Control of radio frequency capacitively coupled plasma asymmetries using Tailored Voltage Waveforms

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Control of radio frequency capacitively coupled plasma

asymmetries using Tailored Voltage Waveforms

Bastien Bruneau

To cite this version:

Control of radio frequency capacitively coupled

plasma asymmetries using Tailored Voltage Waveforms

Control of radio frequency capacitively coupled

plasma asymmetries using Tailored Voltage Waveforms

Bastien Bruneau

Ph.D. thesis defended the 6

of October, 2015, in front of the following jury:

Prof.

Nicholas St John Braithwaite

Open Univ.

Referee

Prof.

Richard van de Sanden

DIFFER

Referee

Prof.

Timo Gans

Univ. of York

Examiner

Prof.

Henri-Jean Drouhin

LSI

Examiner

Dr.

Zolt´

an Donk´

o

WRCP

Examiner

Dr.

Jean-Paul Booth

LPP

Examiner

If you start to cry, look up to the sky Something’s coming up ahead

To turn your tears to dew instead Pink Martini - Hang on Little Tomato

This thesis describes the study of capacitively coupled plasmas excited with Tailored Voltage Waveforms (TVWs) and their use in silicon thin film fabrication for photovoltaic applications. Tailoring the voltage waveforms applied on an electrode of a plasma reactor is a technique, which has been recently developed after the discovery of the electrical asymmetry effect. Com-pared to standard sinewaves, TVWs are composed of a fundamental frequency, in the radio frequency domain, and a number of harmonics. Any waveform can then be approximated by selecting the first components of its Fourier series. It has been shown for example that the ion energy can be controlled by the amplitude asymmetry of the waveform, the ion flux being kept constant.

However, at high pressure, this decoupling weakens. We further explore the reasons behind it, using both simulations and experiments. Based on this insight, we make use of the ability of TVWs to vary the ion energy to understand its effect on silicon thin film deposition. We find well-defined energy thresholds above which we observe, through several characterization tech-niques, a lower nucleation density in the case of microcrystalline silicon growth. This indicates that ions with energy above these thresholds can break the crystalline bonds. We confirm this result by studying low-temperature epitaxial growth on a crystalline silicon wafer. Above the energy threshold, epitaxy breakdown occur, and distinct amorphous columns are observed in cross-section transmission electron microscopy images, which are initiated at locations where the crystal was broken by high energy ions.

Because of unexpected different nucleation densities for sawtooth-up and down waveforms, despite the similar ion energies, we further explore the effect of these waveforms on plasma dis-charges. We find, using simulations in argon, that the asymmetry in the sheaths motion created by these waveforms induces a strong localization of the ionization events close to one electrode, and consequently a strong ion flux asymmetry. This localization is confirmed experimentally with phase-resolved optical emission spectroscopy. Surprisingly, we find that, for a given wave-form, this asymmetry can be reversed for different chemistries (namely CF4 plasmas), because of different electron heating mechanisms. This new asymmetry can be fully controlled via the rise-to-fall ratio of the waveform, i.e. its temporal asymmetry, and therefore opens the path for new applications of TVWs.

Keywords: capacitively coupled plasma, radio frequency, Tailored Voltage Waveforms, ion energy, ion flux, microcrystalline silicon, low-temperature silicon epitaxy, sheath motion, elec-tron heating

Writing the acknowledgments means the end of these three years of PhD. There are many peo-ple I would like to thank for their help and support during these years. I will try to do it in a chronological order.

First and foremost, I would like to thank Erik Johnson, my supervisor, for proposing this PhD subject to me, for the independence he gave me, and for the never-failing enthusiasm in front of new results. Thanks for always being ready to discuss whenever I came knocking at the door unannounced.

I spent these three years at the LPICM, Ecole Polytechnique, and I owe a lot to this research team. Pere Roca i Cabarrocas, head of the lab, for his passion during scientific discussion, and for trying his best - and succeeding - to make the LPICM a nice place to work. I would like to acknowledge the work of Laurence Corbel, Julie Dion, Chantal Geneste, Gabriela Medina and Carine Roger-Roulling for their cheerfulness and their efficiency in all administrative matters. I thank the entire team of the BEER: J´erˆome Charliac, Fr´ed´eric Farci, Cyril Jadaud, Fran¸cois Silva, for helping fixing and improving the many reactors of the lab. I would like to thank also, from LPICM, Rym Boukhicha for her laugh, Jean-Eric Bour´ee for the discussions on ion energy, Pavel Bulkin for the discussions on plasmas or wines, Romain Cariou for having done the longest French PhD and for the best reply to a wedding invitation, Dmitri Daineka, Bernard Drevillon, Martin Foldyna, Ileana Florea, Sofia Gaiaschi, Enric Garcia-Caurel, Farah Haddad, Tang Jian, Frederic Liege, Jean-Luc Maurice, Tang Miao, Soumyadeep Misra, Jean-Luc Mon-cel, Joaquim Nassar, Tatiana Novikova, Eric Paillassa, Garry Rose, Denis Tondelier, Jacqueline Tran, Junkang Wang. Thanks to all these people, LPICM is, indeed, a nice place to work!

I also have to thank the entire team of the ”algecos”, a small part of TOTAL lost far from La Defense. I owe them tons of coffee, but also many cheerful and lively lunches, coffee breaks, and much more. Many thanks to Jean-Fran¸cois Besnier, Fabrice Devaux, Etienne Drahi, a Marseillais in every way, Jara Fernandez, Sergej Filonovich, Nada Habka the diplomat, Ludovic Hudanski the brave (he shares his office with Nada), Gilles Poulain the enthusiastic and Patricia Prod’homme. Many thanks also to the successive PhD students: Kim Ka Hyun, Guillaume Courtois, Coralie Charpentier (for her love of ”potins”), Igor Sobkowicz, Jean-Christophe Dorn-stetter (but definitely not for his musical tastes), Paul Narchi (but not for his jokes), Ronan Leal (but not for his help before 11am), Gw´ena¨elle Hamon (but not for her jokes either), Fabien Lebreton the Russian , and the one who will continue the work on TVWs: Guillaume Fischer, the IPVFien.

The number of pages in this thesis would have been significantly reduced without the col-laborations with other labs. My deepest thanks to Jean-Paul Booth and Trevor Lafleur, from LPP, who taught me everything I know in plasma physics. Next come Deborah O’Connell, Timo Gans, and Arthur Greb from York University, who brought an ultrafast camera with them, together with a strong expertise. I would like to thank also the ”CF4 team”, which was

composed, in addition to the people from LPICM, LPP and York, by Steven Brandt, Edmund Sch¨ungel, and Julian Schulze from West Virginia University, and by Aranka Derzsi, Ihor Ko-rolov, and Zolt´an Donk´o from the Wigner Research Center for Physics. The last collaboration was focused on the study of hydrogen discharges, and included LPICM, LPP, York, but also Savino Longo, from University of Bari, Demetre J. Economou, from University of Houston, and Paola Diomede, from FOM Institute DIFFER. I had a great time working in these collabora-tions and I was able to learn a lot thanks to all these people.

I am grateful to the members of the PhD defence jury, who came from different places in Europe and woke up very early to attend the defence.

1 Introduction 1

1.1 Context . . . 2

1.2 Different types of plasma sources . . . 2

1.2.1 DC plasmas . . . 2

1.2.2 Radio frequency plasmas . . . 3

1.2.3 Microwave plasmas . . . 3

1.3 Applications of plasmas . . . 3

1.3.1 A wide range of applications . . . 3

1.3.2 Microelectronics. . . 4

1.3.3 Photovoltaics . . . 5

1.4 Outline of this thesis . . . 6

References . . . 7

2 Experimental and theoretical background 9 2.1 Radio frequency capacitively coupled plasmas (RF CCP) . . . 10

2.1.1 Generalities about RF CCP . . . 10

2.1.2 Sheaths in radio frequency plasmas . . . 12

2.2 Link between ion flux and ion energy . . . 16

2.2.1 Single frequency capacitively coupled plasmas . . . 16

2.2.2 Dual frequency capacitively coupled plasmas . . . 18

2.3 Particle-in-cell (PIC) simulations . . . 22

2.3.1 The general PIC scheme . . . 22

2.3.2 Limits of the PIC scheme . . . 24

2.4 Characterization techniques . . . 24

2.4.1 Plasma characterization . . . 24

2.4.2 Material characterization . . . 26

References . . . 32

3 Amplitude asymmetry 39 3.1 Tailored Voltage Waveforms (TVWs), a literature overview . . . 40

3.1.1 The origin of Tailored Voltage Waveforms . . . 40

3.1.2 The electrical asymmetry effect . . . 41

3.1.3 Decoupling ion flux and ion energy . . . 45

3.1.4 Multi frequency waveforms . . . 47

3.2 Plasma studies . . . 50

xiv CONTENTS

3.2.2 A light gas: Hydrogen . . . 58

3.2.3 An electronegative plasma: Carbon tetrafluoride (CF4) . . . 61

3.3 Summary and perspectives . . . 67

References . . . 68

4 Effect of TVWs on silicon material deposition 75 4.1 How to estimate ion energy . . . 76

4.1.1 Calculating the plasma potential . . . 76

4.1.2 Verification and limits . . . 78

4.2 Microcrystalline silicon . . . 80

4.2.1 A review on microcrystalline silicon . . . 80

4.2.2 Real-time spectroscopic ellipsometry . . . 84

4.2.3 Surface morphology study . . . 89

4.2.4 Further characterization . . . 91

4.2.5 Growth model . . . 93

4.3 Silicon epitaxy. . . 97

4.3.1 From microcrystalline silicon to epitaxy. . . 99

4.3.2 Impact of ion energy on epitaxial growth . . . 100

4.4 An unexpected change in the chemistry . . . 104

4.5 Summary and perspectives . . . 106

References . . . 107

5 Slope asymmetry 115 5.1 Ion flux asymmetry obtained with sawtooth waveforms . . . 116

5.1.1 Asymmetry observed in simulations . . . 116

5.1.2 Experimental confirmation of the asymmetry . . . 128

5.2 Effect of plasma parameters on the discharge asymmetry . . . 132

5.2.1 Effect of pressure . . . 133

5.2.2 Effect of fundamental frequency . . . 135

5.2.3 Continuous variation of the asymmetry . . . 137

5.3 Temporal asymmetry with H2, a light gas. . . 140

5.4 Temporal asymmetry with CF4, an electronegative gas . . . 142

5.4.1 High pressure regime . . . 143

5.4.2 Low pressure regime . . . 147

5.4.3 Evolution with pressure . . . 149

5.5 Comparison between gas chemistries . . . 151

5.6 Link between ion flux asymmetry and DC self-bias . . . 155

5.6.1 Ideal sawtooth and ion flux ratio . . . 155

5.6.2 Comparison with PIC results . . . 157

5.7 Summary and perspectives . . . 161

References . . . 162

Conclusion 165

Acronym

Unit

Definition

Ap,g m2 Area of the powered, grounded electrode

AFM Atomic Force Microsscopy a-Si:H Hydrogenated amorphous silicon CCP Capacitively Coupled Plasma c-Si Monocrystalline Silicon CVD Chemical Vapor Deposition EAE Electrical Asymmetry Effect

EEPF m−2.s−1.eV−1 Electron Energy Probability Function

ε symmetry parameter

εi Imaginary part of the pseudo dielectric function

epi-Si Epitaxial silicon

η V DC self-bias

FF % Fill factor

φ V voltage applied at the powered electrode ¯

φsp,sg V mean sheath voltage at the powered, grounded electrode

ˆ

φsp,sg V maximum sheath voltage at the powered, grounded electrode

φf

sp,sg V floating potential at the powered, grounded electrode

φb V voltage drop across the plasma bulk

φc V time-varying potential at the center of the discharge

φk V amplitude applied to the kth frequency

φm1,m2 V maximum, minimum of the applied waveform

FWHM Full Width at Half Maximum

γp,g % secondary electron emission coefficient at the electrodes

Γi m−2.s−1 ion flux

HF Hydrofluoric acid

HWCVD Hot Wire Chemical Vapor Deposition

IBE Ion beam energy

IEDF m−2.s−1.eV−1. Ion Energy Distribution Function

ITO Indium Tin Oxide (Tin doped Indium Oxide) Jsc mA.cm−2 Short-circuit current

xviii CONTENTS kiz,exc m−3.s−1 ionization, excitation rate

MCC Monte Carlo Collisions

µc-Si:H Hydrogenated microcrystalline silicon nc-Si:H Hydrogenated nanocrystalline silicon ¯

nsp,sg m−3 mean ion density in the sheath at the powered, grounded electrode

PECVD Plasma Enhanced Chemical Vapor Deposition PIC Particle-In-Cell

PROES Phase Resolved Optical Emission Spectroscopy

PV Photovoltaics

Qmp,mg C maximum charge in the sheath at the powered, grounded electrode

RF Radio Frequency

RTSE Real Time Spectroscopic Ellipsometry SE Spectroscopic ellipsometry

SEM Scanning electron microscopy

sccm cm3_.min−1 _{Standard cubic centimeter per minute}

TEM Transmission Electron Microscopy TVW Tailored Voltage Waveform

Θ rad phase

Voc mV Open-circuit voltage

Vpl V time-averaged plasma potential

VP P V peak-to-peak voltage

ω s−1 angular frequency

ωpi,pe s−1 ion, electron plasma angular frequency

Chapter

1

Introduction

Contents

1.1 Context . . . 2 1.2 Different types of plasma sources . . . 2 1.2.1 DC plasmas . . . 2 1.2.2 Radio frequency plasmas . . . 3 1.2.3 Microwave plasmas . . . 3 1.3 Applications of plasmas . . . 3 1.3.1 A wide range of applications . . . 3 1.3.2 Microelectronics . . . 4 1.3.3 Photovoltaics . . . 5 1.4 Outline of this thesis . . . 6 References . . . 7

2 1.1. CONTEXT

1.1

Context

In 1879, Sir Williams Crook suggested that a fourth state of matter (in addition to the solid, liquid, and gas) should exist, to explain his results on rarefied gas excited with electric dis-charges. However, the word “plasma” was introduced only in 1928 by Irving Langmuir, from General Electric, to describe ionized gases in electric discharge1_{. As reported by Mott-Smith}

in a letter to Nature2_{, the way these ionized gases shone and undulated while transporting its}

constituents (electrons and ions) reminded Langmuir of blood plasma, hence the name.

Why study plasmas? Because plasmas are everywhere. Although it is the least known of the four states of matter, it composes about 99% of all visible matter in the universe. All the stars in the sky are gigantic plasmas. Our sun, of course, is one of them. Lightnings are other well-known examples of plasmas. However, plasmas can be found in rather unexpected places. In certain TVs to begin with, the so-called plasma displays, where light is emitted from very small plasmas, but more generally, in thousand of objects that almost everyone uses in their everyday life. More precisely, plasmas were used in one or several steps of the fabrication of these objects. While the applications of plasmas will be presented more thoroughly later, we can already say that they range from medicine to environmental and waste management, from every type of modern computer to photovoltaic solar cells, from the textile industry to paintings, and this is far from being an exhaustive list.

1.2

Different types of plasma sources

An extremely wide range of plasma sources can be found, since any sources giving enough energy to ionize a gas can be used as a plasma source. Most of the time, electric field, or electromagnetic field, is used3_{. The plasmas produced by these fields can be divided according}

to the frequency of the field, and the mode of power coupling4–7.

1.2.1

DC plasmas

If two electrodes face each other, and a sufficient DC voltage (or, alternatively, a 50 Hz signal, such as used in the electric grid) is applied on one of them while the other is grounded, a plasma can be created. Such plasmas are widely used for lighting with neon tubes or for sputtering targets. However, this source suffers from a major drawback, as it cannot be used whenever a poor conductor covers one of the electrode. As a consequence, other sources should be used for deposition of semiconductor or insulator materials.

1_{I. Langmuir., Proceedings of the National Academy of Sciences of the United States of America, 14: 627–}

637, 1928.

2_{H. M. Mott-Smith., Nature, 233: 219, 1971.}

3_{B. M. Smirnov. Theory of Gas Discharge Plasma. en Springer, Nov. 2014.} 4_{H. Conrads et al., Plasma Sources Sci. Technol., 9: 441, 2000.}

5_{J. Reece Roth. Industrial Plasma Engineering: Volume 1: Principles. en CRC Press, Jan. 1995.}

6_{J. Reece Roth. Industrial Plasma Engineering: Volume 2: Applications to Nonthermal Plasma Processing.}

en CRC Press, Aug. 2001.

1.2.2

Radio frequency plasmas

Radio frequency (RF) refers to a range of frequency from about 1 MHz to 100 MHz. The most classic frequency used is 13.56 MHz. In this frequency ranges, there are two ways of coupling power: capacitively or inductively, leading to two different plasmas:

- Capacitively Coupled Plasmas (CCP), where, similarly to DC plasmas, two electrodes face each other, one being powered with the RF voltage, while the other one is grounded. Because no DC current flows, insulator or semiconductors can be deposited up to large thicknesses with-out any problem. In addition, under certain limitations which are mentioned in more detail in section 2.2.1, these plasmas can be easily upscaled, and reactors of a few square meters exist. However, they are considered as low density plasmas, as the electron density is about 1014_-1016 _m−3_{. This work focuses on this type of plasma.}

- Inductively Coupled Plasmas (ICP), where the power is coupled through a coil surround-ing or adjacent to the plasma, through which an RF current flows. This RF current creates a time-varying magnetic field which is the direct source of energy for the plasma. The electron density in such plasma ranges within 1017_-1018 _m−3_{, and they are therefore considered as high}

density plasmas. However, this source suffers from a difficulty to upscale the reactors beyond 50 cm in diameter.

- The helicon source, where an antenna is used to excite the plasma in the so-called helicon mode. This source provides even greater density, up to 1019 _m−3_{, but its size is limited}

1.2.3

Microwave plasmas

The microwave sources often use antennae, excited at a frequency above one gigahertz. A typ-ical frequency is 2.45 GHz. These antennae create an electro-magnetic field which can easily heat electrons because of its high frequency. These hot electrons can then easily ionize the neutral gas, therefore producing high density plasmas. The difficulty resides in the spatial uni-formity of the plasma, because of the short wavelength associated to this frequency (λ=12 cm for f =2.45 GHz).

When a particular value of magnetic field is used (namely 875 G for a frequency of 2.45 GHz), the electrons enter a resonance which is called the electron cyclotron resonance (ECR), hence creating the so-called ECR plasmas. This resonance allows to create very high density discharges of about 1018m−3 at very low pressure (in the Pascal range). Other types of microwave sources, such as the surface waves or resonant cavities also exist.

1.3

Applications of plasmas

1.3.1

A wide range of applications

4 1.3. APPLICATIONS OF PLASMAS thousandth of atmosphere. Within the articifial, or industrial, plasmas, a first classification can be drawn:

Hot plasmas are characterized by the fact that they are completely ionized (the α coef-ficient, which is the ratio of the ion density over the gas density, is equal to one), and the charged species have high energies of the order of 106_{eV (1 eV=11600 K). These plasmas occur}

in civilian applications, such as nuclear fusion in tokamaks (where the species are confined in strong magnetic fields), or in military applications in nuclear bombs.

Thermal plasmas are usually run at atmospheric pressure. These plasmas are said to be in thermodynamic equilibrium, because the gas is strongly heated by the plasma, and all the par-ticles have about the same temperature of a few thousand Kelvin. These plasmas can be used for waste management using plasma gasification8, or for welding and cutting, taking advantage of the high temperature of the particles.

Cold plasmas are associated with low pressure. Because of this characteristic, there are fewer collisions between the electrons and the heavy particles, and as a consequence the electrons cannot equilibrate with heavy particles. Because electrons are heated more effectively than heavy particles, and because they cannot equilibrate with the latter, their temperature (a few eV) is much higher than the heavy particle temperature (which corresponds to the temperature of the neutral gas, set by the user). The degree of ionization of these plasmas is usually very low (α ' 10−6). Medicine is one application of low temperature atmospheric plasmas. Indeed, one could hardly imagine putting any organ, and even less a whole body, to low pressure. Under the word “medicine”, two well separated domains can be distinguished. First, plasmas can lead to the death of cells exposed to the plasma. Therefore, it can be used to kill living organisms such as bacteria or viruses, and is consequently one of the main methods for sterilization in hospitals. Using plasma-induced cell death, curing cancer, and especially skin cancer, is also currently studied. Secondly, plasmas are also used to increase the biocompatibility of prothesis by modifying their surfaces. Concerning waste management, plasmas can be very interesting because of the very high temperature (a few thousand degree Celsius) that can be attained. As a consequence, most of the materials can be broken down to their most basic elements. These plasmas are widely used in microelectronics and in photovoltaics, and both applications will be described in further details hereafter.

1.3.2

Microelectronics

In 1947, John Bardeen and Walter Brattain, from Bell Laboratories, introduced the first tran-sitor, giving birth to the electronics industry. Less than a century later, this industry can be considered as one of the most important in the world. It relies on integrated circuits (IC), in which transistors are packed densely, and whose number determines how quickly information can be treated, or, if we put it differently, how efficient the system is. As a consequence, there has been, from the beginning, a trend toward the reduction of the size of the transistors in the IC, as limited by the minimum channel length.. From about 30 µm in the sixties, its size was reduced down to a few tens of nanometers nowadays. This reduction followed the famous Moore’s law, which oberves that the density of transistors in the IC doubles every 18 months.

This size reduction was made possible largely due to better use of plasma processes, which represent about one third of the steps needed to fabricate an IC. Interestingly, the different uses of plasma can be separated by the ion energy they require, as depicted in Fig. 1.1:

- For ion energies below '50 eV (1 eV = 11605 K) deposition of material is possible. Indeed, species can be accelerated to the surface where they absorb, diffuse, and create bonds with the surface. The elements to be deposited are contained in the gas mixture which is ionized, and layers from a few nanometres to a few micrometres can be deposited. This technique is called Plasma Enhanced Chemical Vapor Deposition (PECVD). The ion energy must be kept low to avoid damaging the deposited layer, while the ion flux needs to be kept high to increase deposition rate.

- For ion energy between '50 eV and '500 eV, the energy is sufficient to break bonds, therefore material can be removed from a target material. In some cases, very large etching rate differences are obtained on different materials, and this high selectivity can be used to cre-ate features with large aspect ratios, such as trenches, using masks (the mcre-aterial for the mask is chosen so that its etching rate is much lower than the underneath material). In addition, gases which can create reactive species when dissociated in the plasma are used to obtain a synergistic effect with the ions, and increase the etching speed. This is the so-called Reactive Ion Etching (RIE). For this application, an ion energy window is usually targeted. Indeed, with an energy too low, ions are inefficient in etching the material, while an energy too high would severely damage the material below the etched region.

- For ion energies above '500 eV, incoming ions can mechanically eject, or sputter, atoms from the surface. A target material is exposed to a plasma, while a sample on which the target material has to be deposited faces it. The sputtered atoms can then diffuse to the sample. Contrarily to etching, where reactive gases are used, neutral gases such as argon are commonly used in sputtering, with sometimes additional gases to ensure the stoichiometry of the deposited layer. In opposition to PECVD, where the material to be deposited is contained in the injected gases, here the material is supplied in a solid form in the target. It can therefore be used for all materials for which no gas precursor exists. In some cases, some reactive gases are also used, in which case the process is called reactive sputtering.

- For ion energies above '1 keV, ions can penetrate deeply in the material, lose energy in the way, and stop at a given depth. If all the incoming ions have the same energy, they statistically reach about the same depth, and their distribution in the material can be therefore precisely controlled. This is called ion implantation.

1.3.3

Photovoltaics

Whereas the photovoltaic (PV) market was limited to niche applications in the last century, it followed a two-digit annual growth since 2000, with often between 30 and 40% growth9_{. The}

PV market today is largely dominated by crystalline silicon. However, thin film silicon can be an alternative to the high cost of crystalline silicon, since deposition by PECVD is a cheap process. They are however limited by their low conversion efficiency, hence the need to better understand the deposition mechanisms. It should be mentioned that even in the large market of crystalline silicon, plasma may find applications in the deposition of thin films for passivation,

6 1.4. OUTLINE OF THIS THESIS

Fig. 1.1: Different applications of cold plasmas depending on the range of ion energy. anti-reflection coating, or for creating heterojunctions. As a example, the world record today with crystalline silicon was obtained by Panasonic10using heterojunction structure, with a thin amorphous silicon layer deposited by PECVD.

1.4

Outline of this thesis

It was shown in the previous sections that the range of plasma sources and applications is very wide. This thesis focuses on RF capacitively coupled discharges.

In chapter2, the physics of these discharges is presented in more details, together with their major limitations. In the same chapter, the general scheme of the particle-in-cell (PIC) simu-lations, as well as the experimental techniques which have been used in this work, are briefly described.

In chapter 3, the principle and advantages of Tailored Voltage Waveforms (TVWs), which are the core of this thesis, are introduced. TVWs are composed of a fundamental RF frequency and its first few harmonics. Results concerning the effect of the amplitude asymmetry of the waveforms on discharges are presented. Different gas chemistries are studied (namely Ar, CF4,

and H2), and it is shown that the impact of TVWs on a discharge depends strongly on the gas

used.

In chapter 4, the ability of TVWs to vary the ion energy, via the amplitude asymmetry of the waveform, is used to assess the effect of ion energy on silicon thin film growth. It is shown that an unexpected effect occurs when waveforms with significant slope asymmetry are used.

In chapter 5, the effect of the slope asymmetry of the waveform is studied in details, using both simulations and experiments with the three above-mentioned gases. It is shown, here again, that the impact of such waveforms on the discharge strongly depends on the gas chem-istry.

Finally, this work is summarized in the conclusion, and perspectives for further work are proposed.

References

[1] I. Langmuir. Oscillations in Ionized Gases. Proceedings of the National Academy of Sciences of the United States of America, 14: 627–637, 1928. (see p. 2)

[2] H. M. Mott-Smith. History of ”Plasmas”. Nature, 233: 219, 1971. doi: 10 . 1038 / 233219a0(see p. 2)

[3] B. M. Smirnov. Theory of Gas Discharge Plasma. en Springer, Nov. 2014. (see p.2) [4] H. Conrads and M. Schmidt. Plasma generation and plasma sources. Plasma Sources

Sci. Technol., 9: 441, 2000. doi: 10.1088/0963-0252/9/4/301 (see p.2)

[5] J. Reece Roth. Industrial Plasma Engineering: Volume 1: Principles. en CRC Press, Jan. 1995. (see p. 2)

[6] J. Reece Roth. Industrial Plasma Engineering: Volume 2: Applications to Nonthermal Plasma Processing. en CRC Press, Aug. 2001. (see p. 2)

[7] _{M. A. Lieberman and A. J. Lichtenberg. Principles of plasma discharges and} ma-terials processing. Wiley, 2005. (see p. 2)

[8] U. Arena. Process and technological aspects of municipal solid waste gasification. A review. Waste Management , 32: 625–639, 2012. doi: 10.1016/j.wasman.2011.09.025

(see p.4)

[9] REN21 Global Status Report, 2015 (see p. 5)

Chapter

2

Experimental and theoretical background

Contents

2.1 Radio frequency capacitively coupled plasmas (RF CCP) . . . 10 2.1.1 Generalities about RF CCP . . . 10 2.1.2 Sheaths in radio frequency plasmas . . . 12 2.2 Link between ion flux and ion energy. . . 16 2.2.1 Single frequency capacitively coupled plasmas . . . 16 2.2.2 Dual frequency capacitively coupled plasmas . . . 18 2.3 Particle-in-cell (PIC) simulations . . . 22 2.3.1 The general PIC scheme . . . 22 2.3.2 Limits of the PIC scheme . . . 24 2.4 Characterization techniques . . . 24 2.4.1 Plasma characterization . . . 24 2.4.2 Material characterization . . . 26 References . . . 32

10 2.1. RADIO FREQUENCY CAPACITIVELY COUPLED PLASMAS (RF CCP) In this chapter, a short review about the physics of radio frequency capacitively coupled plasmas (RF CCP) is presented, and the effects of the sheath motion are discussed. Then, one of the main challenges in this domain, which is the link between ion flux and ion energy, is highlighted, and the solutions which have been proposed are listed. The general scheme of particle-in-cell (PIC) simulations, which have been used with different chemistries in this work, is presented. Last, experimental characterization techniques, concerning both plasma and material, are described.

2.1

Radio frequency capacitively coupled plasmas (RF CCP)

2.1.1

Generalities about RF CCP

A RF CCP reactor, such as presented in the schematic in Fig. 2.1-a, is generally composed of two electrodes facing each other, while a given gas (or gas mixture) is present between the electrodes. One of the electrodes is grounded, whereas the other is excited with a RF volt-age signal which is supplied by a generator. A matchbox (not shown) is necessary to ensure an optimized power coupling between the generator and the reactor with the plasma running. This power serves to ionize the gas, therefore creating a plasma. As shown in Fig. 2.1-a, the plasma bulk (in grey) is isolated from any electrode surface by a sheath region. The potential profile is represented by a red line. As one can see, the plasma bulk corresponds to a region of nearly constant potential, at a potential which is called the plasma potential Vpl, whereas large

potential drops occur across the sheaths. This is because quasi-neutrality is observed in the plasma bulk, whereas the electron density strongly decreases in the sheath regions. This can be understood by imagining a constant ion (positively charged) and electron density between the two electrodes. As the electrons can respond to the modulation of the electric field (because of their lower mass, and therefore their higher mobility), the electrons close to the electrodes would be quickly lost to the electrodes, leaving behind a region where quasi-neutrality is not respected anymore (the ion density is larger than the electron density). This region would then contain more ions than electrons, and therefore a positive charge, creating an electric field, which means that there is a potential gradient. This creates a potential barrier in front of the electrode, which then repels the electrons back to the plasma regions, and attracts the ions towards the electrode, leading to the establishment of a steady state with zero time-averaged current. Obviously, this vision is quite simplistic, and more information about the complex physics of sheaths can be found in the work and review from Riemann 1–3_.

Figure 2.1-a shows a case where the surface area of the grounded electrode is larger than the powered electrode, because the lateral walls are also grounded. This is the case in most reactors, and this effect becomes larger as the size of the reactor decreases. This is called the geometrical asymmetry, and it is characterized by the ratio of the two electrodes areas, Ap/Ag, where Ap is

the area of the powered electrode, and Ag is the area of the grounded electrode. Because of this

geometrical asymmetry, a DC self-bias voltage, η, appears on the powered electrode, as shown in Fig. 2.1-a (thanks to an external capacitor). Indeed, each sheath can be seen as a capacitor (assuming that the displacement current is much larger than the electron and ion currents4), with the capacitance of the external capacitor being much larger than the capacitance of the

sheaths. Therefore, a voltage divider appears between the two sheaths in the RF domain (not in the DC domain). Because the area of the grounded electrode is larger than the area of the powered electrode (Ag > Ap), then supposing that the impact of the sheath thickness is

small, we obtain that the capacitance of the sheath in front of the grounded electrode is larger (Cg > Cp). As a consequence, the RF voltage amplitude across the sheath at the powered

electrode is larger than across the sheath at the grounded electrode. This remains true in the DC domain since there is a linear relationship between the amplitude of the RF voltage across a sheath and the value of the DC voltage across this sheath5_{. It can be shown}5 _that

η = ( ¯φsp+ ¯φsg), with − ¯ φsp ¯ φsg ∝ Ap Ag q (2.1) with q ≤ 2.5, where ¯φsp and ¯φsg are the mean sheath voltage at the powered and the grounded

electrodes respectively (shown in Fig. 2.1-a), which are taken negative and positive respectively.

Fig. 2.1: (a) Schematic of a geometrically asymmetric RF CCP reactor, and (b) spatio-temporal evolution of the potential between the powered electrode (located at x=0 cm), and the grounded electrode (located at x=2.5 cm), obtained by PIC simulations for an argon plasma, in a geometrically symmetric reactor. The time axis spans one RF cycle.

However, so far we have only considered DC values, whereas one of the main points in using radio frequency is that ions and electrons will behave differently in DC plasmas. Indeed, it can be shown5 _{that a plasma frequency ω}

p can be defined both for the electrons and ions (they are

then written ωpe and ωpi respectively) , where

ωp =

s ne2

m0

(2.2) where n is the density, m is the mass of the electron or ion, e is the charge of an electron, and 0 is the vacuum permittivity. This plasma frequency corresponds to the highest frequency one

species can respond to. According to this equation, light species (such as electrons) can respond to higher frequencies than heavy species (such as ions). In the classic RF CCP, we have

ωpi ω  ωpe (2.3)

12 2.1. RADIO FREQUENCY CAPACITIVELY COUPLED PLASMAS (RF CCP) where ω is the frequency of the applied voltage. Typically, in an Ar discharge with an electron density of 1×1016_m−3_{, the electron plasma frequency f}

pe (fpe = ωpe/2π) would be of about

900 MHz, while the ion plasma frequency fpi would be of about 3 MHz. This means that, at

a classic excitation frequency of 13.56 MHz, the electrons can respond to the instantaneous electric field, while the ions only see the time-averaged quantities. The question is therefore: how can this oscillating behavior be captured by the plasma? Fig. 2.1-b shows the spatio-temporal evolution of the potential between the powered electrode (located at x=0 cm), and the grounded electrode (located at x=2.5 cm) for an argon plasma, in a geometrically symmetric reactor (therefore η = 0). The time axis spans one RF cycle. This was obtained using particle-in-cell (PIC) simulations, with PHOENIX1D code (see section 3.2.1). As one can see, at the powered electrode, the potential evolution follows a sinusoid, which can be expected since this is imposed by the RF generator. Similarly, at the grounded electrode, the potential remains equal to zero, since the electrode is grounded. In the plasma bulk, the potential is always larger than both the potential at the powered electrode and at the grounded electrode. Indeed, if the potential in the plasma bulk is lower than the potential of one electrode, electrons would be attracted to this electrode, and they would all be lost very rapidly. In practice, this effect, which is called field reversal, can be obtained in very specific conditons 6,7_{. Also shown in Fig. 2.1-b,}

is the sheath edge position (with lines). As one can see, the sheath width (i.e. the distance over which the voltage drops) varies within one RF period, indicating that the sheaths will periodically expand and contract. This sheath motion is of primary importance for sustaining the plasma.

2.1.2

Sheaths in radio frequency plasmas

We have seen above that the sheaths oscillate in an RF CCP discharge. In order to study the effect of the sheath motion, the position of its edge should be clearly defined. Unfortunately, there is no general consensus about the definition of the sheath edge8. Among others, one can cite the following criteria:

(1) We initially defined the sheath as the region where a potential drop occurs. A simple criterion would define the sheath edge position as the inflection point of the plasma potential. (2) Following the idea of the first criterion, the sheath edge position can be defined as the point where |ni−ne|/ni ' α where niand neare the ion density and the electron density

respec-tively, with α = 1% or 10% or 50% as a threshold. This criterion presents a major drawback as α has to be set arbitrarily, since no specific value is predicted by any theory. Unfortunately, major changes can be expected when α is modified.

(3) Bohm criterion, where the sheath edge position is the point where the ion speed is equal to the Bohm speed, where Bohm speed is equal to v = (kBTe/Mi)1/2, where kBis the Boltzmann

constant, Te is the electron temperature, and Mi is the mass of the ion1,9–11. In RF discharges,

we can usually consider that the ions in the plasma bulk are in thermodynamical equilibrium with the neutral gas, therefore their temperature Tican be characterized by Ti=Tgas, where Tgas

6_{O. Leroy et al., Journal of Physics D: Applied Physics, 28: 500, 1995.} 7_{A. Salabas et al., Journal of Applied Physics, 95: 4605–4620, 2004.} 8_{R. N. Franklin., Journal of Physics D: Applied Physics, 37: 1342, 2004.}

9_{D. Bohm. The Characteristics of Electrical Discharges in Magnetic Fields. McGraw-Hill, New York, 1949.} 10_{S. B. Song et al., Physical Review E, 55: 1213–1216, 1997.}

is the neutral gas temperature. However, the electrons are not in thermodynamical equilibrium with the gas, and have a temperature Te such that Te  Tgas (usually Te is equal to a few

electron-volts, i.e. a few tens of thousands of Kelvin).

(4) The Brinkmann sheath criterion, which defines the sheath edge position as the point where the integral of the electron density from the electrode to this point is equal to the integral of the ion density minus the electron density, from the plasma bulk to this point12,13. In the simplest model, called the step model and introduced by Godyak and Ghanna14_{, the electron}

density is considered equal to the ion density in the plasma bulk, and zero in the sheath region. The Brinkmann sheath criterion therefore consider an excess of electron (compared to the step model) in the sheath region, which is compensated by a lack of electrons (again, compared to the step model) in the bulk region.

In the following, we will use the Brinkmann criterion to define the sheath edge position, as it allows for an intuitive understanding of the sheath edge, and as it does not depend on an arbitrary choice (unlike criterion 2). Next, we will see first how the sheath motion may have an effect on the electrons, and second its effect on the ions.

2.1.2.1 Effect of the sheaths on electrons

In classic RF CCP discharges, electrons can react to an instantaneous electric field, while the ions only see a DC electric field. Therefore, one can expect a strong effect of the sheath motion on the electrons. Figure 2.2-a shows the spatio-temporal evolution of the electron heating, Se,

in an argon plasma driven at 5 MHz, using again PHOENIX1D code. The electron heating tells us how much the electrons are accelerated. It can be positive when the electrons are acceler-ated, or negative, when the electrons are decelerated. This second case is often called electron cooling. The vertical time axis spans one RF cycle. The white lines represent the position of the sheath edge in front of the powered electrode, and the grounded electrode, as defined by the Brinkmann criterion. As one can see, the sheaths oscillate with a period corresponding to one RF cycle, although their oscillation is not strictly sinusoidal5.

When the sheaths expand toward the center of the discharge, positive heating is observed, because the electrons are accelerated (or heated) by the expanding sheath. Indeed, the sheaths repel the electrons toward the plasma bulk. Therefore, an expanding sheath can accelerate the electrons, whereas a receding sheath would decelerate them. This can be intuitively understood by considering the hardwall model 15, which explains this type of heating (which is called collisionless or stochastic heating, although it has been recently questioned whether this heating is really collisionless16_{) by treating the sheath edge as a rigid barrier that specularly reflects all}

incident electrons. Electrons with a velocity v incident on the moving sheath edge, which has a velocity us, are reflected such that the new velocity vr, is given by:

vr = −v + 2us (2.4)

Because of this condition, electrons reflected when the sheath is moving towards the plasma gain energy, while electrons reflected from the retreating sheath edge lose energy. Such heating,

12_{R. P. Brinkmann., Journal of Applied Physics, 102: 093303, 2007.}

13_{B. G. Heil et al., IEEE Transactions on Plasma Science, 36: 1404–1405, 2008.} 14_{V. A. Godyak et al., Sov. J. Plasma Phys., 6: , 1979.}

14 2.1. RADIO FREQUENCY CAPACITIVELY COUPLED PLASMAS (RF CCP)

Fig. 2.2: (a) Spatio-temporal evolution of the electron heating, Se, and (b) ionization

rate, Kiz, in an argon plasma excited at 5 MHz. The white lines represent the sheath

edge, as obtained by the Brinkmann criterion.

by definition, ”follows” the sheath edge. In can also be seen, in Fig. 2.2-a, that a small negative heating can be observed at the position of the maximum sheath when the sheath collapses (the region is delimited by a black line). This is electron cooling which is due to ambipolar electric field, and is further described in similar conditions in Schulze et al.17_{. It was shown}

that, in electronegative discharges such as CF4, heating by drift-ambipolar electric field can be

dominant because of the low bulk electron density, and therefore the low DC conductivity18–20_.

Figure 2.2-b shows the spatio-temporal ionization rate Kiz associated with the heating

shown in Fig. 2.2-a. It is clear that the two large ionization peaks (denoted “A”) follow the expansion of the sheaths, and therefore occur during the electron heating. This can be easily understood by the fact that electrons are accelerated, therefore the density of electrons with an energy above the ionization threshold (15.8 eV in the case of argon) increases, and subse-quently, the number of ionization events increases. Smaller ionization peaks (denoted “B”) can also be observed within the sheaths, when the sheath expansion is maximum. This is because of secondary electron emission. When ions arrive on the surface, there is a small probability that on neutralisation, the energy released will couple to an electron bound in the surface, leading to electron emission from the electrode. This electron will then be accelerated by the sheath (especially when the sheath is fully expanded), and therefore they can ionize particles within the sheath. This effect strongly depends on the material used for the electrodes, and on pressure. We have therefore already seen three types of plasma modes, depending on the origin of the ionization: (a) ionization due to sheath expansion, (b) ionization due to drift-ambipolar electric field, and (c) ionization due to secondary electrons emission. When sheath expansion is the dominant effect, the discharge is said to be in α-mode, whereas it is said to be in γ-mode when the ionization by secondary electron emission dominates. In the next chapters, we will see how these different modes can impact the discharge.

17_{J Schulze et al., Plasma Sources Science and Technology, 24: 015019, 2015.} 18_{J. Schulze et al., Physical Review Letters, 107: 275001, 2011.}

2.1.2.2 Effect of the sheaths on ions

As mentioned before, RF CCP discharges are supposed to work in a regime where ions only see the DC component of the electric field, and therefore of the potential (see eq. 2.3). This is because, as shown in Fig. 2.3-a (extracted from Chabert and Braithwaite4_{), an ion entering}

a sheath will see the sheath expand and collapse several times during its transit time in the sheath, τi (it is commonly supposed that τi = 1/ωpi). When this is the case, i.e when ω  1/τi,

the ion energy on the electrode does not depend on the phase of the sheath oscillation at which the ion enters the sheath. Therefore, all the ions arriving at the electrode should have the same energy. On the contrary, if the frequency of the applied voltage is decreased, one can reach a point, depicted in Fig. 2.3-b, where the time an ion spends in a sheath depends on the phase at which the ion enters the sheath. If an ion spends less time in a sheath, and therefore experience a lower electric field, it will be less accelerated by the sheath, and therefore its energy when arriving on the electrode will be lower. In this case, characterized by ω ' 1/τi, the ion energy

distribution on the electrode will be larger.

Fig. 2.3: Schematic ion trajectories in a temporally modulated sheath (a) ω ' 5ωpi

: ions approach the sheath boundary close to the Bohm speed, being significantly accelerated when inside the sheath and being repeatedly overtaken by the oscillating sheath, (b) ω ' ωpi : ions approach the sheath boundary close to the Bohm speed,

being significantly accelerated when inside the sheath. Extracted from Chabert and Braithwaite4.

As a result, the ion energy distribution function (IEDF) depends on the ωτi product, as

shown in Fig. 2.4-a, extracted from Panagopoulos and Economou 21. Indeed, when ωτi  1,

the IEDF looks like a bimodal function, with a large energy difference ∆E between the two peaks of the bimodal, whereas when ωτi  1, the IEDF looks almost like a single peak, as ∆E

is negligible. This variation of the ωτi product can be obtained either by changing the

excita-tion frequency, as it was done in K¨olher et al.22_{, or by changing τ}

i, for example by changing

the mass of the ion, as it was done by Coburn and Kay23_{, and shown in Fig. 2.4-b. In this}

figure, the IEDF of different contaminating ions in an argon discharge is displayed, and it can be clearly seen that the heavier the ion, the narrower the IEDF. This is because the ion transit time scales as the square root of the ion mass τi ∝ M

1/2

i 24 (see eq. 2.2). This dependence of 21_{T. Panagopoulos et al., Journal of Applied Physics, 85: 3435–3443, 1999.}

16 2.2. LINK BETWEEN ION FLUX AND ION ENERGY the IEDF shape on the ion mass, as shown by Kuypers and Hopman25, can therefore almost be used as a chemical analysis, where the multiple peaks of the IEDF can be attributed to the different ions in the plasma.

Fig. 2.4: (a) Ion energy distribution function (IEDF) calculated for different values of the ωτi product. Argon pressure is 50mTorr. Extracted from Panagopoulos and

Economou21_{. (b) IEDF of contaminating ions in an argon CCP excited at 13.56MHz.}

Argon pressure is 75mTorr. Extracted from Coburn et al. 23

The first model to account for this change in the IEDF shape was developed by Benoit-Cattin and Bernard26, where they considered a sheath with a constant width s, with an oscillating sheath voltage V (t) = VDC+ VACsin(ωt). They obtained an energy difference between the two

peaks of the IEDF

∆E = 8eVAC 3ωs   2eV_DC Mi 1/2 ∝ ωpi ω V 1/2 DC (2.5)

This suggests that a larger IEDF width can be expected for lighter ions (higher ωpi because

of lower mass) and for lower applied frequency. This model was later refined by Panagopoulos and Economou21 _{to account for the saturation of ∆E at very low and very high frequency. In}

the RF regime, ions can be affected by the sheath motion, when low frequencies (or light ions) are used.

2.2

Link between ion flux and ion energy

2.2.1

Single frequency capacitively coupled plasmas

One of the main challenges in CCP discharges for industrial applications is the strong link between ion flux and ion energy. Indeed, when using single-frequency CCP, the applied volt-age controls both the ion flux (the higher the applied voltvolt-age amplitude, the higher the ion flux), and the ion energy (the higher the applied voltage amplitude, the higher the ion energy). This link between ion flux and ion energy is illustrated in Fig. 2.5-a, extracted from Perret et

al.27,28. Solid lines show the result of a transmission line model described in Chabert et al.29, whereas symbols represent experimental data27_{. It is clear that, for a given frequency, the ion}

flux-energy follows a single line, and therefore that a large part of the two-dimensional ion flux-energy space cannot be accessed. It is possible to change the ion energy at a given ion flux by changing the excitation frequency30_{, however, only discrete values of the ion energy may}

thus be obtained, since only discrete RF frequencies are allowed for industry (namely 13.56 MHz and its harmonics). When the goal is to increase the ion flux, while minimizing the ion energy, as it may be the case in PECVD for thin-film deposition, there is a trend to use very high frequencies (VHF)31–33_{. However, for other applications, an energy window is defined. For}

instance, etching necessitates ion energy higher than 100 eV34 (lower horizontal dashed line) to enhance chemical reactions, but lower than 500 eV35 _{(higher horizontal dashed line) to avoid}

physical damage to the etched surface.

Fig. 2.5: (a) Ion energy vs ion flux at the discharge center for three different frequencies. The argon pressure was 15 mTorr. Solid lines are theoretical curves given by the transmission line model described in Chabert et al.29_{. Symbols are experimental data,}

where the ion flux was obtained with ion flux probes, and the ion energy measured by a retarding field energy analyzer (RFEA). (b) and (c) two-dimensional ion flux uniformity at 150 mTorr and 50W for 13.56 MHz and 60 MHz respectively. The standing wave effect is observed at 60 MHz. Extracted from Perret et al.27,36

Another limitation comes from the non-uniformity issues which appear when the frequency

27_{A. Perret et al., Applied Physics Letters, 86: 021501, 2005.}

28_{A. Hacala-Perret. Effets de la frequence d’excitation sur l’uniformite du plasma dans les reacteurs capacitifs}

grande surface. Theses. Ecole Polytechnique X, June 2004.

29_{P. Chabert et al., Physics of Plasmas, 11: 1775–1785, 2004.} 30_{M. Yan et al., Plasma Sources Sci. Technol., 8: 349, 1999.} 31_{M. Heintze et al., J. Phys. D: Appl. Phys., 26: 1781, 1993.} 32_{S. Oda., Plasma Sources Sci. Technol., 2: 26, 1993.}

18 2.2. LINK BETWEEN ION FLUX AND ION ENERGY is increased, i.e. when the characteristic wavelength of the applied voltage decreases 28,36. Indeed, pushed by the microelectronic industry, and more recently by the growing photovoltaic industry as well, the size of reactors tends to increase 37,38_{. Although the wavelength of the}

applied signal, λ0 usually remains larger than the characteristic length of the reactor, it was

shown by Lieberman et al.39 _{that the presence of a plasma (because of its large, negative}

permittivity) can artificially decrease the effective wavelength λ λ = λ0  1 + d smax −1/2 (2.6) where d is the width of the plasma, and smax is the maximum sheath width. Since we usually

have d  smax, it leads to λ  λ0, the effective wavelength is much shorter than the wavelength

of the applied signal. Therefore, standing waves appear39 _{in reactors much smaller than what}

could be expected on the basis of free space wavelengths, causing non-uniform voltages on the powered electrode 40_{, and consequently cause ion flux uniformity issues}36_{. Figures 2.5-b and c}

show the two-dimensional normalized ion flux distribution in a reactor excited at 13.56 MHz and 60 MHz respectively (corresponding to free space wavelengths of 22 m and 5 m respec-tively), extracted from Perret et al.36_{. The reactor is 0.4 m wide. Whereas the ion flux is rather}

uniform at the lower frequency, significant non-uniformities appear at the higher frequency, with lower ion fluxes further from the center of the discharge. This can be understood from eq.2.6. indeed, considering a plasma width of 2 cm and a maximum sheath width of 1 mm leads to an effective wavelength of about 1 m at 60 MHz. Therefore, the reactor size is not negligible compared to the effective wavelength at this frequency, and non-homogeneities appear.

These non-uniformities have been shown to have detrimental effects on thin-film deposition in the case of PECVD41,42_{. Although some research teams have tried to shape the electrodes}

to counter this effect43–45 _{or to use the so-called ladder-shaped electrode}46,47_{, this remains a}

critical issue to these days and limits the industrial use of very high frequencies.

2.2.2

Dual frequency capacitively coupled plasmas

Figure 2.5-a reminds us that low frequencies allow one to change the ion energy while keeping the ion flux low, and that high frequencies allow one to change the ion flux while keeping the ion energy low. From this observation, one can think of using two frequencies to excite a plasma, where a high frequency signal would set the ion flux, and a low frequency signal would set the ion energy. This is the concept behind dual frequency capacitively coupled plasma.

36_{A. Perret et al., Applied Physics Letters, 83: 243–245, 2003.}

37_{Mao-Jiun J. Wang et al., International Journal of Industrial Ergonomics, 34: 459–466, 2004.} 38_{D. Hrunski et al., Thin Solid Films, 532: 56–59, 2013.}

39_{M. A. Lieberman et al., Plasma Sources Sci. Technol., 11: 283, 2002.} 40_{L. Sansonnens et al., Plasma Sources Sci. Technol., 15: 302, 2006.} 41_{L. Sansonnens et al., Plasma Sources Sci. Technol., 6: 170, 1997.} 42_{A. A. Howling et al., Journal of Applied Physics, 96: 5429–5440, 2004.} 43_{H. Schmidt et al., Journal of Applied Physics, 95: 4559–4564, 2004.} 44_{L. Sansonnens et al., Applied Physics Letters, 82: 182–184, 2003.}

45_{P. Chabert et al., Physics of Plasmas (1994-present), 11: 4081–4087, 2004.} 46_{Yoshiaki Takeuchi et al., Thin Solid Films, 386: 133–136, 2001.}

This idea was first proposed by Goto et al., in 199248, and was later developed by Kitajima et al.49,50_{, who introduced the concept of functional separation. Indeed, each frequency has a}

given ‘function’. Figure 2.6 shows the excitation rate, Λi, from the Ar (3p5) state obtained by

phase resolved optical emission spectroscopy (PROES, more details on this technique will be given in section2.4.1), while a low frequency signal (700 kHz) is applied on one electrode, and a high frequency signal is applied on the other electrode. The frequency of the high frequency signal is 13.56 MHz in Fig. 2.6-a and 100 MHz in Fig. 2.6-b. In both figures, the amplitude of the low frequency signal is varied. In the ideal functional separation case, the excitation rate, and therefore the ion flux, should be independent of the low frequency. It is clear that the excitation rate increases significantly in Fig. 2.6-a, while it only slightly increases in Fig. 2.6-b. This indicates that a better functional separation is obtained when the frequency difference between the two signals is larger.

Fig. 2.6: Effect of the low frequency (700 kHz) bias amplitude on the axial distribution of the net excitation rate of Ar(3p5) in pure argon. The powered electrode is driven

by voltages at: (a) 13.56 MHz, and (b) 100 MHz. Extracted from Kitajima et al.49,50

Indeed, for the functional separation to be effective, the low frequency should not contribute significantly to the electron density, or, similarly, to the electron heating 5_{. At low pressure,}

collisionless heating dominates. It is then proportional to ω2VRF where VRF is the amplitude

of the voltage at ω, as shown by Lieberman51 _{or Gozadinos}52_{. Therefore, for the low frequency}

heating to be negligible, one need ω2

LFVLF  ωHF2 VHF, where LF and HF subscripts stand

for low frequency and high frequency respectively. In addition, the high frequency should not contribute significantly to the ion energy. The ion energy on the electrode can be estimated, in first approximation, as Ei = e(VLF + VHF). Therefore, we should have VLF  VHF. In the

48_{H. H. Goto et al., Journal of Vacuum Science & Technology A, 10: 3048–3054, 1992.} 49_{T. Kitajima et al., Journal of Vacuum Science & Technology A, 17: 2510–2516, 1999.} 50_{T. Kitajima et al., Applied Physics Letters, 77: 489–491, 2000.}

5_{M. A. Lieberman et al. Principles of plasma discharges and materials processing. Wiley, 2005.} 51_{M.A. Lieberman., IEEE Transactions on Plasma Science, 16: 638–644, 1988.}

20 2.2. LINK BETWEEN ION FLUX AND ION ENERGY end, to achieve a complete functional separation, we should have

ω2 HF ω2 LF  VLF VHF  1 (2.7)

Figures 2.7-a and b present the mean ion energy and the ion flux, respectively, for a dual-frequency system, the low dual-frequency being 1 MHz, with varied voltage, and the high dual-frequency being 100 MHz, with constant power of 1200 W.m−2. P.E. and G.E. refer to the powered and grounded electrode respectively. These results were obtained by Boyle et al.53 _{using PIC}

sim-ulations. As expected, the mean ion energy increases with the low frequency voltage, whereas the ion flux is kept relatively constant when the low frequency voltage is increased (Fig. 2.7-b). This indicates that the functional separation is effective in this case, as can be expected by the large ratio between the low and the high frequencies, ωLF/ωHF. Figure 2.7-c shows the

normalized ion current, Γi, as a function of this ratio ωLF/ωHF, for different values of the low

frequency voltage. It is clear that the ion current does not depend on the low frequency voltage when the ratio is small enough. However, when this ratio is larger than 0.1, a sharp increase of ion current is observed, indicating that functional separation is ineffective. These results are therefore consistent with eq. 2.7.

Fig. 2.7: (a) Mean ion energy, and (b) ion flux, as a function of the low frequency (lf) voltage in a dual-frequency plasma. The low frequency is held at 1 MHz, and the high frequency at 100 MHz. The high frequency power is held constant at 1200 W.m2_,

and the pressure is set to 50 mTorr. P.E. and G.E. refer to the powered and grounded electrodes, respectively. (c) Dependence of normalized ion flux, Γion the low frequency,

ωLF . The high frequency, ωHF is held constant at 100MHz and with an applied voltage

of 100V; the low frequency is then varied. The inscription refers to the low frequency voltage amplitude. Extracted from Boyle et al.53

Figures 2.8-a and b show the IEDF of CF+₃ obtained numerically by Georgieva et al 54 _{in a}

single frequency (13.56 MHz) or dual-frequency (at 27 MHz and 2 MHz, 27+2 MHz) discharge, respectively, as a function of the phase within the low frequency period. Due to low pressure used, in the single frequency case (Fig. 2.8-a), the IEDF has a single peak whose position does

not change much with the phase, because the frequency is high compared to the ion charac-teristic frequency (see section 2.1.2.2). Therefore, the time-averaged IEDF would be almost monoenergetic. On the contrary, in the dual frequency case (Fig. 2.8-b), the IEDF is still peaked because of the low pressure, but the position of this peak significantly changes during the low frequency period, because this frequency is close to the ion characteristic frequency. As a consequence, the time-averaged IEDF would be bimodal, with a large ∆E. This is one of the first drawbacks for dual-frequency plasmas: the ion energy can indeed be controlled, but at the price of a large ∆E, which can be strongly detrimental for applications where a mono-energetic IEDF is desired, such as ion implantation.

Fig. 2.8: (a) CF+₃ IEDF at different phases in one HF cycle in the single (13.56 MHz) frequency reactor and (b) at different phases in 2 LF cycles in the dual (27+2 MHz) frequency reactor at pressure 30 mTorr and applied voltage amplitude 700 V. Extracted from Georgieva et al.54

The second drawback of dual-frequency plasmas comes from coupling between low and high frequencies55_{. This process was first predicted from simulations by Turner and Chabert}56_{, who}

found that the collisionless heating obtained in dual frequency plasma is higher than the sum of the heating obtained when each frequency is applied separately. This effect was later confirmed experimentally by Gans et al. 57_{, and by Schulze et al.}58_{, using PROES in a dual frequency}

(27+2 MHz), helium-oxygen (He-O2) discharge. They observed strong excitation dynamics

induced by the low frequency, indicating its strong impact on the electron heating. Therefore, altogether, decoupling the ion flux and ion energy remains difficult, even when using classic dual-frequency techniques.

As a conclusion, dual-frequency excitation presents an interesting alternative to single-frequency excitation, and in certain conditions offers a good decoupling of ion flux and ion

55_{P. Levif. Excitation multifr´equence dans les d´echarges capacitives utilis´ees pour la gravure en}

micro-´

electronique. fr PhD thesis. Ecole Polytechnique X, Nov. 2007.

56_{M. M. Turner et al., Physical Review Letters, 96: 205001, 2006.} 57_{T. Gans et al., Applied Physics Letters, 89: 261502, 2006.}

22 2.3. PARTICLE-IN-CELL (PIC) SIMULATIONS energy. However, this decoupling always comes at the price of a wide distribution of the ion energy. In addition, in some conditions, strong coupling between the frequencies are observed. As a consequence, there is a need for a solution which solves the issue of the ion flux-energy coupling without the drawbacks of dual-frequency excitation. A possible solution is the use of Tailored Voltage Waveforms (TVWs), which are presented in more detail in the section 3.1, and which are the basis of this work.

2.3

Particle-in-cell (PIC) simulations

2.3.1

The general PIC scheme

Simulations using many particles have been used since the very beginning of the digital com-puters (even back to the 1950s59,60). It consists of simulating an environment by using a limited number of superparticles, each superparticle representing a given number of “real” particles. The main interest of particle-in-cell (PIC) simulations, is that it gives information about the distribution functions in phase space. In addition, PIC method usually employs the fundamen-tal equations without too many approximations, allowing it to retain most of the physics.

In the following, the basic principles of the PIC method are presented. Only the electric field E is taken into account, while the magnetic field B is neglected. More information can be found in the book by Hockney61_{, or in the review papers by Birdsall}62 _{and Verboncoeur}63_.

In the general flow of the PIC scheme, such as shown in Fig. 2.9, particles are defined in a continuum space, both in position and velocity, while the field are defined in discrete locations in space (at the grid positions). Both the particles and the fields are defined at discrete times. The particle positions and velocities obey Newton’s second equations of motion

mdv

dt = qE (2.8)

where m is the mass of the particle considered, v is its velocity, q its charge, and E is the electric field, and

dx

dt = v (2.9)

where x is the position of the particle. The electric field is obtained through Poisson’s equation d2φ

dx2 = −

ρ 0

(2.10) where φ is the potential, and ρ is the charge density (ρ = q(ni− ne), where ni and ne are the

ion and electron densities respectively, in the case of positive ions), and E = −dφ

dx (2.11)

Therefore, from Fig. 2.9, we can see the process for one time step as follows:(a) The electric field Ei, defined at each grid point, is interpolated to each particle’s positions, giving the force Fi

ap-plied on each particle. (b) The equations of motion are integrated to give the particle velocities

59_{D. R. Hartree., Applied Scientific Research, Section B, 1: 379–390, 1950.} 60_{O. Buneman., Physical Review, 115: 503–517, 1959.}

61_{R. W. Hockney et al. Computer Simulation Using Particles. en CRC Press, Jan. 1988.} 62_{C.K. Birdsall., IEEE Transactions on Plasma Science, 19: 65–85, 1991.}

v0_i, and their positions xi. (c) The boundaries are treated, with the subsequent losses/gains. (d)

The collisions are treated using a Monte Carlo method64_{. As a consequence of the collisions,}

the velocities v_i0 are modified to vi (new particles may be created by ionization at this point).

(e) The new densities ρi at the grid points are calculated from the particle positions. (f) The

electric field on the grid is calculated using Poisson’s equation and the density on the grid. The time can then be updated by adding a given time step ∆t.

Fig. 2.9: Flow schematic for the PIC scheme, neglecting the magnetic fields. v_i0 and vi represent the particle velocities before and after the collisions. Adapted from

Verboncoeur63

It should be mentioned that, following this scheme, the particle positions and the electric field are defined at a given time step n, then the electric field is used to calculate the velocity at a time n + 1/2, and the velocity itself is used to calculated the particle position at a time n + 1. Therefore, particle positions and electric field on one side, and particle velocities on the other side, are defined at different time steps. This is called the leapfrog scheme, and a schematic is shown in Fig. 2.10. As will be shown later, it is sometimes necessary to know the particle position and velocity at the same time65_{. In this case, the simplest solution is to consider that}

the velocity at a time n is the average of the velocity at a time n − 1/2 and n + 1/2.

Fig. 2.10: Schematic of the leapfrog scheme. The equations of motion are integrated forwards in time. Position at time level n − 1 are updated using velocities at level n − 1/2, velocities at level n − 1/2 are updated using electric field at level n, and so forth. Adapted from Hockney61