Diagnostics and modeling of an inductively coupled radio frequency discharge in hydrogen

Diagnostics and modeling of an inductively coupled radio frequency discharge in hydrogen Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften in der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum von Victor Anatolievich Kadetov aus Moskau Dissertation eingereicht am:.5.4. Tag der mündlichen Prüfung: 9.6.4. Referent: Prof. Dr. rer. nat. Uwe Czarnezki Korreferent: Prof. Dr. rer. nat. Ralf Peter Brinkmann Bochum 4

Contents. Introduction...5. Brief review on ICPs...8 3. Fundamentals... 3.. RF discharge basics... 3... Plasma generation in RF discharges... 3... RF discharges classification...3 3..3. CCP discharges...4 3..4. ICP discharges...7 3.. Diagnostic tools...4 3... Langmuir probe...4 3... Fluorescence-dip spectroscopy...7 3..3. Ion energy and mass analyzer...9 3..4. Emission Spectroscopy...3 4. Theory...34 4.. Analytical model of a hydrogen plasma...35 4... Profiles of plasma potential and plasma density (relative values)...36 4... Electron temperature...38 4..3. Energy balance...4 4..4. Plasma density (absolute values)...43 4.. Electrical model of an ICP source...44 4.. Plasma configuration in a planar ICP...44 4.. Set of equations for the transformer model...47 4..3 Modeling an ICP source...5 4..4 The efficiency of the GEC reference cell as a hydrogen plasma source...57 4..5 Modification of the antenna design...58 4..6 The modified plasma source...63 4.3. Sheath theory...65 4.3.. Static sheath...65 4.3.. Sheath in ICP...67 4.3.3. Ion energy distribution...7 5. Experimental setup...75

5.. Plasma source...75 5... ICP setup...75 5... ICP source characterization...77 5.. Diagnostics...79 5... Langmuir probe measurements...79 5... Laser diagnostics...8 5..3. Measurement of ion energies...83 5..4. Optical measurements...86 6. Results...87 6.. Gas Temperature...87 6... Doppler broadening...87 6... Gas temperature of a hydrogen ICP...88 6.. Plasma bulk parameters...89 6... Electron energy distribution function...89 6... Electron temperature...9 6..3. Plasma density...93 6.3. Sheath dynamics...95 6.3.. Electric field in the ICP sheath...95 6.3.. Time-averaged electric field...95 6.3.3. Time-varied electric field in the sheath...98 6.3.4. Sheath parameters... 6.4. Ion Energy Distribution functions... 6.4.. Energy distribution of the ion flux... 6.4.. Electron temperature at the sheath edge...6 6.4.3. Plasma potential...7 6.5. CCP-ICP mode transition...8 6.5.. ICP emission...8 6.5.. Intensity modulation... 6.5.3. Mode transition... 6.5.4. Mode transition in the pulsed ICP... 7. Summary and conclusions...4 Appendix A List of symbols and abbreviations...7 Appendix B Analytical model of a hydrogen plasma... B.. Particle transport equations...3 3

B... Diffusion approximation...4 B.. Energy transport equation...4 B... Elementary processes in a hydrogen plasma...4 B... Energy loss due to a particle transport...5 B..3. Ohmic heating...5 B..4. The average density and the density at the sheath edge...6 B..5. Local energy balance and induced electric field...7 B.3. One-dimensional model...8 B.3.. B.3.. B.3.3. B.3.4. B.4. Set of dimensionless variables...8 Particle transport equations...9 Profiles of plasma potential and plasma density (relative values)...9 Electron temperature...3 Spatially resolved model in the diffusion approximation...3 B.4.. Plasma density in the entire discharge volume...3 B.5. Absolute value of the plasma density...33 Appendix C Transformer model...34 C.. Inductive power coupling into the plasma...34 C... Structure of an electromagnetic field induced in a vacuum...34 C... Antenna-plasma transformer...36 C..3. Antenna inductance and transformer impedance...38 C..4. Influence of the antenna inductance on current and voltage...39 C..5. Induced currents in conducting materials of the vacuum chamber...39 C..6. Geometrical assumptions about the planar ICP...4 C..7. Electron density...4 C..8. Skin depth...43 C.. Capacitive power coupling into the plasma...43 C... Capacitive power coupling and transformer impedance...43 C... Simplifications in the electric circuit...45 C.3. ICP matching network...46 C.4. Calibration...47 C.4.. Antenna impedance...47 C.4.. Capacitance between the antenna and plasma...48 C.5. Efficiency of power coupling into plasma...49 Literature...5 4

. Introduction The boom in the semiconductor industry has dramatically increased the interest in plasma-assisted material processing. This has prompted a broad range of gas discharge studies, particularly on RF discharges. The inductively coupled plasma (ICP) sources produce high-density, uniform plasmas at low gas pressures and are used for microelectronics production [Hop 9, Eco ]. Although researchers have known about ICPs for over one hundred years [Eck 6], they have not been extensively investigated until recently [Hop 93, Tus 96, Che 96, Lie 98, God 99, Suz, Bog ]. An understanding of plasma physics and surface science is needed to develop plasmaassisted processes. The relationship between the physical and chemical processes occurring in the plasma is generally unclear and complicated. Because of this, processes are often designed using trial and error. The great number of parameters; such as the discharge geometry, gas composition, pressure, incident power, and flow rate; hamper the systematic analysis of experimental data. Therefore, it is important to develop an understanding of the low-pressure ICP in terms of a physical model, which includes the main particularities of the power coupling mechanism, plasma generation and the sheath formation. The model should be simple enough to be calculated analytically, realistic enough to agree qualitatively with measurements, and transparent enough to allow some insight into the physical mechanisms and phenomena occurring in the ICP. A one-component gas simplifies the task by excluding plasma chemistry from the investigation. This orients the plasma model towards a description of electron-molecule interactions. Where as the model of a many-component discharge is more oriented towards the various interactions between atoms, molecules, and radicals. Such complex systems generally only have numerical solutions. In this study, a hydrogen ICP discharge was generated. Hydrogen is widely used for plasma-assisted material processing; it is used as reducing agent for oxidized materials [Got 9], and as a background gas for diamond-like film deposition [Bac 9, Awa 97]. At the time this work was started, there were no investigations on low-pressure ICPs in pure hydrogen. The chief goals of this study were therefore: Design, manufacture, and optimize a hydrogen ICP source. Carrying out a complete set of the plasma diagnostics to study the bulk plasma and the sheath. Model the hydrogen ICP source to gain a better understanding of the possible mechanisms that could be used for controlling the plasma parameters. Analyze the experimental and theoretical results to reconstruct the underlying processes occurring in the RF discharge. 5

The standard GEC reference cell was used as an ICP source [GEC 94, GEC]. In this cell, an additional planar electrode allows more independent control sheath parameters from other plasma parameters. The GEC reference cell is representative of commercial cells used by the semiconductor industry. In order to perform investigations in pure hydrogen, the efficiency of this plasma source was improved. For this purpose, the GEC reference cell was supplied with a novel, multi-spiral antenna. The range of plasma conditions was further extended by pulsing the discharge. For the experimental investigations, Langmuir probe measurements, measurements of the electric field distribution across the sheath, measurements of the mass and energy spectra of hydrogen ions coming to the electrode surface, and time-resolved optical measurements of plasma radiation were carried out. At the time of writing, The fluorescence-dip spectroscopy (FDS) electric field measurements [Cza 98] were the first experimental study of the Debye sheath dynamics in an RF discharge [Cza ]. The physical model of a one-component, molecular gas discharge can be extremely complex. Several research groups have modeled hydrogen discharges numerically. There is a kinetic model for a microwave discharge [Has 99], a particle in cell (PIC) model for an RF parallel-plate reactor [Cap ], and a global model for the deuterium negative ion source experiment (DENISE) [Zor ]. However, a hydrogen plasma is an almost ideal system for analytical modeling: it has one dominate ion species (H 3 + ) at lower electron temperatures, it has a constant collision frequency when expressed as an E-field-to-pressure ratio, and all relevant cross sections are known. Further, this work demonstrated that in the hydrogen ICP the electron energy distribution function (EEDF) is almost Maxwellian. During the course of this study, an analytical model of a hydrogen plasma was developed. The model consists of three modules linked together. Input parameters are the planar cell geometry, the neutral gas pressure, the neutral gas temperature, the incident RF power, and the antenna resistance. The plasma model module allows the calculation of the electron density, the electron temperature, and the effective electron-neutral collision frequency. The electrical model relates the incident RF power to the power dissipated in the discharge either capacitively or inductively by calculating the respective currents in the plasma. Finally, the sheath model calculates the fields, potentials, currents, and the ion energy distribution function. While the former two models are analytical, the latter integrates Poisson s equation numerically. This model is consistent with measurements. Moreover, the analysis of experimental data using the analytical model extends the capabilities of the plasma diagnostics used in the study. For example, the temporal and spatial modulation of Balmer-α emission from the plasma was measured and then used for finding the ICP mode transition. This is a new method which is applicable for both continuous and pulsed discharges. The results from different diagnostics were rigorously compared. The thesis is organized as follows. A brief history of ICP discharges, investigations of these discharges, and its applications is reviewed in Chapter. Chapter 3 introduces the 6

fundamentals of RF plasmas. The first section of Chapter 3 explains the underlying physical principles and contains examples of frequently observed phenomena in RF discharges. The second section describes the analytical methods and the diagnostic tools used in this study. Because of the limited size of the work, a broad variety of historical and scientific facts about RF discharges are not presented here. Therefore, the selected material for Chapters and 3 is subjective. These chapters are not intended as a general review of the subject. A model of the hydrogen ICP is presented in Chapter 4. As was mentioned above, the model consists of three modules. The first two modules, the plasma module and the electrical module, are described in detail in Appendixes B and C. These sections allow one to repeat all calculations from the underling physical laws to the final results. The experimental setup is described in Chapter 5. In addition, this chapter contains the measurement schemes and outlines the calibration of the plasma diagnostics. The neutral gas temperature, the electron temperature, the plasma density, the electron energy distribution (EEDF), the ion energy distribution (IEDF), and the electric field distribution in the sheath were measured. In Chapter 6, these measurements are compared and contrasted with calculations from the ICP model. Chapter 7 summarizes the results and conclusions. The thesis is supported with a list of symbols and abbreviations, Appendix A, and with a bibliography. 7

. Brief review on ICPs The first inductively coupled plasma (ICP) was produced in 884 by J.H. Hittorf by discharging a Leyden jar through a coil surrounding a glass chamber [Eck 6, Mat 3]. At the end of the 9th century, the development of alternating current (AC) sources allowed the stationary generation of ICP discharges. The electrodeless mechanism that sustains ICP generation was disputed until 99 when McKinnen showed that these discharges were usually capacitively sustained by power coupling between the low- and high-voltage ends of the coil. He believed that only at high applied powers and erroneously at high gas pressures the plasma was inductively coupled to the coil [Mac 9]. Mainly due to higher power costs, capacitively coupled plasmas (CCPs) and microwaves were used in place of ICPs for electrodeless discharge technology. In 96, T.B. Reed proposed using an ICP [Ree 6, Reed 6, Ree 6] for atomic emission spectroscopy (AES). In AES, inorganic materials are dissociated and exited using a gaseous discharge. The plasma emissions determine qualitatively the elemental composition of the sample. Direct current plasma (DCP) discharges were used before 964, when Greenfield S. et. al. [Gre 64] and then Wendt R.H. and Fassel V.A. [Wen 65] performed AES using an ICP. The new ICP-AES method was more accurate than the DCP-AES, because impurities from sputtering the walls and electrodes were reduced. ICP-AES has been available commercially since the middle of 97s [Moo 89], and was the most expensive AES technique at the time. In 973, the IBM Corporation experimented with using ICPs for semiconductor manufacturing. Early studies were on ICP-assisted silicon oxidation for the thin-film transistor s (TFT) fabrication on glass for flat panel displays [Pul 73, Pul 74]. They were looking for a low temperature alternative to direct thermal oxidation. A MHz oxygen ICP at about 4 Pa produced similar quality to thermal oxidation. The input power was limited, and kept the gas temperature below 55 K. This study also contained the innovation of using an additional electrode with the ICP source. The silicon substrate was placed on the electrode, and the energy of coming ions was controlled by DC biasing. Widespread use of ICPs in industry began at the end of the 98s after the successful study and commercialization of the helicon discharge [Bos 84]. However, this discharge was developed independently from other ICPs. The interference of ionosphere s whistler waves with telephone communications was firstly observed in 886 [Fuc 57, Sch 98]. In 93s, the phenomenon was intensively investigated [Bar 3, Dar 34, Eck 35]; and in 953, a detailed theory of the whistlers was presented [Sto 53]. Most of the subsequent research on whistler waves was done by solidstate physicists. In 96, P. Aigrain studied the propagation of the whistler waves in materials, and coined the phrase helicon waves [Aig 6]. R.W. Boswell invented a helicon plasma source with waves in the radio frequency (RF) range in 97 [Bos 7]. 8

Because of the unusually high ionization efficiency, the plasma densities achieved in the helicon discharge were almost an order of magnitude higher then in other discharges for comparable gas pressures and input powers. The view that ICPs are inefficient had completely changed. However, normal ICPs operate by capacitive coupling at lower powers and then shift to an inductive coupling mechanism at a threshold power. At this threshold, the total power coupling in the plasma has minimum efficiency. To operate a normal ICP efficiently, it needs to be operated at a power lever much higher than the threshold. ICP reactors are therefore only efficient for high-density plasma sources. Today the ICP discharges are widely used for lighting and technological processes, because of their high charged particle density and their low power losses in the sheath. The inductive power coupling mechanism does not generate a large voltage across the RF sheath, and therefore reduces the interaction of the ions with chamber walls. This reduces chemical reactions on the wall surface, sputtering, and etching in comparison to a CCP. Several contemporary applications of ICPs are presented below. In 99s, the DCP-AES was totally replaced by high-density ICP-AES. Increased power costs were compensated by shorter measurement times. This caused a major change in the techniques, methodology, and apparatus used for AES. These changers are recently reviewed [Mon 9, Bou 3, Hen 3]. Today ICP-AES is widely used for the elemental analysis of metallic aerosols and powders, dusts, and fly ashes. A high-density ICP was also used as an ion source for analytical mass spectrometry (ICP-MS) [Hou 8, Mon 98]. In ICP-MS, the test sample is typically converted to an aerosol and injected into the plasma. An argon ICP that is 9% ionized is used. After vaporization, atomization, excitation, and ionization processes an elemental and isotope analysis is performed using a quadrupole mass spectrometer (QMS). ICP-MS can determine trace elements in air, liquids, and dust [Bal 3]. It is the most sensitive means for detecting trace inorganic metals. For example, in 996, Schultz et al. measured lead (Pb) concentrations in blood from smelter workers using ICP-MS [Sch 96]. In 995, ICP-MS was first used to detect U 35 and U 38 in urine [Vit 95]. In 996 Karpas et al. commercialized a fully automatic version for detecting radionuclide [Kar 96]. They simplified sample preparation by diluting urine in % nitric acid. Other chemical treatments or separations were not needed because of the ICP-MS s sensitivity. In 993, D.P. Myers and G.M. Hieftje combined an ICP with time-of-flight mass spectrometry (ICP-TOF-MS). The TOF-MS acquires data at least two orders of magnitude faster than a quadrupole. The application of ICP-TOF-MS for elemental analysis is reviewed in [Bal 3]. Commercial ICP-TOF-MS instruments have been available since 998. The first elecrodeless fluorescent lamp was developed in 98 [Pro 8]. This lamp was based on a CCP RF discharge, but was not commercially viable due to low energy efficiency. The efficiency was increased later when the CCP was replaced with a lowpressure ICP. The first generation of commercial electrodeless lamps were introduced by Philips (QL) and Matsushita (Everlight) in 99. They were later followed by GE Lighting 9

(Genura) in 994 and OSRAM SYLVANIA (ENDURA, ICETRON) in 998 [Lis ]. All of these products are used for lighting of large areas, where higher applied powers and operated times cannot be met by conventional fluorescent lamps. The electrodeless design allows the use of gases that would normally corrode an electrode, and eliminates wall blackening or the deposition of electrode materials on the wall. The use of high pressure ICPs as a replacement for high intensity discharge (HID) lamps is still being investigated. The major challenge is the matching between the RF power supply and the plasma impedance as it changes through the lamp s starting, ramp up, and stationary modes [Ino 98]. The plasma-assisted thin-film deposition, etching, surface cleaning, oxidation, and hardening play a crucial role in the multibillion-dollar semiconductor industry. The requirements of this industry has and continues to spawn a large variety of different ICP source designs. For example, the efficiency of ICPs generated by the internal and external antennas was investigated by several research groups [Lis 9, Suz 98]. In 995, an ICP source with a neutral magnetic loop inside the plasma cell was developed [Tsu 95]. The presence of magnetic field gradients inside the vacuum chamber increases the power coupling efficiency at low pressures. The etching rate of the neutral loop discharge (NLD) was several times higher than a conventional ICP for otherwise identical conditions. A number of particular improvements for ICP sources can also be found in the literature, and ICP generation is now available for powers as low as several Watts and at gas pressures below. Pa. Up until 99, almost all ICPs had a cylindrical, barrel-like geometry with the antenna wrapped around the diameter. This monopoly ended with the introduction of planar antennas [Ogl 9]. These new discharges had reduced plasma loss, better power coupling, and were more uniform for flat surfaces. Since silicon wafers are flat, planar ICPs are almost exclusively used in the microelectronics industry. It is interesting that early research in using ICPs for spectra-chemistry did not vary the geometry from the original 884 design. They varied the cell dimension, the gas composition, the power, and the pressure; but it was always a cylindrical coil surrounding the reactor vessel. For ICP lamps, cylindrical antennas placed within re-entrant cavities have been used since 936 [Bet 36]. Most innovations in ICP design have been for materials processing. It seems as though this is the most flexible, complicated, and progressive application for ICPs, and an area where many technological requirements still need to be satisfied.

3. Fundamentals 3.. RF discharge basics Since the discovery of RF discharges, a broad variety of experimental data and theoretical facts about this phenomenon have been collected. Today this allows a detailed interpretation of the physical processes occurring in RF plasmas. Of course, all of this material cannot be reviewed in the framework of this dissertation. In this chapter, the underlying physical principles, the fundamentals, and analytical solutions for RF plasma sources operated at low pressures, p < Pa are briefly considered. 3... Plasma generation in RF discharges Gas-discharge plasmas are produced when the electromagnetic energy released in a discharge is sufficient to ionize the gas and maintain the density of charged particles. RF discharges are excited by applying an oscillating electromagnetic field across a discharge gap. Under regulations set up by the International Telecommunication Agreements (ITA) and the Industrial, Scientific and Medical use (ISM), the standard frequency that should be used in commercial RF generators is f RF = 3.56 MHz and its higher harmonics. Systems using non ITA/ISM frequencies must be fully shielded to emissions to ensure that they will not interfere with communications equipment. However, also nonstandard frequencies below this limiting frequency are used [Tus 96]. In a qualitative sense, we can distinguish two operating regimes for RF discharges which are distinguished by the ratio between the exciting-field frequency ω and the electron ω e and the ion ω i plasma frequencies: ω e = nee m ε e, ω i = n e i M i ε (3.) In a low-frequency regime, where ω << ω i << ω e, the motion of all charged particles is governed by electromagnetic-field oscillations. In a high-frequency regime, where ω i << ω << ω e, the spatial positions of ions vary only slightly in time. This simple qualitative picture does not apply to RF discharges at 3.56 MHz. For example, in hydrogen plasmas dominated by H 3 + ions, the condition ω ω i is fulfilled even for ion densities on the order of n i = cm -3. Breakdown of a neutral gas is an avalanche-like process. The gas always contains a small quantity of free electrons produced by cosmic rays. These primary electrons are accelerated by the applied electric field to high energies and collide with neutrals and walls of the chamber producing more charged particles. This gives rise to an avalanche, resulting in the gas breakdown. The final state of the produced plasma and its parameters are determined by a balance between the input and loss power and a particle balance. The processes outlined above is based on energy transfer from electrons accelerated in an RF E-field to the background gas, and is an illustration of the dominant role of electron-

neutral collisions in a plasma. Therefore, it is named collisional or Ohmic heating. The electron elastic collision frequency, v m, is proportional to the gas density and, at a constant gas temperature, to the gas pressure: v m N p. (3.) Fig.3. shows the dependence of the ratio v m /p on the electron energy ε e for hydrogen and argon. In particular, the value of v m /p for hydrogen is constant over a wide range of electron energies: v m = αp, where α = 4.5 7 Pa - s -. (3.3) 4 ν e / p ( 7, s - Pa - ) m 8 6 4 Ar H 4 8 6 4 8 3 36 ε e (ev) Fig.3.. Collision frequency versus electron energy for hydrogen and argon gases [Rai 97]. Fig.3.. Experimental determination of the effective collision frequency in the RF discharge at f RF = 4,8 MHz in mercury [Pop 85]. In RF discharges collisionless or stochastic heating is an additional heating mechanism. In order account for stochastic heating, the effective electron collision frequency, v eff, is used instead of the elastic-collision frequency, v m. Fig.3. shows measurements of v eff in an RF discharge at a frequency of f RF = 4.8 MHz in mercury [Pop 85], and demonstrates that the stochastic heating is more efficient at low gas pressures. Theoretical descriptions of the stochastic heating are still been disputed. Two models are encountered in the literature. One model is based on Fermi acceleration, where electrons gain additional energy due to their reflections from the moving edge of the sheath [Lie 98]. The alternative model is based on the effect of repetitive compression and rarefaction of the electron cloud at the sheath edge [Goz,Goza, Goz, Tur 93]. The stochastic heating in this case is attributed to pressure variations.

3... RF discharges classification Conventionally, RF discharges are classified according to the mechanism of RF power coupling to a plasma, i.e. they are either inductive or capacitive. The RF discharge classifications are shown in Fig.3.3. capacitively coupled plasma (CCP) E-discharge Classification of RF Discharges RF RF discharge applied electro-magnetic field hybrid CCP/ICP H-discharge with with an an additional powered electrode E-mode inductively coupled plasma (ICP) capacitive Fig.3.3. Schematic of conventional RF discharge classifications. H-discharge power coupling hybrid mode capacitive/inductive H-mode inductive For electrodeless RF discharges G.I. Babat [Bab 47] first introduced in 947 the terms E- and H-discharges to differentiate between plasma generation by electric or magnetic fields. The term E-discharge was adopted by V.A. Godiak [God 76] who applied it to capacitively coupled plasma (CCP) sources containing electrodes. In inductively coupled plasma (ICP) sources, the electromagnetic field maintaining a discharge is induced by oscillating currents in the antenna. In addition, antenna can directly generate an electric field, in which case it acts as an electrode. Both the inductive and capacitive mechanisms of power coupling are present in the ICP sources, and there are two modes of operation known respectively as E-mode and H-mode [Lie 94]. Between the E- and H-modes a transition regime, where the dominant power coupling mechanism is difficult to determine, exists. This ICP regime is named the hybrid mode. Besides the purely inductive or capacitive discharges mentioned above, there are hybrid ICP/CCP discharges. In the latter case, the design of an ICP source includes an additional RF-biased electrode typical of CCP sources, so that it becomes possible to independently control the generation of a plasma and the properties of the plasma sheath. 3

3..3. CCP discharges A basic diagram of a simplified CCP cell and its equivalent circuit are shown in Fig.3.4 A discharge is excited between two electrodes which are usually placed inside the plasma cell. Sometimes the surface of either one or both of the electrodes is coated with an insulator. If the electrodes are placed outside the plasma cell, the chamber is made from a dielectric material. Cell sizes are usually chosen to meet the requirements of a particular material processing, and they can be varied within wide limits. The electrode shapes and surfaces can be the same or different. Power Sheath Plasma Sheath R S C S R S R pl L pl R S Fig.3.4. Basic diagram of the simplified CCP cell and its equivalent circuit. R S, R S, C S and C S are resistances and capacitances of the sheaths. R pl and L pl are the resistance and the inductance of the plasma bulk. R pl L pl C S The bulk of the plasma forms in the central region of the cell. The bulk is characterized by quasineutrality, an Ohmic resistance, R pl, and inductance, L pl. Two space change sheaths form near the electrodes. Here electrons are moving in and out periodically while the ions form a practically constant background charge. These time varying sheaths characteristically have high resistances, R S and R S and capacitances, C S and C S : C A d, (3.4) S S S where A S, d S are the sheath area and the sheath gap. In simplified equivalent circuits, the bulk plasma is considered as a resistance, whereas the sheath is considered as a capacitance because of a large difference (in magnitude) between of the complex impedances at frequencies of the RF range: Rpl << ωl pl and RS >> ωcs. (3.5) CCP discharges can be operated in either the α-mode or the γ-mode. The α-mode is the initial state and occurs just after breakdown. As the power applied to the electrodes is slowly increased, the discharge abruptly changes above a threshold power and enters a brighter state, the γ-mode. The discharge behaves as if a secondary breakdown is occurring in the gap. This effect was studied experimentally in [Lev 57, Levi 57]. This experiment shows that the transition from one mode to another is abrupt in hydrogen, where in argon it is more transitional. A physical explanation of α-mode and γ-mode was put forward by the authors of [Rai 97]. When the plasma density is low, a discharge operates in the α-mode. The current through the nonconducting sheath is in fact a displacement current and new charges are produced in the plasma bulk only. As the plasma density increases the current through the sheath increases substantially, and the discharge changes to the γ-mode. Now, the electrons 4

are produced predominantly by secondary ion-electron emission at the electrode. While passing through the sheath, the secondary electrons are accelerated and multiplied by ionizing collisions with neutrals. When they enter the bulk plasma, they contribute efficiently to ionization and excitation of neutral gas particles, and the average plasma temperature is reduced. Taking into account stochastic heating, the classification of CCP discharges was extended by Boeuf et. al. [Boe 9]. For the γ-mode discharge, the authors distinguished a positive column regime where the stochastic heating is negligible and an electron-sheath collision regime where the stochastic heating is essential. The CCP source is powered by an RF-oscillator combined with a matching network. Minimum power loss due to internal resistance of the RF-oscillator, R RF (typically 5 Ω), and due to a power reflection is achieved if the load impedance is equal to R RF. The matching network transforms the plasma complex impedance, Z eff, to R RF. So the circuit operates with maximum efficiency. If the condition Re(Z eff ) R RF is fulfilled, the impedance matching can be provided by a passive matching network. This network consists of two variable capacitors and an inductor. If the plasma resistance is high, Re(Z eff ) > R RF, then the impedance matching becomes a complicated engineering problem. In this case, it would be better to use an RFoscillator with high internal resistance (R RF = kω) instead of an active matching unit. CCPs are often asymmetric, typically with one of the electrodes grounded. If a CCP is symmetrical, it means that both electrodes are powered. Such a circuit complicates realization of the experiment, but allows one to simplify interpretation. If the electrodes in the discharge have different sizes, then sheaths will differ from one another. An additional voltage drop occurs at the smaller, usually powered electrode. This effect, known as self-biasing, was explained in [Lie 94]. It is assumed that the density of the current flowing across the sheath j i is proportional to some power of the voltage across the sheath and inverse proportional to a different power of the sheath thickness. Mathematically, this can be formulated as U m S ji n ds, (3.6) where the powers m and n are specified by the particular sheath conditions. For homogeneous plasma, the current density is the same at both electrodes: j = j. (3.7) S S In this case, equation (3.4) and the Ohm law applied to the impedances of both sheaths give the ratio between the voltage drops across these sheaths: U SCS = U S CS, (3.8) ( A A ) γ U. (3.9) S U S = S S In available theoretical models of colissionless and collisional sheaths the value of the power γ = ( - m/n) - is predicted to lie between.3 and 4. Experimentally γ is often found to be about. 5

The concept of a standardized CCP source was proposed in 988. This furnishes an opportunity to compare results of studies carried out by various research groups. Fig.3.5 shows a schematic diagram of this standard plasma cell, known as the Gaseous Electronic Conference (GEC) reference cell [GEC 94]. Fig.3.5. Schematic diagrams of the plasma cell and the pump-out manifold for the GEC reference CCP source. The standard plasma cell contains two identical plane electrodes with radius r el = 5 cm which are placed parallel to one another at a distance of h cell =.5 cm. Typical discharge parameters are: operating frequency f RF = 3.56 MHz, gas pressure p = - Pa, applied power P = - W, electron density n e = 9 - сm -3, electron temperature T e = - 4 ev. Since a reproducibility of the parameters of a low-pressure plasma depends on the design of the discharge chamber, the identical geometry of plasma cells is important for the standardization of RF discharges. 6

3..4. ICP discharges The alternative form of maintaining an RF discharge is by inductive coupling [Hop 9]. An ICP is used for technologies where a low-temperature, dense and uniform plasma is needed. A typical diagram and the equivalent electric circuit of the ICP source are presented in Fig.3.6 [God 99, Lie 94]. quartz RF power L gas inlet to pump plasma r η Ant D Ant R pl R S C q L L L (e) R C S antenna L pl L matching unit RF generator C S R S R Fig.3.6. Diagram of the ICP cell and the equivalent electric circuit of an antenna-plasma transformer. R and L are the antenna resistance and induction. R, L and L (e) are the plasma resistance, geometrical plasma inductance and inductance due to inertia of electrons. L and L are the interactive inductions between an antenna and a plasma. In an ICP, energy coupling into the plasma is through a transformer whose circuit consist of the antenna and the plasma. The ICP mechanism of capacitive power coupling is equivalent to a CCP. Because of this, we should include the CCP equivalent circuit from Fig.3.4 in the ICP equivalent circuit by connecting the CCP circuit in parallel to the antennaplasma transformer. When an external antenna is used, we should modify the circuit by adding the capacitance of the dielectric separator, C q. The antenna has the inductance, L, and resistance, R. Since the antenna is usually made from a conductor, the following inequality holds: ωl >> R. The plasma serves as the secondary coil of a transformer. In contrast with CCP discharges, the plasma inductance, L, is primarily determined by the geometry of the induced currents. The plasma inductance due to the inertia of electrons L ( e) contributes only insignificantly to the total inductance: L >> L (3.) ( e) The relation between R and ωl depends on many parameters and can vary with changing discharge conditions. As was mentioned above, the ICP discharge can operate in the E- and H-modes, which are distinguished by the dominant power coupling mechanism. In the E-mode, the antenna acts as a powered electrode like in CCP discharges. Capacitive power coupling into a plasma can be illustrated by a simple model proposed in [Lie 94]. In this model, the voltage 7

drop across the sheath can be determined by assuming the plasma potential to be zero. This is a reasonable approach, because the plasma potential is usually much lower than the antenna voltage. The antenna-plasma voltage divider formed by the capacitances of the dielectric separator and the sheath is strongly nonlinear. The dependence of the sheath potential on the antenna potential scales with the fourth power: 4 U.8 ε e U S = en ( ) 4 ub M i hq ε, (3.) where n is the electron density at the sheath edge, u B is the Bohm velocity, and h q is the dielectric-separator thickness [Lie 94]. Energy losses due to particle transfer through the sheath increase as the potential U S increases. Through the energy balance, these additional losses lead to the density decrease. It will be shown later in Subsection 4..3. Iteratively, the sheath potential increases according to equation (3.), and the electron density decreases again. Finally, this leads to a deterioration of conditions favorable for the H-mode. On the other hand, if the plasma density increases or the antenna voltage decreases, the efficiency of capacitive power coupling into the plasma decreases. In the H-mode, ICP discharges are electrodeless. The efficiency of the power coupling into the plasma is determined by the plasma-cell geometry and the plasma parameters, particularly by the conductivity. It is well known that a transformer in which the secondary circuit is open or shorted dissipates the input power totally in the primary circuit. Reasoning from this knowledge, we can distinguish three characteristics inherent to ICP discharges: The plasma cannot be self-ignited using the inductive coupling mechanism alone. The E-mode always precedes the H-mode. Therefore, it is reasonable to consider the formation of the H-mode discharge as a mode transition rather than an ignition process. A maximum efficiency of power coupling into ICP discharges is reached at plasma densities that are one order of magnitude higher than typical plasma densities of CCP discharges. Consequently, a higher incident power is required. Metal near the antenna can couple to it. As a result, a major cause of power loss in the ICP source can be by high conductivity material used to construct the vacuum chamber. The H-mode is dominant for ICP sources. There is an optimum plasma density that ensures a maximum power coupling into the plasma. On one hand, a discharge is maintained by Ohmic heating which increases as the electron density increases: < S Ω > =< j E > ~. (3.) e n e On the other hand, the penetration of the electric field into the plasma is characterized by a skin depth, δ, which decreases as the plasma density increases. Thus, although the efficiency of heating increases, the heating region decreases. A maximum efficiency is achieved when the value of δ is smaller than the plasma cell size, but is still comparable with it. The skin 8

depth in ICP discharges is typically of the order of several centimeters, which is substantially smaller than the RF wavelength (λ RF = m). To analyze the mechanism for the transition between E- and H-modes in ICP discharges, one first assumes that the transition occurs, when the coupling efficiencies of both mechanisms are equal. In this case, the transition should occur at a definite (threshold) power. However, experimental evidence indicates that this interpretation is not always valid. In publications about ICPs, the value of the threshold power varies within rather wide limits. This is typical even when the same gas was used and the discharges were produced in similar plasma sources. Secondly, the presence of some residual features of an alternative power coupling mechanism after the transition leads to the conclusion that a Fig.3.7. Experimental confirmation of a hysteresis at the transition between the modes of an ICP discharge [Loc ]. hybrid CCP/ICP mode is a more adequate description for discharge conditions near the transition of one mode to another. Some research groups have even observed a type of hysteresis. The E-to-H transition occurred at a higher RF power in comparison with the H-to-E transition. For example, Fig.3.7 shows this hysteresis in an experiment carried out by Y. Lokurlu in a hydrogen ICP discharge [Loc ]. The explanation of the hysteresis effect at the transition between modes using the hydrogen ICP model is given in Subsection 4..3. An alternative model which includes nonlinear effects in the energy balance due to electron collisions and multistep ionization can be found in [Tur 99, Elf 98]. The geometry of the electromagnetic field and, therefore, the plasma heating depend heavily on the design of the antenna. Follow the particular requirements of the plasmaassisted material processing, the different antennas have been introduced and successfully used at the time. They are classified using follow types. The antenna may be external or internal. If an external antenna is used, the vacuum chamber must be made of a dielectric material or have a dielectric window. An internal antenna is usually insulated from the plasma by a nonconducting coating. There is also a hybrid type called the quasi-internal antenna. The coil is placed into a dielectric holder, which is then placed in a vacuum chamber [Lee ]. The antenna may consist of a one turn coil or may have a more complicated geometry, which could be difficult to classify. For example, in [Tus 96] the authors proposed a hemispherical antenna with twenty turns which is capable of generating a homogeneous plasma in a relatively large volume. One special type of ICP sources which uses cylindrical antennas is a helicon discharge [Che 96]. A helicon plasma discharge is generated in a cylindrical vacuum cell, has 9

a homogeneous magnetic field, and electrons are heated by Landau damping. The helicon waves have a frequency between ωci << ω << ω ce, (3.3) where ω ci and ω ce are the ion and electron cyclotron frequencies, respectively: eb eb ω ci =, ωce =. (3.4) M m Helicon sources have gained wide acceptance for their high Fig.3.8. Examples of antennas used in helicon discharge. efficiency in comparison with other ICP sources. Several types of helicon antennas [Che 9] are shown in Fig.3.8. There are also many other antenna designs that deserve attention. Industrial devices usually have cylindrical and planar antennas. Cylindrical and planar ICP discharges are named according to the antenna type. ICP sources can also have more than one antenna. Such a design makes it possible to solve several engineering problems at once. For example, in [Col 99] two coaxial antennas were operate at different frequencies f RF = 3.56 MHz and f HF = khz in order to generate a plasma and to heat a substrate simultaneously. In another device, used for sterilization of large tree-dimensional objects, two plane antennas are placed parallel to one another [Mes 3]. This allows to achieve a high plasma density and homogeneity in the radial and vertical directions. i e Fig.3.9. Example of a low induction antenna with several parallel coils [Kim ]. Fig.3.. Example of an ultra-low induction antenna [Wu ]. Oppositely directed currents diminish the antenna inductance. Antennas can be divided into groups depending on the value of their inductance [Men 96]. One group contains antennas in which the direction of current is the same in all coils. This group in turn is subdivided into normal induction antennas containing a single coil and low induction antennas consisting of several parallel coils. An example of a low

induction antenna is presented in Fig.3.9 [Kim ]. The second group contains ultra-low induction antennas. Their inductance is substantially reduced because the current in the adjacent coils flows in opposite directions. An example of this antenna is presented in Fig.3. [Wu ]. Coil designs, such as that shown in the figure, are typically very complicated for low-pressure (p Pa) discharges. Generally, the ICP source efficiency can be optimized using the following principle. Examing equation (3.), the efficiency of capacitive power coupling depends strongly on the antenna voltage. Therefore, by decreasing the antenna voltage and increasing the antenna current, it is possible to reduce the mode transition power. According to the Ohm law, the ratio of voltage U to current I at the antenna is proportional to its impedance which is primarily determined by the antenna inductance Z ωl. This is why a low-inductance antenna operates at lower voltages and higher currents in comparison with a high-inductance antenna. U I ~ L. (3.5) Consequently, by modifying the antenna in order to decrease its inductance, it is possible to achieve optimum operation of the ICP source in the H-mode. Like the CCP source, the ICP device contains a power supply and an impedance matching network. The current in the matching network is higher than in the capacitive discharge. Since the power losses increase in this case, a correct matching of the antenna impedance to the output resistance of a RF generator becomes of crucial importance. Stable discharge parameters can be achieved with the use of a commercial matching network with an auto-matching function. Thus, it is usually the equipment that limits the maximum antenna current. A standard matching network uses the L-matching circuit shown in Fig.3.(a). The device contains two adjustable capacitors. One of them is connected in parallel to the power supply, and is called the load capacitor C load. Another capacitor is connected in series to the antenna, and is called the tune capacitor C tune. The current in the antenna is constant along its length. According to the Ohm law, the potential profile is proportional to the impedance profile. For this reason, a potential along the radius of a planar antenna is described by a parabola. Whereas the potential along the length of a cylindrical antenna is described by a line. The antenna voltage relative to ground can be minimized by correctly tuning the phase shift between potentials at its terminals. Two types of passive matching networks aid in solving this problem. One type is a π-matching unit with interchanged positions of C Tune and L. When this unit is used, at ωl > R RF the antenna voltage is lower in comparison with the L-matching unit. When ωl < R RF, the L-matching unit is preferable. Suzuki et. el. proposed a version of the electric circuit for matching network called an antenna coil capacitance termination [Suz ]. The advantage of this device is illustrated by Fig.3. which shows antenna potential profiles sampled during one RF period. As can be

seen, the maximum value of the antenna potential in the case of a standard L-matching unit doubles in comparison with the unit shown in Fig.3.(b). Z Ant / Z Ant / Z Ant /4 Z Ant V / I V / I -/ Z Ant -/4 Z Ant - Z Ant -r Ant -/r Ant /r Ant r Ant radial position -/ Z Ant -r Ant -/r Ant /r Ant r Ant radial position Z Ant / Z Ant / Z Ant /4 Z Ant V / I V / I -/ Z Ant -/4 Z Ant - Z Ant -/ Z Ant /4h Ant /h Ant 3/4h Ant h Ant vertical position (a) L-matching unit and /4h Ant /h Ant 3/4h Ant h Ant vertical position (b) antenna capacitance termination in the matching unit Fig.3.. Basic equivalent electric circuits and the antenna potential profiles sampled during one RF period for a planar and a cylindrical antennas. It should be mentioned that an antenna terminated by a capacitance has zero potential at the point corresponding to one-half of its inductance, and electron currents in the plasma are not strongly disturbed by the antenna potential. Indeed, the maximum current corresponds to this region because induced electromagnetic field has also a maximum here. In the case of the L-matching unit, the antenna potential at this point is equal to half the applied voltage. On the other hand, the circuit shown in Fig.3.(b) has two disadvantages. First, the inclusion of an additional capacitor connected in series results in additional energy losses in the capacitor itself and its contacts. Second, the rating of each capacitor in this matching network is twice as large as the rating of a capacitor in a L-matching unit, substantially increasing its cost.

Fig.3.. Faraday shield [God 99]. Another way of improving the ICP operation in the H-mode is by using a grounded Faraday shield to isolate the antenna from the plasma. An example of this device is shown in Fig.3. [God 99]. The Faraday shield suppresses the electric field due to the antenna potential, whereas the magnetic field penetrates through radial slits almost without loss. After the transition to the H-mode, the use of the Faraday shield excludes any residual capacitive power coupling to the plasma. However, a current between the antenna and the Faraday shield introduces an additional power loss in the system. With the Faraday shield, there arise two engineering problems: A strong suppression of the initial E-mode makes self-ignition hardly possible. This is acceptable for plasma sources operating in a continuous mode, whereas the use of a Faraday shield in pulsed discharges is a complicated engineering problem. The voltage between the antenna and the Faraday shield limits the maximum incident power because of the danger of high-voltage breakdown. The Gaseous Electronic Conference, GEC, introduced a standard ICP source in 995 [Mil 95]. The GEC reference cell was designed initially as a CCP, and was later modified for use as a planar ICP [GEC]. The modified design, sketched in Fig.3.3, is also by convention. The upper electrode of the CCP was exchanged for a spiral antenna placed inside a metal cylinder with a quartz window with thickness h q = cm. The antenna radius is r ant = 5 cm, and the Fig.3.3. Schematic diagram of the standard ICP cell. number turns is N ant = 5. The antenna is made from a copper tube with an outer diameter of d Cu = 3 mm and is water-cooled. The lower electrode of the CCP was modified by placing a metal plate with radius r el = 8 cm on the top of it. The distance between the quartz window and the lower electrode is h cell = 5 cm. Typical working conditions of the ICP are: frequency f RF = 3.56 MHz, gas pressure p =. - Pa, applied power P = - W, electron density n e = 9 - cm -3, and electron temperature T e = - 5 ev. 3

3.. Diagnostic tools In this section, some of the basics diagnostics used to study RF plasmas are considered: Langmuir probe diagnostic (Subsection 3..), fluorescence dip spectroscopy (Subsection 3..), ion mass and energy analyzers (Subsection 3..3), and time-resolved optical emission spectroscopy (Subsection 3..4). The operational principles, the theory, problems that can arise, and the limits of applicability for these diagnostics will all be discussed. Particular attention will be given to the features of these diagnostics to meet the experiment requirements of RF plasmas studies. 3... Langmuir probe An advantage of Langmuir probe is the simplicity of the measurements, and ability to measure the key plasma characteristics. Probe diagnostics are used extensively in plasma studies, and considered a classical experimental method. The spatial distribution of the plasma potential, density, and electron temperature that are obtained by this technique are commonly used as a reference for other types of plasma diagnostics [Bow 99, Ful, Yoo 99]. In this study, the analysis method invented by Druyvesteyn in 93 was used [Rai 87, Beh 94, Kim 97, Kim, Kor 94, Awa 97, Sch ]. This method calculates the electron energy distribution function (EEDF) from the second derivative of the I-V characteristic of the probe. As an example of the Druyvesteyn method, a Langmuir probe I-V characteristic is shown in Fig.3.4. The probe potential, U p, is plotted as a function of the total electron and probe current (ma) 8 7 6 5 4 3 - P = W p = Pa (H gas) R = 45 mm - -5 - -5 5 5 probe voltage (V) Fig.3.4. Typical example of the I-V characteristic of the Langmuir probe. ion current, I p. At the inflection point in the curve, the I-V characteristic is changes from concave to convex, and occurs approximately at the plasma potential: U p = U pl. In Fig.3.4, the plasma potential is U pl 3 V. The plasma potential is essential the local space charge potential. As a result, when U p = U pl, there will be no electric field between the sheath and the probe and charged particles will not be accelerated or decelerated by it. At the plasma potential, the ion current can practically be neglected, because of the low ion mobility and the absence of thermal equilibrium in the plasma. (The electron temperature is higher than the ion temperature). If the probe is biased less then U pl, the electrons are decelerated and the ions are accelerated in the sheath. For the electrons to get to the probe surface, the kinetic energy at the sheath boundary should be larger than the difference of e(u p -U pl ). The electron current at the probe is decreasing, whereas the ion 4

current is increasing in this case. The dependence of the ion current on the probe potential is approximately: I ( U ) α U U, (3.6) i p ( ) γ = p pl where α and γ are variables. The parameter γ depends on the probe geometry and varies in the range from γ =.5 (for a cylindrical probe) to γ = (for a spherical probe). The floating potential, U f, is lower than U p, and is when the negative electron current and the positive ion current are equal in value. The total probe current is zero: I p = A. The potential of probe satisfying this equality is termed the floating potential. At even lower probe potentials, the probe will be shielded from the electron current, and the value of I p is determined by the ion current alone. By approximating the ion current with (3.6), we can distinguish the ion and 8 electron components of the total probe probe current 7 γ γ P = W I current, as shown in Fig.3.5. It should be i I ( U i = p) ion a= current α( V( U pv pl U) pl ) p = Pa (H gas) 6 R = 45 mm,5 electron current,5< < γ < γ < mentioned that the electron current, which 5 4 3 I i I e we will use below in our analysis, is sufficiently accurate only for the bulk of electrons whose energies are close to the I p mean energy. The current of the low-energy electrons corresponds to the probe potential: current (ma) - - -5 - -5 5 5 probe voltage (V) Fig.3.5. Measured total current, the approximated ion current, and the calculated electron current at the probe. U p U pl. Errors in determining U pl cause introduce large errors into the calculation of the ion current I i (U p ) using (3.6). For negative probe potentials only high-energy electrons contribute to the probe current. However, as can be seen from Fig.3.5, for U p << the value of I e is much smaller than I p and I i. If the electron current is calculated by the difference between the probe current and the ion current, it will introduce a large error in this case too. The electron velocity distribution function (EVDF) is proportional to the second derivative of the electron current with respect to U p in both the velocity space, f e (u e ), and in the energetic space, F e (ε e ): I e p p ( U U ) = f ( u ) = F ( ε ) pl p πe m 3 e S e e e 3 S m e e e, (3.7) where m e is the electron mass, e is the electron charge, ε e = e(u pl - U p ) is the electron energy, u e = (ε e /m e ) / is the electron velocity, and S p is the surface of the probe tip. The inflection point of I e (U p ) and the corresponding plasma potential can be found from the intersection of the second derivative of the electron current I (Up) with the zeroaxis. It should be mentioned that when U p U pl, the second derivative of the ion current I i is substantially smaller than the second derivative of the electron current e I e. This is why the 5

values of I and I are nearly equal, and the plasma potential can be calculated using the e p deflection point in the I-V characteristic Ip(U p ). Since the electron energy distribution function is calculated from the second derivative of the electron current, even small deviations of the electron current from its actual value can result in a large errors. The noise in a measured I-V characteristic is typically removed by using a mathematical filter. The authors of [God 9] examined how the second derivative of the electron current depended on the choice of a mathematical filter and its parameters. They reached the following conclusion: if an I-V characteristic has a relative error δi e, it is possible to filter the function F e (ε e ) without distortion in the energy range from to ε e only if δi e F e (ε e ) / F e (). F e (ε e ) can be filtered over a wider range of energies, but the reliability will be sacrificed. The EEDF is calculated by the formula: f e ( ) = ε F ( ε ) ε. (3.8) e e e The electron density, n e, and the mean electron energy, < ε e >, can be calculated by integrating the EEDF over all electron energies: < ε e ( ε ) e n = f dε. (3.9) e > = n e e ε e e f e e ( ε ) e dε e. (3.) For a Maxwellian EEDF expressed in energy units, the electron temperature is /3 of the mean electron energy. For a non-maxwellian EEDF, the effective electron temperature is generally defined equal to /3 of the mean energy calculated from equation (3.): kt e = < εe 3 >. (3.) The Druyvesteyn method is applicable for I-V characteristics measured in a static sheath. However, an RF sheath has a constant component and a time-varying component that oscillates. When measurements are performed in an IPC discharge, the RF component of the sheath potential should be compensated because the Druyvesteyn theory is unable to account for the oscillations [Coh 96, Gag 7]. refer ence elect rode prob e tip capacitance to current electrical amplifier filter Fig.3.6. Langmuir probe with passive compensation of the RF component of sheath potential. An example of RF compensation is shown in Fig.3.6. In comparison to a standard Langmuir probe, a probe with passive RF compensation has an additional electrode. This electrode is large, and connected with a probe tip through a capacitor. Therefore, the electrode potential will be near the RF floating potential. The probe tip is connected to the 6

power supply through an electrical filter to suppress the RF frequency. During measurement, the probe tip is DC biased through the electrical filter and RF biased by the additional electrode. The probe tip s RF biasing is effective due to the high impedance of the filter. The RF component of the voltage drop across the probe s sheath is suppressed by the RF biasing signal [Hop 87, Par 9, Awa 97, Sch ]. In active compensation, the RF component of the sheath potential is suppressed in a similar fashion. Except that the RF biasing signal is generated by the probe power supply. The wave form of the biasing signal can either be independently synthesized or be an amplification the signal from small additional electrode. For proper compensation, it is critical that the phase of the probe power supply and the RF floating potential of the plasma be synchronized [Fle 96]. 3... Fluorescence-dip spectroscopy Spatial and temporal measurement of the electric field in the sheath of a RF discharge provides valuable information about the plasma and sheath parameters. One method for measuring high electric fields is based on a coherent anti-stokes Raman scattering scheme, however, this is linked to atmospheric pressures [Evs 95, Och 98]. There are also several laser methods for measuring the electric field which are based on the Stark splitting of Rydberg states [Law 94, Gan 86, Alb 93, Heb 94, Boo 94]. The detection of optical-galvanic signals, or the effect of a laser on plasma conductivity, allows the observation of the excitation of Rydberg states and perform sensitive measurements of the electric field. However, the time resolution of this technique is insufficient to carry out measurements in RF discharges. The methods based on the classical laser-induced fluorescence spectroscopy (LIF) are of limited utility because of a rather long lifetime of Rydberg states. For example, the lifetime limits effective LIF field measurements in atomic hydrogen to n = 6 Rydberg state [Boo 94]. The authors of [Cza 98, Cza 99, Czar 99] proposed Fluorescence-Dip Spectroscopy (FDS), a modified LIF method for measuring the electric field from Stark splitting of Rydberg states of atomic hydrogen. This LIF technique is not limited by the lifetime of Rydberg states. In this section, the theoretical foundation of FDS, which was used as the method of measuring the field in the sheath in this study, is developed. Fig.3.7 shows atomic transitions used for FDS in atomic hydrogen. First, hydrogen atoms are excited from the ground state to n = 3 by a two-photon absorption using a 5 nm ultraviolet (UV) laser. This scheme makes it possible to avoid Doppler broadening of the line by using Fig.3.7. Atomic transitions used for FDS in atomic hydrogen. oppositely directed laser beams. The beams are not focused. Since the sheath thickness in an ICP discharge is on the order of mm, which is comparable with the laser beam diameter, 7

hydrogen atoms are excited throughout the sheath. Consequently, the fluorescence of the hydrogen Balmer-α line, 656 nm, can be observed throughout the sheath as well. A second, tunable, infrared (IR) laser is used in the experiment to cause transitions between the n = 3 level and a Rydberg state. If IR laser radiation is tuned to an electric field Stark-split resonance, then electrons will rise from the n = 3 level to the Rydberg state. This causes the n = 3 population to decrease, and will result in a decrease of the Balmer-α line intensity. This dip in intensity makes it possible to reconstruct the transition rate of electrons from n = 3 to the Rydberg state. The electric field strength can then be determined by comparing measured and calculated spectra. Calculations take into account the effect of Stark splitting of both the n = 3 and Rydberg states. An example of calculated spectra for n = 4 Rydberg state is shown in Fig.3.8 [Cza 98]. Peaks in Fig.3.8 correspond to dips in the intensity of the Balmer-α line, where the spectral width of the UV laser is ~ νuv =.5 cm - and the Doppler broadening is of the order of parallel to the electric field. ~ νd =. cm -. The polarization of the IR and UV laser beams is oriented Fig.3.8. Theoretical spectra of the transition between the n = 3 and n = 4 levels of atomic hydrogen as a function of the electric field strength. The polarization of IR and UV lasers is oriented parallel to the electric field [Cza 98]. The n = 4 Rydberg state allows the measurement of electric fields from 5 V/cm to.5 kv/cm. Lower electric fields in the pre-sheath can be measured by using higher principal quantum numbers. For example, n = improves the sensitivity to 5 V/cm. Hence, using n = 4 and n = allows us to reconstruct the time-dependent electric field for the entire sheath. 8

3..3. Ion energy and mass analyzer In this section, we consider the operational principles of a Balzers PPM4, plasma process monitor. This device is either used to measure the ion energy distribution with ions mass being a constant or to measure the ion mass spectra for a constant kinetic energy. The PPM4 scheme is shown in Fig.3.9. Its construction includes an ion-optical system, an ion energy filter, and a quadrupole mass analyzer (QMA). entrance orifice ion-optical system energy filter quadrupole mass analyzer detector PPM4 Fig.3.9. Scheme of the Balzers PPM4, ion energy and mass analyzer. The ion-optical system of the PPM4 is mounted in the front section of the device. It consists of a small entrance orifice and a system of electrostatic lenses. The entrance orifice is electrically isolated and can be biased positively, biased negatively, or floated. In this study, the entrance orifice was grounded. If the bias voltage is lower than the plasma potential, positive ions are extracted from the plasma and enter to the PPM4. A system of electrostatic lenses directs these ions into the entrance of the energy filter, and is designed to transmit % of ions traveling within a small solid angle around of the device axis. An ionization chamber behind the entrance orifice allows neutrals to be analyzed also. The PPM4 uses a cylindrical energy analyzer, shown in Fig.3., as an ion-energy filter. The energy analyzer consists of two axially symmetric cylinders with radiuses r a and r b. The voltage between the cylinders is U p = U b - U a, where U a and U b are the potentials applied to the cylinders with Fig.3.. Schematic of the PPM4 energy filter. radiuses r a and r b. The ions enter into the gap between the cylinders by passing through a hole, F, in the inner cylinder. The ion trajectory is deflected by the electric field produced between the cylinders. The ions can leave the energy filter through a hole, F, in the inner cylinder, if the following equality holds: K ln ( r r ) = eεi U p b a, (3.) Here, ε i is the energy of the ion entering into F, and the factor K depends on the angle between the filter axis and the ion trajectory at its entry into the filter. The maximum energy resolution is achieved when the ion enters at 4 8.5 and K is equal to.3. However, the filter resolution depends weakly on this angle from 4 to 45. When r b / r a = 5/7 for the PPM4, the condition for the ions to pass through the filter can be written as 9

εi eu p =.58 ±.. (3.3) If an ion s energy corresponds to the potential of the inner mirror ε i = eu a it can pass through the filter only if the ratio between the cylinder potentials is U b / U a =.4. By scanning the voltage U a at a fixed ratio U b / U a, it is possible to measure the ion energy. It should be mentioned that, for energy filters consisting of two cylindrical mirrors, the relative energy resolution ε i / ε i is inversely proportional to the distance between F and F and is independent of the ion energy ε i. Since all ions passing through the energy filter have the same energy ε i = eu a, the minimum resolution of the PPM4 is ε i =.3 ev (theoretical limit), and it is a constant over the entire ion energy range. Operation of the QMA, in which ions are selected depending on the mass-to-charge ratio (M i /Q), is shown schematically in Fig.3.. The device consists of four parallel metal rods. Two oppositely placed rods are biased positively (U), whereas the other two are biased negatively (-U). non-resonant ion resonant ion detector ion source ( ωt) V DC and V RF potentials U = U + U cos ~ Fig.3.. Schematic of the quadrupole mass analyzer. The bias is a combination of the constant, U, and RF, U ~, components: ( ωt) U = U + U ~ cos. (3.4) Only ions with a fixed mass-to-charge ratio will be able pass through the QMA. The ion motion trajectory along the axis of the QMA is described by the Mathieu-equations [Veg 3]. A solution to these equations are usually represented as a-q diagrams, where a and q are defined as 8eU a =, r ω ( M Q) i 4eU ~ q =. (3.5) r ω ( M Q) In the a-q diagram in Fig.3. [Bal], there exist two regions in a-q space where an ion has the proper mass-to-charge ratio to transit through the QMA. These regions are designated in the diagram as I and II. Since region II is small in comparison with region I, the mass-tocharge ratio is measured more accurately using region II. However, region II can only be used for measuring light ions. In region II, the values of a and q imply high applied potentials U and U ~. Heavier ions also require a higher bias applied to the rods, and as a result the electric insulation between the QMA rods will be insufficient. i 3

a 3. II 3..8 I.6.4.8.9 3. 3. 3. 3.3 Fig.3.. Solution to the Mathieu-equations represented as a-q diagrams. The first (I) and second (II) regions in the diagram correspond to the parameters of ions that pass through the QMA. Balzers produces QMAs that operate in region II. Their instruments have a high resolution, but are unable to measure ions heavier than helium. For this reason, all standard QMAs, including the PPM4, use region I.A constant ratio between U and U ~ corresponds to a straight line in the diagram: a U =. (3.6) U ( q) q ~ a(q) in equation (3.6) is referred to as the mass scan line. When U / U ~ =, all ions whose mass and charge satisfy the inequality q.95 will pass through the QMA. As the ratio U / U ~ increases, the resolution of the mass analyzer will become better until it reaches its theoretical limit at U / U ~ =.678. The ion mass spectrum can be scanned by either varying the frequency ω at a fixed rod bias U, or by varying the potentials U and U ~ at a fixed ratio. The advantage of these two methods is that measurements of the ion masses are measured with constant resolution. The PPM4 uses the latter method. The range of measured ion masses spans from to 5 a.m.u. Unfortunately, the PPM4 has no software support for the user to calibrate the instrument. All calibrations are done by the manufacturer. The mean energy of ions passing through the analyzer is only the adjustable parameter. q 3..4. Emission Spectroscopy In low-temperature plasmas, light emission is a relaxation of excited states of background species. Since the atomic excitation is predominantly by electron impact, the spectral emissions contain information about the spatial distribution of electrons with enough energy to cause transition [Tur 96]. Most papers on emission spectroscopy analyze the timeaveraged spectral intensity. However, spatial and temporal resolved measurements provide additional information about the plasma formation process and the evolution of RF discharges. Measurements of this kind are described for CCPs in [Rad 95, Toc 9, Wan ], and for ICPs in [Tad 98, Tado 98, Oki 96, Oki 97]. 3

The Balmer-α line can be used for time-resolved emission spectroscopy in hydrogen ICP, because the lifetime of the excited atomic hydrogen is about ns and the RF period is 74 ns. The high intensity of this line also allows us to use short exposure times. It should be remembered that an ICP is maintained by either an E- or H-mode mechanisms. The E-mode is associated primarily with capacitive power deposition into a plasma, and always precedes the inductively coupled H-mode. The mode transition occurs when the efficiencies of the capacitive and inductive mechanisms become equal. The electric field in the E-mode is created by a time-varying voltage applied to the antenna. The motion of electrons in this field resembles the cycle of a mechanical piston. When the antenna potential is positive with respect to the plasma, the electrons move through the sheath from the plasma towards the quartz separator. During the negative half-period, they move away, i.e., from the quartz surface into the plasma where they lose kinetic energy by collisions with neutral species. Consequently, a radiation peak is observed only once during the RF period. The motion of electrons through the sheath and a phase diagram of the sheath potential and plasma heating in the E-mode of ICP are presented in Fig.3.3. For an illustration, consider the Balmer-α intensity measurements for a CCP discharge shown in Fig.3.4 [Rad 95]. Measurements were performed in a GEC reference cell at 3.3 Pa in a 9% argon and % hydrogen gas mixture. Fig.3.4 shows the time behaviour of the Balmer-α line intensity during two RF periods. The time t = corresponds to a maximum electrode voltage, and the distance d is measured from the grounded electrode. From the figure, only one radiation peak appears during a RF period. This is consistent with the capacitive power coupling mechanism described above. -V RF electrode sheath + + - + - + + + plasma - time sheath voltage heating phase time Fig.3.3. A schematic illustration of electron motion through the sheath and a phase diagram of sheath potential and plasma heating in the E- mode of an ICP. Fig.3.4. Intensity of the Balmer-α line in a CCP discharge [Rad 95], shown with a spatial and temporal resolution. 3

H-mode power coupling into a plasma in conventionally similar to power coupling in a transformer, schematically represented by Fig.3.5. Plasma is heated by the electron current induced by the antenna s electromagnetic field. The induced current, and the antenna current, have a harmonic waveform. Since the direction of the induced current is of no consequence for plasma heating, two intensity peaks are observed during a RF period. Fig.3.6 shows optical measurements made in an oxygen ICP in the H-mode [Tad 98]. The power applied to a single-loop antenna was W, and the pressure was 3.3 Pa. Fig.3.6 shows the net excitation rate of the level O 3 (3p 3 P). The radiation intensity can be deduced by integrating the represented data over time. The two peaks in the emission signal during a RF period confirm the dominant role of the inductive power coupling mechanism. antenna E-field plasma I A I Pl B-field current in the plasma heating phase time Fig.3.5.A schematic illustration of the inductive power coupling mechanism in the H-mode of an ICP and a phase diagram of the current and plasma heating. Fig.3.6. Spatial and temporal measurements of the net excitation rate of the level O 3 (3p 3 P) in an oxygen ICP [Tad 98]. Thus, spatially and temporally resolved measurements of plasma emission can serve to identify the ICP mode. Optical spectroscopy is important because it can successfully be used even when other types of mode transition diagnostics cannot. 33

4. Theory This chapter contains the analytical plasma model (Section 4.), the electrical model of an ICP discharge (Section 4.), and the model of the collisionless sheath of an RF discharge (Section 4.3) The combination of the analytical models allows one to perform an analysis of the efficiency of the ICP source and calculate the parameters of the bulk plasma and the sheath plasma. The algorithm for calculations can be clarified using a diagram in Fig.4.. The Analytical ICP Model (Hydrogen) p, p, T gg Plasma Model p, T g, P ξ T e, n e, v eff cell geometry R Electrical Model n e, v eff, P ξ P cap, P ind j cap, j ind PP Sheath Model T e, n, j cap E S, U S j i, j e, j D IEDF Fig.4.. Algorithm for calculating the parameters of an ICP discharge by using the plasma model (Section 4.), the electrical model (Section 4.) and the model of the sheath (Section 4.3). The plasma model is based on the global model [Lie 94] and gives relations between, on the one hand, the pressure in the chamber, p; gas temperature, T g ; power deposited in plasma, P ξ ; and, on the other hand, the density, n e ; electron temperature, T e. In addition, the values of p, T e and n e determine the effective electron collision frequency, v eff, taking into account both Ohmic heating and stochastic heating of the plasma. If the list of specified parameters is supplemented by the antenna resistance, R, then for the calculated values of n e and v eff that determine the resistive and capacitive properties of the plasma, one can calculate the value of incident power, P, in contributions P cap and P ind from capacitive and inductive power coupling into the plasma, respectively. In addition, the electrical model allows one to calculate the current between the plasma and the chamber walls, j cap, and also the current induced in the plasma, j ind. It should be mentioned that in the proposed algorithm, the power P, which is an initial parameter, is calculated from the value of P ξ ; therefore, if necessary, it is possible to perform calculations iteratively, by successively increasing and decreasing the value of P ξ until the specified value of P is obtained. The combination of the plasma model and the electrical model allows one to calculate a characteristic curve of transition between the modes of ICP discharge and to perform an analysis of the efficiency of the ICP source. 34

The electron temperature, T e ; the plasma density at the sheath edge, n ; and the current through the sheath, j cap, which are calculated in the previous stage, are initial parameters for the model of the sheath of an RF discharge. This model allows one to calculate the spatiotemporal distributions of the electric field, E S ; sheath potential, U S ; as well as ion current, j i ; electron current, j e ; and displacement current, j D ; and also to calculate the energy distribution functions of fluxes of H +, H +, and H 3 + ions coming to the electrode. 4.. Analytical model of a hydrogen plasma In this chapter an analytical hydrogen plasma model (AHPM) is discussed. Generally, a self-consistent static plasma model considers the set of equations for particle transport, including the particle and momentum conservations, the electron energy distribution function (EEDF), and the energy balance equation. One can solve this set of equations in a framework of a global plasma model [List 9, Lie 94, Wai 95, Sob 97, Zor, Gud ] which implements the follow simplifications: The electron temperature is a constant throughout the plasma volume. The boundary conditions of the set of equations, which are plasma density and temperature at the sheath edge, are homogenous. Particularly for a hydrogen plasma, the assortment of simplifications can be expanded due to: The EEDF in the hydrogen ICP was previously measured using a Langmuir probe (Subsection 6..). According to these measurements, a Maxwell distribution was assumed throughout. In the hydrogen plasma, all thermal ions H + are almost instantaneously converts into H 3 + due to the very efficient exothermic reaction: H + + H H 3 + + H +.7 ev [Sim 97]. The dominant hydrogen-ion species is H 3 +. In calculations, the ion mass, M i, and ion elastic collision frequency, v M, which is weekly changed at low E/p ratios both can be taking as constants. The set of particle transport equations determines the electron temperature, T e, and the relative plasma density profile, n e /n, which is normalized to the density at the boundary, n. This set of equations has two analytical solutions. An exact one-dimensional solution exists, and a solution which uses a diffusion approximation and assumes a simple, symmetric discharge geometry (spherical, cubic, or cylindrical) exists also. The geometry of the GEC reference cell can be approximated by a cylindrical symmetry. The cylinder height is equal to the cell gap and the radius is the cell radius. The plasma density averaged over the cell volume, n, can be obtained by the energy balance equation. This allows to rescale the relative profile of n e, and to obtain the absolute values. The set of equations of particle transport was solved analytically by using the onedimensional model and the diffusion approximation model assuming a cylindrical symmetry of the cell. The complete solution is presented in Appendix B. Equations in the following sections have an additional label to allow easy reference to Appendix B. 35

4... Profiles of plasma potential and plasma density (relative values) The diffusion approximation neglects the term of the ion inertia in the momentum conservation equation: (B.8) u << v. (4.) After this, the simplified set of particle transport equations describes adequately the main body of the plasma bulk where the ion velocity is much smaller then the Bohm velocity. However, it can falls in the region near the chamber wall where a considerable ion velocity gradient might be present. It is convenient to solve the set of the particle transport equations using dimensionless variables. In the two-dimensional diffusion approximation model, the plasma density distribution can be represented as a product of the axial profile by the radial profile: (B.69) h = n n = h ζ ( ζ) hρ ( ρ) i M, (4.) where n is the plasma density at the sheath edge. ζ = z / Λ and ρ = r / Λ are the coordinates normalized to the ion mean free path at the sheath edge. At a constant electron temperature through the plasma volume, h ρ (ρ) is described by the zeroth-order Bessel function: (B.75) h = ( γρ ρ J ), (4.3) where the parameter γ is determined by the cell size ρ m = r cell / Λ written in the normalized coordinates. A value of γ can be found from the following relation: J ( γρm ) (B.76) γ =. (4.4) J ( γρ ) Fig.4. shows the dependence of the parameter γ on the cell sizes. m The importance of a space resolved modeling is illustrated in Fig.4.3, where the ratio of the plasma density on the discharge axis, h ρ=,to the plasma density averaged over the radius is shown. For most ρ m, h ρ= is twice higher then the average. In a one-dimensional model, this ratio is :.,4,, γ, h ρ = / h ρ,8,6,4, 3 4 5 6 7 8 9,, 3 4 5 6 7 8 9 ρ m ρ m Fig.4.. Parameter γ as a function of the plasma cell radius normalized to the ion mean free path at the sheath edge. Fig.4.3. Ratio of the plasma density on the discharge axis to the plasma density averaged over the radius. 36

The axial profile of the plasma density is described by the following equation: (B.6) * * * * h = cos( κ ζ) + ( κ ) sin( κ ζ), κ = κ γ, (4.5) ζ where κ is the ionization frequency normalized to the ion collision frequency at the Bohm velocity: κ = v i /v M. As was mentioned above, the particle balance equations with the ion inertia term can be solved analytically in the exact one-dimensional model. This model excludes the radial particle transport, and is accurate enough only at low aspect ratios of the plasma cell: ζ m /ρ m. Analytically h ζ (ζ) can be expressed only in an implicit way: (B.5) [ ] +κ h = ( + κ ) ( w + κ ), (4.6) ζ + κ (B.49) w ζ = arctan + arctan ( + w). (4.7) κ κ κ The equations are linked by the ion axial drift velocity, w = u i / u B, normalized to the Bohm velocity. In the limiting case κ = one can express the ion velocity and the density profile as a function of coordinate explicitly: (B.53) w = ζ 4 + ζ ζ. (4.8) This solution applies with reasonable accuracy for finding κ in the region close to the sheath edge: ζ < κ -. The difference between the plasma potential and the potential at the sheath edge can be calculated from the Boltzmann equation: (B.5) = U U = ln( h) ϕ, (4.9) where U = kt e / e is the electron energy. Three examples of the axial plasma density and potential profiles (for the ionization frequencies κ, κ * =., κ, κ * =., and for the limiting case of κ, κ * = ) are presented in Fig.4.4. The position ζ = is the edge of the sheath. h ζ (a) 4 8 6 4 κ, κ* =, κ, κ* = κ, κ* =, exact solution diffusion approximation 4 6 8 4 6 ζ ϕ 3,,5,,5,,5 κ, κ* =, κ, κ* = κ, κ* =, exact solution diffusion approximation, 4 6 8 4 6 Fig.4.4. Axial profiles of (a) normalized plasma density and (b) normalized plasma potential calculated by using the exact model in one dimension (solid line) and the diffusion approximation (dashed line). (b) ζ 37

Fig.4.4 shows the small difference between the profiles calculated by two models, which is a consequence of neglecting the particle inertia term in the diffusion approximation model. For this reason, the exact one-dimensional model is preferable when analyzing the measurements carried out on the discharge axis and especially close to the electrode. 4... Electron temperature As can be seen from diagrams in Fig.4.4, the plasma density and potential profiles depend on the ionization frequency κ. Maximum values of the density and the plasma potential corresponds to the center of a discharge ζ = ζ m. As κ increases, ζ m decreases. The relation between ζ m and the ionization frequency of a plasma in the case of the onedimensional model takes the following form: + κ (B.64) ζ m = arctan, (4.) κ κ in the case of two-dimensional model diffused approximation has the following form: (B.55), (B.63) κ κ * ζ m = arctan, κ = κ γ * *. (4.) The characteristic plasma dimension in the axial direction is determined by the height of the plasma cell: ζ m = h cell / Λ. For a given aspect ratio of the plasma cell ζ m : ρ m, the parameter γ can be found from equation (4.4). For the GEC reference cell, the height to radius ratio is 5 : 3. Fig.4.5 shows the ionization frequency of a plasma as a function of the plasma cell height. On the other hand, the normalized ionization frequency of a hydrogen plasma can be calculated from a ratio of the coefficients of elementary collision processes: ki ( i) 5 (B.65) κ = = < σ j ( u) u >, k =,8 m 3 M s -, (4.) k k where σ j (i) M M j denotes the corresponding cross-sections of the ionization processes in hydrogen, presented in Fig.4.9. Fig.4.6 shows the result of the calculations. Equations (4.)-(4.) determine the relation between the dimensions of a plasma cell and the electron temperature. one-dimension exact solution two-dimension diffusion approximation (ζ m : ρ m = 5 : 3),5, κ κ,5,,5, 4 6 8 4 Fig.4.5. Normalized ionization frequency of a hydrogen plasma versus axial cell dimension normalized to the ion mean free path. ζ m,,,,,3,4,5,6 Fig.4.6. Normalized ionization frequency of a hydrogen plasma versus electron temperature. τ 38

One can pass to dimensionless variables by dividing the thermal energy by one of the energy constants. For example, the ionization energy of hydrogen ε H = 5.4 ev can be used for this purpose. The electron temperature is written as τ = kt e / ε H. The ion mean free path Λ, which was chosen as a normalization factor of the plasma dimensions, depends on the electron temperature T e and on the neutral gas density N in the vacuum chamber. At a given geometry of the plasma cell, the normalization allows us to derive a relation between T e and N: N N (B.66) η = = τ ζ ( κ) m, N εh =. (4.3) M k h Depending on the model used in analysis, the function ζ m (κ) is described by equation (4.) or (4.). Fig.4.7 shows the result of calculations performed for a hydrogen plasma generated in the GEC reference cell. One can see that at high gas pressures and, accordingly, low electron temperatures T e < 3 ev, the calculations performed in the one-dimensional and twodimensional models give similar results. However, there is a substantial deviation between the curves at low pressures and high temperatures, T e > 3 ev. Hence, the effect of the plasma cell geometry on transport processes increases with decreasing gas pressure. For ideal gases, at a given gas pressure in the chamber, the gas density is inversely proportional to the temperature. Consequently, as the temperature increases, the gas density decreases: (B.68) η η = + τ g, ( ) g g () g i M cell τ = T T, (4.4) where η () = η(p = ) and T g () = T g (P = ) are the pressure and temperature in the absence of a discharge, respectively. τ g is the relative increase in the gas temperature in a discharge. As the gas temperature increases, the electron temperature increases only slightly.,8,7,6 one-dimension exact solution two-dimension diffusion approximation (ζ m : ρ m = 5 : 3),8,7,6 η () = τ,5,4,3,, τ,5,4,3,, η () = η () =, - 3 Fig.4.7. Electron temperature versus density of neutral gas consisting of molecular hydrogen H. η,,,5,,5,,5 3, Fig.4.8. Electron temperature versus increase of neutral gas (H ) temperature in discharge for different pressures. τ g Fig.4.8 shows the dependency of the electron temperature on the quantity τ g at several fixed pressures in the chamber. As one can see, the effect of electron heating in the discharge is more pronounced for low gas pressures. 39

4..3. Energy balance The global model accounts for three forms of energy losses coupled into a plasma: the energy loss by all kinds of inelastic electron collisions with molecular hydrogen, the potential energy loss by charges moving in the electric field of the sheath and pre-sheath regions, and the kinetic energy loss due to particles impacting the chamber walls. The energy loss rate, using these processes, can be written in the following form: ( ) ( ) (B.6), (B.35) R ε = R ε + [ e( U U S ) + αkt e ] k i, R ε = ε jk j (4.5) where R ε () is the energy loss through electron collisions in the plasma bulk. U and U S are the potentials of the pre-sheath and sheath regions, respectively. αkt e = Γ ε / Γ is the ratio of the energy flux to the particle flux, i.e., the kinetic energy loss due to one particle impacting the chamber wall. Depending on the choice of the theoretical model, the coefficient α varies from to.5. We use the coefficient α = in our study, because it is appropriate for the geometry of a small plasma cell. To calculate the energy loss R ε () in a hydrogen plasma, one should take into account the polarization scattering and all impact collisions of electrons against a background gas (see Subsection B..). However, because of the low ionization frequency of hydrogen plasma (usually κ ~ - - - ) and the low density of atomic hydrogen (on the order of one percent), the various processes of electron collisions with hydrogen ions and atomic hydrogen is insignificant to the energy balance and can be neglected. For the same reason, electron collisions with hydrogen molecules in excited states, for example, the electron attachment reaction e + H (v 4) (H - ) *, can be neglected too. Eventually, the energy losses are determined only by inelastic electron collisions with molecular hydrogen. These elementary processes are presented in Table 4.. Table 4.. The elementary processes of electron impact collisions with molecular hydrogen. + elementary process e + H X Σ g ) e + H ( v) + e j threshold energy (ev) + + ( ε =5. 4 i + g ( Σ ) e + H ( X Σ g ) e + H ( Σ g, Σu ) + e e + H + H (s ) + e u ε = 8; ε = 6 + + = ) + H + ( v= ) ( v= ) ( v= 3) e H v e ( v,, 3) ε =.5; ε = ; ε =. 5 ( = + * + ( ) e + H ( X Σ g ) e + H ( B Σu p σ) ε B exc =. 37 + * + ( ) e + H ( X Σ g ) e + H ( C Πu p π) ε C exc =. 7 + * + ( E, F ) e H ( X Σ ) e + H ( E, F Σ ) ε =. + g g + * 3 + 3 + 3 e + H ( X Σ ) e + H ( b Σ, a Σ, c Π ) e + H (s ) H (s ) g u g u + ε exc ( b) diss ( Σ i ) = 8.5; ε exc exc ( b) diss i =.7; ε + * * ( ) e + H ( X Σ g ) e + H (sσ g, nl λ Λ) e + H (s ) + H (s) ε Λ diss = 4. 9 e + H + * * * ( X s ( ) ε Π diss Σ g ) e + H (pσ u, nlλ Q Πu ) e + H ( p) + H ( ) = 3 + * e + H ( X Σ g ) e + H ( pσu, n = 3) e + H (s ) + H ( n = 3) = 9 * ( 3) ε n= diss exc ( c) diss =.7 4

Fig.4.9 shows the rate coefficients for the selected reactions as a function of the electron temperature. Most of the coefficients were taken from [Jan 87], whereas the others were calculated by using the cross-sections presented in [Phe 85], and averaging over the Maxwell distribution. -4 e + H (X Σ g + ) e + H + (v) + e ionization -4 excitation of vibrational levels < σv > (m 3 /s) -5-6 e + H (X Σ g + ) e + H + + H (s) + e < σv > (m 3 /s) -5-6 v = v = e + H (v = ) e + H (v =,, 3) v = 3-7 3 4 5 6 7 8 kt e (ev) 9-7 3 4 5 6 7 kt e (ev) 8 9-4 -5 e + H (X Σ g + ) e + H * (B Σ u + p σ) e + H (X Σ g + ) e + H * (C Π u p π) e + H (X Σ g + ) e + H * (E, F Σ g + ) excitation -4-5 e + H e + H (s) + H (s) e + H e + H (s) + H* (s) e + H e + H* (p) + H* (s) e + H e + H (s) + H* (n = 3) dissociation < σv > (m 3 /s) -6 < σv > (m 3 /s) -6-7 3 4 5 6 7 8 9 kt e (ev) -7 3 4 5 6 7 8 9 kt e (ev) Fig.4.9. Rate coefficients of electron impact collisions with molecular hydrogen versus electron temperature. The constant coefficient for the electron momentum transfer to molecular hydrogen: and the mean energy lost per electron collision: (B.5) m = 3( m M ) kte are used for the polarization scattering [Lie 94]. 3 3 km = vm N =.86 m s (4.6) ε (4.7) The value of U in the energy balance equation (4.5) characterizes the mean energy lost by an electron moving from the plasma volume to the sheath edge. The potential is averaged over the plasma density profile: (B.3) ϕ = U U = ζm h ϕdζ ζ ζm h dζ. (4.8) ζ 4

ϕ, ϕ m 5 4 3,, Fig.4.. Potential averaged over the density profile and potential at the center as functions of the normalized ionization frequency. κ ϕ ϕ m The exact value ϕ can be calculated by numerically integrating this formula. However, this procedure can be avoided, since the value of ϕ can be estimated from the plasma potential at the center of a discharge ϕ m, which is described by a simple formula + κ m = ln + (4.9) κ ϕ Fig.4. shows the result of calculations performed in the one-dimensional model. As can be seen from the figure, for a hydrogen plasma, the difference between the potential values ϕ and ϕ m is smaller than 5%. A similar relationship is observed for the twodimensional model. A steady state system implies a local balance between the negative electron and positive ion currents which is assumed to apply locally at the wall. Hence, the voltage drop across the electrode region U S can be determined from the floating-potential condition: (B.38) U S kte M i = ln e πm e. (4.) Fig.4. shows the energetic loss rates R ε, R ε () and the energy loss per electron E T in a hydrogen plasma: (B.37) E T = R k. (4.) ε i - 7 6-3 5 R ε (evm 3 /s) -4 R ε R ε () E T (ev) 4 3 (a) -5 3 4 5 6 7 8 9 kt e (ev) 3 4 5 6 7 8 9 kt e (ev) Fig.4.. Energetic loss rate (a) and energy loss per electron (b) in a hydrogen plasma versus the electron temperature. (b) It should be mentioned that the energy balance could also be calculated without the electron density profile for the average potential U. The authors of [Lie 94] used the constant value U = kt e /. Indeed, the result of calculations presented in Fig.4.6 and Fig.4. shows that this simplification is invalid for temperatures kt e < 5 ev (κ < ). In this case, according to the diagram in Fig.4., the energy loss is predominantly by inelastic electron 4

collisions in the plasma bulk. As a consequence, the energy loss due the particle flux onto the chamber wall is somewhat underestimated in this case. For a hydrogen plasma this effect is of little consequence, and this approach is valid. 4..4. Plasma density (absolute values) One can determine two important average values of the electron density, the density n averaged over the discharge volume V: (B.34) Pξ n = n dv = V VNR and the density n averaged over the discharge surface A S : (B.36) V S AS B ε T (4.) Pξ n = n da = A, (4.3) Su E where P ξ = ξp is the energy coupled into plasma through the antenna and ξ is the energy input efficiency. The calculated value of n is used to calculate the absolute plasma density distribution along the discharge axis. Since the GEC reference cell aspect ratio, h cell : r cell = 5 : 3, favors the axial particle transport, the best way to simulate the density profile is as follows. The twodimensional profile of the diffusion approximation is rescaled using the density averaged over the entire discharge volume, n ; which is obtained from the energy balance equation. It is because this profile accounts for the energy losses due to a particle flux in the radial ambipolar field, while the one-dimensional profile does not. The two-dimension profile gives the average density over the discharge axis, n r=. Finally, the absolute density profile at the discharge axis is calculated using n r= and the relative density profile obtained by the exact one-dimension model. 43

4.. Electrical model of an ICP source As noted in Sections.3 and.4, formal treatment of a plasma source as an electric circuit allows one to investigate different methods for delivery of the electromagnetic field energy into a plasma [Ent 99, Ike ]. In an ICP discharge, the main energy is deposited inductively. Consequently, the equivalent circuit of the discharge is a transformer. Its primary coil is an antenna, and the secondary is the plasma. Accordingly, the electrical model of an ICP discharge is called a transformer model. By measuring the antenna impedance in an ICP, it is possible to estimate certain average plasma parameters, for example, the conductivity and the electron density [Colp 99, Gud 98, Pie 9, Sob 98, Yan 99, Yos, You 99]. The relative simplicity of these measurements, even in the case of a pulsed discharge [Guo ], makes the electrical model a convenient tool for controlling and diagnosing an RF discharge. The analytical hydrogen plasma model (AHPM) considered in Section 4., after being appended with an estimate of the capacitive and inductive energy deposition efficiency, allows one to calculate average plasma parameters and predict a transition between the operation modes of an ICP. The algorithm for calculations is shown in Fig.4.. The basic transformer model, as applied to a planar ICP, is discussed in detail in Appendix C. In this Section, only the outline of the model is presented (Subsections 3.. and 3..). More attention is paid to how to model a hydrogen plasma in the GEC reference cell (Subsection 4..3). The model accurately estimates the efficiency of this plasma source (Subsection 4..4); and as a result, some improvements in the source design have been proposed (Subsections 3..5 and 3..6). 4.. Plasma configuration in a planar ICP The numerically calculated electric field induced in a vacuum in the GEC reference cell is shown in Fig.4.. The field amplitude is normalized to the antenna current. The top of the figures corresponds to the inner surface of a quartz window that separates the antenna from the plasma, whereas the bottom corresponds to the electrode surface. vertical position (mm) 3 4 7.3 8. 5 6.4 5 6 8 E r / I (V/m / A) 7 5 4 vertical position (mm) 3 4 - -7-6 -5-4 -3-7 5 6 4 3 E ϕ / I (V/m / A) 3 4-5 5 - -5 5 - -5 5 radial position (mm) radial position (mm) (a) radial E-field component (b) azimuthal E-field component Fig.4.. The numerically computed electric field induced in a vacuum in the GEC reference cell. 44

Looking at the figures, the radial component of electric field, E r, is smaller than the azimuthal component, E ϕ. In a first approximation, E r can be neglected and thus E E ϕ. The E-field has a characteristic spatial structure. In the radial direction, the field reaches its maximum at r = /3 r ant, where r ant is the antenna radius. The induced electric field is zero on the discharge axis and for r approaching infinity. As an example of the field structure, Fig.4.3 shows the azimuthal component of the E-field in the plane located at the distance h = mm from the antenna surface. The circle of radius r = /3r ant, which is traced in Fig.4.3 by a white dashed line, indicates the maximum of the induced electric field. In the axial direction, the E-field is approximately E / I = 8 Vm - A - near the quartz surface and rapidly decreases as distance from antenna increases. In the presence of a plasma, the electric field decreases more rapidly because of the skin effect. radial position (mm) 5-5 E ϕ / I (V/m / A) -- -- -- 3 3 -- 4 4 -- 5 5 -- 6 6 -- 7 7 -- 8 8 -- 9 E / I (V cm - A - ),7,6,5,4,3, P P ξ = P + P P = P -, P - -5 5 radial position (mm) Fig.4.3. Azimuthal component of the E-field at a distance mm from the antenna surface. The circle of radius r = /3 r ant traced by a white dashed line indicates the field maximum., 3 4 vertical position (mm) Fig.4.4. Axial profile of the squared E-field at r = /3 r ant, where r ant is the radius of the antenna. The squared E-field is proportional to the power dissipated in plasma. 5 The discharge is sustained due to Ohmic heating. Under the assumption of uniform plasma conductivity, the spatial distribution of the absorbed power P ξ is governed by the squared electric field: dpξ r r r (C.) < > =< jee > ~ < E >. (4.4) dv Fig.4.4 shows the axial profile of (E / I ) at the radial distance r = /3r ant from the discharge axis. It is seen that more than half the power deposited in this plasma is absorbed in a layer with a thickness less than cm. The calculation results show that the plasma region heated by the electromagnetic field in a planar discharge can be considered as a single turn of finite size in an equivalent circuit. Its characteristics will be determined by both the design features of the plasma cell and the spatial structure of the field penetrating into the plasma. 45

Fig.4.5 presents the two designs of a plasma cell. The design in Fig.4.5(a) corresponds to a standard GEC reference cell, whereas Fig.4.5(b) corresponds to a modified cell, in which the antenna housing is made of only dielectric material. quartz antenna coil r ant quartz antenna coil r ant S eff h pl h eff h pl metal heated plasma region r pl = r met S eff heated plasma region r eff /3r ant r pl = r eff electrode electrode (a) standard GEC cell (b) plasma cell with dielectric antenna housing Fig.4.5. Schematic of an ICP discharge in the GEC reference cell. A plasma is represented in the form of a turn of uniform conductivity. The vertical cross-section of the plasma turn has the horizontal dimension r pl, which is determined by the cell design, and the vertical dimension h pl being varied depending on the discharge conditions. Electron heating in an ICP has the form of a torus located near the quartz surface. The vertical cross-section of the torus S eff has an outer radius of r pl and a height of h pl. In our case, the inner radius is zero, because there is no decline in the electron density at the discharge axis. The torus radius, r eff, is equal to one-half of r pl, and is determined by the plasma cell design. In the GEC reference cell, r pl is equal to the inner radius, r met, of the metal cylindrical case. When the whole antenna housing is a dielectric, r pl is determined by the structure of E-field. The parameter h pl varies depending on the discharge conditions. While modelling the discharge, a useful approximation is to assume h pl is equal to the skin-layer thickness δ pl. h pl can be calculated by measuring the transformer ratio of the transformer formed by the antenna and plasma. By specifying the plasma geometry, it is possible to calculated inductances used for the transformer model. Computational results are shown in Fig.4.6. Subscripts indicate the coil number, where L is the antenna inductance, L is the plasma turn inductance, and L is the mutual inductance between the antenna and plasma. A superscript (*) marks quantities related to the GEC reference cell when the antenna housing is made of a conducting material. When h pl is higher both the electron density and the mutual inductance between the antenna and plasma is lower. The intrinsic inductance of the plasma will only change slightly. A similar dependencies was observed in the transformer model discussed in [Lis 9, God 98]. However, it should be noted that, in [Lis 9, Gud 98], the plasma configuration and the technique for computing inductances were significantly different from the above discussion. 46

ωl = Ω ωl ωl ωl * = Ω ωl * ωl *,6,5 k k* 8,4 ωl (Ω) 6 4 k,3,, (a) 3 4 5 h pl (mm), 3 4 5 h pl (mm) Fig.4.6. The inductances (a) and transformer ratios (b) computed within the electrical model applied to the GEC reference cell. A superscript (*) marks calculations taking into consideration the high conductivity chamber material. In the electro-technical applications, the ratio k, determined by the following relation between the inductances concerned (C.9) k = L LL, (4.5) is one of the main transformer parameters. As seen in Fig.4.6, the transformer ratio attains its maximum value of about k =.5 at high plasma densities. When the electron density decreases, and when h pl becomes comparable to the plasma cell height, k will decrease by a factor of about. It should be noted that the use of high-conductivity materials in plasma cells results in a reduction in the transformer ratio k. For weakly ionized gases, this effect can greatly reduce the plasma source efficiency. (b) 4.. Set of equations for the transformer model In this model concerned, an ICP is considered as a transformer whose primary coil is an antenna and secondary coil is a plasma. The electrical parameters of the antenna are the resistance, R, and inductance, L, and can be determined during the process of calibrating the experiment. The plasma is characterized by the resistance, R, the geometrical inductance, L, ( e) and an additional inductance, L, arising due to the inertia of electrons [Lis 96]: (C.) ( e) L R = v, (4.6) where v eff is the effective electron collision frequency. v eff can be found by summing up the elastic electron collisions frequency, v m, which corresponds to Ohmic heating of the plasma; and the stochastic frequency, v stoch, which characterizes collisionless heating of the plasma [God 94, God 98, Gody 98, Haa, Kag 96, Kos, Smi 97, Tur 93, Lie 98, Wan 99]: (C.)-(C.4) v = v + v, eff m stoch eff 3 ωωekte vstoch. (4.7) mec The total impedance of the transformer, Z tr, can be calculated by solving Ohm's law for primary and secondary circuits: 47

(C.6) ( ωl ) (C.7) L = ωl ( ωl + ωr ν ) R tr = R + R, (4.8) R + ( ωl + ωr ν ) eff ( ωl ) ω tr eff (4.9) R ( ) + ωl + ωr νeff One should take into account that, in an ICP, there is also a capacitive energy deposition mechanism via the capacitance C ant connected between the antenna and plasma. For an ICP source with an external antenna, this capacitance is formed by the dielectric window and the plasma sheath. The sheath impedance is usually much lower than the dielectric window impedance; hence, the former can be neglected during calculations. The plasma resistance, R pl, which is responsible for the capacitive energy deposition into the plasma, is composed of R Ω and R stoch. These are related to the Ohmic and stochastic heating, respectively [Lie 94]. R pl can be calculated as follows: (C.6), (C.6) R pl m e 8kT = Ω + = e R Rstoch v h +, (4.3) π e ( ) m cell r + π ant r m el e ne where r ant and r el are the radii of the antenna and electrode. Accounting for radial current to the conducting chamber wall, one should multiply R pl, from equation (4.3) by the coefficient W R, which is determined by the geometrical size of the plasma cell: (C.6) W R + w = + w * R * R, w * R rant + r = hcell el rcell ln rant + r el, (4.3) where h cell and r cell are the plasma cell height and radius. In the general case, the formulas for calculating the real, R eff, and imaginary, L eff, parts of the effective antenna impedance in an ICP are complicated: (C.48), (C.56) (C.49), (C.56) R eff ( Rpl + Rtr ) + ( ωltr ) Rpl + Rtr ( ωcant ) ( R + R ) + ( ωc ωl ) Rpl Rtr =, (4.3) tr pl pl tr ant ( R + R ) + ( ωc ωl ) tr ( ωc ωl ) R ωcant + R ωltr + ωltr ωcant ant tr ω L =. (4.33) eff pl In the H-mode of an ICP, the impedances used in equations (4.3) and (4.33), are related as follows: (C.5) R R << ωl << ωc. (4.34) pl tr Simplified expression for the effective antenna impedance can be deduced by using the previous relations: R = R + ωl ωc, (4.35) (C.54) ( ) (C.55) L = ωl ( + ωl ωc ) eff eff tr tr tr tr ant ant tr tr ant ant ω. (4.36) In inverse problem arises while measuring. Namely, it is reconstruction of the plasma parameters from the measured antenna impedance. This can be done in the following way. tr 48

According to equations (4.35) and (4.36), considering only inductive energy input, the transformer impedance can be calculated from the relations: R = R ωl ωc, (4.37) (C.57) tr eff ( eff ant ) (C.58) L = ωl ( ωl ωc ) ω. (4.38) tr eff eff ant Then, one can calculate the transformer ratio k by using equations (4.5), (4.8) and (4.9): (C.) ( Rtr R ) + ( ωl ωltr ) ωl ωl ω ν ( R R ) k =. (4.39) ωl [ ] As is seen in Fig.4.6, knowing the ratio k enables determination of h pl ; and, correspondingly, all the inductances involved in the transformer model. Next, obtaining the plasma resistance R by the formula: (C.) R ωl tr eff tr ( Rtr R ) ( R R ) =, (4.4) ωl ωltr ω νeff tr allows one to determine the conductivity and the plasma density: (C.44) (C.45) n pl π σ pl =, (4.4) R h m ( ω e e = e ν + ν eff eff ) σ pl. (4.4) The effective impedance, Z eff, can be calculated from phase-resolved measurements of the antenna current and voltage. When an auto-matching network is used, Z eff can be obtained by the following formulas: (C.64) (C.65) RRF R eff =, (4.43) R RF ( ωc ) + load RRFCload L eff = +, (4.44) ω C R tune RF ( ωc ) + where C load and C tune are the current capacitances of the variable capacitors in the matching device and R RF is the internal resistance of RF generator (typically 5 Ω). The plasma conductivity and density therefore can be derived from experimental data by employing the electrical model of an ICP. load 49

4..3 Modeling an ICP source The electrical characteristics of the ICP in hydrogen can be derived from the transformer model proposed in the preceding section. To do this would require knowing the electron temperature, T e, and density, n e ; as well as the gas pressure in the chamber, p. The pressure is used in determining the frequency of elastic electron collisions in formula (3.3). The dependence T e (p) can be obtained within the framework of the AHPM discussed in Section 4.. Fig.4.7 presents numerical results performed using the standard GEC reference cell geometry, and without allowance for heating of neutral gas in the discharge (i.e., at T () g = 3 K). The parameters n e and p can be taken as the starting point when modeling an ICP source [Kus 96]. The antenna inductance, L, the resistance, R, and the capacitance, C ant, of the quartz window separating the antenna and plasma can be measured or calculated without a plasma in the chamber. For a GEC reference cell, the parameters are follows: R. 7 Ω, L =.3 H, and C 37 pf. (4.45) µ The resistance of the plasma, R pl, between the antenna and electrode can be calculated using formulas (4.3) and (4.3). Since R pl is inversely proportional to the electron density, n e ; it is reasonable to present the numerical results, shown in Fig.4.8, as the pressure dependence of the product R pl and n e. ant 8 3x 7 7 R Ω n e R stoch n e R pl n e = (R Ω + R stoch ) n e T e (ev) 6 4 R pl n e (Ωm -3 ) 6, p (Pa) Fig.4.7. The electron temperature in a hydrogen plasma versus gas pressure in the GEC chamber. 5, p (Pa) Fig.4.8. The product of the electron density and the plasma resistance versus gas pressure. As is seen in Fig.4.8, at low gas pressures, p <.5 Pa, the main mechanism for sustaining the discharge is stochastic heating. At high pressures, p > Pa, the resistance R Ω significantly exceeds the resistance R stoch, and the process of stochastic heating can be ignored. For pressures within the range.5 - Pa, typical of an ICP discharge, both heating mechanisms are important. According to Fig.4.6, the inductances L and L are determined by how far, h pl, the electromagnetic field induced by the antenna penetrates into the plasma. If the electromagnetic field penetrates a distance shorter than the plasma cell height, h pl < h cell ; then h pl is equal to the skin layer thickness, δ pl. For a homogeneous conducting medium, δ pl 5

depends only on the electron density, n e ; the effective frequency of electron collisions, v eff (4.7); and the applied frequency, ω: (C.46) δ pl c = Im ω e ω ω( ω iv eff ), (4.46) where ω e is the electron plasma frequency (3.). Fig.4.9 shows the dependence of v eff and δ pl on the electron density for different pressures. 5 ν eff (Hz) (a) 9 8 7 p =, Pa p = Pa p = Pa p = Pa 8 9 3 n e (cm -3 ) δ pl (mm) 4 3 p =, Pa p = Pa p = Pa p = Pa 3 n e (cm -3 ) Fig.4.9. The effective electron collisions frequency (a) and skin-layer thickness (b) versus electron density. If the electron density is low enough the electromagnetic field can penetrate as far as the electrode surface. Then h pl will be independent of the discharge conditions and determined by the plasma cell heght: h pl = h cell < δ pl. (b) The resistance R and the plasma inductance that are used in the transformer model, can be calculated from equations (4.4), (4.4) and (4.6). The complex impedance of the transformer composed of antenna and plasma, Ztr, can be obtained from equations (4.8) and (4.9). Numerical results are shown in Fig.4.. L ( e) R tr * (Ω) (a) 8 7 6 5 4 3 8 p =, Pa p = Pa p = Pa p = Pa 9 3 n e (cm -3 ) ωl tr * (Ω) 98 96 94 9 9 p =, Pa 88 p = Pa 86 p = Pa p = Pa 84 8 9 3 n e (cm -3 ) Fig.4.. The real (a) and imaginary (b) parts of the transformer impedance composed of antenna and plasma versus electron density in the discharge. R tr (n e ) has a single maximum which corresponds to the highest energy deposition efficiency into the plasma via the transformer. At an electron densities n e < cm -3, the (b) 5

mechanism for inductive energy deposition becomes inefficient. All the incident power will then be dissipated on the antenna s internal resistance: R tr = R. Further, the transformer inductance is equal to the antenna inductance, L tr = L. However, at high electron densities, n e > cm -3, L tr decreases. When including capacitive coupling of energy into the plasma, the real, R eff, and imaginary, L eff, parts of the complex antenna impedance are given by the equations (4.3) and (4.33). They are shown in Fig.4.; and demonstrate that the behaviour of the effective antenna impedance Z eff is different from that of the transformer impedance Z tr. R eff * (Ω) (a) 9 8 7 6 5 4 3 p =, Pa p = Pa p = Pa p = Pa 8 9 3 n e (cm -3 ) ωl eff * (Ω) 5 5 5 p =, Pa p = Pa p = Pa p = Pa ωl * = Ω 95 8 9 3 n e (cm -3 ) Fig.4.. The real (a) and imaginary (b) parts of the antenna impedance in an ICP discharge versus electron density. The calculations were performed using the transformer model including capacitive energy deposition into the plasma. R eff (n e ) has two maxima that correspond to the optimum conditions for capacitive and inductive coupling of energy deposition into the plasma. These maxima are well resolved at any neutral gas pressures. The electron density corresponding to most efficient E-mode coupling is 3 to 4 orders of magnitude less than the electron density for the most efficient H-mode coupling. Further, incorporating the capacitive energy coupling into the transformer model significantly affects the dependence of the effective inductance, L eff, on the electron density. At the instant the discharge switches on, L eff is equal to the antenna inductance, L, and increases slightly with increasing plasma density. Upon reaching the density corresponding to the ICP mode transition; L eff becomes stable; and does not change with increasing n e. It will begin to decrease when the discharge starts to operate in the H-mode though. The transformer inductance, L tr, always satisfies L tr L ; while the effective antenna inductance, L eff is higher than the antenna inductance, L, for almost all electron densities. The effective antenna impedance, Z eff, is transformed by two variable capacitors in the matching network into the output resistance of the RF generator, R load. Minimum power loss due to internal resistance of the RF-oscillator, R RF (typically 5 Ω), and due to a power reflection is achieved if R load = R RF. In this case, the impedance matching condition is described by equations (4.43) and (4.44). The calculations performed show that the R eff and L eff values in an ICP discharge correspond to the capacitances C load = 47-64 pf and (b) 5

C tune = 8-4 pf. However, in most commercial matching networks, the capacitances of both variable RF capacitors combined are only C = 5-5 pf, which is insufficient for matching the real part R eff of the antenna impedance in an ICP. The addition of 5 pf in parallel to the variable capacitor will enable the network to match the real antenna impedance. R eff resistances.6 Ω and 9. Ω correspond to C load capacitances 5 pf and pf, shown as dash-dot lines in Fig.4.(a). When R eff <.6 Ω or R eff > 9. Ω, there will be an impedance mismatch between R load and R RF. This leads to increased power loss in the internal resistance of an RF generator and a decrease in the efficiency of energy deposition into the load resistance, R load : Pload 4Rload RRF ξ load = =. (4.47) P R + R ξ load (%) 9 8 7 6 5 4 3 R eff = Ω R eff = Ω C load = 5 pf C load = pf 4 6 8 4 6 8 R load (Ω) Fig.4.. Efficiency of the RF power deposition into the load resistance. ( ) load RF ξ load (R load ) is shown in Fig.4.. The dash-dot line shows the maximum efficiency at R RF = 5 Ω when perfect matching is obtained. The two dashed lines in Fig.4. show the maximum mismatch of R load from R RF when the ICP discharge is operating beyond the matching network s range in Fig.4., either. Ω < R eff <.6 Ω or 9. Ω < R eff <. Ω. The variable capacitor in the matching network is either at its maximum or minimum, respectively. As is seen in the figure, the internal dissipation of power in the generator can be as high as %. It should be noted that in commercial matching networks, the option of auto-matching functionality ceases to work for plasma densities corresponding to the ICP mode transition, where R load R RF. However, according to Fig.4., L eff remains constant in this density range, and, therefore C tune will not need adjusting. If the auto-matching function is switched off upon reaching the maximum value of C load, the matching network will be set so that its capacitances are optimized for matching impedance. In an ICP, the power applied to the antenna can be represented as the sum of the three components: (C.7) P = P + P + P, (4.48) loss where P loss is the power loss in the antenna resistance, and P cap is the power deposition by the capacitive energy coupling, and P ind is the power deposition by the inductive energy coupling. In this case, the energy deposition efficiency into the plasma consists of two parts: (C.7) ξ = Pξ P = ξ + ξ, (4.49) cap cap ind ind 53

where P ξ is the power absorbed in plasma, ξ cap is the capacitive energy deposition efficiency, and ξ ind is the inductive energy deposition efficiency. The efficiencies are given by the following relations: ξ (%) (C.75) (C.73), (C.76) p = Pa of H gas 9 8 7 6 5 4 3 eff pl Pcap R + ( ωleff ) Rpl ξ cap = =, (4.5) P R + ( ωc ) R ind 8 9 3 n e (cm -3 ) Fig.4.3. Efficiency of the power coupling in the GEC reference cell versus electron density. ant P ind Rtr R = ( ξcap ) P Rtr ξ =. (4.5) ξ ξ cap ξ ind eff The dependence of the energy deposition efficiency into the plasma on the electron density is shown in Fig.4.3 at a pressure of p = Pa. At low electron densities, an ICP runs in the E-mode, in which case ξ cap > ξ ind. For high pressures, it operates in the H-mode, where ξ cap < ξ ind. The condition ξ cap = ξ ind corresponds to the threshold value of n e = 9 9 cm -3, and is where the transition between the modes occurs. As is in Fig.4.3, the ICP source efficiency is at a minimum in the transition region because capacitive energy deposition is no longer efficient, and the inductive heating of the plasma has not yet become effectual. Here, up to 8% of the incident energy is lost and only % is spent on sustaining the discharge. The plasma source efficiency, ξ(n e ), shown in Fig.4.3 demonstrates a well-known property of ICPs. The electron density will increase not linearly with the incident power during the mode transition. This phenomenon is illustrated in Fig.4.4, where the straight line shows the power P ξ sufficient for creating a specific plasma density. P ξ is calculated using equation (4.) from the analytical model of a hydrogen plasma discussed in Section 4.. In Fig.4.4, there are also several power deposition dependences, P ξ (n e ) = ξp, shown where ξ is the power deposition efficiency calculated using the transformer model. The dependences are plotted in steps of W for the incident power. The intersection point between the straight line and these curves corresponds to the existence conditions for an ICP. The irregular distances between the intersection points reflect the nonlinear dependence of the electron density on the power applied, n e (P). This dependency is plotted in Fig.4.5. The electron density increases faster when the discharge operates in the H-mode rather than the E-mode for an equal increase in incident power. There is no strict definition of the threshold input power P t, at which the transition between the modes of ICP occurs. For example, in [Lie 94], the threshold power was determined by using a linear approximation n e (P) in the H-mode region, where the capacitive mechanism for energy deposition can be neglected because of its low efficiency. Hence, the 54

dashed line in Fig.4.5 corresponds to the inductive energy input. Its intersection with the abscissa axis (power axis) gives the value P t = 6 W. On the other hand, the threshold power can be calculated using a condition of equality between the capacitive and inductive efficiencies of energy deposition into the plasma: ξ cap = ξ ind, as was proposed in [Suz 98]. In this case, the transition between the modes occurs at the electron density n e = 9 9 cm -3 and the power P t = 4 W. The divergence between the threshold powers obtained by different methods is explained by the change in the energy deposition efficiency into the plasma, ξ(n e ), after the H-mode transition. This change was taken into account in [Suz 98]. 35 3 5 8 p = Pa of H gas P ξ (W) 5 5 n e ( cm -3 ) 6 4 3 4 n e ( cm -3 ) Fig.4.4. The balance between the input (curves) and absorbed (straight line) powers in a plasma. The intersection points correspond to the discharge existence condition. n e (cm -3 ) p = Pa of H gas 9, P (kw) Fig.4.6. The electron density in an ICP discharge versus the input power. 5,,,4,6,8,,,4 P (kw) Fig.4.5. The average electron density in a hydrogen plasma calculated by modeling an ICP discharge in a GEC reference cell. For a wider range of input powers, the dependence n e (P), shown in Fig.4.6, can be represented as a composition of three linear sections which are depicted by the dashed lines. The left segment corresponds to the discharge E-mode (at P < 4 W). The middle next one corresponds to the transient region, where the efficiency changes from minimum to maximum value. The third and final section corresponds to the pure H-mode discharge (at P > kw), where efficiency of the energy deposition into plasma is constant at ξ 9%. The width of the transition region depends on the relation between η and η quantities. η is the slope of P ξ (n e ) computed within the framework of the global model, and η is the maximum gradient of the deposited power determined by the energy deposition efficiency into the plasma. As is seen from Fig.4.4, for η > η, the transition region is rather wide. The situation when η ~ η, observed in an argon plasma [Suz 98], is illustrated in 55

Fig.4.7. In this case, the transition region is so narrow that the transition between the modes in an ICP discharge is called a mode jump [Suz 98]. For η < η, there can be several intersection points between the dependences of the input and deposited power. The hysteresis phenomenon is possible here, because the transition from the E-mode to the H-mode occurs at higher power than from H-mode to E-mode [Eng 97]. (a) (a) (b) (b) Fig.4.7. The incident, P tr, and deposited, P dis, powers in plasma (а). The measured and calculated dependences of the electron density on the incident power in argon plasma at p = mtorr for the cases of internal and external antenna (b). The data are taken from [Suz 98]. The transition curve between the modes is shown in Fig.4.8 by a solid line. A criterion for the transition to occur is the equality between the efficiencies: ξ cap = ξ ind. Under the curve, the efficiency of the capacitive mechanism for sustaining the discharge is higher than that of the inductive mechanism; above the curve, the situation is reversed. incident power (kw),,8,6,4,,,8,6,4 H-mode transition region ξ, cap = ξ ind E-mode R eff =,6 Ω,, gas pressure (Pa) Fig.4.8. Mode transition in the ICP discharge in hydrogen (solid line is the transition between E- and H-modes, dash line is the boundary of the transition region). There is no equally evident criterion for determination of the boundary between the transition region and the pure H-mode region. However, one can use the fact that in the transition region the ICP source is unstable because of the limited capabilities of the matching device. It is not possible to transform the antenna impedance in such a way it would be equal to the generator output resistance. When the matching network is able to satisfactorily match the generator and antenna impedances, we consider the plasma in pure H-mode. In our case, the border between the transition region and the region of pure H-mode is determined by the condition R eff =.6 Ω. The corresponding transition curve is shown in Fig.4.8 by the dashed line. By estimating the threshold power from the known n e (P) function (Fig.4.6) and comparing it with the dependence in Fig.4.8, which is obtained using an entirely technical criterion, one observes that both methods lead to the same result: at p = Pa, the threshold power is kw. 56

4..4 The efficiency of the GEC reference cell as a hydrogen plasma source The particular parameters of a low-temperature plasma depend on the design of the source. In this study, the GEC (Gaseous Electronic Conference) reference cell [GEC 94, GEC], shown schematically in Fig.3.3, is used as a prototype. There have been a great number of studies performed by different scientific teams using the GEC standard. Using a GEC reference cell allows the comparison of experimental results with data from literature [Ben 98, Wang 99, Wis 96]. A Cesar 3 RF generator and a VM impedance matching device, both from Dressler GmbH, Germany, were used. power deposited in plasma (W) p = Pa of H gas P ξ P cap P ind P loss 5 4 incident power (W) 3 Fig.4.9. Power balance in the ICP discharge in hydrogen at pressure p = Pa. Calculations are performed for the GEC reference cell geometry. The efficiency of the plasma source can be calculated from the equations (4.) and (4.47) trough (4.5). The calculation results for the total deposited power, P ξ ; the power deposited by the capacitive mechanism, P cap ; the power deposited by the inductive mechanism, P ind ; and the power loss in the antenna resistance are shown in Fig.4.9 for the gas pressure p = Pa. It is seen that the GEC reference cell is efficient in the E-mode for P < 5 W, and efficient in the H-mode for P >.3 kw. For the power range P = 5 - W, the RF generator s power loss P loss exceeds the power P ξ deposited into plasma, i.e. efficiency is less then 5%. These calculation results were confirmed experimentally. While running the discharge in H-mode, it was found that stable operation required a pressure of p = Pa and an applied power of about P = 8 W. This lead to intense heating of the vacuum chamber wall, especially in the vicinity of the antenna. The thermal and chemical processes caused by the heating of the vacuum sealing of the quartz window that separates the antenna from plasma can cause the vacuum seal to fail. Therefore, to carry out measurements in a hydrogen plasma, the design of the GEC reference cell needed to be modified. Two possible improvements to the ICP source design should be noted. Firstly, the ratio of the power deposited inductively to the power loss in the antenna resistance is proportional to the transformer ratio squared, k, for the transformer formed by the antenna and plasma: (C.76) Pind Ploss ~ k, (4.5) where k is given by equation (4.5). As is seen from Fig.4.6, k can be increased by replacing the metal and quartz antenna housing with an entirely quartz antenna housing. This will result 57

in an increase in the energy deposition efficiency of the plasma while in H-mode, since induction losses in the metal are avoided. Secondly, the performance of the plasma source can be characterized by the threshold power, P tr, which corresponds to the mode transition. The relations (3.) and (3.5) show that P tr can be decreased by using a lower inductance antenna. However, any modification of the antenna must not change the induced electromagnetic field s spatial structure. Finally, the GEC reference cell was designed to be used as a standard CCP. It was later modified for use as an ICP also. However, a typical capacitive discharge operates at powers much lower than most inductive discharges. For safety reasons, it is necessary to provide a water- and/or an air-cooling of the GEC reference cell s vacuum chamber. 4..5 Modification of the antenna design In general, the antenna does not contain any expensive components. It is, therefore one of the most attractive components to modify in a plasma source. In this study, the standard GEC reference antenna was replaced by an antenna with lower inductance, L = 6 nh. This was mainly determined by the capabilities of the VM matching device. The standard GEC antenna is a planer spiral, and can be classified as having a normal inductance. The various antenna classifications were discussed in Section 3..4. The group of normal inductance antennas also contains the twin antenna, which is often used in ICPs. Spiral and twin antennas are shown in Fig.4.3. A comparison performed in [Int 96, Men 96] revealed neither has great benefits nor major drawbacks in both designs. Consequently, antenna type choice is determined by the ease of fabrication. (a) (b) Fig.4.3. Spiral (a) and twin (b) antennas. Fig.4.3. Double spiral antenna with three turns in each spiral. Designing an antenna design that minimizes the deviations from axial symmetry of the induced electromagnetic field is very important. These deviations are caused by the geometry of the antenna. Any plane passing through the axis of spiral antenna divides the antenna into two unequal parts, and the field will therefore be asymmetrical. In the case of twin antenna, there are local inhomogeneities in the structure of the induced field near the joints between the turns. 58

The reduction in the antenna inductance can be achieved by decreasing the number of turns without changing the antenna size. It is a very efficient method because the antenna inductance is proportional to the number of turns squared. For example, decreasing the number of turns in the GEC antenna from five to three produces the required inductance of 6 nh. In doing so, the distance between the antenna turns, d turn, will increase to 6 mm, which is larger than the thickness of the quartz window separator, h q = mm. In this case, one can assume that the electromagnetic field induced by this new antenna will have a fine structure near the quartz surface. Consequently, plasma instabilities can develop. This problem can be resolved employing an antenna consisting of multiple spirals connected in parallel, as is shown in Fig.4.3. This antenna belongs to the low inductance group of antennas. The antenna dimensions and parameters of a standard GEC antenna and the actual modified antenna used are presented in Table 4.. Table 4.. The dimensions and parameters of the GEC antenna and double spiral antenna GEC antenna double spiral antenna number of spirals, N coil number of turns per one spiral, N turn 5 3 diameter, d ant (cm), distance between the turns, d turn (mm) 9,4 8, inductance, L (nh) 37 68 impedance at f RF = 3,56 MHz, ωl (Ω) 5,7 The electromagnetic field induced by a planar spiral antenna in vacuum is calculated in Appendix C (Subsection С..). The calculation results of the radial B r /I, vertical B z /I, and azumuthal B ϕ /I components of magnetic field for the GEC and double spiral antennas are shown in Fig.4.3. Here, the top of each plot corresponds to the inner surface of the quartz window, whereas the bottom to the electrode surface. To take into account the effect of antenna inductance on the current flowing through antenna: (C.33) I ~ L, (4.53) the calculated components of the magnetic field are multiplied by the factor: WL ~ RRF ω L, (4.54) where R RF = 5 Ω is a normalizing coefficient. As is seen from Fig.4.3, the spatial distribution of the magnetic fields produced by these antennas are similar. A slight discrepancy (about -5%) occurs only near the dielectric and electrode surfaces. The H-field induced by GEC antenna is stronger by the dielectric, and the H-field induced by the double spiral antenna will be larger near the electrode. In the cell centre, the fields induced by these antennas are equal. For both antennas, B ϕ /I, represents the axial asymmetry of the induced field, and is by an order of magnitude 59

less than B r /I and B z /I. Besides, B ϕ /I is approximately two times lower for a double spiral antenna than for a GEC antenna. This is because the asymmetry of the magnetic field produced by the individual spiral is partially balanced when the fields are superimposed. vertical position (mm) 3 4 -.5 -. -.5 -. -.5 W L B r / I (Gs / A)..5.5..5 vertical position (mm) 3 4 -. -.5 -. -.5.5..5. W L B r / I (Gs / A) 5 - -5 5 radial position (mm) 5 - -5 5 radial position (mm).5.5.5.45..4.35.3 W L B z / I (Gs / A).4..3 W L B z / I (Gs / A) vertical position (mm) 3 4. vertical position (mm) 3 4. 5 - -5 5 radial position (mm) 5 - -5 5 radial position (mm) vertical position (mm) 3 4 5 - - -5 5 4 8 6-4 W L B ϕ / I ( -3, Gs / A) -6 radial position (mm) -8-4 - vertical position (mm) 3 4 5 4-6 6-4 4 - W L B ϕ / I ( -3, Gs / A) - -5 5 radial position (mm) GEC antenna double spiral antenna Fig.4.3. The components of the H-field induced in vacuum by the GEC antenna (left column) and double spiral antenna (right column). Numerical calculations of the radial E r /I and azimuthal E ϕ /I components of electric field are shown in Fig.4.33. The magnitudes of the electric fields induced by both antennas are equal. They differ only by the ratio between the components. The azimuthal component - -4 6

comprises about 8% of the total field for a GEC antenna, E ϕ /E = 8%. In the case of a double spiral antenna E ϕ /E = 95%. Since the power dissipated in plasma is proportional to the azimuthal component of the E-field squared, the efficiency of the double spiral antenna will be higher. vertical position (mm) 3 4..5 3. 3.5 3. 5.5 3.5 5. 4.5 4. W L E r / I (V/m / A).5 3. 3.5 vertical position (mm) 3 4 -.5 -. -.5 -.5 W L E r / I (V/m / A).5.5 5 - -5 5 5.5 - -5 5 radial position (mm) radial position (mm) - -3-4 3 4 W L E ϕ / I (V/m / A) -3 4 3 W L E ϕ / I (V/m / A) 4 vertical position (mm) 3 4 5 - - - -5 5 radial position (mm) vertical position (mm) 3 4 5 - - - - -5 5 radial position (mm) GEC antenna double spiral antenna Fig.4.33. The components of the E-field induced in vacuum by the GEC antenna (left column) and double spiral antenna (right column). The spatial distributions of B z /I, E r /I and E ϕ /I in a horizontal plane passing through the centre of the plasma cell are shown in Fig.4.34, and illustrate the differences in the symmetry of the fields induced the two antennas. The left column corresponds to the GEC antenna, and the right column corresponds to the double spiral antenna. A planar antenna with a single spiral coil will have one internal and one external contact. The current in the antenna can be treated as a sum of the azimuthal current and the current collinear to the line passing through the antenna contacts, i.e., the directed radial current. In this case, the induced E-field is a superposition of the azimuthal and radial fields determined by these currents. This fact accounts for the distributions of E r /I and E ϕ /I calculated for the standard GEC antenna. The E-field of a double spiral antenna is more symmetrical because the radial components produced by the individual spirals partially balance each other. 6

radial position (mm) W L B z / I (Gs / A).55 --.75.35 --.55.5 --.35 5.95 --.5.75 --.95.55 --.75.35 --.55.5 --.35 -.5 --.5-5 -.5 -- -.5 radial position (mm) W L B z / I (Gs / A).55 --.75.35 --.55.5 --.35 5.95 --.5.75 --.95.55 --.75.35 --.55.5 --.35 -.5 --.5-5 -.5 -- -.5 - - (a) - -5 5 radial poasition (mm) - -5 5 radial poasition (mm) radial position (mm) 5-5 W L E r / I (V/m / A) 4 -- 5 3 -- 4 -- 3 -- -- - -- - -- - -3 -- - -4 -- -3 radial position (mm) 5-5 W L E r / I (V/m / A).75 --..5 --.75.5 --.5 --.5 -.5 -- -.5 -- -.5 -.75 -- -.5 - - (b) - -5 5 radial poasition (mm) - -5 5 radial poasition (mm) radial position (mm) -- 8 -- 6 -- 8 5 4 -- 6 -- 4 -- 8 -- 6 -- 8 4 -- 6-5 -- 4 -- - W L E ϕ / I (V/m / A) radial position (mm) 5-5 - W L E ϕ / I (V/m / A) -- 8 -- 6 -- 8 4 -- 6 -- 4 -- 8 -- 6 -- 8 4 -- 6 -- 4 -- (c) - -5 5 radial poasition (mm) - -5 5 radial poasition (mm) GEC antenna double spiral antenna Fig.4.34. The vertical component of the magnetic field (a), the radial (b) and azimuthal (c) components of the electric field in vacuum at the distance mm from the antenna, calculated for the GEC antenna (left column) and double spiral antenna (right column). Like the standard GEC antenna, the double spiral antenna is water-cooled. Water flows through the two external contacts shown in Fig.4.3. Although it is not shown in the figure, these contacts are grounded. A high voltage is applied to the inner contact which is not water-cooled. This prevents the power applied to the antenna from being shunted by the water. 6

4..6 The modified plasma source cell: In this study, the following modifications were made to the standard GEC reference The steel antenna housing and quartz window were substituted with a housing consisting entirely of quartz. The single spiral antenna was replaced by a lower impedance double spiral antenna. In addition to water cooling of the antenna and the electrode, water cooling of the vacuum chamber and air cooling of the antenna quartz housing were added. An electrical model of the modified hydrogen plasma source shown in Fig.4.5(b) was designed according to Subsections 4..-4..4. The dependences of the inductance and transformer ratio of the antenna and plasma on the skin layer thickness are shown in Fig.4.35. By comparing this data with the data in Fig.4.6 from a standard GEC reference cell, one can see that the modification increased the transformer ratio by about %. ωl (Ω) 8 6 4 ωl = 5,7 Ω ωl ωl k,7,6,5,4,3,, 3 4 5 h pl (mm), 3 4 5 h pl (mm) Fig.4.35. The inductances (a) and transformer ratio (b) of the antenna and plasma versus the skin depth for the modified GEC reference cell. After calibrating the double spiral antenna, the following parameters were determined: R.6 Ω, L = 68 nh and C = 47 pf, (4.55) where R is the antenna resistance, L is the antenna inductance, and C ant is the capacitance of the quartz window that separates the antenna and plasma. The dependence of the power deposited into a hydrogen plasma, P ξ, at the gas pressure p = Pa on the incident power, P, is shown in Fig.4.36. As is seen, the changes in the ICP source design increased its efficiency. The incident powers for which the power loss, P loss, exceeds the power deposited into plasma was reduced to the range 6.5 to 4 W. The threshold power for mode transition, P t, was decreased from 4 W to 4 W. ant 63

p = Pa of H gas, power deposited in plasma (W) P ξ P cap P ind P loss 4 4 6,5 incident power (W) Fig.4.36. The power balance in hydrogen discharge at Pa. The calculations are performed for the modified plasma cell. incident power (kw),,8,6,4 H-mode transition region, ξ cap = ξ ind R eff =,6 Ω, E-mode, gas pressure (Pa) Fig.4.37. The transition between the modes of a hydrogen ICP for the modified plasma cell. The mode transition curves for the modified ICP are shown in Fig.4.37. The solid curve shows the incident powers and gas pressures corresponding to when the efficiencies of the capacitive and inductive energy deposition mechanisms are equal, and is therefore also the E- and H-mode transition curve for the ICP discharge. Below the dashed line in Fig.4.37 the matching network is unable to properly match the effective antenna impedance to the RF generator load resistance. The power loss in this region corresponds to a maximum of % at the solid line and approaches zero at the dash line. The dashed line corresponds to a boundary between the transition region and the pure H-mode. 64

4.3. Sheath theory It is possible to mount a planar electrode parallel to the antenna of a planar ICP source. The ion flux to the electrode depends on the plasma density at the sheath edge and the ion energy, both of which can be controlled seperatly. Varying the power applied to the antenna will vary the plasma density. Changing the electrode voltage will change the sheath potential, and therefore the ion energy. Ion energy can be minimized by grounding the electrode. The formation of a collisionless sheath by a grounded electrode is considered in this section. The static sheath model (Subsection 4.3.), the RF sheath model (Subsection 4.3.), and the simple kinetic model (Subsection 4.3.3) of ion movement in the RF sheath will all be discussed. 4.3.. Static sheath n e, n i n i - n e E pre-sheath Debye-sheath sheath sheath edge: u i = u B electron density ion density Fig.4.38. Structure of the sheath. In a static sheath shown in Fig.4.38, we can distinguish three regions: a presheath, a Debye-sheath, and a high-voltage sheath. In the pre-sheath quasineutrality is fulfilled, and the ions are accelerated by an ambipolar electric field driven by the plasma density gradient. The ion velocity u at the sheath edge, where quasineutrality is violated, is defined by the Bohm criterion: where M i is the ion mass. According to Rieman K.U. [Rie 9], the pre-sheath characteristic length, λ ps, is twice as large as the ion mean free path at the sheath edge and can be calculated from the following equation: x u = u = kt M B e i, (4.56) λ ps =, Nσ( u ) (4.57) B where N is the neutral gas density, and σ(u B ) is the cross-section of elastic ion collisions with gas molecules. The energy dependence of the elastic collision cross-section of H 3 + ions, the dominant species in a hydrogen plasma, with H molecules is presented in [Sim 97]. The electric field in the sheath is described by the Poisson equation: E( x) e = x ε ( n ( x) n ( x) ) i e, (4.58) where n i (x) and n e (x) are the ion and electron densities. An inflection point, see Fig.4.38, where the E-field changes from convex to concave separates the Debye and high-voltage sheaths. In the Debye sheath, both electrons and ions are present in densities of nearly the 65

same order of magnitude. In the high-voltage sheath, the field is determined by the positive charge of ions because the electron density is close to zero. The spatial ion density distribution in the sheath can calculated by assuming the ion current as time-independent by the continuity and the energy conservation equations: j i ( x) = eni ( x) ui ( x) = enu, (4.59) where n and u are the plasma density and the ion velocity at the sheath edge. According to equations (4.56) and (4.59), the ion velocity increases and the ion density decreases as the electrode surface is approached: ni ( x) u kte = =, ε i ( x) = kte eu( x) (4.6) n u ( x) ε ( ) i i x where ε i (x) is the energy of an ion moving across the sheath, and U(x) is the potential profile across the sheath. The electron density distribution in the sheath can be approximated with a Boltzmann distribution: n ( U ) = n exp. (4.6) e kt e Boundary conditions for equation (4.58), n and E, can be found by using the AHPM, described in Section 4., and the sheath model proposed in [Rie 9]. The electron temperature and plasma density at the sheath edge, deduced from the AHPM, allow us to determine the electric field at the sheath edge: where λ D is the Debye length. E = kt eλ, (4.6) e 5 3 5 D λ ps The potential, U(x), and field, E(x), profiles in a static sheath can be calculated by integrating Poisson s equation (4.58) and using ion and electron density profiles described by equations (4.6) and (4.6). The sheath thickness, d S, can then be found using the floatingpotential condition (4.). 66

4.3.. Sheath in ICP The physics of the RF sheath is very important for understanding current plasma processing techniques like thin-film deposition and etching. The RF sheath has been extensively studied by the plasma physics community [Ben 98, God 9, Hwa 96, Mil 97, Orl 97, Rai 95, Sob 98, Sob 99, Sob, Ste 96, Tan 99]. A time-dependent (dynamic) potential of the sheath at the electrode surface U S (t) = U(d S,t) can be represented as a sum of a constant and a time-varying component: U ( t ) U U ~ ( t S = + ), (4.63) where U ~ (t) is a harmonic function of the RF field frequency [Rai 95]. In addition to the ion, j i, and electron, j e, currents, the total current at the electrode contains also a displacement current, j D, due to the presence of the RF field. In a one-dimension model the total sheath current at the electrode surface is: j = j + j + j, (4.64) S i e des jd = ε, (4.65) dt where E S (t) is the electric field at the electrode surface. D If the electrode is grounded, the value of U will be the floating-potential determined by the condition that the RF-period-averaged electron and ion currents coming to the electrode should be zero. Hence, for a stationary sheath potential the total current at the electrode is zero. For the inductive power coupling into a plasma, the alternating current induced by the antenna RF field forms a closed loop inside the plasma bulk and consequently does not influences the value of j S. The ICP discharge in the pure H-mode is characterized by a static sheath. A nonzero j S (t), as well as a nonzero U ~, can exist only for capacitive power coupling into a plasma. There is good reason to believe that the ion motion in the sheath of ICP discharge is always time-independent and is described by equation (4.59). For the lower plasma densities corresponding to the E-mode with much higher sheath voltages and large sheath widths, the U S, j i, j e (a.u.).. -. -.4 -.6 -.8 sheath voltage ion current electron current -....4.6.8.. t / t RF Fig.4.39. Time diagrams of potential and currents in the sheath of an RF discharge (displacement current is not shown). ions cannot completely follow the field oscillations, due to their inertia. The ion plasma frequency is much lower than the field frequency: ω i << ω. At higher plasma densities, in the H- mode, the capacitive power coupling into plasma is inefficient, see Fig.4.3. As a result, the ratio U ~ /U decreases, and taking into account the low mobility of ions, we can consider their motion in the sheath as static. Unlike the ions, the electrons in the sheath move synchronously with the RF field. The electron plasma frequency far exceeds the field 67

frequency: ω e >> ω. There will be an electron flux to the electrode surface for only a small partial of the RF cycle when the sheath potential is near its minimum. Whereas the ion flux to the electrode is nearly constant in time. According to the floating-potential condition, the ion current, assumed to be constant, is balanced by the period-averaged electron current: j j =, (4.66) e + i the amplitude of the electron current, j e, far exceeds j i, as shown in Fig.4.39. Assuming a Boltzmann distribution (4.6) for the electrons, the RF-period-averaged electron current at the electrode can be calculated as a Hertz-Langmuir current [Rie 9] including the time-varying potential from equation (4.63): j t = en eu ( t) 8kT S e e ( ) exp kt. (4.67) e 4 πme In this case, a sheath potential satisfying equation (4.66) is calculated from the following relation: eu kt = e M i ln πm e + δ eu ~ ( kt ), δ( kt ) = ln J e e, (4.68) kte where δ(kt e ) is a small correction to the floating-potential equation (4.) and J is the zeroth-order Bessel function. An iterative method can be used to compute the dynamic sheath characteristics of an ICP discharge. First, let us assume that U ~ is known. In this case, the constant component of sheath potential U can be found from equation (4.68). Integrating Poisson s equation (4.58) using the static boundary conditions n and E, we can calculate the ion distribution, n i (x), and the sheath thickness, d S. Poisson s equation is then integrated again using the timeindependent ion density distribution, n i (x), and the time-dependent electron density distribution, n e (x,t), described by equation (4.6): E( x, t) e = x ε ( n ( x) n ( x, t) ) The density at the sheath edge and the potential are: [ h( )] i e. (4.69) n e (, t) = n + t, h <<, h = and U ( x, t) = U ( x) + U ~ ( x, t). (4.7) The potential, U(x,t), and E-field, E(x,t), are calculated from this second integration with the requirement that the potential at the electrode surface, U(d S,t), is equal to U S (t) from equation (4.63). Taking into account that the potential near the sheath edge is small, and ( kte ) eu kte exp eu ~ + ~ Poisson s equation (4.69) have the form: U ( x) e = x ε, the time-independent and time-varying components of { n ( x) n exp[ U ( x) ]} i kt e and (4.7) U ~ ( x, t) en eu ~ ( x, t) = h( t) +, (4.7) x ε kt e 68

The boundary conditions in the second integration must also be time-varying: U ( ) =, U () = E, and x U () (4.73) ~ U ~ (, t) =, = Eϑ( t), ϑ <<, ϑ =. x E is the static electric field at the sheath edge (4.6). Equation (4.7) is Poisson s equation for the static sheath. The general solution for the time-varying components of the potential and the E-field near the sheath edge (eu ~ /kt e << ) are: kte U ~ ( x, t) = Eλ Dϑ( t)sinh( x λ D ) + h( t) [ cosh( x λ D ) ] and e (4.74) kte E~ ( x, t) = Eϑ( t)cosh( x λ D ) h( t) sinh( x λ D ). eλ (4.75) It should be mentioned that the calculated E-field at a distance x > λ D from the sheath edge will be the same, no matter which of the two boundary conditions, the density or the electric field, is varied. For example, for the two limiting cases when h = or when ϑ =, the sheath potential at the electrode surface for d S >> λ D will be the same if and only if: E ~ / (E ϑ ) time-varied boundary conditions: E-field variation plasma density variation,,,5,,5,,5 3, x / λ D Fig.4.4. Time-varied component of E-field near the sheath edge. D kte Eϑ ( t) = h( t). (4.76) eλ D Both cases for the time-varying component of E-field are shown in Fig.4.4. Hence, when calculating U(x,t) and E(x,t), one of the timevaried boundary conditions, n e (,t) or E(,t), can be chosen to be constant in time. According to equations (4.6) and (4.76), the density at the sheath edge varies much less then the electric field: h = (λ D /λ ps ) 3/5 ϑ << ϑ. For calculating the E-field and potential in the ICP sheath, n e was chosen to be constant and E (t) was used for the time-varying boundary conditions. Within the region shown in Fig.4.4 the approximate analytical calculation (4.7)-(4.75) agrees perfectly with a numerical integration of the original Poisson s equation (4.69). The displacement current j D (t) was calculated by substituting the intensity of electric field on the electrode surface E S (t) = E(s,t) into equation (4.65). Thereafter, the current through the sheath, j S (t), can be calculated as a sum of the ion current, j i, from equation (4.59), the electron current, j e (t), from equation (4.67), and the displacement current, j D (t). The transformer model can be used to estimate the RMS current through the sheath: j RMS = R pl ξ cap ( S + S ) el P wall. (4.77) 69

In this equation ξ cap is the capacitive power deposition efficiency; P is the incident power; R pl is the plasma resistance from equation (4.3); S el = πr el, and S wall = πr cell h cell are the areas of the electrode and cylindrical wall of the vacuum chamber. If the RMS current does not coincide with the total sheath current: j =, (4.78) S j RMS then the our initial condition, time-varying potential U ~, should be changed. The higher U ~ is, the higher j S will be, and vice versa. Results of the dynamic sheath model for a hydrogen ICP in the modified GEC reference cell are presented in Fig.4.4-Fig.4.43. The initial discharge parameters are: incident power P = 3 W, gas pressure p = Pa, and gas temperature T g = 3 K. According to the AHPM and the transformer model discussed in Sections 3. and 3., the electron temperature is kt e =.8 ev, the plasma density at the sheath edge is n = 6.6 9 cm -3, and the RMS current is j S =.6 A/m. The RF-period-averaged E-field at the sheath edge, calculated from equation (4.6), is equal to E =. V/cm. Fig.4.4 shows the temporal evolution of the electric-field and potential profiles for these conditions. The RF-period-averaged values are solid lines, whereas the maximum and minimum values are dashed lines. The coordinate x = corresponds to the sheath edge. The sheath thickness, d S, is.5 mm, approximately nine time the Debye length λ D. The sheath is characterized by a relatively low (for RF discharges) electric field E ~ 5 V/cm, and a low sheath potential, U S ~ - V. E (V/cm) 35 3 5 5 5 (a) E (V/cm) 9 E =, V/cm,,,4,6,8, time / period,,,4,6,8,,,4 x (mm) U (V) - -4-6 -8 - - -4 U S (V ) -6-8 - - -4 U S (t) = -,7 + 4,5sin(ωt) -6,,,4,6,8, time/ period -6,,,4,6,8,,,4 Fig.4.4. Electric field (a) and potential (b) distributions in the sheath of an ICP discharge. Solid lines correspond to the RF-period-averaged values, and dashed lines correspond to the maximum and minimum values. The ion and electron density distributions in the dynamic sheath of an ICP are shown in Fig.4.4. Solid lines in the figure correspond to the static ion density profile and the period-averaged values of the electron density. Dashed lines show the maximum and minimum values of electron density. As can be seen in the figure, electrons are present throughout the sheath. Consequently, this sheath is a Debye sheath. Fig.4.43 shows a phase diagram of the sheath currents in an ICP discharge. The electron diffusion current is calculated from equation (4.67). Owing to the Boltzmann factor, (b) x (mm) 7

this current is not sinusoidal, its maximum corresponds to a minimum potential in the sheath that is observed during the RF period. The electron current through the sheath is present only during the first half of the RF period, during the second half it is close to zero. The static ion current, j i, calculated from equation (4.59) is equal to. A/m. The waveform of the displacement current, j D, calculated from relation (4.65) is nearly sinusoidal and has an amplitude of 7.7 A/m. The total current through the sheath, j S, has a waveform that is typical for RF discharges [Cza 99]. n i, n e ( 9, cm -3 ) 7 6 5 4 3 ions electrons,,,4,6,8,,,4 x, (mm) Fig.4.4. Ion and electron density distributions in the sheath. Solid lines show the static ion density profile and the period-averaged electron density, and dashed lines show the maximum and minimum electron densities. sheath current (A/m ) - - electron ( j e ) ion ( j i ) displacement ( j D ) total sheath ( j S ) -3,,,4,6,8, time / period Fig.4.43. Phase diagrams of currents in the lowvoltage sheath of an ICP discharge: electron and ion currents, displacement current, and total current in the sheath. Capacitive and inductive discharges can be distinguished by the proportion of the displacement current to the charged particle current. In a CCP discharge, a main contributor to the total current at the electrode is the displacement current, whereas the ion and electron currents are dominant in the ICP discharge, as seen from Fig.4.43. 7

4.3.3. Ion energy distribution As the electric field in the sheath E(x,t) is known, we can determine the ion energy distribution function (IEDF). The IEDF can be calculated using a simple kinetic model [Mis ]. It is first assumed that ions enter the sheath at a uniform rate according to the phase of the RF field, and each ion has the Bohm velocity. The force exerted on the ion in the electric field E(x,t) is dui F = M i = ee( x, t). (4.79) dt Equation (4.79) can be solved iteratively. First, the distance from the sheath edge to the electrode surface is divided into intervals x. The intervals are chosen small enough that the electric force acting on the ion can be assumed to be constant. In this case, the time it takes for the ion to traverse x can be found from: The change in the ion velocity will be ( ee( x, t) M ) t x = u t +. (4.8) i ( ee( x t) M ) t Hence, the initial conditions for the next iteration would be i u =,. (4.8) i i x = x + x, t = t + t, ui = ui + ui. (4.8) When ion reaches the electrode surface its energy will be added to a histogram of the ion energy distribution. For example, consider the motion of H +, H + and H 3 + hydrogen ions through the ICP sheath for which the E-field was calculated in the previous section (Fig.4.4). Histograms of ion energy distributions computed using the kinetic model are displayed in Fig.4.44. According to the Bohm criterion, ions entering the sheath have an energy kt e / =.4 ev. The histograms in Fig.4.44 show that the RF-period-averaged energy gained by the hydrogen ions in the sheath amounts to.7 ev. This corresponds to the constant component of sheath potential U. Note that each histogram in the figure has two characteristic maximums corresponding to the minimum and maximum ion energies. The greater the ion mass is, the shorter the distance between maximums will be. As the ion mass increases, the mobility will decrease. Then ions moving in the sheath will therefore be unable to follow the rapidly varying electric field. The lightest ion possible is H +. Its maximum energy deviation from the mean is ε i = ±.7 ev, which is less than the component due to the fluctuating sheath potential eu ~ = 4.5 ev. The ion energy distribution function can be measured using an ion energy analyzer. A PPM4 ion energy and mass analyzer, which measures a flux of ions with given energy and mass, was used in this study. In order to model PPM4 measurements, a calculated histogram should be multiplied by the incoming ion velocity, and then convoluted with a function which represents the response of the energy analyzer. The analyzer signal corresponding to the H 3 + ion flux is approximately: 7

I + = j i πr, (4.83) H 3 PPM where j i is the ion current, r PPM is the analyzer entrance radius. Since the proportion between H +, H + and H 3 + ion densities is a priory unknown, estimates for all hydrogen ions were made using H 3 + coefficients. It should be mentioned that the measurements show that the contribution of H + and H + to the ion flux is about two orders of magnitude lower then the H 3 + flux. The energy distributions of hydrogen ion fluxes shown in Fig.4.45 were calculated using the histograms Fig.4.44. ions (a.u.) 5 5 5 5 5 5 energy (ev) H + H + H 3 + 9 3 4 5 Fig.4.44. Histograms of energy distribution of hydrogen ions calculated using the kinetic model. ion current ( -, A) 5 4 3 H + H + H 3 + 4 6 8 4 6 8 energy (ev) Fig.4.45. Flux of hydrogen ions onto the electrode surface as a function of energy. Fig.4.46 shows how the energy distribution of H 3 + flux onto the electrode surface varies depending on the power applied to the ICP discharge at a fixed gas pressure. 6 5 4 3 ion current (na) - 8 6 4 8 ion energy (ev) 6 4,6,8,,4 incident, power (kw), Fig.4.46. Energy distribution of the H 3 + ion flux for various values of incident power for a fixed gas pressure of Pa. 73

At lower incident powers, corresponding to the E-mode, the ion energy distribution has two maximums, and the mean ion energy gained in the sheath far exceed the 9.6 ev of the static sheath floating potential, see equation (.39). A dashed line in Fig.4.46 represents the energy that an ion passing just through the static sheath would have. At higher incident powers, the discharge changes to the H-mode, and both the distance between two ion flux maximums and the mean ion energy will decrease. If the power increases further, the sheath will become static and the energy of incoming ions will be uniform. As one should expect, the ion flux onto the electrode will increase with power. At powers P > 5 W, the flux intensity is approximately linear with respect to the power, I i (P). The energy distributions of the H 3 + ion flux for several gas pressures at 3W incident power are shown in Fig.4.47.,8,5 ion current (na),6,4,,,8,6,4, 5 Pa 7 Pa Pa 4 Pa 3 Pa Pa,5 Pa, Pa,5 Pa, 5 5 5 3 35 4 ion energy (ev) ion current (na),4,3,, Pa Pa 5 Pa Pa, 5 5 5 ion energy (ev) Fig.4.47. Energy distributions of the H 3 + ion flux for various values of gas pressure and fixed applied power P = 3 W. Fig.4.37 demonstrates for identical discharge conditions as Fig.4.47 that the discharge is in the H-mode in the pressure range.7 Pa < p < Pa, whereas the E-mode occurs at lower and higher pressures. In Fig.4.47, the maximum monoenergetic ion flux from the plasma is at p = 3 Pa. The efficiency of inductive power coupling into plasma is apparently a maximum at this pressure. The two peaks which appear in the ion flux distribution at higher and lower pressures indicate that there is an RF sheath potential associated with the capacitive power coupling mechanism.the mean ion energy is a function of the electron temperature and the constant component of sheath potential U. The minimum value of the mean ion energy was observed at Pa and not 3 Pa, because the electron temperature increases with decreasing pressure. 74

5. Experimental setup In this section, the experimental setup is briefly described (Section 5.), the measurement scheme (Section 5.) is presented, and the calibration of the measuring equipment is outlined. 5.. Plasma source A modified GEC reference cell was used for the experiments. The modifications of the plasma cell are described in Section 4.. In addition, an ion mass and energy analyzer was integrated into the planar electrode. The experimental setup is described in Subsection 5... The measurement results of the hydrogen ICP efficiency, the self-ignition curve, and the mode transition curve are presented in Subsection 5... 5... ICP setup The modified GEC reference cell is shown schematically in Fig.5.. Hydrogen bleeds into the vacuum chamber () through four inlets () mounted at its upper flange. The gas flow is regulated between to sccm by a Tylan 9 flow-controller. The chamber has two separate paths to a Pfeifer TC6 turbopump (3). When the discharge is not being used, there is a direct connection between the chamber and the pump(4a). This maximizes the conductance between the pump and the chamber and brings the chamber to a lower pressure of -5 Pa. When the discharge is operating, the chamber is pumped through small holes in the chamber bottom and a variable diameter diaphragm (4b). In this case, the flow and the gas pressure in the chamber will be uniform, which is necessary for stable operation of the plasma source. Energy is coupled into the plasma using a double spiral antenna (5) through a quartz cylinder (6) that separates the antenna from the vacuum. The antenna is powered by a Cesar 3 RF generator (7) and a VM impedance matching network (8). A planar aluminum electrode (9) is mounted on a specially designed holder () for the Balzers PPM4 plasma process monitor (). The PPM4 is positioned so that its entrance orifice is at the electrode centre and is located on the electrode surface within. mm. It is pumped by a Pfeifer TMU65SG turbopump (), and is operated by a QMS4 controller (3). The quadrupole ion-mass analyzer incorporated in the PPM4 is powered by a high-voltage generator (4), and the ion flux passing through it is measured by an electrometer (5). The vacuum chamber (6a), the antenna (6b), and the electrode (6c) are watercooled. The capacitors of the matching network (8) and the quartz cylinder (6) are air-cooled (7). 75

7 8 7a 7b 6b 5 4a 6 6a 3 4b 3 5 9 6c 4 - vacuum chamber - gas supply 3 - Pfeifer TC6 turbopump 4 - pumping path (a - direct pumping, b - pumping through a diaphragm) 5 - double spiral antenna 6 - quartz antenna housing 7 - Cesar 3 RF generator 8 - VM impedance matching unit 9 - grounded electrode - holder of the ion energy and mass analyzer - Balzers PPM4 plasma process monitor - Pfeifer TMU65SG turbopump 3 - PPM4 controller (energy and mass chassis) 4 - high voltage generator 5 - electrometer 6 - water-cooling supply (a - chamber cooling, b - antenna cooling, c - electrode cooling) 7 - air-cooling stream (a - entrance, b - exit) Fig.5.. Experimental setup. 76

5... ICP source characterization A hydrogen ICP source can operate in the E-mode, the transition regime, and the H-mode (see Section 4.). The E-mode exists at low incident powers. If the power is increased past the mode transition threshold, the plasma shifts into the transition regime. In this regime, the effective antenna resistance is too low for the VM matching network for perfect matching. However, the maximum power loss does not exceed %. The pure H- mode exists at high incident powers, where the matching network properly matches the effective antenna impedance and the RF generator. The limited capability of the VM does not affect the stability of the Cesar 3 RF generator in the transition region, because the maximum reflected power of the generator can be as high as 4% of the forward power. This allows stable discharge operation throughout the transition regime. The mode transition threshold can be measured by finding the lowest RF generator power at which the auto-matching network is still reliable. incident power (kw),,,8,6,4, measurement T g = T g (),, gas pressure (Pa) Fig.5. shows the measured (open circles) and calculated (lines) mode transition curves. Numerical results which use both the AHPM and the electrical model is shown as a solid line. At pressures below Ра, there is good agreement between the experiment and the model. When pressures are above Ра, the computed threshold power is higher than the measured one. It is likely that this divergence is related to an increase in temperature of the background gas, which was not taken into account during the calculations. The gas temperature was assumed to be a constant 3 K. At a fixed pressure, the increase in the gas temperature in the discharge causes a decrease in the molecule concentration. This will then cause a reduction in the electron collision frequency, and the mode transition curve will shift towards higher pressures. To illustrate this process, let us assume that the increment in the gas temperature, T g, is proportional to the power deposited into the plasma, P ξ : T g = T g () ( + αp ξ ) Fig.5.. Mode transition curves. Open circles show the experimental results. Lines show the result of calculations excluding (solid line) and including (dashed line) the neutral gas heating. ( + α ) () g = T Pξ, (5.) T g where α is a proportionality factor. The computation shown by the dashed line in Fig.5. was performed with α =. W - and is in better agreement with the experiment for a wide range of gas pressures p = (.7 - ) Pa. However, both the value of α and the linear dependence of the gas temperature on the power deposited into the plasma are assumptions. More accurate calculations that account for the gas heating in discharge can be performed if 77

T g is measured in advance. The general effect of gas heating was confirmed qualitatively by laser Doppler measurements at Pa (see Section 6.). At pressures below.7 Pa, the mean free path of H molecule: where σ H =.7-9 m Λ = kt g pσ, (5.) H H is the gas kinetic cross section, becomes comparable with the plasma cell height: Λ H h cell. Under these conditions, the hydrogen molecules arrive at the chamber wall without undergoing collisions, and thus the particle flow to the wall is no longer diffusive. It seems that there is no effective heating of the gas in the discharge at lower pressures, and that the solid curve in Fig.5. describes the experimental data better. incident power (kw)...8.6.4. area IV area II mode transition self-ignition area I area III.. gas pressure (Pa) Fig.5.3. The modes of an ICP discharge in hydrogen. Area I: H-mode, discharge self-ignition is possible; area II: H-mode, discharge self-ignition is impossible, area III: E-mode; area IV: discharge is impossible. The discharge self-ignition curve and the mode transition curve in Fig.5.3 divide the possible gas pressures and incident powers into four areas. At pressures and powers corresponding to area I, the hydrogen ICP discharge operates in the H-mode and can be pulsed. At lower pressures, area II, discharge pulsing becomes impossible. Within area III, the power is lower than in area I and the discharge operates in the E-mode. For pressures and powers in area IV, operation of the plasma source is no longer possible. H gas 78

5.. Diagnostics In the course of experimental work, Langmuir probe measurements (Subsection 5..), measurements of the electric field distribution across the sheath (Subsection 5..), measurements of the mass and energy spectra of hydrogen ions coming to the electrode surface (Subsection 5..3), and time-resolved optical measurements of plasma radiation (Subsection 3.5.4) were carried out. 5... Langmuir probe measurements The arrangement of the Langmuir probe in the vacuum chamber and a functional diagram of its instrumentation are shown in Fig.5.4. The probe head was a commercially produced APS3 (AG Prof. Dr. P. Awakowicz) and was mounted on one of the four CF4 flanges of the vacuum chamber. The probe tip () was a tungsten wire of diameter d p = 5 µm and length l p = 5 mm. The high-frequency component of the sheath potential was compensated using the passive method explained in Subsection 3... An additional electrode () was placed cm from the probe tip and had an area of S p = 3 cm. The probe was constructed so it could move from the chamber wall to the center of the discharge. The plasma s radial profile was therefore able to be measured at a fixed height of h =,5 cm from the electrode surface. Fig.5.4 contains a photograph of the probe in the plasma cell made by a CCD camera with the electrode surface illuminated by a diode laser. The scale of the photograph is shown by superimposing an image of a ruler photographed with identical optical parameters. The APS3 probe system is designed for measurements in reactive RF plasmas. In order to avoid damage to the probe, the I-V characteristic can be measured in as short as ms. However, the short duration of probe measurements imposes limits on the resolution as low as 8 bits, which is insufficient for a detailed analysis of the plasma parameters. Better results were obtained in our study by making all I-V characteristic measurements using a National Instruments multifunction PCI input/output (I/O) board. (3) The PCI-635E has a multichannel 6-bit analog input (AI) and -bit analog output (AO). The AO of the PCI- 635E is connected to the input of a x voltage amplifier (4), which is then connected to the Langmuir probe. The probe supply circuit also contains a low thermal noise, precision resistor, R pr, and an RF filter (5) for suppressing 3.56 MHz and its first two harmonics. The voltage drop, U R, across R pr and the probe potential, U p, are measured by two AI channels of the PCI-635E. The signals are measured through an AI3 optical isolated amplifier and an A buffered voltage attenuator that are incorporated into a National Instruments SC-345 signal conditioning box (6). The PCI-635E measures the potential relative to the vacuum chamber s ground, which introduces noise into the measured signal. Because of this, the I-V characteristic of the probe cannot be measured with the maximum 6-bit resolution of the PCI-635E. The 79

accuracy was improved by repeatedly measuring and then averaging the probe potential and current. In this case, the measurement errors U p and I p are the RMS deviations from the mean, and are inversely proportional to the square root of the number of measurements, N p : U pr ~ I pr ~ N p -/. The maximum measurement frequency when using two AI channels of the PCI-635E is khz. Thus, if the I-V characteristic consists of 5-7 measurement points and lasts a minute, the total number of measurements of U p and I p will be N p 4. Under these conditions, the noise level in reduced enough to allow a measurement resolution better than 4-bits. probe position electrode position 4 voltage amplifier x AO 3 R pr AI3 AI current 5 A AI voltage 6 SC-345 PCI-635E. probe tip. additional electrode 3. PCI-635E multifunction I/O board 4. voltage amplifier 5. electrical filter 6. SC-345 signal conditioning box Fig.5.4. The arrangement of the Langmuir probe in the vacuum chamber and a functional diagram of its instrumentation. The analysis of I-V characteristics was discussed previously in Subsection 3... Experimental data obtained in our study will be presented and discussed below in Section 6.. 5... Laser diagnostics As discussed earlier in Subsection 3.., the electric field of an ICP sheath can be measured using FDS, which is based on the Stark splitting of atomic levels. Fig.5.5 illustrates measurement of the spectra for atomic hydrogen transitions from n = 3 to the Rydberg level. This scheme allows us reconstruction the E-field in the sheath with a resolution better than 5 V/cm. The spatial resolution of FDS is limited by the optical properties of the CCD system used for optical detection and is 58 µm. The time resolution is determined by the laser pulse duration and is 5 ns. The hydrogen plasma is maintained in the vacuum chamber () by an RF generator () coupled through an antenna (3). The repetition rate of the Continuum PL8 Nd:YAG laser pulses, f SU, used in the experiment was Hz. The pulses were synchronized to the 8

phase of the RF generator by a synchronization unit (4), which essentially divides the RF frequency into longer periods. The phase shift of the laser pulse relative to the RF period is controlled by the DG535 delay generator (5). One channel of the DG535 is used to trigger the Nd:YAG laser (6). This 535 nm laser is used to pump the Continuum ND6 blue 65 nm (7) and the Continuum ND6 red 68 nm (8) dye lasers, shown in Fig.5.5. delay settings CCD camera gate CCD camera image 3,56 MHz SU Hz DG535 5 4 M RF power M5 F L gate (c) 8 antenna voltage D (a) (b) 4 measured delays 3 9 to lasers PC M4 choker M9 3 M8 Q-switch Nd:YAG M Blue laser M 6 8 Red laser M6 - vacuum chamber - Cesar 3 RF generator 3 - double spiral antenna 4 - synchronization unit (SU) 5 - Stanford DG535 delay generator 6 - Continuum PL8 Nd:YAG laser (53 nm) 7 - Continuum tunable dye laser ND6 with THG crystal ( blue 5 nm) Fig.5.5. Scheme of the fluorescence dip spectroscopy (FDS). 7 9 M3 F3 P F M7 8 - Roper Scientific PI-MAX CCD camera 9 - main PC - light choker - Continuum ND6 tunable dye laser ( red 68 nm) - Rahmann cell (hydrogen at p = 3 bar) 3 - Tektronix TDS64A oscilloscope 4 - photodiode The blue laser is passed through a nonlinear BBO crystal, which generates the third harmonic of the incident radiation. The output signal has a wavelength of 5 nm. The red laser radiation is shifted into the IR region by a Raman cell (9) filled with hydrogen at 3 bar. A dielectric mirror M7 and a color plan filter F separate the first Stokes component from the pump beam and other radiation components. An additional polarizer P improves the beam 8

polarization. The blue and red laser beams are directed by the mirrors M-M4 and M6-M into the plasma cell just grazing the electrode. The optical paths of the blue and red laser beams are nearly equal. The spectra of the hydrogen n = 3 and Rydberg level transitions are measured as follows. Initially, the red beam is blocked by a choker (). After the blue beam passes through the vacuum chamber, the mirror M5 reflects beam back upon itself into the vacuum chamber again. Thus, two oppositely directed beams meet in the plasma cell and excite hydrogen atoms from the ground state to n = 3 via two-photon absorption. The wavelength of the blue is tuned to maximize the fluorescence of atomic hydrogen at the Balmer-α wavelength (the transition from the level n = 3 to n = ). The laser beams are directed oppositely in order to provide Doppler free excitation of the absorption line. The pulsed Balmer-α radiation is recorded with the Roper Scientific PI-MAX CCD camera () equipped with a 656.6 nm colored glass filter. The CCD camera is gated by a ns signal from DG535 to reduce the background plasma radiation. The total exposure time needed depends on the laser power and the hydrogen neutral density in the discharge. In our measurements the exposure time was about min corresponding to 6 laser shots. This image of the laser excited Balmer-α radiation in the sheath is then stored in a computer (). Next, the light choker is opened. The IR radiation of the red laser enters the plasma cell and depletes the n = 3 population by exciting transitions to one of the Rydberg levels. As a result, the Balmer-α radiation decreases. Hence, by scanning the red laser s wavelength and comparing the Balmer-α intensities in the presence and absence of the red laser, the spectra of atomic hydrogen transitions between n = 3 and the Rydberg level is measured. The incident IR beam is attenuated to the optimum intensity by a grey-filter F3. It should be mentioned that laser beams are polarized parallel to each other and to the sheath electric field. Since a single FDS spectra contains to points, and it takes approximately one minute to measure a single point; experiments usually last between two and three hours. Special care must therefore be taken, because the laser can drift thermally over this period of time. The problem is in synchronizing the measurements. The synchronization is corrected by making the following measurements with a Tektronix TD64A oscilloscope (3) and correcting the settings of the DG535 delay generator: (a) The RF phase is measured by a Tektronix P65A high voltage probe on the antenna. (b) A photodiode (4) allows us to determine when the laser beam enters the plasma cell by measuring the scattered radiation from the edge of the diaphragm D. (c) The timing of gate signal of the CCD camera is measured. The timing of (b) and (c) are compared relative to (a) and then used to correct synchronization signals of the DG535 every five seconds by a computer (). The temperature of atomic hydrogen in the discharge can be measured from the Doppler broadening of the n = to n = 3 line. In this case, the blue beam is not reflected back into the plasma cell, and the red beam is switched off. The absorption line profile at the Balmer-α wavelength is measured by scanning the blue laser wavelength (7). 8

5..3. Measurement of ion energies The PPM4 ion energy and mass analyzer is incorporated into the planar electrode, as shown in Fig.5.. The potential of the electrode containing the PPM4 extraction hood is zero. The PPM4 measures correctly, if particles passing through the analyzer do not collide. In this case, the gas pressure in the PPM4 should not exceed -3 Pa which corresponds to a chamber pressure of Pa. In this study, most of the PPM4 measurements were performed at a chamber pressure of Pa, and a PPM4 pressure of 7.6-4 Pa. Before using the PPM4, two preliminary procedures should be performed. First, it is necessary to calibrate the ion energy filter. The ion energy filter calibration using a PPM4 ionization chamber is described in the manual [Balz, p.5]. Fig.5.6 shows the energy spectrum of hydrogen ions produced by an electron beam in neutral hydrogen. Since the gas is at room temperature, the average energy is near zero. The energy scale should also be calibrated, if the ion fluxes have nonzero average energy [Balz, p.55]. Second, it is necessary to adjust the entrance optics of the PPM4, which accelerates and focuses the incoming ion beam. This is achieved by adjusting the INTRO, ENTRY, and FOC potentials to maximize the PPM4 signal. However, it was noticed that adjusting the entrance optics for hydrogen influences not only the signal, but also the shape of the IEDF. detector current ( -, A),7,6,5,4,3,, H + CENTR = 4 MIRR = 7 INFL = 7,5, -, -,8 -,6 -,4 -,,,,4,6,8, ε i (ev) Fig.5.6. Energy spectrum of hydrogen ions produced by an electron beam in neutral hydrogen at room temperature. p = Pa (H gas) detector current, ( -, A),6,4,,,8,6,4, H 3 + ENTRY = ITRO = FOC =, 4 6 8 4 6 8 ε i (ev) Fig.5.7. Energy distribution function of H 3 + ions in hydrogen ICP at Pa and 3 W measured for INTRO, ENTRY, and FOC set to zero. P = 3 W, p = Pa (H gas) Unfortunately, neither the PPM4 manual not the literature does provide any special recommendations or comments for such a procedure. For adjusting the PPM4, the following algorithm was chosen and tested. Initially, INTRO, ENTRY, and FOC are set to zero. Incoming plasma ions will be collimated into a beam by the entrances of the analyzer and energy filter. Hence, only the ions moving collinearly to the discharge axis are measured, resulting in low sensitivity measurements. Under these conditions, measurement of the distribution function is possible for only H 3 + ions which are the dominant species in a hydrogen plasma. The measured shape of the ion energy distribution will be believable and will be undistorted by the entrance optics. Measurement of a H 3 + flux is presented in Fig.5.7. Next, the INTRO, ENTRY, and FOC potentials are chosen to optimize the sensitivity of the 83

PPM4 sufficiently to allow measurement of H + and H + ion fluxes while also minimizing the distortion of the H 3 + ion energy distribution function relative to measurements made when the potentials were set to zero. Fig.5.8 shows the energy distribution functions of hydrogen ions measured with two different settings of the potentials. Results shown in the left column in Fig.5.8 use identical settings to those used in the official test report (key Nr.333) of the analyzer. In the right column, settings were determined by our novel calibration of the PPM4. detector current, ( -, A) 9 8 7 6 5 4 3 H + ENTRY = 8 ITRO = 3 FOC = 4 6 8 4 6 8 P = 3 W, p = Pa (H gas) detector current, ( -, A) 3,5 3,,5,,5,,5 H + ENTRY =,5 ITRO = FOC = 5, 4 6 8 4 6 8 P = 3 W, p = Pa (H gas) ε i (ev) ε i (ev) detector current, ( -, A) 4, 3,5 3,,5,,5,,5 H + ENTRY = 8 ITRO = 3 FOC =, 4 6 8 4 6 8 P = 3 W, p = Pa (H gas) detector current, ( -, A),,8,6,4, H + ENTRY =,5 ITRO = FOC = 5, 4 6 8 4 6 8 P = 3 W, p = Pa (H gas) ε i (ev) ε i (ev) detector current, ( -8, A) 9 8 7 6 5 4 3 H 3 + ENTRY = 8 ITRO = 3 FOC = 4 6 8 4 6 8 ε i (ev) P = 3 W, p = Pa (H gas) detector current, ( -9, A),4,,,8,6,4, H 3 + ENTRY =,5 ITRO = FOC = 5, 4 6 8 4 6 8 testreport (key Nr-333) calibration of PPM4 Fig.5.8. Energy distribution of the hydrogen ion flux measured with two different potential settings for the PPM4entrance optics. Results shown in the left column are obtained using the manufacture s settings from testreport (key Nr.333). Results in the right column are obtained using the settings determined while calibrating the analyzer. ε i (ev) P = 3 W, p = Pa (H gas) 84

The two maxima present in the ion energy distributions functions in a collisionless sheath are due to the time-varying potential. Theoretically, the distance between the peaks should be shorter for heavier ions. The distribution function in the left column is contrary to this, and casts some doubts on the correctness of the settings provided by the manufacturer. In should be mentioned that the shape of the distribution function will be affected by the potential settings of the entrance optics only if the lighter ions are being measured. This is illustrated in Fig.5.9, which shows measurements of an argon ICP in the H-mode with an incident power of W at a pressure of Pa for both settings. The shapes for both settings are equal, the only difference is in the signal intensity. detector current, ( -9, A) (a),6,4,,,8,6,4, Ar + ENTRY = 8 ITRO = 3 FOC =, 4 6 8 4 6 8 ε i (ev) P = W, p = Pa (Ar gas) detector current, ( -, A),5,,5,,5 Ar + ENTRY =,5 ITRO = FOC = 5 4 6 8 4 6 8 testreport (key Nr-333) calibration of PPM4 Fig.5.9. Energy distribution functions of Ar + ions measured with two different potential setting for the PPM4: (a) manufacture s settings from testreport (key Nr.333) and (b) settings generated while calibrating the analyzer., (b) ε i (ev) P = W, p = Pa (Ar gas) 85

5..4. Optical measurements Fig.5. shows the arrangement and configuration of the apparatus used for measuring the spatiotemporal distribution of the Balmer-α emission in a hydrogen plasma. delay settings CCD camera gate CCD camera image 3,56 MHz gate SU khz DG535 4 RF power RF power F L 6 antenna voltage antenna voltage 8 7 5 3 - vacuum chamber - Cesar 3 RF generator 3 - double spiral antenna 4 - synchronization unit (SU) 5 - Stanford DG535 delay generator 6 - Roper Scientific PI-MAX CCD camera 7 - main PC 8 - Tektronix TDS64A oscilloscope Fig.5.. Scheme of space and phase resolved optical measurements. The hydrogen plasma is maintained in the vacuum chamber () by an RF generator () coupled through an antenna (3). The Roper Scientific PI-MAX CCD camera was synchronized to the phase of the RF generator by a synchronization unit (4), which essentially divides the RF frequency into longer periods. A repetition rate of khz and a gate width of ns were used when CCD camera acquiring a data by the plasma emission. The exposure time needed depends on the plasma emission, and was typically a few tenth of a second corresponding to several hundred superimposed images. The phase shift of the gate signal relative to the RF period is controlled by the computer (7) and the Stanford DG535 delay generator (5). A phase resolved movie was recorded for two RF periods by shifting the camera gate ns after each image was acquired. The experiment s synchronization was monitored by a Tektronix TDS64A oscilloscope (8) that recorded both the RF phase and the timing of the gate signal. 86

6. Results In this section, a study of an ICP hydrogen discharge using various diagnostics is presented and compared with the theoretical predictions of the models described in Section 4. 6.. Gas Temperature In Subsection 5.., it was assumed that the neutral species in a hydrogen plasma are heated. At a fixed pressure, an increase in the gas temperature leads to a decrease in the molecular density and, accordingly, to a decrease in the frequency of electron collisions. For this reason, the models in Sections 4. and 4. needed to take into account the gas heating in an ICP. 6... Doppler broadening Two-photon absorption of 5 nm UV laser radiation by hydrogen atoms excites them from the ground state to n = 3. For a hydrogen atom with a velocity component parallel to the laser beam, the Doppler effect results in a shift of the wavelength needed for this transition. The thermal motion of the gas particles causes a broadening of the absorption-line profile. Since this depends on the particle velocity distribution, the broadening can be used to deduce the gas temperature: where M is the proton mass and absorption line of atomic hydrogen, ~ kt =, (6.) g Mc 8ln ~ νd νγ νd is the Doppler broadening of the equivalent two-photon ν = γ 975 cm. The profile of this absorption line intensity (a.u.) 8 6 4 measurement Gauss fit ν γ =.48 cm - ν UV =.5 cm - ν D =.46 cm - -3 - - 3 ν (cm - ) Fig.6.. Experimental profile of the two-photon absorption line corresponding to the transition from the ground state to the excited state n = 3. (The abscissa represents twice the frequency shift of the laser at 5 nm). T g = 443 K corresponds to the transition from the ground state to n = 3, and can be detected by measuring the fluorescence of the 656 nm Balmer-α line when scanned with a tunable UV laser. Fig.6. shows a typical absorption profile for a hydrogen ICP at Pa and an incident power of 3 W. As it seen from the figure, the line fits well to a Gaussian profile and has a full width at half-maximum (FWHM) of ~ ν =.48 cm γ. The Doppler broadening of the absorption line is: ~ ~ ν = ν γ, (6.) D ν UV 87

where ν =.5 cm << ν ~ γ ~UV is the laser spectral width (FWHM) at 5 nm. The neutralgas temperature calculated using the spectra in Fig.6. is 443 K. 6... Gas temperature of a hydrogen ICP The UV laser beam was aligned to pass through the discharge axis.5 cm above the grounded electrode, at the same height where the Langmuir probe measurements were performed. No variation in the gas temperature was observed radially. Fig.6.(a) shows the gas temperature in the ICP discharge as a function of the incident power at Pa. Fig.6.(b) shows the dependence of the gas temperature on the pressure at a fixed incident power of 3 W. 5 gas temperature (K) 8 6 4 measurement T g = T g () ( + αp) 4 6 8 p = Pa (H gas) gas temperature (K) 4 3 measurement polynomial fit 5 5 5 3 (a) incident power (W) (b) pressure (Pa) Fig.6.. Gas temperature in the ICP as a function of the incident power (a) and gas pressure (b). The gas temperature, T g, in Fig.6. increases proportionally to the incident power: T g () = T + g ( αp), (6.3) where T g () = 3 K is room temperature, P is the incident power, and α =.5 kw - is a constant of proportionality. The gas temperature varies only slightly over a wide range of gas pressures. Solid lines in the figure show a fit to the measured values of the gas temperature by a linear function in Fig.6.(a) and by a fourth degree polynomial in Fig.6.(b). These fits allow us to estimate a value of T g we can use in the analytical hydrogen plasma and transformer models. Ideally these measurements should have been selected to a wide range of powers and pressures. However, since they are quite involving we restricted ourself to the data shown above and made extrapolation for other parameter sets in the model. P = 3 W (H gas) 88

6.. Plasma bulk parameters Probe measurements were carried out in the hydrogen ICP discharge to study the H-mode. The design of the probe, its calibration, and the experimental arrangement are described in Section 4.. The analysis of I-V characteristic of the probe is considered in Subsection 3.5.. Here, only the results of measurements are presented, and compared with the theoretical calculations. 6... Electron energy distribution function Fig.6.3(a) shows the electron velocity distribution functions (EVDFs) for various values of the incident power calculated using the Druyvesteyn method. In the course of the measurements, the gas pressure in the chamber was always Pa. As is seen from the figure, all of the distribution functions in semilogarithmic charts are linear and therefore Maxwellian. The slope of these functions is determined by the electron temperature and varies only slightly over a wide range of incident powers. In other words, the electron temperature is approximately independent of the incident power. Which is in agreement with the analytical hydrogen plasma model (AHPM) presented in Section 4.. Fig.6.3(b) shows the EVDFs measured at various gas pressures for a fixed incident power 3 kw. As can be seen from the figure, the distribution functions for pressures between 5 to Pa are Maxwellian. At lower pressures, the slope of the distribution function is larger at higher energies. The cross section for inelastic collisions of an electron with a hydrogen molecule is large, and it is likely that the population of electrons above threshold energy is being depleted by the processes listed in Table 4.. Further, at 3 W and Pa, the discharge operates already at the E-mode transition, which might modify the distribution function. All of the EVDF s in Fig.6.3 are Maxwellian below molecular hydrogen s ionization energy, 5.4 ev. 6 6 5 4 3 EVDF (ev -3 / m -3 ) 5 4 3 EVDF (ev -3 / m -3 ) (a) 5 5 energy (ev) 5 3 5 8 6 5 4 incident power (W) 5 5 energy (ev) 5 3 5 5 pressure (Pa) Fig.6.3. Electron velocity distribution functions obtained by the Druyvesteyn method for various incident powers at Pa (a) and for various gas pressures at 3 W (b). (b) 89

At higher pressures, the non-maxwellian nature of the EVDFs is not apparent. At lower pressures, EVDF can be measured for energies up to 3 ev. As the pressure increases, the electron temperature decreases and the slope in the EVDF becomes steeper. The resolution of the probe measurement depends upon the signal to noise ratio. The measurement of the EVDF at high energies for higher pressures becomes impossible because the signal is too small. 6... Electron temperature Fig.6.4 shows the measured and calculated electron temperature in the ICP as a function of the incident power (a) and pressure (b). In order to estimate how an increase in the gas temperature affects the electron temperature, calculations were performed which took into account (solid lines) and did not take into account (dashed lines) the gas heating. The gas temperature was measured instead of calculated (see Section 6.), because these calculations would have been beyond the scope of this study. 4, 3,5 3,,5 6, 5, 4, measurement T g = T g () ( + αp) T g = T g () kt e (ev) (a),,5,,5 measurement T g = T g () ( + αp) T g = T g (), 4 6 8 P (W) p = Pa (H gas) kt e (ev) 3,,,, 4 6 8 4 6 8 Fig.6.4. Electron temperature in the ICP discharge as a function of the incident power (a) and gas pressure (b). Open circles show measurements data and lines show calculations which took into account (solid line) and did not take into account (dashed line) the gas heating. Fig.6.4(b) shows the pressure dependence. The model predicts the measured data quite accurately, although systematically higher values are calculated. The electron temperature rises at low pressures due to increased particle loss to the wall. As may be seen from Fig.6.4(a), the electron temperature, T e, increases with the incident power. This is adequately described by the calculations that account for the increase in the gas temperature. However, the measured T e is % lower than its calculated value. This may be attributed to the global model s assumption of a spatially uniform T e. Fig.4.(a) shows that, the electron energy losses at low T e in the bulk of a hydrogen plasma far exceed the energy losses due to the particle flux to the chamber wall. In this case the dissipated energy, dp ξ /dv, due to Ohmic heating by induced currents balances losses by the inelastic collisions of electrons: dp (b) ξ < > =σ pl < Eind > = nenrε dv r ( ) p (Pa), (6.4) P = 3 W, (H gas) 9

where σ pl is the plasma conductivity, E ind is the induced electric field, R ε () is the energetic loss rate from equation (4.5), n e is the electron density, and N is the neutral-gas density. The electric field and the energetic loss rate can be found by substituting σ pl from equation (4.4) into equation (6.4): m ( ω + v ) < E R e m () () ind > = NR ~ ε ε, (6.5) vme where m e is the electron mass, v m is the electron collision frequency, and ω is the electromagnetic field frequency. According to (6.5), the energetic loss rate R ε () of the local energy balance is proportional to the square of the induced electric field. The nonuniform spatial distribution of this field in the plasma cell is responsible for the nonuniform temperature in the ICP discharge. Fig.6.5 shows the measured electron temperature radially in the hydrogen ICP at an incident power of 3 W and a gas pressure of Pa. The measurements were performed.5 cm from the electrode surface. Experimental data are compared with calculations in the figure by presenting calculations that take into account or ignoring the gas heating as solid and dashed lines. In both cases, the calculated value exceeds the measurements. The T e profile has a small dip on the axis of the plasma cell, and has a maximum temperature at a radius of 4 cm or /3 of the antenna radius. As was shown in Subsection 4.., the E-field has its maximum at the same position. 3,5, 3, kt e (ev),5,,5, antenna radius electrode radius cell radius measurement,5 T g = T () g ( + αp) () T g = T g, -4-4 6 8 radial position (mm) Fig.6.5. Measured electron temperature radially.5 cm from the electrode surface in the hydrogen ICP at 3 W and Pa. Lines show calculations that take into account (solid line) or ignoring (dashed line) the gas heating. (V/cm) < E ind > /,5,,5 deduced from T e < E ind > / = E ind () (in vacuum) < E ind > / = E () ind exp(-x / δ ) pl, -4-4 6 8 radial position (mm) Fig.6.6. Calculations of the E-field s radial profile. Open circles show calculations using the measured electron temperature, solid and dashed lines show ab initio calculations either taking into account or ignoring the absorption of the electromagnetic field in the plasma. The radial profile of the E-field strength in the ICP can be determined in two ways. First, this profile can be calculated from equation (6.5) by using the measured T e. Second, the profile can be calculated for a particular antenna geometry by assuming the plasma conductivity to be uniform (see Appendix C). The AHPM, and the transformer model, show that the RMS antenna current in a hydrogen ICP at 3 W and Pa is 3.4 A, when the 9

plasma skin depth is cm. Calculations of the E-field are presented in Fig.6.6. Open circles / in the figure show < E ind > calculated using T e from probe measurements, whereas the / solid and dashed lines show the radial profile of < E ind > calculated ab initio either taking into account or ignoring the absorption of the electromagnetic field. Since there is no E-field on the discharge axis, the high T e surrounding the axis can only be maintained by electron flow from the regions where the E-field is present and the plasma heating is occurring. As the temperature and the E-field strength difference between the regions increases, the energy transferred by the electron flow increases as well. Because of this, for r >> /3r ant, where r ant = 6 mm is the antenna radius, the radial profiles of the E-field are almost identical, whereas for r < /3r ant they differ substantially. Further, and probably most important, the probe tip has a size of 5 mm and integrates over even a larger region. In addition, the probe might not pass exactly through the discharge axis. Both effects will smooth out the temperature dip in the center. Fig.6.7 shows the E-field s vertical profile. The calculations were performed at a radius of /3r ant, where the radial gradient of T e is a minimum and we expect the local power balance given by equation (6.4) to hold. A dashed line in the figure indicates the E-field in vacuum, and the solid line indicates the E-field in the plasma. The vertical position zero corresponds to the inner surface of the dielectric window, where the strength of the induced field is five times larger than in the region where probe measurements were done. 7 8 < E ind > / (V/cm) 6 5 4 3 E ind () (in vacuum) E ind () exp(-x / δ pl ) probe position kt e (ev) 7 6 5 4 3 measurement deduced from < E ind > / T g = T () g ( + αp) () T g = T g 5 5 5 3 35 4 45 5 vertical position (mm) Fig.6.7. Calculations of the E-field s vertical profile at /3 of the antenna radius. Solid and dashed lines show calculations either taking into account or ignoring the absorption of the electromagnetic field in the plasma. 5 5 5 3 35 4 45 5 vertical position (mm) Fig.6.8. Electron temperature at /3 of the antenna radius calculated assuming a local power balance (dashed-dotted line) and for the AHPM taking into account (solid line) and ignoring (dashed line) the gas heating. Open circle shows the measured temperature. The electron temperature calculated assuming a local power balance (6.4) is shown by a dashed-dotted line in Fig.6.8. Also shown in the figure is the experimental value of T e (open circle) and T e calculated taking into account (solid line) and ignoring (dashed line) neutral gas heating. As is seen from Fig.6.8, values of T e from the AHPM exceed the measurements because there is a gradient of T e along the discharge axis. 9

6..3. Plasma density Fig.6.9 shows the plasma density as a function of incident power (a) and pressure (b). Open circles represent measurements, and lines show calculations taking into account (solid line) and ignoring (dashed line) the gas heating in the discharge. The calculations of the analytic model agree well with the measurement results except for pressures greater than Pa. At pressure above this value, the local plasma density on the discharge axis and.5 cm from the electrode surface increases but the calculations predict that this density should decrease. 3,, n e (, cm -3 ),5,,5,,5 measurement T g = T g () ( + αp) T g = T g (), 4 6 8 p = Pa (H gas) n e (, cm -3 ),,8,6,4, measurement T g = T g () ( + αp) T g = T g (), 4 6 8 4 6 8 (a) P (W) (b) p (Pa) Fig.6.9. Plasma density as a function of the incident power (a) and gas pressure (b). Open circles represent the Langmuir probe measurements, and lines show calculations taking into account (solid line) and ignoring (dashed line) the gas heating. Several examples of the radial profiles of n e and T e can be seen in Fig.6. and Fig.6.. T e and n e were measured for various incident powers with a fixed gas pressure of Pa (a) and also for various pressures with a fixed incident power of 3 W (b). Solid lines show n e profiles calculated assuming a uniform T e profile. The increase in the gas temperature was taken into account in the calculations. P = 3 W (H gas) n e (, cm -3 ) (a),5,,5,,5 P = W P = 4 W P = W, -4-4 6 8 radial position (mm) p = Pa (H gas) n e (, cm -3 ),,,8,6,4, p = Pa p = Pa p = Pa, -4-4 6 8 radial position (mm) Fig.6.. Plasma density s radial profile for various incident powers (a) and for various gas pressures (b). Open points show the Langmuir probe measurements and lines show the AHPM calculations taking into account the increase in the gas temperature. (b) P = 3 W (H gas) 93

The measurement results coincide with the calculations when the incident power is lower than 8 W and the gas pressure is between and Pa. For powers above 8 W, the measured n e profile has a triangular shape. This is shown in Fig.6.(a) for a power of kw. The deviation of the measured profile from the zeroth-order Bessel function (4.3) is probably a consequence of the inhomogeneity of the induced electric field at high plasma densities more pronounced. 3,5 3, 8 7 kt e (ev) (a),5,,5,,5 P = W P = 4 W P = W, -4-4 6 8 radial position (mm) p = Pa (H gas) kt e (ev) 6 5 4 3 p = Pa p = 5 Pa p = Pa p = Pa -4-4 6 8 radial position (mm) Fig.6.. Measured electron temperature s radial profile for various incident powers (a) and for various gas pressures (b). (b) P = 3 W (H gas) The electron density at the center of the discharge is somewhat lower for pressures below Pa. This may be attributed to a centrifugal force acting on the electrons accelerated by the induced electric field [Suz 98]. On the contrary, the electron density at the discharge center increases substantially for pressures above Pa. At higher pressures the electron temperatures is lower, and it is likely that the local power balance (6.4) becomes dominant. Since the majority of the power input into the plasma is lost through inelastic electron collisions, the particle flux onto the chamber wall can be neglected. A contraction of the discharge will then be observed. Since the plasma volume is smaller than the cell, the electron density in the discharge should be higher for the same input power as predicted by equation (4.). 94

6.3. Sheath dynamics The electric field near the surface of the grounded electrode was measured by the FDS technique explained in Subsection 3... The experimental setup was already described in Subsection 5.., so only the measurement results and a comparison with theoretical calculations are discussed in this section. 6.3.. Electric field in the ICP sheath Fig.6. shows the phase-resolved spatial distribution of the electric field strength obtained using FDS. E (V/cm) 6 8 4,8 ns, ns 9, ns 8,4 ns electrode position -,,,,4,6,8,, x (mm) P = 3 W, p = Pa H gas E (V/cm) 6 8 4 37,6 ns 46,8 ns 56, ns 65, ns electrode position -,,,,4,6,8,, x (mm) (a) sheath contraction (b) sheath expansion Fig.6.. Phase-resolved spatial distribution of the electric field in the hydrogen ICP sheath measured using FDS. To look at the phase resolved electric field strength, measurements were made at eight different times within one period and were 9. ns apart. The time t = ns coincided with the antenna s positive voltage maximum. The point x = mm was chosen to be the preliminary sheath edge, which we defined to be where the E-field approached a minimum. Of course, this point is strongly depending on the spatial resolution, x = 57.6 µm, and on the E-field sensitivity, E min = 5 V/cm. The precise determination of the point where actually ions reach the Bohm velocity is critical and will be explained in the following chapter. The electrode surface was at x =.5 mm. As can be seen from Fig.6., the E-field is measured approximately.9 mm into the sheath. FDS measurements closer to the electrode were prevented by the scattering of laser radiation and by the reflection of Balmer-α emissions from the electrode surface. P = 3 W, p = Pa H gas 6.3.. Time-averaged electric field The time evolution of the spatially resolved potential can be determined by integrating the measured E-field: x U( x, t) = E( x, t) dx. (6.6) 95

When the electron temperature is known, the normalized ion, n i (x)/n, and electron, n e (x,t)/n, densities can be found from equations (4.6) and (4.6), where n is the plasma density at the sheath edge. There it is assumed that the ion current to the electrode is constant in time (static) and the electron energy distribution obeys the Boltzmann equation. The static ion density is calculated using the time-averaged sheath potential: U(x) = < U(x,t) >. Since the charge densities are related to the E-field by the Poisson s equation, we can fit a calculated E(x,t) to the measured data by substituting n i (x)/n and n e (x,t)/n into equation (4.69), varying the values of T e and n. A conventional least-square method is used to choose the optimum values of T e and n. σ (T e,n ) is the sum of squares of the difference between the left and right sides of the Poisson s equation: σ X, T n i ( Te ) ne( Te ε E, n +. (6.7) x, t= n n e x x x, t x, t ( Te n ) = ) Here the result is taken over all spatial and temporal data points. It is shown in Fig.6.3. As can be seen from the plot, σ (T e,n ) is minimized when kt e =. ev and n = 8.5 9 cm -3. In statistics, the region of uncertainty for a least-square method is usually defined by: σ edge =.σ. (6.8) min This region is shown in Fig.6.3 by a discontinuous white line. From this, the maximum errors in the temperature and density are: (kt e ) = ±.6 ev and n = ±.7 9 cm -3. If either T e or n is known in advance, the accuracy of the calculation can be improved. For example, if the value of the electron temperature from the probe measurements is used: kt e =.4 ev, the error in calculating the plasma density at the sheath edge can be reduced by a factor of three: n = (9.6 ±.6) 9 cm -3. Unfortunately, the presence of an axial gradient in the electron temperature prevents this. In further analysis, we will use the value kt e =. ev. It should be mentioned further that for a single parameter, n or T e, the minimum of σ and the related error both can be found analytically. kt e (ev) 3,,8,6,4,,,8,6,4,, 5 6 7 8 9 3 4 5 n ( 9 cm -3 ) Fig.6.3. Sum of squares of the difference between the left and right sides of the Poisson s equation. The sum is minimized when kt e =, ev and n = 8.5 9 cm -3. E (V/cm) 3 5 5 5 measurement E =,5 V/cm electrode position fit to measurement -,,,,4,6,8,, x (mm) Fig.6.4. Time-averaged electric field in the ICP sheath. Solid line shows E-field calculated by integrating the Poisson equation with the best fit for T e, n, and E. P = 3 W, p = Pa H gas 96

Beside the statistical errors the most critical part in the above analysis is the choice of the sheath edge position, i.e. the point where ions reach the Bohm velocity and n e = n i = n. A following iterative procedure for determination of this point out of the measurements is proposed. First, T e and n are determined for a selected position of the sheath edge. Second, with these parameters, the time-averaging E-field distribution is calculated by integrating the static Poisson s equation (4.58) for U = V and various boundary condition, E, until a best fit to the measured time-averaging electric field is found. The best fit for the preliminary choice of the sheath edge at E =.5 V/cm is shown in Fig.6.4. These two steps are repeated for various positions of the sheath edge, and result is shown in Fig.6.5-Fig.6.6. kt e (ev) n ( 9, cm -3 ),3,,,,9,8 8,8 8,7 8,6 8,5 8,4 8,3 sheath edge 8, -, -,5 -, -,5,,5,,5, position of the sheath edge (mm) Fig.6.5. T e and n as functions of the sheath edge s position. E (V/cm) 8 6 4 8 6 4 fit to measurement data deduced from n, T e sheath edge -, -,5 -, -,5,,5,,5, position of the sheath edge (mm) Fig.6.6. E-field at the sheath edge. The sheath edge s position is given by the intersection of curves on the plot. According to the static sheath theory proposed in [Rie 9], T e, n and E will satisfy equation (4.6) only at the sheath edge. The calculated values of T e, n and E are shown in Fig.6.5 and Fig.6.6 by solid lines with open circles. The E-field strength, E, is then calculated by substituting the T e and n into equation (4.6) and is shown by a solid line with filled circles in Fig.6.6. Only at the intersection of the curves equation (4.6) is in agreement with the measurement and therefore this corresponds to the sheath edge,.7 mm, relative to the initially chosen position. The electron temperature, plasma density, and electric field strength at the sheath edge are: kt e =.5 ev, n = 8.55 9 cm -3 and E =.3 ev. The result shows that the error due to the incorrect determination of the sheath edge is generally much smaller then the statistical uncertainty of a least-square method used in the analysis. It is true even if a maximum error of ± pixels is assumed. The static E-field strength E(x) and potential U(x) in the sheath are presented in Fig.6.7. The static ion density distribution n i (x) in the sheath, calculated using equation (4.6), is shown in Fig.6.8. Open circles on the plots show the measured E-field, solid lines show the result of integrating the Poisson equation, and the dotted lines at mm and.4 mm indicate the sheath edge and the electrode surface. 97

< E > (V/cm) 3 5 5 5 < E > < U > - -,,,,4,6,8,, x (mm) Fig.6.7. Time-averaged E-field and potential in the sheath. Open circles and triangles show the measured E-field, solid lines show the result of integrating the Poisson s equation, and the dotted lines at mm and.4 mm indicate the sheath edge and the electrode surface. - -4-6 -8 < U > (V) ion density ( 9, cm -3 ) 9 8 7 6 5 4 3,,,4,6,8,, x (mm) Fig.6.8. Ion density distribution in the sheath. Open circles show the measurements, solid line shows the calculations and dotted line indicates the electrode surface. P = 3 W, p = Pa H gas 6.3.3. Time-varied electric field in the sheath The time-varying E-field distribution in the sheath, E(x,t), is calculated by integrating the Poisson s equation (4.69). In this case, a time-varying boundary condition E (t) is selected to fit E(x,t) best to the measured electric field (see Subsection 4.3.). The results are presented in Fig.6.9. 3 3 5 fit to measurement 5 fit to measurement E (V/cm) 5 5,8 ns, ns 9, ns 8,4 ns,,,4,6,8,, x (mm) (a) sheath contraction P = 3 W, p = Pa H gas E (V/cm) 5 5 37,6 ns 46,8 ns 56, ns 65, ns,,,4,6,8,, x (mm) (b) sheath expansion Fig.6.9. Time-varying E-field in the sheath. Open circles show the E-field measured using FDS, solid lines show the E-field calculated by integrating the Poisson equation with time-varying boundary conditions. The boundary conditions are selected to fit the calculations best to the measurements. The time dependent sheath potential, U S (t), can be calculated by integrating these fitted E-fields for the corresponding time-period over the entire sheath, shown in Fig.6.(a) by open circles. This fit is needed in order to extrapolate the measurements to the region close to the electrode. P = 3 W, p = Pa H gas 98

Alternatively, U S (t) can be obtained directly from the static E-field by employing the sheath model discussed in Section 4.3. In this model, the fluctuating potential U S (t) is represented as a sum of the stationary component U and the fluctuating component U ~ (t) (4.63). U ~ (t) is a harmonic function and U obtained by integrating the static E-field. In Fig.6.7, U is equal -8. V. Then, the amplitude of U ~ (t) is calculated from the RF floating potential (4.68). When the electron temperature is.5 ev, U ~ = 3. V results. The sheath potential calculated in this way is shown in Fig.6.(a) by solid curve The time-varying E-field strength at the sheath edge is shown in Fig.6.(b). The circles show the results of the best fit to the measured E-field and the solid curve shows the value corresponding to the harmonic sheath potential, U S (t). The accuracy of the sheath model of Section 4.3 is shown by the agreement of the values of E (t) and U S (t) obtained by different methods. -5,7-6,6-7,5 U S (V) (a) -8 U S = -(8, + 3. -9 cos(ωt)) V - - - 3 4 5 6 7 8 t (ns) P = 3 W, p = Pa H gas E (V/cm) (b),4,3,,, < E > =,3 V/cm 3 4 5 6 7 8 t (ns) P = 3 W, p = Pa H gas Fig.6.. Time dependent sheath potential (a) and E-field at the sheath edge (b). Open circles show calculations using the E-field measurements for the corresponding time-period over the entire sheath. Solid lines show numerical results obtained from the time-averaged E-field by employing the sheath model (see Section 4.3). It should be noted that in the dynamic sheath simulation, the time-averaging potential was calculated in a first approximation by the static Poisson s equation. It allowed, also in a first approximation, to determine the static ion density distribution. With this, the dynamic electric field and potential were calculated. In a recent step, these potentials were timeaveraged and an improved time-averaging potential and an improved static ion density distribution were obtained. Then, in principle, the simulation procedure can be iteratively repeated all over. However, it is turned out that the corrections in the second iteration were already negligibly small. 99

6.3.4. Sheath parameters The ICP sheath electric field and potential profiles in Fig.6. were obtained by integrating the Poisson s equation (4.69). In Fig.6.(a), the boundary conditions for the integration were determined from the FDS E-field measurements. In Fig.6.(b) the boundary conditions for the integration were determined using the ICP model and by the iterative process explained in Subsection 4.3.. In latter case, kt e = 3. ev and n =.8 cm -3 were determined by the AHPM which takes into account neutral gas heating, and j S =.57 A/m was determined by the transformer model. 35 3 potential - 4 36 potential - 5-4 3-4 E (V/cm) 5-6 -8 U (V) E (V/cm) 4 8-6 -8 U (V) - - 5 E-field -4,,,4,6,8,, x (mm) (a) experiment - 6 E-field -4,,,4,6,8,, x (mm) (b) theoretical model Fig.6.. E-field and potential in the sheath calculated by integrating the Poisson equation. Boundary conditions were determined from the E-field measurements (a) and simulated by the iterative process explained in Subsection 4.3. (b). Solid curves show time-averages, dashed curves show extremes. The static ion density and the time dependant electron density are show in Fig.6.. Phase diagrams of the currents flowing through the sheath are shown in Fig.6.3. - 9 n i, n e ( 9, cm -3 ) 8 7 6 5 4 3 electrons ions,,,4,6,8,, x (mm) (a) experiment n i, n e ( 9, cm -3 ) 8 6 4 electrons ions,,,4,6,8,, x (mm) (b) theoretical model Fig.6.. The static ion density and the time-varied electron density in the ICP sheath.

sheath current (A/m ) - - total ( j ) S electron ( -3 j ) e ion ( j ) i displacement ( -4 j ) D,,,4,6,8, time/period (a) experiment sheath current (A/m ) - - total ( j ) S electron ( -3 j ) e ion ( j ) i displacement ( j ) D -4,,,4,6,8, time/period (b) theoretical model Fig.6.3. Phase diagrams of the electron, ion, displacement, and total currents flowing through the ICP sheath. As can be seen from the figures, the hydrogen ICP model of Chapter 3 adequately describes both the spatial structure and the temporal evolution of the sheath. Generally, the difference between the calculated and measured values of the electric field, the sheath potential, the sheath thickness, the ion density, the electron density, the total current, and the displacement current does not exceed 4%. The difference between the calculated and measured values of the ion and electron currents is 7%. This is because the AHPM assumes a constant electron temperature and overestimates the electron temperature at the sheath edge.

6.4. Ion Energy Distribution functions The ion energy distribution functions of the H +, H +, and H 3 + ions arriving at the electrode can be measured with a PPM4 energy analyzer. Its basic design is described in Subsection 3..3. The experimental setup and the calibration procedure are discussed in Subsections 4.. and 4..4. 6.4.. Energy distribution of the ion flux The measured and calculated energy distributions of the H +, H +, and H 3 + fluxes to the grounded electrode versus the applied power at Pa are shown in Fig.6.4. The method for calculating the energy distribution of an ion flux accounting for the neutral gas heating is considered in Subsection 4.3.3. Fig.6.4 shows that at an applied power of 5 W there is a distinct ICP mode transition. The three measured and calculated energy distributions of the ion flux differ greatly when the discharge operates in the E-mode, whereas in the H-mode they are very similar. This is perhaps because the probability that these ion experience inelastic collisions is increased by the higher sheath potential and larger thickness of the sheath in the E-mode. Moreover, collision energies above ev are needed to dissociate H 3 + ion efficiently. The H + and H + ion fluxes are approximately.5% and % of the H 3 + ion flux. As a result, the energy distributions of the H + or H + ions from the plasma will be altered if only a small percentage of the H 3 + ions traveling through the sheath is converted into H + or H +. This seems to explain the irregular shape and large intensity of the H + distribution in the E-mode. On the other hand, the dependence of the H + ion flux on the applied power agrees well with the collisionless sheath model. It is likely that H + is not efficiently produced during the dissociation of H 3 +. There is only a minor peak at 6 ev in the H + distribution which might indicate H + formation in the sheath during E-mode. In both the E- and H-modes, the measured energy distribution of H 3 + has the two distinctive peaks characteristic for a collisionless sheath and which are caused by the oscillations of the sheath potential. It should be noted that, for high ion fluxes the space charge can cause aberrations in the QMA of the PPM4 resulting in a decreased signal. This might explain why the measured intensity of the H 3 + ion flux increases more slowly than the simulation predicts. The discharge is in the E-mode at powers below W. As a result, the average energy of the H 3 + ions is determined by the high sheath potential and is approximately - 5 ev. As the power increases towards the transition threshold of 5 W, the efficiency of the capacitive coupling mechanism decreases. This decrease is accompanied by reduction in the steady, U, and fluctuating, U ~, components of the sheath potential. Consequently, both the average ion energy and the difference between its maximum and minimum values decrease. In Section 4.3 it was shown that if the applied power to a discharge in the inductive mode is increased, U will be hardly affected, and U ~ will gradually decreases. This power dependency is demonstrated by the energy distributions of the hydrogen ion fluxes shown in

35 3 5 5 ion energy (ev) 5 5 + H 4 3 35 3 5 5 ion energy (ev) -9 - ion current (, A) 5 + H - 5,5 + H3,,5,,5, 35 3 5 5 ion energy (ev) 5 8 + H3 6 4 -,5 35 3 5 5 ion energy (ev) 5 -, 8 6 4 incident power (W) 7 6 5 4 3 - - -3 8 6 4 incident power (W) -9-9 5 ion energy (ev) detector current (, A) 35 3 5-8 6 4 incident power (W) 7 6 5 4 3 - - -3 8 6 4 incident power (W) -9 ion energy (ev) -,5 5 5 incident power (W) + ion current (, A) 3 H ion current (, A) 4 detector current (, A),5,,4,,,8,6,4,, -, -,4 -,6 8 6 4 incident power (W) -, detector current (, A),5 + - H detector current (, A) Fig.6.4. When the discharge is operating in the H-mode, the average ion energy will remain nearly constant, and only increases from 9 to ev for a wide range of powers as a consequence of the gas heating discussed earlier. As the power increases, the distance between the peaks of the distribution function decreases and disappears at 6 W. - 35 3 5 5 ion energy (ev) 5 8 6 4 incident power (W) (a) experiment (b) theoretical model + + + Fig.6.4. Energy distributions of the H, H, and H3 fluxes to the grounded electrode versus applied power at Pa. Fig.6.5 shows the measured and simulated energy distribution of the ion fluxes for the E-mode at W (a) and the H-mode at 3 W (b). The solid curves show measurements, whereas the dashed and dashed-and-dotted curves show numerical results 3

where E(x,t) corresponds to Fig.6. in case of 3 W. E(x,t) was obtained by integrating Poisson s equation (4.69) with different boundary conditions. In Fig.6.(a), boundary conditions were determined using the FDS E-field measurements, and in Fig.6.(b), the boundary conditions were determined using the AHPM and the transformer model. The calculated energy distributions were normalized to the corresponding measurements. When the electric field distribution in the sheath is known, the simple kinetic model considered in Subsection 4.3.3 accurately simulates the energy distribution of the ion flux. detector current ( -3, A) detector current ( -, A) 7 6 5 4 3 H + measurement simulations 5 5 5 3 35 4 5 4 3 H + ion energy (ev) 5 5 5 3 35 4 ion energy (ev) measurement simulations P = W, p = Pa (H gas) P = W, p = Pa (H gas) detector current ( -, A) detector current ( -, A) 3,5 3,,5,,5,,5 measurement simulations from E-field measurement, 4 6 8 4 6 8,,8,6,4, measurement simulations from E-field measurement ion energy (ev), 4 6 8 4 6 8 ion energy (ev) H + H + P = 3 W, p = Pa (H gas) P = 3 W, p = Pa (H gas) detector current ( -, A) 5 4 3 measurement simulations H 3 + 5 5 5 3 35 4 ion energy (ev) (a) E-mode at W P = W, p = Pa (H gas) detector current ( -9, A),4,,,8,6,4, measurement simulations from E-field measurement, 4 6 8 4 6 8 ion energy (ev) (b) H-mode at 3 W Fig.6.5. Energy distribution of the ion fluxes. Solid curves show measurements, dashed and dashed-and-dotted curves show numerical results obtained using the E-fields in Fig.6. by employing the simple kinetic model (see Subsection 4.3.3). H 3 + P = 3 W, p = Pa (H gas) 4

The divergence between the measurements and simulations is largest when the discharge is in the E-mode because the transformer model used ignores power loss in the high voltage sheath at the antenna (see Section 4.). The measured energy distribution of the H 3 + ion flux allows one to correct the sheath thickness used for the FDS E-field analysis. In this study, the optical resolution of the CCD camera enabled determination of the sheath thickness with an accuracy x = 57.6 µm. From the FDS measurements, shifting the electrode position by x is equivalent to the shifting the sheath potential by U =.3 V. U can be determined from the average ion energy with equation (4.6). Since, the average ion energy measured by the PPM4 was 9.3 ±.3 ev, the sheath thickness could be corrected in principle. However, quite good agreement between the mean ion energy obtained from the electric field measurement and the PPM4 data within. ev is found. The dependence of the measured and calculated energy distribution of the H 3 + ion flux on the gas pressure for a fixed power of 3 W is shown in Fig.6.6. H 3 + 5, 4, 3,,,, detector current ( -9, A) H 3 +,5,,5, ion current ( -9, A) 5 5 ion energy (ev) 5 3 5 5 -, -, pressure (Pa) 5 5 ion energy (ev) 5 3 5 5 -,5 pressure (Pa) (a) experiment (b) theoretical model Fig.6.6. Energy distribution of the H 3 + ion flux versus the gas pressure for a fixed power of 3 W. It is seen that the measured energy distribution agrees with the simulation. The highest ion fluxes to the electrode surface occur between and 5 Pa. The energy distributions in this pressure range have a single-peak and therefore the sheath is static. Both decreases and increases in the pressure leads to sharp reduction in the ion flux, and the energy distribution of the ion fluxes becomes double-peaked. This indicates that the sheath potential has become dynamic. The average energy of the ions arriving from plasma has a minimum between 5 and Pa. 5

6.4.. Electron temperature at the sheath edge Assuming a collisionless sheath, the energy distribution of the ion fluxes can be used to determine the electron temperature. According to equation (4.6), the average ion energy is related to the static component of the sheath potential: < ε > = kt e eu i. (6.9) Half of the distance between the peaks of the energy distribution is lower or approximately equal to the amplitude of the fluctuating component of the sheath potential: ε i eu ~. (6.) The ions are not able to follow the electric field oscillations because of their inertia, and the U ~ obtained by equation (6.) will be slightly underestimated. The most accurate way to determine the amplitude of the fluctuating component of the sheath potential is to examine the energy distribution of light ions like H +. By substituting the numerical values of expressions (6.9) and (6.) into the RF floating potential, equation (4.68), one can find approximately the electron temperature at the sheath edge. The electron temperature as a function of the applied power and gas pressure is shown in Fig.6.7(a) and (b), respectively. Ion energy analyzer measurements are shown by triangles and Langmuir probe measurements are shown by circles. A square is used to show the electron temperature at the sheath edge deduced from the FDS E-field measurements. The solid curve corresponds to the electron temperature calculated using the AHPM. The different electron temperature measurements agree with each other. The reasons for the divergence between the measurements and calculations are already discussed in Subsection 6... kt e (ev) (a) 4, 3,5 3,,5,,5,,5 ion energy analyzer probe measuremets E-field measuremet by FDS global model, 4 6 8 P (W) p = Pa (H gas) kt e (ev) 6 5 4 3 ion energy analyzer probe measuremets E-field measuremet by FDS global model 4 6 8 4 6 8 Fig.6.7. The electron temperature as function of the power (a) and gas pressure (b). Open symbols (triangles, circles, and squares) show the measurements, solid curves show the calculations using the AHPM. (b) p (Pa) P = 3 W (H gas) 6

6.4.3. Plasma potential The AHPM of Section 4. can be used for comparing experimental results obtained with different techniques. As an example, the plasma potential at the sheath edge measured using FDS and the PPM4 can be compared to the plasma potential.5 cm above the electrode measured by the Langmuir probe. Fig.4.4 represents the axial profile of U pl simulated for different ionization frequencies, κ, using the one-dimensional AHPM. It can be shown that the ionization can be neglected for area near the sheath edge. For κ <<, this is equivalent to a region smaller a /3 of the discharge gap. During the measurements, the probe was positioned clearly within this region, at.5 cm above the electrode surface. This appreciably simplifies the calculations of the plasma potential from equations (4.6)-(4.9): () kte ( ) U = U ln ζ + ζ + 4ζ pl pl, (6.) e where U pl () is the plasma potential at the sheath edge; ζ = (h d S ) / Λ is the normalized distance from the sheath edge, d S, to the probe position, h; and Λ is the ion mean free path at the sheath edge. In the H-mode, d S << h and can be neglected in the calculations. Fig.6.8 shows U pl () and U pl versus the applied power and gas pressure. Direct measurement are plotted using open symbols, and filled symbols are used to plot U pl calculated.5 cm above the electrode using equation (6.). The dashed curve shows the static floating potential, U pl (), calculated using equation (4.), and the solid curve represents U pl calculated by substituting U pl () into equation (6.). U pl (V) (a) -8-6 -4 - - -8 U pl () U pl -6 probe measurements -4 ion energy analyzer - E-field measurement by FDS global model 4 6 8 P (W) p = Pa (H gas) U pl (V) - - -8-6 -4 - - -8 U pl () U pl -6 probe measurements -4 ion energy analyzer - E-field measurement by FDS global model 4 6 8 4 6 8 Fig.6.8. Potentials at the sheath edge and.5 cm above the electrode versus the incident power (a) and the gas pressure (b). Open symbols (triangles, circles, and squares) show the measurements. Closed symbols (triangles and squares) show numerical results obtained from the measurements by equation (5.). Solid and dashed lines represent the calculations using the AHPM. The agreement between the plasma potential U pl calculated using the AHPM, the measured sheath potential U pl (), and the probe measurements in Fig.6.8 confirm the validity of the AHPM of Section 4.. (b) p (Pa) P = 3 W (H gas) 7

6.5. CCP-ICP mode transition As was mentioned in Subsection 3..4, the time-resolved measurement of the Balmer-α emission intensity allows one to discriminate between the E- and H-modes of a hydrogen ICP. In this study, the emission intensity was spatially and temporally measured with a CCD camera. This setup is described in Subsection 5..4. 6.5.. ICP emission The temporal evolution of the Balmer-α emission intensity from a hydrogen plasma is shown in Fig.6.3. The CCD images of the ICP discharge operating in the E-mode, the H-mode, and at the mode transition during one RF period are shown in three separate columns from the left to the right. The top of each image corresponds to the inner surface of the quartz window and the bottom corresponds to the electrode surface. The starting time coincides with the maximum of the negative antenna voltage. The CCD camera gate time was ns for these images. U / U max, I / I max,,5, -,5 -, antenna voltage antenna current -,5 3 4 5 6 7 t (ns) Fig.6.9. Time evolution of the antenna voltage and current. The images in the left column were made when the discharge was in the E-mode, and had an applied power of W with a gas pressure of Pa. At 37 ns, a thin glowing zone is seen near the surface of the quartz window. Looking at the temporal evolution of the antenna voltages in Fig.6.9, we can see that the voltage amplitude reaches its positive maximum at this instant. As the antenna potential decreases, electrons will move away from the quartz surface into the plasma bulk. This motion lasts until the antenna voltage reaches its maximum negative voltage at 74 ns. The images show the gradual shift of the intensity s maximum towards the discharge center. The overall maximum of the emission intensity occurs at 65 ns. After 74 ns, the antenna potential begins to increases. The electrons will then move in the opposite direction back towards the antenna, and will progressively charge the quartz surface. During this process, the emission intensity decreases. Thus, the emission intensity of the Balmer-α line will have only one peak for a discharge in the E-mode. A peak of emission intensity close to the quartz surface corresponds to the effect of field reversed. Details about this feature that is particular for hydrogen RF discharges can be found in the literature [Cza 99]. In the H-mode, there are two peaks in the emission intensity at 9 ns and 56 ns. They coincide with the maximum amplitude of the antenna current. It should be noted that, the heating region of the hydrogen plasma in the H-mode is torus shaped due to the induced electric field s structure (see Figs.3.3 and 3.3). 8

E-mode W, Pa H-mode 3 W, Pa hybrid mode 3 W, 4 Pa 4 9 4 9 3 8 time (ns) 3 37 4 47 5 56 6 65 7 75 x x 3 x6 intensity (a.u.) Fig.6.3. Temporal evolution of the Balmer-α emission intensity from a hydrogen plasma. The CCD camera gate time was ns for the images. 9

The images in the right column of Fig.6.3 show three emission maximums during a single RF period. There are two peaks in the plasma volume and one peak in the narrow zone near the quartz window s surface. In addition to the purely E- and H-modes of the ICP, there is also a transitional hybrid mode which has emission intensities characteristic of both. Therefore, the spatially and temporally resolved measurement of emission intensity allows to determine the E-mode, the H-mode, and the hybrid mode of an ICP discharge. 6.5.. Intensity modulation The ICP s mode was determined by examining the total emission intensity during one RF period in two sub regions. Area was located in the plasma volume at the induced electric field s maximum and area was on the discharge axis near the quartz window. They are shown by the white lines in Fig.6.3. Since the plasma emission intensity in the E-mode is lower than that in the H-mode by an order of magnitude, the intensity modulation, I M (t), was investigated instead of the absolute intensity, I(t): where T is the RF period. I M I( t) ( t) =, T I( t) dt T area area area area (6.) (a) E-mode (b) H-mode Fig.6.3. Balmer-α emission from a hydrogen ICP for the E- and H-modes. White lines show areas used for determination of the ICP s mode by examining the emission intensity during one RF period. 6.5.3. Mode transition Fig.6.3 shows the mode transition of a Hydrogen ICP by comparing the Balmer-α emissions to the applied power. The gas pressure was Pa, and I M (t) was monitored during two RF periods. For powers less than 5 W, there is only one emission intensity peak during an RF period, and the ICP is clearly operating in the E-mode. At powers above 5 W, there are two emission intensity peaks during each RF period. This corresponds to the H-mode. This threshold between the discharge modes is clearly seen in both area and area, and was also observed with the ion energy analyzer, see Fig.6.4(a). Another interesting fact that can be learned by examining the figure is that even though the emission intensity of a discharge in the H-mode increases as the power increases, the intensity modulation does not change. Also, the intensity modulation in the E-mode is much higher than in the H-mode.

6 6 p = Pa (H gas), 8 6 4,5, time / period,5, 4 4 p = Pa (H gas) 4 incident 3 power (W), 8 6 4,5, time / period,5 (a) area, 4 intensity modulation (%) 8 intensity modulation (%) 8 incident 3 power (W) (b) area Fig.6.3. Intensity modulation of the Balmer-α emission for the hydrogen ICP versus the incident power at the gas pressure of Pa. The Balmer-α emission intensity from the hydrogen ICP versus pressure at an applied power of 3 W is shown in Fig.6.33. The mode transition, as determined by the number of the plasma emission peaks during a single RF period, occurred at. and 35 Pa. However, the mode transitions observed by varying the pressure were not as clearly pronounced as was for varying the applied power. The intensity modulation indicates that the discharge was in the hybrid mode between 3 and 4 Pa. Apparently, capacitive coupling is more pronounced near the quartz surface (area ). Fig.6.33 also shows that IM increases as the gas pressure is increased. 5 p = Pa (H gas) 5, 5,5, time / period,5, (a) area 6 5 4 3 pressure (Pa) 5 p = Pa (H gas) intensity modulation (%) 5, 5,5, time / period,5, 6 5 4 3 preesure (P (b) area Fig.6.33. Intensity modulation of the Balmer-α emission for the hydrogen ICP versus the gas pressure power at the incident power of 3 W.

power (W) 8 6 4 minimum powers at which the matching unit is still able to automatically match phase resolved Balmer-α emission spectroscopy, pressure (Pa) Fig.6.34. Mode transition in the ICP. Solid line with open circles shows the antenna impedance s measurements, and triangles show the results of the phase-resolved emission spectroscopy. H gas Fig.6.34 shows the minimum powers (versus pressure) at which the matching unit is still able to automatically match. It was shown in Subsection 5.. that the mode transition occurs at approximately these powers and pressures. The power and pressure mode thresholds obtained using time-resolved emission spectroscopy are shown as triangles. As seen in Fig.6.34, these two different experimental methods agree with each other. 6.5.4. Mode transition in the pulsed ICP Since the matching network cannot automatically match a pulsed discharge, the mode transition cannot be determined by the antenna s impedance change. On the other hand, the temporal resolution of the Langmuir probe is not sufficient to resolve the rapid change in the plasma density. In contrast, the optical method is not restricted in this way. As an example, Fig.6.35 shows I M (t) for a pulsed ICP with an applied power of 3 W and a gas pressure of Pa. The pulsing frequency was khz with a duty cycle of 5%. The time was measured from the leading edge of the power pulse to the antenna. Immediately after switching on the power the discharge operates in the E-mode. However, one can see that I M gradual increases as time passes. At about 4 µs, a transitional hybrid mode begins and lasts for about µs. This 6 µs period of switching on corresponds to about RF periods. 6 8 4 intensity modulation (%) 6 8 4 intensity modulation (%),,5, time / period,5, 5 5 time (µs) 5,,5, time / period (a) area (b) area,5, 5 5 time (µs) 5 Fig.6.35. Intensity modulation of the Balmer-α emission for a pulsed ICP with the applied power of 3 W and the gas pressure of Pa. The pulsing frequency is khz with a duty cycle of 5%.

Fig.6.36 also shows I M for a pulsed ICP, but for a lower applied power of W. As expected, the switch-on period increased with a decrease in the applied power. The duration of the preliminary E-mode was about 5 µs, and the duration of the transition hybrid mode of discharge was about µs. At 5 µs the amplitude of the two peaks corresponding to the H-mode are not equal. This suggests residual capacitive power coupling to the plasma. In this case as well as in the above the capacitive mode survives longer close to the quartz window (area ) as might be expected. 5 5 intensity modulation (%) 5 5 intensity modulation (%),,5, time / period,5, 3 4 time (µs) 5,,5, time / period (a) area (b) area,5, 3 4 time (µs) 5 Fig.6.36. Intensity modulation of the Balmer-α emission for a pulsed ICP at W and Pa. 3

7. Summary and conclusions The aim of this study was to increase the understanding of hydrogen ICPs by using complex diagnostics and by developing a physical model that includes the mechanisms of the power coupling, plasma generation and the sheath formation. When this work was started there were no investigations on low-pressure ICP sources in pure hydrogen. The GEC reference cell was selected for developing this new plasma source. However, first characterization of the hydrogen plasma clearly illuminated its low efficiency. In order to carry out measurements, the standard cell geometry was modified. The steel antenna housing and quartz window were substituted with a housing consisting entirely of quartz. In this study, a novel, planar antenna design with multi-spiral coils was developed. In the modified GEC reference cell, the antenna inductance was reduced and the symmetry of the induced electro-magnetic field was improved by using the double-spiral antenna (see Subsection 4..5). The matching of the antenna impedance to the generator output resistance was automated to ensure the stability of the plasma source and the reproducibility of its parameters. Normally Faraday shields, commonly used to suppress the electric field produced by the antenna potential, are needed for ICP reactors. The optimized plasma cell design did not need a Faraday shield, and measurements showed that, the incoming plasma ions had low energies over a wide range of incident power. Indicating that the plasma was operating in the pure H-mode. Removing the Faraday shield allowed the pulsing of the ICP at a repetitionfrequency range that was limited by only the frequency range the RF generator. By the time these studies were completed, research groups at several German universities had constructed similar ICP sources based on our prototype: AG Prof. Dr. H. F. Döbele, Universität Duisburg-Essen; AG Prof. Dr. V. Buck, Universität Duisburg-Essen; AG Prof. Dr. J. Winter, Ruhr Universität Bochum; AG Prof. Dr. A. von Keudell, Ruhr Universität Bochum; AG Prof. Dr. J. Engemann, Bergische Universität Wuppertal; Centre for Interdisciplinary Plasma Science, Max-Planck-Institut für Plasmaphysik. The experiments performed on the hydrogen ICP and its numerical modeling yielded many interesting results. For example, plasma bulk parameters were measured using a Langmuir probe. The I-V characteristics were analyzed using the Druyvesteyn method (see Subsections 3.. and 6..). These measurements show that the EEDF in the H-mode is approximately Maxwellian up to energies of 5 ev, sufficiently above the 5.4 ev ionization level of molecular hydrogen. This allowed the use of a Maxwellian distribution in our models. 4

Measurements as well as calculations show that the electron temperature is related strongly to the neutral gas density (see Section 4. and 6.). The electron temperature increased from ev to 5 ev as the gas pressure was reduced from Pa to Pa. The incident RF power had almost no direct effect. However, measurements showed a linear increase in the neutral gas temperature with an increase in the applied power. The heating of the gas probably occurs by electrons being accelerated by the RF field and then transferring their energy to the background gas. The gas temperature in ICP discharge was measured spectroscopically using the Doppler broadening of the ground state to n = 3 (.54 nm) transition line. At Pa and. kw incident power, the measured gas temperature was 9 K. Using the ideal gas law, the neutral gas density should be three time lower than a gas at the same pressure at room temperature. This predicts a significant increase in the electron temperature with power, and was observed in the experiment and the model. The measurements and calculations for the energy relaxation length show that the electron temperature is inhomogeneous. The analytical hydrogen plasma model (AHPM) assumes a homogeneous temperature, and as a result it systematically overestimates the electron temperature by about.5 ev. In the H-mode the plasma density scales linearly with power. To calculate the proper slope, the effect of gas heating and the power coupling efficiency between the antenna and the plasma should be taken into account. The efficiency is a function of the plasma density and is about 8 % at Pa with an incident power of. kw, but is only about % at 5 W. The mode transition threshold power is defined as the minimum power for which the matching network functioned automatically. These definition was later confirmed by optical and ion energy measurements. Between 3 Pa and Pa, the threshold was about 5 W. This agreed well with calculations where the threshold is defined as when the power deposited capacitively and inductively is equal. The spatiotemporal distributions of the electric field strength in the sheath at a power of 3 W and at Pa were determined by using FDS to measure the Stark splitting of Rydberg n = 4 and n = levels of atomic hydrogen. At the time of writing, it was the first measurement of Debye sheath dynamics in an RF discharge. The electric fields measured in the sheath agree well with the model and allowed determination of the density at the sheath edge, the electron temperature, the sheath potential, and the current density (see Sections 4.3 and 6.3). The energy distributions of H +, H +, and H 3 + fluxes at the grounded electrode were measured using a PPM4 ion energy and mass analyzer. In H-mode, the ion energy distributions show the double peak structure characteristic of residual capacitive coupling. With increasing power, the plasma density increases and the capacitive coupling decreases. Eventually distribution becomes almost mono-energetic. The calculated energy distributions had same shape and parameter scaling, but they had a higher mean energy because of the overestimated electron temperature (see Section 6.4). 5

In addition, the time evolution of Balmer-α emissions from the plasma was measured throughout an RF period using a high-speed CCD camera. In this study, the modulation of the plasma emission was novelly used for finding the mode transition in both continuous and pulsed discharges (see Section 6.5). The various diagnostics can be developed and improved by comparing their results. For example, the electron temperature measurements from the Langmuir probe and from the ion energy spectra of H + (see Subsection 6.4.) agree well. Measurements of the E-field by FDS can determine the boundary between the pre-sheath and the Debye sheath directly by applying the static sheath theory [Rie 9]. This extends the capabilities of the FDS (see Subsection 6.3.). Using the example of FDS measurements again, it was shown that both the electron temperature and the plasma density at the sheath edge could be found from the E-field distribution (see Subsection 6.3.). Further, the ICP model allows one to find the plasma potential at the sheath edge and to extrapolate the density and potential, to different positions along the discharge axis. The numerical result agrees well with the Langmuir probe measurements and with the IEDF measurements (see Subsection 6.4.). The AHPM (Section 4.), the electrical model of an ICP discharge (Section 4.) and the model of the RF discharge sheath (Section 4.3) are all consistent with measurements. The analysis has shown the applicability of these models to the hydrogen ICP. The diagnostics and analytical methods used in this study can be applied not only for hydrogen ICPs but also for general investigations on different plasma sources. The simplicity and clarity of the model assists in the control and the optimization of ICP sources. This model may be used in the development of new RF discharges and plasma diagnostics in our group. For example, the ICP model was used for designing a hyper-thermic atomic hydrogen beam source and a planar neutral loop discharge. In this study, the efficiency of the hydrogen plasma source does not attain its maximum using the. kw RF generator. The ICP model predicts that an increase in the frequency will increase the capacitive and the inductive power coupling and allow the maximum efficiency to be reached using a. kw generator. Therefore, a new 7 MHz, ICP source for a GEC reference cell was recently developed. There are two important tasks which should be considered for any future ICP discharge experiments or models. First, a Boltzman solver for the EEDF and a gas temperature module are needed to complete the ICP model. In this study, the measured EEDFs and gas temperatures were used for calculations. A Boltzman solver would expand the applicability of the model to different one-component plasmas. Second, more research into the physics of pulsed ICP discharges is needed because the transient processes that occur during the on/off switching are poorly understood. The ICP model should be also modified to include these physical mechanisms and to be able to correctly describe the transient phenomena. 6

Appendix A List of symbols and abbreviations A S a B B r B z B ϕ C ant C load C S C tune c D<number> d a d b d Cu d p d S d turn E E E r E S E T E ϕ e F<number> f AI f pulse f RF f SU f t h h cell h pl h q h ρ h ζ area of the sheath parameter in the Mathieu-equations magnetic induction radial component of the magnetic induction axial component of the magnetic induction azimuthal component of the magnetic induction capacitance between the antenna and plasma load capacitance in the matching unit sheath capacitance tune capacitance in the matching unit speed of light in vacuum diaphragm diameter of the inner cylinder in the energy filter diameter of the outer cylinder in the energy filter outer diameter of a copper tube diameter of the probe tip sheath length distance between an antenna turns E-field strength electric field strength at the sheath edge radial component of the induced electric field electric field strength at the electrode surface energy loss per electron azimuthal component of the electric field elementary charge filter repetition frequency of the analog input card pulsing frequency frequency of the RF generator frequency of experiment s synchronization threshold frequency dimensionless plasma density gap of a plasma cell vertical dimension of the heated plasma region modeled as a torus thickness of a quartz window radial profile of the dimensionless plasma density axial profile of the dimensionless plasma density 7

I emission intensity I current in the antenna I current induced in the plasma I e current of electrons to the probe tip I i current of ions to the probe tip I M modulation of the emission intensity I p Langmuir probe current j e electron current j D displacement current j i ion current j S current through the sheath K parameter of the ion energy filter k Boltzmann constant; or transformer coefficient k i rate coefficient of the ionization by an electron impact collision k M transport rate coefficient L antenna inductance L interactive inductance between an antenna and a plasma L geometrical inductance of the plasma ( e) L plasma inductance due to inertia of electrons L eff effective inductance of an antenna and plasma L pl inductance of the plasma between electrodes L tr inductance of the antenna-plasma transformer l p length of the probe tip M<number> mirror M i ion mass m e electron mass N gas density N normalization factor of the gas density N ant number turns in the antenna N p number of performed measurements n plasma density; or principal quantum number n electron density at the sheath edge n e electron density n i ion density P incident power P cap power capacitively coupled into a plasma P ind power inductively coupled into a plasma P load power deposited into the load resistance P loss power loss threshold power P t 8

P ξ power coupled into a plasma p gas pressure p PPM pressure in the ion energy and mass analyzer chamber p t threshold pressure Q ion charge q parameter in the Mathieu-equations R antenna resistance R plasma resistance R eff effective resistance of an antenna and plasma R load load resistance of a RF generator R pl plasma resistance between electrodes R pr resistance in the Langmuir probe supply R RF internal resistance of RF generator R S sheath resistance R stoch plasma resistance related to the stochastic heating R tr resistance of the antenna-plasma transformer R ε energy loss rate () R ε energy loss through inelastic collisions in the plasma bulk R Ω plasma resistance related to the Ohmic heating r radial coordinate r ant antenna radius r eff radial position of the maximum induced electric field r el radius of an electrode r met radius of a metal antenna holder r pl radial dimension of the heated plasma region modeled as torus S eff vertical cross-section of the heated plasma region modeled as torus S el electrode surface S p area of the probe tip S wall area of the cylindrical wall of the vacuum chamber < S Ω > local power dissipated in plasma by the Ohmic heating T RF period T e electron temperature T g gas temperature () T g room temperature t time U potential U electron energy divided on an elementary charge U voltage applied to an antenna U a potential on the inner cylinder in the energy filter U b potential on the outer cylinder in the energy filter 9

U p U pl () U pl Langmuir probe potential plasma potential plasma potential at the sheath edge U R U S U U U ~ u u u B u i V W R W L w x Z Z eff Z tr z α ~ ν D voltage drop on the resistance R pr sheath voltage potential averaged over the plasma density profile constant component of the sheath edge potential time-varied component of the sheath edge potential velocity ion velocity at the sheath edge Bohm velocity ion velocity volume normalization factor of an resistance normalization factor of an inductance normalised drift velocity coordinate on an axis antenna impedance effective impedance of an antenna and plasma complex impedance of the antenna-plasma transformer azimuthal coordinate constant; parameter; or fit parameter spectral width of a Doppler broadening ~ ν UV spectral width of a UV laser skin depth of a plasma δ pl ε ε ε e ε i ε H ε t ε norm γ γ e η η () η η ϑ ϕ energy; or permittivity of material permittivity of free space electron energy ion energy threshold energy of an ionization process in hydrogen threshold energy energetic normalization factor constant; parameter; or fit parameter amplification factor of the secondary electron emission by electrons dimensionless gas density dimensionless gas density at the room temperature slope of the energy loss curve maximum gradient of the power transfer efficiency curve dimensionless electric field strength azimuthal coordinate; or dimensionless potential

ϕ m dimensionless potential in the plasma cell center ϕ dimensionless potential averaged over the plasma density profile κ ionization rate Λ mean free path of ions at the sheath edge Λ H mean free path of the hydrogen molecule λ wavelength λ / full width at the half-maximum λ D Debye radius λ ps characteristic thickness of the pre-sheath λ RF RF wavelength µ permeability of free space v vibrational quantum number ~ equivalent two-photon absorption line of atomic hydrogen ν γ v eff v M v m v stoch ρ σ j σ H σ pl τ τ g ω ω ce ω ci ω e ω i ξ ξ cap ξ ind ξ load ζ ζ m effective collision frequency of electrons including stochastic heating elastic collision frequency of ions elastic collision frequency of electrons stochastic frequency dimensionless radial coordinate cross-section of an elementary process gas-kinetic cross section of hydrogen plasma conductivity dimensionless electron temperature dimensionless gas temperature radio frequency electron cyclotron frequency ion cyclotron frequency electron plasma frequency ion plasma frequency efficiency of the power coupling efficiency of the capacitive power-coupling mechanism efficiency of the inductive power-coupling mechanism efficiency of the power transfer from RF generator into the load resistance dimensionless coordinate on the axis dimensionless coordinate of the plasma cell center

Appendix B Analytical model of a hydrogen plasma In order to describe a plasma mathematically various approaches are possible. One of the most precise ways is to solve the Boltzmann equation for each individual species and the Maxwell equations for the electromagnetic field simultaneously at all spatial points in the discharge. However, this approach is usually by far too demanding and often instead of the full Boltzmann equation the first three moments are used. The set of fluid equations comprises the continuity equation, the momentum conservation equation, and the energy conservation equation. Nevertheless, the transport coefficients and the collision rates still depend on the particular form of the distribution functions. A further simplification is often made by using a Maxwellian distribution, at least for finding the transport coefficients. Still, the remaining set of equations for the electron and ion temperatures, the plasma density, and the electromagnetic field can not be decoupled and can be solved only by time-expensive iterative numerical procedures. These problems can be avoided if one assumes: a homogeneous EEDF, a constant plasma density at the sheath edge, and a constant ion temperature (cold ions) over the entire plasma volume. Especially assumption of a homogeneous EEDF becomes critical when the energy relaxation length is much shorter then the typical dimension of the discharge. This is clearly a function of pressure. In molecular gases, one can expect generally that the assumption fails at lower pressures than in noble gases. Models based on these assumptions are called global models because they allow a global (volume average) balance between the power deposited in a plasma and the energy loss processes [List 9, Lie 94, Wai 95, Sob 97, Zor, Gud ]. Finally, it is assumed that the electron motion is governed only by the electric field and not by the magnetic field. The advantage of the global model is that it can be solved analytically and therefore it provides some inside into the discharge mechanisms. An exact analytical solution for the plasma density is only possible in a one-dimensional geometry, i.e. neglecting radial transport. Radial transport can be taken into account in a diffusion approximation for cylindrical symmetry of the plasma cell. In the diffusion approximation, the inertia term is neglected in the momentum conservation equation. In the model presented below, both solutions are combined in order to include twodimensional density distributions and to calculate the plasma density at the probe position and at the sheath edge for comparison with the experiment.

B.. Particle transport equations The electron density profiles and the electron temperature are determined by the continuity and the momentum conservation equations. In this section, the relevant equations are introduced. The solutions are derived in Sections B.3 and B.4. Electron and ion densities in the plasma are linked by the quasineutrality condition and described below by a single plasma density, n = n e = n i. The continuity equation for the ions is: r ( n u i ) = nvi, (B.) r where u is the ion velocity and vi is the ionization frequency. i The stationary momentum conservation equation is r r r ee kti n r ( ui ) ui = vm ui, (B.) M M n i i where E r is the electric field, M i is the ion mass, and v M is the ion elastic collision frequency. The electric field is a combination of the induced electric field and the ambipolar field. The induced electric field in a planar ICP has predominantly only an azimuthal component and does not contribute to the radial and the axial transport in a cylindrical symmetry so that it can be neglected in this context. The ambipolar electric field is given by r µ ekte + µ ikti n kte n E =, e( µ µ ) n e n (B.3) i e where µ e is the mobility for electrons, µ i is the mobility for ions. This field is a consequence of the condition of equal electron and ion fluxes. Since in low-temperature plasmas µ e >> µ i and kt i << kt e, the more simple form of equation (B.3) is used here and the diffusion term in the momentum equation (B.) can be neglected comparing to the drift term: r r kt n r. (B.4) M n e ( ui ) ui = vm ui i The simple form of equation (B.3) is equivalent to assuming a Boltzmann distribution for electrons: ( eu ) n = n exp kt e, (B.5) where U is the difference between the plasma potential and the potential at the sheath edge and n is the plasma density at the sheath edge. The global model assumes that the voltage drop across the sheath and the plasma density at the sheath edge are constants. The ion velocity at the sheath edge, u, is determined by the Bohm criterion: u = u = kt M B e i. (B.6) u is a constant because of the global model s assumption of a constant electron temperature. Therefore, the set of particle transport equations have the following boundary conditions: n = const and u = const. (B.7) 3

B... Diffusion approximation In the diffusion approximation, the left hand side of equation (B.4), which is the so-called ion inertia term, is neglected. This approach is valid as long the velocity gradients are not too strong: u i << v M. (B.8) Since ions are accelerated mostly in regions close to the sheath edge, the diffusion approximation generally fails here. The ion velocity is in the diffusion approximation given by: r kte ui = M v Inserting this relation into equation (B.) yields: i M kte n = M v i M n. (B.9) n v n. i (B.) The analytical integration of equation (B.) assuming a simple, symmetric discharge geometry allows us to determine the electron density profile. The diffusion approximation describes adequately most of the plasma bulk. B.. Energy transport equation The average electron density and the density at the sheath edge are determined by the energy balance equation. This equation expresses the balance between the power coupled to the electrons by the electromagnetic field and the power dissipated by collisions and particle transport. In the stationary case, the energy balance equation has the form: r r r j E ( n < εu > ) n < εq >. (B.) e = e c The left hand is the Ohmic heating, the first term on the right side of equation (B.) is the particle transport and the second term is the energy loss due to electron impact collisions. The brackets indicate average over the EEDF. B... Elementary processes in a hydrogen plasma The energy loss terms due to particular inelastic collisions are calculated by: k j < εqc > j = Nk jε j, (B.) = ε j ε f ( ε) σ j ( ε) dε. (B.3) m Here σ j (ε) is the cross-section for the collision process j, f(ε) is the EEDF, ε j is the threshold energy, and N is the neutral gas density. It should be mentioned that only collisions with H in the ground state are considered. The corresponding cross-sections and energies of these processes can be found in [Jan 87, Phe 85]. e 4

For a constant (velocity independent) collision frequency, v m, the energy loss in elastic collisions is generally described by k = v N and (B.4) m m m ( m M ) kte Contribution of all processes gives for the following loss term: ε = 3. (B.5) n < εqc > = nnr, (B.6) R ) () ε ( ε = ε jk j (B.7) where R ε () is the energy loss rate due to electron collisions in the plasma bulk. j The collision process probability is proportional to its cross-section, σ j (u). It is defined as a ratio of the collision frequency to the flux of colliding particles: v j ( u) σ j ( u) =, (B.8) N u where N j is density of colliding particles and u is their relative velocity. j B... Energy loss due to a particle transport The energy loss due to particle transport depends on the particular form of the EEDF. For a Maxwellian electron velocity distribution, one calculates in the form of the two-term approximation: r < εue > 5 r = kte. (B.9) < u > e However, as will be shown below, in the global model, the particle flux to the wall is the relevant quantity. In the sheath, the isotropic Maxwellian distribution no longer holds. Often a truncated Maxwell distribution, i.e. no reflection of electrons from the wall, is assumed. In this case, the energy loss per one electron is: r < εue > r = kte. (B.) < u > e The difference between the two very different cases is quite small, only %, and is not significant in the general calculations. Since, the global model deals with particle fluxes at the wall, the ratio from equation (B.) is used. B..3. Ohmic heating The Ohmic heating term is given by: r r SΩ = je E. (B.) In an ICP, the total electric field is a sum of the ambipolar field due to the electron density gradient and the field produced by the antenna: 5

r r r E = E a + E ind. (B.) The electron current is therefore the sum of the electron diffusion current and the induced current: r r r j = j + j. (B.3) e De ind Since the diffusion current is directed radially and axially toward the chamber walls and the induced current is directed azimuthally and forms a closed loop inside the plasma bulk, these currents are perpendicular to each other, and the mixed products in equation (B.) are zero: r r r r SΩ = jde Ea + jind Eind. (B.4) The term related to the ambipolar field is r r r r r j E = en < u > U = en < u > U eu n < u >. (B.5) De a e ( ) ( ) With the help of the continuity equation (B.) this leads r r r j E = en < u > U ennk De a e ( ) U e i e, (B.6) where k i is the ionization rate coefficient. The energy dissipation caused by the induced electric field and current equals to the power coupled to the electrons: r dpξ jind E r ind =. (B.7) dv Here P ξ is the mean power coupled to the plasma during one RF period; P = ξp, where ξ is the power coupling efficiency and P is the time-averaging incident power. This temporal average is valid because the energy relaxation time is of order µs, and is by far longer than the RF period, 74 ns. ξ B..4. The average density and the density at the sheath edge The power deposited in the plasma is found by substituting (B.6), (B.), (B.6) and (B.7) into the power balance equation (B.) and then by integrating: P dpξ = nn ε jk dv j ξ = j + euk i + r ], (B.8) [( kt eu ) n < u > nn ε jk j + euki dv + n < ue > V j e ( kt ) e eu A e r r da, (B.9) where V is the plasma cell volume and A is the cell s surface area. The integration using boundary conditions (B.7) yields: P nn k euk ξ = ε j j + i V + n u( kte eu A )A, (B.3) j where n is the density average over V and n is the density averaged over A: 6

n = V V n dv and n = n da. A (B.3) The potential U is averaged using the normalized profile of the electron density: A n U = U dv V n (B.3) V This implies that U is the mean energy lost by an electron while moving from the plasma to the sheath edge. U A is the potential at the wall. A relation between n and n can be derived applying Gauss s theorem to equation (B.): n u A = nkinv. (B.33) By substituting (B.33) into (B.3), we determine the electron density averaged over the entire plasma volume: Here R ε is the energetic loss rate: Pξ n =. (B.34) VNR ε [ e( U U A ) + kt e ] k i R = ε k + ε j j. (B.35) j The average plasma density at the sheath edge is then: where E T is the energy loss per electron: n E Pξ =, (B.36) Au E T = R. (B.37) T ε k i It should be noted that the assumptions of a homogeneous electron temperature over the entire plasma volume and constant boundary conditions was essential for the integration. Further, not the sheath potential but the potential difference, U - U A, is the relevant quantity. The steady state ion current is equal to the mean electron current onto the chamber wall and the sheath potential is given by the floating potential condition: U S kte M i = ln e πm e. (B.38) If the sheath edge is selected as the reference point where U =, then U A = U S. U is calculated in following Sections B.3 and B.4. B..5. Local energy balance and induced electric field If the electron diffusion current is much smaller than the induced current, then the energy loss due to the particle flux can be neglected. In this case, the local energy balance will depend entirely upon inelastic collisions of electrons with neutral species in the background gas: 7

dpξ ( ) 5 ( ) = nn Rε + kte ki nnrε. (B.39) dv In hydrogen, equation (B.39) is applicable if T e 4 ev, when the energetic loss rate R ε is primarily determined by R ε (). The deposited energy in the plasma is determined by Ohmic heating: dp r ξ = j dv ind r E ind = Im( σ pl ) E ind. (B.4) Langmuir probe measurements of T e allow one to deduce the induced electric field in the plasma by using the following equation: where σˆ is () E ind = NRε σˆ, (B.4) ( σ ) Im σ ˆ =, (B.4) v pl e vm = n mevm ω + σ pl is the complex plasma conductivity, m e is the electron mass, ω is the electric field frequency, and v m is the electron elastic collision frequency. m B.3. One-dimensional model The set of the one-dimensional particle transport equations can be solved analytically. This solution allows the determination of the electron density profile on the discharge axis if the radial transport is negligible. B.3.. Set of dimensionless variables It is convenient to solve the set of the particle transport equations using dimensionless variables: () () (3) (4) (5) (6) n h =, n is a plasma density at the sheath edge - electron or ion density n u i w =, u is an ion velocity at the sheath edge (B.6) - ion velocity ub z ζ =, Λ v u B Λ = is a mean free path at the sheath edge - coordinate on the axis vm i κ =, = κ( T e ) vm E ε =, E E U κ - ionization rate u M ivm u = is a drift field leading to u - electric field µ e = i U kt ϕ =, U = e is the energy of electrons - potential e 8

(7) (8) τ = kt e, ε H N η =, N N εh =5.4 ev is the ionization energy of hydrogen εh = is the normalization factor, M i k M l pl k M is the rate coefficient of ion elastic collisions and l characteristic plasma dimension pl is the - plasma temperature - neutral gas density B.3.. Particle transport equations The particle transport equations written in dimensionless variables take the form: ζ () ( hw) = κ h w () w = ε w ζ (3) ε = h h ζ h - continuity equation (B.) (B.43) - momentum conservation (B.) for kt e >> kt i (B.44) - Boltzmann distribution (B.5) (B.45) (4) = κ h - diffusion approximation (B.) (B.46) ζ The boundary conditions at ζ = are: h = and w =. (B.47) B.3.3. Profiles of plasma potential and plasma density (relative values) Solving the set of equations (B.43)-(B.45) and (B.47), one obtains the plasma potential, ϕ, and the plasma density, h: (B.43) (B.44), (B.45) h w = κ h ζ w ζ h w = w w h ζ ζ w κ w w = w w ζ ζ ζ = w (B.44), (B.48) w κ + w + κ dw = κ w κ + w = ζ w w arctan + arctan ( + w) κ κ w κ + w = w + w = w + w ζ w ε w ε = ( + κ ) w (B.48) (B.49) (B.5) 9

ϕ = ζ εdζ = w~ w~ ( ζ) ( ) = w = ζ + κ + κ ε = ~ dw~ ln w w + κ (B.5) +κ + κ h = exp( ϕ) =. (B.5) w + κ The plasma density is found from equations (B.49) and (B.5) linked through the variable w. An explicit form of h(ζ,κ) exists only for κ =, or for the diffusion approximation (B.8). In general, an explicit form of w(ζ,κ) does not exist (see (B.49)). When κ =, w is determined by the following expression: w = ζ 4 + ζ ζ. (B.53) The set of the particle transport equations for the diffusion approximation (B.46) is solved similarly to the above algorithm: (B.43) (B.43), (B.46) ζ = w (B.43), (B.45), (B.46) The drift velocity is: ϕ = ζ h = κ h ζ w h = w h ζ w ζ w = κ ζ + w (B.54) dw w = arctan + arctan κ + κ (B.55) w κ κ ε = εdζ = w~ w~ h h ζ ( ζ) ( ) = w = ζ + κ ε = ~ dw~ ln w w + κ / ε = w (B.56) (B.57) + κ exp( ) h = ϕ = (B.58) w + κ αβ α + β arctan ( α) + arctan( β) = arctan ( ) ( κζ) ( κζ) ( + w) κ tan κζ =. κ w κ tan w = κ. κ + tan (B.59) Finally, by substituting (B.59) into equation (B.58), the diffusion profile of the plasma density assumes its familiar form: B.3.4. Electron temperature h = cos( κζ) + ( κ) sin( κζ). (B.6) ζ The aspect ratio, the height to the radius, is 5:3 for the GEC reference cell. Because of this, the charged particle flux to the chamber wall is mainly axial, and the characteristic plasma dimension is approximately the plasma cell gap: 3

l pl h cell. (B.6) The position where the plasma density is maximum at the discharge axis, ζ m, is determined by the profile h(ζ,κ) in equations (B.49) and (B.5). The global model assumes that the electron temperature is constant throughout the entire plasma volume. Since the ionization frequency is determined by the electron temperature and is also a constant, ζ m will be at the center of the discharge: l pl ζm =. (B.6) Λ At the discharge center, the ambipolar field reverses its direction and at this position the velocity of charged particles will be zero: ( ) = w. (B.63) ζ m A relation between κ and ζ m is derived by substituting (B.63) into equation (B.49): + κ ζ m = arctan. (B.64) κ κ Therefore, κ can be calculated using the characteristic plasma dimension. On other hand, the dependence of κ on the electron temperature can be found by using the ionization cross-sections for hydrogen: κ () τ v i ki < σi ( u) u = = = >, v k k (B.65) M M where k i = v i / N is the ionization rate coefficient, v M is the elastic collision frequency of ions at Bohm velocity, and k M = v M / N. Calculation of these rate coefficients is discussed in more detail in Subsection B... The electron temperature is determined by comparing the values of κ found from equations (B.64) and (B.65). In terms of dimensionless variables, the relation between the neutral density and the electron temperature takes the form: In accordance with the ideal gas equation: M + κ η = τ arctan. (B.66) κ κ the temperature dependence of η at a fixed gas pressure is: p = NkT g, (B.67) () Tg () g η =, (B.68) ( ) η T + T where η () = η(p = ) is the pressure and T g () = T g (P = ) is the gas temperature in the absence of a discharge. T g = T g - T g () is the increase in the gas temperature in a discharge. g 3

B.4. Spatially resolved model in the diffusion approximation Generally, the plasma density through the entire cell volume is obtained by solving the three-dimensional particle transport equations numerically. In diffusion approximation, one can solve these equations analytically assuming a simple symmetry of the discharge. A planar ICP in a GEC reference cell is approximately cylindrical. B.4.. Plasma density in the entire discharge volume (9) Analogous to ζ, a radial, dimensionless coordinate is introduced: r u ρ =, where B Λ = is a mean free path at the sheath edge - radial coordinate Λ v M If the plasma density is represented as a product of the longitudinal and radial profiles: ( ζ ρ) = h ζ ( ζ h ( ρ) h, ), (B.69) then equation (B.) can be written in terms of dimensionless variables: hζ hρ + + κ ρ hζ ζ ρhρ ρ ρ ρ =. (B.7) Since equation (B.7) has independent variables, it can be separated into an axial equation: and a radial equation: ζ ρ ρ where γ is the constant of separation. h = ( κ γ ) h ζ ρ The solution to equation (B.7) is: ρ ζ hρ = γ hρ, ( ρ) = A J ( γρ) + A ( γρ) ρ Y (B.7) (B.7) h, (B.73) where А and А are constants. If the boundary conditions are: h ( ρ = ) =, ( ρ = ρ ) = ρ w, (B.74) an acceptable physical solution for h ρ (ρ) is a zeroth-order Bessel function: ρ J ( ρ = ) = Y A ρ J ( ρ = ) ( ρ = ) = = ρ ( ρ) = J ( γρ) ρ m = A = γ is obtained by substituting equation (B.8) into (B.79): h. (B.75) 3

hρ h ρ ρ ( ρ = ρ ) m J = γ J ( γρm ) ( γρ ) m =. (B.76) As a result of separating the coordinates, the longitudinal component of the plasma density, equation (B.7), has the same form as the one-dimensional model, see equation (B.46). This equation can be solved for the ionization frequency, κ * : κ κ * ζ m = arctan, κ = κ γ * *. (B.77) For κ <<, two eigenwert equations have the same form. Still, as one can see, the ionization frequency, κ, is higher than in the one-dimensional model. The energy loss due to the radial particle flux in the ambipolar field included in the two-dimensional model is responsible for the higher electron temperature. B.5. Absolute value of the plasma density If the electron temperature is known, the absolute value of the electron density can be calculated using equations (B.34) and (B.35). These equations are valid not only for the onedimensional case, but also for two- and three-dimensional models. In equations (B.34) and (B.35), the potential U has to be known for finding the absolute value of the plasma density. This quantity can be now calculated for the twodimensional model in the diffusion approximation and for the exact one-dimensional model. Since h(ζ,κ) was not derived explicitly for the exact one-dimensional model, the potential is averaged over the plasma density profile as follows: U U = ϕ w= ( κ) = ϕ( w, κ) h( w, κ) dw h( w, κ) ζ w= w w= w= ζ dw. (B.78) w 33

Appendix C Transformer model An electric discharge model treats a plasma source as an electrical circuit. The equivalent circuit of an ICP source is a transformer. In this transformer model, the primary coil is the antenna, and the secondary is the plasma. Using the antenna impedance as an input parameter, the transformer model allows one to estimate the average plasma conductivity, the average electron density, and the power coupling efficiency. The results of these calculations strongly depend on the plasma source geometry. In this section, a transformer model of an ICP source for a GEC reference cell is discussed [GEC, Mil 95]. C.. Inductive power coupling into the plasma Power is deposited into a plasma by inductive and capacitive mechanisms. In the inductive mechanism, currents flowing in the antenna generate an electromagnetic field. This field penetrates into a plasma, and induces electron currents in the same the way an transformer induces a current in the secondary. Power is deposited into the plasma by Ohmic and stochastic heating, which are a result of the interaction of the electron current and the electric field. C... Structure of an electromagnetic field induced in a vacuum The penetration of the electromagnetic field into the plasma determines the geometry of the heating region. Therefore, the field induced in vacuum can be used as a first estimate of the second coil geometry in the transformer model. The magnetic field in vacuum can be calculated by using the Biot-Savart law: r r r db µ dl A = 3 I π r, (C.) where I is the antenna current and 4 r A A is the vector from the antenna current segment, dl r, to where the magnetic field is being calculated. Maxwell s equation, written in the differential form, gives the relation between the magnetic and electric fields: r r r B E =. (C.) t The harmonic behavior of the electromagnetic field implies that the amplitude of the E-field is proportional to the amplitude of the rotor-vector of B-field. The electric field in vacuum produced by the current through dl r can then be calculated by using: r de dl r µ = ω r. (C.3) I 4 π r A 34

According to the superposition principle, the amplitudes of the magnetic and electric fields can be calculated by integrating equations (C.) and (C.3) along the entire length of the antenna. It is convenient to use cylindrical coordinates for this integration. z r r r r y r B r ϕ r A O A h x r A = r e + h e r (C.4) r B radius-vector of d l r x ϕ angle between rr r B and e x r, h position of point A in the cell region Fig.C.. Cylindrical coordinates used to calculate the electromagnetic field in vacuum induced in a planar plasma cell. r r r Using coordinates { er, eϕ, ez} as shown in Fig.C., the components of the electromagnetic field are calculated as following: B I r µ = 4π πn h rb cosϕ ϕ µ dϕ, = 3 r I 4π E I r A µ = ω 4π πn πn B A h rb sinϕ Bz µ dϕ, = 3 r I 4π A πn E rb sinϕ ϕ µ r dϕ, = ω r I 4π B πn r r B z B cosϕ r 3 A r B dϕ, (C.5) cosϕ dϕ, E z =, (C.6) r where r B = r B (ϕ) is the equation describing the shape of a planar antenna and r A is the distance from the antenna to where the field is being calculated: r ( r r cosϕ) + ( r sin ϕ) h A B B + A =. (C.7) Due to the cylindrical symmetry of the cell, the components of the electromagnetic field satisfy: B ϕ << B r, B ϕ << B z and E r << E ϕ. Hence, the quantities B ϕ and E r can be excluded from consideration. In this case, the relationship between the amplitudes of the electric and magnetic fields can be determined by using Faraday s law in integral form. The E-field can be calculated from a known vertical component of the H-field: E I ϕ r ω Bz = r dr, (C.8) r I If the azimuthal E-field is known, one can reverse this and calculate the H-field. The vertical and radial components of the magnetic field are determined by using the differential form of Faraday s law: ωb z = r r ( re ) ϕ, ω B r E = z ϕ. (C.9) Ohmic heating is an important mechanism for maintaining plasmas. The power absorbed locally in a plasma is proportional to the square of the induced field: dp r r r ξ = jee ~ E. (C.) dv The E-field s maximum in the axial direction is at the surface of the dielectric separator. It rapidly decreases with the distance from the antenna. With a plasma present, the 35

depth to which the electric field can penetrates into the discharge decreases. This causes energy to be absorbed inside a thin layer (skin depth) near the dielectric surface. If a spiral antenna or a twin antenna with equidistant coils is used, the E-field is maximum at r = /3 r ant, where r ant is the antenna radius. At the center of the discharge and at a large distance from the axis, the induced electric field vanishes. Thus, the plasma heating region is an annular region of limited size. C... Antenna-plasma transformer The equivalent electric circuit of the ICP is shown in Fig.C.. The antenna in the model has active resistance, R, and inductance, L. The plasma forms the transformer s secondary coil, and is modeled by specifying a resistance, R ; an inductance, L ; and an additional inductance due to the inertia of electrons, L. L is calculated from the following equation: ( e) ( e) L = R veff, (C.) where v eff is the effective electron collision frequency. In addition to the Ohmic heating due to electron collisions with neutrals, there is also a collisionless energy transfer mechanism from the electromagnetic field to the plasma in the skin layer. This stochastic heating mechanism is significant if the interaction time of electrons with the skin layer is much shorter than the time interval between collisions. Stochastic heating is taken into account by replacing the frequency of elastic collisions of electrons with an effective collision frequency: v = v + v, (C.) eff m stoch where v stoch is the stochastic frequency. There are two self-consistent models for stochastic heating in the literature. The first is based on the Fermi acceleration mechanism [Lie 98]. According to this model, v stoch can be calculated from the equation: 3 3 4 ωωekte ωωekte v =.84 (C.3) stoch π mec mec In the other model the electron cloud is repetitively compressed and expanded as a result of the oscillation of the direction of the E-field [Tur 93]. This model predicts the following frequency: 3 6 3 3 ωωekte ωωekte.78 mec mec π vstoch =. (C.4) As can be seen from equations (C.3) and (C.4), both models give the same result. It should be mentioned that v stoch depends only slightly on the pressure, whereas v m is proportional to the gas pressure. This implies that Ohmic heating is the dominant mechanism at higher pressures and that stochastic heating is dominant at lower pressures. The transformer impedance, Z tr, can be determined by manipulating equations (C.5). ( e) 36

RF power L R L L L (e) L R ( R + iωl ) L Z Fig.C.. Equivalent electrical circuit of the ICP. ( e) ( R + iωl + iωl ) tr = L = U The real, R tr, and imaginary, ωl tr, parts of Z tr are: R I I iωl = R e tr I I + iωl = U iωl tr I = (C.5) R, L resistance and inductance of the antenna R, L, L e resistance and inductance of the plasma L, L inductance between the antenna and the plasma ( ωl ) tr = R + R, (C.6) R ( ) + ωl + ωr νeff ( ωl ) ω Ltr = ωl ( ωl + ωr νeff ). R ( ) (C.7) + ωl + ωr νeff The complex impedance of the antenna and the transformer can be determined experimentally. The set of equations (C.6) and (C.7) still contain three unknown variables: L, L and R. Because of this, it is convenient to use dimensionless variables: α = R ωl and = R ωl α, (C.8) where the subscript refers to the coil number. The transformer coefficient, k, is defined by the inductance ratio: k = L. (C.9) LL Hence, the transformer model calculates α and k using the following equations: R R tr α =, (C.) ωl ωltr ω νeff ( Rtr R ) ( Rtr R ) + ( ωl ωltr ) ωl ωl ω ν ( R R ) k =. (C.) ωl [ ] tr eff Thus, equations (C.) and (C.) are closed with respect to the dimensionless variables α and k. tr 37

C..3. Antenna inductance and transformer impedance k: Equations (C.6) and (C.7) can be written in terms of dimensionless variables α and Rtr ωl ωl ωl = α + + tr = k α k α ( α + ω veff ) ( α + ω veff ) ( α + ω v ) = α + k = k f L + eff f R ( α, ω v ) eff ( α, ω v ) eff, (C.), (C.3) where f R (α,ω/v eff ) and f L (α,ω/v eff ) are functions of the positive variables α and ω/v eff. The real R tr and imaginary L tr parts of the transformer impedance in equations (C.) and (C.3) are normalized to the inductance of the antenna. max(f R ),5,4,3,,,,, ω /υ m Fig.C.3. Maximum of the function f R (α,ω/v eff ) versus the parameter ω/v eff. The function f R (α,ω/v eff ) has a single maximum at α = ( + ω/v eff ) -/. The dependence of this maximum on ω/v eff is shown in Fig.C.3. As can be seen from the figure, the maximum of f R (α,ω/v eff ) is limited to /. As the pressure decreases, the maximum of f R (α,ω/v eff ) also decreases. f L (α,ω/v eff ) has no extremes for positive values of α. The maximum of this function does not depend on the value of the parameter ω/v eff and is equal to at α =. From the aforesaid, one draws the conclusion that, the antenna-plasma transformer impedance for all ICP discharges is limited by the following inequalities: R tr ω L < α + k, (C.4) ω ωl > k. (C.5) L tr Antennas for ICP sources are usually made of low-resistance materials, where R << ωl. Therefore, the value of α is negligible. For example, for a standard GEC antenna α.7. The transformer coefficient, k, is always below by definition. Moreover, when an external antenna is used, k usually does not exceed /. Consequently, the normalized impedance can be estimated as: R tr ωl <. and (C.6) ωl tr ωl >.7. (C.7) This means that R tr does not exceed % of ωl. At the same time, the transformer and antenna inductances are nearly equal: L tr L. Note that the limitations imposed by inequalities (C.6) and (C.7) apply to all ICP discharges with external antennas, regardless of the gas pressure, gas composition, and incident power! In general, the phase shift between the antenna voltage and current is also limited: 38

o ( ω R ) ϕ = arctan L > 75. (C.8) tr tr The phase shift is slightly lower than π/. Hence, it is reasonable to suppose that the amplitude of the current and voltage in an ICP antenna is quite high. C..4. Influence of the antenna inductance on current and voltage The influence of the antenna inductance on the power coupling into a plasma can be simply demonstrated. First, assume that a spiral antenna is replaced with a new one such that the number of turns is reduced, but the overall size and number of spirals remains the same. Neglecting the fine structure of the induced field near the dielectric separator, an electromagnetic field produced by this new antenna that is identical to the previous antenna. Although the antenna was changed, the value of the transformer coefficient between the antenna and plasma remained the same: k = const. (C.9) If resistive energy losses in the antenna are neglected (R = Ω), then all of the incident power will be absorbed by the plasma. Consequently, at a fixed incident power ( ωl ) const P = I Rtr = U Rtr + tr =, (C.3) and the generated plasma density must remain the same. In this case, the plasma conductivity, and consequently α, will remain the same: α = const. (C.3) From equations (C.), (C.3), and conditions (C.9)-(C.3), it follows that the antennaplasma transformer impedance is proportional to the antenna inductance: R tr ~ L, L tr ~ L. (C.3) The influence of the antenna inductance on the amplitudes of antenna voltage and current can be written in the following form: I ~ L, (C.33) U ~ L. (C.34) According to relations (C.33) and (C.34), the voltage decreases and the current increases as the antenna inductance decreases. C..5. Induced currents in conducting materials of the vacuum chamber The poor efficiency of many ICP sources can be attributed to the design of the vacuum chamber. For example, in a standard ICP GEC reference cell, the antenna is placed in a metal cylinder with a quartz window, and there are large power losses due to coupling to the metal cylinder. If the Ohmic heating of the chamber wall is neglected by assuming that the chamber material has zero resistance (R 3 = Ω), then the ICP can be modeled as a transformer with an additional secondary that is shorted. This equivalent electric circuit is 39

shown in Fig.C.4. The transformer impedance can be determined by solving the set of equations (C.35). RF power R L L L 3 L L 3 L 3 (e) L R R 3 = Ω ( R + iωl ) Z ( e) ( R + iωl + iωl ) ( R + iωl ) tr 3 = U 3 I I I 3 iωl + iωl = R e tr I 3 I I + iωl iωl + iωl iωl tr 3 3 I 3 I 3 I 3 = U iωl = I = (C.35) R, L resistance and inductance of the antenna R, L, L e resistance and inductance of the plasma R 3, L 3 resistance and inductance of the antenna housing L, L 3, L 3 mutual inductances Fig.C.4. Modified electrical circuit of antenna-plasma transformer including the influence of highconductive chamber materials. If the heating of the chamber walls can be neglected, the real R tr and imaginary L tr parts of the impedance take the form: tr R tr * ( ωl ) * ( ) ωl + ωr νeff * * ( ) ( ωl ) ωl + ωr νeff * ( R + ωl + ωr ν ) = R + R and R + * ω L = ωl, where the inductances are calculated as follows: L * = L ( ), L L ( ) * k3 L k =, LL = 3, * = 3 3 L k k L 3 k 3 =, and LL 3 The antenna-plasma transformer coefficient can be calculated from: k eff (C.36) (C.37) k k L, (C.38) L 3 k 3 =. (C.39) LL3 * * L k k3k3 = =. (C.4) * * LL k3 k3 Note that the form of the solutions with and without conductive walls is the same. C..6. Geometrical assumptions about the planar ICP The geometry of a plasma determines the unknown inductances in its transformer model. Typically a geometry is specified, and then the plasma inductance, L, and the mutual inductance of the antenna and plasma, L, are calculated numerically. Once these inductances are determined, the set of equations (C.5) can be used to quantitatively estimate plasma parameters such the volume-averaged conductivity, σ pl, and the electron density, n e. A homogeneous, annular, conducting medium of fixed size is the simplest discharge geometry that can be assumed [Pie 9]. This geometry is appropriate for an ICP fluorescent 4

lamp if its antenna is encased in a small bulb, and is surrounded by the discharge. In this case, the plasma dimensions are independent of the incident power and the gas pressure. This discharge geometry does not suitably describe larger plasmas. Because the transformer ratio in equation (C.) can be determined experimentally, it can be used to check the validity of this geometry. If the transformer ratio is independent of the power and pressure, then it may be assumed that this geometry is applicable. Otherwise, a more sophisticated model of the plasma geometry is needed. In this thesis, it is proposed that the sizes of the transformer secondary be restricted to the plasma region where an electron current is induced. Using this assumption, the plasma geometry is taken as the region where the electromagnetic field penetrates. The plasma will be modeled as an annular, homogeneous conductor [God 97]. For a planar ICP, the structure will be torodial. A toridal plasma geometry can be specified by three parameters: radius, r eff ; position at the discharge axis, h eff ; and cross section, S eff. The cross section is approximately the product of the radial and vertical dimensions: S eff ~ r pl h pl. Since the plasma is next to the dielectric separator, the vertical height and vertical position of the plasma are interdependent: h pl = h eff. If the density does not have a minimum on the discharge axis, then a similar relation can be written for the plasma radius: r pl = r eff. The model proposed in [Suz 98] should be used if the last assumption is not satisfied for a particular plasma geometry. First, consider cells where there are no conductive materials near the antenna which could disturb the electric field structure. The induced electron current and induced electric field have their maximum at the same position. The calculations of Subsection C.. show that the maximum E-field in a vacuum, at the surface of the dielectric window is almost /3 of the antenna radius. As the distance from the dielectric window increases, the radial position of the maximum gradually increases. For the homogenous conductor approximation, a torus with radius, r eff = /3r ant, can be used. Direct measurements of the magnetic field by B-dot probes performed in planar ICP sources [Haa 97, Lie 94, Mey 95, Mey 96] support the use of this approximation. If an antenna is placed inside a metal cylinder, i.e. in the GEC reference cell, it is possible that the plasma geometry can be reduced to the inner radius of the cylinder, r met. Fig.C.5 shows a diagram of the electric-field penetration when an antenna is placed in dielectric and conducting cylinders. The radial dimension of the plasma, r pl, is fixed by the cell geometry, whereas h pl depends upon the plasma parameters. The electron density affects how far the electromagnetic field can penetrate into the plasma, and is usually shorter than the height of the plasma cell: h pl < h cell. If the electron density is low enough, the electromagnetic field will be able to propagate through the whole cell to the electrode surface: h pl = h cell. 4

quartz antenna coil r ant quartz antenna coil r ant S eff h eff h pl h pl heated plasma region r eff /3r ant r pl = r eff metal heated plasma region r pl = r met S eff electrode electrode (a) plasma cell with dielectric antenna housing (b) standard GEC cell Fig.C.5. Schematic diagrams of the electric-field penetration region: the radius r pl is determined by the plasma cell geometry, the value of h pl is varied depending on the operating conditions. Choosing the plasma geometry allows the determination of all inductances for the transformer model. As an example, consider the annular plasma. The plasma cross sectioned into N successive elements parallel to each other. The inductance between i-th and j-th elements can be calculated by the following formula: r r µ dri dr j Lij = π r r. (C.4) 4 li l j i j The total inductance of the system, L, is approximated by summing the inductances of each individual element: L = N N i N j L ij (C.4) The mutual induction between the antenna and the plasma, L, can be calculated from the following: r r N µ dri dr Li = π r r, L = L i. (C.43) 4 N lant li The inductances in the transformer model are calculated similarly. i i C..7. Electron density The conductivity of a homogeneous conductor is a function of its resistance and dimensions: πreff π σ pl = =. R S R h eff pl (C.44) On the other hand, if the electron density is known, the plasma conductivity can be determined [Rai 97]: e νeff + νeff e σ pl = ne, (C.45) m ( ω ) where the electron elastic collision frequency has been replaced with the effective collision frequency, v eff, to account for stochastic heating. 4

C..8. Skin depth The plasma heated by induced currents can be modeled a homogeneous conductor with a variable height, h pl. h pl depends on how deep the electromagnetic field penetrates into the plasma, which is known as the skin layer depth, δ pl. For a homogeneous conducting medium, δ pl can be calculated using: δ pl c = Im ω e ω ω( ω iv eff ), (C.46) where ω e is the electron plasma frequency (3.). The skin depth depends on the electron density and the effective electron collision frequency. Since there is a correspondence between h pl and δ pl, a comparison of h pl and δ pl allows the validation of the model for plasma geometry in Subsection C..6. If it is assumed that h pl δ pl, the model can determine the efficiency of the ICP source. This will be developed in detail in the following sections. C.. Capacitive power coupling into the plasma In ICPs, energy is deposited in two ways. In the previous section, inductive coupling was considered. In capacitive coupling, the oscillating potential on the antenna couples energy into the plasma. The antenna acts as an electrode. The transformer model can be modified to include this capacitive coupling. C... Capacitive power coupling and transformer impedance The total energy input into a plasma is a superposition of two parallel processes. They can be represented as a capacitive and an inductive impedance in parallel. An equivalent electric circuit is shown in Fig.C.6. The circuit corresponding to the capacitive coupling includes the capacitance of the dielectric separator, the resistance and capacitance of the plasma sheath, and the capacitance and inductance of the plasma bulk. Considering all of these quantities, the real, R pl, and imaginary, C ant, impedance can be determined. Since the capacitance is uniform along the antenna, the length averaged impedance can be determined from equation (C.47). RF power C ant R pl L tr R tr Z eff = R ( R i ωc )( R + iωl ) tr pl + R pl + i ant ( ωl βωc ) tr tr C ant is used. The total tr ant (C.47) R pl resistance of plasma C ant effective capacitance between antenna and plasma R tr, L tr impedance of antenna-plasma transformer Fig.C.6. Equivalent electric circuit of capacitive and inductive energy couplings in the ICP. The real R eff and imaginary L eff parts of the effective impedance Z eff from equation (C.47) have a complicated form: 43

R eff R R ( R pl + Rtr ) + ( ωltr ) R pl + Rtr ( ωcant ) ( R ) ( ) pl + Rtr + ωcant ωltr ant + R plωltr + ωltr ( ωcant ωltr ) ω ( R + R ) + ( ωc ωl ) pl tr eff =, (C.48) Rtr ωc Cant ω L =. (C.49) pl tr Inductive energy coupling dominates in the H-mode, and most of the current flows through transformer elements L tr and R tr. Hence, for the circuit shown in Fig.C.6, the following relations apply: R pl R tr << ωl tr ant << ωc They allow the simplification of expressions (C.48) and (C.49): R eff eff tr ant ( + ωltrωcant ) ( + ωl ωc ) tr. (C.5) R, (C.5) ωl ωl. (C.5) tr It should be mentioned that in relations (C.5) and (C.5) the effective impedance Z eff does not depend on R pl, and they are valid only if the capacitive coupling can be neglected. tr ant power R ω L ω power R ω L ω C ω R ω L ω C ω R ω L ω C ω Fig.C.7. Equivalent electric circuit of a transmission-line. If R pl is negligibly small, Z eff can be calculated with the formula for the impedance of a transmission-line, see Fig.C.7: where Z eff ω ω ω ic + ω ( l ( R il )( )) VRF R + il = = tanh ω icω, (C.53) I RF R ω = R Tr l, Cω = ωc ant l, Lω = ωl Tr l are the resistance, capacitance, and inductance, respectively. The real, R eff, and imaginary, L eff, parts of Z eff are found from equation (C.53), and have the form: R eff eff tr ( + ωltrωcant ) ( +. ωl ωc ) = R and (C.54) ωl = ωl 5. (C.55) tr From equations (C.5)-(C.5) and (C.54)-(C.55), the average capacitance between the antenna and the plasma can be found: tr ant C =. (C.56) ant C ant In general, the effective antenna impedance of ICP discharges can be calculated from equations (C.48), (C.49), and (C.56). In experiments, however, the inverse problem is usually encountered; Z eff is measured and the transformer impedance, Z tr, for inductive coupling is deduced. In the H-mode, Z tr can be calculated from the following equations: R tr ( ωleff ωcant ) (. ωl ωc ) tr = Reff, (C.57) ωl = ωl 5. (C.58) eff eff ant 44

C... Simplifications in the electric circuit The complete equivalent circuit for capacitive power coupling in a RF discharge is shown in Fig.C.8. However, this circuit can usually be simplified for ICPs by excluding elements whose contribution to the total impedances is insignificant. power antenna quartz sheath plasma sheath electrode C q R S C S R cp L cp R S C S C q capacitance of the dielectric separator R S, R S resistances of the sheaths C S, C S capacitances of the sheaths R cp, L cp resistance and inductance of the plasma bulk Fig.C.8. Diagram and the equivalent electric circuit of capacitive power coupling in RF discharge. In the simplified model, the plasma bulk is considered to be a conducting medium that is predominantly resistive, ωl cp << R cp ; whereas the electrode sheaths are predominantly capacitive, R S >> /ωc S, R S >> /ωc S. In addition, the capacitance of the dielectric separator is significantly smaller than the sheath capacitance in plasma cells with an external antenna. The impedance of the equivalent circuit shown in Fig.C.8 is dominated by the plasma resistance and the dielectric separator s capacitance: Rpl R cp, ωcant ωcq. (C.59) R cp includes components corresponding to Ohmic, R Ω, and stochastic, R stoch, plasma heating and can be calculated using equation (..36) in [Lie 94]: R pl m e 8kT e = R Ω + R = stoch vmhcell +, (C.6) e AS ne πme where A S is the electrode sheath area. For a planar ICP discharge, this area is: rant + rel AS = πreff, reff =, (C.6) where r ant and r el are the antenna and electrode radii. The current to a conductive chamber wall is accounted for by multiplying R pl by the weighting factor W R, which is dependant on geometry of the plasma cell: W where r cell is the cell radius. R + w = + w * R * R, w * R r = h eff cell r ln r cell eff, (C.6) 45

C.3. ICP matching network An ICP power source contains an impedance matching unit. The power loss due to the internal resistance R RF of the RF generator (usually R RF = 5 Ω) is minimized only if the complex antenna impedance is properly converted to R RF. The main principles can be illustrated by matching an RF generator (f RF = 3.56 MHz, R RF = 5 Ω) to an external antenna using a conventional L-matching network, shown in Fig.C.9. The matching unit consists of two adjustable capacitors. One of them is connected in parallel to the power supply and is referred to as a load capacitor C load. Another capacitor is connected in series to the antenna. This element is referred to as a tune capacitor C tune, and almost completely compensates for the phase shift between the current and voltage at the antenna due to inductance L eff. C load and C tune can be calculated from equations (C.63). RF generator C load C tune L eff R eff Fig.C.9. Electric circuit of a standard L-matching unit. i Zload = Reff Re( Zload ) = R Im( Zload ) = ( i ωctune + iωleff + Reff ) ωcload + i( ωc ωc + ωl ) RF load = 5Ω tune eff (C.63) The maximum capacitance of commercial available, adjustable RF capacitors without water cooling is 5 pf. For most matching networks C load = C tune = 5-5 pf. This limitation of available capacitors limits the range of impedances that can be matched. The following expressions are found by solving equations (C.63) for the real, R eff, and imaginary, L eff, impedances: RRF R eff = and (C.64) R RF ( ωc ) + Load RRFCLoad L eff = +. (C.65) ω C R Tune RF ( ωc ) + The values of R eff and L eff satisfying the impedance matching condition are shown in Fig.C. and Fig.C.. In the E-mode, R eff is equal to the antenna resistance, R. After the H-mode transition, R eff increases and is bounded by relations (C.4) and (C.54). For ICP discharges generated using an external antenna, R < R eff < Ω. As can be seen in Fig.C., the corresponding capacitance needed for impedance matching is above 47 pf. Consequently, additional capacitance should be added in parallel with an adjustable capacitor. By adding 5 pf, the resistance range for which it is possible to match is shifted from 9-5 Ω to 3-9 Ω. R eff cannot be matched to R RF over a wide range. For example, to convert the resistance R =.5 Ω, typical for a small ICP antenna; a nf variable capacitor is needed, which exceeds C load in available commercial matching units. This explains why the mode transitions of ICP discharges are difficult to investigate experimentally. Load 46

R eff (Ω) 8 6 4 8 6 4 matching R eff L eff (nh) 8 6 4 8 6 C Load = 5 pf C Load = pf matching L eff GEC antenna coil antenna coil with minimum induction region of typical values of R eff 4 3 4 5 6 7 8 9 C Load (pf) Fig.C.. Matched resistance R eff. By adding a fixed capacity in parallel to C load, the resistance of R eff can be shifted toward smaller values. 3 4 5 C Tune (pf) Fig.C.. Matched inductance L eff. A maximum value of the antenna inductance is determined by the value of C tune in the matching unit. It should be mentioned that, the matching of the imaginary part of Z eff, using boundary conditions (C.5) and (C.55), is only little below the antenna s self inductance L. As was shown in Subsection C..4, low-inductance antennas are preferable for ICP operation. According to a diagram in Fig.C., using capacitance C tune = 5 pf, the antenna s self inductance can be as low as 6 nh, which correspond to ωl = 5 Ω at 3.56 MHz. C.4. Calibration The inductance L and resistance R of the antenna as well as the capacitance C ant between the antenna and plasma used in the transformer model can be calibrated. C.4.. Antenna impedance R depends not only on the geometry and material of the antenna, but also on the quality of the antenna surface. This is because the RF current flows in a thin skin layer. In addition, because of the small value of R, the resistance of antenna contacts can substantially influence the total resistance. Because of this, the calculated value of R is usually smaller than the actual resistance. L depends primarily on the antenna geometry, and can be calculated accurately by integrating numerically (C.4). The antenna impedance can be measured using the circuit presented in Fig.C.. For these measurements, an RF generator, a 5 Ω resistor, and a variable capacitor, C tune, from the matching network were employed. C tune was adjusted so that the antenna voltage, U, was minimized. When this happens, the capacitive impedance, /ωc tune, compensates fully for the antenna inductance, ωl. Consequently, R and L can be calculated from equations (C.66) and (C.67). 47

power R C tune L R U U Fig.C.. Measurement of the antenna impedance. R (min) U = R (min) U U ωl = ωc tune (C.66) (C.67) C.4.. Capacitance between the antenna and plasma As was mentioned previously, the capacitance between the antenna and plasma is primarily determined by the dielectric separator: C ant C q. The minimum and maximum values of C q can be estimated by using the following models. In [Lie 94], it is assumed that C q is equivalent to a capacitance between two parallel conducting wires with radius r wire = η/, and is calculated from the following equation: C (min) q πεεl = cosh (h r q wire, ) (C.68) where h q is the thickness of the dielectric separator, ε is the dielectric permittivity (for quartz ε = 5), l is the length of the copper tube, and η is the smallest radius of the copper tube used in the antenna. Since the capacitance between the antenna coils is ignored in the calculations, this model only estimates the minimum Cq. On the other hand, C q s maximum is bounded by the capacitance of a plane capacitor with same radius as the antenna: antenna quartz metal plate Fig.C.3. Capacitance between the antenna and a metal plate C r =, (C.69) (max) πεε max q hq where r max is the antenna radius. The value of C q can also be measured in a simple experiment illustrated by (C.3). One of the capacitor plates is the antenna. Another plate is a metal plate situated on the opposite side of the dielectric. Actually, for typical dimensions of experimental plasma cells, these two solutions differ only slightly. 48

C.5. Efficiency of power coupling into plasma The incident RF power, P, dissipated by the effective antenna resistance consists of three components: P = P + P + P, (C.7) loss cap where P loss is the power loss due to the antenna s resistance, P cap is the power capacitively coupled to the plasma, and P ind is the power inductively coupled to the plasma. Let P tr be the power dissipated by the transformer resistance excluding the power capacitively coupled into the plasma: P tr = P loss + P ind = = P - P cap. The power coupling efficiency into a plasma is a ratio of the absorbed power to the incident power: Pξ ξ = = ξ P cap + ξ ind ind, (C.7) where P ξ is the absorbed power; ξ cap is the capacitive coupling efficiency, and ξ ind is the inductive coupling efficiency. The values of ξ cap and ξ ind can be found from the following equations: ξ ind P = P ind cp RF Pcap I Rpl ξ cap = =, (C.7) P I R = ( ξ cap P ) P ind tr = eff ( ξ ) cap P + P loss ind. (C.73) The relation between the RF current, I RF, and the current flowing through the capacitance between the antenna and the plasma, I cp, can be determined from Ohm s law: RF power I RF ( R + iωl ) Cant R pl L tr I cp R tr I tr eff eff I RF = VRF ( R i ωc ) I = V pl ant cp RF I cp total RF current I cp current flowing through antenna-plasma capacitance I tr current flowing through antenna-plasma transformer Fig.C.4. Capacitive and inductive power coupling mechanisms. (C.74) The capacitive coupling efficiency in an ICP discharge can be calculated as follows: eff pl R + ( ωleff ) R pl ξ cap =. (C.75) R + ( ωc ) R ant The dimensionless expression for R tr in equation (C.) is used to calculate the ratio P P : loss ind P P loss ind = R tr R R α = k α + α eff + ω ν eff. (C.76) 49

for both limiting values of parameter α (α and α ), the ratio P P tends to infinity: P loss P ind. This implies that the efficiency of inductive energy coupling into plasma decreases at low and high plasma densities. All the incident power will go towards heating the antenna rather than the plasma. This resembles the behavior of a transformer with an open or shorted secondary. Between these two limits of α, there is an optimum plasma density which maximizes the inductive coupling efficiency: ( + ω ν ) α = eff. (C.77) An ICP s operating mode depends on its dominant power coupling mechanism. An ICP will operate in the E-mode at low densities, because ξ cap > ξ ind. As the plasma density increases, the ratio between ξ cap and ξ ind changes. When ξ cap < ξ ind the discharge changes to the H-mode. Using the condition ξ cap = ξ ind, the plasma density corresponding to the transition between the modes can be determined. Note that during the mode transition, the discharge will not operate efficiently, because both ξ cap and ξ ind are small. The effective electron collision frequency can be determined experimentally by measuring the antenna impedance. The power applied to the antenna is varied until the coupling efficiency is maximized. In this case, it is reasonable to neglect the capacitive power coupling into the plasma. The measured antenna impedance is then used to calculate v eff from equations (C.) and (C.77). loss ind 5

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Danksagung An dieser Stelle möchte ich allen danken, die zum Gelingen meiner Arbeit beigetragen haben. Zuallererst möchte ich meinen tiefsten Dank meinem Doktorvater Herrn Prof. Dr. U. Czarnetzki aussprechen, der mich zu jeder Zeit mit Rat und Tat unterstützt hat. Unter seiner Anleitung habe ich viel gelernt und wertvolle Erfahrungen gesammelt. Bei Herrn Prof. Dr. H.F. Döbele bedanke ich mich für die Möglichkeit, den experimentellen Teil meiner Arbeit in seiner Arbeitsgruppe an der Universität Essen durchführen zu können. Mein besonderer Dank gilt Dipl.-Phys. Dirk Luggenhölscher. Seit meiner Ankunft in Deutschland konnte ich immer auf seine Hilfe zählen, sowohl in wissenschaftlichen als auch in organisatorischen Fragen. Ich möchte mich bei den Technikern der AG Döbele, Jürgen Leistikow und Carola Fischer, bedanken, deren Hilfe wesentlich zum Gelingen der Arbeit beigetragen hat. Im letzten Teil meiner Arbeit hatte ich Schwierigkeiten mit der korrekten Schreib- und Ausdrucksweise in der englischen Sprache. Deshalb möchte ich Brian Heil für das mühsame Korrekturlesen danken. Außerdem danke ich den Mitgliedern des Instituts für Laser- und Plasmaphysik (Universität Essen) und des Instituts für Experimentalphysik V (Ruhr- Universität Bochum) für ihre Hilfsbereitschaft und tatkräftige Unterstützung. Der Abteilung für Niedertemperaturplasmaoptik (P.N. Lebedev Physics Institute) und der Abteilung für Plasmaphysik (Moscow Engineering Physics Institute) gilt mein Dank für ihr Interesse an meiner Arbeit. 59

Curriculum Vitae Personal Data Name: Last name: Victor Kadetov Birthday:.6.97 Birthplace: Family status: Nationality: Moscow single Russian Education 979 987 study in secondary school 889 (Moscow) 987 989 study in high school 54 (Moscow) 989 995 student in Moscow Engineering Physics Institute (State University) Diploma: Investigations on radiations from cryogenic liquids induced by plasmas 995 998 research assistant in Department of Plasma Physics, Moscow Engineering Physics Institute (State University), Russia 998 999 research assistant in Department of Optics for Low-Temperature Plasma, P.N. Lebedev Physical Institute, Russia 999 research assistant in AG Prof. Dr. H.F. Doebele, University Essen, Germany 4 research assistant in Institute for Experimental Physics V, Ruhr-University Bochum, Germany 6