Electron heating in capacitively coupled radio frequency discharges

Transcription

1 Electron heating in capacitively coupled radio frequency discharges Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften in der Fakultät für Physik und Astronomie der Ruhr-Universität Bochum von Dipl.-Phys. Felix Julian Schulze aus Mülheim an der Ruhr Bochum 29

2 2 Dissertation eingereicht am: Tag der Disputation: Gutachter: Prof. Dr. U. Czarnetzki 2. Gutachter: Prof. Dr. R. P. Brinkmann

3 Contents 1 Introduction 1 2 Fundamentals Capacitively coupled radio frequency discharges DC self bias generation in capacitive RF discharges The Plasma Series Resonance Electron heating in capacitive RF discharges Diagnostics and simulations Diagnostics Phase Resolved Optical Emission Spectroscopy Fluorescence Dip Spectroscopy in krypton Langmuir probe measurements Measurements of ion energy and ion flux Current and voltage measuremts Simulations Hybrid fluid-monte Carlo simulation Particle in Cell simulation Results Geom. symmetric single frequency discharges Experimental setup Excitation dynamics Geom. asymmetric single frequency discharges Experimental setup Results at intermediate pressure (1 Pa) Results at low pressures (1 Pa,.5 Pa) Electron beams in geometrically asymmetric capacitive RF discharges at low pressures Electric field reversals during sheath collapse in geometrically asymmetric single frequency discharges at low pressures Classical geom. symm. dual frequency discharges Experimental setup Excitation dynamics Frequency coupling i

4 ii CONTENTS Electric field reversals during sheath collapse in geometrically symmetric dual frequency discharges at intermediate pressures Secondary electrons Electron dynamics under different discharge conditions Electrically asymm. dual frequency discharges Experimental setup Generation of a variable DC self bias in geometrically symmetric discharges Separate control of ion energy and ion flux at the electrodes via the Electrical Asymmetry Effect Optimization of the Electrical Asymmetry Effect in dual frequency discharges Excitation dynamics in electrically asymmetric discharges Self-excited non-linear plasma series resonance oscillations in geometrically symmetric discharges The Electrical Asymmetry Effect in multi frequency discharges Quality of separate control of ion energy and flux - comparison to alternative methods Geom. asymmetric hybrid CCP-ICP discharges Experimental setup Excitation dynamics Conclusions 21 6 Outlook 25

5 Abstract Electron heating, mechanisms of plasma generation, and electron dynamics in different types of capacitively coupled radio frequency (CCRF) discharges relevant for industrial applications are investigated by a combination of different experimental diagnostics, simulations and models. Geometrically symmetric and asymmetric single frequency discharges operated at low pressures, geometrically symmetric dual frequency discharges operated at two substantially different frequencies and two similar frequencies (fundamental and second harmonic with variable phase shift between the driving voltages) as well as hybrid capacitively/inductively coupled (CCP-ICP) RF discharges are studied. Electron heating is found to be strongly affected by phenomena characteristic for a certain discharge type, that do not occur in another. At low pressures the generation of highly energetic electron beams by the expanding sheath is observed. Such beams propagate through the entire plasma bulk and are reflected at the opposing plasma boundaries, if the electron mean free path is long enough. An analytical model demonstrates that these beams lead to an enhanced high energy tail of the electron energy distribution function and are, therefore, closely related to stochastic heating. The concept of Non-Linear Electron Resonance Heating (NERH) in asymmetric discharges is verified experimentally by measurements of the RF current to the chamber wall. Results are compared to an analytical Plasma Series Resonance (PSR) model and a global model. The PSR leads to a faster sheath expansion and, therefore, enhances the generation of highly energetic electron beams. The novel diagnostic of Fluorescence Dip Spectroscopy (FDS) using krypton as a probe gas to measure electric fields in the plasma boundary sheath space and phase resolved is applied for the first time to an asymmetric single frequency discharge. The results are compared to a fluid sheath model. Via this diagnostic the sheath movement is studied in detail and the theoretically assumed quadratic charge voltage relation of the sheath is verified experimentally. Electric field reversals in single and dual frequency discharges are investigated experimentally, by a Particle in Cell (PIC) simulation as well as by an analytical model. It is demonstrated that such field reversals are caused by collisions of electrons with the neutral background gas and/or electron inertia. In classical dual frequency discharges operated at substantially different frequencies the nature of the frequency coupling is understood based on experimental and simulation results. It is demonstrated that this frequency coupling limits the separate control of ion energy and flux at the electrode surfaces. Different modes of electron heating such as α- and γ-mode are studied and compared to single frequency discharges. A novel type of geometrically symmetric dual frequency discharge operated at a fundamental frequency and its second harmonic with variable phase shift between the driving voltage waveforms is investigated in detail. The recently theoretically predicted Electrical Asymmetry Effect (EAE) [BGH8], [1] is verified experimentally

6 ii CONTENTS for the first time. The EAE allows the generation of a variable DC self bias even in geometrically symmetric discharges. It is demonstrated experimentally as well as by a PIC simulation that this effect allows efficient separate control of ion energy and flux at the electrodes at a better quality compared to classical dual frequency discharges within a broad range of discharge conditions. For the first time, self-excited non-linear PSR oscillations of the RF current are observed in simulations of a geometrically symmetric discharge. Such oscillations and NERH can be switched on and off as a function of the phase between the driving voltages via the EAE. Electron dynamics in electrically asymmetric discharges are studied experimentally as well as by a PIC simulation and found to work differently compared to classical CCRF discharges. The excitation dynamics are understood in the frame of a simple analytical model. Finally, coupling mechanisms and PSR oscillations of the RF current in geometrically asymmetric hybrid CCP-ICP RF discharges are investigated experimentally.

7 Chapter 1 Introduction Figure 1.1: Trench etch (.2 µm wide and 4 µm deep) in single crystal Si [2]. Non-equilibrium (T e T i ) low temperature (k B T i.3 ev, T e >> T i ) plasmas are of paramount importance for a variety of plasma processing applications ranging from chip and solar cell manufacturing to the creation of biocompatible surfaces. Amongst others such plasma discharges are used for etching and deposition processes on microscopic scales [2 8], ion [9] as well as neutral particle sources [1, 11] and for medical applications such as sterilization [12]. Capacitively coupled radio frequency (CCRF) discharges, reviewed briefly in chapter 2, are frequently used for most of these applications. They are standard tools for etching processes such as physical/chemical sputtering or ion assisted/inhibited etching as well as for deposition processes such as Plasma Enhanced Chemical Vapor 1

8 2 CHAPTER 1. INTRODUCTION Deposition (PECVD), sputter deposition and Plasma Immersed Ion Implantation (PIII). CCRF discharges can be operated at pressures ranging from about.1 Pa to atmospheric pressures and at powers ranging from a few W to several kw. Typically the plasma density is n e = n i cm 3 in the bulk. Here n e, n i is the electron and ion density, respectively. The electron temperature k B T e is typically of the order of a few ev, whereas the ion temperature T i is close to room temperature. Due to the low electron and high ion inertia as well as the externally applied RF voltage a sheath develops adjacent to each plasma boundary. The sheath width is time modulated within the RF period. In this region of positive space charge ions are accelerated towards the boundary surface and electrons are repelled during most of the RF period. Figure 1.2: Experimental demonstration of the effectiveness of ion-enhanced plasma etching of Si in XeF 2 gas with energetic Ar + ion bombardment of the substrate compared to chemical etching of Si by XeF 2 gas only and physical sputtering of Si by energetic Ar + ion bombardment only [3]. If a substrate is located on one of the electrodes, anisotropic ion etching of deep trenches without unwanted thermal damage of the wafer can be performed as one example for various plasma processes CCRF discharges are used for (see figure 1.1). In contrast to non-plasma processes the temperature of the heavy particles (neutrals and ions) is close to room temperature. Despite this low heavy particle temperature the gas is still ionized and chemically active species are produced by the much

9 hotter electrons, e.g. by dissociation of an inert molecular gas. In the sheath ions are accelerated to high energies perpendicular to the electrodes. Therefore, at low pressures most ions hit the wafer vertically and strongly anisotropic etching can be performed. Compared to chemical etching in a neutral gas and pure physical sputtering by a single ion species, plasma assisted etching yields sufficiently higher etching rates. This is demonstrated by a measurement of Coburn and Winters [3] of the Si etch rate in XeF 2 gas only, XeF 2 gas with energetic Ar + ion bombardment simulating plasma assisted etching, and Ar + ions only simulating physical sputtering (see figure 1.2). Despite the enormous relevance of such processes for industrial applications they are usually not understood. Fundamental phenomena such as electron heating and mechanisms of plasma sustainment are not understood or at least not agreed upon. However, without detailed comprehension of power coupling under industrially relevant conditions plasma processes can only be optimized via trial and error, but not based on a solid scientific basis. Therefore, in this work electron heating and dynamics in various different types of CCRF discharges relevant for applications are investigated by a combination of different experimental diagnostics, simulations and models. Based on this synergistic approach the physical nature of different electron heating mechanisms in CCRF discharges will be understood and the industrially most relevant problem of separate control of ion energy and flux will be solved based on the theoretically predicted Electrical Asymmetry Effect (EAE) [BGH8], [1], [ZD9a, UC9, JS9b, ZD9b]. The thesis is structured in the following way: In the next chapter fundamentals of CCRF discharges such as the different types of discharges investigated, their typical experimental realization, the generation of a DC self bias as well as existing models of electron heating will be discussed. In the third chapter the diagnostics and simulations used in this work will be introduced. In chapter 4 the results will be presented: Electron heating in each type of CCRF discharge introduced in chapter 2 will be discussed separately pointing out similarities and differences to other types of capacitive discharges. In the fifth chapter conclusions will be drawn and in the final chapter an outlook towards potential future projects based on the results of this work will be given. The references are divided into two separate lists: The first comprises all publications of the author, that are summarized in this thesis. The second list contains all other references. 3

10 Chapter 2 Fundamentals 2.1 Capacitively coupled radio frequency discharges Grounded electrode Sheath chamber wall Plasma bulk RF generator Sheath Powered electrode Impedance matching Figure 2.1: Typical experimental realization of a capacitively coupled RF discharge. Capacitively coupled radio frequency discharges consist of two typically plane parallel electrodes of either the same or different surface areas. The electrodes are located within a vacuum chamber in direct contact with the plasma or isolated from the plasma by a dielectric. The chamber wall is usually grounded. By connecting one electrode to a RF generator via an impedance matching network one of the electrodes is driven by a RF high voltage waveform φ of the following form: 4

11 2.1. CAPACITIVELY COUPLED RADIO FREQUENCY DISCHARGES 5 φ(t) = n k=1 φ k cos(2πf k t + θ k ) n N (2.1) The other electrode is grounded. f k is the frequency, φ k the amplitude, and θ k the phase shift of the k-th voltage waveform applied to the discharge. In this work almost all applied voltage waveforms are phase-locked, i.e. they are multiples of a fundamental frequency. An exception is the hybrid CCP-ICP RF discharge discussed in chapter 4.5. If n = 1 or n = 2 (f 1 f 2 ) the discharge is operated as a single or a dual frequency discharge, respectively. If n > 2 the discharge is operated as a multi frequency discharge. In case of dual or multi frequency discharges both electrodes can be powered simultaneously at different frequencies. Such a scenario is not discussed in this work. The externally applied radio frequencies typically range between 1 MHz - 1 MHz. The matchbox matches the impedance of the load (plasma) to the output impedance of the source (RF generator), which is typically 5 Ω. If the surface area of the powered electrode A p is equal to the surface area of the grounded electrode A g (including all grounded surfaces in contact with the plasma) the discharge is geometrically symmetric. If A p A g the discharge is geometrically asymmetric. A sketch of a CCRF discharge is shown in figure 2.1. In order to provide an experimental platform for comparing measurements in CCP/ICP RF discharges a common reactor geometry was introduced at the Gaseous Electronics Conference - the so called GEC cell [13, 14]. A slightly modified version of a GEC cell is used for most measurements performed in this work. Detailed reviews of CCRF discharges can be found in the books of Lieberman and Lichtenberg [2] as well as Raizer [15]. Due to the difference between electron mass m e and ion mass m i (m i /m e >> 1) the ion inertia is much higher than the electron inertia. Therefore and since electrons have much higher energies than ions, electrons will be lost to the walls quickly, if the plasma gets in contact with a boundary wall. As a consequence a sheath of positive space charge will develop adjacent to the wall. In the sheath an electric field accelerates ions towards the wall and repels electrons confining them in the discharge. In the sheath quasi-neutrality is violated. In the quasi-neutral bulk an ambipolar field couples ion and electron diffusion. This ambipolar diffusion leads to axial ion density profiles in the bulk similar to the ones shown in figure 2.2. If a voltage waveform of frequency ω RF = 2πf RF is applied to one electrode externally, the charged particle densities in the sheath region will react to this time dependent potential. If the applied radio frequency as well as the ion mass are both high enough and the ion density is low enough, so that ω RF > ω pi = ((e 2 n i )/(ε m i )) 1/2, where e is the elementary charge and ε is the dielectric constant, the ions will not be able to follow the fast potential variations and will react only to the time averaged electric field in the sheath. This assumption is true for most CCRF discharges, but can be violated for discharges operated at very low frequencies or in light gases such as H 2 or He. Such a scenario is investigated in chapter 4.3. Due to the low electron mass the electron plasma frequency ω pe = ((e 2 n e )/(ε m e )) 1/2 is typically of the order of GHz and significantly higher than the applied radio frequency. Therefore,

12 6 CHAPTER 2. FUNDAMENTALS density [cm -3 ] 2,x1 11 1,5x1 11 1,x1 11 5,x1 1 n e n i,,,2,4,6,8 1, 1,2 Distance from powered electrode [cm] Figure 2.2: Time averaged axial ion and electron density profiles in a geometrically symmetric CCRF discharge (Particle in Cell simulation). The time averaged sheath width defined by equation 2.2 is s.2 cm at both electrodes. electrons can follow the fast potential variations and the width of the sheath region is time modulated within one RF period. This is shown schematically in figure 2.3. According to Brinkmann [17] the sheath region can be divided into 3 different zones, the unipolar zone at the electrode, where the electron density is zero, the quasi-neutral zone, where the electron density nearly equals the ion density, i.e. the bulk, and a transition zone (see figure 2.3). The position of the sheath edge s is located in the transition zone and is defined by the following criterion [17]: s n e (z)dz = d/2 s (n i (z) n e (z))dz (2.2) where d is the electrode gap. Ions enter the sheath at velocities higher or equal to the Bohm velocity u B = (k B T e /m i ) 1/2. Within the presheath the ions are accelerated to this velocity. In the sheath the ions are strongly accelerated by the high sheath electric field. Typically, ionization in the sheath and temporal changes of the ion density can be neglected. Due to particle continuity under these conditions the ion flux is conserved ( (n i u i )/ z = ). Thus, the density of the accelerated ions decreases monotonically towards the electrode surface. This is shown in figure 2.3. Detailed reviews about sheaths were written by Riemann [18, 19]. In typical CCRF discharges the length of the quasi-neutral bulk L is significantly larger than the time averaged sheath width s so that the following ordering of length scales applies:

13 2.1. CAPACITIVELY COUPLED RADIO FREQUENCY DISCHARGES 7 Figure 2.3: Time resolved charged particle densities in the sheath region adjacent to an RF driven electrode [16]. λ D s L λ S (2.3) here λ D = ((ε k B T e )/(en e )) 1/2 is the debye length and λ S is the skin depth. At moderate electrode sizes and plasma densities the skin depths is larger than the electrode gap and electromagnetic phenomena such as skin and standing wave effects can be neglected [2 23]. This assumption is justified for all CCRF discharges investigated in this work. If the plasma density is particularly low and if the sheath voltage is high, e.g. at very low neutral gas pressure, the time averaged sheath width s can become comparable to the bulk length. Such a scenario is discussed in chapter 4.2. For CCRF discharges operated in heavy gases at reasonably low plasma densities the following ordering of frequency scales applies: ω pi ω RF ω PSR ω pe (2.4) Here ω PSR is the Plasma Series Resonance (PSR) frequency, which will be introduced in chapter 2.3. For plasmas operated in light gases and/or at high plasma densities ions might be able to follow the RF potential variations. This can significantly affect the heating dynamics as will be discussed in chapters and The total current in the discharge j tot = j d +j c, where j d is the displacement current and j c is the conduction current, is assumed to be mainly displacement current in the sheath and mainly conduction current in the bulk. At very high RF frequencies this approximation might be critical, since the displacement current in the bulk might no longer be negligible compared to the conduction current. Depending on the choice of electrode surface areas (chamber geometry), electrode gap, applied voltage waveforms, gas mixture, pressure, power, and potential com-

14 8 CHAPTER 2. FUNDAMENTALS binations with other discharge types there is a variety of different types of CCRF discharges. Each type provides unique features useful for particular applications. Some examples are: Frequency: Dual frequency discharges operated at substantially different frequencies [24 29] are used to separately control ion energy and ion flux. Such separate control is essential for many applications, since the ion flux determines the throughput of a given process and the ion energy controls the etching and deposition processes at the wafer s surface. An alternative to dual frequency discharges for separate control of ion energy and flux are hybrid combinations of capacitive and inductive RF discharges [3, 31]. Due to the inductive coupling higher plasma density can be achieved in such hybrid discharges compared to dual frequency CCRF discharges. Chamber geometry: Strongly geometrically asymmetric discharges are often used to generate a strong DC self bias (see chapter 2.2) to accelerate ions to high velocities in the sheath. Electrode size: Large area electrodes are required to process large area wafers. Gas mixture: The choice of the processing gas is essential for chemical processes in the plasma itself and at surfaces in contact with the plasma. Depending on the gas the discharge might be electropositive or electronegative [32, 33]. Electrode gap: Microdischarges are often CCRF discharges with particularly small electrode surface areas and electrode gaps. Amongst others such discharges are used for medical applications. Pressure: CCRF discharges can be operated at low pressures down to less than 1 Pa for anisotropic etching processes [2] or at high pressures, e.g. microscale atmospheric pressure plasma jets (µ AP P J). High pressure microdischarges can provide high radical densities without the need of expensive vacuum systems [34 37]. In this work electron heating and mechanisms of plasma sustainment are investigated in several different types of low pressure macroscale capacitive RF discharges and found to work differently depending on the kind of CCRF discharge. Electron heating is found to be strongly affected by phenomena characteristic for a certain type, that do not occur in another. The thesis is structured according to the different discharge types investigated to provide a straightforward outline. The following types of CCRF discharges are investigated: Geometrically symmetric single frequency discharges (chapter 4.1) Geometrically asymmetric single frequency discharges (chapter 4.2)

15 2.1. CAPACITIVELY COUPLED RADIO FREQUENCY DISCHARGES 9 Geometrically symmetric dual frequency discharges operated at substantially different phase locked frequencies (chapter 4.3) Geometrically symmetric dual frequency discharges operated at similar (fundamental + second harmonic with variable phase), phase locked frequencies (chapter 4.4) Geometrically asymmetric hybrid CCP-ICP RF discharges (13.56 MHz MHz) (chapter 4.5)

16 1 CHAPTER 2. FUNDAMENTALS 2.2 DC self bias generation in capacitive RF discharges Figure 2.4: Sketch of a CCRF discharge (left) and location of the net positive charge in the discharge Q at the phase of complete sheath collapse at the grounded (middle) and powered electrode (right) [UC9]. In typical CCRF discharges power is applied to one electrode through a blocking capacitor, which is part of the matchbox. Thus, there is no net charge transport across the discharge time averaged over one RF period, i.e. the uncompensated net positive charge in the discharge Q must be constant on time average. Within the RF period, however, Q changes as a function of time (see chapter 4.4.2, Small Angle Effect): During one RF period the electrodes are permanently bombarded by positive ions and positive charge is lost from the discharge. At each electrode this positive charge is compensated by electrons that leave the discharge only at the time of sheath collapse at the respective electrode. At each electrode within one RF period the number of ions leaving the discharge equals the number of electrons lost from the discharge in case of single positively charged ions. Figure 2.4 shows the sketch of a typical setup of a CCRF discharge. The discharge consists of a quasi-neutral plasma bulk and two sheaths, one adjacent to each electrode. The uncompensated net positive charge in the entire discharge Q is located in the sheaths. The voltage balance for such a CCRF discharge is (see chapter 2.3, [1],[UC9]): φ + η = φ sp + φ sg (2.5) where φ is the applied RF voltage, η the DC self-bias and φ sp, φ sg are the sheath voltages at the powered and grounded electrode, respectively. Here the floating potential and the voltage drop across the bulk are neglected and the voltage drop across the discharge is assumed to be determined only by the sum of both sheath voltages. A PIC simulation has demonstrated that this assumption is justified at low pressures (see chapter [ZD9a]). At two distinct phases within one RF period one sheath is completely collapsed at one of the electrodes (see figure 2.4). If the sheath is collapsed at the grounded electrode, the applied RF voltage φ will be minimum ( φ = φ m2 ), no voltage will drop across the sheath at ground (φ sg = ) and

17 2.2. DC SELF BIAS GENERATION IN CAPACITIVE RF DISCHARGES 11 maximum voltage will drop across the sheath at the powered electrode (φ sp = ˆφ sp ). Vice versa if the sheath is collapsed at the powered electrode, the applied RF voltage will be maximum ( φ = φ m1 ), no voltage will drop across the sheath at the powered electrode (φ sp = ) and maximum voltage will drop across the sheath at ground (φ sg = ˆφ sg ). The voltage balance of equation 2.5 at these two distinct phases is: φ m2 + η = ˆφ sp (2.6) φ m1 + η = ˆφ sg (2.7) The maximum sheath voltages ˆφ sp and ˆφ sg can be calculated by integrating Poisson s equation at the corresponding phase: ˆφ sp = 1 2eε ˆφ sg = 1 2eε ( Qmp A p ( Qmg A g ) 2 I sp n sp (2.8) ) 2 I sg n sg (2.9) Here Q mp,mg is the maximum charge in the sheath at the powered and grounded electrode, respectively, n sp,sg is the mean ion density in the respective sheath and I sp,sg is the respective sheath integral. I s = 2 1 p s (ξ)ξdξ (2.1) with ξ = x/s m and p s (ξ) = n i (z)/ n i. Here s m is the maximum sheath width. Under the assumption of a temporally constant total charge Q = Q mg = Q mp the ratio of equations 2.8 and 2.9 yields: ε = ˆφ sg ˆφ sp = ( Ap A g )2 n sp n sg I sg I sp (2.11) ε is called the symmetry parameter [1]. Using equations 2.11, 2.6, and 2.7 the following expression for the DC self bias is derived, that depends only on the extremes of the applied RF voltage waveform and the symmetry parameter [1]: η = φ m1 + ε φ m2 (2.12) 1 + ε In the frame of this model three fundamental assumptions are made: (i) The overall net positive charge in the discharge Q is assumed to be temporally constant. As will be discussed in detail in chapter this assumption is true to a good approximation, however, it does not exactly hold. Due to the time modulated electron loss to the electrodes Q changes with time within one lf period. The consequences of this charge dynamics on the DC self bias will also be discussed in chapter

18 12 CHAPTER 2. FUNDAMENTALS (ii) Each sheath collapses completely at least once per RF period, i.e. all charge is located in one sheath at phases of maximum/minimum applied voltage. This assumption is verified by a PIC simulation. (iii) The voltage drop across the plasma bulk is neglected. This is well justified under low pressure conditions as will be demonstrated by a PIC simulation in chapter According to equation 2.12 a DC self bias can be caused either geometrically by different electrode surface areas (A p A g ) and/or electrically by applying a voltage waveform to the discharge with φ m1 φ m2. The former is well known for many years [38 41], whereas the latter has recently been discovered by Heil et al. in an argon discharge and is called the Electrical Asymmetry Effect (EAE) [BGH8] [1] [ZD9a, UC9, ZD9b, JS9b]. Longo et al. [42] verified the EAE in a hydrogen discharge by a self-consistent kinetic model. For instance, in a single frequency discharge a voltage waveform φ = φ sin (ω RF t) with maximum φ m1 = φ and minimum φ m2 = φ is applied to one electrode. If A p A g (geometrically caused DC self bias), the symmetry parameter ε will be zero according to equation 2.11 and following equation 2.12 η = φ. If A p = A g, ε = 1 and η =. If A p A g, ε = and η = φ. Figure 2.5: Exemplary voltage waveform φ(t) = 315 (cos(2πft) + cos(4πft)), where f = MHz, θ =, and φ lf = φ hf = 315 V, applied to a geometrically symmetric discharge for two RF periods (solid black line). The absolute values of the positive and negative extremes are different. Therefore, a DC self bias develops under these conditions [UC9]. If a temporally symmetric voltage waveform is applied to one electrode, that contains one or more even harmonics of the fundamental frequency, the sheaths in front of the two electrodes will necessarily be asymmetric even in a geometrically symmetric discharge [BGH8]. In a dual-frequency discharge this is achieved optimally by driving one electrode at a phase locked fundamental frequency and its second

19 2.2. DC SELF BIAS GENERATION IN CAPACITIVE RF DISCHARGES 13 harmonic, e.g MHz and MHz, i.e. the following voltage waveform is applied to one electrode in a geometrically symmetric discharge: φ(t) = φ lf cos(2πft + θ) + φ hf cos(4πft) (2.13) with f = MHz. θ is the phase angle between the harmonics. Figure 2.5 shows such a voltage waveform ( φ lf = φ hf ) as well as the fundamental cosine function and its second harmonic for θ =. Each of the two cosine functions is harmonically symmetric ( φ(ϕ + π) = φ(ϕ), ϕ = ωt), but the sum of the two is not. The sum is symmetric with respect to ϕ = π ( φ(ϕ) = φ( ϕ)). The absolute values of the positive and negative extremes are different. Therefore, according to equation 2.12 at θ = a DC self bias will be generated (ε 1), although A p = A g. applied RF voltage (a.u.) = cos( ) + cos(2* ) S S AS = /4 AS = = /2 = / (rad) Figure 2.6: RF voltage waveform according to equation 2.13 at different phase angles θ [UC9]. By tuning the phase angle θ from to 9 the symmetry of the applied RF voltage waveform is changed from symmetric (S, θ = ) to anti-symmetric (AS, φ(ϕ) = φ( ϕ), θ = 45 ) and back to symmetric (S, θ = 9 ). This is shown in figure 2.6. In case of an anti-symmetric waveform the absolute values of the positive and negative extremes are the same and no DC self bias will develop. At θ = 9, η 9 = η. Already based on this simple graphical analysis it is obvious that by adjusting the phase between the applied voltage harmonics the DC self bias can be changed (assuming ε = 1). The EAE leads to the generation of a DC self bias as a function of the phase between the applied voltage harmonics in geometrically symmetric as well as asymmetric discharges. As will be demonstrated in chapter the DC self bias depends

20 14 CHAPTER 2. FUNDAMENTALS almost linearly on the phase angle and the role of the electrodes (powered and grounded) can be reversed. At low pressures the EAE is self-amplifying due to the conservation of ion flux in the sheaths (see chapter 4.4.2). By tuning the phase, precise and convenient control of the ion energy at the electrodes can be achieved [BGH8] [1] [ZD9a, UC9, ZD9b, JS9b]. The EAE, its application to separately control ion energy and flux, and other applications are investigated in this work in chapter The Plasma Series Resonance sheath at powered electrode elastic collisions electron inertia bulk plasma sheath at grounded electrode Figure 2.7: Equivalent circuit of a CCRF discharge. Under the conditions defined by equations 2.3 and 2.4 a CCRF discharge can be identified with an equivalent circuit in the frame of a global model [43 53]. This equivalent circuit is shown in figure 2.7: The RF generator is represented by an ideal voltage source, which generates one or more sinusoidal voltage waveforms. The DC self bias η is represented by the bias capacitor. The sheaths adjacent to each electrode are identified with non-linear capacitors. The charge voltage-relation of the sheath is non-linear, since the sheath width changes as a function of the applied voltage in contrast to a capacitor with fixed distance between anode and cathode. Assuming a quasi-static ion density in the sheath (n i (z) = n s ) integration of Poisson s equation yields:

21 2.3. THE PLASMA SERIES RESONANCE 15 φ s s = e ε n s s (2.14) Here φ s is the voltage drop across the sheath. In the same approximation the surface charge density σ is: σ(t) = e s(t) n i (z)dz e n s s(t) (2.15) Substitution of equation 2.15 in 2.14 yields the quadratic charge voltage relation for a matrix sheath [2]: 1 φ s (t) Q 2eε n s A 2 s (t) 2 (2.16) s Here A s is the area of the electrode adjacent to the considered sheath and Q s is the charge in this sheath. The result of equation 2.16 is a result of the assumption of a static ion density profile. In general higher order non-linearities will occur [46, 47, 49 53]. Constant or linear terms do not occur, since the sheath voltage and sheath electric field have to vanish at the phase of sheath collapse, when Q s = (neglecting the low floating potential). However, the quadratic non-linearity will always be dominant. In the frame of this work it will be demonstrated that the charge-voltage relation of the sheath is quadratic to a good approximation by an experiment ([JS8b, JS7a], see chapters and 4.2.3) as well as by a PIC simulation (see chapter 4.4.6). The bulk is identified with a series of an inductance representing electron inertia and a resistance representing elastic electron-neutral collisions. If ω RF ω pe the displacement current in the bulk must not be neglected. In this case a capacitor in parallel to the series of inductance and resistance must be implemented in the equivalent circuit [2, 54 57] and a parallel resonance can occur. At the corresponding resonance frequency the plasma impedance is maximal and the current is minimal. However, in this work ω RF ω pe and the lumped circuit model of figure 2.7 can be used. In this picture the impedance of the electrode is neglected [58]. The voltage drop across a discharge, which is either strongly asymmetric or of cylindrical or spherical geometry, is then given by a generalized form of the PSR equation of [48]: ( η n + φ n = q 2 + ε(q t q) 2 2 q + 2 τ + κ q ). (2.17) 2 τ Here q = Q sp /Q m is the positive charge in the sheath at the powered electrode normalized to the maximum space charge Q m = A p 2eε n s ˆφsp in this sheath at the phase of maximum negative sheath voltage ˆφ sp = φ m2 φ m1 1+ε (equations 2.6 and 2.12). q t (t) is the total net positive charge in the discharge normalized by Q m. ε is the symmetry parameter defined by equation κ = ν/ω is the normalized electron-neutral elastic collision frequency. η n = η/ ˆφ sp and φ n = φ/ ˆφ sp are the DC self bias (equation 2.12) and the applied voltage (equation 2.1) normalized by

22 16 CHAPTER 2. FUNDAMENTALS the maximum voltage across the sheath at the powered electrode, respectively. All temporal derivatives are performed with respect to τ = ω t where ω = γ ω p is the plasma frequency reduced by a dimensionless geometry factor γ = s m A/LA p < 1 with s m = 2ε ˆφsp /en s. A is an effective value for the discharge area [48], L the bulk length, and n s the effective density in the sheath. Due to its analogy to a series of a capacitor, an inductance, and a resistance a self-excited series resonance can occur in CCRF discharges, if they are strongly asymmetric (ε < 1) and operated at low pressures typically below about 1 Pa. Then the bulk part is dominated by the inductive component (κ small) and the PSR oscillations are not completely damped out by collisions. At the PSR frequency ω PSR high frequency oscillations of the RF current of the order of typically 1 MHz are self-excited by the sheath non-linearity. At this frequency the current flowing through the plasma is maximal and the impedance is minimal. In case of a strongly asymmetric (ε = ) single frequency discharge ( φ 1 = φ /2, Ω = ω RF /ω ) equation 2.17 can be solved analytically with good accuracy. In this case equation 2.17 reduces to [48]: With ( ) 2 ( Ωτ sin = q 2 2 q τ + κ q ). (2.18) 2 τ q = q + q 1 (2.19) where q is the solution of the PSR equation without bulk part, equation 2.18 can be solved. The solution for q is: q = Ωτ sin 2 (2.2) With j = q / τ this yields a sawtooth current waveform typical for CCRF discharges shown in figure 2.8 (dash-dotted line) for Ω =.1 and κ =.2. Substitution of equation 2.19 in equation 2.18 and neglecting all non-linear terms in q 1 yields: ( ) 2 q 1 τ + κ q τ + q q q 1 = τ + κ q (2.21) 2 τ Equation 2.21 describes a non-linear damped oscillator in q 1 under an external periodic force. The numerical solutions of equation 2.21 as well as the numerical solution of the full PSR equation 2.18 are shown in figure 2.8 as dashed and solid line, respectively. The linearization in q 1 mainly leads to a small phase shift between the two solutions. Obviously, q 1 leads to damped high frequency current oscillations superimposed on the sawtooth current only resulting from q. According to equation 2.2 the second derivative of q corresponds to a δ-function. If all other terms on the RHS are neglected compared to the δ-function, equation 2.21 becomes:

23 2.3. THE PLASMA SERIES RESONANCE 17 normalized current 1,5 1,,5, -,5-1, -1,5-2, -2,5 numerical solution: linearized equation numerical solution: exact equation =.1 p =.2-3, normalized time Figure 2.8: Numerical solution of the PSR equation 2.18 (solid line), equation 2.18 without bulk part (dash-dotted line) and the linearized PSR equation 2.21 (dashed line) [48]. 2 q 1 τ + κ q 1 2 τ + q q 1 = Ωδ(τ) (2.22) This is transformed into the following problem to be solved: 2 q 1 τ 2 + κ q 1 τ + q q 1 = with q 1 () = and Using q 1 (τ) = f(τ) exp ( κ 2 τ) yields: q 1 () τ = Ω (2.23) 2 f τ 2 + Ω2 PSRf = with f() = and Ω 2 PSR = q κ2 4 q = sin ( Ωτ 2 f() = Ω (2.24) τ ) (2.25) Equation 2.24 describes an oscillator oscillating at the frequency Ω PSR, which is a function of q and, therefore, time dependent. The maximum frequency appears in the middle of the RF period: Ω PSR,max = ω PSR,max ω = 1 ω PSR,max s L ω pe (2.26) Equation 2.26 is an order of magnitude formula for the maximum PSR frequency. Equation 2.24 can be solved analytically as discussed in reference [48]. The essential point for the self-excitation of non-linear PSR oscillations is the quadratic sheath non-linearity q 2 in equation If ε = 1 (total symmetry) the non-linearity

24 18 CHAPTER 2. FUNDAMENTALS q 2 vanishes and a simple harmonic oscillator equation in q results. Then q oscillates like φ without higher harmonics. Therefore, self-excited PSR oscillations due to quadratic non-linearities cannot occur in symmetric discharges. In symmetric CCRF discharges there is still a series resonance corresponding to a series of two linear capacitors, an inductance, and a resistance [54 57, 59, 6]. However, this resonance will only be excited, if the discharge is driven at the resonance frequency. Oscillations of the current at frequencies higher than the driving frequency cannot be excited, since the quadratic non-linearities of both sheaths necessarily cancel out in symmetric discharges [2]. In case of a completely asymmetric discharge (equation 2.22) the δ-function represents the external force driving the damped non-linear oscillator. This means, that the sudden change of sign of j is the major source for the excitation of the PSR oscillations. With increasing discharge symmetry the DC self bias gets smaller and the δ-function changes to a smooth function with finite width and height. Physically this means, that both the sheath collapse and expansion happen more slowly and the PSR oscillations are excited less strongly. PSR oscillations of the RF current are theoretically known to enhance ohmic and stochastic heating (see chapter 2.4) in CCRF discharges by Non-Linear Electron Resonance Heating (NERH) [46, 47, 49 53]. Under resonant conditions this can lead to higher electron densities and etch rates [58]. Experimental investigations of NERH in a geometrically asymmetric single frequency discharge will be performed in this work (chapter 4.2). Furthermore, NERH in geometrically symmetric, but electrically asymmetric discharges will be investigated by a PIC simulation (chapter 4.4.6). In ICP discharges operated in E-mode non-linear self-excited PSR oscillations are also observed. PSR oscillations disappear, when the discharge jumps into H-mode. Thus, the occurence/disappearance of PSR oscillations in ICP discharges can be used as an identifier for the E- to H-mode transition [61]. In this work self-excited non-linear PSR oscillations in hybrid CCP-ICP RF discharges will be investigated experimentally in chapter 4.5. Experimentally PSR oscillations of the RF current can be detected by a small current sensor implemented in the side wall of the discharge chamber, that picks up a fraction of the RF current. Such sensors are commercially used as non-intrusive diagnostic and process monitoring technique, that detects small changes of the electron neutral collision frequency and, therefore, changes of the gas composition or density. The diagnostic is known as Self-Excited Electron Resonance Spectroscopy (SEERS) [43 45].

25 2.4. ELECTRON HEATING IN CAPACITIVE RF DISCHARGES Electron heating in capacitive RF discharges Figure 2.9: Electron effective collision frequency ν eff (theory and experiment) and electron-neutral collision frequency ν as a function of neutral gas pressure in a mercury vapor CCRF discharge [62]. In CCRF discharges the voltage drop across the discharge is the sum of applied RF voltage and DC-self bias (equation 2.17). In low pressure discharges, such as investigated in this work, most of the voltage drops across the sheaths and only a small fraction of the voltage drops across the bulk. The sheath voltage drops across a short distance of spatially non-uniform ion density (see figure 2.3). Therefore, in the sheath the electric field is high (typically several hundred V/cm) and nonuniform ( E ). In the plasma bulk the electric field is much lower (typically a few 1 1 V/cm) and can be approximated to be spatially uniform. Deviations from a spatially uniform electric field profile occur near the sheaths, since the plasma density decreases towards the walls. As the current has to be the same everywhere in the discharge at a given time and since the current is purely conduction current in the bulk, the electric field, that drives the current, must be higher in regions of low density. In a spatially uniform electric field, that oscillates perpendicular to the electrodes harmonically in time, E(t) = Re (E e iωt ), electrons periodically gain and loose energy in the absence of collisions without any net energy gain. Such an electric field is similar to the field in the plasma bulk. Here only the direction perpendicular to the electrodes is considered. In the direction parallel to the electrodes the electric field is assumed to be zero. Generally (including collisions), the time averaged power

26 2 CHAPTER 2. FUNDAMENTALS per unit volume absorbed by electrons in such a spatially uniform harmonically oscillating electric field is: p ohm = 1 T T j tot (t) E(t)dt (2.27) here T is the duration of one RF period and j tot = ε E/ t + enu is the total current density, which is the sum of displacement and conduction current density. The electron velocity u e is obtained by solving the equation of motion for electrons: du e (t) m e = ee(t) m e ν m u e (2.28) dt here ν m is the electron-neutral collision frequency. Equation 2.28 is solved for ( u e (t) = Re e ) 1 E e iωt (2.29) m e iω + ν m Substitution of equation 2.29 into equation 2.27 yields: p ohm = 1 2 E 2 Re (σ p ) = 1 2 j 2 Re (σ p ) 1 (2.3) here σ p = n e e 2 /m e (ν m + iω) is the plasma conductivity and j = σ p E. Equation 2.3 shows, that in such a spatially uniform harmonically oscillating electric field power can be transferred to electrons only via collisions (if ν m =, p ohm = ). Via elastic electron-neutral collisions energy gained by an electron in the oscillating field is transferred into a direction perpendicular to the field and is not lost during the reversal of the electric field. This mechanism to transfer energy to electrons is called ohmic heating and mainly takes place in the bulk [2, 63]. In 1985 Popov and Godyak accurately measured the total RF power dissipated in a CCRF discharge in mercury vapor as a function of neutral gas pressure by measuring the applied voltage, discharge current and phase shift between current and voltage [62]. If the heating of electrons was purely due to ohmic heating, one could calculate an effective collision frequency ν eff from the dissipated power, if the current is known (similar to equation 2.3). This effective collision frequency would have to be identical with the electron neutral collision frequency ν. The result is shown in figure 2.9 (circles - experiment). The actual electron-neutral collision frequency was calculated from the neutral gas pressure and assumed to be independent of the electron mean energy (solid line in figure 2.9). Obviously, at low pressures there is a big difference between ν eff and ν. Consequently, at low pressures there must be another collisionless heating mechanism, that transfers energy to electrons effectively. Besides collisional ohmic heating net energy can also be transferred to electrons by collisionless interactions of electrons with the spatially non-uniform temporally oscillating electric fields in the plasma sheaths. Godyak and Lieberman adopted the idea of Fermi Acceleration - a concept originally proposed by Fermi [64] to explain the generation of cosmic rays - to describe the interaction of electrons with the moving sheath edge as a collision of a particle with an infinitely massive moving

27 2.4. ELECTRON HEATING IN CAPACITIVE RF DISCHARGES 21 Laboratory frame: Before collision After collision n n e= n i n e= ni n n e= n i n e= ni u u r u s e - e - u s s(t) x s(t) x Frame, where the sheath is at rest: Before collision After collision n n e= n i n e= ni n n e= n i n e= ni u = u - us u =-u + u r s e - e - s x s x Figure 2.1: Collision of an electron with the moving sheath edge in the hard wall model. wall. When the electron collides with the collapsing sheath, it looses energy, whereas it gains energy, when it collides with the expanding sheath. As the probability for collisions of an electron with the expanding sheath is higher, net energy is transferred to electrons by this mechanism. This concept is called Hard Wall Model [62, 63, 65] and illustrated in figure 2.1. Assuming an initial electron velocity u, the velocity u r after the collision, is calculated by moving to a frame, in which the sheath edge is at rest (see figure 2.1). In this frame the electron s energy is conserved resulting in the same absolute velocity after the collision (opposite direction). Transferring back to the lab frame one obtains: u r = u + 2u s (2.31) Here u s is the velocity of the sheath edge. The number of electrons per unit area, colliding with the sheath in a time interval dt and a velocity interval du, is (u u s )f s (u, t)dudt. Here f s (u, t) is the normalized electron velocity distribution function at the sheath edge. The resulting power transfer ds stoch per unit area is: ds stoch = 1 2 m e(u 2 r u 2 )(u u s )f s (u, t)du (2.32) Substitution of equation 2.31 into equation 2.32 and integration of the resulting equation yields:

28 22 CHAPTER 2. FUNDAMENTALS S stoch = 2m e u s u s (u u s ) 2 f s (u, t)du (2.33) In the frame of a homogeneous discharge model [63] various simplifying assumptions are made to solve equation If a spatially uniform plasma density, a Mawellian EEDF and a sinusoidal sheath movement are assumed, the time averaged power per unit area transferred to electrons via stochastic heating is: S stoch = 1 2 m e ū e e 2 n e j 2 (2.34) here ū e is the mean electron velocity and j is the current density. According to equation 2.3 the time averaged power per unit area transferred to electrons via ohmic heating is: S ohm = 1 m e ν m l j 2 e 2 2 (2.35) n e here l is the bulk length. For a symmetric discharge (2 sheaths) the sum of the time averaged power per unit area transferred to electrons via ohmic and stochastic heating is: m e S tot = 1 j 2 e 2 2 (ν m l + 2ū e ) = 1 n e 2 m e lν eff e 2 n e j 2 (2.36) ν eff = ν m + 2ū e (2.37) l is an effective collision frequency, which comprises an additional term independent of collisions, that represents stochastic heating (the electron bounce frequency 2ūe ). l Popov and Godyak [62] calculated this effective collision frequency from measurements of the electron mean energy (ν eff - theory in figure 2.9). Their calculations reproduce the measured effective collision frequency, which is substantially higher than the electron neutral collision frequency at low pressures. These measurements clearly demonstrate the relevance of stochastic heating at low pressures. In the homogeneous discharge model used to solve equation 2.33 various assumptions were made, that do not exactly hold in a realistic discharge: (i) the density is nonuniform due to acceleration of ions towards the electrode, (ii) the EEDF is usually non-maxwellian, (iii) the sheath motion is non-sinusoidal due to the ion density profile in the sheath and/or non-linear effects (see chapter 2.3). In order to calculate the stochastic heating power more correctly equation 2.33 was solved in the frame of a self-consistent sheath model by Wood et al. neglecting non-linear effects for u s u e and u s u e. The result is an enhancement of the stochastic heating compared to the homogeneous model [66]. According to equation 2.36 at high pressures ohmic heating is dominant, whereas at low pressures stochastic heating is the most important heating mechanism. At intermediate pressure there is a heating mode transition, that was verified experimentally by Godyak et al. [67 7]. At low pressures the EEDF is Bi-Maxwellian,

29 2.4. ELECTRON HEATING IN CAPACITIVE RF DISCHARGES 23 i.e. the high energy tail is enhanced by stochastic heating, with overall low mean electron energy. At high pressures the EEDF is Druyvesteyn-like with overall high electron mean energy. These measurements were recently verified theoretically by a hybrid fluid-monte Carlo simulation of Heil et al. [16, 71]. With the same simulation Heil et al. calculated the EEDF with and without sheaths using the same electric fields in both cases. The result is that the sheaths have a net cooling effect, i.e. with sheaths the mean electron energy is lower. This is caused by losses of energetic electrons, which cross the sheath and are lost to the electrodes [72]. The loss of energetic electrons to the electrodes is not included in the original Hard Wall model. In the same work it was demonstrated that the Hard Wall model describes the interaction of energetic electrons with the sheath correctly as long as these electrons are not lost to the electrodes. However, the interaction of low energetic electrons with the sheaths is not described correctly. But low energetic electrons are usually confined in the discharge by the ambipolar field and do not substantially interact with the sheaths. The exact mechanism of collisionless heating is not completely understood yet and an important topic of current research. Besides the Hard Wall Model alternative concepts to describe collisionless heating in CCRF discharges have been proposed: Turner and Gozadinos et al. proposed that electrons are heated by pressure effects. Within a fluid theory they demonstrated that electrons are compressed on one side of the discharge, when the sheath expands, and rarefied on the other side, where the sheath collapses at the same time. Due to finite electron thermal conductivity this pressure heating transfers energy to electrons, which is on average not zero, but positive [73 75]. Pressure heating does not result from a direct interaction of electrons with the sheath and is not a fluid representation of stochastic heating described by the Hard Wall model, but a distinct mechanism. Thus, a combination of both mechanisms might describe collisionless heating correctly. Kaganovich et al. performed kinetic studies of stochastic heating in CCRF discharges [76 8]. Mussenbrock and Brinkmann et al. demonstrated theoretically that non-linear selfexcited PSR oscillations, such as described in chapter 2.3, drastically enhance both ohmic and stochastic heating in geometrically strongly asymmetric CCRF discharges operated at low pressures [46, 47, 49 51]. They suggested that this Non-linear Electron Resonance Heating (NERH) should be included in models describing electron heating. Ziegler et al. investigated the temporal structure of NERH in the frame of a global model [52, 53]. NERH is investigated experimentally and by a PIC simulation in this work (see chapters 4.2 and 4.4.6). Using a PIC simulation Vender et al. investigated the effect of electron heating in a geometrically symmetric hydrogen single frequency discharge (f = 1 MHz) on the spatio-temporal ionization at low pressures of 2.66 Pa (see figure 2.11). Dark regions in figure 2.11 correspond to high ionization and white regions correspond to low ionization (in the sheath regions). During phases of sheath expansion the generation of highly energetic electron beams was observed, that penetrate into the plasma bulk. During sheath collapse additional ionization located at the sheath edge and caused by a local electric field reversal was observed [81, 82].

30 24 CHAPTER 2. FUNDAMENTALS Figure 2.11: Spatio-temporal ionization (PIC simulation) in a single frequency discharge operated in hydrogen at 2.66 Pa (2 mtorr), φ 1 = 1 kv and f = 1 MHz [81, 82]. Wood investigated the generation of highly energetic electron beams during sheath expansion in more detail using a PIC simulation of a geometrically symmetric argon discharge operated at.4 Pa (3 mtorr), MHz, and φ 1 =.5 kv (figure 2.12). At low pressures, if the electron mean free path is long enough, electron beams generated by the expanding sheath on one side propagate through the entire discharge and are reflected back into the bulk by the opposing sheath (figure 2.13). Depending on the phase, when the beam hits the opposing sheath, resonance heating was observed. The generation of electron beams by the expanding sheath at low pressures and the spatio-temporal ionization/excitation in various types of CCRF discharges will be studied experimentally in this work (see chapters 4.1 and 4.2.4).

31 2.4. ELECTRON HEATING IN CAPACITIVE RF DISCHARGES 25 Figure 2.12: Spatio-temporal electron velocity in a geometrically symmetric single frequency discharge operated in argon at.4 Pa (3 mtorr) with 1 cm electrode gap. Only electrons with kinetic energies above 15.7 ev are counted [83]. Belenguer and Boeuf studied the ionization space and phase resolved in a single frequency geometrically symmetric helium discharge operated at 4 Pa and 3.2 MHz by a fluid model [84]. Figure 2.14 shows spatio-temporal plots of the ionization at different amplitudes of the applied RF voltage resulting from their model calculations. At low voltage ionization is observed during sheath expansion and collapse. At high voltage amplitudes a third ionization mechanism occurs at phases of maximum sheath voltage: When the voltage drop across a sheath is high, secondary electrons are created by ions impinging on the electrode and/or UV radiation at the electrode surface and are accelerated out of the sheath by the high sheath potential. If the pressure is high and the sheath is big (high voltage amplitudes) secondary electrons are multiplied by ionizing inelastic electron neutral collisions inside the sheath. The number of secondary electrons created per ion impact at the electrode strongly depends on the electrode material and surface conditions. The secondary electron emission coefficient γ describes the number of secondary electrons generated per ion impact. Typically γ ranges between.1 and.4 [85, 86]. If ionization due to sheath expansion and sheath collapse dominates the ionization, the discharge is operated in α-mode. If secondary electrons dominate the ionization, the discharge is operated in γ-mode. Figure 2.14 shows the transition from α- to γ-mode operation with increasing amplitude of the applied RF voltage waveform at high pressures. It should be noted that there can be a substantial difference between the heating mechanism, that dominates the overall power dissipated to electrons in the discharge, and the heating mechanism, that dominates the ionization, i.e. sustains the discharge. A discharge can be operated in a regime, in which collisional ohmic heating dominates (at high pressures). However, the ionization can still be dominated by interaction of electrons with the sheath, i.e. stochastic heating. This is caused by the fact, that a lot of power is dissipated by ohmic heating, however, it is distributed

32 26 CHAPTER 2. FUNDAMENTALS Figure 2.13: Paths of beam electrons for electrode gaps of 13, 1 and 7.5 cm. The dotted lines correspond to the electron sheath edge [83]. to a lot of electrons, since the electron density is high in the bulk. Therefore, the energy gain per electron via ohmic heating is too low to cause ionization, although ohmic heating dominates. Vice versa at the sheath edge the electron density is typically one order of magnitude lower than in the bulk and, therefore, the energy gain per electron via stochastic heating can be much higher than the energy gain per electron via ohmic heating in the bulk, although the overall power dissipated via stochastic heating is lower. Experimentally the ionization/excitation dynamics in CCRF discharges can be investigated by Phase Resolved Optical Emission Spectroscopy (PROES). Using this diagnostic the electron impact excitation from the ground state into specifically chosen energy levels of probe gases is calculated out of the measured space and time resolved emission using a collisional radiative (CR) model (see chapter 3.1.1). Assuming similar cross sections for excitation and ionization the excitation then probes

33 2.4. ELECTRON HEATING IN CAPACITIVE RF DISCHARGES 27 Figure 2.14: Relative spatio-temporal ionization in a single frequency discharge operated in helium at 4 Pa, f = 3.2 MHz and different RF voltage amplitudes (plot (a): 12 V, plot (b): 25 V, plot (c): 4 V) [84]. the ionization. In this work it will be shown that this assumption is not generally true (see chapter 4.3.5). For the first time Makabe et al. investigated electron dynamics in a single frequency discharge by PROES [33, 87, 88]. They observed the ionization dynamics predicted by the simulations described above experimentally. In dual-frequency discharges operated at substantially different frequencies electron heating was investigated theoretically by Turner and Chabert [89 91] as well as Kawamura et al. [92]. Boyle et al. predicted the opportunity to separately control ion energy and flux in these discharges under certain conditions [27, 28, 93, 94]. Donkó and Petrović [24, 25] as well as Lee [26, 95] and Kitajima [29] investigated this separate control in more detail. Excitation/ionization dynamics in dual-frequency discharges are investigated experimentally and using PIC simulations for the first time within this work [TG6, JS7b, JS7b, ZD9a, UC9, JS9a, JS9b]. Excitation/ionization dynamics in geometrically strongly asymmetric single-frequency discharges operated at low pressures and NERH are also investigated for the first time in this work [JS8b, BGH8, JS8a, JS8d, JS8c, JS7a, ZD9a, ZD9b].

34 Chapter 3 Diagnostics and simulations 3.1 Diagnostics In this work various diagnostics are used to get access to a variety of plasma parameters essential for the understanding of electron heating in different types of CCRF discharges with high spatial and temporal resolution. Each of these diagnostic techniques is introduced shortly in this section Phase Resolved Optical Emission Spectroscopy Figure 3.1: Principle of Phase Resolved Optical Emission Spectroscopy [JS8b]. In order to measure the emission from specifically chosen rare gas levels space and phase resolved a fast-gateable (gatewidth of a few ns) ICCD camera (Andor Istar and Roper PI-MAX) with a high repetition rate ( 2 khz) is synchronized with 28

35 3.1. DIAGNOSTICS 29 the applied RF voltage waveform (see figure 3.1). In case of phase locked dual frequency discharges the camera is synchronized with the low frequency RF voltage waveform. The emission is observed through an adequate optical filter with two dimensional spatial resolution line integrated in the direction of the line of sight [JS8b, TK5]. In order to perform phase resolved measurements the internal delay generator of the ICCD camera sets a certain delay between the trigger and the camera gate. The signal is acquired at a certain phase during several thousands of RF periods. Then the delay is increased and the next phase is scanned. As only CCRF discharges with homogeneous emission parallel to the electrodes are investigated, all images are binned in horizontal direction in order to reduce the noise resulting in one dimensional spatial resolution along the discharge axis. Typically a temporal resolution of about 5 ns and a spatial resolution of about.5 mm are achieved. From the measured spatio-temporal emission the electron impact excitation from the ground state E,i (t) is calculated space and time resolved to eliminate the influence of the lifetime of the observed level. The population dynamics of an excited state i with a population density n i is described by the following rate equation: dn i (t) dt = n E,i (t) + m n m E m,i (t) + c A ci n c (t) A i n i (t) (3.1) Here n is the population density of the ground state. The term n m E m,i (t) represents excitation from metastable levels m of population density n m. In this context E m,i (t) is the electron impact excitation function for excitation from the metastable level m into the observed level i. The term A ci n c (t) describes additional population of level i due to cascades from higher levels c. n c is the population density of the respective cascade level and A ci the decay rate for transitions from the cascade level c into level i. A i is an effective decay rate, that takes into account reabsorption of radiation and radiationless collisional de-excitation, so called quenching: A i = 1 τ = k A ik g ik + q k q n q (3.2) Here τ is the effective lifetime of level i and A ik is the transition probability of spontaneous emission from level i to k. In equation 3.2 reabsorption of radiation is included by introducing escape factors g ik, which reflect the probability of one photon originating from the transition from level i to k to leave the plasma without being reabsorbed. Reabsorption plays an important role in case of the overpopulated ground state. In comparison to other optically thin transitions, the effective transition rates into the ground state can be neglected (g ik = ). In general, reabsorption can reduce the effective decay rate or increase the effective lifetime of one state resulting in an apparent metastable state. Quenching is represented by the sum of all products of the density n q of all collision partners and the corresponding quenching coefficients k q. In a de-excitation process, caused by quenching, energy is not lost by radiation, but is transferred into energy of both collision partners. Thus,

36 3 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS quenching increases the effective decay rate A i, reducing the lifetime and emission from a certain level i [96, 97]. Equation 3.1 is part of a system of coupled differential equations comprising rate equations for all cascade and metastable levels. Furthermore, atomic and molecular data like quenching coefficients and decay rates need to be known. Thus, the determination of the electron impact excitation function E,i (t) is generally difficult. In order to simplify equation 3.1 specific levels must be chosen, for which some population channels can be neglected. The criteria that must be fulfilled are the following: 1. low population due to cascades 2. low population due to excitation from metastable levels 3. knowledge of optical transition rates 4. enough intensity 5. no superposition with other emission lines 6. short lifetime in order to temporally resolve the RF period (typically 74 ns) 7. influence of quenching is generally low One energy level, that fulfills these criteria well, is Ne 2p 1. It has a short lifetime of only 14.5 ns [98], which allows access to excitation dynamics within the RF period of typically about 74 ns. The threshold energy for electron impact excitation from the ground state is 19 ev. The contribution of cascades to the population of this C level is particularly low (see table 3.1). In table 3.1, denotes the contribution D of cascades in relation to direct excitation of the corresponding state as they were determined from electron beam experiments [99 11]. The influence of quenching is generally low under the low pressure conditions investigated in this work. Neglecting cascade contributions and excitation out of metastable states the rate of electron impact excitation from the ground state E,i (t) is determined from the measured emission [12]: ( ) E i, (t) = 1 dṅ ph,i (t) + A i ṅ ph,i (t) (3.3) A ik n dt with ṅ ph,i (t) = A ik n i (t) (3.4) Here ṅ ph,i (t) is the measured number of photons per unit volume and time. n is unknown and, therefore, only relative values of the excitation rate are calculated. Using equation 3.3 a spatio-temporal excitation matrix is calculated from the measured spatio-temporal emission matrix.

37 3.1. DIAGNOSTICS 31 The three rare gas states mainly used for PROES in this work are listed in table 3.1. The observed lines fulfill the criteria described above. The excitation thresholds of these states cover an energy interval from 11.7 ev to 19. ev. Therefore, PROES is only sensitive to highly energetic electrons. However, these electrons are particularly interesting, since they cause ionization and sustain the discharge. State λ [98] E [98] τ A ik C D [nm] [ev] [ns] [ 1 s ] Kr 2p [12] [11] Ar 2p [98] [1] Ne 2p [98] [99] Table 3.1: Table of emission lines, that are used for the investigation of electron dynamics, including the characteristic data of each line. The diagnostic of PROES was used by many authors to study electron dynamics in CCRF discharges. For the first time de Rosny et al. performed PROES measurements and found, that the electron impact excitation into specifically chosen atomic energy levels is time modulated within the RF period in a CCRF discharge operated at MHz [13]. Various other authors then found different excitation maxima within the RF period in different types of single frequency CCRF discharges caused by, a.o., the expanding and collapsing sheath, secondary electrons, and heavy particle collisions [33, 87, 88, 12, ]. PROES measurements in dual frequency discharges were performed by the author of this thesis [JS7b, TG6], [114] as well as by O Connell et al. [115]. A detailed description of the PROES method can also be found in reference [JS9d]. PROES is based on time dependent measurements of the population densities of specifically chosen excited rare gas states. A time dependent model, based on rate equations, describes the dynamics of the population densities of these levels. First order cascade contributions can be included [18, 114], [JS7b]. Based on this model and the comparison of the excitation of several different rare gas states, with different excitation thresholds (trace rare gas optical emission spectroscopy) several plasma parameters can be determined space and phase resolved [116]: 2E E i, (t) = n e σ i (ɛ) f e (ɛ)dɛ (3.5) m e Here σ i (ɛ) is the energy dependent electron impact excitation cross section for excitation from the ground state into level i and f e (ɛ) is the EEDF. If E i, (t) is known from this model for different energy levels, parameters such as the electron temperature, electron density as well as electron energy distribution functions can be determined based on equation 3.5, which then corresponds to a set of equations for a certain number of unknowns (plasma parameters). For this analysis an adequate ansatz for the EEDF depending on the individual discharge must be made and the cross sections for electron impact excitation from the ground state must be

38 32 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS accurately known for all energy levels involved. For rare gases such as argon, neon, krypton, and xenon these cross sections are provided by Lin et al. [99 11]. The number of energy levels, that must be compared, depends on the number of plasma parameters to be determined, which in turn depends on the ansatz for the EEDF, e.g. Maxwellian EEDF, Bi-Maxwellian EEDF, etc. Details of this technique are described elsewhere [18, 114], [JS7b] Fluorescence Dip Spectroscopy in krypton nd'[3/2] nd'[5/2] 5p' 2 [3/2] 2 E= E= Stark-split Rydberg states nm (n = 7...5) 826 nm fluorescence two-photon excitation 2x 24 nm I fl 5s' 2 [1/2] 1 measured spectrum: 1 S ground state dip Figure 3.2: Excitation scheme in krypton. Using a laser the intermediate state 5p 2 [3/2] 2 is excited by two photons out of the ground state. The fluorescence light from the transition to 5s 2 [1/2] 1 is depleted, when the second step excitation is resonant with a (Stark-split) Rydberg state. In order to measure electric fields in the sheath of CCRF discharges the Starksplitting of high Rydberg states of krypton was investigated experimentally and theoretically [TK7], [117]. Using the resulting Stark maps the electric fields within the sheath of a geometrically strongly asymmetric single frequency discharge are measured by Fluorescence Dip Spectroscopy (FDS) space and time resolved within the RF period in this work (see chapters and 4.2.3). The technique is applied to krypton as a probe gas for the first time. Similar methods for electric field measurements have already been developed and applied to other molecular and atomic probe gases like BCl [32, 118], NaK [119], CS [12], He [16, ], H [16, 117, 124, 125], Ar [117, ] and Xe [131, 132]. Knowledge of the field allows the determination of voltages, charge densities and currents.

39 3.1. DIAGNOSTICS 33 A small admixture of krypton as a probe gas can be used to measure electric fields in arbitrary gas mixtures. This will be demonstrated in this work. Using krypton as a probe gas has several advantages compared to other gases: (i) Krypton is an inert rare gas and, therefore, little intrusive. This is a strong advantage compared to molecular gases such as H 2 and N 2. (ii) The first excitation step is an excitation out of the ground state and not from a metastable state such as for helium and argon. In krypton the laser wavelength used for the first excitation step is nm. In helium and argon excitation out of the ground state requires vacuum UV wavelengths. Thus, excitation from metastable states is used for these gases. If the excitation starts from a metastable state, a calibration without plasma will not be possible, since the metastable states will not be populated. However, the calibration must be performed at known electric fields. Therefore, metastables must be produced by a remote plasma source [128]. Furthermore, metastables are quenched effectively at high pressures. The excitation scheme used for the FDS measurements performed in this work is shown in figure 3.2. The 1 S to 5p 2 [3/2] 2 transition is pumped at nm (two-photon excitation). The fluorescence from the 5p 2 [3/2] 2 state to the 5s 2 [1/2] 1 state at 826 nm is monitored. A second tunable dye laser is used to excite from the 5p 2 [3/2] 2 state to high Rydberg states, which are shifted due to the Stark effect [TK7]. Scanning the wavelength yields the dip spectrum. The frequency-doubled output beam (532 nm) of a seeded Nd:YAG laser (Continuum, 92) with a pulse width of 7 ns and pulse energy of 9 mj at a repetition rate of 2 Hz is split by a ratio of 2:1 to pump two tunable double-grating dye lasers (Radiant Dyes, Narrow Scan). The first one emits a fundamental wavelength of nm, which is frequency-tripled for the two-photon excitation step (24.13 nm, 3 mj). The fundamental of the second dye laser ( nm) is focused into a tube filled with 1 bar of H 2 gas, converting it into the desired wavelength range of nm by Stimulated Anti-Stokes Raman Scattering at approximately 15 µj. A filter and dielectric mirrors block both the fundamental and Stokes component. The second laser beam is switched on and off by a mechanical shutter, so that drifts of the intensity (on the timescale of seconds) can be compensated. Both beams are guided collinearly into a modified GEC reference cell. The Q-switch of the Nd:YAG pump laser is synchronized with the RF generator (13.56 MHz) using a frequency divider (2 Hz) and a delay generator (Stanford, DG-535). Moreover, the gate of an ICCD camera (Princeton Instruments) is synchronized with the laser pulse. The variable delay between Q-Switch and RF generator allows phase resolved measurements within the RF period. The spatial resolution of this diagnostic is about 5 µm. The temporal resolution corresponds to the temporal width of the laser beam of about 7 ns. A wavelength scan of the second dye laser over a certain wavelength interval with a stepwidth of 3 pm is performed in case of every measurement at a certain phase. The wavelength interval is chosen corresponding to the respective Rydberg state and its potential shift. The fluorescence light is accumulated on the CCD for several seconds at each wavelength of the second dye laser and two images are taken, one with the shutter being open and one with the shutter being closed.

40 34 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS An edge filter in front of the camera is used, that blocks all wavelengths below 78 nm. Then a difference image is calculated in order to determine the fluorescence decrease. Pixels in the horizontal direction are binned in order to reduce the noise. The spatial resolution is achieved by using a laser beam that covers 2 mm of the sheath and by moving the entire discharge chamber up and down leaving the laser beams at their original position. Using this technique the entire sheath region is scanned. Finally, the wavelength positions of the dips are identified at each spatial position and at each phase, respectively. The resulting shifts are then compared to the database of Stark shifts of Rydberg states in krypton [TK7] and the electric fields are determined spatially and temporally resolved. Different Rydberg levels with different field sensitivities are used depending on the electric field strength in the sheath at a given spatial position and phase (see table 3.2). For these measurements only nd -levels are used because of their distinctive dips [TK7] d'[5/2] d'[5/2] [cm -1 ] d'[3/2] 2,[5/2] 2 [cm -1 ] d'[3/2] 2,[5/2] E [V/cm] E [V/cm] d'[5/2] d'[5/2] 3 [cm -1 ] d'[3/2] 2,[5/2] 2 [cm -1 ] d'[3/2] 2, [5/2] E [V/cm] E [V/cm] Figure 3.3: Measured (rectangles) and numerically calculated (dotted lines) energy shifts of four exemplary nd [5/2] 3 levels. The calculated intensities are indicated by the sizes of the grey circles [TK7]. Figure 3.3 shows the energy shifts of four examplary nd [5/2] 3 levels. The intensity is represented by the size of the grey circles [TK7]. The intensity is high at low

41 3.1. DIAGNOSTICS 35 electric fields and decreases with increasing electric field. Depending on the principal quantum number each nd state starts to be shifted at different threshold electric field strengths. Due to the decreasing intensity each nd state can only be used for measurements of electric fields within a distinct sensitivity range. Table 3.2 shows which nd levels are used in which range of electric fields in the sheath. As the field sensitivity ranges of some Rydberg states overlap, the field measurements could be cross-checked using different Rydberg states for measuring electric fields within the small range of overlapping field sensitivities. Rydberg state Electric field [V/cm] 15d d d d d d d d 8-15 Table 3.2: nd levels and respective field sensitivities used for measurements of the electric field in the sheath of a geometrically asymmetric single frequency discharge in this work.

42 36 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS Langmuir probe measurements Current [ma] I-V Characteristic Second Derivative d 2 I/dV 2 [ma V -2 ] Probe Bias [V] -.1 Figure 3.4: Current-voltage characteristic measured in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated in krypton at 1 Pa and 8 W using a Scientific System Smart-Probe. The probe tip is located 2.5 cm above the powered electrode at a radial distance of 4 cm away from the discharge center. Langmuir probes - named after their inventor Irving Langmuir [133, 134] - are one of the oldest and most commonly used plasma diagnostics. A Langmuir probe is essentially a conducting wire biased at a variable voltage, which is put into the plasma. The current drawn from the plasma is measured as a function of the bias voltage. The result is a current-voltage characteristic. Detailed information on Langmuir probes can be found elsewhere [ ]. An example for such a current-voltage characteristic and its second derivative is shown in figure 3.4. This measurement was performed in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated in krypton at 1 Pa and 8 W using a commercial Scientific System Smart-Probe. The probe tip is located 2.5 cm above the powered electrode at a radial distance of 4 cm away from the discharge center. The probe is equipped with a linear drive unit to perform radially resolved measurements at a given vertical distance from the powered electrode. The electron density, electron mean energy and electron energy distribution function (EEDF) are then obtained temporally averaged and radially resolved from the measured current-voltage characteristic, which can be divided into three different regions: Ion saturation current: At strongly negative bias voltages almost no electrons reach the probe. The measured current is purely ion current.

43 3.1. DIAGNOSTICS 37 Electron retarding region: With increasing voltage more electrons reach the probe. In this region the electron current increases exponentially. The floating potential φ f is the potential, at which electron and ion current balance, i.e. at which zero net current flows to the probe. The plasma potential φ b is the potential of the plasma, at which electrons are no longer repelled from the probe. Electron saturation current: At voltages higher than the plasma potential electrons are accelerated towards the probe. In principle a Langmuir probe acts as an energy filter for electrons, if a voltage scan is performed and the current is measured as a function of the applied voltage. Although the practical realization of Langmuir probe measurements is relatively simple, a complex model for the determination of the EEDF is required. In this work the Druyvesteyn method [14] to determine the EEDF is used. If the EEDF is Maxwellian, i.e. if the electrons are in thermal equilibrium, the Boltzmann relation for the electron density in the probe sheath holds: n e (U) = n,e e eu k B Te (3.6) Here U is the probe voltage and n,e is the electron density in the bulk. In this case the electron current I e in the electron retarding region is: I e = 1 4 en ea 8kB T e πm e e eu k B Te (3.7) here A is the surface area of the probe. On a semi-logarithmic scale the electron current yields a straight line according to equation 3.7. From the slope the electron temperature can be determined. Knowing the electron temperature the electron density can be determined from the vertical position of this line. This technique is, however, only applicable if the EEDF is Maxwellian. Another way to determine the electron temperature in case of a Maxwellian EEDF is the integration of the current-voltage characteristic from the floating potential φ f to the plasma potential φ p. Within this voltage interval the electron retarding current increases exponentially in case of a Maxwellian EEDF: I e = en ea 8kB T e e e(u φp) k B Te (3.8) 4 πm e Integration of equation 3.8 from the floating potential to the plasma potential yields: φp φ f I e du = en ea 8kB T φp e 4 πm e φ f e e(u φp) k B Te du = en ( ea 8kB T e k B T e 1 e 4 πm e e e(φ f φp) k B Te ) en ea 8kB T e k B T e 4 πm e e (3.9)

44 38 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS In equation 3.9 the exponential function is neglected, since usually e(φ f φ p ) k B T e. Using equation 3.9 the electron temperature can be calculated: k B T e = ( en e A 4 8kB T e πm e ) 1 φp φ f I e du (3.1) Knowing the electron temperature the electron density can be calculated using equation 3.8. This technique is applied by the Smart Probe Software to automatically calculate electron density and electron temperature from the measured current-voltage characteristics. However, this technique is not applicable in case of non-maxwellian EEDF such as observed in this work. Using the Druyvesteyn method the EEDF, electron density and electron mean energy can be determined without the assumption of a specific shape of the EEDF. However, several other assumptions are made: The electron mean free path is larger than the width of the sheath adjacent to the probe, i.e. the probe sheath is collisionless. This assumption is well justified in the low pressure CCRF discharges investigated in this work. The DC voltage between probe and plasma drops entirely across the probe sheath, i.e. the plasma is not disturbed by the probe. All electrons that reach the probe are absorbed by the probe, i.e. no electrons are reflected and no secondary electrons are generated. The probe is cylindrical with a radius that is much smaller than the probe length, i.e. electrons moving parallel to the probe axis do not contribute to the electron current. In the electron retarding region the electron conduction current density j e = I e /A can be expressed by the integral over the electron velocity distribution function (EVDF) g(u e ) in spherical coordinates: Θmin 2π j e = e u 3 eg(u e ) cos Θ sin ΘdφdΘdu e (3.11) u min Here u e cos Θ is the projection of the electron velocity u e on the direction perpendicular to the probe surface. cos Θ min = u min /u e is the relation between minimum electron velocity u min and minimum azimuthal angle Θ min. Electrons with velocities perpendicular to the probe surface smaller than u min do not reach the probe. Integration of equation 3.11 with respect to φ and Θ yields: ( ) j e = eπ u 3 e 1 u2 min g(u u min u 2 e )du e (3.12) e Introduction of a change of variable ɛ = m 2e u2 e under the assumption of an isotropic EVDF and expressing u min by the potential difference between probe and plasma V = φ p U yields:

45 3.1. DIAGNOSTICS 39 j e = 2πe3 m 2 V ( ɛ 1 V ) g [u e (ɛ)] dɛ (3.13) ɛ Differentiating equation 3.13 twice with respect to V yields the Druyvesteyn formula [14]: d 2 I e dv 2 = 2πe3 m 2 Ag [u e(ɛ)] (3.14) According to equation 3.14 g [u e (ɛ)] can be directly determined from the second derivative of the measured current voltage characteristic. The EEDF f e (ɛ) is defined by: f e (ɛ)dɛ = 4πv 2 g(u e )dv (3.15) ( ) 3/2 2e f e (ɛ) = 2π ɛ 1/2 g [u e (ɛ)] (3.16) m e Using equation 3.16 to eliminate g from equation 3.14 yields: f e (V ) = 2m ( ) 1/2 e 2e V 1/2 d2 I e (3.17) e 2 A m e dv 2 According to equation 3.17 the EEDF f e (ɛ) is not directly proportional to the second derivative of the measured current voltage characteristic, but depends additionally on V 1/2. The so called electron energy probability function f p (ɛ) (EEPF) is now introduced: f p (ɛ) = V 1/2 f e (ɛ) (3.18) The EEPF is directly proportional to the second derivative of the measured current voltage characteristic. For a Mawellian EEDF f e (ɛ) = n e 2 (πk B T e ) 3/2 ɛ1/2 e ɛ/k BT e (3.19) the corresponding EEPF is a straight line on a semi-logarithmic plot. From the slope of this line the electron temperature can directly be determined. For identifying a specific kind of EEDF (Maxwellian, bi-maxwellian, Druyvestian, etc.) the EEPF is much more convenient to use than the EEDF, since different kinds of distribution functions can be distinguished more easily. It should be noted that an integration of the EEPF (typically in units of cm 3 ev 3/2 ) with respect to energy does not yield the electron density. Only an integration of the EEDF (typically in units of cm 3 ev 1 ) yields the electron density. For the Druyvesteyn method to determine the EEDF only the electron current is needed. The probe, however, measures the sum of ion and electron current. Therefore, the ion current must be subtracted [141].

46 4 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS In this work a Scientific Systems Smart Probe is used for measurements of plasma parameters in a geometrically strongly asymmetric single frequency discharge operated at low pressures. The probe system automatically calculates the EEPF using the Druyvesteyn method. The electron density and electron temperature are automatically calculated from the current-voltage characteristic under the assumption of a Maxwellian EEDF using the algorithm described before. As the EEDF is non- Maxwellian under the conditions investigated the electron density and electron mean energy were calculated manually. From the measured EEPF the EEDF is calculated according to equation Then the electron density n e and electron mean energy < ɛ > are determined by integration of the EEDF: n e = f e (ɛ)dɛ (3.2) < ɛ >= 1 ɛf e (ɛ)dɛ (3.21) n e The resulting electron densities and electron mean energies are used as input parameters for a hybrid fluid-monte Carlo model of CCRF discharges (see chapter 3.2.1). If uncompensated Langmuir probe measurements are performed in symmetric CCRF discharges, the measured current-voltage characteristics will be distorted by the RF modulation of the plasma potential, since the potential drop between probe and plasma will not be temporally constant. In strongly asymmetric CCRF discharges the plasma potential is temporally constant and this problem does not occur. In symmetric discharges such a distortion is caused by the non-linearity of the current voltage characteristic in the electron retarding region. In order to avoid such distortions the probe potential needs to follow the RF modulations of the plasma potential. This can be achieved either by a passive [67, 142, 143] or active [67, 144] RF compensation of the probe. In case of a passive RF compensation an electrode located inside the plasma close to the probe tip samples the RF oscillations of the plasma potential and couples this potential oscillation capacitively to the probe wire. As a result the probe potential oscillates with the plasma potential and the voltage drop between probe and plasma is constant at all times. Typically directly behind the probe tip LC parallel resonant circuits filter out voltage oscillations at the fundamental RF frequency and its first harmonics. One filter is used for filtering out each harmonic. In case of an active compensation the RF voltage is sampled at the electrode. The phase and amplitude of the sampled signal are then adjusted such that it reproduces the modulation of the plasma potential. The probe is then biased with this signal.

47 3.1. DIAGNOSTICS Measurements of ion energy and ion flux Figure 3.5: Schematic setup of the Plasma Process Monitor 422 [145]. In this work ion fluxes and ion flux-energy distribution functions are measured in a geometrically symmetric electrically asymmetric dual frequency discharge by a Balzer Plasma Process Monitor 422 (PPM)[145]. The PPM 422 is a combination of an ion energy and an ion mass filter. Its setup is shown schematically in figure 3.5. The PPM can either be used to measure ion flux-energy distribution functions at fixed ion mass or to measure ion mass spectra at fixed ion energy. Here the PPM is only operated in the former mode, i.e. as an energy filter at fixed ion mass. For the investigations performed in this work the PPM is connected to the grounded electrode of a geometrically symmetric CCRF discharge through an extraction hole of diameter 1µm. The extraction hole is grounded. Some ions accelerated towards the grounded electrode by the sheath potential enter the PPM through the extraction hole and pass the ionization chamber (see figure 3.5). The ions are focused on the entrance of the energy filter by a system of ion optical lenses [145]. The PPM can also be used to detect neutral particles. In the neutral particle detection mode the neutrals entering the PPM through the extraction hole are ionized by electron impact ionization inside the ionization chamber. A hot biased filament in the ionization chamber provides the energetic electrons required for ionizing the neutral particles. In the ion detection mode applied in this work the filaments are not used. Figure 3.6 shows the schematic setup of the Cylindrical Mirror Analyser (CMA) used for ion energy filtering inside the PPM 422. It consists of two axially symmetric cylinders with radii 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 inner and outer cylinder, respectively. Ions entering the CMA at energy ɛ i through the entrance hole F 1 leave the CMA through the exit hole F 2 only if the following criterion is fulfilled:

48 42 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS Figure 3.6: Schematic setup of the Cylindrical Mirror Analyser (CMA) used for ion energy filtering inside the Plasma Process Monitor 422 [145]. K = eɛ i U p ln(r b /r a ) (3.22) K depends on the angle between the filter axis and the ion trajectory at its entry into the filter. The maximum energy resolution and the best focusing is achieved, if ions enter the CMA at and K = In case of the PPM 422 r b /r a = 15/7 and the citerion for ions to pass through the filter is: ɛ i = 1.72 U p (3.23) The CMA is set to the transition energy ɛ i = U a and equation 3.23 yields: U b U a = 2.4 (3.24) By scanning the voltage U a at a fixed ratio U b /U a = 2.4 ion flux-energy distributions are measured. The resolution of these measurements is about.3 ev [145]. Figure 3.7: Schematic setup of the Quadrupole Mass Analyser (QMA) used for filtering ions of a specific mass to charge ratio inside the Plasma Process Monitor 422 [145].

49 3.1. DIAGNOSTICS 43 After passing the CMA energy filter the ions enter a Quadrupole Mass Analyser (QMA) used to filter out ions at a specific mass to charge ratio. The schematic setup of the QMA is shown in figure 3.7. The standard QMA consists of four parallel molybdenum rods with 8 mm in diameter and 2 mm in length. Two oppositely placed rods are biased positively (+U), whereas the other two are biased negatively ( U) with U = U + U cos (ωt), where U and U are constants and ω is the RF frequency used for the QMA. The ion motion trajectory along the axis of the QMA is described by the Mathieu-equations [146]. Only ions with an appropriate ratio of mass to charge achieve stable trajectories in the high frequency field and pass the mass filter. In this way ions with specific mass to charge ratios are filtered out of the incoming ion beam. Behind the QMA the remaining ion beam is reflected by 9 and detected by a Secondary Electron Multiplier (SEM). In this work single positively charged ions at constant mass of 4 amu (argon ions) are filtered out by the QMA. At this fixed mass energy scans are performed using the CMA. The internal pressure of the PPM must be lower than 1 3 Pa in order to avoid collisions of ions within the PPM that would distort the measurements. This reduces the maximum pressure in the chamber to about 2 Pa. At this chamber pressure the pressure inside the PPM is reduced to about 1 3 Pa by the PPM internal pumps. The PPM is calibrated carefully with respect to its energy scale as well as the shape of the measured ion flux-energy distribution functions. The energy scale is calibrated by creating ions at known energy (1 ev) in the ionization chamber of the PPM by setting the potential of the ionization chamber to 1 V. Low extraction voltages (1 V) and low emission currents (1 ma) are used to avoid space charges and potential shifts [145]. To ensure a correct shape of the distribution function all potentials of the ion optics focusing the incoming ion beam into the energy analyzer are initially switched off. Therefore, only ions moving collinearly to the discharge axis are detected. Under these conditions in a Hydrogen plasma H 3 + ions (dominant ion species) were detected. Although the measured intensity is very low, the shape of the distribution function is not affected by the focusing optics and is, therefore, correct. Then the potentials of the ion focusing optics are optimized to maximize the signal intensity, but still maintain the original shape of the distribution function (measured with all potentials set to zero). The calibration procedure required to ensure a correct shape of the measured ion flux-energy distribution functions is described in detail elsewhere [147].

50 44 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS Current and voltage measuremts The RF current flowing through the plasma and the RF voltage drop across the discharge are key quantities for understanding basic discharge dynamics as well as for modeling and simulating CCRF discharges. In this work the RF voltage drop across the plasma is measured by a LeCroy high voltage probe, that measures the potential in front of the powered electrode relative to ground. The amplitudes and - in case of dual frequency discharges operated at two similar phase locked frequencies - the relative phase between the individual harmonics of the measured RF voltage waveforms are used as input parameters for PIC simulations (see chapter 3.2.2) and for a hybrid fluid-monte Carlo simulation (see chapter 3.2.1) of CCRF discharges. In case of a geometrically strongly asymmetric single frequency discharge the RF current density to the grounded chamber wall is measured by a SEERS current sensor (Plasmetrex) [43 45] integrated into the grounded chamber wall. This sensor picks up a fraction of the overall RF current. The resulting RF current waveforms are used for the analytical PSR model described in chapter 2.3 as well as for calculations of the surface charge density at the powered electrode and the sheath motion in a geometrically strongly asymmetric single frequency discharge. In principal the RF current could also be measured directly in front of the powered electrode in the plasma bulk by a Rogowski coil. This method was tested in the frame of this work. A Rogowski coil was put into a circular glass tube, which was inserted into the plasma. The inner part of the tube was under atmospheric pressure. However, the presence of the glass tube strongly disturbed the plasma and lead to plasma instabilities [JS8e]. Therefore, in this thesis only current measurements performed by a SEERS sensor are discussed.

51 3.2. SIMULATIONS Simulations In order to obtain a more detailed insight into the physical mechanisms of electron heating in various types of CCRF discharges most experimental results are compared to simulations. Two different kinds of simulations are used, a hybrid fluid-monte Carlo simulation [16, 71] and a Particle in Cell simulation complemented with a Monte Carlo treatment of collision processes (PIC/MCC) [24, 25], [ZD9a, ZD9b, JS9a, JS8a]. The hybrid fluid-monte Carlo simulation is performed by Heil and the PIC simulation is performed by Donkó. The former is used to investigate electron dynamics in geometrically strongly asymmetric single frequency discharges. Using the PIC simulation electron heating in geometrically symmetric dual frequency discharges is investigated. Both are introduced in this section Hybrid fluid-monte Carlo simulation Figure 3.8: Overall numerical scheme of the hybrid fluid Monte-Carlo simulation [16]. The hybrid fluid-monte Carlo simulation consists of three different modules: A boundary sheath module, an equivalent circuit module, and a Monte-Carlo module (see figure 3.8). The boundary sheath and equivalent circuit modules are closely intertwined and were both developed by Brinkmann [17, 148]. Both modules iteratively calculate the electric field due to displacement current in the sheath and the Fourier components of the RF current space and time resolved. The resulting fields and RF current are then used as input parameters for the Monte-Carlo module, which calculates the ohmic electric field, the field due to conduction current, and

52 46 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS the EEDF space and time resolved. The simulation is not self-consistent. Plasma parameters are not calculated based on particle and energy balance equations, but based on input parameters. Besides the discharge geometry, neutral gas species, -pressure and -temperature as well as the applied voltage waveform, the electron temperature and plasma density in the bulk need to be specified by the user in advance. The plasma density in the bulk is assumed to be constant. The uncompensated net positive charge in the discharge is assumed to be temporally constant, i.e. charge dynamics such as discussed in chapter cannot occur in the simulation. In its current form the simulation considers only one atomic gas species and singly charged ions. In the following part of this section every module will be introduced shortly. The fluid sheath module The fluid sheath module calculates the electric field in the sheath based on 4 simplified fundamental equations: The ion continuity equation, the ion momentum balance equation, Poisson s equation, and the electron momentum balance equation. The ion density profile is assumed to be stationary ( n i / t =, see equation 2.4) and ionization in the sheath is neglected. Under these assumptions the ion continuity equation is reduced to: n i u i = Ψ i (3.25) where Ψ i is the constant ion flux, which is calculated by an analytical diffusion model in advance. The ion momentum balance equation is simplified by assuming that the ions only react to the time averaged electric field < E > and that the pressure tensor can be replaced by an isotropic pressure based on the ion temperature T i : u i u i z = e m i < E(z) > k BT i m i 1 n i n i z ν i(u i )u i (3.26) here ν i (u i ) is the velocity dependent collision frequency for momentum transfer. In case of the electron momentum balance equation it is assumed, that the electrical force on the electron fluid is compensated by the pressure gradient, i.e. all inertia and collision terms as well as magnetic fields are neglected in the momentum balance equation. Furthermore, the pressure tensor is replaced by an isotropic pressure based on the electron temperature T e. Under these assumptions the electron momentum balance equation yields the Boltzmann relation for the electron density: n e = n,e e eφ k B Te (3.27) Neglecting electromagnetic effects [23] the one dimensional Poisson equation is valid: 2 φ z 2 = E z = e ε (n i n e ) (3.28)

53 3.2. SIMULATIONS 47 Equations form the basis of the sheath module. Due to the approximations made in the frame of their derivation certain effects cannot be observed in this simulation. As electron inertia and collisions are neglected in the electron momentum balance equation 3.27, the ohmic electric field - the field due to conduction current - does not result from the sheath module. The ohmic electric field is calculated by the Monte-Carlo module. For the same reason electric field reversals during sheath collapse caused by either electron inertia and/or collisions of electrons with the neutral gas cannot be observed in the frame of the fluid sheath module in its present implementation. However, as will be shown below, the calculation of the ohmic electric field as a correction to the fluid sheath model yields terms, that lead to a field reversal. Using two coordinate changes [149] and Brinkmann s matched asymptotic expansion around the sheath edge s(t), defined by equation 2.2, to approximate a solution of the Boltzmann-Poisson equation (combination of equations 3.27 and 3.28) the electric field can be calculated [17] taking into account the first two corrections to the Hard Wall Model [62, 63, 65] resulting from this matched asymptotic expansion. The temporally averaged electric field is substituted into equation Using equation 3.25 the ion density is replaced by the ion velocity and the ion flux to obtain the main equation, which is solved for the ion velocity [15, 151]. In order to calculate the ion velocity Q = Idt is needed. Q is obtained from the equivalent circuit module. Using equation 3.25 the ion density is calculated from the ion velocity. Then the time averaged electric field and the charge voltage relation φ(q, Q) are calculated. Using equations 3.27 and the expression for the electric field [17] the time resolved electron density and electric field are obtained, respectively. The problem is, that the equivalent circuit module requires the charge voltage relation of the sheath as input parameter to calculate I and finally Q. The charge voltage relation is calculated by the sheath module, which in return requires Q from the equivalent circuit module as an input parameter. Thus, the sheath and equivalent circuit modules are used iteratively to calculate the RF current and the sheath charge voltage relation. The equivalent circuit module In the equivalent circuit model an equation similar to equation 2.17 is solved for the electron conduction current I: L p di(t) dt with + RI(t) + φ sg ( ) ( Idt + φ sp ) Idt + R ap I(t) + R ag I(t) φ RF (t) = (3.29) L p = m el πn e e 2 r p r g (3.3)

54 48 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS R = m elν m πn e e 2 r p r g (3.31) R ap,ag = 2 S stoch j 2 c A (3.32) Here R ap,ag are anomalous sheath resistances, which are used in order to take into account stochastic heating, S stoch is the time averaged power per unit area transferred to electrons via stochastic heating defined by equation φ sg ( Idt ) and φsp ( Idt ) are the charge voltage relations for the sheath adjacent to the grounded and powered electrode, respectively. Both are calculated by the sheath module, which uses Q resulting from the equivalent circuit module as an input parameter. The Monte Carlo module The Monte Carlo module uses the electric fields and the RF current calculated by the sheath and equivalent circuit module as input parameters. The EEDF and EVDF are constructed by dividing the electron phase space into a series of cells and determining the time each electron spends in a particular cell. Elastic hard sphere collisions and inelastic collisions are included. A correction to the electric field resulting from the sheath module is calculated. The ohmic electric field - the field due to conduction current - is not obtained from the sheath module, since electron inertia and collisions are neglected in the electron momentum balance equation. In the Monte Carlo module the ohmic electric field E ohm,b in the bulk, where the current is purely conduction current, is calculated based on the electron momentum balance equation neglecting the inertia term ( u ) u [16, 71]: E ohm,b (z, t) = m ( ) e jc e 2 n e t + ν cj c (z, t) (3.33) In the sheath the current is not purely conduction current, but partly or totally displacement current. Therefore, equation 3.33 cannot be applied in the sheath. Using a multi timescale analysis a corrected form of equation 3.33 is determined, that is valid in the sheath and is used to calculate the ohmic electric field in the sheath E ohm,s [16, 71]: E ohm,s (z, t) = m ( ) e n e (z, t) jc e 2 n i (z) n i (z) t + ν cj c (z, t) (3.34) As discussed in [16, 71], in the frame of an expansion of the electric field in terms of ω RF /ω pe the ohmic electric field corresponds to a first order correction of the zero order electric field resulting from the fluid sheath model. As will be shown in chapter this first order correction corresponds to all terms that yield a reversed field except the one related to the neglected non-linear inertia term ( u ) u in the electron momentum balance equation.

55 3.2. SIMULATIONS 49 Integration of the space and time resolved EEDFs resulting from the Monte Carlo module above a certain energy threshold yields the density of electrons with energies above this threshold. For a threshold of 12 ev this integration qualitatively corresponds to spatio-temporal excitation profiles such as measured by PROES using energy states with a threshold for excitation similar to 12 ev. In chapter measured spatio-temporal excitation profiles in a geometrically strongly asymmetric single frequency discharge are compared to the EEDFs resulting from the Monte Carlo module above 12 ev Particle in Cell simulation Assign charges to grid positions Monte Carlo: Check for collisions, and new particles Calculate electric field at grid points (Poisson equation) time Check for boundaries: remove particles (secondary emission) Weight field to particle positions (calculate forces) Advance particles (equation of motion) new velocities and positions Figure 3.9: Principle of PIC simulations complemented by a Monte-Carlo treatment of collision processes [152]. The principle of a PIC simulation is shown in figure 3.9: A spatial grid with a division of about z λ D is defined. The time step t is chosen in accordance with ω pe t <.2 to resolve electron oscillations at the electron plasma frequency and to ensure that electrons do not cross more than one grid point per t. Additionally an electrode gap d is defined. A voltage waveform is applied to one of the electrodes and a few particles are initially randomly distributed between the electrodes. At each time step the charge of each particle is assigned to the nearest two grid points

56 5 CHAPTER 3. DIAGNOSTICS AND SIMULATIONS weighted by the distance. Solving Poisson s equation the electric potential is calculated from the charge density ρ. Solving the particle equations of motion with the electric fields weighted by the distance to the particle position yields the new particle velocities and positions. At each time step t collisions are taken into account via a Monte-Carlo method [152]. Then the time is increased by t, and the same procedure starts again. This technique is continued for typically a few thousand RF periods until the number of superparticles remains approximately constant, i.e. until the simulation has converged. The PIC simulation code used in this work was developed by Donkó. It is a one-dimensional (1d3v) bounded plasma particle-in-cell model, complemented with Monte Carlo treatment of collision processes (PIC/MCC). The electrodes are assumed to be infinite, planar and parallel. One of the electrodes is driven by a voltage corresponding to equation 2.1. Here investigations are restricted to geometrically symmetric dual-frequency discharges. The simulations are carried out both for helium and argon gases. The collision processes considered for electrons are elastic scattering, excitation of background gas atoms to several energy levels, as well as ionization. The cross sections for these processes have been taken from [153] and [86], respectively, for helium and argon. For the positive ions elastic collisions with the gas atoms have been divided into an isotropic and a backward part [154]. The cross sections for these collisions have been taken from [154, 155]. Excitation and ionization from metastable levels are neglected. The number of superparticles in the simulations is typically 1 5, which is expected to provide acceptable accuracy, although in the case of argon slight dependence of the results on the number of superparticles has been observed even in this range [156]. Donkó s studies show that simulations of helium discharges (helium shows no Ramsauer effect) are less susceptible to this problem. The electrodes reflect the electrons arriving at their surface with a pre-defined probability, usually taken to be α =.2. The emission of secondary electrons from the electrodes is also taken into account. It will be demonstrated in this work, that secondary electrons can play an important role in dual-frequency discharges [157], [JS9a]. Typical secondary yields used in this work range between γ =.1 and γ =.45, which are reasonable for Si electrodes [85]. The simulations provide the spatio-temporal distributions of several discharge characteristics. The distribution of the calculated electron impact excitation is in direct correspondence with the light intensity distributions observed by PROES. In the simulation a total excitation rate is calculated using the sum of several cross sections for excitation into individual levels. In this way the calculated excitation rate does not correspond to the excitation into a real energy state, but into a notionally lumped state. Other quantities, like the space- and time-resolved electric field, particle velocities, and currents as well as EEDFs, aid the interpretation of the experimental data and the understanding of the discharge physics.

57 Chapter 4 Results In this chapter electron heating in different types of CCRF discharges is discussed based on experimental as well as simulation and model results. The diagnostics and simulations introduced in chapter 3 are combined to get access to a variety of plasma parameters crucial for the understanding of electron heating and mechanisms of plasma sustainment. This synergistic approach is complemented with different analytical models to isolate and identify the essential physics causing individual phenomena in capacitive RF discharges. The chapter starts with a discussion of electron heating in geometrically symmetric single frequency discharges, which is well understood based on simulation results of Vender [81, 82], Wood [83], as well as Belenguer and Boeuf [84]. Experimental results verifying their simulations, particularly the generation of beams of highly energetic electrons by the expanding sheath and their reflection at the opposing sheath at low pressures predicted by Wood [83], will be presented. In the following sections electron dynamics in increasingly more complex and less well understood discharges will be discussed: In the second section electron heating in geometrically strongly asymmetric single frequency discharges operated at low pressures will be investigated with particular focus on self-excited non-linear PSR oscillations of the RF current [43 45, 48] and the recently theoretically predicted NERH [46, 47, 49 53]. Such discharges operated under low pressure conditions are frequently used for technological applications [2], since a strong DC self bias is generated by the geometric asymmetry, which is used to achieve highly energetic vertical ion etching. In the third section classical geometrically symmetric dual-frequency discharges operated at two substantially different phase locked frequencies are investigated with particular focus on frequency coupling and different modes of electron heating. Such discharges are frequently used for technological applications to separately control ion energy and ion flux at the wafer surface [27, 28, 93, 94]. Limitations of this separate control caused by the frequency coupling and similarities as well as differences to α- and γ-mode operation of single frequency discharges will be outlined. The fourth section focuses on a novel type of CCRF discharge, a geometrically symmetric dualfrequency discharge operated at two similar frequencies - a fundamental and its second harmonic - with fixed, but variable phase shift between the harmonics. The 51

58 52 CHAPTER 4. RESULTS EAE - theoretically predicted by Heil et al. [1] - will be verified experimentally. It will be demonstrated that separate control of ion energy and flux at the electrodes can be achieved in an almost ideal way via the EAE. The effect will be optimized and it will be shown, that PSR oscillations and NERH can be switched on and off as a function of the electrical asymmetry in geometrically symmetric discharges. In the fifth section electron heating in geometrically asymmetric hybrid CCP-ICP discharges will be discussed again with focus on coupling effects and separate control of ion energy and flux. 4.1 Electron heating in geometrically symmetric single frequency discharges In this section electron dynamics in geometrically symmetric single frequency discharges operated at either MHz or MHz is investigated by PROES (see chapter 3.1.1). Simulation results of Belenguer and Boeuf [84] as well as Vender [81, 82] and Wood [83] will be verified experimentally. It will be demonstrated that beams of highly energetic electrons are generated by the expanding sheath at both electrodes. With decreasing pressure these electron beams propagate further into the plasma bulk and are reflected at the opposing sheath, if the mean free path for electrons is long enough. The results presented in this section are the basis of later investigations of more complicated CCRF discharges Experimental setup Glass cylinder Optical filter Bulk plasma Grounded electrode Powered electrode High voltage probe ICCD Camera Matching Amplifier Function generator Sync. (PROES) Figure 4.1: Experimental setup: Geometrically symmetric single frequency capacitive RF discharge (13.56 MHz/27.12 MHz). Figure 4.1 shows the experimental setup. A function generator (Agilent 3325A)

59 4.1. GEOM. SYMMETRIC SINGLE FREQUENCY DISCHARGES 53 is used to generate a low amplitude RF voltage waveform (13.56 MHz or MHz). The voltage waveform is amplified by a broadband amplifier to amplitudes of about 9-33 V, matched and applied to the bottom electrode located in a vacuum GEC reference cell [13]. A grounded electrode is located at a distance d from the powered electrode. The radius of both stainless steal electrodes is 5 cm. The electrode gap d is variable. Measurements are performed with d = 1 cm, d = 2.5 cm, and d = 5 cm. The plasma is shielded from the outer grounded chamber walls by a glass cylinder. Therefore, the discharge is geometrically symmetric. The voltage drop across the plasma relative to ground is measured by a LeCroy high voltage probe in front of the powered electrode. The RF period average of this voltage waveform yields the DC self bias. The emission from a specifically chosen neon state (Ne2p 1 ), that fulfills the criteria listed in chapter (see table 3.1), is measured space and phase resolved at nm by either an Andor Istar or a Roper PI-MAX ICCD camera synchronized with the RF voltage waveform in combination with an interference filter. The RF frequency is reduced to 2 khz by a frequency divider, since the maximum repetition rate of both cameras is 2-3 khz. The discharge is operated in different mixtures of argon and neon. In case of the Andor Istar camera the minimum gate width is 4.2 ns. Measurements of the emission in a MHz discharge (d = 1 cm) are performed with 5 ns gate width using this camera. In a MHz discharge (d = 5 cm) as well as a MHz discharge (d = 2.5 cm) measurements are performed with the Roper camera at the minimum gate width of 2 ns. Images are taken at different phases within the RF period with a step width equal to the gate width. Typical exposure times are about 1 s. The resulting images are binned in horizontal direction and combined to an emission matrix providing one dimensional spatial resolution perpendicular to the electrodes of about.5 cm. From the emission the excitation is calculated by equation Excitation dynamics Figure 4.2 shows the spatio-temporal excitation resulting from the measured spatiotemporal emission in a geometrically symmetric single frequency discharge operated at MHz, 95 Pa, 2 W, and an electrode gap of 1 cm. Due to similar cross sections for excitation and ionization the excitation is assumed to probe the ionization. Figure 4.3 shows the measured voltage drop across the discharge. As the discharge is symmetric under these conditions, the DC self bias is zero and the voltage drop across the discharge corresponds to the applied RF voltage waveform according to equation In the simulations of Belenguer and Boeuf [84] a geometrically symmetric discharge is operated in helium at 4 Pa, while the experiment is operated mainly in argon at much lower pressure of 95 Pa. However, due to the significantly higher cross section for electron-neutral elastic collisions in argon compared to helium [15] at energies typical for electrons causing excitation (E 15 ev) the electron mean free path under the conditions investigated experimentally here is even shorter compared to the conditions used in the simulation. Under the assumption of a neutral gas temperature T g 3 K (room temperature) the electron

60 54 CHAPTER 4. RESULTS Figure 4.2: Spatio-temporal excitation into Ne2p 1 in a geometrically symmetric single frequency discharge operated in argon with 25 % neon admixture at MHz, 95 Pa, 2 W, and d = 1 cm U [V] t [ns] Figure 4.3: Voltage drop across the discharge measured by a high voltage probe in front of the electrode under the same conditions. mean free path for electron-neutral elastic collisions at such high electron energies is λ Ar,95 P a, 25 cm in Argon at 95 Pa and λ He,4 P a, 35 cm in Helium at 4 Pa. Due to the discharge symmetry similar, but 18 phase shifted (φ sp (ωt) = φ sg (ωt+

61 4.1. GEOM. SYMMETRIC SINGLE FREQUENCY DISCHARGES 55 π) for symmetric sheaths), excitation patterns are observed in front of the powered and grounded electrode, respectively: Shortly after the phase of sheath collapse, i.e. when the sheath expands rapidly, strong excitation is observed. This excitation is caused by highly energetic electrons accelerated by the expanding sheath. Due to the high pressure of 95 Pa the electron mean free path is so short, that these highly energetic electrons do not penetrate far into the bulk. Thus, the excitation caused by sheath expansion heating is localized at the sheath edge. At the phase of maximum sheath voltage another less pronounced excitation maximum is observed. In analogy to previously performed simulations of the ionization dynamics [84] this maximum is caused by secondary electrons created by ion impact at the electrode surface. The γ-electrons are accelerated out of the sheath by the repelling sheath potential. At high pressure and RF voltage amplitudes secondary electrons are multiplied by collisions in the sheath. These experimental results verify the predictions of Belenguer and Bouef [84] (see figure 2.14). Under these conditions the discharge is operated in a transition mode between α- and γ-mode, at which sheath expansion heating dominates the excitation dynamics. However, secondary electrons also contribute to the excitation. In the simulations weak ionization during sheath collapse caused by a local electric field reversal was observed. Under the conditions investigated experimentally here no such excitation due to a field reversal is found. Czarnetzki et al. [16] developed an analytical model, that shows, that field reversals during sheath collapse at high pressures are caused by collisions of electrons with the neutral background gas. These collisions hinder electrons from following the fast collapsing sheath to fill up the ion matrix and lead to the generation of a field reversal. For a strong field reversal a large electron/ion current and a low electron mobility is required. Thus, the generation of a field reversal strongly depends on the choice of the processing gas. Detailed investigations of field reversals during sheath collapse are performed in chapters and Under such high pressure conditions collisional ohmic heating is the dominant heating mechanism. Nevertheless, the excitation/ionization dynamics are dominated by interactions of electrons with the boundary sheath. Most power is dissipated via collisional heating, but distributed to many electrons in the bulk. Therefore, each electron does not gain sufficient energy to cause excitation/ionization, i.e. to sustain the discharge, via collisional ohmic heating. Although the power dissipated by collisionless heating is lower, it is distributed to typically an order of magnitude fewer electrons. Thus, an individual electron gains enough energy to cause excitation/ionization. Figure 4.4 shows the measured spatio-temporal excitation in a geometrically symmetric single frequency discharge operated in neon at MHz, 4.5 Pa, 1 W, and an electrode gap of 2.5 cm. Figure 4.5 shows the measured voltage drop across the discharge. Although the electrode areas are identical, a DC self bias of η 32 V is generated. This DC self bias is caused by the bad aspect ratio between electrode radius r and gap d at d = 2.5 cm (r/d = 2) and the related capacitive coupling between the glass cylinder and the outer grounded chamber wall. This capacitive

62 56 CHAPTER 4. RESULTS distance from powered electrode [cm] t [ns] Exc. rate [a.u.] Figure 4.4: Spatio-temporal excitation into Ne2p 1 in a single frequency discharge with equal electrode surface areas operated in neon at MHz, 4.5 Pa, 1 W, and d = 2.5 cm. The dotted lines indicate the sheath movement and the arrows indicate trajectories of highly energetic electron beams generated by the expanding sheath. U [V] t [ns] Figure 4.5: Voltage drop across the discharge measured by a high voltage probe in front of the electrode under the same conditions. The horizontal dashed line indicates the DC self bias. coupling effectively enlarges the grounded surface and, therefore, leads to an ad-

63 4.1. GEOM. SYMMETRIC SINGLE FREQUENCY DISCHARGES 57 ditional asymmetry. At d = 1 cm (r/d = 5) this additional asymmetry is greatly reduced (see figure 4.3). Such an effective asymmetry of geometrically symmetric discharges was observed before by Coburn et. al. [38] and modeled by Lieberman and Savas [41]. However, the asymmetry is relatively small ( η = η/ φ RF 28 %) and two well-pronounced sheaths develop, one adjacent to each electrode. The sinusoidal motion of the sheath edge is indicated by dotted lines in figure 4.4. In contrast to the high pressure conditions of figure 4.2 the sheath is significantly larger, since the ion density in the sheath is lower. The maximum sheath width at the powered and grounded electrode is estimated from the spatio-temporal excitation profiles to be about s p.87 cm and s g.77 cm, respectively. These low pressure conditions are similar to conditions investigated by Vender [81, 82] and Wood [83] in PIC simulations of geometrically symmetric CCRF discharges (see figures 2.11, 2.12, 2.13). Here these simulations are verified experimentally for the first time. During phases of sheath expansion the generation of beams of highly energetic electrons, that penetrate into the plasma bulk, is observed at each electrode, respectively. In contrast to the high pressure conditions investigated before, the mean free path for energetic electrons at these low pressures is long enough, so that the excitation is not purely localized at the sheath edge, but a trajectory of a highly energetic electron beam is observed. These trajectories are indicated by dashed arrows in figure 4.4. Under these conditions the electron mean free path is λ Ne 4.5 Pa 2 cm, which agrees well with the overall length of the beam trajectory in figure 4.4. After about one mean free path the beam diverges due to collisions and dephasing of faster and slower electrons in the beam. However, the energy of the beam electrons is not lost: The randomization of the directions of the individual beam electrons velocities corresponds to a non-local heating of electrons. Due to the slight discharge asymmetry the excitation caused by the electron beam originating from the sheath expansion at the powered electrode is slightly stronger than the excitation caused by the electron beam originating from the sheath expansion at the grounded electrode. In order to sustain the discharge at even lower pressures to increase the electron mean free path the discharge must be operated at larger electrode gaps at a given power due to Paschen s curve. Figure 4.6 shows the spatio-temporal excitation into Ne2p 1 in a single frequency discharge with equal electrode surface areas operated in neon at MHz, 2 Pa, 14 W, and d = 5 cm. Figure 4.7 shows the measured voltage drop across the discharge under the same conditions. As the electrode gap is bigger, the parasitic capacitive coupling between glass cylinder and grounded outer chamber wall causes a stronger negative DC self bias of η 25 V compared to the conditions of figure 4.5. Nevertheless, there is still a well pronounced modulated sheath adjacent to the grounded electrode. The maximum sheath widths are about s p 1.85 cm and s g 1.1 cm, respectively. Due to the longer electron mean free path of λ Ne 2 Pa 5 cm, the beam electrons generated by the sheath expansion at the bottom electrode propagate further and hit the collapsing sheath at the other side. Similar to Wood s simulation results (see figure 2.13) the beam is reflected back into the bulk by the opposing sheath. When the reflected beam hits the fully expanded

64 58 CHAPTER 4. RESULTS distance from powered electrode [cm] t [ns] Exc. rate [a.u.] Figure 4.6: Spatio-temporal excitation into Ne2p 1 in a single frequency discharge with equal electrode surface areas operated in neon at MHz, 2 Pa, 14 W, and d = 5 cm. The dotted lines indicate the sheath movement and the arrows indicate trajectories of highly energetic electron beams generated by the expanding sheath. U [V] t [ns] Figure 4.7: Voltage drop across the discharge measured by a high voltage probe in front of the electrode under the same conditions. The horizontal dashed line indicates the DC self bias. sheath at the powered electrode, it is reflected back again, before it is randomized by collisions after about one mean free path.

65 4.1. GEOM. SYMMETRIC SINGLE FREQUENCY DISCHARGES 59 Certainly, resonance effects of this Electron ping pong between two modulated sheaths depending on the electrode gap size could be observed such as predicted by Wood [83]. This might be an interesting topic for future experimental investigations.

66 6 CHAPTER 4. RESULTS 4.2 Electron heating in geometrically asymmetric single frequency discharges In this section electron (stochastic) heating including non-linear self-excited PSR oscillations of the RF current [43 45, 48] and the recently theoretically predicted NERH [46, 47, 49 53] in a geometrically strongly asymmetric single frequency discharge operated at low pressures is investigated. The results presented in this section can be found in references [JS8b, BGH8, JS8a, JS8d, JS8c, JS7a, TK7]. Various plasma parameters are investigated in detail applying different diagnostics, analytical models and simulations. Knowledge of the time and space resolved electric field in the sheath of CCRF discharges is essential for the understanding of basic mechanisms such as electron heating. It allows the determination of voltages, charge densities and currents. Furthermore, the field itself and its spatial as well as temporal evolution is the cause of electron stochastic heating. Therefore, a novel technique of FDS in krypton for space and phase resolved electric field measurements was developed [TK7] (see chapter 3.1.2) and is applied to a CCRF discharge for the first time. The results are compared to a fluid sheath model [17, 148] (see chapter 3.2.1). Furthermore, excitation dynamics, voltage, current and other plasma parameters such as electron density, electron mean energy and EEDF are studied under various conditions and compared to theoretical results. The focus lies on the comparison of two specific sets of conditions, for which detailed investigations applying all diagnostics are performed. Those are measurements in a geometrically strongly asymmetric krypton discharge at an intermediate pressure of 1 Pa, 8 W and at a low pressure of 1 Pa, 8 W, respectively. These two different pressures are chosen, since the PSR effect and NERH are more pronounced at low pressures. At 1 Pa PSR oscillations of the current start to be observable and at 1 Pa they are clearly pronounced. At higher pressures no high frequency oscillations of the RF current are observed. Thus, at high pressures (> 1 Pa) NERH does not play an important role, whereas it strongly contributes to electron heating at lower pressures. Krypton is chosen as gas, since the electric field in the sheath is measured space and phase resolved by FDS using krypton as probe gas. Special attention will be paid to electron beams generated by the expanding sheath and enhanced by the PSR effect. Their impact on the time averaged isotropic EEDF and their relation to stochastic heating will be studied. Finally, electric field reversals during sheath collapse in geometrically strongly asymmetric CCRF discharges will be investigated Experimental setup Figure 4.8 shows the experimental setup used for the investigation of electron (stochastic) heating in geometrically strongly asymmetric single frequency discharges operated at low pressures [JS8b]. The chamber is a modified standard GEC cell [14]. The metal cylinder surrounding the ICP antenna and the dielectric window are replaced by a monolithic quartz housing. This quartz cylinder acts as a floating surface in the discharge. Here the inductive coupling is not used. Similar to the

67 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 61 ICP Antenna Quartz Current sensor Bulk plasma Langmuir probe Laser beam High voltage probe ICCD Camera Optical filter Matching RF power supply Figure 4.8: Experimental setup: Geometrically strongly asymmetric GEC reference cell [14] including all diagnostics (FDS, PROES, Langmuir probe, current and voltage measurements). The arrows indicate the direction of the current flow [JS8b]. investigations performed in a geometrically symmetric discharge (see chapter 4.1) a low amplitude RF voltage waveform (13.56 MHz) is generated by a function generator, amplified, matched, and applied to the bottom electrode. The whole grounded chamber wall now acts as grounded electrode. Therefore, the discharge is strongly geometrically asymmetric (A p A g ) and almost the entire voltage drops across the sheath at the powered electrode. The electrode radius and gap between electrode and quartz are both 5 cm. Several diagnostics are applied: (i) FDS in krypton (chapter 3.1.2): Two laser beams enter the discharge collinearly to the bottom electrode from the side. An ICCD camera (Princeton Instruments) synchronized with the laser pulses in combination with an optical filter monitors the fluorescence light. Figure 4.9 shows an example of such an electric field measurement in an argon discharge with 1 % krypton admixture at the phase of maximum sheath expansion [JS8b]. It is important to note, that FDS in krypton can be performed in other gas mixtures than pure krypton discharges admixing only a small amount of krypton as a probe gas. (ii) PROES (see chapter 3.1.1): An ICCD camera (Andor Istar) is synchronized with the RF voltage waveform via a frequency divider. A transition from the Kr2p 5 -state at nm is observed using an optical interference filter. The lifetime of 21.5 ns is short enough to resolve electron dynamics within one RF period at MHz (T RF = 74 ns). (iii) Langmuir probe measurements (see chapter 3.1.3): In order to characterize the plasma in terms of electron density, electron mean energy, and EEDF a Langmuir probe (SmartProbe, Scientific Systems) is inserted into the discharge from the side. Radial scans through the entire discharge volume are performed 2.5 cm above the powered electrode well outside the sheath. (iv) Voltage measurements (see chap-

68 62 CHAPTER 4. RESULTS E [V/cm] ns Distance from powered electrode [mm] Figure 4.9: Space resolved electric field in the sheath of a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated in an argon (9 %) - krypton (1 %) mixture at 1 Pa and 8 W at the phase of maximum sheath expansion [JS8b]. ter 3.1.5): Simultaneously to every other measurement, the RF voltage relative to ground is measured at the output of the matching box directly in front of the powered electrode using a high voltage probe (LeCroy). (v) Current measurements (see chapter 3.1.5): The relative current to the chamber wall is measured simultaneously by a SEERS current sensor (Plasmetrex) [44] integrated into the side wall of the GEC cell (see figure 4.8).

69 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES Results at intermediate pressure (1 Pa) Current and voltage measurements I [a.u.] Pa t [ns] [V] Figure 4.1: Comparison between measured (blue line) and modeled (black line) current in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated in krypton at 45 Pa, 8 W. The measured RF voltage drop across the discharge (red line) is also shown [JS8b]. Figures 4.1 and 4.11 show the measured RF current to the chamber wall and the measured voltage drop across the discharge in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated in krypton at 45 Pa, 8 W and 1 Pa, 8 W [JS8b]. The measured currents are compared to the result of an analytical PSR model described in chapter 2.3 [48] using the parameters Ω and κ shown in the figure. At both pressures the measured current is well reproduced by the analytical model. The voltage drop across the discharge is sinusoidal at all pressures investigated. Due to the strong geometric discharge asymmetry (A p A g ) the DC self bias corresponds to a good approximation to the amplitude of the applied voltage waveform (see chapter 2.2) and the voltage drop across the discharge is mostly negative. The RF current to the wall is sawtooth-like without any high frequency oscillations at 45 Pa. At higher pressures the current is sinusoidal. At low pressures the current starts to deviate from this sinusoidal form, which is usually assumed in models of stochastic heating in CCRF discharges [63, 65, 73 75, 78, 8] except the recent works on NERH [46, 47, 49 53]. At 1 Pa some small high frequency oscillations superimposed on the sinusoidal RF current become visible. These oscillations are caused by the PSR effect. However, the pressure and, therefore, the collision frequency at 1 Pa is still quite high and the oscillations are damped fast. At lower pressure these oscillations become significantly stronger.

70 64 CHAPTER 4. RESULTS 1 Pa I [a.u.] -1 =.225 = [V] t [ns] -4-5 Figure 4.11: Comparison between measured (blue line) and modeled (black line) current in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated in krypton at 1 Pa, 8 W. The measured RF voltage drop across the discharge (red line) is also shown [JS8b]. Langmuir probe measurements n e [cm -3 ] 8x1 9 7x1 9 6x1 9 5x1 9 4x1 9 3x1 9 2x1 9 1x Radial distance from centre position [mm] Figure 4.12: Radially resolved measurement of the electron density in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge (krypton, 1 Pa, 8W) 2,5 cm above the electrode [JS8b]. Figures 4.12 and 4.13 show radially resolved profiles of the electron density and the electron mean energy in the plasma bulk of a strongly geometrically asymmetric

71 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 65 < > [ev] Radial Distance from centre position [mm] Figure 4.13: Radially resolved measurement of the electron mean energy in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge (krypton, 1 Pa, 8W) 2,5 cm above the electrode. The solid line is not a fit, but is only used for reference [JS8b]. single frequency (13.56 MHz) discharge operated in krypton at 1 Pa and 8 W 2.5 cm above the powered electrode [JS8b]. The maximum density is of the order of n e cm 3 and the electron mean energy is of the order of < ɛ > 3 ev. Both profiles show shapes typical for CCRF discharges. 1x1 1 EEDF*E -1/2 [cm -3 ev -3/2 ] 1x1 9 1x1 8 1x1 7 1x1 6 1x E [ev] Figure 4.14: Measured EEDF E 1/2 in the radial centre of a pure Kr discharge (1 Pa, 8W) 2,5 cm above the electrode [JS8b].

72 66 CHAPTER 4. RESULTS Figure 4.14 shows the measured EEDF E 1/2, in the radial center of the discharge [JS8b]. It has a Bi-Maxwellian shape typical for CCRF discharges at low pressures [67, 69 71, 135, 139], [BGH8]. The enhancement of the high energy tail can be interpreted as influence of stochastic heating, that is efficient at low pressures. Besides the mean energy of all electrons the electron temperature T c of the low energetic part of the EEDF and the electron temperature of the high energetic part T h can be determined out of the respective slopes of the EEDF E 1/2. In the radial center those temperatures are T c 1 ev and T h 5 ev. The Bi-Maxwellian shape of the measured EEDF is reproduced well by the hybrid fluid Monte Carlo simulation described in chapter As only qualitative comparisons are performed here, it is important to note, that Bi-Maxwellian EEDFs are observed in the experiment and the simulation. Electric field measurements E [V/cm] ns 21.5 ns 39.5 ns 57.5 ns Distance from powered electrode [cm] Figure 4.15: Spatio-temporal evolution of the electric field in the sheath of a geometrically strongly asymmetric single frequency (13.56 MHz) discharge at 1 Pa and 8 W in krypton [JS8b]. The markers are experimental (FDS) and the solid lines theoretical results (fluid sheath model). Figure 4.15 shows the spatial and temporal evolution of the electric field in the boundary sheath of a geometrically strongly asymmetric single frequency (13.56 MHz) discharge at 1 Pa and 8 W in krypton at the powered electrode [JS8b]. 4 different phases within the RF period are investigated. The electric fields in the sheath shortly after the collapse at 7.5 ns, during the phase of sheath expansion (21.5 ns), shortly after the phase of maximum sheath expansion (39.5 ns) and during the sheath collapse (57.5 ns) are shown. The maximum sheath width at the phase of

73 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 67 full sheath expansion can be identified to be s 1 Pa 5 mm. The phases of sheath expansion and collapse are clearly visible. In particular there is a big jump of the sheath edge between 7.5 ns and 21.5 ns. The solid lines in figure 4.15 are the result of the fluid sheath model (see chapter 3.2.1) [17, 148] using experimentally obtained input parameters such as pressure, applied voltage, electron density, and electron temperature. Good agreement between experiment and simulation is found. Following Poisson s equation a steeper decrease of the electric field in the sheath (n e = cm 3 ) corresponds to a higher ion density. Therefore, the ion density is lowest at the electrode and increases towards the sheath edge due to flux conservation. Close to the sheath edge highly energetic electrons can enter the sheath (n e cm 3 ) and the absolute value of the slope decreases again. However, in this range the electric fields are close to the sensitivity limit of about 5 V/cm. Qualitatively this agrees very well with the spatial shape of the electric field distribution shown in figure High V-Probe Measurement sp [V] t [ns] Figure 4.16: Comparison between measured RF voltage drop across the discharge (high voltage probe and sinusoidal fit) and sheath voltage determined out of the integration of the measured electric fields at different phases [JS8b]. In order to check the reliability of the electric field measurements the voltage drop across the discharge is measured at the output of the matchbox using a high voltage probe. Under these low pressure conditions the voltage drop across the bulk is negligible. Due to the strong geometric asymmetry there is essentially no voltage drop across the sheath at ground. Thus, the measured voltage drop corresponds to a good approximation to the voltage drop across the sheath at the powered electrode φ sp. The measured electric fields are integrated over the entire sheath at each phase, at which FDS measurements are performed to obtain the momentary sheath voltage from the field measurements. The comparison of both ways of determining the momentary sheath voltage is shown in figure 4.16 [JS8b] and very good agreement is found verifying the correctness of the field measurements.

74 68 CHAPTER 4. RESULTS 25 2 sp 1/2 [V 1/2 ] ,,2,4,6,8 1, 1,2 1,4 [1-6 C m -2 ] Figure 4.17: Relation between sheath voltage φ sp and surface charge density at the electrode σ at 1 Pa and 8 W [JS8b]. In the sheath (n e = cm 3 ) Poisson s equation is: de dz = e ε n i (4.1) Integration of equation 4.1 from to the momentary sheath edge s(t) yields: E el = e s(t) n i dz (4.2) ε where E el is the electric field at the electrode. The surface charge density σ at the electrode is defined as: σ = e s(t) n i dz (4.3) Substitution of equation 4.3 into equation 4.2 yields a direct relation between the electric field at the electrode and the surface charge density: σ = ε E el (4.4) Using equation 4.4 and the measured electric fields at the electrode at different phases, the surface charge density σ can be determined at different phases. The result is plotted against φ sp where φ sp is the momentary sheath voltage. The result is shown in figure 4.17 [JS8b]. Obviously, φ sp σ 2 (4.5)

75 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 69 This relation is an important fundamental assumption for the one dimensional PSR model described in chapter 2.3 (equation 2.16) [48] and has now been verified experimentally for the first time. Excitation dynamics Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] Figure 4.18: Spatio-temporal plot of the excitation into Kr2p 5 close to the powered electrode in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated at 1 Pa and 8 W in krypton. The arrow indicates the trajectory of a beam of highly energetic electrons, that is accelerated by the expanding sheath [JS8b]. Figure 4.18 shows the spatio-temporal excitation into Kr2p 5 close to the powered electrode under the same conditions as before [JS8b]. An absolute phase calibration between the FDS and PROES measurements is performed, so that the time scales in figures 4.15 and 4.18 are identical. Based on figure 4.18 the sheath width can be estimated to be 5 mm. This agrees well with the maximum sheath width s 1 Pa 5 mm determined from the electric field measurements. During the phase of fastest sheath expansion (7.5 ns ns, see figure 4.15) the most intense excitation is observed. At this phase a beam of highly energetic electrons is generated by the expanding sheath, that penetrates into the plasma bulk (dashed arrow in figure 4.18). During the time interval from 7.5 ns to 21.5 ns almost all electrons, that have been located in the sheath region before, are pushed out of the sheath by the increasing electric field. The maximum excitation does not occur directly at the sheath edge (5 mm), but after about one electron mean free path. Further inside the bulk the excitation decreases and the highly energetic directed beam electrons loose energy and their directed character through collisions. Similar phenomena have been observed in hydrogen [12, 18, 124] before. In case of the investigations performed in hydrogen, electron beams could not directly be observed,

76 7 CHAPTER 4. RESULTS since the pressure was too high and the beams were immediately stopped behind the sheath. However, from these investigations at low pressures, the nature of this excitation mechanism in terms of electron beams is now obvious. From the slope of the beam trajectory the electron drift velocity can be estimated. Under these conditions the drift velocity is v d,1p a m/s. Distance from powered electrode [cm] 2. Density of electrons above 12 ev [1 4 cm -3 ] time [ns] Figure 4.19: Integral of the EEDF above 12 ev as it results from the hybrid fluid Monte Carlo simulation (chapter 3.2.1) of a symmetric discharge in argon at 9.3 Pa and 6.7 cm electrode gap [16], [JS8b]. Figure 4.19 shows the result of the hybrid fluid Monte Carlo simulation [16] [JS8b] in terms of the integration of the calculated EEDF above 12 ev in a symmetric discharge in argon operated at 9.3 Pa [67]. Only the fraction of the discharge close to the bottom powered electrode is shown here. Similar to the experiment, trajectories of beams of highly energetic electrons are observed, that are generated by the expanding sheath. As the simulated discharge is symmetric, an electron beam is also generated during the sheath expansion phase at the grounded electrode 18 phase shifted in comparison to the beam generated at the powered bottom electrode (not shown in figure 4.19). In contrast to the experiment in the simulation a sinusoidal sheath motion is observed due to the fact that a symmetric discharge geometry is used. In the experiment the discharge is asymmetric and the PSR leads to very fast sheath expansions and non-sinusoidal sheath motions at low pressures. The same effect is even more pronounced at 1 Pa.

77 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES Results at low pressures (1 Pa,.5 Pa) Current and voltage measurements,4,2, 1 Pa -1-2 I [a.u] -,2 -,4 -, t [ns] [V] Figure 4.2: Comparison between measured (blue line) and modeled (black line) current in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated in krypton at 1 Pa and 8 W. The measured RF voltage (red line) is also shown [JS8b]. Figure 4.2 shows a comparison between the measured current to the grounded chamber wall (SEERS sensor implemented into the side wall of the GEC chamber) and the modeled current using the analytical PSR model described in chapter 2.3 [48] at 1 Pa and 8 W in krypton [JS8b]. The measured RF voltage drop across the discharge is also shown. At high pressures (45 Pa) the current is nearly sinusoidal, as it is usually assumed in most studies, and the agreement with the PSR model is good (see figure 4.1). At an intermediate pressure of 1 Pa small high frequency oscillations superimposed on the sinusoidal shape become observable (see figure 4.11). At low pressures (1 Pa) the temporal characteristics of the current are completely different compared to 45 Pa. There are strong high frequency oscillations superimposed on the sinusoidal shape. These high frequency oscillations are caused by the PSR effect and have been observed theoretically before [46 53]. The current oscillations are well reproduced by the analytical PSR model under the corresponding conditions. These high frequency oscillations lead to a faster sheath expansion and, therefore, enhance the generation of beams of highly energetic electrons. The PSR oscillations of the RF current clearly indicate that the discharge is operated under conditions, at which NERH plays an important role. The enhancement of electron heating is quantified by a comparison to a global model of Ziegler et al. [53]. In this work an equivalent circuit model similar to the one described in chapters 2.3 and is used (see figure 2.7 and equation 2.17). The current I and the total

78 72 CHAPTER 4. RESULTS accumulated dissipation [a.u.] Non-resonant dissipation Resonant dissipation t [ns] Figure 4.21: Phase accumulated power dissipated to electrons within two RF periods resulting from a global model [53] of a krypton discharge. In the model PSR oscillations are switched on (resonant dissipation) and off (non-resonant dissipation) by setting the bulk impedance to zero. accumulated dissipation [a.u.] Krypton 1 Pa, 8 W t [ns] Figure 4.22: Phase accumulated power dissipated to electrons within two RF periods resulting from the experiment operated under the same conditions as before. instantaneous power dissipation P (t) are calculated: P (t) = L p ν eff I 2 I 2 (4.6) with L p given by equation 3.3 with r p = r g and ν eff by equation From P (t) the total accumulated power dissipated to electrons P (t) is calculated:

79 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 73 P (t) = 1 T RF t P (t )dt (4.7) In this model the potential of the discharge to resonate can be removed by setting the bulk impedance to zero in equation 2.17, i.e. by removing electron inertia and collisions. Such a non-resonant model corresponds to the traditional approach towards electron heating neglecting the PSR effect. Figure 4.21 shows the accumulated power dissipated to electrons as a function of time within two RF periods resulting from the resonant and non-resonant model, respectively. The calculations are performed for a voltage amplitude 5 V, MHz, an electron density n e = cm 3, T e = 5 ev, krypton, an electrode radius of 5 cm (powered electrode, A p A g ), 1 Pa, and T g = 3 K. Figure 4.22 shows the experimentally determined accumulated dissipated power resulting from an integration of the square of the measured current shown in figure 4.2. As only a small fraction of the total current is measured and the experimental conditions are different from the model assumptions, only a qualitative comparison is possible. However, excellent qualitative agreement between experimental and model results is found. As the PSR oscillations and NERH can be switched off in the model, the influence of NERH on electron heating can be studied: Obviously electron heating is strongly enhanced, when the PSR is switched on. In both, experiment and model, this enhancement takes place at the moment of sheath collapse, when the PSR is excited (see chapter 2.3). In the model calculations the total accumulated power is enhanced by a factor of about 2 by NERH. 8 sheath voltage [V] π 2π 3π 4π phase Figure 4.23: Voltage drop across the sheath at the powered electrode resulting from the global model with (solid line) and without (dashed line) PSR oscillations. The PSR oscillations are switched off in the model by setting the bulk impedance to zero. Figure 4.23 shows the voltage drop across the sheath at the powered electrode resulting from the global model with (solid line) and without (dashed line) PSR oscillations. PSR oscillations of the current, and consequently the charge in the sheath,

80 74 CHAPTER 4. RESULTS lead to damped high frequency oscillations of the sheath voltage and, consequently, the sheath width. Such a temporal modulation of the sheath voltage is similar to classical dual frequency discharges operated at substantially different frequencies. The difference is, that the hf modulation of the sheath voltage is caused by an externally applied hf voltage waveform in case of a dual frequency discharge, whereas the hf modulations are self-excited in case of the single frequency discharge investigated here. Another difference is the fact, that the hf modulations of the sheath voltage are damped here. Similar results were obtained by Czarnetzki et al. using the analytical PSR model introduced in chapter 2.3 [48]. These oscillations are also observed experimentally and found to affect the spatio-temporal excitation dynamics at low pressures (see figure 4.32). Langmuir probe measurements 2.1x x1 9 n e [cm -3 ] 1.5x x1 9 9.x1 8 6.x Radial distance from centre position [mm] Figure 4.24: Radially resolved measurement of the electron density in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge (krypton, 1 Pa, 8W) 2,5 cm above the electrode. Figure 4.24 shows a radially resolved measurement of the electron density in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge (krypton, 1 Pa, 8W) 2,5 cm above the electrode. The maximum electron density is n e cm 3. Figure 4.25 shows a comparison between an experimentally obtained EEDF E 1/2 measured close to the radial center of the discharge at 1 Pa and 8 W 2.5 cm above the powered electrode and a theoretically calculated EEDF E 1/2 using an analytical model, that is explained in chapter [JS8b, JS8c]. The agreement is good and a Bi-Maxwellian shape is observed in both experiment and theory. At 1 Pa the electron mean energy is < ɛ > 5.1 ev, the temperature T c of the low energetic part of the EEDF is T c 2.3 ev and the electron temperature of

81 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 75 EEDF * E -1/2 [cm -3 ev -3/2 ] E [ev] Figure 4.25: EEDF E 1/2 in the radial center of a geometrically strongly asymmetric single frequency (13.56 MHz) discharge (krypton, 1 Pa, 8W) 2,5 cm above the electrode [JS8b,JS8c]. the high energetic part T h is T h 7 ev. Bi-Maxwellian EEDFs are also observed using the hybrid Monte Carlo simulation of a symmetric discharge at similar pressures [16, 71]. Electric field measurements Figure 4.26 shows the spatial and temporal evolution of the electric field in the boundary sheath of a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated at 1 Pa and 8 W in krypton at the powered electrode [JS8b, JS8c]. The markers correspond to measured fields using FDS and the solid lines to the result of the fluid sheath model described in chapter using experimentally determined input parameters. Good agreement between experiment and theory is found. Similar to the results obtained at 1 Pa several different phases are investigated. Due to the lower pressure the maximum sheath width is bigger than in the 1 Pa case (see figure 4.15) and the electric field in the sheath is lower. Based on figure 4.26 the maximum sheath width can be estimated to be s 1 Pa 1 cm. Similar to the case of higher pressure the fast sheath expansion between 7.5 ns and 21.5 ns as well as the sheath collapse are clearly observable. At 1 Pa and 8 W the voltage measured by a high voltage probe close to the electrode is again compared to the momentary sheath voltage resulting from an integration of the measured electric fields over the sheath. This comparison is shown in figure 4.27 [JS8b] and very good agreement is found. Furthermore, the proportionality of

82 76 CHAPTER 4. RESULTS E [V/cm] ns 21.5 ns 39.5 ns 57.5 ns Distance from powered electrode [cm] Figure 4.26: Spatio-temporal evolution of the electric field in the sheath of a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated at 1 Pa and 8 W in krypton. The markers are experimental (FDS) and the solid lines theoretical results (fluid sheath model) [JS8b,JS8c]. 6 5 High V-Probe Measurement 4 sp [V] t [ns] Figure 4.27: Comparison between measured RF voltage (high voltage probe and sinusoidal fit) and sheath voltage determined out of the integration of the measured electric fields at different phases [JS8b]. the sheath voltage φ sp to the surface charge density σ is investigated. The surface charge density is again calculated out of the electric field at the powered electrode using equation 4.4. The result is the same as before (see figure 4.28 [JS8b]). The

83 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES sp 1/2 [V 1/2 ] ,,1,2,3,4,5,6,7 [1-6 C m -2 ] Figure 4.28: Relation between sheath voltage φ sp and surface charge density at the electrode σ at 1 Pa and 8 W [JS8b]. sheath voltage is proportional to the square of the surface charge density also at lower pressures. E [V/cm] Distance from powered electrode [mm] Figure 4.29: Time averaged electric field (with a polynomial fit) at 1 Pa and 8 W in a pure krypton discharge [JS8b]. Based on the time resolved measurements of the electric field shown in figure 4.26 the time averaged electric field in the sheath is calculated by averaging the measured electric fields at different positions in the sheath over one RF period. The result is shown in figure 4.29 [JS8b]. Spatial integration of the time averaged field yields the

84 78 CHAPTER 4. RESULTS [V] Distance from powered electrode [mm] Figure 4.3: Time averaged potential in the sheath determined from figure 4.29 at 1 Pa and 8 W in a pure krypton discharge [JS8b]. time averaged potential φ in the sheath at different positions (figure 4.3 [JS8b]). As the ions in the discharge are much heavier than the electrons, they only react to this time averaged potential. Based on the conservation of ion energy and flux the following expression for the ion density n i (z) in the sheath as a function of the time averaged potential φ(z) is derived [2]: n i (z) = n ( 1 ) 1 2 2e φ(z) m i u 2 B (4.8) Here n is the ion density in the bulk, which is equal to the electron density, and u B is the Bohm-velocity, at which the ions enter the sheath. The result in terms of the space resolved ion density in the sheath is shown in figure 4.31 [JS8b]. The ion density is cm 3 at the electrode and increases by one order of magnitude towards the sheath edge located 1 cm away from the electrode due to flux conservation. The absolute value at the sheath edge of n i cm 3 is the same as the electron density in the bulk (n e cm 3 ) measured by a Langmuir probe (see figure 4.25). A similar ion density profile in the sheath was found by Ziegler et al. using a global model of a dual-frequency discharge [52]

85 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 79 2.x x1 15 n i [m -3 ] 1.x x distance from powered electrode [mm] Figure 4.31: Time averaged ion density in the sheath at 1 Pa and 8 W in a pure krypton discharge [JS8b]. Excitation dynamics Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] Figure 4.32: Spatio-temporal plot of the excitation into Kr2p 5 close to the powered electrode in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated at 1 Pa and 8 W in krypton [JS8b]. In analogy to the investigations performed at 1 Pa and presented in chapter 4.2.2, the effect of electron heating in terms of the spatio-temporal excitation, probed

86 8 CHAPTER 4. RESULTS through the Kr 2p 5 -state, is also investigated at 1 Pa. The result is shown in figure 4.32 [JS8b]. The maximum sheath width obtained from the spatio-temporal contour plot of the excitation is s 1 Pa 1 cm. This is again in good agreement with the electric field measurements (see figure 4.26). Similar to the situation at a higher pressure of 1 Pa (see figure 4.18) the generation of a highly energetic electron beam is observed. From the slope of the corresponding trajectory the propagation velocity of the beam-electrons can be estimated to be v d,1p a m/s. In comparison to the observed drift velocity at higher pressures (v d,1p a m/s) the beam electrons are substantially faster at lower pressures. This is caused by the bigger sheath width and the PSR effect, which plays an important role at 1 Pa. Both arguments cause a faster sheath expansion. The corresponding drift energy at 1 Pa is E d,1p a 1.5 ev, whereas the drift energy at 1 Pa is E d,1p a 11 ev. Because of this difference in energy, that is deposited in the plasma by beam electrons, it is well understandable, that stochastic heating is dominant at low pressures. In contrast to the situation at 1 Pa there are additional excitation maxima during the later part of the RF period (2 ns - 74 ns), that occur after the tilted maximum caused by the electron beam. Those straight maxima are directly correlated with the high frequency oscillations of the RF current due to the PSR effect, that are well pronounced at 1 Pa in contrast to the 1 Pa case, where no additional excitation maxima are observed. This phenomenon will be discussed in more detail in section at even lower pressure. The solid orange line in figure 4.32 is the sheath width as a function of time. It is the result of the following algorithm using data from current and electric field measurements: The values of the current shown in figure 4.2 do not correspond to the actual current in the discharge, but are proportional to it. Consequently, the following relation between measured current density j meas and actual current density j is valid: j = C j meas (4.9) where C is a constant. The surface charge density at the electrode surface σ at a certain time τ m within the RF period can be calculated out of the current density: τm τm σ(τ m ) σ = jdt = C j meas dt (4.1) where σ is the surface charge density at the beginning of the RF period at τ =. Due to electron inertia the sheath collapse at low pressures does not take place at τ =, when the applied voltage is zero, but is slightly phase shifted and takes place later. Therefore, σ is not zero, but has some unknown value. Under these conditions the electrode is assumed not to charge up positively (negative σ) at any phase within the RF period (no field reversal during sheath collapse). Therefore, a constant is subtracted, so that σ during the entire RF period. This constant corresponds to σ. In fact, σ is always positive during the entire RF period, since the sheath never collapses completely due to the dynamic floating potential φ f [48]:

87 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 81 φ f = k BT e 2e [ ( ) ( mi 2πeφ ln ln 2πm e k B T e )] (4.11) For an electron temperature of 2.3 ev, a krypton mass of M = kg and an amplitude of the applied voltage φ 26 V the floating potential is φ f 4.1 V. This is small in comparison to the applied voltage and, therefore, neglected here. Equation 4.11 is based on the assumption of sinusoidal sheath voltages [48]. Due to the PSR effect the sheath voltages are not sinusoidal at 1 Pa in an asymmetric discharge. Therefore, equation 4.11 only yields an approximate result for the floating potential. The surface charge density at different phases is known from figure Using equation 4.1 the constant C is determined to be C and the actual current in the discharge is known. The sheath velocity u s (t) in dependence of time is given by: u s (t) = j(t) en s (t) (4.12) where n s (t) is the ion density at the sheath edge, that is equal to the electron density at this point due to quasi-neutrality. j(t) is known from equation 4.9 and n s (t) can be determined out of current and electric field measurements..8.6 [1-6 C/m 2 ] t [ns] Figure 4.33: Surface charge density at the powered electrode in a pure krypton discharge at 1 Pa and 8 W as a function of time [JS8b]. σ can also be determined out of the integration of the charge density in the sheath over the sheath: σ = e s(t) n s dz (4.13)

88 82 CHAPTER 4. RESULTS [1-6 C/m 2 ] s [mm] Figure 4.34: Surface charge density at the powered electrode in a pure krypton discharge at 1 Pa and 8 W as a function of sheath width [JS8b]. The surface charge density at the electrode is assumed to be equivalent to the overall charge in the sheath. Figure 4.33 shows σ as a result of equation 4.1 and figure 4.34 as result of equation 4.13 [JS8b] u s (t)*1 5 [m/s] t [ns] Figure 4.35: Velocity of the sheath edge as a function of time within one RF period in a pure krypton discharge at 1 Pa and 8 W [JS8b]. The following procedure is then applied in order to determine n s (t) and, consequently, u s (t):

89 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES s(t) [mm] t [ns] Figure 4.36: Position of the sheath edge as a function of time within one RF period in a pure krypton discharge at 1 Pa and 8 W [JS8b]. At a certain phase σ is determined from figure Then, this value is used to determine the sheath width s at this phase from figure Finally, this sheath width is used to determine n s at this phase from figure Knowing n s (t) the sheath velocity u s (t) is calculated using equation One RF period is now separated into discrete time steps and a linear interpolation between the sheath velocities at different phases is performed. Based on this interpolation s(t) is determined. The results are shown in figures 4.35, 4.36, and 4.32 (orange line), respectively [JS8b]. The PSR oscillations of the current are reflected in the sheath velocity and width. Following the Hard Wall model [63, 65] (see chapter 2.4) the propagation velocity of the electron beam should be twice the sheath velocity during the time interval of sheath expansion. The value of u d m/s, determined from the tilt of the beam trajectory in figure 4.32, is a time averaged value. The corresponding value for the drift velocity resulting from this algorithm (figure 4.35) is also of the order of m/s. The left plot of figure 4.37 shows a result of the hybrid Monte Carlo simulation described in chapter in terms of the integration of the calculated EEDF E 1/2 above 12 ev in a geometrically symmetric single frequency (13.56 MHz) discharge operated in argon at 2.7 Pa [16, 67, 71], [JS8b,JS8c,BGH8]. Again, only the fraction of the discharge close to the bottom electrode is shown. Similar to the experiment trajectories of beams of highly energetic electrons are observed, that are generated by the expanding sheath also at lower pressures. Although the qualitative agreement with the experimental observation (see figure 4.32) is good, a quantitative comparison is difficult. The simulation is inherently

90 84 CHAPTER 4. RESULTS Figure 4.37: Left: Time and space resolved integral of the EEDF above 12 ev as it is obtained from the hybrid Monte Carlo simulation described in chapter 3.1 of a geometrically symmetric single frequency (13.56 MHz) discharge operated in argon at 2.7 Pa (d = 6.7 cm) using input parameters from [67] [16, 71], [JS8b,JS8c,BGH8]. Right: Comparison between measured (Figure 18 in [67]) and calculated EEDF E 1/2 under these conditions [16, 71], [JS8b,JS8c,BGH8]. one-dimensional, while the geometry of the experiment is two-dimensional. Therefore, data from an effectively one-dimensional experiment performed by Godyak are used for a direct comparison of the EEDFs in the right plot of figure 4.37 [16, 67, 71], [JS8b,JS8c,BGH8] (p = 2.7 Pa, n e = cm 3, T e =.74 ev, d = 6.7 cm, j = 3 ma [67]). The simulation reproduces the experimentally obtained EEDF cm 2 well and, therefore, yields realistic results. It should be noted, that using experimental data from the same source at different discharge conditions yields similarly good agreement throughout [16, 71]. In the simulation interaction between charged particles is excluded. This strongly supports the hypothesis of electron beams in comparison to wave effects.

91 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 85 Results at.5 Pa I [a.u.] Distance from powered electrode [cm] I 2 [a.u.] t [ns] Figure 4.38: Correlation between current, space and phase resolved excitation into Kr2p 5 and square of the current (deposited power) in a geometrically strongly asymmetric single frequency (13.56 MHz) krypton discharge operated at.5 Pa and 8 W [JS8b]. Current and PROES measurements are also performed at.5 Pa and 8 W in pure krypton. If the pressure is even lower than 1 Pa, the excitation related to the PSR oscillations of the RF current waveform is clearly pronounced. Figure 4.38 [JS8b] shows the correlation between current I, excitation and deposited power (I 2 ) at.5 Pa and 8 W in a pure krypton discharge. At the beginning of one RF period a direct correlation between the three graphs indicated by vertical lines is observed. At the phases of maximum excitation, extrema of the current and maxima of the deposited power are observed. The first extremum of the current corresponds to the fast initial expansion of the sheath. Consequently, the generation of an electron beam is observed at this phase. When these beam electrons hit the opposing quartz cylinder, it is charged negatively and, consequently, a sheath develops in front of it. In contrast to the tilted excitation maximum caused by the electron beam the following maxima are straight. Their formation might be understood based on the following hypothesis: At the phases of straight excitation maxima there are sheaths at the quartz and at the powered electrode. Consequently, electrons in the plasma bulk are confined within a potential well. At relatively low energies this potential can be assumed to be harmonic. In such a harmonic potential well, the oscillation period of the confined low energetic electrons does not depend on the spatial oscillation amplitude.

92 86 CHAPTER 4. RESULTS Therefore, the observed excitation maxima are straight and maximum excitation is observed in the discharge center, where the absolute value of the potential is minimum and the kinetic energy is maximum. The correlation between excitation maxima and current extrema might show, that additional energy is transferred to the oscillating electrons via the PSR oscillations of the sheath edge at the bottom powered electrode. Once these PSR oscillations stop, the amplitude of the excitation maxima decreases, because the energy loss via collisions is no longer compensated. This scenario might be similar to Electron Bounce Resonance Heating described in [158] with the difference, that here the Bounce Resonance Frequency is not the RF, but the PSR frequency. However, this argumentation remains a hypothesis, which needs to be tested and verified against experiments and/or simulations. Similar measurements have been performed in hydrogen at 1.2 Pa and 6 W before [113]. These results are in good agreement with PIC simulations of the power density performed by Vender [159]. It should be noted that in this model the electron oscillation leading to the observed excitation pattern is perpendicular to the quartz surface. The current amplitude is measured outside the gap between electrode and quartz. Therefore, the measured current corresponds to electron oscillations parallel to the quartz and electrode surface. Although both directions are coupled, they do not necessarily have to be identical Electron beams in geometrically asymmetric capacitive RF discharges at low pressures In the previous sections the generation of beams of highly energetic electron beams by the expanding boundary sheath has already been observed by PROES. In this section the generation of such electron beams as a function of pressure, reflections at the plasma boundaries and their impact on the time averaged isotropic EEDF, such as measured by Langmuir probes, is studied in detail. Figures 4.39, 4.4, and 4.41 show spatio-temporal plots of the excitation into Ne2p 1 within one RF period at various pressures and in different gas mixtures [JS8c, JS8d]. At the beginning of each RF period, when the sheath expands fast, the generation of a beam of energetic directed electrons is observed that penetrates into the plasma bulk. Here, the term highly energetic directed electrons is understood as an anisotropic electron velocity distribution with a strong drift component in (or near) the direction of the discharge axis. Its propagation velocity is determined from the slope of the arrows shown in figures 4.39, 4.4, and However, the analysis involves an inherent uncertainty due to the relatively broad excitation pattern. At all pressures discussed here, the propagation velocities range around m/s (17.8 ev) with a variation of ± m/s being about equal to the uncertainty. This energy is substantially higher than the electron mean energy of about 5 ev under similar conditions (see chapter 4.2.3). Under the conditions investigated here, the PSR leads to nonsinusoidal RF currents. High frequency oscillations of the measured current waveform of the order of 5 MHz

93 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 87 Distance from powered electrode [cm] 5 Exc. rate [a.u.] t [ns] Figure 4.39: Phase and space resolved excitation into Ne2p 1 in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated at 5 Pa, 8 W in krypton with 1 % neon admixture [JS8d]. Current and voltage waveforms look similar as shown in figure Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] Figure 4.4: Phase and space resolved excitation into Ne2p 1 in a geometrically strongly asymmetric single frequency (13.56 MHz) discharge operated at 2 Pa, 8 W in neon [JS8d]. Current and voltage waveforms look similar as shown in figure are observed (see figure 4.41). Correspondingly, the sheath expands more rapidly than in regimes of sinusoidal RF currents. The beam occurs exactly in the interval between the first two zeros of the current ( t = 14 ns). This clearly indicates its

94 88 CHAPTER 4. RESULTS Distance from powered electrode [cm] I [a.u.] Exc. rate [a.u.] U[V] Figure 4.41: Phase and space resolved excitation into Ne2p 1 in a geometrically strongly asymmetric single frequency discharge (13.56 MHz,.2 Pa, 9% krypton, 1 % neon). The measured voltage drop across the discharge and RF current to the chamber wall are also shown [JS8c,JS8d]. generation by the fast sheath expansion caused by the PSR effect. Therefore, the PSR effect enhances the generation of such electron beams and leads to the observed spatio-temporal excitation. With decreasing pressure, the electron mean free path increases, and these energetic directed electrons can propagate further into the plasma until they hit the opposing quartz cylinder (see figure 4.8). This cylinder acts as a floating surface and charges up negatively, as it is bombarded by energetic electrons. Consequently, a sheath develops in front of the quartz that reflects following electrons back into the plasma. It should be noted that the floating potential at the quartz is high enough to reflect the energetic beam electrons (E 19 ev). Assuming a Maxwellian EEDF, the static floating potential would already be higher than 2 V. Here, the EEDF is strongly non-maxwellian. Therefore, the floating potential is even higher. At very low pressures (see figure 4.41), the reflected electrons hit the fully expanded sheath at the powered electrode, where they are reflected again back into the bulk. Under the conditions of figure 4.41 the electron mean free path for elastic electron-neutral collisions is about 8 cm. This agrees quite well with the overall length of the visible beam trajectory. Such beams and their reflections have been predicted by PIC simulations of Vender [81, 82] and Wood [83], but have never been detected experimentally until now. Similar to the dynamics in geometrically symmetric discharges (see chapter 4.1) this electron ping pong effect leads to an effective heating of the

95 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 89 plasma at low pressures, since the high energy of the beam electrons is deposited in the plasma to a great extent. In symmetric discharges PSR oscillations are not possible and the sheath expansion as well as the related beam velocity are expected to be lower. Maxima of the excitation are observed at the positions where the beam is reflected, since incoming and reflected beam overlap at these positions. After some time the beam diverges and is no longer observed in the spatio-temporal plots of the excitation. This is probably caused by collisions and dephasing of faster and slower electrons in the beam. Finally, the impact of such highly energetic directed electron beams on the time averaged isotropic EEDF in the bulk, such as measured by a Langmuir probe, is investigated by a simple analytical model [JS8b, JS8c]. Such a measured EEDF in a krypton discharge operated at 1 Pa is shown in figure It has a Bi- Maxwellian shape typical for CCRF discharges at low pressures. The high energy tail is enhanced by stochastic heating [67]. It is reproduced qualitatively by an analytical model (red line in figure 4.25), that takes into account the influence of a beam of energetic electrons, that is generated during a short fraction of the RF period (τ/t =.15). As this model is strongly simplified and does not consider various processes such as confinement of energetic electrons, it must not be expected to reproduce experimental results quantitatively. In this model the overall EEDF is assumed to be the sum of a time independent isotropic part f, an anisotropic part f 1 attributed to the ohmic current (two-term approximation) and a beam part f beam : f = f + f 1 cos θ + f beam with cos θ = v z v For f beam, a shifted Maxwellian shape is assumed: (4.14) f beam ( v) = αf [ ( v u) 2 ] Θ( v u) (4.15) with the Heavyside function Θ( v u), considering only positive electron velocities, and α = ns n, where n s is the electron density in the collapsed sheath and n the density in the bulk (n s n ). All velocities are normalized to the thermal velocity. u is a normalized drift velocity corresponding to twice the sheath velocity [63, 65]. In order to reproduce the measured EEDF, the beam part is averaged over the full solid angle and the EEDF is time averaged using experimentally determined input parameters. As a reasonable analytical approximation u is assumed to be a constant u for t τ = 11 ns and zero otherwise. The result is: < f >= f (v 2 )(1 + αh(v)) (4.16) h(v) = 1 + τ T ( e u2 e2vu o ) 1 1 2vu (4.17) For the conditions investigated in chapter (strongly geometrically asymmetric single frequency krypton discharge, 1 Pa, 8 W, MHz) typical values for the input parameters for this model can be estimated based on experimental results

96 9 CHAPTER 4. RESULTS applying various diagnostics: The ion density n s at the electrode is known from the electric field measurements to be n s cm 3 (see figure 4.31). n cm 3 is known from Langmuir probe measurements (see figure 4.25, α.2). τ/t.15 is known from PROES (see figure 4.32) and current measurements (see figure 4.2). The beam velocity is estimated based on the PROES measurements (see figure 4.32) to be m/s. The thermal velocity is known from the electron temperature of the cold electrons measured by a Langmuir probe (see figure 4.25) kt e,cold 2.3 ev ( u 2.5). Figure 4.42: Left: Analytically calculated time averaged isotropic EEDF for two different beam velocities [JS8,JS8c]. Right: EEDF at two different time intervals and time averaged (u = 2.5) [JS8b,JS8c]. Figure 4.42 shows the time averaged EEDF as it results from the model assuming two different beam velocities u = 1 and u = 2.5 [JS8b, JS8c]. An enhancement of the tail is observed only for u > 1, i.e. for beam velocities exceeding the thermal velocity of the cold bulk electrons. This result clearly shows, that the nature of stochastic heating is closely related to electron beams. Figure 4.42 also shows the temporal variation of the EEDF resulting from this model [JS8b, JS8c]. As the scattering of the beam is neglected here, the EEDF only deviates from a Maxwellian shape during the fraction of the RF cycle, when the beam propagates Electric field reversals during sheath collapse in geometrically asymmetric single frequency discharges at low pressures Until now the generation of highly energetic electron beams by the expanding sheath and the effect of self-excited non-linear PSR oscillations of the RF current on electron

97 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 91 heating in geometrically strongly asymmetric single frequency discharges operated at low pressures have been discussed. In this section an additional source for energy gain of electrons during sheath collapse will be investigated: If the discharge is operated under conditions, at which the sheath collapses particularly fast, i.e. if the sheath width is very large for a given duration of the RF period, a reversed electric field is generated, that accelerates electrons towards the electrode. This field reversal causes additional excitation during sheath collapse. Excitation dynamics Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] Figure 4.43: Phase and space resolved excitation into Ne2p 1 in a geometrically strongly asymmetric single frequency (13.56 MHz) neon discharge operated at 1 Pa, 8 W [JS8a]. Figures 4.43 and 4.44 show spatio-temporal plots of the excitation into Ne2p 1 at 1 Pa and.5 Pa within one RF period in a geometrically strongly asymmetric single frequency (13.56 MHz) neon discharge [JS8a]. At the beginning of the RF period, a beam of highly energetic electrons is generated by the expanding sheath in both cases. The beam trajectory is indicated in the figures by arrows. In contrast to discharge conditions discussed before the maximum sheath width is much bigger (2.5 cm at 4 ns). This is caused by the lower ion density in neon compared to argon or krypton at the same power. The RF current is still non-sinusoidal. Consequently, the velocities of sheath expansion and collapse are much higher. The electron beam is reflected at the opposing quartz surface such as discussed in chapter The trajectory of the reflected beam is clearly visible at.5 Pa. In contrast to higher pressures, at 1 Pa and.5 Pa an additional source of excitation is observed during sheath collapse. Similar to the trajectory of the electron beam generated by the expanding sheath, the excitation structure during sheath collapse is also tilted.

98 92 CHAPTER 4. RESULTS Distance from powered electrode [cm] 5 Exc. rate [a.u.] t [ns] ,7 Figure 4.44: Phase and space resolved excitation into Ne2p 1 in a strongly geometrically asymmetric single frequency (13.56 MHz) neon discharge operated at.5 Pa, 8 W [JS8a]. However, it is tilted into the opposite direction indicating an acceleration of electrons towards the electrode. Sato and Lieberman measured a reversed electric field at the sheath edge during the phase of sheath collapse at similar pressure (.3 Pa) in argon [16]. In this work the field reversal disappeared with increasing pressure. According to their results the excitation observed here during the phase of sheath collapse is caused by a local field reversal at the sheath edge caused by electron inertia. Similar to their investigations, the effect vanishes with increasing pressure. Figures 4.45 and 4.46 show spatio-temporal plots of the excitation into the n = 3 state of atomic hydrogen (Balmer-α emission) in a geometrically strongly asymmetric single frequency (13.56 MHz) H 2 discharge operated at 8 W at different pressures of 1 Pa and.5 Pa [JS8a]. At 1 Pa an excitation maximum close to the electrode during the phase of sheath collapse is observed. At.5 Pa two excitation maxima during the phase of retreating sheath are observed, one close to the electrode and one further inside the plasma at an earlier phase. The spatio-temporal evolution of the electric field and the excitation into the n = 3 state of atomic hydrogen (Balmer-α emission) in the sheath of a H 2 CCRF discharge at 8 Pa were measured before by Czarnetzki et al. [16]. In this work an excitation maximum close to the electrode during the phase of sheath collapse was observed, which could directly be identified with a field reversal. However, these measurements were performed at much higher pressures, where the regime is collisional, whereas it can be assumed to be collisionless at.5 Pa. At these high pressures the generation of the field reversal was understood by an analytical model [16]: It is caused by collisions of electrons with the neutral background gas, that hinder electrons from following the fast collapsing sheath. In order to ensure flux balance of electrons and ions at each electrode a re-

99 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 93 Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] ,3 5,9 3,4 1, Figure 4.45: Phase and space resolved excitation into the n = 3 state of atomic hydrogen (Balmer-α emission) in a geometrically strongly asymmetric single frequency (13.56 MHz) H 2 discharge operated at 1 Pa, 8 W [JS8a]. Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] ,7 6,8 5, Figure 4.46: Phase and space resolved excitation into the n = 3 state of atomic hydrogen (Balmer-α emission) in a geometrically strongly asymmetric single frequency (13.56 MHz) H 2 discharge operated at.5 Pa, 8 W [JS8a] versed field must enhance the electron loss. Vender and Boswell explicitly predicted two different field reversal mechanisms, which lead to an increased electron loss to ensure current continuity. According to their simulation, [159] a field reversal at the electrode surface occurs, if the wall (electrode) potential rises and the plasma potential lags behind. They also mention a second mechanism leading to a field

100 94 CHAPTER 4. RESULTS reversal close to the sheath edge during sheath collapse: If the sheath collapses so fast that electrons cannot follow and compensate the ion space charge, the changing wall potential drives a field reversal within the ion sheath. Both mechanisms are observed experimentally here (figure 4.46). Analytical model to describe the field reversal In this section a simple fluid model will be developed, that describes the field reversal in the different pressure regimes qualitatively [JS8a]. It is based on an analytical model developed by Czarnetzki et al. [16] to describe field reversals at high pressures caused by collisions of electrons with the neutral background gas. This model is now extended to include electron inertia effects and, therefore, also describes field reversals at low pressures. The model is limited to those regions where the condition of quasi-neutrality is fulfilled and the current dominated by electron conduction current. Displacement current is not included. It does not describe how the sheath is filled with electrons. The momentum balance equation for electrons in a discharge without magnetic fields is given by: [ ] u m e n + ( u ) u = ene t p m e nν c u (4.18) where n = n e = n i is the plasma density, u is the electron velocity, and ν c is the collision frequency for elastic collisions of electrons with the neutral background gas. As only the electron motion perpendicular to the electrode (defined as z direction) is relevant here and only gradients in this direction occur, equation 4.18 reduces to n [ ] u t + u u z = ene m e k BT e m e n z nν cu (4.19) with p = k B T e n. Here u and E are the z-components of u and E, respectively. Substituting the current density j = enu into equation 4.19 assuming a stationary ion density profile yields: j t + j u z = e2 ne m e The continuity equation is given by: + ek BT e m e n z ν cj (4.2) n t + (nu) = (4.21) z A simple analysis based on the global particle balance shows that the ionization rate in the sheath can be neglected in the continuity equation, if the sheath width is much smaller than the bulk length and the ion density in the sheath is much lower than in the bulk. Therefore, there is no source term in equation 4.21 used to describe the phase of the field reversal. Under the assumption of a stationary density profile ( n = ) equation 4.21 reduces to: t

101 4.2. GEOM. ASYMMETRIC SINGLE FREQUENCY DISCHARGES 95 u z = j n en 2 z (4.22) Substitution of equation 4.22 into equation 4.2 yields the following expression for the electric field at the bulk side of the sheath edge: E = m ( e j ne 2 t + ν cj + (j 2 jth) 2 1 ) n en 2 z (4.23) with jth 2 = e2 n 2 k B T e m e. The first and third term on the RHS of equation 4.23 correspond to an electric field due to electron inertia. The first term is attributed to temporal changes of the electron current and the third one to the electron velocity and its spatial changes. The second term corresponds to collisions and the fourth one to diffusion. The first three terms yield a reversed field accelerating electrons towards the electrode, whereas the fourth (diffusion) term yields a field, that repels electrons from the electrode. The fourth term also results from the zero order electric field calculated in the frame of the fluid sheath model, i.e. ambipolar fields result from the fluid sheath model in its current implementation. The first and second term correspond to the ohmic electric field in the bulk calculated in the frame of the Monte Carlo module as a first order correction to the electric field resulting from the fluid sheath module (see chapter 3.2.1). The third term does not result from this simulation due to the negligence of an inertia term in the electron momentum balance equation. Equation 4.23 clearly shows that, depending on the discharge conditions, a field reversal in capacitive discharges can either be caused by electron inertia (low pressures), collisions of electrons with the background gas (high pressures), or a combination of both (intermediate pressures). Each of these mechanisms can prevent electrons from instantaneously following the collapsing sheath. If electrons cannot follow the retreating sheath by diffusion, a reversed field builds up that accelerates electrons towards the electrode in order to keep a constant current and to compensate the ion flux to the electrode within one RF period. Under the low pressure conditions investigated in this section the field reversal is clearly caused by electron inertia. It should be noted, that at such low pressures and in asymmetric discharges the current can be non-sinusoidal due to the PSR effect. Figure 4.47 shows phase resolved measurements of the RF current and voltage in neon under the conditions investigated experimentally here [JS8a]. The high frequency oscillations of the RF current at such low pressures are not completely damped until the end of the RF period. Due to the PSR effect the spatial oscillation of the sheath within one RF period is non-sinusoidal. It leads to a faster expansion and collapse of the sheath and has, therefore, probably an important influence on the occurence of a field reversal. Furthermore, in asymmetric discharges, such as the single frequency discharge investigated here, there is a DC bias, which is nearly equal to the RF amplitude of the voltage, if the discharge is strongly asymmetric. This DC bias affects the field reversal at the powered electrode, since it leads to bigger sheath widths at the driven electrode. Consequently, the sheath collapses

102 96 CHAPTER 4. RESULTS I [a.u.] Current Voltage t [ns] U[V] Figure 4.47: Measured current and voltage in a geometrically strongly asymmetric single frequency (13.56 MHz) neon discharge operated at 1 Pa and 8 W [JS8a]. faster in asymmetric than in symmetric discharges at a given RF voltage amplitude and plasma density. With increasing velocity of the collapsing sheath the reversed field needed to keep a spatially constant current also increases. If there is a DC bias the ion density at the powered electrode will decrease. Consequently, the electron density at the powered electrode at the phase of sheath collapse will also be lower. Therefore, a given ion flux to the electrode will be compensated less effectively, if there is a DC bias, and a higher reversed field is needed to ensure this compensation. Here the fluid model is only used to qualitatively explain the generation of a field reversal during sheath collapse in a single frequency discharge operated at low pressure. In principle it could also be used to model the reversed field quantitatively, if the input parameters needed for the model (ion density, electron current density, collision frequency, electron temperature) were provided either by an experiment or by a simulation. Any simulation would, however, have to take into account the discharge asymmetry and the excitation of the PSR. This might be a topic for future investigations. In this work this model will be used to quantitatively describe field reversals in geometrically symmetric dual-frequency discharges operated at substantially different frequencies (see chapter 4.3.4) using input parameters from a PIC simulation.

103 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES Electron heating in geometrically symmetric dual frequency discharges operated at substantially different phase locked frequencies In this section electron heating in geometrically symmetric dual frequency discharges operated at substantially different phase locked frequencies is investigated experimentally, by a PIC simulation, and by analytical models. The focus lies on an analysis of frequency coupling and different modes of discharge operation (α-, γ-, and hybrid mode) in different gases in an industrial Lam Exelan discharge. Similarities and differences to the respective modes in single frequency discharges will be outlined. Such discharges are frequently used for applications to separately control ion energy and ion flux at the electrode surfaces. The idea is, that the ion flux is mainly controlled by the hf component, since electron heating is more efficient at high frequencies. The ion energy is assumed to be mainly controlled by the lf component, since the applied lf voltage amplitude is typically higher than the hf voltage amplitude. The PIC simulations are performed by Zoltan Donkó in collaboration with the author. Some of the simulation and model results are compared to experimental results of phase and space resolved investigations of the excitation dynamics in a Lam Exelan discharge obtained in the frame of the diploma thesis of the author [114]. These experimental results are the only available data that can be used for comparison with the simulation and model results performed in this work. This comparison to experimental results is essential for the interpretation of the PIC results. Therefore, some of these experimental results will be discussed shortly. The results of this section can be found in references [JS9a, JS8a, JS7b, TG6] Experimental setup The reactor used for these investigations is a modified Lam Exelan industrial dualfrequency discharge with plane parallel electrodes separated by a gap of 12 mm (see figure 4.48, [JS7b]). A true summation of two RF-voltages is applied to the bottom electrode, whereas the top electrode is grounded. The two electrodes are made from single-crystal silicon of radius 11 mm. The upper electrode is water cooled. The lower electrode consists of a 2 mm diameter bare silicon wafer clamped to an electrostatic chuck (7V). The gap between wafer and chuck is filled with helium at 1.33 kpa (1 Torr) to improve heat transfer. An annular silicon ring lies co-planar and immediate outside of the wafer, completing the lower electrode to a total radius of 11 mm. The plasma is confined in the radial direction by adjustable quartz rings (11 mm internal dimension), shielding it from the chamber walls. Furthermore, the pressure between the rings and the wall is too low, so that no breakdown should occur. Thus, the discharge is geometrically symmetric. In order to provide access to the chamber for optical diagnostic techniques the original visual access windows are extended and the confinement rings modified. In front of one flange a section of

104 98 CHAPTER 4. RESULTS Figure 4.48: Experimental setup used for PROES measurements in a geometrically symmetric dual-frequency discharge (1.937 MHz MHz) [JS7b]. the rings is cut and replaced by an optical quartz block, through which photons can leave the reactor and enter the optical detection system (see figure 4.49, [JS7b]). Gas is introduced into the chamber through a showerhead built into the upper electrode, and exits the plasma region radially through the confinement rings. The pump-channel outside the confinement rings connects to a high-conductance manifold of an Osaka TC44 helical groove pump, backed by a two stage rotary pump. For these experiments, the discharge is operated at P 27 = 8 W and P 2 = 2 W at p = 65 P a. The RF voltage waveforms are fixed in frequency at MHz and MHz with a common phase reference (f hf = 14 f lf ). The gas mixtures are helium (72 %) and O 2 (19 %) with a 9 % admixture of tracer gases (neon, argon, krypton) used as reference gases for PROES (see table 3.1). Helium is chosen as the main constituent, as it does not sputter silicon and is the lightest rare gas, for which most data is known. Oxygen is used in order to avoid deposition of silicon on the optical block as it reacts with the sputtered particles resulting in quartz-like (transparent) deposition. Furthermore, it stabilizes the discharge, because a pure helium discharge is difficult to ignite due to its high ionization energy. The plasma emission is focused on the entrance slit of a Carl Zeiss PGS 2 spectrometer (3 nm - 9 nm) by a lens. The variable slit width is set to 5 µm for this experiment. The resolution is 8.18 pm/pixel on the camera resulting in 4 nm/512 pixels. In the spectrograph the light is dispersed and then detected by a fast gateable ICCD camera (PicoStar HR, LaVision), that is triggered and synchronized with the low frequency RF voltage waveform. Using this camera, gate widths of a few 1 ps can be realized. In this work the minimum temporal gate width used for phase resolved measurements is 4.1 ns. Due to the camera s high repetition rate of

105 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 99 Figure 4.49: [JS7b]. Modification of the confinement rings including the optical block up to 1 MHz, every RF period can be used for phase resolved measurements of the plasma emission Excitation dynamics In order to investigate electron heating in geometrically symmetric dual-frequency discharges operated at substantially different frequencies a PIC simulation (see chapter 3.2.2) is performed under conditions similar, however not identical to the experiment described in the previous section. The simulation yields access to a variety of plasma parameters with high spatial and temporal resolution. For simplicity, helium is chosen as a model gas, since it is the main constituent of the experimentally used gas mixture. The applied frequencies and discharge geometry are identical to the experimental conditions. One of the electrodes is driven by a voltage φ(t) = φ hf sin(2πf hf t ) + φ lf sin(2πf lf t ) (4.24) The applied low and high frequency voltages are φ lf = 8 V and φ hf = 55 V, respectively. A gas pressure of 65 Pa (same as in the experiment), a gas temperature of 4 K, a secondary electron emission coefficient of γ =.45 [85] and an electron reflection coefficient of α =.2 are used as input parameters. Figure 4.5 shows the voltage drop across the discharge as a function of time within one lf period resulting from the simulation. As the PROES measurements (see figure 4.51) do not start at t = (equation 4.24), the timescale of the simulation is

106 1 CHAPTER 4. RESULTS U [V] t [ns] Figure 4.5: Voltage drop across the discharge as a function of time in a geometrically symmetric dual frequency discharge operated at MHz and MHz in helium (PIC). The applied voltage amplitudes are φ lf = 8 V and φ hf = 55 V. 1,2 Exc. rate [a.u.] Distance from powered electrode [cm] 1,,8,6,4,2, 1 2 t [ns] U PE [a.u.] t [a.u.] , 52,5 1, Figure 4.51: Experimentally determined phase and space resolved electron impact excitation rate into Ne2p 1 in the Exelan at 65 Pa and sketch of the voltage drop across the discharge (P lf = 2 W, P hf = 8 W) [JS7b,JS8a,JS9a]. shifted by 1 ns in order to synchronize the timescales of simulation and experiment (t = t 1 ns, in figure 4.5). If this synchronization is performed, the temporal trends of the excitation resulting from the simulation and the experiment agree well.

107 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 11 As the discharge is symmetric, the DC self bias is zero and, thus, similar, but 18 phase shifted, sheath dynamics are observed at each electrode. Distance from powered electrode [cm] U PE [a.u.] 1,2 1,,8,6,4,2, 1 2 t [ns] t [a.u.] Exc. rate [a.u.] ,5 65, 32,5 Figure 4.52: Phase and space resolved excitation as it results from the PIC simulation (helium, 65 Pa, φ lf = 8 V, φ lf = 55 V) [JS8a,JS9a]. Figure 4.51 shows a spatio-temporal plot of the experimentally determined excitation into Ne 2p 1 in the Exelan under the conditions described in the previous section [JS7b, JS8a, JS9a]. The spatio-temporal excitation profile is complex and a strong coupling of both frequencies is already obvious. The nature of this frequency coupling will be discussed in the following section. The relative phase of the RF potential at the bottom electrode is also shown qualitatively. Figure 4.52 shows the total spatio-temporal excitation as it results from the PIC simulation [JS8a, JS9a]. Good qualitative agreement between experiment and simulation is found. In the experiment the processing gases are different than in the simulation. Particularly, oxygen ions, which might play an important role in the experiment, are not taken into account in the simulation. The total helium excitation rate (simulation) is compared to the excitation rate into a specific neon energy level (experiment). Although the corresponding cross sections are not the same, their shapes are still sufficiently similar to justify this comparison [99, 153]. Certainly the claim of this comparison cannot be a quantitative reproduction of details of the experimentally observed excitation profiles. Nevertheless, the gross spatio-temporal structure of the measured excitation is reproduced well qualitatively by the simulation taking into account only helium gas. The excitation dynamics seems to be dominated by processes generally typical for dual-frequency discharges operated at substantially different frequencies. These processes will be discussed in detail in the following section.

108 12 CHAPTER 4. RESULTS Distance from powered electrode [cm] 1,2 Exc. rate [a.u.] 1,,8,6,4,2, t [ns] Figure 4.53: PIC simulation: Spatio-temporal excitation within the first half of one low frequency RF period [JS8a,JS9a]. Distance from powered electrode [cm] 1,2 Exc. rate [a.u.] 1,,8,6,4,2, t [ns] Figure 4.54: PIC simulation: Spatio-temporal excitation within the second half of one low frequency RF period [JS8a,JS9a]. Figures 4.53 and 4.54 show two different temporal sections of figure 4.52 (PIC simulation) within the first and second half of one lf period, respectively [JS8a, JS9a].

109 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 13 Distance from powered electrode [cm] 1,2 Exc. rate [a.u.] 1, 36,,8 27,,6 18,,4 9,,2, t [ns] Figure 4.55: Experiment: Spatio-temporal excitation within the first half of one low frequency RF period [161]. Distance from powered electrode [cm] 1,2 Exc. rate [a.u.] 1, 36,,8 27,,6 18,,4 9,,2, t [ns] Figure 4.56: Experiment: Spatio-temporal excitation within the second half of one low frequency RF period [161]. Figures 4.55 and 4.56 show two different temporal sections of figure 4.51 (experiment) within the first and second half of one lf period, respectively. Excellent agreement is found between simulation and experiment. The excitation maxima are tilted to the right during the first half, whereas they are tilted to the left during

110 14 CHAPTER 4. RESULTS the second half. In analogy to the results of PIC simulations in single frequency discharges (see chapter 4.1 and 4.2) the tilt of the excitation maxima is a consequence of a retardation caused by beams of highly energetic electrons generated by the expanding sheath. Obviously, electron beams are generated by the expanding sheath at the bottom electrode during the first half of one lf period and at the top electrode during the second half. The tilts of the arrows in figures yield a beam velocity of about m/s (corresponding to approximately 2 ev energy). This agrees fairly well with the threshold energy for excitation used in the simulation. This result is also in good agreement with a previous detailed experimental determination of plasma parameters [JS7b]. The question why electron beams are only generated at a given electrode during distinct phases of one lf period is closely related to the frequency coupling. Figure 4.57: Experimentally determined phase and space resolved optical emission from the He3 3 S state in the Exelan at 65 Pa in a helium-oxygen discharge with low temporal resolution of ns averaging over the hf dynamics. The distance from the powered electrode is normalized to the electrode gap of 1.2 cm [TG6]. Figure 4.57 shows the measured spatio-temporal emission from the He3 3 S state at 76.5 nm in the Exelan within one lf period [TG6]. The discharge is operated in a helium-oxygen mixture at 65 Pa. The flow rates of helium and oxygen are f he = 15 sccm and f O2 = 1 sccm, respectively. These measurements are performed with low temporal resolution of ns. Thus, dynamics within the hf period are not resolved. Two double peak structures can be easily identified: two peaks (indicated as 2 and 27) at the bottom electrode at different phases and two peaks (indicated as 27 and 2 ) close to the top electrode at the same phases. The two peaks at each electrode are separated by half a low frequency period. In principle, these peaks can also be identified in figures 4.51 and 4.52, if the excitation is averaged

111 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 15 over one hf period. Separate power variations of both frequency components show a diagonal correlation of the peaks. With increasing 27 MHz power the peaks indicated as 27 and 27 increase in relation to the other two peaks, whereas for increasing 2 MHz power the peaks indicated as 2 and 2 increase correspondingly. This result shows, that the 2 MHz component does affect the ionization/excitation and, thus, separate control of ion energy and flux is expected to be limited. The physical origin of this diagonal correlation is understood in the frame of the frequency coupling discussed in the next section. Figure 4.58: Normalized modulation of the spatially integrated emission, in single frequency (2 MHz only) and dual frequency operation as a function of the lowfrequency phase [TG6]. Figure 4.58 shows the temporal modulation of the emission, spatially integrated and normalized to the phase averaged emission, for single (2 MHz only) and dual frequency operations [TG6]. The emission maxima, of dual and single frequency operation, are clearly separated by distinct phase shifts of about 45, i.e. the emission maximum in the dual frequency case occurs 45 earlier than in the single frequency case. This phenomenon is again explained by the frequency coupling (see next section). Figures 4.59 and 4.6 show sections of figures 4.51 and 4.52, respectively, directly in front of the powered bottom electrode during the first half of one low frequency period [162], [JS8a, JS9a]. Figure 4.59 shows the experimentally determined excitation.4 mm in front of the bottom electrode. The main maxima correspond to the beam trajectories shown in figures 4.51 and However, in this profile more detailed structures are observed. Between two major excitation maxima, that occur at the phases of high frequency sheath expansion, additional maxima are detected. These weaker maxima occur at distinct phases of high frequency sheath collapse. The PIC simulation yields similar results. Figure 4.6 shows the excitation as it results from the simulation at two positions in front of the bottom electrode. At

112 16 CHAPTER 4. RESULTS 6 Excitation [a.u.] t [ns] Figure 4.59: Experimentally determined excitation into Ne2p 1 at a distance of.4 mm in front of the bottom powered electrode during the first half of the low frequency period. The circles indicate excitation caused by a localized field reversal [162], [JS8a,JS9a]. Excitation [a.u.] mm.4 mm t [ns] 4, 3,5 3, 2,5 2, 1,5 1,,5, -,5 Excitation [a.u.] Figure 4.6: Spatio-temporal excitation at two different positions close to the powered electrode as it results from the PIC simulation [JS8a,JS9a]..4 mm only the smaller maxima corresponding to the encircled maxima in the experimental plot are observed. At 1.5 mm only the maxima caused by the hf sheath expansion are visible. Similar to the experimental results the maxima observed at

113 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 17.4 mm and 1.5 mm are phase shifted. In the experiment, the spatial resolution is only about 1 mm. Therefore, both mechanisms are observed at the same position in the experiment and are spatially resolved only in the simulation. Compared to the collisionless single frequency case at low pressures, discussed in chapter 4.2, excitation at the phase of collapsing sheath is also observed in a dualfrequency discharge in a collisional regime at higher pressures (65 Pa). Due to this analogy the excitation might also be caused by a local field reversal. Based on the analytical model developed in chapter (equation 4.23) this field reversal is either caused by a collisional drag force on the electrons and/or electron inertia. Both mechanisms prevent electrons from advancing into the sheath. The analytical model to describe the field reversal will be applied quantitatively to this dual frequency case using input parameters from the simulation in chapter This additional excitation is again only observed at a given electrode during one half of the lf period, i.e. it is also affected by the frequency coupling Frequency coupling In this section the physical nature of the frequency coupling in symmetric dualfrequency discharges operated at substantially different frequencies is discussed. Distance from powered electrode [cm],3,25,2,15,1,5, t [ns] Ion density [1 16 m -3 ] Figure 4.61: Spatio-temporal plot of the ion density in front of the bottom powered electrode as it results from the PIC simulation under the same conditions as before [JS9a]. The solid black line corresponds to the sheath width s calculated using equation 2.2. Figure 4.61 shows the ion density space and phase resolved in front of the bottom powered electrode as it results from the PIC simulation [JS9a]. The oscillation of the sheath width s (solid black line) is also shown. The sheath width as a function of time is calculated by equation 2.2. Analogical results are obtained close to the

114 18 CHAPTER 4. RESULTS top grounded electrode. In dual-frequency discharges the sheath can be considered to be a superposition of lf and hf component. In this picture the hf sheath oscillates around the position of the lf sheath edge. The total sheath oscillation is the sum of lf and hf oscillations. Under the conditions investigated here, the velocity of the total sheath oscillation is dominated by the hf component (f hf = 14f lf ). However, the spatial position of the hf sheath oscillation is determined by the lf voltage. In CCRF discharges the ion density is high in the plasma bulk and decreases monotonically towards the electrode due to flux continuity. The sheath width s is a function of the ion density. At a given sheath voltage the sheath width is big, if the ion density at the position of sheath oscillation is low. As the ion density decreases towards the electrode, the lf sheath voltage component determines the local ion density at the position of hf sheath oscillation. Consequently, it determines the amplitude of the hf sheath oscillation at a given hf sheath voltage via controlling the position of the lf sheath edge. Due to this coupling of both frequencies the hf sheath oscillates in a region of low ion density during lf sheath collapse. Therefore, maximum excitation is observed during the period of lf sheath collapse at phases of fastest hf sheath expansion (figures 4.51 and 4.52). This coupling of both frequencies explains why electron beams are only generated during one half of the lf period at a given electrode (see figures ). Only during one half the lf sheath width is small and the hf sheath oscillates in a region of low ion density and expands fast. It also explains the diagonal correlation of the emission maxima in figure The peaks 2 and 2, corresponding to minimum low frequency sheath voltages, are strongly dependent on the 2 MHz power, since at these phases the sheath expansion velocity is particularly sensitive to changes of the spatial structure of the sheath. The peaks 27 and 27 correspond to excitation at phases of high low frequency sheath voltage. This energy gain, through high-frequency oscillations, is less dependent on the 2 MHz power since the spatial structure of the sheath is not as relevant. The frequency coupling also explains why excitation in dual frequency discharges operated at substantially different frequencies occurs about 45 earlier than in single frequency discharges (low frequency only, see figure 4.58). Under these conditions in both discharge types maximum excitation is observed at phases of fastest sheath expansion. In dual frequency discharges the sheath expands fastest at phases of lf sheath collapse, while in single frequency discharges (low frequency only) the (lf) sheath needs to expand at maximum velocity to cause strongest excitation. This happens about 45 later. Finally, the frequency coupling explains why excitation caused by a localized field reversal during sheath collapse is only observed at a given electrode during one half of the lf period. Only when the sheath collapses fast, electrons cannot follow the retreating sheath (due to collisions with the neutral background gas and/or electron inertia) and a reversed field is generated, that accelerates electrons towards the electrode to keep a constant current and to compensate the ion flux to the electrode within one lf period. Figure 4.61 shows that the ion density in the sheath is time modulated under the

115 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 19 conditions investigated here. It is modulated with the low as well as with the high frequency. The modulation of the ion density with the high frequency is most pronounced during the second half of one lf period, when the lf sheath is expanded. This temporal modulation is a consequence of the choice of a light gas (helium) and the high ion density. For a typical ion density in the sheath n i,s = m 3 the local ion plasma frequency in the sheath is f pi,s = 1 ω 2π pi,s = 1 21 MHz. 2π e 2 n i,s ε m He This is comparable to the applied high frequency. Therefore, the ions can react even to potential changes induced by the high frequency component. If a heavier gas is used and if the ion density in the sheath remains the same, the ions will react more slowly to potential changes in the sheath. However, the local ion plasma frequency depends only on m 1/2. For oxygen ions, which might play an important role in the experiment discussed above, ω pi,s will be reduced by a factor of less than three. Therefore, the ion density will probably not be modulated by the high frequency (27 MHz), but still by the low frequency (2 MHz). In case of even heavier gases (e.g. argon) the ion density will not be modulated at all (see chapter 4.3.6). The temporal modulation of the ion density in the sheath can be described quantitatively by a simple analytical model [JS9a] based on the continuity equation, Poisson equation and an equation describing the ion motion in the sheath: n i t + z (n iu i ) = (4.25) E z = en i ε (4.26) u i = c E (4.27) Equation 4.25 is the one dimensional continuity equation, where u i is the ion velocity. Ionization in the sheath is neglected (see figures 4.51 and 4.52). Equation 4.26 is the one dimensional Poisson equation. Here n e = is assumed in the sheath. Equation 4.27 describes the ion motion in the sheath. It is based on the assumption, that ion inertia can be neglected and the ion velocity adjusts itself instantaneously to the electric field. The square root relation applies only at sufficiently high E/p, which is fulfilled well within the sheath [163]. Neglection of the ion inertia terms is justified at sufficiently high collision rate (pressure) and slow field changes. A rough estimation shows that these conditions are fulfilled marginally here. However, as shown below, applying this simplification still captures the major physics involved. For helium c 56 (kg/mc) 1/2 [163]. As the applicability of equations is limited to the sheath region, this model is only valid inside the sheath (about 27 ns - 5 ns, z < s, see figure 4.61). Combining equations the following expression for the temporal modulation of the ion density in the sheath is derived: n i t = c ( E n i z en2 i 2ε E ) (4.28)

116 11 CHAPTER 4. RESULTS Based on flux continuity and Poisson s equation the electric field is expressed by the sheath potential U in dependence of z, where z is the distance from the electrode: E = 5 3 s 5/6 (s z) 1/3 U (4.29) Equation 4.29 is based on the assumption of a quasi-static ion density and a constant sheath width. Both is, of course, not exactly true. The electric field resulting from equation 4.29 agrees well with the electric field resulting directly from the simulation inside the sheath at the powered electrode during the second half of one lf period. dn i /dt [1 23 m -3 s -1 ] Model PIC t [ns] Figure 4.62: Comparison between the temporal modulation of the ion density.15 cm in front of the bottom electrode during the second half of one lf period as it results from the PIC simulation (red line) and from the analytical model (black line, equation 4.28) [JS9a]. Figure 4.62 shows a comparison between the modulation of the ion density.15 cm in front of the bottom electrode during the second half of one lf period as it results from the PIC simulation (black line) and from the analytical model (red line, equation 4.28) [JS9a]. For this comparison the ion density and the electric field directly resulting from the simulation are used as input parameters in equation Equation 4.29 is not used. Good agreement between model and simulation results is found. Deviations are found at phases, when s.15 cm, since the model is only valid in the sheath. Similar results are obtained at different positions in the sheath. Using this analytical model the physical mechanism leading to the strong hf modulations of the ion density only during the second half of one lf period are understood, since the contributions of the individual terms in equation 4.28 can be distinguished. Figure 4.63 shows these contributions for a given position in the sheath [JS9a]. Obviously, both terms are out of phase and have different signs. The ion density increases with time, if the absolute value of the ion velocity and the gradient of the

117 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 111 Different terms [1 23 m -3 s -1 ] Term 2. Term Sum t [ns] Figure 4.63: Contribution of the first and second term on the RHS of equation 4.28 to the temporal modulation of the ion density.15 cm in front of the bottom electrode and sum of the two terms [JS9a]. ion density are high. It decreases, if the ion density and the gradient of the ion velocity are high. If equation 4.29 is used to determine the electric field from the sheath potential assuming a constant sheath width and if a temporally constant ion density and ion density gradient in equation 4.28 are assumed, the effect of the temporal modulation of the ion velocity and its spatial gradient caused by the modulated sheath potential on the temporal modulation of the ion density can be demonstrated. The modulation of the sheath potential is high only during the second half of one lf period. Under these assumptions the modulation of the sheath potential yields a similar modulation of the ion density as observed in figure 4.63, since it affects the two terms in equation 4.28 inversely. However, the amplitude of the modulation under these assumptions is only about 5 % of the amplitude observed in figure The remaining part is caused by the temporal modulation of the ion density and its spatial gradient. However, the modulation of the ion velocity and its spatial gradient caused by the modulated sheath potential have a strong impact on the modulation of the ion density. During the first half of one lf period the modulation of the sheath potential is much lower and the sheath is collapsed completely during about 5 % of the first half. Therefore, the ion density is less time modulated during this half period.

118 112 CHAPTER 4. RESULTS Electric field reversals during sheath collapse in geometrically symmetric dual frequency discharges at intermediate pressures In this section the physical origin of the excitation observed experimentally as well as in the PIC simulation at phases of sheath collapse in symmetric dual frequency discharges operated at substantially different frequencies (see figures 4.59 and 4.6) will be explained by a localized field reversal based on simulation results. It will be demonstrated that such field reversals also occur in other gases than pure helium. The physical causes of this field reversal will be identified based on an explicit application of the analytical model described in section to the helium dual frequency discharge investigated here. Input parameters for the model are taken from the PIC simulation. Simulation results Distance from powered electrode [cm] 1,2 1,,8,6,4,2, t [ns] Figure 4.64: Phase and space resolved electric field in a dual frequency helium discharge operated at 65 Pa as it results from a PIC simulation. The electric field is given in units of kv/m [JS8a]. Figure 4.64 shows the result of the PIC simulation in terms of a spatio-temporal plot of the electric field under the conditions of figure 4.6 (helium, 65 Pa, φ lf = 8 V, φ lf = 55 V) [JS8a]. A modulation of the sheath electric field with both the high and low frequency is observed. The colour scale is chosen such that brown corresponds to a reversed field at the bottom electrode and dark green to a reversed field at the top electrode. Except for the phases of field reversal the electric field at the bottom electrode is negative and the electric field at the top electrode is positive.

119 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 113 Distance from powered electrode [cm],3,25,2,15,1,5, t [ns] Figure 4.65: Zoom into figure 4.64 within the area indicated by the white rectangle. The electric field is given in units of kv/m [JS8a]. Figure 4.65 shows the indicated area in figure 4.64 enlarged, i.e. a spatio-temporal plot of the electric field close to the bottom powered electrode during the first half of one low frequency period under the same conditions [JS8a]. At phases of collapsing high and low frequency sheath there is a local reversal of the electric field at the sheath edge. A reversed field is indicated by the white color and clearly observable in this plot. At the same phases additional excitation is observed (see figures 4.59 and 4.6). Therefore, this excitation seems to be caused by a local field reversal. The same effect is observed at the top electrode during the second half of one lf period, since the discharge is symmetric. Figure 4.66 shows a spatial profile of the electric field under these conditions at the specific phase of τ = 25 ns (red line in figure 4.65) [JS8a]. At this phase the low and high frequency sheath collapse simultaneously. Close to the bottom electrode, the field is positive (field reversal), and close to the sheath edge it shows a minimum (negative field). Towards the bulk it approaches zero. The maximum observed value of the reversed field is about 13 kv/m directly at the electrode. Figure 4.67 shows a spatio-temporal plot of the electron heating in the Exelan as it results from a PIC simulation under the same conditions as before [JS8a]. Here the electron heating is defined as power dissipated to electrons: p(t) = j e (t)e(t), (4.3) where j e is the electron conduction current density and E the electric field. Red corresponds to heating and blue to cooling (negative heating). Heating is generally observed at phases of hf sheath expansion, when beams of highly energetic electrons are generated by the expanding sheath. However, positive heating is also observed

120 114 CHAPTER 4. RESULTS 15 ns 12 E [kv/m] ,,1,2,3,4,5,6 Distance from powered electrode [cm] Distance from powered electrode [cm] Figure 4.66: Electric field in dependence of the distance from the bottom powered electrode in a dual frequency helium discharge operated at 65 Pa at a specific phase of 25 ns (red line in figure 4.65) as it results from a PIC simulation [JS8a]. 1,2 El. heating [a.u.] 1 1, 8 6,8 4,6 3 1,4-1,2-3 -5, t [ns] Figure 4.67: Spatio-temporal plot of the electron heating in a dual frequency helium discharge operated at 65 Pa as it results from a PIC simulation. Phases, when a local field reversal is observed, are indicated by circles [JS8a]. at phases, when hf and lf sheaths collapse simultaneously and a reversed field is observed (circles in figure 4.67). The deposited power at these phases leads to

121 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 115 the experimentally observed excitation at phases of field reversal (see figures 4.59 and 4.6). The observed positive heating in the sheath at phases of full lf sheath expansion is caused by secondary electrons. Distance from powered electrode [cm],3,25,2,15,1,5, t [ns] Figure 4.68: Spatio-temporal plot of the electric field close to the bottom powered electrode in a dual frequency argon discharge operated at 13 Pa during the first half of one low frequency period (PIC simulation). The electric field is given in units of kv/m [JS8a]. Such a field reversal is not only observed in helium dual-frequency discharges, but also under different conditions in other gases. Figure 4.68 shows the space and time resolved electric field close to the bottom electrode during the first half of one lf period in a dual-frequency argon discharge at 13 Pa as it results from the PIC simulation [JS8a]. The applied frequencies and geometry are the same as before. The applied voltages are φ lf = 1 V and φ hf = 5 V. A gas temperature of 5 K, a secondary electron emission coefficient of γ =.25 and an electron reflection coefficient of α =.2 are used as input parameters. The white areas in figure 4.68 correspond to reversed fields again. Figure 4.69 shows a spatial profile of the electric field at τ = 27 ns in argon [JS8a]. Again a reversed field at the bottom electrode is observed. The electric field shows a minimum (negative field) close to the sheath edge and approaches zero towards the bulk. The local minimum of the electric field at the sheath edge corresponds to an ambipolar field. As will be discussed in detail below by an analytical model, the heavy ions cannot follow the collapsing sheath as fast as the electrons. Consequently, an ambipolar field builds up to couple electron and ion motion. A similar local extremum of the electric field at the position of maximum sheath width during the phase of sheath collapse is observed in the simulation of a single

122 116 CHAPTER 4. RESULTS E [V/m] ns -5,,1,2,3,4,5,6 Distance from powered electrode [cm] Figure 4.69: Electric field in dependence of the distance from the bottom powered electrode at 13 Pa at a specific phase of 27 ns (red line in figure 4.68) [JS8a]. frequency discharge modeled by the fluid sheath model described in chapter Figure 4.7 shows three spatially resolved electric field profiles close to the powered electrode in a symmetric single frequency argon discharge operated at MHz and 1 Pa [JS8a]. The time averaged field (solid black line) and the field at the phase of maximum sheath expansion (dotted blue line) as well as at the phase of minimum sheath expansion (dashed red line) are shown. The simulation is performed in a cylindrical discharge with equal electrode areas of 5 cm 2 and an inter electrode spacing of 6 cm. An electron temperature of 2.5 ev, a gas temperature of 35 K, and a RMS voltage of 19 V are used as input parameters. Absolute values of the electric field are plotted in figure 4.7 on a logarithmic scale in order to emphasize the local extremum during sheath collapse. All fields are negative under these conditions. No reversed field is observed. However, similar to the dual-frequency case a local extremum at maximum sheath width during the sheath collapse, an ambipolar field, is observed.

123 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES Average Electric Field Minimum Sheath Extension Maximum Sheath Extension E [V/cm] 1 1,,1,2,3,4,5,6,7,8 Distance from powered electrode [cm] Figure 4.7: Absolute values of spatially resolved electric fields close to the powered electrode in a single frequency discharge in argon at 1 Pa calculated using a fluid sheath model [17, 148]. The time averaged field (solid black line) and the field at the phase of maximum sheath expansion (dotted blue line) as well as at the phase of minimum sheath expansion (dashed red line) are shown [JS8a]. Physical origin of the field reversal In this section the analytical fluid model introduced in chapter is explicitly applied to a dual frequency discharge operated in helium at 65 Pa (same conditions as discussed before) to understand the physical origin of the observed field reversal (see figures 4.64 and 4.65). These discharge conditions are investigated by a PIC simulation and similar conditions are investigated experimentally. Equation 4.23 is applied to a given set of input parameters provided by the PIC simulation. These input parameters are the electron current density and ion density profiles as well as the collision frequency (ν c s 1 ) and electron temperature (T e 2.6 ev). This approach yields a detailed understanding of the cause of an observed field reversal, since different mechanisms can be separated in the model. A field reversal is observed during different time intervals within one lf cycle (see figures 4.64 and 4.65), when lf and hf sheath collapse simultaneously. During these time intervals additional excitation is observed close to the bottom powered electrode (see figures 4.59 and 4.6). For example, a strong field reversal is observed between the phases of 22 ns and 34 ns. In the following, this time interval is discussed in detail. Figures 4.71, 4.72, 4.74, and 4.75 show a comparison between the electric field resulting from the analytical fluid model (equation 4.23, black line 1) using input

124 118 CHAPTER 4. RESULTS E [V/m] ns 2 1 E FM 2 E PIC 3 (n i - n e )/n i ,,1,2,3,4,5-2,6 Distance from powered electrode [m] (ni -n e )/n i [%] Figure 4.71: Spatial profile of the electric field in front of the powered electrode as it results from the analytical fluid model (equation 4.23, black line 1) and the PIC simulation (red line 2) in a dual-frequency discharge in helium at 65 Pa at 23 ns. The spatial profile of the relative deviation from quasi-neutrality is also shown (blue line 3, right scale) [JS8a]. parameters from the PIC simulation and the electric field directly resulting from the PIC simulation (red line 2) [JS8a]. The figures show spatial profiles of the electric field at different phases within one hf period (23 ns, 24 ns, 25 ns, 33 ns). Good agreement between the fluid model and PIC simulation is found as long as the condition of quasi-neutrality is fulfilled (blue line 3). At 23 ns (figure 4.71) the sheath is retreating, however it is not yet fully collapsed. In the sheath, where quasi-neutrality is violated, the fluid model is not applicable. As the sheath collapses electrons are transported from the bulk towards the electrode in order to fill up the ion matrix. Since the ion density close to the electrode is lower than in the bulk a higher electric field is needed in the sheath region compared to the bulk region in order to keep a constant current. This field is generated by a local negative charge excess at the sheath edge, which is clearly visible in figure 4.71 (23 ns).

125 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 119 E [V/m] ns 1 E FM 2 E PIC 3 (n i - n e )/n i (ni -n e )/n i [%] -6-1,,1,2,3,4,5,6 Distance from powered electrode [m] Figure 4.72: Spatial profile of the electric field in front of the powered electrode as it results from the analytical fluid model (equation 4.23, black line 1) and the PIC simulation (red line 2) in a dual-frequency discharge in helium at 65 Pa at 24 ns. The spatial profile of the relative deviation from quasi-neutrality is also shown (blue line 3, right scale) [JS8a]. Figure 4.72 (24 ns) shows a comparison between electric field profiles obtained from the fluid model (black line 1) and the simulation (red line 2) one nanosecond later. Now the sheath is fully collapsed and the condition of quasi-neutrality is fulfilled almost everywhere. Only at the sheath edge an excess of negative charges is observed again. High values of the reversed field (6 kv/m) are observed and well reproduced by the fluid model. Figure 4.73 shows the respective contributions of the individual terms of equation 4.23 to the total electric field at this phase [JS8a]. The black line 1 in figure 4.72 (24 ns) is the sum of these four terms. The origin of the negative field between the bulk and the sheath edge is clearly electron diffusion. As electrons are much lighter than the He ions, they can follow the collapsing sheath faster by diffusion. The ions cannot follow and the electrons move away from them. This leads to an ambipolar field between the negative charge excess close to the sheath edge and the positive charge excess in the bulk. The amplitude of this ambipolar field is determined by the decay length of the ion density 1 n i. n i The reversed field at 24 ns is caused z by both electron inertia and electron collisions with the neutral background gas to similar extends. At the other phases discussed here the relative contributions of the individual terms to the electric field are similar. Obviously, the discharge is operated in a transition regime between the inertially and collisionally dominated regime. Under these conditions both mechanisms contribute equally to the field reversal. At much lower pressures, such as the single frequency case discussed in

126 12 CHAPTER 4. RESULTS E [kv/m] 4, 3,5 Inertia term I 3, 2,5 2, 1,5 1,,5, -,5,,1,2,3,4,5,6 Distance from powered electrode [m] E [kv/m] 4, 3,5 Collisional term 3, 2,5 2, 1,5 1,,5, -,5,,1,2,3,4,5,6 Distance from powered electrode [m] E [kv/m] 1,8 1,6 Inertia term II 1,4 1,2 1,,8,6,4,2, -,2,,1,2,3,4,5,6 Distance from powered electrode [m] E [kv/m] Diffusion term -7,,1,2,3,4,5,6 Distance from powered electrode [m] Figure 4.73: Spatial profiles of the different terms of equation 4.23, that contribute to the electric field in front of the powered electrode at 24 ns. Inertia term I corresponds to the first term in equation 4.23, Collision term to the second, Inertia term II to the third and Diffusion term to the fourth [JS8a]. chapter 4.2.5, the field reversal can only be caused by electron inertia. At much higher pressures collisions will be responsible for a field reversal. One more nanosecond later, at 25 ns, the reversed electric field is even stronger. At this phase an interesting phenomenon is observed: As shown in figure 4.74 (25 ns) quasi-neutrality is violated in the vicinity of the electrode and the fluid model is no longer applicable. However, there is no conventional sheath consisting of an excess of positive charges. At this phase the electrode charges up positively due to ion bombardment and to compensate the ion current. Consequently, a sheath filled with electrons is observed at this phase. A similar phenomenon was observed by Vender theoretically in a single frequency discharge [159]. At the end of the time interval, when a field reversal is observed, the (ion) sheath expands again. The corresponding electric field profiles at a phase of 33 ns are shown in figure Shortly after this phase, a beam of energetic electrons is generated

127 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 121 E [V/m] ns 1 E FM 2 E PIC 3 (n i - n e )/n i 3-1,,1,2,3,4,5,6 Distance from powered electrode [m] (ni -n e )/n i [%] Figure 4.74: Spatial profile of the electric field in front of the powered electrode as it results from the analytical fluid model (equation 4.23, black line 1) and the PIC simulation (red line 2) in a dual-frequency discharge in helium at 65 Pa at 25 ns. The spatial profile of the relative deviation from quasi-neutrality is also shown (blue line 3, right scale) [JS8a]. by the expanding sheath that penetrates into the plasma bulk.

128 122 CHAPTER 4. RESULTS ns 4 3 E [V/m] E FM 2 E PIC 3 (n i - n e )/n i -6 3, -1,1,2,3,4,5,6 Distance from powered electrode [m] 2 1 (ni -n e )/n i [%] Figure 4.75: Spatial profile of the electric field in front of the powered electrode as it results from the analytical fluid model (equation 4.23, black line 1) and the PIC simulation (red line 2) in a dual-frequency discharge in helium at 65 Pa at 33 ns. The spatial profile of the relative deviation from quasi-neutrality is also shown (blue line 3, right scale) [JS8a]. Model to describe the ambipolar field during sheath collapse In argon, the field reversal is much less pronounced (see figures 4.68 and 4.69). However, an ambipolar field that couples electron and ion motion is also observed. Figure 4.76 shows the spatial profile of the electric field in argon close to the powered bottom electrode during the phase of sheath collapse at 27 ns as it results from a PIC simulation (red line 2) [JS8a]. In this case the discharge is operated at 65 Pa. The applied frequencies and geometry are the same as before. The applied voltages are φ lf = 5 V and φ hf = 22 V. A gas temperature of 6 K, a secondary electron emission coefficient of γ =.1, and an electron reflection coefficient of α = are used as input parameters. In order to verify the hypothesis, that the extremum present in figure 4.76 is caused by an ambipolar field, the field is calculated by a model based on the assumption of an ambipolar field using the ion density profile and electron temperature (T e = 1, 4 ev) resulting from the PIC simulation as input parameters. The ambipolar field E amb is given by [2]: E amb = D i D e µ i + µ e n i n i D e µ e n i n i (4.31) Here D i and D e are the ion and electron diffusion coefficients, respectively, and µ i and µ e are the ion and electron mobilities, respectively. The electron diffusion

129 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 123 E [kv/m],5, -,5-1, -1,5-2, -2,5-3, -3,5-4, E amb 2 E PIC 3 (n i - n e )/n i,,1,2,3,4,5,6 Distance from powered electrode [m] (ni -n e )/n i [%] Figure 4.76: Electric field as a function of the distance from the bottom powered electrode in a dual frequency argon discharge operated at 65 Pa at a specific phase of 27 ns as it results from a model to describe the ambipolar field (black line 1) and PIC simulation (red line 2). The relative deviation from quasi-neutrality is also shown [JS8a]. coefficient and mobility are generally given by [2]: D e = 4π v 4 3n e ν m (v) f edv (4.32) µ e = 4πe v 3 df e dv 3mn e ν m (v) dv (4.33) Here ν m (v) is the effective collision frequency for momentum transfer for electrons and f e the isotropic part of the electron distribution function f e. Generally ν m (v) depends on velocity and, therefore, the above integrals must be solved explicitly to get D e and µ e. However, in case of Maxwellian distribution functions D e and µ e are related by the Einstein relation [2]: D e = kt e µ e e (4.34) Figure 4.77 shows the normalized time averaged electron distribution function in the plasma bulk as it results from the PIC simulation under the conditions discussed here [JS8a]. As a good approximation the distribution function is Maxwellian and the Einstein relation is, therefore, used to calculate the ambipolar field (equation 4.31).

130 124 CHAPTER 4. RESULTS f(e)/(n e E 1/2 ) [ev -3/2 ] E [ev] Figure 4.77: Normalized time averaged electron distribution function in the centre of the discharge as it results from the PIC simulation (argon, 65 Pa) [JS8a]. The result of this calculation is shown in figure 4.76 (black line 1). As the electric field profile obtained from the simulation is reproduced well, the local extremum of the electric field near the sheath edge can be identified with an ambipolar field. In a single frequency discharge ambipolar fields during the phase of sheath collapse are also observed (figure 4.7).

131 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES Secondary electrons Until now only electron heating caused by the expanding and collapsing sheath in dual frequency discharges operated at substantially different frequencies has been discussed. However, it is well known from single frequency discharges that secondary electrons generated at the electrode surface can contribute significantly to electron heating and excitation/ionization dynamics [84] (see chapter 2.4). The effect of secondary electrons on the excitation/ionization dynamics in dual frequency discharges is discussed in this section based on simulation results. Effect of secondary electrons on the spatio-temporal excitation In order to identify the influence of secondary electrons on the excitation a PIC simulation of an argon discharge operated at 65 Pa is performed with and without taking secondary electrons into account. φlf = 5 V, φhf = 22 V, T g = 6 K, γ =.1 and α = are used as input parameters for the simulation taking into account secondary electrons. φlf = 5 V, φ hf = 71 V, T g = 6 K, γ = and α = are used for the simulation neglecting secondary electrons. Distance from powered electrode [cm] 1,2 1,,8,6,4,2, 1 2 t [ns] U PE [a.u.] % (?Y) t [a.u.] Exc. rate [a.u.] Figure 4.78: Spatio-temporal excitation in an argon discharge operated at 65 Pa including secondary electrons (γ =.1) [JS9a]. Figure 4.78 shows the spatio-temporal excitation including secondary electrons (γ =.1) and figure 4.79 shows the spatio-temporal excitation without secondary electrons (γ = ) [JS9a]. In both cases the excitation is modulated by the high fre-

132 126 CHAPTER 4. RESULTS Distance from powered electrode [cm] 1,2 1,,8,6,4,2, 1 2 t [ns] U PE [a.u.] % (?Y) t [a.u.] Exc. rate [a.u.] Figure 4.79: Spatio-temporal excitation in an argon discharge operated at 65 Pa without secondary electrons (γ = ) [JS9a]. quency. Therefore, these maxima are clearly not caused by secondary electrons, but by the expanding sheath. However, secondary electrons cause a modulation of the excitation with twice the low frequency. Figures 4.78 and 4.79 clearly show this effect. Without secondary electrons (figure 4.79) there is no such modulation. Secondary electrons are produced at the electrode surface by ion bombardement and are accelerated into the plasma bulk by the sheath potential. They can be multiplied ionizing through collisions, if the mean free path is shorter than the sheath width. Therefore, secondary electrons contribute to the excitation at phases of high sheath potential. The sheath potential is dominated by the low frequency component and is maximum at each electrode once per lf period. The phases of maximum sheath potential at each electrode are 18 phase shifted (symmetric discharge). Therefore, secondary electrons originating from the top electrode contribute to the excitation during the first half of one lf period (7 ns - 2 ns). During the second half (33 ns - 46 ns) secondary electrons originating from the bottom electrode contribute to the excitation. In between the contribution of secondary electrons is lower causing a modulation of the excitation with twice the low frequency. The hf component might lead to small high frequency modulations of the excitation due to secondary electrons, which cannot be identified in this case. Secondary electrons might also contribute to the excitation observed in a helium discharge operated at 65 Pa in a similar way (see figures 4.51 and 4.52). However, in this case sheath expansion heating dominates.

133 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 127 Under these conditions and taking into account secondary electrons the discharge is operated in a hybrid mode. Electrons accelerated by both, sheath expansion and secondary electrons, contribute to the excitation dynamics. At lower pressures, the discharge is rather operated in α-mode and mostly sheath expansion and collapse (field reversal) contribute to the excitation. However, α-mode operation of dualfrequency discharges differs significantly from α-mode operation of single frequency discharges, since the frequency coupling plays an important role. Besides the frequency coupling described in the previous sections, secondary electrons can also limit separate control of ion flux and ion energy in dual-frequency discharges operated at substantially different frequencies. If secondary electrons play an important role, the low frequency component will contribute significantly to ionization, since it dominates the sheath potential under conditions typical for industrial applications. γ-mode operation Distance from powered electrode [cm] 1,2 1,,8,6,4,2, 1 2 t [ns] U PE [a.u.] % (?Y) t [a.u.] Exc. rate [a.u.] Figure 4.8: Phase and space resolved excitation as it results from the PIC simulation of a dual-frequency discharge operated at MHz and MHz in helium at 12 Pa [JS9a]. With increasing pressure the mean free path for electrons decreases. At 65 Pa (see figures 4.51 and 4.52) the mean free path for highly energetic electrons (E > 4 ev) in helium is about 4.3 mm [15]. The maximum sheath width under these conditions

134 128 CHAPTER 4. RESULTS is about 3 mm (see figure 4.61). Therefore, most of the secondary electrons leave the sheath without collision and, therefore, without multiplication. At higher pressure of 12 Pa the mean free path is only 2.3 mm ( φ lf = 28 V, φ hf = 14 V, T g = 4 K, γ =.45 and α =.2). This is similar to the sheath width under these conditions. Therefore, secondary electrons are multiplied in the sheath and the excitation/ionization is dominated by secondary electrons (see figure 4.8 [JS9a]). The modulation of the excitation due to secondary electrons by the hf component is significantly less pronounced than the modulation of the excitation by the high frequency caused by sheath expansion heating. Ionization vs. excitation Distance from powered electrode [cm] 1,2 1,,8,6,4,2, 1 2 t [ns] U PE [a.u.] % (?Y) t [a.u.] Exc. rate [a.u.] Figure 4.81: Spatio-temporal excitation in an argon discharge operated at 65 Pa ( φ lf = 5 V, φ hf = 22 V, T g = 6 K, γ =.1 and α = ) [JS9a]. Another interesting phenomenon related to secondary electrons is observed in the simulation: In experiments only the excitation can be calculated from the space and time resolved plasma emission from a specifically chosen rare gas level measured by PROES (see chapter 3.1.1, equation 3.3). However, in order to investigate mechanisms of plasma sustainment, the ionization needs to be known. In this context it is usually assumed in experiments, that the excitation probes the ionization [112, 115, 164]. Usually this assumption is true, since electrons, that have enough energy to ionize, typically also excite. However, if secondary electrons play an important role, the situation can be different. Secondary electrons can be so energetic

135 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 129 Distance from powered electrode [cm] 1,2 1,,8,6,4,2, 1 2 t [ns] U PE [a.u.] % (?Y) t [a.u.] Ion. rate [a.u.] Figure 4.82: Spatio-temporal ionization in an argon discharge operated at 65 Pa ( φ lf = 5 V, φ hf = 22 V, T g = 6 K, γ =.1 and α = ) [JS9a]. (up to several hundred ev), that the cross section for ionization at these high energies is high, whereas the cross section for excitation is low [86]. Figures 4.81 and 4.82 show an example for such a scenario in argon [JS9a]. Under these conditions the mean energy of electrons causing excitation is about 3 ev and the mean energy of electrons causing ionization is about 9 ev. These values are obtained from a statistical analysis of the simulation data. At 9 ev the cross section for ionization is high and the cross section for excitation is low. At about 3 ev the cross section for excitation is maximum. Therefore, in contrast to usual assumptions the excitation (figure 4.81) does not probe the ionization (figure 4.82) under these conditions. The excitation is dominated by electrons accelerated by the expanding sheath and the ionization is dominated by secondary electrons.

136 13 CHAPTER 4. RESULTS Electron dynamics under different discharge conditions Distance from powered electrode [cm] 1,2 1,,8,6,4,2 Argon, 65 Pa, MHz,,,2,4,6 T/T lf,8 1, U PE [a.u.] % (?Y) T/T lf Exc. rate [a.u.] Distance from powered electrode [cm] Argon, 65 Pa, MHz,4 Ion density,35 [1 16 m -3 ] 5,,3 4,4,25 3,8,2 3,2 2,6,15 2,,1 1,4,5,9,3,,,2,4,6,8 1, T/T lf Distance from powered electrode [cm] 1,2 1,,8,6,4,2,,,2,4,6 T/T lf,8 1, U PE [a.u.] Helium, 65 Pa, MHz T/T lf Exc. rate [a.u.] Distance from powered electrode [cm],3,25,2,15,1,5 Helium, 65 Pa, MHz,,,2,4,6,8 1, T/T lf Ion density [1 16 m -3 ] Distance from powered electrode [cm] 2, 1,8 1,6 1,4 1,2 1,,8,6,4,2 Argon, 1 Pa, MHz,,,2,4,6 T/T lf,8 1, U PE [a.u.] % (?Y) T/T lf Exc. rate [a.u.] Distance from powered electrode [cm] Argon, 1 Pa, MHz,4 Ion density,35 [1 15 m -3 ],3 2,5 2,3,25 2,1,2 1,9,15 1,7 1,5,1 1,3,5 1,1,9,,,2,4,6,8 1, T/T lf Figure 4.83: Left: Spatio-temporal plots of the excitation resulting from the PIC simulation in different gases, at different pressures and frequencies [JS9a]. Right: Spatio-temporal plots of the ion density under the same conditions (only the sheath region at the bottom powered electrode is shown). The vertical dashed lines mark the phases of strongest excitation [JS9a].

137 4.3. CLASSICAL GEOM. SYMM. DUAL FREQUENCY DISCHARGES 131 In this section excitation dynamics in geometrically symmetric dual frequency discharges operated in different gases, at different pressures and frequencies is discussed. Figure 4.83 shows spatio-temporal plots of the excitation and the ion density for three different sets of conditions [JS9a]. The first scenario corresponds to the argon discharge operated at MHz and MHz and 65 Pa, that was introduced in section (see figures 4.78 and 4.79). The second scenario (helium, 65 Pa, MHz and MHz) is the same as discussed in section (see figure 4.52). The third scenario corresponds to an argon discharge operated at 1 Pa, similar frequencies of MHz and MHz, 2 cm electrode gap, φ = 3 V, γ =.1 and α =.2. The later case will be discussed in detail in chapter 4.4. The vertical dashed lines in figure 4.83 mark the phase of strongest excitation. Under each set of conditions maximum excitation is observed at different phases within one lf period indicating different excitation dynamics. In an argon discharge operated at 65 Pa and substantially different frequencies maximum excitation at the bottom electrode is observed at about 2 ns (T/T lf.39). As the argon ions are much heavier than the helium ions, the ion density close to the bottom electrode is hardly time modulated. Maximum excitation is observed at phases of fastest sheath expansion (electron beams). Due to the frequency coupling fastest sheath expansion takes place, when the hf sheath oscillates in regions of low ion density. In a helium discharge operated at 65 Pa and substantially different frequencies (1.937 MHz MHz) the ion density at the electrode is minimum during the lf sheath collapse, since the light helium ions can follow the slow modulation of the lf sheath potential. Secondary electrons contribute to the excitation in the entire plasma bulk at phases of maximum sheath potential at either electrode. Thus, the spatiotemporal excitation profile is a superposition of excitation caused by secondary electrons and excitation caused by beam electrons accelerated by the expanding (and collapsing) sheath. If the discharge is operated at substantially different frequencies the hf sheath expansion velocity is generally much higher than the lf sheath expansion velocity (d φ hf /dt >> d φ lf /dt). Therefore, excitation is dominated by the hf component and the lf component mainly contributes indirectly via the frequency coupling to the excitation, if secondary electrons can be neglected. At similar frequencies this is different. Figure 4.83 also shows the excitation in an argon discharge operated at MHz and MHz at 1 Pa. Here maximum excitation occurs at phases of simultaneous sheath expansion and both frequencies contribute directly to the excitation via similar contributions to the sheath expansion velocity. Although the discharge is geometrically symmetric, the spatio-temporal excitation is asymmetric (stronger excitation at the bottom electrode compared to the top electrode). This phenomenon is caused by an electrical asymmetry discussed in detail in chapter 4.4.

138 132 CHAPTER 4. RESULTS 4.4 Electron heating in geometrically symmetric dual frequency discharges operated at similar phase locked frequencies In this section electron heating in geometrically symmetric dual frequency discharges operated at similar phase locked frequencies, namely a fundamental and its second harmonic with variable phase shift, will be studied. The focus lies on an investigation of the recently discovered Electrical Asymmetry Effect (EAE) [165]. In chapter it will be demonstrated for the first time experimentally and by a PIC simulation, that a variable DC self bias is generated via the EAE even in geometrically symmetric discharges. The physical mechanisms causing this variable DC self bias will be analyzed and understood in detail. In the next section it will be shown, that the EAE can be utilized to achieve separate control of ion energy and flux at the electrode surfaces in an almost ideal way under a broad range of discharge conditions. Limitations of this separate control caused by the frequency coupling in conventional dual frequency discharges operated at substantially different frequencies (see chapters 4.3 and 4.4.8) are avoided using this concept. In chapter the EAE is optimized by choosing optimum voltage amplitudes of each harmonic to induce the strongest possible electrical asymmetry in dual frequency discharges to change the ion energy over a wider range. Then, excitation dynamics in electrically asymmetric, geometrically symmetric dual frequency discharges are investigated and found to work differently depending on the pressure and compared to classical CCRF discharges. In chapter it will be demonstrated that non-linear self-excited PSR oscillations of the RF current waveform can be induced also in geometrically symmetric discharges via the EAE. Before it was believed that such PSR oscillations and NERH are a phenomenon purely restricted to geometrically asymmetric discharges. By tuning the phase the PSR oscillations and NERH can be turned on and off. In chapter the EAE in multi frequency discharges is shortly discussed theoretically. Finally, the quality of separate control of ion energy and flux at the electrodes via the EAE is compared to alternative concepts such as classical dual frequency discharges. The results presented in this section can be found in references [ZD9a, JS9b, UC9, ZD9b, JS9c] Experimental setup Figure 4.84 shows the experimental setup used for the investigations of electron heating in combination with the EAE in a geometrically symmetric dual frequency discharge operated at two similar phase locked frequencies [JS9b, JS9c]. Two synchronized function generators (Agilent 3325A) are used to generate the phase locked MHz and MHz voltage waveforms. The phase angle between these harmonics can be adjusted via the frequency generators. Each voltage waveform is then amplified individually by a broadband amplifier and matched individually (hf +

139 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 133 PPM Extraction hole Glass cylinder Optical filter Bulk plasma Grounded electrode Powered electrode High voltage probe ICCD Camera Filter (lf block) Filter (hf block) hf Match lf Match hf Generator lf Generator Sync. + variable phase Sync. (PROES) Figure 4.84: Experimental setup used for the investigations of electron heating and the EAE in a geometrically symmetric dual frequency discharge (13.56 MHz MHz) [JS9b,JS9c]. lf match). Behind each matchbox a filter blocks the other harmonic (lf + hf block). Each filter is a cable, whose length corresponds to an odd multiple of a quarter of the wavelength of the voltage waveform to be blocked. This voltage waveform will be blocked due to destructive interference of the incoming and reflected waveform ( λ = λ/2). Behind the filters the two voltage waveforms are added and applied to the bottom electrode. The radius of both electrodes (powered and grounded) is 5 cm. The gap between the electrodes is variable. Here gaps of 1 cm and 2.5 cm are used. Both electrodes are located in a vacuum GEC cell. The plasma is shielded from the outer grounded chamber walls by a glass cylinder. Therefore, the discharge is geometrically symmetric. It is mainly operated in argon at low powers of a few W. The ion energy and ion flux at the grounded electrode are measured by a Balzer Plasma Process Monitor 422 (PPM) [145] (see chapter 3.1.4). Some ions accelerated towards the grounded electrode by the sheath potential enter the PPM through an extraction hole of diameter 1 µm. The extraction hole is grounded. Using the PPM energy scans of ions at fixed mass (argon ions) yield ion flux energy distribution functions at the grounded electrode. The PPM is calibrated carefully with respect to its energy scale as well as the shape of the measured ion flux-energy distribution functions as described in chapter In order to measure the emission from a specifically chosen neon state (Ne2p 1 ) space and phase resolved 1 % Neon are admixed to the discharge only for the optical measurements (see chapter 3.1.1). The emission at nm is measured by an Andor Istar ICCD camera synchronized with the low frequency voltage waveform in combination with an optical interference filter. The temporal resolution of these

140 134 CHAPTER 4. RESULTS measurements is 5 ns. Images are taken at different phases within the lf period (step width of 5 ns). The resulting images are binned in horizontal direction and combined to an emission matrix providing one dimensional spatial resolution perpendicular to the electrodes of about.5 cm. From the emission the excitation is calculated using equation 3.3. A voltage waveform of the shape of equation 2.13 is applied to the bottom electrode. Here f = MHz. For the experiments presented in chapters and the amplitudes of the two voltage waveforms are chosen to be identical with an accuracy of ±3 V in order to use similar conditions as in [1]. In chapter different amplitudes are used to optimize the EAE. Due to limited mechanical access to the powered electrode the voltage drop across the discharge in terms of the superposition of both harmonics is measured by a LeCroy high voltage probe about 1.5 m in front of the powered electrode. The RF period average of the measured voltage yields the DC self bias. The amplitude of each individual harmonic is determined by a Fourier analysis of the measured superposition. Due to reflection on the cable the voltage amplitudes and the phase θ between the harmonics are different at the electrode and at the original position in front of the electrode, where the voltage is measured during plasma operation. As the voltage amplitudes and phase at the electrode must be known, when the plasma is switched on, the following calibration procedure is performed: When the chamber is vented (no plasma), the voltage is measured directly at the electrode and at the position, where the voltage is measured during plasma operation. From a comparison of these two measurements the phase shift and the calibration factors for both voltage amplitudes are determined. When no plasma is ignited, the electrode corresponds to an open end. Therefore, this calibration is only accurate and reliable, if the electrode acts similar to an open end also when the plasma is ignited. This is checked by carefully estimating the capacitive reactance of the cable to ground and the plasma impedance: The capacitive reactance of the cable to ground (about 7 Ω) was found to be significantly smaller than the plasma impedance (about 55 Ω), which is dominated by the capacitive reactance of the sheaths (the reactance and resistance of the bulk are negligible). The sheath capacitance was estimated based on the sheath width obtained from optical emission measurements. By calculating the reflection factor r = (Z p Z L ) / (Z p + Z L ), where Z p is the plasma impedance and Z L is the wave impedance of the cable, with and without plasma (open end) the error regarding the measured phase shift between the applied voltage harmonics caused by this calibration is determined to be about θ 6. This error is significantly smaller than the typical step width of the voltage measurements (15 -steps). It is caused by the fact that the plasma does not exactly, but to a very good approximation, correspond to an open end during plasma operation. In principle the measured data can be corrected for this small error. However, due to the comparably large step width of the voltage measurements it is neglected in most cases, if changes of the DC self bias on the scale of 15 are discussed. Only in case of the analysis of the Small Angle Effect, that leads to a shift of the phase of strongest DC self bias by about 7, and the optimization of the EAE the measured data is corrected for this phase shift.

141 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES Generation of a variable DC self bias in geometrically symmetric discharges In this section the generation of a variable DC self bias via the EAE in a geometrically symmetric discharge is investigated by a PIC simulation and the experiment described in the previous section. Previous results obtained by the non self-consistent Brinkmann fluid simulation (see chapter 3.2.1) and by an analytical model (see chapter 2.2) by Heil et al. [1] are verified. The PIC simulation shows small differences compared to the fluid simulation and the analytical model, which are attributed to a kinetic effect. This effect is investigated in detail. Particle in Cell simulation First, the focus lies on the scenario investigated by a fluid simulation and an analytical model before [1]. Direct comparisons between PIC and previously obtained results are performed. In the second part a more general investigation of the DC self bias generated by the EAE at different pressures and electrode gaps is performed. The first set of conditions investigated corresponds to a geometrically symmetric dual frequency discharge operated in argon at MHz and MHz at 2.66 Pa (2 mtorr) with an electrode gap of d = 6.7 cm, similar to conditions of an experimental investigation by Godyak and Piejak in a single frequency discharge [67]. A voltage waveform φ with a shape specified by equation 2.13 and with amplitudes φ lf = φ hf = 315 V is applied to the discharge. A neutral gas temperature T g = 35 K, a secondary electron emission coefficient γ =.1 and an electron reflection coefficient α =.2 are used. Figure 4.85 shows the voltages across the sheaths at the powered electrode (φ sp ) and at the grounded electrode (φ sg ) as well as the total voltage across the discharge (φ tot ) according to equation 4.35 (solid black line) at θ = within one low frequency period [ZD9a]. A DC self bias η = 213 V builds up at this phase angle. These voltages agree well with those resulting from the analytical model introduced in chapter 2.2 (figure 11 in [1]). In the analytical model the total voltage across the discharge is assumed to be the sum of the voltages across both sheaths: φ tot = φ sp + φ sg (4.35) The total voltage across the discharge is the sum of the applied voltage, φ, and the DC self bias, η: φ tot = φ + η (4.36) Figure 4.85 shows that the assumption corresponding to equation 4.35 is correct under the conditions investigated here: At the low pressure of 2.66 Pa (2 mtorr), the voltage drop across the plasma bulk is negligible and the total voltage across the discharge is indeed the sum of both sheath voltages. Figure 4.86 shows the ion density profiles in front of each electrode as calculated by the PIC simulation at θ = [ZD9a]. The dashed lines correspond to the maxi-

142 136 CHAPTER 4. RESULTS 6 Voltage [V] V sg (t) V total (t) -4-6 V sp (t) t [ns] Figure 4.85: Total voltage across the discharge, φ tot = φ+η, (green circles) and sum of both sheath voltages (solid black line), voltage across the sheath at the powered electrode, φ sp, voltage across the sheath at the grounded electrode, φ sg, and DC self bias, η, as they result from the PIC simulation at θ = [ZD9a]. The results agree well with model calculations performed before [1]. mum sheath edge at the respective electrode within one period of the fundamental frequency calculated based on equation 2.2. It clearly shows that the sheath widths at each electrode are different and that the discharge is, therefore, asymmetric, although the reactor is geometrically symmetric. The symmetry of a discharge is characterized by the symmetry parameter ε defined by equation In the analytical model the ratio of the sheath integrals is assumed to be unity [1]. The PIC simulation verifies this assumption at all phase angles investigated. Figure 4.87 shows the ratio of the sheath integrals I sg and I sp as a function of the phase angle θ [ZD9a]. The ratio is close to unity (within ±5 %) and is relatively insensitive to changes of the discharge conditions due to a change in θ. Therefore, under the assumption of temporally constant uncompensated net positive charge Q in the discharge the symmetry parameter in a geometrically symmetric discharge (A p = A g ) only depends on the ratio of the mean ion densities in the respective sheath: ε = n sp n sg (4.37) The symmetry of the applied voltage waveform can be changed by changing the phase angle θ. This leads to different absolute values of the positive and negative extremes of the applied voltage and, therefore, to a DC self bias varying with the phase angle between the two harmonics (equation 2.12). A finite bias leads to different sheath voltages, which then - at low pressures - lead to different sheath densities. Different sheath densities cause the symmetry parameter ε to deviate

143 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 137 Ion density [cm -3 ] 8x1 9 6x1 9 4x1 9 2x1 9 Powered electrode Grounded electrode,,1,2,3,4,5,6,7,8,9 Distance from electrode [cm] Figure 4.86: Time averaged ion density profiles in front of each electrode at θ =. The dashed lines show the maximum sheath edge at each electrode within one lf period calculated by equation 2.2 [ZD9a]. from unity. This yields an even stronger self bias and self amplifies the effect. Figure 4.88 shows the symmetry parameter ε as a function of θ calculated by the PIC simulation and the fluid simulation of [1] [ZD9a]. The symmetry parameter is a nearly linear function of the phase angle. Small deviations between PIC and fluid simulation are found. The minimum of ε calculated by the PIC simulation is found at θ = 7.5, whereas the fluid simulation finds a minimum at θ =. In the fluid simulation Q is temporally constant. As will be demonstrated later, Q changes with time within one lf period. This causes the difference of ε and finally also of the DC self bias resulting from the fluid and PIC simulation. Figure 4.89 shows the effect of a variation of the phase angle θ in equation 2.13 on the DC self bias η as it results from the PIC simulation, as well as from the fluid simulation and from the analytical model, respectively [ZD9a]. The self consistent PIC simulation essentially verifies the result of the models: The DC self bias is a nearly linear function of the phase angle between the applied frequencies (in the intervals θ 9 and 9 θ 18, respectively). The results from the fluid simulation and the analytical model on one hand and the PIC simulation on the other hand are basically identical. Nevertheless, both curves differ by a small phase shift of about 7. While the fluid simulation confirms the analytical prediction that extrema of the bias are reached at phase angles of and 9, respectively, the PIC simulation finds extremes at 7.5 and The origin of this Small Angle Effect (SAE) is related to the temporal changes of Q within one lf period and is discussed in detail later. Figure 4.9 shows the DC self bias normalized to the sum of the amplitudes of the applied voltage [ZD9a].

144 138 CHAPTER 4. RESULTS I sg / I sp 2, 1,8 1,6 1,4 1,2 1,,8,6,4,2, [Degree] Figure 4.87: Ratio of the sheath integrals I sg and I sp as a function of the phase angle θ. The ratio is relatively insensitive to changes in the sheath caused by a change of θ and can be approximated to be unity [ZD9a]. 1,6 1,5 1,4 1,3 1,2 1,1 1,,9,8,7,6,5 PIC simulation Fluid simulation [Degrees] Figure 4.88: Symmetry parameter defined by equation 2.11 as a function of the phase angle θ (black line (squares) - PIC simulation, blue line (triangles) - fluid simulation [1]) [ZD9a]. η = η φ lf + φ hf (4.38) as a function of the phase angle θ at different pressures of 4 Pa ( φ lf = φ hf = 1 V), 1 Pa ( φ lf = φ hf = 3 V), and 1 Pa ( φ lf = φ hf = 12 V). The

145 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES [V] -1-2 sheath model analytical model PIC simulation [Degree] Figure 4.89: DC self bias η calculated by the Brinkmann sheath model (solid black line, [1]), the analytical model (red markers, [1]) and the PIC simulation (blue markers and solid line) [ZD9a]. Norm,5,4,3,2,1, -,1 -,2 -,3 -,4 -,5 4 Pa 1 Pa 1 Pa [Degree] Figure 4.9: DC self bias normalized to the sum of the amplitudes of the applied voltage as a function of the phase angle Θ at different pressures. The dashed lines correspond to the normalized bias η = 7 resulting from the analytical model for 32 equal voltage amplitudes of [1] assuming ε = 1 [ZD9a]. strongest normalized bias η = 7 resulting from the analytical model for equal 32 voltage amplitudes of [1] assuming ε = 1 is also shown (dashed lines). Obviously, the EAE is strongest at low pressures. At high pressures the normalized DC self

146 14 CHAPTER 4. RESULTS bias and, consequently, the degree of discharge asymmetry is smaller, since the self amplification of the EAE vanishes with increasing pressure, because the sheaths get more collisional (ε 4 Pa.6, ε 1 Pa.8, ε 1 Pa 1 for θ = ). Experiment [V] t [ns] min = = 9 Figure 4.91: Voltage drop across the discharge at θ = and θ = 9 (argon, 2 Pa, φ = 76 V, 1 cm electrode gap). The voltage amplitudes of each harmonic are 62 V, 63 V, and 76 V for the measurements at 4 Pa, 1 Pa, and 2 Pa, respectively [JS9b]. Figure 4.91 shows the measured voltage drop across the discharge described in chapter at θ = and θ = 9 [JS9b]. The voltage waveforms are reconstructed from the first and second harmonic of the measured voltages applying the calibration procedure described in chapter Higher harmonics are not observed. The discharge is operated in argon, at 2 Pa, φ lf = φ hf = 76 V, and an electrode gap of 1 cm. Although the discharge is geometrically symmetric, a DC self bias η, indicated in figure 4.91, is generated. This DC self bias is caused by the EAE and is observed experimentally here for the first time. Until now the generation of a DC self bias has only been observed in geometrically asymmetric CCRF discharges [38 41]. Figure 4.92 shows the DC self bias at different pressures and electrode gaps as a function of θ neglecting the error θ caused by the phase calibration procedure. As predicted by previous models and simulations [1] the DC self bias changes almost linearly as a function of θ due to the EAE. At a large electrode gap of 2.5 cm the bias is mainly negative, whereas at a gap of 1 cm maximum and minimum DC self bias are almost identical. Due to the bad aspect ratio between electrode radius r and gap d at d = 2.5 cm (r/d = 2) the capacitive coupling between the glass cylinder max

147 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 141 [V] 6 4 Pa, 2.5 cm 1 Pa, 2.5 cm 4 2 Pa, 1 cm [Degrees] Figure 4.92: DC self bias as a function of θ at different pressures and electrode gaps. The voltage amplitudes of each harmonic are 62 V, 63 V, and 76 V for the measurements at 4 Pa, 1 Pa, and 2 Pa, respectively. and the outer grounded chamber wall effectively enlarges the grounded surface and, therefore, leads to an additional asymmetry. At d = 1 cm (r/d = 5) this additional asymmetry is greatly reduced. This effective asymmetry of geometrically symmetric CCRF discharges was observed before by Coburn et. al. [38] and modeled by Lieberman and Savas [41]. In order to perform measurements in a geometrically symmetric CCRF discharge without substantial parasitic capacitive coupling between glass cylinder and outer chamber wall, the discharge must be operated with a small electrode gap, e. g. 1 cm. However, due to Paschen s curve at small electrode gaps the discharge cannot be operated at low pressures at reasonably low powers, which can only be applied in this experiment. In order to perform measurements at low pressures, which are particularly relevant for industrial applications, the electrode gap had to be increased to 2.5 cm. However, in this case the capacitive coupling to the outer walls causes an additional constant negative bias, which shifts the curves of figure 4.92 to more negative values. In order to avoid this problem a discharge, that can be operated at low pressures, with high aspect ratio would be needed. This could be realized by choosing electrodes of large radius.

148 142 CHAPTER 4. RESULTS The Small Angle Effect [V] Pa, U lf =U hf =5V [Degree] Figure 4.93: Measured DC self bias η as a function of the phase θ in argon at 1 Pa and φ lf = φ hf = 5 V. The measured data is corrected for the error caused by the phase calibration procedure described in chapter ( θ 6 ). Until now the error caused by the phase calibration of the voltage measurements of about θ 6 has been neglected (see chapter 4.4.1). If this error is corrected, the measured DC self bias reproduces the SAE already observed in the PIC simulation, i.e. the phase of minimum DC self bias is not, but about 8. Figure 4.93 shows the measured DC self bias as a function of θ in a pure argon discharge operated at 1 Pa and φ lf = φ hf = 5 V. During the measurement it was carefully checked that θ 8 is indeed the phase of minimum bias with an accuracy of about ±1. The SAE is observed in the PIC simulation (see figures 4.89 and 4.88) as well as in the experiment (see figure 4.93). However, it is not observed in the analytical model and the fluid sheath simulation (see figure 4.89). In both, the analytical model and the fluid simulation, the overall net positive charge Q is assumed to be constant. In reality, however, this is not the case.

149 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 143 sg [V] 4 2 = 2,5 tot [1-7 Cm -2 ] 9,9 9,6 9,3 9, sp [V] t [ns] sg [V] 4 2 = 7,5 tot [1-7 Cm -2 ] 9,9 9,6 9,3 9, sp [V] t [ns] Figure 4.94: Sheath voltages φ sg, φ sp and the total charge in the discharge per area σ tot as a function of time within one lf period resulting from the PIC simulation of an argon discharge at θ = 2.5 and θ = 7.5 (d = 6.7 cm, p = 2.66 Pa (2 mtorr), φ lf = φ hf = 315 V, γ = ). The vertical dashed lines indicate the phases of minimum sheath voltage at the grounded electrode within one lf period. Figures 4.94 and 4.95 show the sheath voltages at each electrode (top: Grounded electrode, bottom: Powered electrode) as well as the total charge in the discharge per area σ tot (middle plot) as a function of time within one lf period at θ = 2.5, θ = 7.5,

150 144 CHAPTER 4. RESULTS sg [V] 4 2 = 12,5 tot [1-7 Cm -2 ] 9,9 9,6 9,3 9, sp [V] t [ns] Figure 4.95: Sheath voltages φ sg, φ sp and the total charge in the discharge per area σ tot as a function of time within one lf period resulting from the PIC simulation of an argon discharge at θ = 12.5 (d = 6.7 cm, p = 2.66 Pa (2 mtorr), φ lf = φ hf = 315 V, γ = ). The vertical dashed lines indicate the phases of minimum sheath voltage at the grounded electrode within one lf period. and θ = 12.5, respectively (PIC simulation). The vertical dashed lines indicate the phases of minimum sheath voltage at the grounded electrode. At t = ns the sheath collapses at the powered electrode. The simulations are performed in pure argon (d = 6.7 cm, p = 2.66 Pa (2 mtorr), φ lf = φ hf = 315 V, γ = ). The discharge conditions are the same as used in the analytical model and the fluid simulation (figures 4.88, 4.89). The uncompensated charge in the discharge is positive at all times, since more ions than electrons are located in the plasma. Obviously, the assumption of a temporally constant charge is approximately, however, not exactly true: The total charge per area, σ tot, fluctuates by about 1 %. Ions are continuously lost from the discharge at a constant rate due to the ion fluxes to both electrodes. Only at the short phases of sheath collapse at each of the electrodes electrons are lost and the total positive charge increases again. The loss rate of σ tot can be estimated based on the following model: The one dimensional continuity equations for electrons and ions are: n e t + z (n eu e ) = n e ν iz (4.39) n i t + z (n iu i ) = n e ν iz (4.4)

151 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 145 here ν iz is the ionization frequency. Subtraction of equations 4.39 and 4.4 yields: t (n i n e ) + z (n iu i n e u e ) = (4.41) Under the assumption of n e = at the electrodes, integration of equation 4.41 over the entire discharge length yields: σ tot t + [(n i u i ) L (n i u i ) ] = (4.42) Under the assumption of equal ion densities n i,el and absolute values of the ion velocities u i,el at the electrodes equation 4.42 yields: σ tot t = 2n i,el u i,el (4.43) max tot [a.u.] tot min T RF /2 time T RF Figure 4.96: Sketch of the charge dynamics assumed in the frame of the analytical model used to estimate σ tot. One sheath collapse at each electrode within one RF period with equal time intervals T RF /2 in between are assumed. The short times of sheath collapse, when electrons leave the discharge, are neglected. The spatially averaged ion density in the sheath n i can be approximated by the ion density at the electrode n i n i,el [1]. Assuming one sheath collapse at each electrode with equal time intervals T RF /2 between two sheath collapses, i.e. a single frequency symmetric discharge (see figure 4.96), the order of magnitude of the relative fluctuation of σ tot can be estimated: σ tot σ tot = u i,el s max T RF (4.44)

152 146 CHAPTER 4. RESULTS here σ tot is the time average value of σ tot, σ tot = σ max σ min (see figure 4.44), s max is the maximum sheath width of the considered sheath, and T RF is the duration of one RF period ( σ tot = s max n). Based on the assumption of a Matrix sheath s max is given by: s max = 2ε ˆφs e n (4.45) Here ˆφ s is the maximum sheath voltage. Based on the conservation of ion energy in the sheath the ion velocity at the electrode can be calculated from the temporal average of the sheath potential φ s ˆφ s /2: u i,el = e ˆφ s m i (4.46) Substitution of equations 4.45 and 4.46 into equation 4.44 yields the following result for argon, an applied RF frequency of MHz, and a typical ion density in the sheath of n cm 3 taken from the PIC simulation: σ σ tot 2π ω pi,s ω RF, 1 (4.47) here ω pi,s and ω RF correspond to the ion plasma frequency in the sheath and the applied RF frequency, respectively. Equation 4.47 is independent of the electron temperature and agrees well with the PIC simulation results. The sheath dynamics and particularly the total number of sheath collapses at both electrodes are affected by the choice of the phase angle θ. This is shown in figures 4.94 and 4.95: For all phase angles the sheath collapses once at the powered electrode at t = ns. However, the number of sheath collapses at the grounded electrode changes as a function of θ. At θ = 2.5 the sheath collapses twice at the grounded electrode at times indicated by the vertical dashed lines in the top plot of figure Thus, electrons are lost to the grounded electrode twice per lf period. With increasing θ the minimum sheath voltage at the phase, when the sheath at ground collapses the second time (second vertical dashed lines in figures 4.94 and 4.95), increases until no electrons can cross this potential barrier at θ = 12.5 and electrons are only lost to the grounded electrode during the first sheath collapse. Figure 4.97 shows the minimum voltage drop across the sheath at ground at the phase of the second sheath collapse. The vertical dashed line in figure 4.97 marks the phase angle of 12.5, when no electrons reach the grounded electrode anymore at this time within the lf period. The number of electrons lost to each electrode within one RF period has to compensate the number of ions lost to the same electrode within one RF period. Consequently, if the sheath at ground collapses only once, all electrons required to compensate the ion flux to this electrode will leave the discharge during this phase of sheath collapse. If the sheath at ground collapses twice, the electron flux to the

153 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES (2) sg,min [V] , 2,5 5, 7,5 1, 12,5 15, 17,5 2, [Degree] Figure 4.97: Minimum voltage drop across the sheath at ground at the phase of the second sheath collapse indicated by the second vertical dashed lines in figures 4.94 and 4.95 as a function of θ. The discharge conditions are the same as in figures 4.94 and The vertical dashed line marks the phase angle of 12.5, when no electrons reach the grounded electrode anymore at this time within the lf period. grounded electrode will be distributed between the two sheath collapses. Consequently, the total charge in the discharge at the phase of the first sheath collapse at ground will be higher, if the sheath at ground collapses only once compared to two sheath collapses. This causes the maximum charge in the sheath at the powered electrode Q mp to increase as a function of θ, while the maximum charge in the sheath at ground Q mg remains approximately constant. The ratio of Q mg to Q mp is determined from the strongest electric fields at each electrode (PIC simulation) using equation 4.4. Figure 4.98 shows the square of this ratio as a function of θ. Obviously, the ratio is generally not unity such as assumed in the analytical model and the fluid simulation. It decreases as a function of θ due to the charge dynamics. This decrease particularly affects the calculation of the symmetry parameter ε (equation 2.11). Knowing the correct ratio of the maximum charges in both sheaths from figure 4.98, the symmetry parameter resulting from the fluid simulation performed under the same discharge conditions (blue line in figure 4.88) can be corrected for each θ: ε corr = ε fluid ( Qmg Q mp ) 2 (4.48) Figure 4.99 shows the result of this correction. The corrected symmetry parameter reproduces the symmetry parameter directly resulting from the PIC simulation to a good approximation. Obviously, the difference of the symmetry parameters resulting from the fluid and PIC simulations is caused by the charge dynamics.

154 148 CHAPTER 4. RESULTS 1,15 1,1 (Q mg /Q mp ) 2 1,5 1,,95, [Degree] Figure 4.98: Square of the ratio of the maximum charge in the sheath at ground and the maximum charge in the sheath at the powered electrode as a function of θ obtained by the PIC simulation of an argon discharge (d = 6.7 cm, p = 2.66 Pa (2 mtorr), φ lf = φ hf = 315 V, γ = ). 1,6 1,5 1,4 1,3 1,2 1,1 1,,9,8,7,6,5 sheath model PIC corrected [Degree] Figure 4.99: Symmetry parameter ε resulting from the fluid simulation (blue line), the PIC simulation (black line), and resulting from equation 4.48 (red line). If the symmetry parameter directly resulting from the PIC simulation, which is well approximated by the corrected ε from the fluid simulation, is used as an input parameter in the analytical model, the DC self bias calculated by the analytical model is essentially identical with the DC self bias resulting from the PIC simulation, i.e. the SAE is reproduced (see figure 4.1). In conclusion, the SAE is caused by

155 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES PIC Model ( PIC) 1 [V] [Degree] Figure 4.1: DC self bias η calculated by the analytical model using ε from the PIC simulation as an input parameter. the charge dynamics in the discharge, which is affected by the choice of θ (one or two sheath collapses at ground).

156 15 CHAPTER 4. RESULTS Separate control of ion energy and ion flux at the electrodes via the Electrical Asymmetry Effect In this section it will be demonstrated by a PIC simulation as well as experimentally, that the Electrical Asymmetry Effect, described in the previous section, allows efficient separate control of ion energy and ion flux at the electrode surfaces in an almost ideal way under a broad range of discharge conditions. By adjusting the phase shift θ the mean ion energy can be changed by a factor of about two by changing the DC self bias, while the ion flux remains constant within ±5%. Particle in Cell simulation First, the separate control of ion energy and ion flux at the electrodes is investigated under the same conditions used before by a fluid simulation and an analytical model. The conditions correspond to a geometrically symmetric dual frequency discharge operated in argon at MHz and MHz at 2.66 Pa (2 mtorr) with an electrode gap of d = 6.7 cm, φ lf = φ hf = 315 V, T g = 35 K, γ =.1, and α =.2. The DC self bias as a function of θ is shown in figure E [ev] [Degree] Ion flux [a.u.] Figure 4.11: Ion flux-energy distributions at the powered electrode as a function of the phase angle θ calculated by the PIC simulation under the conditions mentioned in the text [ZD9a]. Figures 4.11 and 4.12 show the effect of varying the phase angle θ on the ion fluxenergy distribution function at each electrode [ZD9a]. By changing θ from to 9 the maximum ion energy at each electrode can be changed by a factor of about three. Furthermore, the role of each electrode can be reversed. This change is caused by the change of the DC self bias (see figure 4.89). Figures 4.11 and 4.12 agree well with the distribution functions reported in [1]. The local maxima of the distribution functions at low energies are caused by ions, that undergo charge exchange collisions

157 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES E [ev] [Degree] Ion flux [a.u.] Figure 4.12: Ion flux-energy distributions at the grounded electrode as a function of the phase angle θ calculated by the PIC simulation under the conditions mentioned in the text [ZD9a]. in the sheath. At this low pressure the sheath is almost collisionless and many ions hit the electrode with high energies corresponding to the time averaged sheath potential. This is the origin of the peak at high energies. Ion Flux [1 14 cm -2 s -1 ] Powered electrode Grounded electrode [Degrees] Figure 4.13: Ion fluxes at the powered and grounded electrode as a function of the phase angle θ [ZD9a]. Figure 4.13 shows the ion flux at both electrodes as the phase angle θ is varied from to 9 [ZD9a]. The ion flux is constant within ± 5%, while the maximum ion

158 152 CHAPTER 4. RESULTS n i,center [1 16 m -3 ] 3, 2,7 2,4 2,1 1,8 1,5 1,2,9,6,3, [Degrees] Figure 4.14: Ion density in the discharge center as a function of the phase angle θ [ZD9a]. energy changes by a factor of three as θ changes (see figures 4.11 and 4.12). The observed stability of the ion flux is within the range of tolerance for most industrial applications [7, 166]. Figure 4.14 shows the ion density in the discharge center. The ion density is basically constant as the phase angle changes from to 9 [ZD9a]. Based on the above results this technique easily allows to control the ion energy separately from the ion flux by keeping the applied voltage constant and changing the phase angle θ. The ion flux can be adjusted by the voltage amplitude at fixed phase angle θ. In the PIC simulation the voltage is an input parameter and the voltage amplitude is kept constant. However, in experiments the applied power and not the voltage is usually set externally. Therefore, it is important to examine how the absorbed power changes as a function of θ, while the voltage amplitude is kept constant. Figures 4.15 and 4.16 show the space and phase resolved power density dissipated to electrons and ions for θ = and θ = 9 (strong DC self bias) [ZD9a]. The power density dissipated to electrons and ions p e,i is defined as: p e,i = j e,i E (4.49) Here j e,i is the current density of the respective particle species and E is the electric field. Again the asymmetry of the discharge due to the EAE is obvious. At θ = most power dissipation takes place at the bottom powered electrode. At θ = 9, the role of the electrodes is reversed via the EAE and most power dissipation takes place at the top grounded electrode. Electrons are accelerated at the sheath edge by the expanding sheath at both electrodes. The high frequency oscillations of the power density dissipated to the electrons during the initial sheath expansion at the powered electrode at θ = ( ns - 2 ns) and at the grounded electrode at θ = 9

159 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 153 Distance from powered electrode [cm] Distance from powered electrode [cm] = t [ns] = t [ns] Power density (electrons) [1 4 W m -3 ] Power density (electrons) [1 4 W m -3 ] Figure 4.15: Spatio-temporal plots of the power density dissipated to electrons at θ = (top) and θ = 9 (bottom) [ZD9a]. (2 ns - 4 ns) are caused by the PSR effect. Here the PSR effect is observed in a geometrically symmetric discharge for the first time. In a completely symmetric discharge the non-linearities of both sheaths cancel (see equation 2.17) and the PSR effect cannot be observed. Before the EAE was discovered, it had been believed that the only way to make a discharge asymmetric is via changing the electrode sizes. However, via the EAE asymmetry can also be achieved electrically and, therefore, the PSR effect can be observed also in geometrically symmetric discharges. This phenomenon will be discussed in detail in chapter Most of the power is absorbed by the ions, mainly inside the sheaths, where the ions are accelerated towards the electrodes by the strong sheath electric field.

160 154 CHAPTER 4. RESULTS Distance from powered electrode [cm] Distance from powered electrode [cm] = t [ns] = t [ns] Power density (ions) [1 4 W m -3 ] Power density (ions) [1 4 W m -3 ] Figure 4.16: Spatio-temporal plots of the power density dissipated to ions at θ = (top) and θ = 9 (bottom) [ZD9a]. The mean power density absorbed by electrons and ions, p e,i, results from an integration of the space and time resolved dissipated power density shown in figures 4.15 and 4.16: p e,i = 1 Tlf d p e,i dz dt (4.5) d T lf Here T lf is the duration of one lf period. The result for the total dissipated power as well as the electron and ion components are shown in figure 4.17 [ZD9a]. The absorbed power density is essentially constant and does not differ from its mean value by more than about 6%. This means, that keeping the applied voltage amplitude

161 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 155 Absorbed power density [ kw m -3 ] Electrons Ions Total [Degrees] Figure 4.17: Power absorbed by electrons and ions and total power absorbed as a function of the phase angle θ [ZD9a]. constant in the simulation corresponds to a good approximation to keeping the power constant. The small modulations of the absorbed power reflect the small modulations of the ion flux (figure 4.13). Therefore, the ion flux might change even less, if the power is kept constant. The question why the power dissipated to electrons and, consequently, the ion flux remain constant can be answered by an analysis of the current dynamics within one lf period as a function of θ: Figure 4.18 shows the square of the electron conduction current in the discharge center resulting from the PIC simulation and q 2 resulting from the analytical model (equation 4.63) as a function of time and θ within one lf period. In the analytical model ɛ from the PIC simulation is used as input parameter. Details of the algorithm used to calculate q 2 from the analytical model of the EAE are discussed in chapter Due to the change of θ the current dynamics change. However, the mean power dissipated to the electrons averaged over one lf period is proportional to the integral of p e (equation 4.49). Consequently, equation 4.7 yields: p e TRF I 2 dt (4.51) The result of equation 4.51 using data from the PIC simulation and the analytical model (figure 4.18) as input parameters is shown in figure The current dynamics do change within one lf period as a function of θ, however, time averaged over one lf period the power dissipated to electrons and, therefore, the ion flux remain fairly constant.

162 156 CHAPTER 4. RESULTS 3, PIC simulation p e [a.u.] 2,5 2, 1,5 1,,5, t [ns] [Degree] 3, Analytical Model p e [a.u.] 2,5 2, 1,5 1,,5, t [ns] 153 [Degree] Figure 4.18: Square of the electron conduction current in the discharge center (PIC, top plot) and q 2 resulting from the analytical model (bottom plot, equation 4.63) as a function of time and θ. Efficient separate control of ion energy and flux is also possible under different discharge conditions. In the following, results from the PIC simulation are discussed for a gap of 2 cm and pressures of 4, 1 and 1 Pa, for voltages of 1, 3 and 12 V. The conditions are otherwise the same as previously discussed. Figure 4.11 shows the ion flux-energy distributions resulting from the DC self bias generated via the EAE at these different pressures at each electrode [ZD9a]. The left column shows the ion flux-energy distributions at the indicated pressure at the powered electrode and the right column shows the ion flux-energy distributions at the grounded electrode. Due to the strong variable DC self bias at low pressures of

163 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 157 p e [a.u.] 1,2 Model PIC 1,,8,6,4,2, [Degree] Figure 4.19: Mean power dissipated to electrons calculated based on equation 4.51 (PIC + analytical model). 4 Pa and 1 Pa the maximum ion energy is changed by a factor of about three by changing the phase angle from to 9. As discussed before, at the higher pressure of 1 Pa the DC self bias is smaller (no self-amplification of the EAE, ε 1). At 1 Pa the flux-energy distribution function is exponential. However, the mean ion energy can still be changed by tuning the phase angle. Figure shows the mean ion energy < ɛ i >= ɛ max ɛf(ɛ)dɛ/γ i as a function of the phase angle θ at 1 Pa [ZD9a]. The mean ion energy is changed by a factor of about 1.5 by tuning the phase from to 9 at this pressure. Figure shows the ion flux as a function of the phase angle θ at the three different pressures investigated [ZD9a]. At the lower pressures of 4 Pa and 1 Pa, when a strong DC self bias is generated via the EAE (self-amplification of the EAE, see figure 4.9) and the maximum ion energy is changed by a factor of three by changing the phase angle (see figure 4.11), the ion flux is constant within ±1%. At the higher pressure of 1 Pa the ion flux changes more significantly (±3%) and separate control of ion energy and flux is therefore limited. At higher pressures, secondary electrons become more important, since they are confined in the discharge volume and multiply themselves through ionization in the sheaths. The generation of secondary electrons is very sensitive to the sheath voltage and consequently also to the DC self bias. Since the DC self bias changes with θ, ionization due to secondary electrons might also change with θ.

164 158 CHAPTER 4. RESULTS 4 Pa E [ev] Degree Ion flux [a.u.] 4 Pa E [ev] Degree Ion flux [a.u.] 1 Pa E [ev] Degree Ion flux [a.u.] 1 Pa E [ev] Degree Ion flux [a.u.] 1 Pa E [ev] Degree Ion flux [a.u.] 1 Pa E [ev] Degree Ion flux [a.u.] Figure 4.11: Ion flux-energy distributions at the powered (left column) and grounded (right column) electrodes as a function of θ at 4 Pa, 1 Pa and 1 Pa for an electrode gap of 2 cm [ZD9a]. Experiment Figures and show measured ion flux energy distribution functions at the grounded electrode at 4, 1, and 2 Pa ( φ lf = φ hf = 62 V, 63 V, and 76 V, respectively) at different electrode gaps as a function of θ [JS9b]. The measurements are performed with a PPM connected to the grounded electrode (see chapter

165 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 159 < i > [ev] 1,5 1, 9,5 9, 8,5 8, 7,5 7, 6,5 Powered electrode Grounded electrode [Degrees] Figure 4.111: Mean energy of ions hitting the electrode as a function of θ at 1 Pa [ZD9a]. IonFlux [1 15 cm -2 s -1 ] Pa Powered Electrode Grounded Electrode 1 Pa 1 Pa Degrees] Figure 4.112: Ion Flux as a function of θ at 4 Pa, 1 Pa and 1 Pa for an electrode gap of 2 cm [ZD9a] ). Excellent qualitative agreement with results of the PIC simulation and a hybrid fluid Monte Carlo simulation [1] is found. Generally, at a given phase angle the distribution functions measured here agree well qualitatively with ion energy distribution functions observed in CCRF discharges before [ ]. Figure shows the mean energy < ɛ i > of ions hitting the grounded electrode as a function of θ under the three different sets of conditions investigated [JS9b].

166 16 CHAPTER 4. RESULTS 4 Pa, d = 2.5 cm E [ev] 1 Pa, d = 2.5 cm E [ev] Degree Degree Ion flux [a.u.] Ion flux [a.u.] Figure 4.113: Measured ion flux energy distribution functions as a function of the phase angle θ between the applied voltage harmonics at the grounded electrode at 4 Pa ( φ lf = φ hf = 62 V) and 1 Pa ( φ lf = φ hf = 63 V) at an electrode gap of 2.5 cm [JS9b]. Here ɛ max is the maximum ion energy and f(ɛ) is the ion flux energy distribution function shown in figures and Γ i is the ion flux at the grounded electrode resulting from an integration of f(ɛ) from to the maximum ion energy. Due to the variable DC self bias induced by the EAE the mean ion energy at the electrode increases by a factor of about 2 as θ is changed from to 9 under all conditions investigated. At the same time the ion flux Γ i at the grounded electrode remains constant to a good approximation. This is shown in figure [JS9b] for all conditions investigated.

167 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES Pa, d = 1 cm E [ev] Degree Ion flux [a.u.] Figure 4.114: Measured ion flux energy distribution functions as a function of the phase angle θ between the applied voltage harmonics at the grounded electrode at 2 Pa ( φ lf = φ hf = 76 V) at an electrode gap of 1 cm [JS9b]. < i > [ev] Pa, 2.5 cm 1 Pa, 2.5 cm 2 Pa, 1 cm [Degree] Figure 4.115: Mean energy of ions hitting the grounded electrode as a function of θ at 4, 1, and 2 Pa at different electrode gaps [JS9b]. Based on figures these measurements clearly demonstrate that separate control of ion energy and flux at the electrodes can indeed be achieved in a simple and almost ideal way via the EAE.

168 162 CHAPTER 4. RESULTS Ion flux [a.u.] Pa, 2.5 cm gap 1 Pa, 2.5 cm gap 2 Pa, 1 cm gap [Degree] Figure 4.116: Ion fluxes at the grounded electrode as a function of θ at 4, 1, and 2 Pa at different electrode gaps [JS9b] Optimization of the Electrical Asymmetry Effect in dual frequency discharges Until now the investigations of separate control of ion energy and flux via the EAE have been restricted to a superposition of two RF voltage waveforms (fundamental + second harmonic) with identical amplitudes of each harmonic. Thus, two fundamental questions most relevant for applications remain: (i) What is the optimum choice of amplitudes for each voltage harmonic to induce the strongest electrical discharge asymmetry? (ii) How sensitive is the DC self bias generated by the EAE to changes of the amplitudes of each voltage harmonic? The former question is most relevant for applications, since a higher electrical discharge asymmetry provides the opportunity to change the ion energy over a wider energy range by adjusting the phase. The latter question is most important, since a high degree of process stability is required for applications, i.e. the ion energy should not change drastically, if the amplitudes of both applied voltage harmonics are changed by a few percent. In this section the investigations of the EAE are extended to different lf and hf voltage amplitudes and the above questions are answered: A geometrically symmetric dual-frequency discharge operated at MHz and MHz with variable phase shift θ between the voltage harmonics is again investigated experimentally, by a PIC simulation and by an analytical model. The amplitude of the MHz harmonic, φ lf, is kept constant and the amplitude of the MHz component, φ hf, is varied between φ hf 2 φ lf, i.e. the ratio between high and low frequency voltage amplitude is varied. The DC self bias normalized to the sum of both voltage amplitudes is measured and calculated (by a PIC simulation and an analytical model) as a function of this amplitude ratio A = φ hf / φ lf. It will be demonstrated that (i) the

169 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 163 strongest normalized DC self bias is not generated at equal voltage amplitudes such as used before, but at A < 1 and that (ii) the normalized DC self bias is relatively insensitive to changes of the voltage amplitudes around its optimum value providing a high degree of process stability. Neglecting the Small Angle Effect (see chapter 4.4.2) the analytical model of the EAE introduced in chapter 2.2 predicts the most negative DC self bias at θ =. For this phase angle the extrema φ m1 and φ m2 of equation 2.13 are calculated for different voltage amplitude ratios A and then substituted into equation 2.12 to calculate η. At A = 1/4 a transition occurs: for A < 1/4 equation 2.13 has one maximum and one minimum per low frequency period. For A > 1/4 equation 2.13 still has only one maximum, but two minima of the same modulus per low frequency period. The maximum φ m1 and minimum φ m2 of equation 2.13 are found at the following values ϕ 1 and ϕ 2 for ϕ(t) = 2πft with f = MHz, respectively [JS9c]: A 1 4 : φm1 : ϕ 1 = 2kπ, k Z φ m2 : ϕ 2 = (2k + 1)π, k Z (4.52) A 1 4 : φm1 : ϕ 1 = 2kπ, k Z ( φ m2 : ϕ 2 = arccos 1 ) + 2kπ, k Z 4A (4.53) Substitution of equations (4.52) and (4.53) into equation (2.13) yields φ m1 and φ m2, which are then substituted into equation (2.12) to calculate η. Then the DC self bias normalized by the sum of the voltage amplitudes is: η η = = φ lf + φ hf }{{} 1 εf(a), (4.54) 1 + ε θ= { f(a) = 1 1 A, A A A + 1, A 1 8A 4 The normalized bias η depends on ε and f(a). This function f(a) has a minimum at A = 1 and increases only slowly for larger values of A. On the other hand 2 the absolute value of η increases monotonically with decreasing ε. Thus, if ε = 1 (high pressure conditions), this analytical form yields a maximum bias η = 1 at 4 A = 1 and not at A = 1. For ε 1 a calculation of η is more complicated, since ε 2 itself depends on the DC self bias due to the self-amplification of the EAE. At low pressures the sheath is collisionless and, thus, a higher DC self bias leads to faster ions at one electrode and, consequently, to a smaller mean ion density within the sheath adjacent to this electrode due to flux conservation. This effect changes the symmetry of the discharge and yields an even stronger DC self bias. The symmetry parameter ε is calculated by the PIC simulation and is used as an input parameter

170 164 CHAPTER 4. RESULTS for the analytical model. In this way η is calculated as a function of A at 6, and 1 Pa (d = 2.5 cm). At 1 Pa ε = 1 is assumed, which is well justified by the simulation results, and η is calculated as a function of θ and A by the analytical model. 1,1 symmetry parameter 1,,9,8 6 Pa 1 Pa,7 1 Pa,,4,8 1,2 1,6 2, amplitude ratio A Figure 4.117: Symmetry parameter ε (calculated by the PIC simulation) as a function of the voltage amplitude ratio A at θ = and at pressures of 6, 1, and 1 Pa (φ lf = 1 V, d = 2.5 cm) [JS9c]. Figure shows the symmetry parameter ε as a function of the voltage amplitude ratio A resulting from the PIC simulation at θ = for 6, 1, and 1 Pa (d = 2.5 cm, φ lf = 1 V) [JS9c]. Due to the self amplification of the EAE ε is significantly smaller than unity at low pressures for voltage ratios A, at which a strong DC self bias is observed. At 1 Pa ε is essentially unity, since the self amplification is greatly reduced. The η values obtained from the analytical model along with those determined directly from the PIC simulation are shown in figure 4.118, for θ =, d = 2.5 cm, and pressures of 6, 1, and 1 Pa [JS9c]. Generally, excellent agreement is found. At all pressures the self bias vanishes at A =, since the discharge is operated as a geometrically symmetric single frequency discharge (low frequency only). With increasing A the absolute value of the normalized bias η increases until it reaches a flat maximum at A max < 1 (A max.7 at 6 Pa and 1 Pa, A max.6 at 1 Pa). The value of A max.6 at 1 Pa agrees well with the model prediction of A max =.5 for ε = 1 (see figure 4.117). The difference between these A max values at low and high pressures is caused by the self-amplification of the EAE at low pressures and the related dependence of ε on A (see figure 4.117) shifting the maximum of η due to equation (4.54). For the same reason η is stronger at lower pressures (28.5 %, 27 %, 24.5 % at 6, 1, and 1 Pa, respectively). For A the normalized DC

171 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 165 a) self bias [%] model PIC,,4,8 1,2 1,6 2, amplitude ratio A b) self bias [%] model PIC,,4,8 1,2 1,6 2, amplitude ratio A c) self bias [%] model PIC,,4,8 1,2 1,6 2, amplitude ratio A Figure 4.118: Normalized DC self bias η as a function of the amplitude ratio A for θ = resulting from the analytical model and the PIC simulation ( φ lf = 1 V) at 6 Pa (a), 1 Pa (b), and 1 Pa (c). The vertical dashed lines indicate the maxima of η at A = A max [JS9c].

172 166 CHAPTER 4. RESULTS self bias η decreases monotonically towards zero, since for very high values of A the geometrically symmetric discharge is again essentially driven by only one frequency (high frequency only). For applications the most relevant result is the flatness of the maximum of η observed at all pressures. Around the maximum the normalized self bias changes by less than 1 %, if A is changed by 15 %. In non-normalized quantities at 6 Pa this corresponds to a change of 1 V of the amplitude of the hf component ( φ,lf = 1 V, φ,hf = 7 V), which causes a change of the self bias of only about 2.5 V (DC self bias of 48.7 V at A =.7). This means that the normalized DC self bias generated by the EAE is very insensitive to small changes of the amplitudes of the applied voltage harmonics. Thus, a high degree of process stability is expected, if the ion energy is controlled via the EAE. In order to generate the strongest variable electrical asymmetry, to have optimum control of the ion energy, the discharge should be operated at A max rather than at different voltage amplitude ratios. Figure shows the normalized DC self bias η as a function of θ for A = (single frequency discharge), A =.6 (strong η), and A = 2 at 1 Pa, d = 1 cm, and φ lf = 5 V resulting from the analytical model assuming ε = 1 (plot a), the PIC simulation (plot b) and the experiment (plot c) [JS9c]. By changing θ from to 9 η is changed from η( ) to η(9 ) = η( ) for all values of A investigated, i.e. the role of both electrodes is reversed. In contrast to the analytical model and the PIC simulation in the experiment the DC self bias is not exactly zero at A =, since the capacitive coupling between the glass cylinder and the outer grounded chamber wall causes a small effective discharge asymmetry even at this short electrode gap. Figure 4.12 shows the ion density in the discharge center as a function of A and θ resulting from the PIC simulation at 1 Pa, 1 cm electrode gap and φ lf = 5 V [JS9c]. With increasing amplitude ratio A more power is applied to the discharge, since the hf voltage amplitude is increased, and, consequently, the ion density and the ion flux increase as a function of A at constant phase angle θ. By increasing A from to 2 the ion density is increased by one order of magnitude. The ion density remains approximately constant within about ±1 % as a function of θ at constant A.

173 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 167 a) self bias [%] 2 A = A =.6 A = phase angle [degree] b) self bias [%] A = A =.6 A = phase angle [degree] c) self bias [%] 2 A = A =.6 A = phase angle [degree] Figure 4.119: Normalized DC self bias η as a function of θ for A = (single frequency discharge), A =.6 (strong η), and A = 2 at 1 Pa, d = 1 cm, and φ lf = 5 V resulting from the analytical model assuming ε = 1 (a), the PIC simulation (b) and the experiment (c) [JS9c].

174 168 CHAPTER 4. RESULTS n i [1 15 m 3 ] ,,4,8 1,2 1,6 2, amplitude ratio A [Degree] Figure 4.12: Ion density in the discharge center as a function of A and θ (PIC: 1 Pa, φ lf = 5 V, d = 1 cm) [JS9c]. Figure shows the normalized DC self bias η as a function of θ and A at 1 Pa and 1 cm electrode gap, resulting from the analytical model, the PIC simulation, and the experiment [JS9c]. In simulation and experiment identical conditions are investigated. In the model ε = 1 is used, which is well justified by the PIC simulation (see figure 4.117). Again, an excellent agreement between the results of the model, the simulation, and the experiment is found. At all phase angles θ a dependency of η on A, similar to the ones shown in figure 4.118, is observed. At fixed amplitude ratio A dependencies of η on θ qualitatively similar to the ones shown in figure are found. This result shows that a high degree of process stability can be expected at all phase angles θ, if the discharge is operated at A = A max and that the ion energy can be controlled effectively via the EAE at all amplitude ratios A investigated.

175 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 169 a) , 1,6 1,2,8,4, self bias [%] amplitude ratio A phase angle [degree] b) , 1,6 1,2,8,4, self bias [%] amplitude ratio A phase angle [degree] c) , 1,6 1,2,8,4, self bias [%] amplitude ratio A phase angle [degree] Figure 4.121: Normalized DC self bias η as a function of θ and A at 1 Pa resulting from the analytical model (assuming ε = 1) (a), the PIC simulation (b) and the experiment (c). d = 1 cm, φ lf = 5 V. The color scale at the top of the figure shows the normalized DC self bias in percent [JS9c].

176 17 CHAPTER 4. RESULTS Excitation dynamics in electrically asymmetric discharges Distance from powered electrode [cm] 2, 1,6 1,2,8,4, t [ns] Exc. rate [a.u.] 5,2 4,8 4,1 3,4 2,8 2,1 1,4,7, Distance from powered electrode [cm] 2, 1,6 1,2,8,4, t [ns] Exc. rate [a.u.] 5, 4,4 3,8 3,1 2,5 1,9 1,3,6, Distance from powered electrode [cm] 2, 1,6 1,2,8,4, t [ns] Exc. rate [a.u.] 4,9 4,4 3,8 3,1 2,5 1,9 1,3,6, Distance from powered electrode [cm] 2, 1,6 1,2,8,4, t [ns] Exc. rate [a.u.] 4,9 4,4 3,8 3,1 2,5 1,9 1,3,6, Figure 4.122: Spatio-temporal plots of the total excitation rate (PIC) of argon atoms at 1 Pa, φ lf = φ hf = 12 V, and d = 2 cm at different phase angles θ calculated by the PIC simulation. In this section excitation dynamics in electrically asymmetric, geometrically symmetric dual frequency CCRF discharges are investigated at high pressures (1 Pa and 6 Pa) and low pressures (2.66 Pa). Figure shows spatio-temporal plots of the total excitation rate of argon atoms at 1 Pa, φ lf = φ hf = 12 V, and d = 2 cm at different phase angles θ (, 3, 6, and 9 ) resulting from the PIC simulation. The applied voltage waveforms normalized to its amplitude φ n = φ/ (2 φ ) lf at these phase angles are shown in figure The DC self bias as a function of θ is shown in figure 4.9 for these discharge conditions. Similar to other types of CCRF discharges excitation caused by the expanding sheath is observed. Due to the change of the applied voltage waveform with θ the sheath dynamics and, consequently, also the excitation dynamics change as a function of θ. At θ = and θ = 9 the excitation maxima at the bottom and top electrode are similarly strong, whereas at θ = 3 and θ = 6 the excitation maximum at the

177 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 171 1,,5 ~ n, -,5-1, time = = 3 = 6 = 9 T RF Figure 4.123: Applied voltage waveform normalized to its amplitude φ n as a function of time within one lf period at different phase angles θ. bottom electrode is significantly stronger than the maximum at the top electrode. In this sense the excitation is symmetric at and 9 and asymmetric at 3 and 6. Here the spatio-temporal excitation profiles are only shown from - 9 in 3 -steps. The strongest asymmetry of the excitation maxima at the top and bottom electrode is found at 45. From 9-18 a similar change of the excitation dynamics as a function of θ is observed with the difference, that the excitation maximum at the top electrode is stronger than the maximum at the bottom electrode. Similar results are observed experimentally at a relatively high pressure of 6 Pa. Figure shows spatio-temporal plots of the excitation into Ne2p 1 in an argon discharge with 1 % Neon admixture ( φ lf = φ hf = 76 V, and d = 1 cm) at different phase angles θ (, 3, 6, and 9 ). The temporal resolution of these measurements is about 5 ns (gate with). Small differences to the simulation results shown in figure might be explained by the lower pressure. In particular, at θ = the experimentally observed spatio-temporal excitation profile at 6 Pa is not as symmetric as the excitation resulting from the simulation at 1 Pa. This might be caused by the fact that ε is not unity for all θ at 6 Pa. The effect of the symmetry parameter on the symmetry of the excitation will be discussed later in this section. At high pressures the strongest asymmetry of the excitation is not found at the phase angle θ at which the strongest DC self bias is generated (see figure 4.9). The strongest bias is generated around θ =. However, at this phase angle the excitation maxima at the top and bottom electrode are similarly strong. The strongest asymmetry of the excitation is observed at θ = 45, when the DC self bias vanishes. This is substantially different to classical CCRF discharges operated at high pressures, in which the DC self bias is generated by a geometric asymmetry (no EAE).

178 172 CHAPTER 4. RESULTS Distance from powered electrode [cm] 1,,8,6,4,2, t [ns] Exc. rate [a.u.] 11,5 1,2 8,9 7,6 6,3 5, 3,7 2,4 1,1 Distance from powered electrode [cm] 1,,8,6,4,2, t [ns] Exc. rate [a.u.] 11, 9,8 8,5 7,3 6, 4,8 3,5 2,3 1, Distance from powered electrode [cm] 1,,8,6,4,2, t [ns] Exc. rate [a.u.] 11, 9,8 8,5 7,3 6, 4,8 3,5 2,3 1, Distance from powered electrode [cm] 1,,8,6,4,2, t [ns] Exc. rate [a.u.] 11, 9,8 8,5 7,3 6, 4,8 3,5 2,3 1, Figure 4.124: Measured spatio-temporal plots of the excitation into Ne2p 1 in an argon discharge with 1 % Neon admixture at 6 Pa, φ lf = φ hf = 76 V, and d = 1 cm at different phase shifts θ. In these discharges the asymmetry of the spatio-temporal excitation is strongest, when the strongest DC self bias is generated, which causes the maximum sheath voltages at both sides to become different. As a consequence of this, the sheath expansion velocity at one side will be decreased, while the expansion velocity on the other side will be increased. Here this is not the case: At high pressures the maximum sheath voltages are the same on both sides for all phase angles θ (ε = 1). Under these conditions the ion density profiles at both electrodes are the same and the sheath expansion velocities are determined only by the temporal change of the applied voltage waveform, which changes as a function of θ. The excitation dynamics at different pressures are understood in the frame of a simple analytical model: Generally, the excitation rate E i is: E i = n e v ex dvv 3 σ < g e > Ω (4.55) here g e is the EVDF. Similar to the investigations on electron beams (chapter 4.2.4) in CCRF discharges the EVDF is typically displaced by a velocity u due to the

179 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 173 acceleration of electrons perpendicular to the electrodes by the moving sheath. In this case the excitation rate is: E i = n e v ex dvv 3 σ < g e (( v u) 2 ) > Ω (4.56) here v is the electron velocity, g e (( v u) 2 ) is the EVDF displaced by a velocity u, and < g e > Ω = g e dω yields the average of the distribution function over the full solid angle. Such a displaced EVDF can be expanded [17], where the dimensionless smallness parameter by which successive orders scale is α = u/v th with v th = 2k B T e /m e being the thermal electron velocity or an equivalent velocity for non-maxwellian distributions. After averaging the expanded distribution function over the full solid angle, all odd order terms vanish: < g e (( v u) 2 ) > Ω g e (v 2 )+ < 1 2 ( e u v ) 2 g e (( v u) 2 ) u= > Ω u 2 + O(u 4 ) (4.57) This expansion yields g e as a polynomial in u 2. Substitution of equation 4.57 into equation 4.56 taking into account only terms until the second order in u yields: ( E i = n e dvv 3 σg e (v 2 ) + u 2 dvv 3 σ < 1 ) v ex v ex 2 ( e u v ) 2 g e (( v u) 2 ) u= > Ω (4.58) If higher order terms play an important role, this model gets significantly more complex. The total excitation can be split into a time independent and a time dependent fraction: E i = Ēi + Ẽi(t) (4.59) here Ēi = n e v ex dvv 3 σg e (v 2 ) is the temporally constant fraction of the excitation and Ẽi(t) = u 2 n e v ex dvv 3 σ < 1 ( e 2 u v ) 2 g e (( v u) 2 ) u= > Ω is time dependent, since it is a function of u 2, which is strongly time modulated. In spatio-temporal plots of the excitation such as figure Ēi corresponds to a temporally constant background at a given spatial position. If this background is subtracted from the total excitation at this spatial position, only Ẽi(t) will remain. In this way the time modulated fraction of the strongest excitation at the bottom powered and top grounded electrode, Ẽ i,p (t a ) and Ẽi,g(t b ), respectively, are determined from the PIC simulation at times t a and t b, respectively. Assuming equal densities on both sides the ratio only depends on the square of the ratio of the electron drift velocities at the bottom powered and top grounded electrode, u p (t a ) and u g (t b ), respectively: Ẽ i,p (t a ) Ẽ i,g (t b ) u p(t a ) 2 u g (t b ) 2 (4.6) The ratio u p (t a ) 2 /u g (t b ) 2 can also be calculated based on an analytical model:

180 174 CHAPTER 4. RESULTS Neglecting the voltage drop across the bulk and using the quadratic charge voltage relation of both sheaths the voltage balance equation is [1]: φ = η + φ = q 2 + ε (q t q) 2 (4.61) In equation 4.61 all quantities are normalized to the amplitude of the applied RF voltage waveform. q t is the total net charge in the discharge. Solving equation 4.61 for q yields: ( εq t + εqt 2 (1 ε) η + φ(t) ) q(t) = (4.62) 1 ε here q t is assumed to be temporally constant. Differentiation of q yields the current density j(t) = Q q(t) = enu(t) with Q = A sp (2eε n sp φ /I sp ) 1/2 ( φ = φ lf = φ hf ): q(t) = e nu(t) = 1 φ(t) Q 2 ε ( ) (4.63) qt 2 1 ε ε φ(t) + η Equation 4.63 directly yields the ratio u p (t a ) 2 /u g (t b ) 2 assuming that the ion density is temporally constant and approximated by the mean ion density in the respective sheath. Furthermore, q t = 1 is assumed: Ẽ i,p (t a ) Ẽ i,g (t b ) ( ) 2 up (t a ) = 1 u g (t b ) ε 2 ( φ(ta ) φ(t b ) ) ε ε 1 1 ε ε ( φ(tb ) + η) ( φ(ta ) + η) (4.64) To a good approximation the last fraction on the RHS of equation 4.65 is unity, even if ε is small. For ε = 1 it is exactly unity: Ẽ i,p (t a ) Ẽ i,g (t b ) 1 ε 2 ( ) 2 φ(ta ) (4.65) φ(t b ) At high pressures ε = 1 and equation 4.65 reduces to: Ẽ i,p (t a ) Ẽ i,g (t b ) ( ) 2 up (t a ) = u g (t b ) ( ) 2 φ(ta ) (4.66) φ(t b ) According to equation 4.66 the ratio of the maximum electron energies at each electrode should only depend on the first temporal derivatives of the applied voltage waveform at the respective times at high pressures. Figure shows the ratio of the time modulated fraction of the excitation maxima close to the powered and grounded electrode resulting from the PIC simulation (figure 4.122) and from the analytical model (equation 4.66) as a function of θ at 1 Pa. The model reproduces the observed change of the spatio-temporal excitation profiles with θ qualitatively and, therefore, demonstrates that at high pressures

181 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 175 E ~ ~ i,p (t a )/E i,g (t b ) 2,4 2, 1,6 1,2,8,4, Model PIC [Degree] Figure 4.125: Ratio of the time modulated fraction of the excitation maxima close to the powered and grounded electrode resulting from the PIC simulation (figure 4.122) and from the analytical model (equation 4.66) as a function of θ at 1 Pa. (ε = 1) the extrema of the temporal derivative of the applied voltage waveform determine the excitation dynamics. In addition to the correct prediction of the ratio of the excitation maxima at both electrodes, the phases within one lf period, when the excitation maxima occur at each electrode can also be calculated by this model: In case of identical amplitudes of both harmonics the applied voltage waveform and its first two temporal derivatives are: φ = φ (cos (ϕ + θ) + cos (2ϕ)) (4.67) φ = φ (sin (ϕ + θ) + 2 sin (2ϕ)) (4.68) φ = φ (cos (ϕ + θ) + 4 cos (2ϕ)) (4.69) here φ is the identical amplitude of both applied voltage harmonics and ϕ = 2πft with f = MHz. The extrema of equation 4.68 are found at: ϕ n = n π 2 + π 4 + n (4.7) with n cos ( n π 2 + π 4 + θ) sin ( n π 2 + π 4 + θ) + 8 ( 1) n (4.71)

182 176 CHAPTER 4. RESULTS here n is a natural number. Within one lf period local extrema of the first derivative of the applied voltage waveform are found at the following four different phases ϕ n : ϕ = π ( π ) 8 cos 4 + θ π 4 ϕ 1 = 3π ( π ) 8 sin 4 + θ 3π 4 ϕ 2 = 5π 4 1 ( π ) 8 cos 4 + θ 5π 4 ϕ 3 = 7π 4 1 ( π ) 8 sin 4 + θ 7π 4 (4.72) t n [ns] o [Degree] Figure 4.126: Times within one lf period, when extrema of the first derivative of the applied voltage waveform occur (equation 4.72), as a function of θ. Figure shows the times within one lf period, when extrema of the first derivative of the applied voltage waveform occur (equation 4.72), as a function of θ. According to equation 4.72 the time, when an individual extremum is found, essentially does not change as a function of θ. These times predicted by the model agree well with the spatio-temporal excitation profiles resulting from the PIC simulation (see figure 4.122) and the experiment (see figure 4.124). In contrast to the phase, when an individual local extremum is found, the extremum itself and the time, when the global extremum occurs, do change. Figure shows the extrema of the first derivative of the applied voltage waveform as a function of θ. Positive values, i.e. φ(ϕ 1 ) and φ(ϕ3 ), correspond to an expanding sheath at the grounded electrode, whereas negative values, i.e. φ(ϕ ) and φ(ϕ2 ), correspond to an expanding sheath at the powered electrode. Similar to the results

183 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 177 d ( n )/dt [V/ns] ~ ~ d /dt, d ~ /dt, 1 ~ d /dt, d ~ /dt [Degree] Figure 4.127: Extrema of the first derivative of the applied voltage waveform as a function of θ. of the simulation and the experiment the global minimum is φ(ϕ ) for all θ, i.e. strongest excitation at the powered electrode always happens at the same time t a 1 ns within one lf period for θ 9. Furthermore, the absolute value of this global minimum and, therefore, also the maximum excitation at the bottom electrode essentially do not change as a function of θ. However, the global maximum is found at different times within one lf period depending on θ and its absolute value does change with θ. At θ = and θ = 3 φ(ϕ3 ) corresponds to the global maximum and strongest excitation at the grounded electrode is observed at t b 65 ns. At θ = 6 and θ = 9 φ(ϕ1 ) corresponds to the global maximum and strongest excitation at the grounded electrode is observed at t b 29 ns. Figure shows the maximum of the time modulated excitation at the bottom and top electrode resulting from the PIC simulation and the analytical model at 1 Pa for θ 9. As predicted by the model, the maximum of the excitation at the bottom powered electrode is almost independent of θ, whereas the maximum excitation at the top grounded electrode changes as a function of θ with a minimum at θ = 45 and maxima at θ =, 9. At low pressures the excitation dynamics work substantially different. Figure shows spatio-temporal plots of the total excitation rate of argon atoms at 2.66 Pa, φ lf = φ hf = 315 V, and d = 6.7 cm at different phase angles θ (, 3, 6, and 9 ) resulting from the PIC simulation. The DC self bias and the symmetry parameter ε as a function of θ are shown in figures 4.89 and 4.88 for these discharge conditions, respectively. Again the excitation dynamics are dominated by sheath expansion heating. Due to the lower pressure and longer electron mean free path, the electron beams generated by the expanding sheaths propagate further into the plasma bulk. However, in

184 178 CHAPTER 4. RESULTS ~ E max [a.u.] PIC Bottom Model Bottom PIC Top Model Top [Degree] Figure 4.128: Maximum of the time modulated fraction of the excitation at the bottom powered and top grounded electrode as a function of θ at 1 Pa resulting from the PIC simulation and the analytical model (PIC simulation data taken from figure 4.122, model data taken from figure using equation 4.66). contrast to high pressures the spatio-temporal excitation profiles are now clearly asymmetric at θ = and θ = 9 (symmetric excitation at high pressures) and almost symmetric at θ = 6 (asymmetric excitation at high pressures). At low pressures the ratio of the maximum electron energies at both sides of the discharge is no longer purely determined by the temporal derivative of the applied voltage waveform, since the ion density profiles in the sheaths at both electrodes are no longer identical (see chapter 4.4.2, self amplification of the EAE). Similar to the nature of the frequency coupling in dual frequency discharges operated at substantially different frequencies the sheath will expand faster, if the ion density at the position of the sheath edge is lower. For this reason, stronger excitation is observed adjacent to the sheath with the lower mean ion density. Furthermore, at low pressures the generation of a DC self bias causes the maximum sheath voltages to be different. This mechanism also shifts the phase angle of strongest asymmetry of the spatio-temporal excitation profiles to phases of stronger DC self bias, i.e. lower θ, compared to the high pressure case. This effect is reproduced by the analytical model using equation 4.65 with ε taken from the PIC simulation (see figure 4.13, blue solid line and black solid line). Due to the effect of the DC self bias on the ion density profiles in both sheaths maximum asymmetry of the spatio-temporal excitation profiles is observed at lower θ compared to the high pressure case. At both pressures investigated the number of electrons above a certain energy threshold is assumed to be the same at both electrodes. At high pressures of 1 Pa this assumption is certainly justified, however, at low pressures of 3 Pa the ion density

185 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 179 Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] 16, 14, 12, 1, 8, 6, 4, 2,, Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] 13,9 12,3 1,5 8,8 7, 5,3 3,5 1,8, Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] 12,4 1,8 9,3 7,7 6,2 4,6 3,1 1,5, Distance from powered electrode [cm] t [ns] Exc. rate [a.u.] 15,9 14, 12, 1, 8, 6, 4, 2,, Figure 4.129: Spatio-temporal plots of the total excitation rate of argon atoms at 2.66 Pa, φ lf = φ hf = 315 V, and d = 6.7 cm at different phase angles θ calculated by the PIC simulation. profiles at both electrodes can be different depending on the DC self bias. However, if the model result is corrected by the ratio of the mean ion densities in both sheaths at low pressures by multiplying equation 4.65 with ɛ as an approximation of the different number of electrons, the agreement with the PIC simulation is much worse (see figure 4.13, red dashed line). Figure 4.13 also shows the model result obtained by multiplying equation 4.65 with ɛ 2, i.e. without taking into account different mean ion densities adjacent to the electrodes (green dashed line).

186 18 CHAPTER 4. RESULTS ~ ~ E i,p (t a )/E i,g (t b ) 3,5 3, 2,5 2, 1,5 1,,5, Model ( -2 ) PIC Model ( ) Model ( ) [Degree] Figure 4.13: Ratio of the time modulated fraction of the excitation maxima close to the powered and grounded electrode resulting from the PIC simulation (figure 4.129, blue solid line), from the analytical model using equation 4.65 (black solid line), from the model using equation 4.65 multiplied with ɛ to take into account the number of energetic electrons on both sides (red dashed line), and from the model using equation 4.65 multiplied with ɛ 2 (green dashed line) as a function of θ at 2.66 Pa Self-excited non-linear plasma series resonance oscillations in geometrically symmetric discharges As described in chapter 2.3 the phenomenon of self-excited non-linear plasma series resonance oscillations of the RF current can be understood in the frame of a simple voltage balance [48]. The applied RF voltage φ and the self bias η have to be balanced by the sum of the two sheath voltages and the voltage across the bulk φ b. Using proper normalization the PSR equation is [48]: η + φ = q 2 + ε(q t q) 2 + φ b. (4.73) here q is the unbalanced charge in the sheath at the powered electrode, q t is the total unbalanced charge in the discharge, ε (A p /A g ) 2 ( n sp / n sg ) is the symmetry parameter, and φ b = 2 ( 2 q/ t 2 + κ q/ t). Regarding φ b the first term represents electron inertia and the second term the resistive part. The experimentally verified (see chapter 4.2) non-linear charge voltage relation of the sheath q 2 and the inertia term d 2 q/dt 2 in equation 4.73 form a non-linear oscillator, which leads to highfrequency harmonics (more than an order of magnitude higher than the driving RF frequency) of the current with large amplitudes, if (i) collisional damping is sufficiently low, i.e. at low pressures, and (ii) the non-linearity q 2 in equation 4.73 does not cancel out. In case of a conventional geometrically symmetric discharge the

187 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 181 electrode areas A p, A g and the mean ion densities in the respective sheath n sp, n sg are equal and ε = 1. In this case the individual sheaths are still non-linear, but the non-linearity cancels out for the sum of both sheath voltages and PSR oscillations due to even non-linearities cannot be observed. On the other hand, if the grounded area is infinitely larger than the area of the powered electrode, ε will approach zero. In this case the non-linearity will not cancel. Clearly this condition is optimum for the generation of PSR oscillations. For this reason, PSR oscillations have been observed only in strongly geometrically asymmetric discharges until now. The EAE now provides a means to achieve ε 1 even in geometrically symmetric discharges so that PSR oscillations become possible [ZD9b]. At low pressures the variable DC self-bias generated by the EAE leads to different mean ion densities in both sheaths and, therefore, to ε 1. In this way an asymmetry in the sheaths is created electrically instead of geometrically. The results presented in this section are obtained by a PIC simulation of a geometrically symmetric discharge. As in the previous sections one electrode is driven by a voltage waveform of the shape of equation 2.13 with amplitudes φ lf = φ hf = 1 V. The electrode gap is L = d = 2.5 cm. Figure 4.131: (a) Self-bias voltage and (b) temporally averaged ion density profiles as a function of the phase angle θ at 3 Pa [ZD9b]. The calculated DC self-bias voltage, the symmetry parameter, and the temporally averaged ion density profiles are displayed in figure as a function of the phase angle θ, for a pressure of 3 Pa [ZD9b]. By adjusting θ the position of the plasma bulk and the sheath lengths can be changed. This control of the position of the density profile provides a unique opportunity for controlling the relative fluxes of radicals (created in the plasma bulk) to the electrodes. Experimentally, at low pressures the position of the plasma bulk is also found to change as a function of θ. This is observed as a change of the vertical position of the region of strong emission

188 182 CHAPTER 4. RESULTS between the electrodes, i.e. the plasma bulk, as a function of θ. Figure 4.132: Spatio-temporal distribution of the electron heating rate at θ = 7.5 (strongest self-bias η), at (a) 1 Pa, (b) 5 Pa, and (c) 3 Pa. (d) shows a zoomed part of (c). Units of electron heating are 1 5 Wm 3. (e) Spatio-temporal distribution of the electron impact excitation rate in units of 1 22 m 3 s 1 (θ = 7.5, p = 3 Pa) [ZD9b].

189 4.4. ELECTRICALLY ASYMM. DUAL FREQUENCY DISCHARGES 183 Figure shows the development of PSR oscillations with decreasing pressure, at fixed voltage amplitude ( φ lf = φ hf = 1 V) and phase angle (θ = 7.5, phase of strongest DC self bias, SAE) [ZD9b]. At a high pressure of p = 1 Pa the PSR oscillations are strongly damped by collisions and the well-known pattern of electron heating/cooling is observed. At 5 Pa slight modulations appear at both the heating and cooling regions at both electrodes during the first quarter of the low frequency period. These modulations become very pronounced at 3 Pa. Finer details are revealed in figure 4.132(d). The values of the symmetry parameter ε are displayed, too. Figure also shows: (i) the heating of electrons due to an electric field reversal near the powered electrode during sheath collapse (at t/t.85 in figures 4.132(a)-(c), see chapters and 4.3.4) and (ii) the formation of electron beams during sheath expansion (tilted excitation maxima in figure 4.132(e), see chapters ). Both phenomena are related to the rapid expansion and constriction of the sheath. Figure agrees qualitatively well with experimental investigations of NERH described in chapter 4.2. Figure illustrates how the PSR modulations of the electron heating are turned on and off as a function of the electrical discharge asymmetry induced and controlled by the EAE [ZD9b]. The control parameter is the phase angle θ between the driving voltages (see figure for η and ε). Compared to the case of θ = (strong electrical asymmetry) PSR oscillations become less pronounced at θ = 3 and practically vanish at θ 51.8, where the bias voltage η vanishes as well, and the electrical asymmetry is minimum. At phase angles above θ 51.8 the bias becomes positive and PSR oscillations move to the second quarter of the RF period, with NERH occurring near the grounded electrode. The PSR oscillations lead to a rapid expansion of the sheath, which is much faster than just by the fundamental RF frequencies. Thereby, fast beams of electrons traveling from the sheath region towards the plasma bulk are created. The density of these fast electrons equals the sheath density and is much lower than the bulk density. However, their density can exceed by far the density of the bulk electrons at comparable energy. Thereby, ionization can be strongly enhanced by the PSR effect. This enhancement is not necessarily adequately represented in the temporally averaged heating rate. It is a purely kinetic effect: in fact, few hot electrons are more efficient in ionizing than many warm electrons. Here, the energy gain of beam electrons by the PSR oscillations is well visualized in the heating rate plots. Electrons gain energy only within the sheath region, where large fields are necessary to drive the current with a low density of electrons. In the bulk the electron density is much higher, the field is much lower, and no substantial energy gain occurs. However, the fast electrons from the sheath continue ballistically into the bulk, where they become isotropic after about one mean free path. The trajectories in the bulk are visible in the excitation rate that probes energetic electrons. The opportunity to induce and control self-excited non-linear PSR oscillations of the RF current via the EAE should be investigated experimentally in the future. By making geometrically asymmetric discharges electrically symmetric using the EAE it might also be possible to avoid PSR oscillations in geometrically asymmetric

190 184 CHAPTER 4. RESULTS Figure 4.133: Electron heating/cooling rate (in units of 1 5 Wm 3 ) as a function of the phase angle θ indicated at the right side of the panels, together with the corresponding values of the symmetry parameter ε (p = 3 Pa) [ZD9b]. discharges operated at low pressures. Figure reveals the dynamics of the PSR oscillations induced by the EAE. Figure (a) shows the total charge per area, σ tot. Similar to the results discussed