2p2 107496.463 ) Figure A.1 shows a typical the optical emission spectroscopic scan from 3500 A to 8000 A. The strongest emission lines are located between 7000 A to 8 00 0 A. Table A .2 shows the spectroscopic data for the used transition lines. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table A.2 Transition Lines Spectroscopic Data (from N IS T Atomic Spectra Data) Wavelength E, Aul Eu Ji - Ju Uncertainty of Au, 9i “ 9u (10s s'1) A ir (A) (cm'1) (cm'1) 4300.101 93750.5978 116999.3259 1 -2 3 -5 3.77E-03 25% 6032.127 105462.760 122036.0704 3 -4 7 -9 2.46E-02 25% 6043.223 105617.270 122160.1502 2 -3 5 -7 1.47E-02 25% 6965.431 93143.7600 107496.4166 2 -1 5 -3 6.39E-02 25% 7067.218 93143.7600 107289.7001 2 -2 5 -5 3.80E-02 25% 7147.042 93143.7600 107131.7086 2 -1 5 -3 6.25E-03 25% 7272.936 93750.5978 107496.4166 1 -1 3 -3 1.83E-02 25% 7503.869 95399.8276 108722.6194 1 -0 3 -1 4.45E-01 25% 7514.652 93750.5978 107054.2720 1 -0 3 -1 4.02E-01 25% 7635.106 93143.7600 106237.5518 2 -2 5 -5 2.45E-01 25% 176 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.0000B06 Emission Strength (a.u.) 1.0000EMM 1.0000EMM 1.0000EMJ3 1.0000EMJ2 ----------------------------------------------------------------------------------------------------------------------------------------------3000 4000 5000 6000 7000 8000 Wavelength (A) Figure A .l A Typical Optical Emission Scan of Microwave Argon Plasma at 680W, 5 Torr, and 250 seem flow rate (Data taken on 04/16/98) 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. APPENDIX B PROGRAM LISTING B.1 2-D Argon Fluid Model The codes for the 2-D argon fluid model were written in the W A TC O M FO R TR A N format. A short description of the source codes is as follows. B.1.1 Description of the Source Codes ecrver.for: This file contains all the routines used for generating output files for either checking or saving data. extpow.for: This file initializes the MWPD for the fluid model at the beginning of solution. After the first iteration with the EM model, it reads the MWPD from the output file generated in the EM model, interpol.for: This file interpolates the results in the 2-D fluid model between the meshes of the fluid model and EM model. plasana.for: This file is not routinely used in the code during the solution of the equations. It analyzes different source and fluxes terms in the conservation equations. The heat flux terms are generated by this subroutine, plasbloc.for: This files treats the boundary conditions in the blocked off region situated in the computation domain. plascalc.for: Main driver for the computation of the coefficients of the quasi-linear algebraic system resulting from the descretization of the transport equation and their boundary conditions. It contains the calling sequences for the computation of the Peclet numbers, Fluxes term coefficients, sources terms, and so on. plaschem.for: This file calculates the chemistry source terms and their Jacobian if needed. plasclax.fon 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This file computes the coefficients resulting from the boundary conditions at the reactor axis. plasclin.for: This file computes the coefficients resulting from the boundary conditions at the reactor inlet. plasclw.for: This file computes the coefficients resulting from the boundary conditions at the reactor walls. plasco.for: Secondary driver for the computation of the coefficients resulting from the discretization of the flux terms in the transport equations. plascoO.for: Routine which computes the coefficients resulting from the discretization of the flux terms of the transport equations plascomp.for: This file calculates some plasma physical characteristics needed in the computation: plasma molar mass, species mass fraction, species molar concentration, species and total density, etc. plascons.for: This file contains the routines that force the electrical neutrality by correcting the electron mole fraction, and maintain pressure to be constant in the reactor by correcting H2 mole fraction, plascvhs.fon This file splits the fluxes terms of enthalpy into two terms: the temperature dependant part of the enthalpy is treated as a pseudo-convective term while the one related to the formation enthalpy at a reference temperature T0 is treated as a source term. The resulting discretization coefficients are then calculated in the same file, plasgsri.for & plasgsrj.for: These routines perform the Gauss-Siedel Line relaxation for all the conservation equation, plasgsri.for relaxes in the Oi (or Oz) direction while plasgslrj.for relaxes in the Oj (or Or) direction, plasinitfon This is the initialization routine, plasmain.fon 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Main driver of the plasma module. plasnrex.for: This routine computes the energy exchange source terms between the different plasma energy modes and during chemical reactions. plaspecl.fon This file contains the routine for the computation of the Peclet number which is important for setting up the numerical scheme for the differentiation of the conservation equations, plasphys.fon Secondary driver for the computation of all the physical properties of the plasma at each grid point: enthalpy, energy exchange rates, chemistry rate, etc. plasprpt.for: Main driver for the computation of all the physical properties and the transport coefficients of the plasma at each grid point. plasread.for: This routine reads all the data of related plasma physical constants and the plasma chemistry. plasreru.for: This routine enables the program to restart the computation starting from previously relaxed fields. plasres.for: This file contains the main relaxation loop where all other routines are called. Thus it is the most important file among ail the fifes used in the fluid model. plasrest.for: This file contains the routines for the calculation of the residues of the conservation equations and the rates of the changes with the temperatures and specie concentrations during the relaxation, plassour.for: In this routine, the final form of the source terms is calculated. These source terms are 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. linearized by splitting the positive part from the negative part, plastran.fon This routine calculates the transport coefficients for the two/three temperature nonequiiibrium flow. rddat.for: This routine calculates all the parameters related to the reactor geometry. B.1.2 Input Data Files Practically all the input data files are read in the routine 'plasread.for1. T he data files are of three types: - Those dealing with the plasma chemistry and physics; - Those dealing with geometry and input power spatial distribution; - Those dealing with the numerical procedures. (Note: All the data are in MKS units, that is, kg, m, sec, J, etc.) A short description of the input data files used in the 2-D fluid model is as follows: • Plasma Chem istry and Physics plasspec.dat: This file gives the chemical species that may be present in the plasma. Each data line is preceded by some comments in French. The data deals with the nature of species (atoms, molecules, ions, etc.), molar mass, charges, specific heats ratio (Cp/Cv), thermodynamic data, transport coefficient data for collision integrals calculation, etc. plasreac.dat: This file contains the plasma chemistry model: total number of reactions, the number and the indices of the reactions where electrons significantly dissipate their energy (line 1), 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the energy threshold of these reactions (lines 2 and 3). Then the following information is present for each reaction (as shown in the example below): 1. The reaction equation; 2. Three integers indicating if the reaction involves a third body, the reaction is reversible and it follows Arrhenius law. 3. The activation mode of the reaction: gas temperature, electron kinetics mode, vibrational mode, or the mixed vibration-gas mode, (the last mode does not play any role for Ar and H2 plasmas. It was done for N2 plasma). 4. Curve-fitted coefficients for rate constant calculations. For electron mode activated reactions, the curve-fitted equation comes from the solution of the electron Boltzmann equation. Otherwise, the coefficients correspond to the Arrhenius form. 5. LHS Stochiometric coefficient for all the species. 6. RHS Stochiometric coefficient for all the species. Exam ple: T1 e- + H2 = 2e- + H2+ 000 0 100 -.236880D+04 ,394732D-K)3 .187693D+07 -677875D+10 .673533D+13 -.163752D+02 10000000 1 0 0 0 0 0 1002 plassurf.dat: This file gives: • The number of catalytic reactions; • Recombination coefficients on the reactor wall; • Recombination coefficient on the substrate surface; • Then for each recombination reaction we have three lines: 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • The species; • LHS Stochiometric coefficient of the recombination reaction; • RHS Stochiometric coefficient of the recombination reaction. plasnrje.dat: This file gives curve-fitted coefficients for the elastic cross-section for calculations of the energy transfer between the translation modes of electron and atoms/molecules; • Geometrical Data maillage.dat: This files contains the computational grid information: 1. Number of grids in the z direction; 2. Number of grids in the r direction; 3. An integer indicating if the cavity has a cylindrical (1) or Cartesian (0) geometry; 4. An angle for axisymetrical computation; 5. The z positions of the grid points; 6. The r positions of the grid points; 7. Integers indicating blocked off region (solid region). This is done to treat the complicated geometry or reactors with substrate in stagnation point configuration without using generalized grid system (Note that the grid may be, and is in fact, not regular). • MWPD Input powtim.out: This file gives the microwave power density absorbed at each grid point. • Num erical Data plasmain.dat: 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The significance of different symbols is given in the comments of the file. Some of the important things to note are the following: The first line contains general information on the discharge conditions: pressure, wall and substrate temperatures. It also contains the values of temperatures (electron gas and vibration) which have to be used during the initialization procedure. These values, especially for Te, are of prime importance for the convergence of the calculation. The general requirement for the electron temperature initial values is to put a high value in the discharge region (where the absorbed power density is high) and a low value elsewhere. The next nine lines contain the initial values inside and outside the discharge region as well as the inlet values for species molar fraction. Here it is also required to initialize the electron molar fraction with a high value (10'7-10"5) in the discharge. Note: Set IECOU to 0 because the present version of the code does not treat the convection. numecou.dat: Most of the numerical data contained in this file should not be changed except in the case with serious convergence problems. The only exception is the integer IREP that has to be set to 0 for a new calculation and to 1 to continue the calculation for better convergence. IREP is the sixth data (integer) in the second line. Generally IREP is changed routinely from one simulation to another. plasnum.dat: This file is one of the most important files. It contains the relaxation parameters. The values chosen for these parameters will make the computation successful or not. The first line contains: NITPM: the number of total iterations (between 400 and 2000, depending on the discharge conditions, is sufficient to achieve the convergence); NITPD: the interval of iterations for the calculated field and other flow characteristics to be saved; 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. NITPE: the calculated field will be saved each NITPD iterations; NRED: put NRED equal to NITPM; XRED: not important for the moment; RPLM & TCPLM: Total residue and convergence rate for all the plasma transport equations (not used); The remaining lines give the relaxation factors used during the numerical solution of the total energy equation, the electron energy equation, the vibration energy equation and the species equations. For example, the relaxation factors for the total energy equation are: RTTM, TCTTM, NGSTTM, FRTTO, FMFRTT, FRTTM, FRGTTU, FRGTTO The most important parameters that will have to be changed are: TCTTM: maximum allowed rate of change of the gas temperature, which is the unknown of the total energy equation; FRTTO: Initial relaxation factor for the total energy equation; FRTTM: Maximum relaxation factor for the total energy equation. Important Notes: The convergence of the code depend of the choice of these three parameters (for each equation): - The initial relaxation factor FRTTO has to be quite low to prevent divergence of the code in the early stage of the relaxation. The initial fields in general are far from the converged one and the conservation equation may show a lot of stress and high instability during the first 100 iterations. - The maximum relaxation factor must not exceed 1.0. In practice a value of 0.5-0.8 is OK. Instability may appear above these values. - The maximum allowed rate of change is very important. This parameter has two main tasks: • To prevent the instability during the first stage of the relaxation; 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • When the convergence starts to be achieved, small oscillations of the plasma temperatures and species mole fraction will take place if the relaxation factors are maintained high (0.5 for example). The residues of the equation will also slightly oscillate, due to the strong non-linearity and stiffness of the system. A method to check the convergence validity is to adopt a smaller value for TCTTM and to restart the calculation from the approximately converged field. If the convergence is OK, a strong decrease of the residues will be observed. numnrj.dat: Not important (used during the development phase). B.1.3 Output Files Practically almost all the output files are generated by the subroutines in the file 'ecrver.for*. The routinely used output files are: 'champs.out1, 'champs.in1, 'plasnum.out', 'therplas.out' and 'speplas.out'. The other files w ere used only during the development phase or when changing the grid meshes, champs.out & champs.in These files are used to save the currently calculated fields and to restart the calculation, respectively. Suppose that a calculation has been completed for 400 iteration and the convergence was not completely achieved. The currently calculated filed for 400 iteration is saved in ’champs.out'. To continue the iteration: - Copy 'champs.out' to 'champs.in' - Change the number of iteration NITPM in 'plasnum.dat' (for example if NITPM is changed from 400 to 800, the code will run until 800 starting from 400. - Change IREP from 0 to 1 in the file 'numecou.dat'. This tells the code to read the initial Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. data in the file 'champs.in'. plasnum.out: This file is very important for tracking and checking the convergence. For each temperature and each species, it reports the rate of change, the residue of the corresponding equation, and the currently used relaxation coefficients, plasemf.in: This file provides detailed information about the plasma discharge, which is the input file for the EM model, therplas.out & speplas.out These two files report the temperature distribution (therplas.out) and species mole fraction fields (speplas.out), respectively, interpolmp.out, interpolmu.out & interpolmv.out The solution of the plasma equations makes use of staggered grid systems (insuring the treatment of convective or quasi-convective terms). These files were used to check the accuracy of the interpolation between the three grid systems: main, Oz staggered and Or staggered grid systems. Jacmp.out, Jacmu.out & Jacmv.out Check the accuracy the Jacobian of the geometrical transformations from non-regular (physical) to regular (indices) grid (for spatial derivative computation) for the main and the staggered grids. mailp.out, mailu.out & mailv.out Check the behavior of the grid meshes for the main and the staggered grids, plassouhs.out Report the quasi-source term which appears after the discretization of the total energy equation and which corresponds to the part of the diffusion of enthalpic species involving only the species formation enthalpy, plasco.out 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Report the values of the coefficients of the quasi-linear system resulting from the discretization of the transport equations. B.1.4 How to Run the 2-D Fluid Model To run the 2-D fluid model with W A T C O M IDE (Integrated Development Environment), the procedure listed beiow should be followed: I. Make sure all the source codes and input data files are in the same directory. For the source codes, the common block files should also be included in the same directory. II. Open the WATCOM IDE. III. If the project has already created, go to the T ile ' menu and choose ‘open project’. Then browse the directories and choose the project name. IV. If this is a new project, go to the tile’ menu and choose ‘new project’. The system will prompt you for the project name. The default target environment' is 'Win32 (NT/95/32s)’. Set the ‘Image Type’ to ‘Windows Executable (.exe)’. Then click ’OK’. A window for the new project will pop out. V. In the blank space for the source file, click the right button of the mouse, a menu will pop up. Then go to ‘new source’. Choose the source codes needed for this project. VI. Right click the mouse again and go to ‘source options’. Choose 'Fortran Compiler Switches’. Then click on the ' » ’ button until '9 Application Type Switches’. Set this switch to ‘Default Windowed Application’. In the source file window, the ‘[sw]’ sign will be shown with the files. VII. Go to Target’ menu and select ‘make’. Or click ‘F4’ on the keyboard to compile the source codes. If ail the data files are set correctly and the compilation is passed successfully, you then can run the program by clicking ‘ctri-R’. 188 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. VIII. To start a new calculation, make sure the parameter IREP in ‘numecou.dat’ is set to 'O'. Otherwise, set the value to ‘1 ’ to read in the MWPD profile from the EM model. IX. The codes will be coupled with the EM model. For better convergence, ten or more iterations between these two models may be needed. For each iteration, make sure the 'champs.out’ file has been copied to ‘champs.in’. 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. B.2 2-D Electromagnetic Model The codes for the 2-D EM model were also written in the W A TC O M FO RTR AN format. Following is a short description of the codes used in the 2-D EM Model. interfield.for: This file interpolates the electrical field information from EM model simulation grids to the fluid model simulation grids. interflu.for: This file interpolates the electron number density, electron temperature, gas temperature and other information from the 2-D fluid model to the EM model. interpwd.for: This file interpolates the absorbed microwave power density distribution from the EM model to the fluid model. main2.for: This is the main control program for the 2-D EM model. The fluid model can be called in this program if needed. max.for: This routine uses the FDTD method to solve for the electrical field in the simulation domain. It is the most important file in the EM model. readem.for: This file reads in the EM simulation information from ‘EMGEOM.IN’. The input files for the EM model are: plasemf.in: This file contains all the plasma discharge results calculated from the 2-D fluid model, which are needed to start the EM model simulation. EMGEOM.IN: This file provides the geometrical information for the EM simulation region. 190 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. The output files are: powtim.out: This file contains the MWPD information that will be used in the 2-D fluid model, eltim.out: This file contains the electrical field information, geo.flu: This file provides the geometric data for the fluid model, geo.max: This file provides the geometric data for the EM model. ver_fd.out: This file contains the fluid model data. The purpose of this file is to verify the 2-D fluid results used in the EM model are consistent with the original results. maxpwd.save: This file saves the MWPD information that is in Origin format, maxfield.save: This file saves the electric field data that is in Origin format. Other data files used in the EM model are either for the debugging purpose or to save the intermediate results during the calculations. To run the 2-D EM model, the same instruction as in the 2-D fluid model should be followed. The microwave power can be changed in the ‘main2.for’ file. The param eter in ,main2.for’ representing microwave pow er example, to set the microwave power to 30 W, the line should is 'watt'. be like: watt = 30.0 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For The iteration number for the EM model simulation can be changed in ‘m ax.for\ The parameter to change is ‘niter1. M ore iteration will have better results for the EM simulations. However, the EM model is coupled with the fluid model. To achieve a converged solution, more iteration between these two models is needed. Thus more calculation time with only one simulation may turn out to be less effective. 192 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. B.3 2-D CRM Due to the fact that E LE N D IF (the Boltzmann Solver) was written in Fortran 90 format, the 2-D CRM codes were written in either Fortran 90 format or C format. The interfaces between the Fortran 90 codes and C codes w ere then implemented to ensure the correct data flows. The source codes of the 2-D CRM are as follows: arcrm.c: This file contains all the routines for the argon collisional-radiative model. It reads EEDF from the Boltzmann solver. The excited state number densities and the absorbed microwave power densities are calculated in this file. This file can also provide the reaction rates directly from the CRM, instead of from the curve-fitted rates, am ovalue.c: Due to the fact that the Boltzmann solver cannot handle extremely low electron number densities (<1e16 m'3) and E/N values (<0.01 Td), this routine is added to avoid calling the Boltzmann solver. Then the excited state number densities are set to zeroes. No reaction rates are calculated. crm2d.f: This file is the main driver for the 2-D CRM. The plasma parameters are read in this routine. ebyn_0.f: This is the Boltzmann solver used in 2-D CRM. The source code is from ELENDIF. In this file, the CRM routine is called after the EEDF is calculated. The input files for the 2-D CR M are: ar2.inp: 193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This is the input file for argon CRM. In this file, some switches are specified to check the physical meanings. Some of the data in this file may be overwrite inside the routines. cvdlib.dat: This file is a library file for the CVD processes. Most species used in CVD processes can be found in this file, elencom.inc & in_outinc: These two files are used for ELENDIF. Some common blocks are specified here, eltim.out: This file contains the electric field information from the 2-D EM model. The data is not directly used in the 2-D CRM. Input file for ELENDIF, the Boltzmann solver. This files specifies some default values for the electron number density, electron temperature, etc. This file also specifies time step and other parameters for ELENDIF to achieve a converged solution. An example is as follows: Arcase names of chemical species, fractional concentrations. & ionization fractions: Ar 1.0 0.0 / End_Species / End_Pops / Press_Torr 5 / gas pressure in Torrs T_Electron 1.2 / initial electron temperature in eV N_Electron lel2 / in cm-3 d Max_Tsteps 100 / maximum number of time steps d N_Cycles 1 / number of AC cycles to run the calculation Max_Tsteps 100000 / maximum number of time steps c/Time_Step l.e-11 / time step in seconds (AC case) Time_Step l.e-7 / time step in seconds N_Print 50 / print every 5 time steps DC 10.0 0.0 0.0 / 50 Td DC electric field c/AC le-1 2.54e9 0.0 / 50 Td DC electric field Du_Umax 0.20 20.0 / 100 grid points between 0 and 20 eV End Data / sigma.dat: Another library file containing plasma physics and chemistry data for most species used 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in plasma processes, speplas.out: Input file of 2-D CRM. This actually is the output file from 2-D fluid model, which contains the spatial species distributions. thermplas.out: Input file of 2-D CRM. This is the output file from 2-D fluid model, which contains the spatial temperature distributions. The output files are: amew.out: Output file for argon CRM for debugging and tracing the program. excite.dat: This file contains the spatial excited state number densities, fek.out: This file contains the information of EEDF calculated by the Boltzmann solver. NU.out: Used in the development phase. This file contains the calculated collision frequency used for curve-fitting of the electron-neutral energy exchange term. Out: Output file of ELENDIF. This file contains almost all the information from the Boltzmann % solver. Pabs2d.dat: This file contains the predicted absorbed microwave power from 2-D CRM. rate.dat: This file contains the information of the rates used in 2-D fluid model, which are calculated directly in 2-D CRM and can be used in the fluid model to reduce uncertainties caused by curve-fitting. \ v e r files: 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These files are for verification. To run the 2-D CRM, the following procedure should be followed: 1. Make sure all the input files and library files are copied to the working directory. 2. Compile the source code as follows: f90 -c crm2d.f f90 -c ebyn_0.f cc -c arcrm.c cc -c amovaIue.c f90 -o YOUR_PROGRAM crm2d.o ebyn_0.o arcrm.o amovalue.o 3. Run YOUR_PROGRAM (the executable program). 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Numerical and Experimental Investigation of Nonequilibrium Microwave Argon Plasmas Abstract of dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy By Yunlong Li, B.S.M.E., M.S.M.E. Xian Jiaotong University, Xian, China, 1992, 1995 May 2000 University of Arkansas, Fayetteville Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This abstract is approved by: DISSERTATION DIRECTORS: Dr. l^ rry Roe -----------Dr. Matthew Gordon Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Optical emission spectroscopy (O E S) and absorption m easurements have been used in tandem with numerical models to characterize microwave argon plasmas. A W A V E M A T (model M P D R -3135) microwave diamond deposition system was used to generate an argon plasma at 5 Torn. Three excited state number densities (4p, 5p, and 5d) were obtained from the O E S measurements. However, the numerical predictions match the experimental data only with changed coefficients. The numerical results also showed that only a small amount (2-5% ) of the metered microwave energy (680±40 W ) w as absorbed by the argon plasma. An energy balance study showed that the energy absorbed by the argon plasma was far less than the metered power, in agreement with the pseudo-1-D predictions. A global energy balance study showed that the bulk of the energy (3 2 0 ± 8 0 W ) was dissipated in the base-plate/applicator water-cooling line. Only a small amount of energy (about 10W ) was dissipated into the substrate. A controlvolume heat transfer model was constructed for a better understanding. Combining results from the global energy balance study and the control-volume heat-transfer model, only 10-60W of the microwave energy can be absorbed by the argon plasma. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A 2-D self-consistent fluid model coupled with a 2-D electromagnetic model was constructed for the microwave argon plasmas. High non-uniformity is observed in the microwave argon plasma. The 2-D fluid model also provides information of the heat fluxes along the belljar walls and substrate surface. Comparing with results of the energy balance study, only the cases with microwave power less than 50 W can have the predicted heat transfer rates consistent with the energy balance study. Further experimental work includes absorption m easurements of the m etastable state (4s) number density. A 2-D Collisional-Radiative Model was developed along with the 2-D fluid model to predict the spatial excited state number densities. Good agreement was achieved between the 2-D predictions and the OES/absorption measurements. 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CRM Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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