Low Pressure Plasmas and Microstructuring Technology
Gerhard Franz
Low Pressure Plasmas and Microstructuring Technology
123
Prof. Dr. Gerhard Franz Professor for Applied Physics Munich University of Applied Sciences Department of Precision- and Microengineering Engineering Physics 34 Lothstrasse 80335 M¨unchen Germany
[email protected] www.gerhard-franz.org
ISBN 978-3-540-85848-5 e-ISBN 978-3-540-85849-2 DOI 10.1007/978-3-540-85849-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009921817 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my wife
Preface
Over the last forty years, plasma supported processes have attracted ever increasing interest, and now, all modern semiconductor devices undergo at least one plasma-involved processing step, starting from surface cleaning via coating to etching. In total, the range of the treated substrates covers some orders of magnitude: Trenches and linewidths of commercially available devices have already passed the boundary of 100 nm, decorative surface treatment will happen in the mm2 range, and the upper limit is reached with surface protecting layers of windows which are coated with λ/4 layers against IR radiation. The rapid development of the semiconductor industry is inconceivable without the giant progress in the plasma technology. Moore’s law is not carved into stone, and not only the ITRS map1 is subject to change every five years but also new branches develop and others mingle together. Moreover, the quality of conventional materials can be improved by plasma treatment: Cotton becomes more crease-resistant, leather more durable, and the shrinking of wool fibers during the washing process can be significantly reduced. To cut a long story short: More than 150 years after the discovery of the sputtering effect by Grove, plasma-based processes are about to spread out into new fields of research and application [1]—no wonder that the market for etching machines kept growing by an annual rate of 17 % up to the burst of the internet bubble, and it took only some years of recovery to continue the voyage [2]. To realize a multi-color flat panel display measuring 16 square feet, a total of 300 000 LEDs is required. 1994, just in front of the launching of the blue LED, the fraction of LEDs (which effectively emit in the visible range) amounted to about 1/3 of the total compound semiconductor market of $4.4 billion. Four years later, the fraction of so-called high-brightness LEDs had been grown by 64 % (total market 1997: $7.1 billion), and the number of substrates which were subject to epitaxial processes to produce LEDs had been increased by almost 300 % (Fig. 1). This was conservatively updated to an estimated amount of 1
International Technological Roadmap for Semiconductors
VII
VIII
Preface
MBE 5%
LPE 72%
MOVPE 54%
MOVPE VPE 13% 15% 1994: 5550 m
2
LPE 31% VPE 10% 2
1998: 14,190 m
luminescence [lm/W]
InGaAlP/GaAs red/orange
10
1
10-1
AlGaAs/GaAs red
GaP/ZnO/GaP red
GaAsP:N/GaP red/yellow GaP:N/GaP green
GaAsP/GaAs red
GaAs/GaP
1960
1970
1980 year
GaN/InGaN on sapphire
SiC on SiC
1990
2000
Fig. 1. From 1994 to 1998, the area which had been epitaxially grown for LEDs rised by about 300 % from 5 550 m2 to 14 190 m2 (about 11/2 soccer pitches). Simultaneously, the main interest moved from relatively simple techniques (liquid-phase epitaxy) to highly sophisticated gas-phase epitaxy. The etching techniques covered the whole palette from sewing via wet etching up to dry etching methods, and here from hard gunfire by an argon ion beam to subtle etching by generating a reactive plasma of high density but consisting of ions with relatively low energy. This is associated with a steep increase in colorful brillance or efficacy.
$1 billion in 2003 [3]. Despite the deep crisis which was even aggravated by the assassination on Sep 11, the market had recovered. Alone the market for high-brightness LEDs has hit the volume of the total LED market of 1997 [4] (in numbers of LEDs shipped: 20 billion units in 2004, 30 billion units in 2006), and the next landmark has been reached just now: Almost every third LED is used for automotive purposes, most of them belong to the category high-brightness [5]. (Figs. 2). Progress is made most obvious by a flashback. In 1978, the costs for a complete wafer fab amounted to about $20 mio., the narrowest geometries were about 2 and 3 microns, and the yield remained at ridiculous 10 %. Some statements from the first number of Semiconductor International from the year 1979:
Preface
IX 30
7.5
total LED usage [%]
LED market [$ billions]
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5.0
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2006 year
2008 2010 2012
20
10
0 2000
2002
2004 2006 year
2008
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Fig. 2. 2007, the market for LEDs alone has hit the total volume of III/V compound semiconductors of 1997. In particular, the market for automotive LEDs has grown by more than 30 % in the last five years. By the end of 2008, almost every third LED is expected to be part of an automotive illumination with a growing fraction of high-brightness LEDs [4] [5].
• Progress in the area of photoresist and masks provided, deep-UV lithography will be the next technological step [6]. • Plasma etching will become a very important technology for pattern transfer, and plasma deposition shows the same auspicious propects . . . [7]. • Many of the problems what design is concerned can get over provided – a less efficient chip is accepted by the market, – test functions are integrated on the chip, – which is synonymous not to strive for the highest packing density [8]. This was 1978 [9]: • minimum of the structures by about 2.5 μm; • the design was at a maximum of eight masks; • clean room class between 100 and 1 000; • first application of steppers for processing of 4" wafers; • gate electrodes with doped polysilicon; • first operation of lasers for annealing purposes;
X
Preface • minimum gate oxide thickness: 100 ˚ A; • wet etching still dominates, plasma etching as high-end technology is operated only in areas where undercut is not tolerable.
For the public, the computer market was entirely dominated by dinosaur computers, but the first programmable calculators were offered by HewlettPackard and Texas Instruments, the overwhelming part of light was generated by Edison’ evanescent bulb, and “cold light” was just a dream of some scientists. Ion Beam Etching with Ar (DC) anisotropic etching low selectivity poor etch rate sputter yield at ≈ 60 ◦ poor efficiency massive damage
MW-CCP 2.45 GHz ashing with O2 very soft etching very low etch rates no anisotropy
RF-Sputtering 13.56 MHz option w. magnetron with Ar sputtering of dielectrics
downstream
reactive @
@ @
@ capacitive @ coupling @ @
capacitive coupling
MW
@ @ R ? @ reactive sputtering CCP-RF oxides from metals @ capacitive “hot” electrode @ coupling @ R @ CCP-RIE: (ME) “ RIE” 13.56, 27.12 MHz ne ≥ 109 /cm3 antenna anisotropic, selective etch.@ coupling @ prone to high damage @ helicon R discharges @ static 6 13.56 MHz ne ≥ 1012 /cm3 introduction of static resonant excit. magnetic fields
? RF-Ion Etching with Ar for large areas
RF ? Ion Beam Etching with ICP-RF mainly 2 MHz reactive, soft anisotropic processes
RF
@ static @ @ R @
? ICP-RIE 2 or 13.56 MHz non-resonant excit. ne ≥ 1012 /cm3 reactive, soft, anisotropic very high etch rates
? ECR-RIE 2.45 GHz resonant excitat. ne ≥ 1012 /cm3 reactive, soft, anisotropic very high etch rates
Fig. 3. Flow Diagram for Mutual Development of Excitation Methods and Reactive Processes
Preface
XI Sputtering RF DC-Magnetron t
t IBC Dielectrics, Metals Diamond, DLC
Plasma Coating
t PECVD Dielectrics, Metals Diamond, DLC
t
Ion-Plating Dense Metals PECVD, Plasma Enhanced Chemical Vapour Deposition: p ≥ 1000 mTorr (130 Pa): sample on grounded electrode; IBC, Ion Beam Coating; Ion Plating: p < 1 mTorr, evaporation of very dense metal on a sample atop a powered electrode;
Fig. 4. Various dry etching methods. They mainly differ in the excitation method.
Today, the line widths of the most advanced devices are less than 1 000 ˚ A, but the main issues in nanometer devices are still the patterning of the gate (MOS devices) or the mirror or facet (semiconductor lasers). The end of the common planar technology although further miniaturized is predicted to be reached by less than a decade. But perhaps the prognostic power will be comparable with the score of the prognosis for the world-wide oil deposits which should be depleted always “in the next fifteen years”, who knows? Therefore, we consider to address again a broad readership for the next edition of this book. After a short phenomenological introduction, the various methods to generate charged carriers are extensively discussed, keeping in mind that low temperature plasmas have been subject of intense research for now more than a century which comprises the methods to characterize the plasmas. In the second half, the technological techniques of surface refinement are discussed in two lengthy chapters, and we become again aware of the mutual challenges of surface treatment and plasma technology. Detailed derivations are compiled in a special chapter, including the detailed but tedious algebra. It has become customary to spell acronyms which are introduced just recently in capitals which are gradually transformed to small letters, and it is a matter of taste to decide whether this time is already passed by. Since this book is mostly directed to plasma beginners, I preferred to apply capitals throughout.
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Preface CCP-IE RF: PE, RIE, MERIE MW: PE t
ICP-IE
t
Plasma Etching
t
IBE
RIBE CAIBE
t
MW-RIE ECR-RIE CCP, Capacitively Coupled Plasma; ECR, Electron Cyclotron Resonance, downstream; ICP, Inductively Coupled Plasma, downstream; MW, Micro Wave (2.45 GHz); PE, Plasma Etching: p > 75 mTorr (10 Pa): sample on grounded electrode; IE, Ion Etching, RIE, Reactive Ion Etching: p <50 mTorr (7 Pa), sample on powered electrode; IBE, Ion Beam Etching; MERIE, Magnetically Enhanced Reactive Ion Etching: RIE; electrons are suppressed to reach the sample’s surface by means of a magnetic field; CAIBE, Chemical Assisted Ion Beam Etching; RIBE, Reactive Ion Beam Etching.
Fig. 5. Various dry coating methods. They mainly differ in the excitation method.
It is simply impossible to name all the collegues who discussed certain issues extensively or courteously passed me figures and micrographs (cf. Chap. 15). For more than a decade, I am now used to meet most of them during the annual conference of the AVS, feeling the reality of the scientific community. But in particular, I would like to thank Peter Awakowicz, Rod Boswell, Francis Chen, John Coburn, Vince Donnelly, Demetre Economou, David Graves, Michael Klick, Michael Lieberman, Ivo Rangelow, and Peter Unger. I would like to thank also the team at Springer-Verlag for the pleasant cooperation, and last but not the least my wife for her infinite patience over all the years. Munich, January 27, 2009
Gerhard Franz
Contents
1 Introduction
1
2 Collisions and cross sections 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Elastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Cross section . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Cross section and mean free path in kinetic gas theory. . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.2 Cross section and mean free path in plasmas. . 2.2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Elastic collisions between electrons and neutrals . . . . . . . . . 2.3.1 The total cross section . . . . . . . . . . . . . . . . . . . 2.3.2 Differential cross section . . . . . . . . . . . . . . . . . . 2.3.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3.1 Cross section for the interaction between a point charge and an induced dipole. . . . . . . . . . . 2.3.3.2 Ramsauer effect. . . . . . . . . . . . . . . . . . 2.3.4 The frequency of elastic collisions between electrons and neutrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Cross section and rate constant for argon . . . . . . . . . 2.4 Elastic collisions between heavy particles . . . . . . . . . . . . . 2.5 Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Inelastic collisions between electrons and heavy particles 2.5.1.1 Experimental methods. . . . . . . . . . . . . . . 2.5.1.2 Cross section. . . . . . . . . . . . . . . . . . . . 2.5.1.3 Rate constant for ionization. . . . . . . . . . . . 2.5.1.4 Electron attachment. . . . . . . . . . . . . . . . 2.5.1.5 Total collision cross section. . . . . . . . . . . . 2.5.2 Inelastic collisions between heavy particles . . . . . . . . 2.5.2.1 Charge transfer. . . . . . . . . . . . . . . . . . 2.5.2.2 Resonant charge transfer. . . . . . . . . . . . . 2.5.2.3 Penning ionization. . . . . . . . . . . . . . . . .
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3 The plasma 3.1 Direct current glow discharge . . . . . . . . . . . . 3.1.1 Phenomenology . . . . . . . . . . . . . . . . 3.1.2 V-I characteristic . . . . . . . . . . . . . . . 3.2 Temperature distribution in a plasma . . . . . . . . 3.3 Neutralization of charges in an undisturbed plasma 3.4 Potential variation in the plasma . . . . . . . . . . 3.4.1 The low-voltage plasma sheath . . . . . . . 3.4.1.1 Approximation of first order. . . . 3.4.1.2 Approximation of second order. . . 3.4.1.3 Approximation of third order. . . . 3.5 Temperature and density of the electrons . . . . . . 3.5.1 Elastic scattering and thermalization . . . . 3.5.2 Energy loss . . . . . . . . . . . . . . . . . . 3.5.3 Electron temperature . . . . . . . . . . . . . 3.5.4 Electron density . . . . . . . . . . . . . . . . 3.6 Plasma oscillations . . . . . . . . . . . . . . . . . .
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4 DC discharges 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 4.2 Gaseous generation of carriers . . . . . . . . . . . 4.2.1 Townsend’s equation . . . . . . . . . . . . 4.2.2 The primary ionization coefficient . . . . . 4.3 The normal cathode fall . . . . . . . . . . . . . . 4.3.1 The secondary ionization coefficient . . . . 4.3.2 V-I characteristic . . . . . . . . . . . . . . 4.3.2.1 Matrix sheath. . . . . . . . . . . 4.3.2.2 Child-Langmuir sheath. . . . . . 4.4 The abnormal cathode fall . . . . . . . . . . . . . 4.4.1 Discussion of Townsend’s approximation . 4.5 Negative glow and positive column . . . . . . . . 4.6 Ionization . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Ionization in the negative glow . . . . . . 4.7 Loss of carriers . . . . . . . . . . . . . . . . . . . 4.7.1 Free diffusion . . . . . . . . . . . . . . . . 4.7.2 Ambipolar diffusion coefficient . . . . . . . 4.7.3 Modified boundary . . . . . . . . . . . . . 4.7.4 Diffusion processes in the positive column
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2.6
2.5.3 Collisions between photons and molecules Generation of secondary electrons at surfaces . . . 2.6.1 Electrons . . . . . . . . . . . . . . . . . . 2.6.2 Ions . . . . . . . . . . . . . . . . . . . . . 2.6.3 Photons . . . . . . . . . . . . . . . . . . .
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Contents 4.8 4.9 4.10 4.11
Anodic region . . . . . . Hollow cathode discharge Similarity laws . . . . . Conclusion . . . . . . . .
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5 High-frequency discharges I 5.1 Phenomenological introduction . . . . . . . . . . . . . . . . . . 5.2 Generation of carriers . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Operating frequency and the EEDF . . . . . . . . . . . . . . . . 5.4 Loss mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Recombination . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Attachment in electronegative gases . . . . . . . . . . . . 5.4.4 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Microwave discharges: model for breakdown . . . . . . . 5.5.1.1 Frequency. . . . . . . . . . . . . . . . . . . . . . 5.5.1.2 Pressure. . . . . . . . . . . . . . . . . . . . . . 5.5.1.3 Oscillation amplitude. . . . . . . . . . . . . . . 5.5.1.4 Low gas pressure. . . . . . . . . . . . . . . . . . 5.5.1.5 Breakdown. . . . . . . . . . . . . . . . . . . . . 5.5.1.6 Conclusion. . . . . . . . . . . . . . . . . . . . . 5.6 Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 EEDF and the electric field . . . . . . . . . . . . . . . . 5.6.2 Collision frequency . . . . . . . . . . . . . . . . . . . . . 5.7 High-frequency coupling: qualitative approach . . . . . . . . . . 5.8 High-frequency coupling: quantitative approach . . . . . . . . . 5.8.1 Absorption circuit . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Eliminator circuit . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Coupled parallel circuits . . . . . . . . . . . . . . . . . . 5.8.3.1 Transformer coupling. . . . . . . . . . . . . . . 5.8.4 Capacitive and inductive coupling . . . . . . . . . . . . . 5.8.5 Dual circuit of the capacitively coupled plasma . . . . . 5.8.5.1 First approximation (symmetric discharge). . . 5.8.5.2 Second approximation (asymmetric discharge). 5.9 Matching networks . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Coaxial cable . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1.1 Characteristic impedance. . . . . . . . . . . . . 5.10.2 Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.3 Mode patterns in transmission lines . . . . . . . . . . . . 5.10.3.1 Coaxial cable. . . . . . . . . . . . . . . . . . . . 5.10.3.2 Waveguide. . . . . . . . . . . . . . . . . . . . . 5.10.4 Electrode . . . . . . . . . . . . . . . . . . . . . . . . . .
103 103 105 110 111 111 113 114 115 116 120 121 121 122 123 123 124 124 124 125 126 130 131 132 133 134 137 137 138 138 139 141 143 144 145 145 145 145 146
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Contents
5.11 Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6 High-frequency discharges II 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Electric fields across the sheaths . . . . . . . . . . . . . . . . . . 6.3 Current-voltage characteristic at one electrode . . . . . . . . . . 6.4 Sheath potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Symmetric system . . . . . . . . . . . . . . . . . . . . . 6.4.2 Asymmetric system . . . . . . . . . . . . . . . . . . . . . 6.4.2.1 Sheath potential theorem. . . . . . . . . . . . . 6.4.2.2 Calculation. . . . . . . . . . . . . . . . . . . . . 6.4.3 Resistive Coupling . . . . . . . . . . . . . . . . . . . . . 6.5 Power input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Sheath heating . . . . . . . . . . . . . . . . . . . . . . . 6.5.1.1 Displacement current heating. . . . . . . . . . . 6.5.1.2 Ohmic heating. . . . . . . . . . . . . . . . . . . 6.5.1.3 Stochastic heating. . . . . . . . . . . . . . . . . 6.5.2 Two regimes of power transfer . . . . . . . . . . . . . . . 6.6 Spatial distribution of charged carriers . . . . . . . . . . . . . . 6.7 Dual-frequency discharges . . . . . . . . . . . . . . . . . . . . . 6.7.1 Frequency dependence of plasma density . . . . . . . . . 6.7.2 Mutual influence of two electrodes . . . . . . . . . . . . . 6.7.2.1 Narrow gap. . . . . . . . . . . . . . . . . . . . . 6.7.2.2 Wide gap. . . . . . . . . . . . . . . . . . . . . . 6.8 Collisional sheaths . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1.1 Elastic scattering and resonant charge-transfer. 6.8.1.2 Elastic scattering. . . . . . . . . . . . . . . . . 6.8.1.3 Modulation. . . . . . . . . . . . . . . . . . . . . 6.8.2 Computer simulations . . . . . . . . . . . . . . . . . . . 6.8.3 Hybrid sheath model . . . . . . . . . . . . . . . . . . . . 6.8.3.1 Ions. . . . . . . . . . . . . . . . . . . . . . . . . 6.8.4 Measurements and modellings . . . . . . . . . . . . . . . 6.8.4.1 IEDF in the sheath. . . . . . . . . . . . . . . . 6.8.4.2 IEDF in the sheath of the powered electrode. . 6.9 DC discharges and capacitively coupled RF plasmas . . . . . . . 6.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
151 151 157 160 163 163 165 165 166 169 172 175 175 175 176 181 182 185 185 186 187 187 190 194 195 198 198 201 205 206 206 206 207 209 212
7 High-frequency discharges III 7.1 Introduction . . . . . . . . . . . . . . . . . . . 7.2 Inductively coupled plasma . . . . . . . . . . . 7.2.1 Transformer Model . . . . . . . . . . . 7.2.2 Power input for inductive coupling . . 7.2.2.1 Plasma resistance and plasma
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7.2.2.2 Coupling between coil and plasma. 7.2.2.3 Primary circuit. . . . . . . . . . . . 7.2.3 Limits of power input . . . . . . . . . . . . . 7.2.4 Top coil configuration . . . . . . . . . . . . 7.2.4.1 E-mode and H-mode. . . . . . . . . 7.2.5 Modeling of ICP discharges . . . . . . . . . 7.2.6 Conclusion . . . . . . . . . . . . . . . . . . . 7.3 Generation of plasmas supported by magnetic fields 7.3.1 R´esum´e of the properties of HF discharges . 7.3.2 Whistler waves . . . . . . . . . . . . . . . . 7.3.2.1 Phenomenology. . . . . . . . . . . 7.3.2.2 Dispersion and absorption. . . . . . 7.4 Helicons in a bounded plasma . . . . . . . . . . . . 7.4.1 Introduction . . . . . . . . . . . . . . . . . . 7.4.2 Dispersion and wave fields . . . . . . . . . . 7.4.3 Antenna coupling . . . . . . . . . . . . . . . 7.4.4 Operation . . . . . . . . . . . . . . . . . . . 7.4.5 Experiments . . . . . . . . . . . . . . . . . . 7.4.6 Summary . . . . . . . . . . . . . . . . . . . 7.5 Electron cyclotron resonance . . . . . . . . . . . . . 7.5.1 The electric field and the diffusion length . . 7.5.2 Coupling of microwaves . . . . . . . . . . . . 7.5.3 Electron cyclotron resonance heating . . . . 7.5.4 Electron cyclotron resonance reactors . . . . 7.5.4.1 Waveguide applicator. . . . . . . . 7.5.4.2 Cavity applicator. . . . . . . . . . 7.5.4.3 Conclusion. . . . . . . . . . . . . . 7.6 Comparison of high-density discharges . . . . . . . 8 Ion beam systems 8.1 Introduction . . . . . . . . . . . . . . . . . . . . 8.2 Plasma sources . . . . . . . . . . . . . . . . . . 8.2.1 Kaufman source . . . . . . . . . . . . . . 8.2.2 HF sources . . . . . . . . . . . . . . . . 8.2.2.1 Design of a grid optics with RF 8.2.2.2 Boundary voltage. . . . . . . . 8.3 Grid optics . . . . . . . . . . . . . . . . . . . . 8.3.1 Configuration and potential adjustment . 8.3.2 Screen grid . . . . . . . . . . . . . . . . 8.3.2.1 Kaufman source. . . . . . . . . 8.3.2.2 RF source. . . . . . . . . . . . 8.3.3 Accelerator grid . . . . . . . . . . . . . . 8.4 Qualitative treatment of beam extraction . . . . 8.4.1 Extraction without a grid . . . . . . . .
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XVIII 8.4.2 Extraction with one grid . . . . . . . . . 8.4.3 Extraction with two or more grids . . . . 8.5 Quantitative treatment of beam extraction . . . 8.5.1 Current density . . . . . . . . . . . . . . 8.5.1.1 Derivations from Child’s law. . 8.5.1.2 Grid transparency. . . . . . . . 8.5.2 Focusing and divergence . . . . . . . . . 8.5.3 Conclusion . . . . . . . . . . . . . . . . . 8.5.4 Three-grid ion beam source . . . . . . . 8.5.5 Four-grid ion beam source . . . . . . . . 8.6 Neutralization . . . . . . . . . . . . . . . . . . . 8.6.1 Principle of operation . . . . . . . . . . . 8.6.2 Neutralization elements . . . . . . . . . . 8.7 Process optimization . . . . . . . . . . . . . . . 8.7.1 Maximum power and substrate damage . 8.7.2 Discharge voltage and substrate damage 8.7.3 Power and grid current . . . . . . . . . . 8.7.4 Electron backstreaming . . . . . . . . . . 8.7.5 Current density of the ion beam . . . . . 8.7.6 Uniformity . . . . . . . . . . . . . . . . .
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279 280 280 280 280 282 283 285 285 288 288 288 290 291 291 292 292 293 294 294
9 Plasma diagnostics 9.1 Langmuir probe . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Conditions for performance . . . . . . . . . . . . . . . . 9.1.3 Characteristic of the Langmuir probe . . . . . . . . . . . 9.1.3.1 Principle of the measuring technique. . . . . . . 9.1.3.2 Extraction of plasma parameters. . . . . . . . . 9.1.4 Plasma potential . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Principle of the double probe . . . . . . . . . . . . . . . 9.1.6 Principle of the asymmetrical double probe . . . . . . . . 9.1.7 Determination of potentials in high-frequency discharges 9.1.8 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.9 Probe radius . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.10 Thin sheath: space charge limited current . . . . . . . . 9.1.10.1 Positive ions. . . . . . . . . . . . . . . . . . . . 9.1.10.2 Electrons. . . . . . . . . . . . . . . . . . . . . . 9.1.10.3 Finite electron temperature. . . . . . . . . . . . 9.1.11 Thick sheath: Orbital Motion Theory (OML Theory) . . 9.1.11.1 Electron saturation current. . . . . . . . . . . . 9.1.11.2 Electron current in the retarding-field region. . 9.1.11.3 Current transition at plasma potential. . . . . . 9.1.11.4 Ion current. . . . . . . . . . . . . . . . . . . . . 9.1.11.5 Summary. . . . . . . . . . . . . . . . . . . . . .
299 300 300 300 303 303 304 306 307 309 310 313 313 315 315 315 316 318 320 324 324 325 325
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9.2
9.3 9.4
9.5 9.6 9.7
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9.1.12 Electron temperature and plasma potential . . . . . . 9.1.13 Influence of a magnetic field . . . . . . . . . . . . . . 9.1.14 Measurements . . . . . . . . . . . . . . . . . . . . . . 9.1.14.1 Grounding problems. . . . . . . . . . . . . . 9.1.14.2 Determination of the characteristic. . . . . . 9.1.14.3 Electron temperature and plasma potential. 9.1.15 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . Self-Excited Electron Resonance Spectroscopy . . . . . . . . 9.2.1 Non-linear response between voltage and current . . . 9.2.2 Technical realization . . . . . . . . . . . . . . . . . . 9.2.3 Inherent properties . . . . . . . . . . . . . . . . . . . 9.2.3.1 Electronic plasma density. . . . . . . . . . . 9.2.3.2 Frequency of momentum transfer. . . . . . . 9.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . Impedance analysis . . . . . . . . . . . . . . . . . . . . . . . Optical emission spectroscopy (OES) . . . . . . . . . . . . . 9.4.1 Electron temperature with OES . . . . . . . . . . . . 9.4.1.1 Corona model and its validity. . . . . . . . . 9.4.1.2 Direct electronic excitation. . . . . . . . . . 9.4.1.3 Parametrization of the cross section. . . . . 9.4.1.4 Details of the evaluation. . . . . . . . . . . 9.4.1.5 Chosing the right electron distribution. . . . 9.4.1.6 RF power. . . . . . . . . . . . . . . . . . . . 9.4.1.7 Corona model: limits of applicability. . . . . 9.4.2 Plasma gas temperature . . . . . . . . . . . . . . . . 9.4.2.1 Features in noble and inert gases. . . . . . . R´esum´e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Electronegative Plasmas . . . . . . . . . . . . . Capacitively coupled plasmas . . . . . . . . . . . . . . . . . 9.7.1 Electrical considerations . . . . . . . . . . . . . . . . 9.7.2 Plasma density . . . . . . . . . . . . . . . . . . . . . 9.7.3 Electron temperature . . . . . . . . . . . . . . . . . . 9.7.4 Power dissipation . . . . . . . . . . . . . . . . . . . . 9.7.4.1 Effective collision frequency. . . . . . . . . . 9.7.4.2 Plasma gas temperature. . . . . . . . . . . . Inductively coupled plasmas . . . . . . . . . . . . . . . . . . 9.8.1 General remarks . . . . . . . . . . . . . . . . . . . . 9.8.2 Influence of the reactor geometry . . . . . . . . . . . 9.8.3 Electron temperature . . . . . . . . . . . . . . . . . . 9.8.4 Plasma density . . . . . . . . . . . . . . . . . . . . . 9.8.5 Density of neutrals . . . . . . . . . . . . . . . . . . . 9.8.6 Plasma gas temperature . . . . . . . . . . . . . . . . 9.8.7 Modeling . . . . . . . . . . . . . . . . . . . . . . . . 9.8.8 Negative Ions . . . . . . . . . . . . . . . . . . . . . .
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Contents 9.8.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 373 9.8.10 Interaction between Plasma Bulk and Surfaces . . . . . . 373
10 Plasma deposition processes 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Sputter deposition systems . . . . . . . . . . . . . . . 10.2 Sputtering kinetics . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Target processes . . . . . . . . . . . . . . . . . . . . . 10.2.1.1 Sputtering yield. . . . . . . . . . . . . . . . 10.2.1.2 Energy distribution of the sputtered atoms. 10.2.2 Transport processes: energy distribution of the atoms 10.2.3 Substrate processes: Film formation . . . . . . . . . . 10.3 Target topography . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Historical review . . . . . . . . . . . . . . . . . . . . 10.3.1.1 Roughness-induced mechanism. . . . . . . . 10.3.1.2 Contamination-induced mechanism. . . . . . 10.3.2 Comparison of topographical mechanisms . . . . . . . 10.4 Sputtering conditions . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Electrical properties . . . . . . . . . . . . . . . . . . 10.4.2 Temperature control of the substrate . . . . . . . . . 10.4.2.1 Temperature measurement. . . . . . . . . . 10.4.2.2 Temperature control. . . . . . . . . . . . . . 10.4.3 Contamination . . . . . . . . . . . . . . . . . . . . . 10.4.3.1 Target and purity requirements. . . . . . . . 10.4.3.2 Contamination by argon. . . . . . . . . . . . 10.4.3.3 Contamination by other gases. . . . . . . . 10.4.3.4 Reactive sputtering. . . . . . . . . . . . . . 10.4.3.5 Bombardment with other particles. . . . . . 10.5 Sputtering with bias techniques . . . . . . . . . . . . . . . . 10.5.1 Deposition rate and film composition . . . . . . . . . 10.5.2 Further film properties . . . . . . . . . . . . . . . . . 10.5.3 Mechanisms of bias sputtering . . . . . . . . . . . . . 10.5.4 Homogenity of coating at rectangular steps . . . . . . 10.5.5 Mechanical tension and substrate bias . . . . . . . . 10.6 Sputter deposition of multicomponent films . . . . . . . . . . 10.6.1 Target processes . . . . . . . . . . . . . . . . . . . . . 10.6.2 Between target and substrate . . . . . . . . . . . . . 10.6.3 Substrate processes . . . . . . . . . . . . . . . . . . . 10.6.4 Preparative aspects . . . . . . . . . . . . . . . . . . . 10.7 Sputtering systems with increased plasma density . . . . . . 10.7.1 Magnetically improved sputtering systems . . . . . . 10.7.1.1 Theory. . . . . . . . . . . . . . . . . . . . . 10.7.1.2 Technological issues. . . . . . . . . . . . . . 10.7.2 Triode systems . . . . . . . . . . . . . . . . . . . . .
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Contents 10.7.3 Ion plating systems . . . . . . . . . . . . . . . . . . 10.8 Plasma Enhanced Chemical Vapour Deposition (PECVD) 10.8.1 Instantaneous mass spectrometry . . . . . . . . . . 10.8.2 Diamond-like coatings (DLCs) . . . . . . . . . . . . 10.8.2.1 Applications. . . . . . . . . . . . . . . . . 10.8.2.2 Diamond electronics. . . . . . . . . . . . . 10.9 Ion beam deposition (IBD) . . . . . . . . . . . . . . . . . .
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11 Plasma etch processes 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sputter etching . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Reactive etch processes . . . . . . . . . . . . . . . . . . . . . 11.4 General dependence on independent properties . . . . . . . . 11.4.1 Substrate temperature . . . . . . . . . . . . . . . . . 11.4.1.1 Introduction. . . . . . . . . . . . . . . . . . 11.4.1.2 Etchrate and its temperature dependence. . 11.4.2 Gas composition . . . . . . . . . . . . . . . . . . . . 11.4.3 Gas pressure and RF power . . . . . . . . . . . . . . 11.4.4 Electrode geometry . . . . . . . . . . . . . . . . . . . 11.4.5 Gas flow effects and the loading effect . . . . . . . . . 11.4.5.1 Gas flow. . . . . . . . . . . . . . . . . . . . 11.4.5.2 Loading effect. . . . . . . . . . . . . . . . . 11.4.6 Transport effects and reactor design . . . . . . . . . . 11.5 Characteristics of dry etching . . . . . . . . . . . . . . . . . 11.5.1 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Selectivity . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Mask effects . . . . . . . . . . . . . . . . . . . . . . . 11.5.3.1 Erosion. . . . . . . . . . . . . . . . . . . . . 11.5.3.2 Faceting. . . . . . . . . . . . . . . . . . . . 11.5.3.3 Metal masks and trilevel photoresist. . . . . 11.5.3.4 LER and CD. . . . . . . . . . . . . . . . . . 11.5.4 Redeposition and sidewall passivation . . . . . . . . . 11.5.5 Microfeatures . . . . . . . . . . . . . . . . . . . . . . 11.5.5.1 Trenching. . . . . . . . . . . . . . . . . . . . 11.5.5.2 Shadowing. . . . . . . . . . . . . . . . . . . 11.5.5.3 Microloading. . . . . . . . . . . . . . . . . . 11.5.5.4 Aspect-Ratio Dependent Etching (ARDE). . 11.5.6 Charging effects . . . . . . . . . . . . . . . . . . . . . 11.5.7 High-end etching processes with high density plasmas 11.5.8 Towards nanostructures . . . . . . . . . . . . . . . . 11.6 Special features of ion beam etching . . . . . . . . . . . . . . 11.6.1 Applications . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Ion beam assisted etching: IBAE or CAIBE . . . . . 11.7 Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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439 439 442 443 446 447 447 448 448 449 451 452 452 455 457 464 465 465 466 466 466 467 469 472 474 474 475 476 478 481 483 488 488 490 493 494
XXII 11.8 Process control . . . . . . . . . . . . . . . . . . . . 11.8.1 Impedance of a discharge . . . . . . . . . . . 11.8.2 Ellipsometry . . . . . . . . . . . . . . . . . . 11.8.3 Optical emission spectroscopy . . . . . . . . 11.8.4 Interferometric methods . . . . . . . . . . . 11.8.4.1 Metals and dielectrics. . . . . . . . 11.8.4.2 Semiconductors. . . . . . . . . . . 11.8.5 CCD controlled laser interferometry . . . . . 11.8.6 Mass spectrometry . . . . . . . . . . . . . . 11.8.6.1 Conventional mass spectrometry. . 11.8.6.2 Glow discharge mass spectrometry. 11.8.7 Problems during in-situ monitoring . . . . . 11.8.8 Conclusion . . . . . . . . . . . . . . . . . . .
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12 Etch Mechanisms 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Quantitative calculation with Langmuir’s theory . . . . . . 12.3 . . . and ion etching? . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . 12.4 Etching of Si and its compounds with F-containing gases . 12.4.1 Experimental observations . . . . . . . . . . . . . . 12.4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Chemical selectivity . . . . . . . . . . . . . . . . . . 12.4.4 PECVD vs. RIE . . . . . . . . . . . . . . . . . . . 12.4.5 Anisotropy and the so-called Bosch process . . . . . 12.5 Etching of Si and its compounds with Cl-containing gases . 12.5.1 Thermodynamical selectivity . . . . . . . . . . . . . 12.5.2 Chemical selectivity . . . . . . . . . . . . . . . . . . 12.5.3 Doping effect . . . . . . . . . . . . . . . . . . . . . 12.6 Etching of III/V-compounds . . . . . . . . . . . . . . . . . 12.6.1 Why avoid fluorine? . . . . . . . . . . . . . . . . . 12.6.2 Etching with chlorine . . . . . . . . . . . . . . . . . 12.6.2.1 Etchrate and its temperature dependence. 12.6.2.2 Chlorine sources. . . . . . . . . . . . . . . 12.6.3 Sidewall passivation . . . . . . . . . . . . . . . . . . 12.6.4 Chemical selectivity . . . . . . . . . . . . . . . . . . 12.6.5 Kinetics of chlorine etching . . . . . . . . . . . . . . 12.6.6 Methane-hydrogen process . . . . . . . . . . . . . . 12.7 Combination of various etch methods . . . . . . . . . . . . 12.8 Hazards associated with chlorine etching . . . . . . . . . . 12.9 Simulation of dry etching processes . . . . . . . . . . . . .
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13 Outlook
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Contents 14 Advanced Topics 14.1 Electron Energy Distribution Functions (EEDFs) . . . . . . 14.1.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . 14.1.2 External field as small disturbance . . . . . . . . . . 14.1.2.1 Elastic collisions. . . . . . . . . . . . . . . . 14.1.2.2 Inelastic collisions. . . . . . . . . . . . . . . 14.1.3 Approximate solutions of the Boltzmann equation . . 14.1.3.1 High frequencies. . . . . . . . . . . . . . . . 14.1.3.2 Maxwellian distribution. . . . . . . . . . . . 14.1.3.3 Margenau distribution. . . . . . . . . . . . . 14.1.3.4 Druyvesteynian distribution. . . . . . . . . 14.1.4 Frequency effects . . . . . . . . . . . . . . . . . . . . 14.2 Sheath and presheath . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Conditions . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Presheath . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Charge density across the sheath . . . . . . . . . . . 14.2.5 Approximations . . . . . . . . . . . . . . . . . . . . . 14.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Plasma oscillations . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Dispersion relation . . . . . . . . . . . . . . . . . . . 14.3.2 Landau damping . . . . . . . . . . . . . . . . . . . . 14.4 Capacitive coupling for collisionless sheaths . . . . . . . . . . 14.4.1 The symmetric case . . . . . . . . . . . . . . . . . . . 14.4.1.1 Introduction. . . . . . . . . . . . . . . . . . 14.4.1.2 Assumptions. . . . . . . . . . . . . . . . . . 14.4.1.3 Spatial sheath structure. . . . . . . . . . . . 14.4.1.4 Carrier density and sheath potential. . . . . 14.4.1.5 Sheath dynamics. . . . . . . . . . . . . . . . 14.4.2 The asymmetric case . . . . . . . . . . . . . . . . . . 14.5 Motion in a magnetic field . . . . . . . . . . . . . . . . . . . 14.5.1 The magnetic bottle . . . . . . . . . . . . . . . . . . 14.5.2 Modification of diffusion . . . . . . . . . . . . . . . . 14.6 Dispersion in a HF plasma . . . . . . . . . . . . . . . . . . . 14.6.1 Cutoff and skin depth . . . . . . . . . . . . . . . . . 14.6.2 Complex properties . . . . . . . . . . . . . . . . . . . 14.7 Whistler waves . . . . . . . . . . . . . . . . . . . . . . . . . 14.7.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . 14.7.1.1 Formula of Appleton and Hartree. . . . . . 14.7.1.2 Cutoff and resonance. . . . . . . . . . . . . 14.7.1.3 Dispersion relation. . . . . . . . . . . . . . . 14.7.1.4 R- and L-waves. . . . . . . . . . . . . . . . 14.7.1.5 Dispersion relation for arbitrary directions. 14.7.2 Bounded plasma . . . . . . . . . . . . . . . . . . . .
XXIII
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
567 567 567 567 568 568 569 569 570 571 571 575 576 576 577 579 580 582 583 583 586 589 591 591 591 592 593 598 600 601 604 604 610 613 613 617 623 623 623 628 629 631 633 637
XXIV
Contents 14.7.2.1 14.7.2.2 14.7.2.3 14.7.2.4 14.7.2.5 14.7.2.6 14.7.2.7 14.7.2.8
Introduction. . . . . . . . Extended Drude equation. Wave equation. . . . . . . Cylindrical confinement. . Dispersion relation. . . . . Azimuthal wave fields. . . Radial modes. . . . . . . . Conclusion . . . . . . . .
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637 638 639 642 646 647 651 658
15 Reference list of figures
661
16 Symbols, abbreviations, acronyms
663
References
669
Register
705
1 Introduction
From everyday life, we are familiar with discharges; consider atmospheric discharges which we observe in the form of lightning, and which equalize potential differences of some millions of volts within milliseconds. Very high temperatures can be generated with electric arcs, which were described for the first time by Humphry Davy in the year 1813. Two years later, they were applied to melt and fuse refractory metals [10]. The temperatures are 4 700 K at the electrodes and up to 7 000 K in the arc itself. With increased current densities, temperatures of up to 50 000 K could be reached. These are the surface temperatures of blue-white glowing O-stars (our sun, a G-type, has a surface temperature of about 6 000 K [11]). Bushels of pink-purple sparks can be seen at sharply pointed structures (ship masts or spires) during thunderstorms. This electroluminescent coronal discharge is well known as St. Elmo’s fire. It was Benjamin Franklin who in 1749 proved this phenomenon to be electric in nature. Glow discharges occur in diluted gases (low pressure discharges) at small current densities; they are used in sodium vapor lamps and fluorescent lamps; the former replaced normal incandescent lamps for street lighting from the 1960s, the latter still characterize the nightly atmosphere of famous places (e. g. Piccadilly Circus or Times Square). A short glance in the Encyclopædia Britannica, and we find under “glow discharge”: Self-sustained gas discharge with cold electrodes, and “discharge” itself means the evening out of various potentials. A very simple glow discharge can be generated by fusing two electrodes into a glas tube (the typical Geissler tube tube has a length of about 50 cm) with a pumping-port. When we start evacuating the tube and simultaneously apply a voltage of about 10 kV, we will recognize a light apparition in the tube below a pressure of some Torr (some hundreds of Pa). We see a glowing serpent between the electrodes which enlarges its cross section with further falling pressure until it fills the whole cross section of the tube. At pressures below about 1 Torr (1 kPa), the Crooke’s or Hittorf’s dark space appears in the area adjacent to the cathode, whereas glowing zones appear in the rest of the tube. Further decrease of pressure causes the dark space to grow until it fills the whole tube at pressures around 10 mTorr (1 Pa). Now, the greenish fluorescence of the glass tube remains as the last glowing apparition. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 1,
1
2
Introduction
The gas itself is partly ionized, its degree of ionization is typically some tens of ppm or ≤10−4 . At a neutral density of 1014 cm−3 or at a pressure of some mTorr (1/2 Pa), this would mean a charge density of less than 10−10 cm−3 (sum of electron density and ion density, which must always be of equal size). In 1928, it was Langmuir who denoted plasma a gas which is ionized to a large extent [12], and therefore, its charge density is called plasma density. In this nomenclature, this plasma will fall into the category tenuous plasma or low-density plasma. It conducts electric currents and can easily be influenced by external electric or magnetic fields. On the other hand, the plasma bulk will remain neutral, and this state is denoted quasineutral. It is the simultaneous occurrence of long-range and short-range forces which distinguishes a plasma from a neutral gas, which can be described, at least in principle, with the kinetic gas theory and the corresponding refinements (various potentials of interaction to take into account short-range forces). Both types of matter have some properties in common; among them is the chaotic behavior of their constituents: The neutrals and charged particles move on randomly scattered trajectories. 1018
shock waves -6 cm high pressure 10 controlled fusion arcs alkali 10 metal fusion -4 cm 0 experiments plasmas 1 low pressure arcs 1012 glow photosphere -2 m discharges 10 c 9 flames
np [cm-3]
15
10
106
0 cm 10
corona
ionosphere
103 100 -2 10
2 cm 10
interstellar space
van-Allen belt interplanetary 4 m space 0 c
1
10-1
100
101 102 k BTe [eV]
103
104
Fig. 1.1. Electron density in cm−3 for various plasmas as a nction of the electron temperature in eV. The straight lines are the Debye lengths.
105
Since the plasma density continuously diminishes by loss mechanisms such as recombination within the plasma and diffusion out of the glowing plasma volume with subsequent wall reactions, an external power source is required to keep a state of equilibrium. This source can be thermal in nature (flame or stars) as well as electric (gaseous discharge) or radiative (ionosphere). Plasmas are stable within an enormous pressure range beginning at 10−4 Pa in the ionosphere (120−400 km height), 0.1 μbar (0.01 Pa) in ion beam sources, up to 1011 bar in the interior of the stars (Fig. 1.1). Since the 1950s, there have been tremendous efforts to keep plasmas stable in nuclear fusion systems. The discharge pressure predominantly determines the properties of the plasma and hence its applications via the mean free path (MFP). Some examples are:
Introduction
3
• Geiger-M¨ uller counter (p ≈ 100 mbar). • Glow lamps (1 mbar − 30 mbar). • Fluorescent lamps (0.01 − 1 mbar). • Plasma ion sources (1 μbar − 1 mbar). In these few paragraphs, we have introduced several important properties for the understanding of a plasma. For the generation of carriers, some electrodynamical theories have to be applied to understand the creation of electric and magnetic fields in the plasma tube, whether they are static or time-dependent, and second how the single electron will respond to the action of these fields. Therefore, the point of view will vary between continuum theory and focus on a single particle, especially when collisions between particles are concerned by which carriers are generated and by which the energy which has been gained by the electrons will be dissipated into the plasma. Because collision phenomena are essential in plasmas but disturb the didactic track from introducing phenomena and explaining them in an inductive way to eventually reach a state of comprehensive view, they are placed at the beginning of this treatise.
2 Collisions and cross sections
Elastic collisions are of paramount importance for Ohmic heating and energy dissipation into the plasma; inelastic collisions are responsible for all the electronic excitation processes. For both processes, the deductive approach is chosen. By electronic and ionic impact, electrons can be released from surfaces; this process is required for the carrier avalanche in DC discharges, but has regained interest for stabilizing processes in production reactors whose inner surface is subject to various (mostly unintended) coating reactions.
2.1 Introduction Plasmas are chaotic gaseous systems which contain electrically charged carriers and which maintain themselves by collisions of electric carriers with neutrals. These impacts lead to further carriers, but other energy-consuming processes, such as optical transitions, can be triggered as well. These collisions are subsumed as inelastic. Nearly all of the gaseous ionizations are due to collisions between slow heavy particles and rapidly moving electrons. If an electron falls short of the required kinetic energy to excite a heavy particle (atom or molecule), it can but exchange momentum with its counterpart, and we denote this process an elastic collision. This process dominates all other possible collisions since only a very small fraction of electrons have kinetic energies which exceed the threshold of excitation. However, due to the unfavorable mass ratio of individual neutrals/ions over electrons, only a very small portion of energy is exchanged. The lack of efficiency causes an athermal plasma (Chap. 3): Electrons and heavy particles (sometimes even ions alone and neutrals alone) are in separate thermal equilibrium, and the temperatures differ tremendously. In the case of electron trapping by a neutral molecule or ion, respectively, the electron density is reduced. This process is denoted electron attachment; it will become effective beyond pressures of about some Torr. Below this limit, diffusion processes dominate the loss of electrons: The particles exhibit random walks, caused by stochastic collisions. When the reactor wall becomes a partner of a collision process, the charge is terminated. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 2,
5
6
2 Collisions and cross sections
With elastic collisions, only a very small fraction of kinetic energy is exchanged between particles which significantly differ in mass, which leads to only a small increase of kinetic energy of the translational energy of the gas constituents, i. e. a small rise in gas temperature. The electron kinetic energy is significantly reduced by inelastic collisions, which cause excitations of molecular rotations, vibrations and eventually electronic transitions. It is this excitation which generates ions, radicals and other dissociation products; furthermore, metastable species can be formed. When these inelastic collisions are possible, they dominate all other processes because of their large losses of energy. All these species can again react among themselves, but they can also relax. An electronic transition is often combined with radiation of light (UV/VIS range), which is specific to the ambient conditions (cf. Chap. 9). Our interest in these processes is twofold: Elastic scattering is mainly responsible for energy transfer and dissipation of energy; by inelastic scattering, optical levels can be occupied or neutrals are excited to ions, and electrons are generated. For electron generation, we distinguish between certain classes which are denoted primary processes, i. e. thermal emission and field emission, and secondary processes: • Generation of secondary electrons at the cathode and the walls by particles abundant in energy (γ-reaction): photons, electrons incident to the anode, ions incident to the cathode. • Ionization by impact by particles abundant in energy in the negative glow and the dark spaces of either electrode (impact by electrons: α-reaction, impact by ions: β-reaction). Primary processes can be neglected in glow discharges. Ions are produced only by gaseous reactions; otherwise, they can be generated by thermal emission, however, with very low yield [13].
2.2 Elastic collisions 2.2.1 Cross section In an approximation to first order, the cross section for elastic scattering is adapted from kinetic gas theory, which regards molecules as hard spheres, and its fourfold area is defined as the cross section; we begin with the relation between mean free path λ and particle density n 1 . (2.1) nσ The channel of a molecule with cross section σ as inner diameter is now exposed to an electron beam with flux I0 , which is shot into an ensemble of λ≈
2.2 Elastic collisions
7
non-interaction particles. After having covered a certain distance x, the flux has been attenuated to I = I0 e−μx
(2.2)
according to Beer and Lambert with μ a reciprocal length which can be interpreted as mean free path λ following Eq. (2.1) 1 = nσ (2.3) λ with n the density of the molecules and σ the cross section of the specific interaction. For a fundamental understanding of the cross section, the integral attenuation of an electron beam, its angular and eventually its energetic dependence are required. 2.2.1.1 Cross section and mean free path in kinetic gas theory. The time τ between two collisions is of the order of τ ≈ λ/ < v > with < v > the mean thermal velocity. The mean free path λ itself depends on particle density n and cross section σ according to Eq. (2.3) because it roughly scales with the target area of the molecule. In the simple approach of kinetic gas theory, this cross section is the only one considered and is the cross section for elastic scattering (see [14]). In this approximation, the cross section for elastic scattering is independent of temperature (energy) and is tabulated in units of A), πa20 —the cross section of the hydrogen atom with a0 Bohr’s radius (0.529 ˚ 8.82 × 10−17 cm−2 [sometimes the averaged collision number Pc across 1 cm for A2 ]. a gas at 1 Torr (133 Pa) and 0 ◦ C is found; the relation is σ = 0.283Pc in ˚ A typical value for λ is 4.5 cm for nitrogen at 1 mTorr (0.2 Pa). In fact, the cross section weakly depends on temperature (kinetic energy): With decreasing temperature (reduced molecular speed), it rises slowly due to prolongation of the interaction. For example, by reducing the temperature by 200 ◦ C from +100 ◦ C to −100 ◦ , σel rises by 30 % for nitrogen and oxygen. 2.2.1.2 Cross section and mean free path in plasmas. This temperature dependence is caused by weak intermolecular forces (polarization), which exhibit a weak r−6 dependence for the attractive part of the Lennard-Jones potential ΦLJ . In plasmas, however, we have bare charges which lead to intense, farreaching interactions which can be described analytically only in some simple cases. To begin with, we introduce some experimental details for atoms and electrons: • Atoms – In a sputter system operated with argon at 50 mTorr (7 Pa), the mean free path of the argon atoms is approximately 1.54 mm (25 A2 ). meV Ar, σtot = 26 ˚
8
2 Collisions and cross sections • Ions – The mean free path of ions is considerably smaller. λ for Ar+ ions of medium velocity in argon at 30 mTorr (4 Pa) and 4 eV is about 0.87 mm and rises for 10 eV ions to 1 mm and for 100 eV atoms to 1.4 mm. – In ion beam sources which are driven at significantly lower pressures, they are larger by orders of magnitude: For 400 eV Ar+ ions and a gas pressure of 10−4 Torr (13 mPa), λ equals 51.6 cm. • Electrons – λe of fast electrons in argon [16 eV, which is at the maximum of σ (26 ˚ A2 )] amounts to about 2.5 mm at a pressure of 50 mTorr (7 Pa), corresponding to about 7.5 mm at 20 mTorr (3 Pa). ˚2 ) show a – Slower electrons (Ekin ≈ 4 eV) with a smaller σ (9.7 A longer λe (at 20 mTorr about 13 mm) which applies also to rapid electrons (λe for 100 eV electrons is about 75 mm at 20 mTorr in argon). The frequencies for elastic collisions between electrons and neutrals yield, according to νm = nσv: ∗ for 1 Torr (133 Pa, 3.54 × 1016 cm−3 ) 3.8 × 109 s−1 , ∗ for 100 mTorr (13.3 Pa) 382 × 106 s−1 and ∗ for 10 mTorr (1.3 Pa) 38 × 106 s−1 .
2.2.2 Definitions First, we consider a beam of monoenergetic particles orientated in a parallel direction with respect to the x-axis and exhibiting a flux of N particles per cm2 sec (Fig. 2.1). z dW
Nt
q j
x Np
y
Fig. 2.1. Definition of the differential cross section. Np projectiles collide with Nt targets and are deflected into dΩ, the differential solid angle element.
2.2 Elastic collisions
9
In the case of hitting an annular target with area 2πb db (b: inner diameter, b + db: outer diameter), we define the differential scattering cross section for solid angle dΩ = 2π sin ϑdϑ: dσ(v, ϑ) b db = , dΩ sin ϑ dϑ and the total cross section (“c” for collision) σc (v) = 2π
π dσ(ϑ) 0
dΩ
(2.4)
dΩ,
(2.5)
which, in turn, does not allow any statement of the angular dependence. Furthermore, we define the frequency of collision according to νc (v) = nσc (v)v,
(2.6)
the mean free path again, but decorated with a subscript “c” λ≈
1 nσc (v)
(2.7)
1 . pλ
(2.8)
and the probability of scattering Pm =
The change of momentum is in the center-of-mass system (Fig. 2.2):
q b
CM
r j
Fig. 2.2. Collision between two spheres in the center-of-mass system. b: scattering parameter, θ, ϕ: scattering angle, CM: center of mass.
Δp = mv(1 − cos ϑ) :
(2.9) ◦
Collisions in the forward direction are minimally weighted (cos 0 = 1), those in the backward direction (cos 180◦ = −1), however, are considered most prominent: The diffusion is slowed down most effectively by collisions which are orientated backwards. For the treatment of the transfer of energy and momentum, the most important property is the cross section for momentum transfer, which is σc (v), but weighted by the factor 1 − cos ϑ:
10
2 Collisions and cross sections π dσ(ϑ)
(1 − cos ϑ)dΩ, dΩ along with the frequency of momentum transfer: σm (v, ϑ) = 2π
(2.10)
0
v (2.11) = νc 1 − cos ϑ. λ The cross section of momentum transfer σm can be extracted from the total collision cross section when its angular dependence is known. For a head-on collision (ϑ = 180◦ = π), we calculate, using the laws of conservation for energy and momentum, the maximum amount of transferred energy to (m the mass of the pushing sphere, M the mass of the stroken sphere) νm =
EM,kin =
4mM Em,kin (m + M )2
(2.12)
which is often rewritten using Langevin’s energy loss parameter L=
2m m+M
(2.13)
eventually yielding [15] 2M (2.14) Em,kin . m+M It can easily be shown that the maximum of this function is given for m = M : Collisions between ions with high kinetic energy and slowly moving parent atoms are very effective (cf. Sect. 2.5.2). On the other hand, the energy transfer between an electron and a heavy neutral is negligible (me /mAr ≈ 70 000). EM,kin = L
2.3 Elastic collisions between electrons and neutrals 2.3.1 The total cross section Measurement of the total cross section (sum of all possible scattering processes) was first performed by Ramsauer using the instrument which is sketched in Fig. 2.3. By exposing a plate of zinc sulfide to UV light, electrons are released at point A by the photoelectric effect. Without collisions, they are forced into orbital motions when a perpendicular magnetic field is applied. They pass all the slots from S1 to S5 until they hit the collector C. With elastic collisions, however, electrons are deflected and cannot pass; with inelastic collisions (without deflection), kinetic energy is lost, the orbital diameter increases and these electrons fail to reach the collector as well. Under a certain pressure p, the electron current is measured in B (current i) and C (current I). For known distance s between S4 and S5 , the cross section of absorption can be evaluated from the currents i and I at two different pressures p1 and p2 according to Beer’s law with α = nσ:
2.3 Elastic collisions between electrons and neutrals S2
Fig. 2.3. Ramsauer’s apparatus to measure the total cross section [16]. First, electrons are released from a slab made of ZnS. By applying a magnetic field orientated normally to the plane of the paper the electrons are forced in orbital motions. Their diameter is given by the distance between slots S2 and S5 . Scattering, however, will lead to losses in the Faraday cages B and C.
S1
hn
S3 S4
A S5
B
11
C
i = Ie−αps (p1 − p2 )αs = ln
(2.15)
I1 i 2 , I2 i 1
(2.16)
and for the lightest targets (hydrogen and helium), a hyperbolic dependence as pictured in Fig. 2.4, is obtained. For zero energy, the curve saturates at a certain constant level. 12
2
6
s [10
cm ]
8
-16
10 helium
Fig. 2.4. Experimental cross sections for the elastic collision of electrons through helium as function of electron energy according to [17].
4 2 0 0
10 20 electron energy [eV]
30
2.3.2 Differential cross section We consider now the total cross section and the cross section of momentum transfer, which can be evaluated after having measured the angular depen-
12
2 Collisions and cross sections
dence of the scattering amplitude according to Eq. (2.10). It was again Ramsauer (this time together with Kollath) who applied the measuring principle sketched in Fig. 2.5 [18]. A reactor contains a filament that emits electrons which are accelerated across an electric field. These electrons are deflected by gas molecules and are caught in Faraday cages and eventually measured.
electrometer with Faraday cups
filament grid system
reactor
collector
Fig. 2.5. Sketch of the apparatus developed by Ramsauer and Kollath to determine the angular dependence of the cross section for elastic scattering of electrons after [18].
Some general guidelines are: • At sufficiently high electron energies, we find an intense zero beam: Almost no deflection and then a gradual decline for large scattering angles. • Approaching low electron energies, this picture becomes more complicated. Maxima and minima occur which resemble X-ray diffractograms of fluids (Fig. 2.6). For the first time, this was observed by Bullard and Massey in argon. They explained these patterns by diffraction of electron waves at symmetrically scattering atoms, comparable with the pattern of light waves which are scattered at spheres of similar dimension [19]. However, for the quantitative interpretation of this phenomenon, a quantum mechanical treatise is required. • The complexity of this pattern rises with increasing atomic number of the scattering atom. For elastic scattering of electrons of low energy, a successful collision is assumed to have occurred for a deflecting angle of 90◦ . According to Eq. (2.11), νm then equals νc . For the noble gases, both dependencies (on velocity and on angle) have been measured; hence, σc (v) and σm (v, ϑ) can be compared. For low
2.3 Elastic collisions between electrons and neutrals
13
normalized scattering intensity
1.00
0.75
4.00 eV
0.50 2.80 eV
Fig. 2.6. Scattering of electrons at argon atoms for very c low electron energies [20] ( J. Wiley & Sons, Inc.).
2.00 eV
0.25 1.15 eV
0.00 0
45 90 135 scattering angle [°]
180
energies, the agreement is satisfactory (argon) or almost perfect (helium, Figs. 2.7).
25 argon
6 20
sc sm
4
s [10-16cm2]
s [10-16cm2]
helium
sc sm
15 10
2 5
0
0
20 40 electron energy [eV]
60
0 0
20 40 electron energy [eV]
60
Fig. 2.7. Comparison between total cross section σc and cross section for momentum transfer σm for the lightest noble gas helium and the medium-light noble gas argon, which behaves entirely differently for low electron energies [20]. For helium, σm exceeds σc for low electron energies: The scattering in the backward direction becomes more important than forward scattering.
2.3.3 Modeling 2.3.3.1 Cross section for the interaction between a point charge and an induced dipole. For potentials which decrease with r−4 (i. e. for forces which decline with r−5 ), there exists a simple correlation between the frequency of collisions and number density, which was pointed out by Maxwell [21].
14
2 Collisions and cross sections
An electron is moving in a field of a potential Φ which declines with r−n (Fig. 2.2). Introducing polar coordinates yields 2 = v 2 = v2 + v⊥
∂r ∂t
2
+ r2
∂φ ∂t
2
,
(2.17)
with v and v⊥ the two components of the velocity v which are orientated in parallel or in perpendicular fashion with respect to the radius vector r, we find for energy and angular momentum m0 m0 v 2 +Φ= · E= 2 2
∂r ∂t
2
+r
2
∂φ 2 ∂t
+ Φ = const,
(2.18)
∂φ = const. (2.19) ∂t With b the scattering parameter, the distance between the two particles under consideration (for lacking interaction, the trajectory of the electron would be a straight line) we get for the angular momentum L = m0 r2
∂φ = m0 bv0 , (2.20) ∂t with the boundary condition for the initial energy E0 (initial velocity v0 ) for vanishing potential Φ = 0 m0 r2
m0 v02 . 2 Since the time-dependent derivations of the coordinates are E0 =
∂φ ∂r ∂φ ∂r ∂r = v0 br2 ∧ = · = · v0 br2 , ∂t ∂t ∂φ ∂t ∂φ
(2.21)
(2.22)
we obtain
∂r ∂φ
2
E0 b2
E0 b2 + Φ = E0 . r4 r2 From Eq. (2.23), the angular dependence of r (∂r/∂φ) yields r2 ∂r =± ∂φ b
+
b2 Φ 1− 2 − r E0
(2.23)
1/2
,
(2.24)
with the minus sign for approaching particles, the plus sign for disappearing ones. This function exhibits a minimum at ∂r/∂φ = 0, when the argument of the square root vanishes. With Φ = ar−n and b = cr, we find for the minimum distance rmin n rmin =
2a m0 v02 (1
− c2 )
.
(2.25)
2 , Applying the hard sphere model, the minimum distance rmin equals to σscatt = πrmin and we can write it down with combined constants:
2.3 Elastic collisions between electrons and neutrals σscatt =
15
A . v 4/n
(2.26)
A , v
(2.27)
For n = 4, this yields the simple form σscatt =
which is the potential which will form between a point charge and an induced dipole with moment μind (with α the polarizability): 1 1 μind = αE ∧ Epot = − μ · E = − αE 2 . 2 2
(2.28)
For a Coulomb potential, we calculate the potential energy to 1 e2 Epot = − α 04 . 2 r
(2.29)
With this derivation, we have obtained the important result that for a force with a declining behavior according to r−5 , e. g. a point charge in a neutral gas which exerts polarization effects, the specific conduct of the EEDF (and hence ve ) does not matter at all. In this very special case, the frequency of elastic collision between electrons and neutrals νm (and also the mean free path λ) will become a√single function of the number density [21].1 σ will then scale with 1/v ∝ 1/ Ekin (Fig. 2.8).
12
s [10-16 cm 2]
10 8
Fig. 2.8. Experimental and calculated cross sections for the elastic collision of electrons through helium as a function of electron kinetic energy according to [17].
6 4 2 0
0
10
20
30
E [eV]
1 For the interaction between point charges Coulomb’s law holds: The potential drops according to 1/rn=1 . In this case, σscatt = A/v 4 , which determines the generalized resistance η in the case of high-density plasmas (cf. Sect. 14.7.2).
16
2 Collisions and cross sections
2.3.3.2 Ramsauer effect. For electron energies larger than some tens V, the cross section decreases gradually; it scales with the inverted ionization potential and is proportional to the polarizability (to a first order approximation: the atomic number) since scattering takes place at the bound electrons of the atoms. Very low values of the scattering cross section are caused by weak interaction with the higher noble gas atoms (Ramsauer effect, Fig. 2.7.2 [22]). This effect was explained for the first time using quantum mechanical methods by Allis and Morse using the partial wave method. According to this theory, the cross section of elastic scattering is composed of partial cross sections, which are denoted s scattering for l = 0, p scattering for l = 1 etc. (ηl is the partial phase shift of the waves with angular momentum L = l(l + 1)¯ h, p=h ¯ k): σ0 =
σl ∧ σl =
l
4π (2l + 1) sin2 ηl . k2
(2.30.1)
The affiliated potentials become deeper but simultaneously more short-ranged— the atomic radius grows by about 150 % when going from helium (0.49 ˚ A) to xenon (1.24 ˚ A) leading to a ratio of atomic volumes of about 16, whereas the number of electrons increases by a factor of 27! For slow electrons (k → 0 ∧ λ → ∞), L becomes very small, and for the noble gas atoms argon and heavier, only the component of zero order, i. e. s scattering, remains of importance: 4π sin2 η0 . (2.30.2) k2 η0 → π at k = 0 ⇒ σ0 = 0. Hence, a head-on collision (ϑ = π) between an electron and an atom cannot be observed. For the heaviest noble gas atoms (Xe and Rn) this effect is much more distinct than for the lighter ones (Ar and Kr) and vanishes for He and Ne. σ0 =
2.3.4 The frequency of elastic collisions between electrons and neutrals For the special cases of hydrogen and helium, the collision frequency simply scales with discharge pressure (number density): νm = const × p0 , which follow the equation with p0 = 273/T × p [p in Torr (in Pa), Figs. 2.9]: • He: νm = 2.31 × 109 p0 (3.07 × 1011 p0 ); • H2 : νm = 5.93 × 109 p0 (7.89 × 1011 p0 ). This does not hold true for the heavier noble gases (at least for neon, we see a fairly good approximation with an asymptotic behavior) and a strange conduct is observed for chlorine. Here, the maximum is foreshadowed by a minimum at very low energies in Cl· and Cl2 (Fig. 2.10). Following Massey, we estimate the mean free paths for electrons λe in argon according to Eq. (2.7) [25]. For typical electron energies of several electronvolts, we calculate a λe of several centimeters (Figs. 2.11).
2.3 Elastic collisions between electrons and neutrals
6
25 H2
n/p [10 9 sec Torr] -1
n/p [10 9 sec Torr] -1
5 4 3
He
2 1 0 0
17
5
10
15 20 Ekin [eV]
25
30
35
Xe Kr
20
Ar
15 10 5 0 0
Ne
5
10
15 20 Ekin [eV]
25
30
35
Fig. 2.9. Frequency of elastic collisions between electrons and neutrals for hydrogen and helium, and the heavier noble gases [23].
nm [sec -1]
108
107
Cl2
Cl
106 10-2
10-1
100 101 Ekin [eV]
102
Fig. 2.10. Frequency of elastic collisions between electrons and neutrals for Cl· and Cl2 after [24]; discharge pressure 10 mTorr.
2.3.5 Cross section and rate constant for argon By Eqs. (2.6) + (2.11), the cross section for a specific scattering process is related to the probability of its occurrence. This frequency ν depends on the density of target molecules, and reducing ν to the unit density yields the rate constant k, k = < ve > σ
(2.31.1)
in particular for the momentum transfer km : km = < ve > σelast .
(2.31.2)
This relation allies atomic properties which must be evaluated by scattering experiments with macroscopic kinetic parameters, with the mean electron velocity being the connecting link. The directed energy E from the scattering experi-
18
2 Collisions and cross sections
100 argon
10 1
1 mTorr 5 mTorr 10 mTorr
0.1 0.01 0
100 mTorr
3
mean free path [cm]
mean free path [cm]
100
argon
10
0.5 eV 1.0 eV 1.6 eV 2.4 eV 4.3 eV 8 eV 12 eV
1
50 mTorr
6 9 12 electron energy [eV]
0.1 0
15
10
20 30 40 pressure [mTorr]
50
60
Fig. 2.11. Calculated mean free paths of low-energy electrons (below the ionization threshold) in argon after [25].
ments has to be replaced by the thermal energy E for a discharge. Most frequently, a Maxwellian distribution is assumed but especially for low plasma densities (E = kB Te ), however, this assumption has turned out to be highly questionable (Sect. 14.1). The connection between σ and k is pictured for argon in Figs. 2.12. The data for σelast have been compiled from Nakanishi and Szmytkowski [26, 27]. 30 100
s
-1
3
10
-1
k
1
-2
10
-6
k
s
k [10 cm sec ]
1 10
selast [10-16 cm2]
20
10
k [10 -6 cm3sec-1]
selast [10-16 cm2]
2
10-3 0.1
0
0
25
50
75
electron energy [eV]
0 100
0.01
0.1
1
10
10-4 100
electron energy [eV]
Fig. 2.12. Argon: total cross section for elastic scattering σelast and the corresponding rate constant k = σelast × < ve > after [25] − [27].
2.4 Elastic collisions between heavy particles The scattering among heavy particles exhibits a remarkably distinct maximum in the forward direction [28]. Measuring such an angular dependence is almost hopeless. For energies of a few electronvolts, the requirements for angular resolu-
2.4 Elastic collisions between heavy particles
19
tion are difficult to meet, and these demands increase with rising momentum of the projectiles. The accuracy necessary to measure the total elastic cross section for argon is for thermal atoms 0.70◦ , those with 1 000 K temperature require an accuracy of 0.30◦ , which is even sharper for atoms of 1 eV kinetic energy (11 600 K, 0.11◦ ). These difficulties are reinforced for the measurement and calculation of the differential cross section which means the determination of the fraction of molecules which hit target molecules and are subsequently scattered into the solid angle dΩ = 2π sin θdθ. In Table 2.1, the results of calculations for the differential cross section of helium against a beam of protons for two different scattering potentials (Coulomb and Hartree, a simple pseudopotential) are compiled [29].
Table 2.1. Differential cross sections dσ(ϑ)/dΩ per unit solid angle for protons of kinetic energy of 110 eV in helium in units of a20 .
0 12 28 34 57 80 114 137 167
dσ(ϑ)/dΩ Hartree potential Coulomb potential 9 × 103 ∞ 7.85 124.0 2.00 6.10 0.72 2.85 0.21 0.40 0.08 0.12 0.04 0.04 0.03 0.02
This forward direction is more distinct for larger kinetic energies of the projectiles (coming from low energies, only elastic scattering processes can occur). Normally, the determination of the angular dependence is confined to the range without the zero beam. One of the pioneering experiments was carried out by Berry and Cramer in neon and argon [30] − [33] (Fig. 2.13). We note that the cross section remains at several ˚ A2 up to kinetic energies of about 500 eV. Hence, an ion beam is almost entirely dissipated after having passed a distance of only 1 cm at high pressures (cf. Sect. 6.8). However, some advanced, indirect methods are available, e. g. measuring the complex conductivity (Sect. 5.6), and the line width of the electron cyclotron resonance, which is mainly determined by the collisions between electrons and neutrals [34] (Sect. 7.5).
20
2 Collisions and cross sections
4 A
s[10-16 cm2]
3
A - 250 eV B - 500 eV C - 750 eV D - 1000 eV E - 1250 eV F - 1500 eV G - 1750 eV
B
C D 2 E F G
1
0 0
45
90 135 scattering angle [°]
180
Fig. 2.13. Cross section of Ar+ ions in argon as a function of scattering angle [30].
2.5 Inelastic collisions 2.5.1 Inelastic collisions between electrons and heavy particles During inelastic collisions, internal grades of freedom are excited by transformation of kinetic energy of the colliding projectiles into internal energy levels of the target. For electron impact, the most important reactions are with A∗ an excited molecule: • Excitation in upper molecular states (vibration levels or electronic states, [Eq. (2.32)]. • Dissociation into radicals (which is very effective in discharges through electronegative gases [35], [Eq. (2.33)]. • Dissociation into ions [Eq. (2.34)]. • α-ionization [Eq. (2.35)]. • Ionizing dissociation [Eq. (2.36)] and • Electron attachment (dissociative or electron attachment, respectively), for electronegative gases [Eq. (2.37)]. e− + A2 −→ A∗2 + e−
(2.32)
e− + A2 −→ 2 A∗ + e−
(2.33)
e− + A2 −→ A+ + A− + e−
(2.34)
2.5 Inelastic collisions
21 − e− + A2 −→ A+ 2 + 2e
(2.35)
e− + A2 −→ A+ + A + 2 e−
(2.36)
e− + A2 −→ A− + A∗ .
(2.37)
2.5.1.1 Experimental methods. In atomic gases, dissociation reactions do not take place; one of the “classical” reaction is the ionization of mercury vapor by electron impact (Franck-Hertz experiment, Figs. 2.14). In a triode tube
anode grid filament 0.5 V
_ 20 V
+
400 14.7 V 9.8 V
200
sion [10-16 cm2]
anode current [a. U.]
30
300
4.9 V
100
0
0
2
4
6 8 10 12 anode voltage [V]
14
16
20
10
0 10
12 14 electron energy [eV]
16
Fig. 2.14. Franck-Hertz experiment. Top: experimental setup, bottom: the anodic current (LHS) and the cross section of ionization (RHS) as functions of acceleration voltage (electron energy) in mercury vapor [36]. At the ionization potential of 4.9 eV and its multiples, the characteristic exhibits distinct deviations from the expected behavior, and the cross section does not exhibit a smooth behavior [37].
filled with mercury vapor, the electrons which are emitted by a glowing cathode are accelerated through a uniform field to a grid with the anode plate placed directly behind the grid. Those which have passed it will be slowly decelerated by elastic collisions with gas atoms, but in general, we observe a slight increase of the anodic current with growing anodic voltage. However, when the ionization potential of mercury (4.9 eV) has been reached, the I(V ) characteristic
22
2 Collisions and cross sections
drops sharply. This phenomenon is caused by inelastic collisions of the electrons with mercury atoms which are ionized by electron impact. A small retarding field between grid and anode causes the anodic current to vanish at this value, because the electrons have less than 4.9 eV kinetic energy and are not able to move against this retarding potential: Hgvapor + e− −→ Hg+ + 2 e− .
(2.38)
This setup has been improved by Maier-Leibnitz [38] and further by Schulz and Fox who employed a retarding potential difference method [39]. They concluded that the electron energy distribution function, EEDF, in the original Frank-Hertz experiment was very broad, too broad to obtain an accurate measurement of the excitation probability. For that purpose, the energy of the bombarding electrons as well as the energy of those that have lost energy in inelastic collisions must be known. Therefore, they constructed an energy filter system which made use of also in the determination of the ion energy distribution function IEDF (cf. Chap. 6). By means of this three-grid optics, it is possible to extract a nearly monoenergetic beam of electrons (Figs. 2.15).
filament
F1
F2
F3
0 collision chamber
F [a.u.]
retarding potential aperture F3 F1 F2
collector
x [a. u.]
Fig. 2.15. By employing the retarding potential difference method, Schulz and Fox could significantly improve the accuracy of the determination of the inelastic scattering cross section [39].
The first grid is positively biased by approximately 3 V with respect to the filament and serves to draw electrons. The second grid is slightly negative with respect to the filament, the electrons must run uphill against a retarding potential, and as a result, the EEDF is chopped off. It is only the high-energy fraction of the electrons which is able to pass. The third grid shows the same potential as the first grid. Since the EEDF can be sensitively influenced by variation of Φ2 , Φ2 + ΔΦ2 , we obtain two values for the collector current which can be solely referred to the electrons with energy ΔΦ2 . By variation of the retarding potential Φ2 , grid 2 also varies the voltage to the collision chamber which determines the collector current, i. e. the EEDF.
2.5 Inelastic collisions
23
2.5.1.2 Cross section. The cross section for an inelastic scattering process exhibits a threshold which is followed by a steep rise (several orders of magnitude) to reach a maximum. This is sharply peaked for optical transitions at energies which are close to the threshold, and a little broader for ionizations at about 100−150 eV, which is consistent with the de Broglie wavelength of the electrons of approximately 1 ˚ A, the typical diameter of the targets where the electron waves are expected to be deflected most intensely. For higher energies, we find a slow decrease since the time of interaction between the collision partners is gradually shortened (Fig. 2.16 in linear scale, Figs. 2.17 in logarithmic scale).
cross section [10-16cm2]
3
2 argon hydrogen
1
0
10
100
1000
Fig. 2.16. Typical energy dependence of the cross section for inelastic scattering between electrons and molecules for argon and hydrogen.
electron energy [eV]
The cross section of ionization has been comprehensively described by Bethe and Salpeter [40, 41]. According to Born’s approximation, the total cross section of a discrete excitement into state n with energy En of the excited state can be approximated by
1 4πa20 z 2 C A ln(4 B T ) + + O , (2.39) T T T2 with a0 Bohr’s radius, z the charge and T the kinetic energy of the projectile, mostly given in Rydberg (= 13.6 eV), irrespective of the nature of the projectile. A is the squared matrix element of the specific excitation, and the B, C are constants which will be generated during the integration [42], and the factor 4 in the logarithmic summand which is left for historical reasons. The third term is a very small correction, and most frequently, the second term is neglected as well, and we arrive at the following simplified approximation [43, 44], which is known as Bethe’s asymptotic cross section [41]: σn =
4πa20 z 2 [A ln(4 B T )] . (2.40) T Unfortunately, this theory holds best for large kinetic energies of the projectiles. However, for ionization and excitation processes by electron impact, the energy σn =
24
2 Collisions and cross sections
dependence just above threshold is of extreme importance. To describe this regime properly, in particular the steep rise beyond threshold and the maximum, Wannier developed a theory for single ionization which states that in the energy range near threshold, the probability of ionization rises exponentially with the energy in excess of the ionization energy (more correctly: raised to the 1.127th power) [45, 46], which had been improved by Geltman [47]. Lotz introduced an equation with three parameters (a, b and c), which approximates the cross sections of most of the interesting gases within a deviation of less than ±10% to experimental data [44] (Fig. 2.17.2):
1 21P
0.1
1
2S
3
0.01
0.001 10
ln(T /En ) T 1 − b exp −c −1 T En
cross section [10-17 cm 2]
cross section [10-17 cm2]
σn = a
2P
(2.41)
10
1 argon hydrogen
0.1
0.01 10
100 1000 electron energy [eV]
.
100 electron energy [eV]
1000
Fig. 2.17. Typical dependence of the inelastic cross section of molecules and electrons on the electronic kinetic energy. Note the double logarithmic scale to make the steep increase visible. LHS: energy dependence of the cross section for excitation into various optical levels of helium, according to the formula of Mityureva and Smirnov [48]. RHS: Lotz’s theory for ionization of hydrogen and argon, experimental data for argon are from one of the most cited papers by Rapp and Englander-Golden [49].
The main difference between ionization and excitation into optical levels by electron impact is caused by the electrostatic interaction of three charges after the collision process. For the first time, this problem has been addressed by Wannier ([46], also cf. [50], Fig. 2.17.1). As a main result, the maximum for excitation into optical levels is located very close to the threshold, whereas for ionization, the difference in energy between maximum and threshold will be no less than a factor of 4 in excess of the ionization energy [48]. The graphs of Figs. 2.17.1 are plotted applying an approximation containing three adjustable parameters σ0 , φ, and γ:
σn = σ0
T − En T
−γ
T −1+φ En
.
(2.42)
2.5 Inelastic collisions
25
In summary, we find the cross section for ionization by electronic impact to rise sharply from zero at a specific threshold by several orders of magnitude until it reaches a maximum at approximately 100 eV because the de Broglie wavelength of about 1.2 ˚ A fits the atomic dimensions. For further increases of electron energies, the cross section gradually decreases. At 500 eV, it has typically weakened by a factor of about 2 compared with its maximum value (Fig. 2.18).
Hg
sion [10-16 cm2]
10
air Ar Ne
1
H2 He
0.1 10
100 1000 electron energy [eV]
Fig. 2.18. Cross section for ionization by electron impact for various gases in double-logc Oxford arithmic scale [51] ( University Press).
10000
2.5.1.3 Rate constant for ionization. Employing Eqs. (2.31), we ally the frequency of ionization with the rate constant for this process, pictured in Figs. 2.19 for ionization of argon, with a Maxwellian distribution assumed. In fact, it is the rate constant we are interested in to calculate the carrier generation. Especially at the threshold, the slope of the k(E) curve repeats the steep increase of the cross section. In low temperature plasmas, it is but these electrons in the high-energy tail that are responsible for the maintenance of the discharge. 2.5.1.4 Electron attachment. Electron attachment plays a significant role in discharges through electronegative gases. One of the main processes is dissociative electron attachment, e. g.: O2 + e− −→ O− + O,
(2.43)
SF6 + e− −→ SF− 5 + F,
(2.44)
a two-body collision leading to the formation of a negatively charged oxygen ion and and oxygen atom, or a very special case, the formation of a SF− 5 ion and a fluorine atom.2 2
It is this ion which captures almost all the electrons at very low discharge pressures.
26
2 Collisions and cross sections
3
Lotz Li
2
1 argon
0
0
250
500
750
1000
1000 k s
1
100
0.1
0.01 10
electron energy [eV]
10
argon
100 electron energy [eV]
1000
rate constant [10-11 cm3sec-1]
cross section [10-16 cm 2]
cross section [10-16 cm 2]
10
1
Fig. 2.19. Ionization by electron impact for argon. LHS: Two different formulae, according to Lotz and Li, yield an almost indistinguishable result [44, 52]. RHS: For cross section and rate constant, a Maxwellian distribution for the electrons assumed. The calculation is based on the Lotz formula [44].
normalized cross section
1.0 Schulz Craggs et al.
0.8 0.6 0.4 0.2 0.0 4
6 8 10 electron energy [eV]
12
Fig. 2.20. Normalized cross section of electron attachment and formation of O− ions. The asterisks were evaluated by Schulz, the circles are due to measurements by Craggs et al. The maximum (Schulz) has been found at 1.25 × 10−18 c J. Wicm2 and 6.7 eV [53] ( ley & Sons, Inc.).
Atoms are not able to capture electrons in a two-body collision process because the laws of conservation of momentum are violated. Therefore, the cross sections are very low (in the order of 10−18 cm2 ), but exhibit sharp maxima at electron energies of a few eV (Fig. 2.20). The density of negative ions can far surpass that of the electrons which changes the characteristics of the discharge completely. As is evident from Fig. 2.22, the cross section for dissociative attachment exhibits two maxima for oxygen. The first can be referred to reaction (2.43), whereas the maximum at higher energies is due to the reaction e− + O2 −→ O− + O+ + e− .
(2.45)
2.5 Inelastic collisions
27
One of the most prominent cases is that of BCl3 as an additive in discharges through chlorine. In principle, three reaction paths are possible [54]: BCl3 + e− −→ BCl∗3 ,
(2.46.1)
BCl3 + e− −→ BCl2 + Cl− ,
(2.46.2)
BCl3 + e− −→ BCl + Cl− 2,
(2.46.3)
BCl∗3
the metastable parent anion, which can be effectively generated by with extremely low energy electrons up to 0.1 eV [54]. Although the pyramidal BCl− 3 ion is expected to be long lived, (Petrovic et al. detected this anion in a mass spectrometer, which indicates a lifetime in the microsecond range [55]), due to its small electron affinity of 0.33 eV, it will be easily discharged by electron impact. All of the discharges which are dealt with in the following chapters exhibit higher mean electron energies. Therefore, BCl− 3 anions are most likely to elude detection.3 Tav et al. have estimated the cross section for dissociative electron attachment to BCl3 to be ≤ 5 × 10−18 cm2 [according to Eqs. (2.46.2) and (2.46.3)] and for electron attachment to BCl3 according to Eq. (2.46.1) to 4.5 × 10−15 cm2 . 2.5.1.5 Total collision cross section. Since the numerous possibilites of excitation exclude each other, their probabilities must be added up, and we can define a total cross section for all possible excitations [57]: σtot =
Pi σi (v),
(2.47)
i
with Pi the probability and σi the differential cross section for reaction i. For example, the maximum of the total cross section for electron impact for argon A2 and can be found just above the ionization amounts to 26 × 10−16 cm2 = 26 ˚ threshold of 15.76 eV. This means an electronic mean free path of 1.5 mm at a discharge pressure of 50 mTorr (7 Pa, n = 2 × 1015 cm−3 ). For low electron energies, the cross section for elastic scattering almost equals the total scattering cross section since the velocities of the electrons do not suffice for atomic or molecular excitation. Especially for the noble gases, numerous measurements have been performed and modeled. A compilation of these measurements is pictured in Fig. 2.21. It was Myers who split up the total cross section σO2 ,tot into its separate parts for oxygen (Fig. 2.22) [58]. For energies up to approximately 50 eV, the 3 However, Gottscho and Gaede succeeded in generating and observing this anion in low-frequency discharges at 50 kHz where the electron density will completely relax during each half cycle leaving ample time for the slow electrons to generate the BCl− 3 anion [56]. Using a non-invasive method (laser photodetachment), they could detect this anion before the electrons were reheated in the next half cycle.
28
2 Collisions and cross sections
40 Xe Kr
s[10-16 cm2]
30 20
Ar
10 Ne
0
0
2
4 6 8 electron velocity [V1/2]
10
Fig. 2.21. Experimental scattering total cross sections of the heavy noble gases for low electron velocities. The almost complete transparency at very low velocites is striking; the distinct maximum corresponds well with the ionizac Oxtion energy Eion [25] ( ford University Press). Compare with Fig. 2.7.2.
cross section for elastic scattering is the most important contribution to the cross section for ionization.
A
-15
10
G
s [cm2]
10-17
C
B
B F
C E
10-19
E
D
D G
C
10-21 0.1
1
10
Fig. 2.22. The single contributions of the total cross section for electronic impact were split up by Myers for O2 [58]. The letters denote the contributions as follows: A: elastic scattering, B: rotatory excitation, C: oscillatory excitation, D: excitation to singlet-oxygen, E: dissociative attachment (two maxima!), F: excitation to upper electronic states, G: ionization.
E [eV]
2.5.2 Inelastic collisions between heavy particles To begin with, the literature covering the low-energy range, which is of paramount importance in glow discharges, is widely scattered, in particular when we compare this range with the countless number of papers dealing with energies exceeding 1 keV. This is mainly caused by the low cross sections for inelastic processes. Hence, elastic scattering will still dominate the interaction between the gas constituents. The main reactions between heavy particles are: • Generation of electrons by ionic impact [so-called β-ionization, Fig. 2.23, Eq. (2.48)].
2.5 Inelastic collisions
29
• Excitation to electronic levels [Eq. (2.49)]. • Electron stripping [Eq. (2.50)]. • Simple charge transfer [Figs. 2.25 − 2.28, Eq. (2.51)]. • Double charge transfer [Eq. (2.52)].
+
Ar in Ar
sion [10-16 cm2]
10
H2+ in H 2 Ne+ in Ne
He+ in He
1
0.1 2 10
103 ion energy [eV]
104
Fig. 2.23. β-ionization: cross section of ionization by ions of the parent gas after [59]. The values are not corrected by ionizations caused by secc Oxford ondary electrons ( University Press).
+ + − A+ 2 + B −→ A2 + B + e ,
(2.48)
A+,∗ + B −→ A+ + B∗ ,
(2.49)
A+ + B −→ A2+ + B + e− ,
(2.50)
B+ + A −→ A+ + B,
(2.51)
A+ + B −→ A− + B2+ .
(2.52)
As we can see from Fig. 2.23, the threshold for ionization by ionic impact is relatively low but amounts to about twice the ionization potential. The classical theory of J.J. Thomson yields cross sections equal in size for equal velocities of ions and electrons. The same result is obtained with quantum mechanics with the restriction that the impact of the colliding particles is large compared with their pair potential (Born’s approximation). Choosing the abscissa scale skilfully (in fact, it is the momentum of the colliding particle), we can find a similar behavior between the ionic and the electronic cross section, however, dilated by some orders of magnitude for the heavy particles (Fig. 2.24). The
30
2 Collisions and cross sections
sion [10-16 cm2]
electron energy [eV] 100 1000
Hooper et al., 1962 Smith, 1930 Tozer, Craggs; 1930 Bleakney, 1930 Lampe et al., 1957
1
0.1
1 proton energy [MeV]
Fig. 2.24. Comparison of the ionization cross sections of argon for protons (dashed) and c J. Wielectrons (solid) [60] ( ley & Sons, Inc.).
cross sections for ionization (β-ionization) do not reach hogh values until the ions have been accelerated to velocities which are comparable to those of the electrons for α-ionization. Hence, for energies up to several hundreds of eV, the ionic cross section is smaller by about two orders of magnitude. For the same kinetic energy, the ionic momentum is considerably greater, which leads to very short de-Broglie wavelengths. Hence, even for inelastic collisions, there are hardly any diffraction effects. 2.5.2.1 Charge transfer. For charge transfer, an almost complete exchange of momentum is of importance. Consider a rapidly moving ion colliding with a slow molecule (Fig. 2.25). This collision can result in a slowly moving ion and a rapid neutral, and the components of velocity are conserved. In the case that both colliding particles have the same mass, this process is very efficient, and this collision is denoted (symmetric or resonant charge transfer, Fig. 2.26). We expect the cross section for this process to be comparable with that for elastic collision. To begin with, we assume A = B, i. e. the ionization potentials should be differ by the amount ΔEion : Eion,B = Eion,A + ΔEion .
(2.53)
Massey discussed this charge transfer by means of the correspondence principle [62]. For a particle velocity v smaller than that of the bounded electrons, they will adapt to this disturbance without an electron transfer (so-called adabatic approximation). We find the time of collision τ to scale inversely with the frequency ν of an electron oscillation with τ ≈ b/v (b the so-called adiabatic parameter):
2.5 Inelastic collisions
31
15 H+, Kr
s [10-16 cm2]
12 9
H +, Xe
6
Fig. 2.25. Cross section for the asymmetric charge transfer of protons in noble gases after [61].
3 0
0
5
10
15
20
25
V1/2
1 ΔEion aΔEion a = ∧ν = =⇒ v = . (2.54) v ν h h When the velocity of the particles exceeds this threshold, no adiabatic collision will take place but an ionization will occur. The larger the difference in ionization potential ΔEion , the higher the energetic threshold to become a competitive process. On the other hand, the time for interaction declines with rising energy which eventually reduces the probability of ionization: A maximum has been evolved (Fig. 2.26). τ=
s [10-16 cm2]
100
Fig. 2.26. Principal energy dependence of the two mechanisms for charge transfer. in double-logarithmic scale, the resonant charge transfer linearly scales with energy, whereas the asymmetric charge transfer exhibits an energetic maximum.
10
symmetric charge transfer asymmetric charge transfer
1
1
10
100
1000
E [eV]
2.5.2.2 Resonant charge transfer. For a resonant charge transfer with ΔEion = 0, we expect a monotonous increase in the cross section for ionization for lower energies in the adiabatic approximation. Albeit the correspondence principle predicts this conduct qualitatively correctly, more extensive descriptions of the height of the maximum (e. g. ΔEion /Ekin ) are definitely impossible.
32
2 Collisions and cross sections
The principal shape of the energetic dependence of the cross section is depicted in Fig. 2.26. It can be described by the formula σ = (a − b ln v)2
(2.55)
with a, b empirical constants and v the relative velocity of the colliding particles [63]. For a resonant reaction, there exists a linear relation between the cross section and the logarithmic kinetic energy (Fig. 2.27).
Firsow formula Gilbody, Hasted (1956) Flaks, Solovev (1958) Gustafsson, Lindholm (1960)
sCT [10-16 cm2]
60
40
Fig. 2.27. Resonant charge transfer: Ar+ fast + Arslow −→ Ar+ + Ar fast ; comparison of slow experimental data with theory c J. (Firsow formula) [64] ( Wiley & Sons, Inc.).
20
0 0
20
40 v [eV1/2]
60
80
The cross section reaches a maximum for an ion in its parent gas at slow velocities. In this energy range, it can deliver the main contribution to the total cross section (Fig. 2.28). The electron transfer mostly happens without any exchange of momentum: The generated ions exhibit negligible velocities, and the scattered atoms alter their direction only in a very narrow scale; hence, the scattering cone exhibits a very low opening angle [65]. The measuring of the charge transfer was made possible by an apparatus which consists of a magnetic field which deflects the ion beam and focuses it on to the sample. To prevent multiple ionizations, the cross section of the reactor has to be made sufficiently thin. The ions produced are sucked off by an electric field which is directed perpendicular to the reactor. The ions are measured by a Faraday cage which is negatively biased to avoid distortions by secondary electrons (so-called condensor method) [66]. Ternary collisions occur relatively rarely; the ratio of the probabilities of a two-body collision over a three-body collision is about thousand [67], therefore reactions + 2A+ + B −→ A+ 2 +B
(2.56)
(recombination) are not very likely to happen. Since the number of collisions scales with the squared pressure p (at constant temperature: With the squared number density n), the probability for recombination increases with rising pressure (cf. Sect. 4.7).
2.5 Inelastic collisions
33
50
50
s [10-16 cm2]
s [10-16 cm2]
s (total) s (resonant charge transfer) s (elastic collision)
helium
25
0 0
5
10 v [eV1/2]
15
neon
25
20
0 0
5
10 v [eV1/2]
15
20
s (total) s (resonant charge transfer) s (elastic collision)
75
s [10-16 cm2]
s (total) s (resonant charge transfer) s (elastic collision)
argon
50
25
0 0
5
10 v [eV1/2]
15
20
Fig. 2.28. Cross sections for the processes of elastic scattering and symmetric (resonant) charge transfer of He+ in He, Ne+ in Ne, Ar+ in Ar [31] − [33]. At low energies, the resonant charge transfer can deliver the main contribution to the total cross section.
2.5.2.3 Penning ionization. Among the numerous reactions which can occur between heavy particles, we should mention the Penning ionization, collisions of second order. This mechanism becomes important in discharges of reactive gases which are doped with noble gases (cf. Sects. 10.4.3, 10.6.4 and 11.8.6) and consists of ionization by metastable species. For ionization by collisional impact, the metastable level must be higher than the ionization potential of the neutral molecule or atom. Since the return to the ground level from a metastable level is forbidden by selection rules, these levels exhibit considerable lifetimes. The metastable levels for neon (16.62 and 16.7 eV) and those of argon (11.55 and 11.72 eV) are higher than the ionization potentials of most of the gaseous elements and gaseous compounds. By doping a discharge of argon with neon, metastable neon atoms can ionize argon
34
2 Collisions and cross sections
atoms. The state of metastability is denoted with an asterisk (cf. also Fig. 4.4, enlargement of α). Ne∗ + Ar −→ Ne + Ar+ + e− .
(2.57)
The cross sections at thermal energies are somewhat larger than the gas kinetic cross sections (≈ 10−15 cm2 ), therefore, its probability approaches unity, and this process becomes dominant very quickly [68, 69] and will be manifested in a rise of the discharge current [70]. Mierdel discussed consecutive staircase processes which cause an additional ionization after a collision of two metastables [71]; this effect would become more probable with rising density of the metastable atoms. Ar∗ + Ar∗ −→ Ar+ + Ar + e− .
(2.58)
Furthermore, ionization of metastables can happen by electron impact: Ar∗ + e− −→ Ar+ + 2 e− .
(2.59)
Since the energy difference from the metastable level to ionization has shrunk from 15.76 eV to only 4.21 eV, considerably more electrons are able to succeed in an ionization. Although the density of the metastable species is very low, Ingold has estimated this mechanism to be the most important ionization source in discharges of Hg/Ar [72]. 2.5.3 Collisions between photons and molecules As final collision processes between individual particles, we must consider impacts of photons (Fig. 2.29). Again, the threshold of ionization has to be exceeded. But, in contrast to the impacts of charged particles, we observe for photons the occurrence of a steep absorption edge of several orders of magnitude at E = Eion to decline precipitously beyond this sharp maximum. For example, the maximum of the photoelectric excitation is found for argon at 15.5 eV and exhibits a value of about 0.36 × 10−16 cm2 . Hence, the maxima for the yield of ionization for electrons, ions and photons are located at very different values.
2.6 Generation of secondary electrons at surfaces 2.6.1 Electrons The impact of electrons incident on a surface causes the following reactions: • Elastic scattering, the energy does not change but the momentum, very high contribution in the spectrum.
2.6 Generation of secondary electrons at surfaces 40
M2 M3 Ar
s [10-18 cm 2]
30
M1
20
10
Ne
L1
K
Fig. 2.29. Cross section for photo absorptions of c Sprinnoble gases [73] ( ger-Verlag).
L2 L 3
He
0 20
35
40
l [nm]
60
80
• Inelastic scattering, both energy and momentum are changed, small contribution in the spectrum. • Towards low energies, the second rise peaking at energies between 5 and 10 eV is caused by secondary electrons (Fig. 2.30).
1.00
f(E)
0.75
0.50
0.25
0.00 0
50
100 Ese [eV]
150
Fig. 2.30. Energy distribution of secondary electrons, f (E)se , for 160 eV electrons incident on a shiny gold surface (normalized to the low-enc Oxergy maximum) [74] ( ford University Press).
The yield of secondary electrons depends sensitively on • The energy of the electrons incident on the surface, which causes a maximum at medium energies. At low energies, the electrons are rather absorbed, at higher energies, they are scattered elastically. • The chemical composition of the topmost layer: It rises with increasing density and work function with the well-known periodic dependencies. The yield increases with growing fill-up of the electronic shells. When the filling of a new shell has begun before an inner shell has been completely filled, we observed discontinuities, in particular for half-occupied shells.
36
2 Collisions and cross sections • The quality of the surface: the shinier, the higher the yield.
For many metals with a clean surface the yield for secondary electrons, δ, equals unity (Fig. 2.31). For insulating layers, the values for δ can reach 15 (to measure δ, momentum methods are applied, however, the errors are rather large). 2.0
1.5
Ag
d
W Cu
1.0
Mo
0.5 Be
0.0
0
500 1000 electron energy [eV]
1500
Fig. 2.31. Coefficients for secondary emission, δ, for several metals as a function of the energy of the primary electrons incident on the surface.
In insulating materials, the conduction band lacks electrons almost entirely whereas the valence band is nearly filled. Hence, the primary electrons can generate secondary electrons only by interaction with valence band electrons. But these electrons gain a higher amount of kinetic energy than comparable metal electrons, which enhances their probability of reaching the surface and eventually leaving the substrate. For the same reason, the yield increases with flatter angles of incidence. To differentiate between the electrons, the coefficient for backscattering η refers to those electrons with energy between 50 eV and the energy of the electrons incident on the surface, and the coefficient for secondary electrons δ refers to those electrons having energy between 0 and 50 eV. 2.6.2 Ions Ions incident on the surface can release electrons as well, and this mechanism is of considerable importance in DC discharges and also plays a role in RF discharges which are capacitively coupled. The energy distribution of the secondary electrons responds relatively flatly to the kinetic energy of the incident ions and exhibits a broad maximum between 5 and 10 eV [75] (Fig. 2.32). The distributions for Ei ≤ 200 eV are very similar; γi rises steeply for growing energies, and a maximum becomes visible at very low electron energies. Eventually, for the highest ion energies, the energy distribution of the secondaries peaks close to zero.
2.6 Generation of secondary electrons at surfaces 30 25
1000
10
100 200
15 1000
10 5 0 0
Eion - 2 Wa
n(Ei) [10-3]
20
He+
Mo
Fig. 2.32. Energy distribution of electrons which are released from a surface being atomically cleaned by He+ ions with various kinetic energies. The dashed lines without symbols separate the Auger processes from the other mechanisms which release secondary elecc The American trons [75] ( Physical Society).
E(He+) [eV] 10 40 100 200 600 1000
40
600
10
5 10 15 electron energy [eV]
37
20
An electron can be released only when the sum of the kinetic energy of the ion incident on the surface and the ionization potential exceeds twice the value of the work function Wa : For every electron emitted from the surface, a second one is required to neutralize the incident ion. Ekin (e− ) ≥ Eion − 2 Wa .
(2.60)
This condition is remarkably violated by neon which can be attributed to an additional Auger process [76], and which is not observed when the electron is released by Ne ions of 10 eV kinetic energy. This strange phenomenon is explained by the discharging of ions of higher energy in close vicinity to the surface. Afterwards, these atoms can generate rapidly moving electrons in a process denoted Auger relaxation. The yield of this process, however, exceeds that of the Auger neutralization. Typical values for the yield for secondary electrons, γ, vary between 0.05 and 0.1 for the heavy noble gases incident on molybdenum and tungsten (Fig. 2.33). The distinct energetic minimum of the number of secondary electrons for He+ is absolutely striking. This is repeated for the heavier noble gas, however, to a weaker extent, which is due to a larger penetration depth with rising kinetic energy, and the electrons which are now released are attenuated during their course to the surface. This reduction in secondary electrons is overcompensated by the growth of the yield at further growing kinetic energies. For neon, this effect is covered by the above mentioned Auger relaxation.4 Although the curves of yield are very similar, the electron yield is • always higher for molybdenum than for tungsten, 4 This is valid for ion energies which are typical for glow discharges. Positive ions moving very rapidly (α-particles with energies of 1 MeV) can release up to thirty electrons; their energy amounts then to some keV.
38
2 Collisions and cross sections
0.30
Ne+
gi [electron/ion]
0.25
He+
0.20
W Mo
0.15 Ar+
0.10
Kr+
0.05
+
Xe
0.00 0
250
500
750
1000
Fig. 2.33. Comparison of the yield for secondary electrons, γi , for the ions of the heavy noble gases incident on pure molybdenum and tungsten c The American [75, 77] ( Physical Society).
ion energy [eV]
• and depends on the ionization energy of the incident ion: The lower its ionization potential the less the yield of secondary electrons. This is due to the low work function of molybdenum—for constant energy of the incident ions, the energy of the secondary electrons rises and enhances the probability of being released. On the other hand, Fermi energy and density of states at the Fermi level (this equals the number of conduction electrons divided by the Fermi energy in the free-electron model) energy and the number of released electrons: A falling Fermi energy causes a larger kinetic energy of the emitted electrons, simultaneously, the density of state goes up. Furthermore, Hagstrum investigated the influence of the surface quality on the yield for secondary electrons [78]. Fig. 2.34 impressively illustrates the difficulties obtaining reliable data for δ and γ (γ as anisotropic property additionally depends on the crystal orientation). 0.10 W +
Ar
gi [electron/ion]
0.08
N2/W
Ar+
0.06
W +
Kr
0.04 Kr+
0.02 Xe+
0.00 0
N2/W W
250
500 750 ion energy [eV]
N2/W
1000
Fig. 2.34. γi as a function of kinetic energy of the heavy noble gas ions for pure tungsten and tungsten covered with a c mono layer of nitrogen [78] ( The American Physical Society).
2.6 Generation of secondary electrons at surfaces
39
But also the energy distribution is altered by the contaminated surface: The fraction of rapid electrons is reduced. This is not only caused by a change in the work function but also by fundamental alterations of electron generation mechanisms with the exception of Auger processes. 2.6.3 Photons The electron yield caused by photons, triggered by the Einsteinian photo effect, oscillates between 100 and 1000 ppm in the near UV (NUV) (dependent on the work function Wa ) rising to values of up to 0.5 in the vacuum UV (VUV) (Figs. 2.35 and 2.36).
0
10
2015 10
5
E [eV]
Pt
10
potassium on platinum
Au
_ _
g hn
_
_
-3
10-6
100
200
300 400 l [nm]
500
600
Fig. 2.35. Yield for photoelectrons γhν of gold and platinum as functions of wavelength (lower abscissa) and energy (upper abscissa). γhν additionally depends on the polarization of the incoming light [79]. ⊥ and : E of linearly polarized light is orientated in perpendicular or in parallel fashion, respectively, with rec Oxford spect to the surface ( University Press).
E [eV] 30 0,20
25
20
15
Fig. 2.36. The yield for photoelectrons depends sensitively on the pretreatment of the surface [80]: squares: untreated tungsten electrode; triangles: 5 min heated at T > 1 000 ◦ C and 10 Torr; circles: heated up to reproc ducibility at T > 1 000 ◦ C ( J. Wiley & Sons, Inc.).
0,10
g
hn
0,15
0,05
0,00 40
60
l [nm]
80
100
40
2 Collisions and cross sections
Two main mechanisms have been identified which contribute to the steep increase in yield: First, most of the radiation in the long-wavelength range is reflected, and second, due to conservation of momentum, the excitation of free electrons in the valence band by a two-electron process seems very unlikely to happen. Going to short wavelengths, the reflectivities of the metals begin to decline to very low values (with simultaneous rise in transparency), and it becomes more likely to excite valence electrons. All these observations add experimental proof to the suggestion that this process consists of two steps which both belong to the activation type with a certain threshold, generating the electrons as well as their escaping from the solid. Measuring the work function and the various coefficients γ, δ and η remains a great challenge still, since the yield depends strongly on the purity (volume property), but even more sensitively on the contamination (surface property). Theoretically, the main problem is the exact calculation of the barrier which has to be overcome (model potentials etc., cf. [81]). Although the cross sections for photoionization are extremely small at low energies, this does not mean that they are without significance in glow discharges. In DC discharges, the charged carriers exhibit kinetic energies of several thousands of eV, therefore, photons of high energy are generated by Auger processes.
3 The plasma
3.1 Direct current glow discharge In this chapter, we will give a qualitative description of a very simple discharge, will describe some experiments, and will evaluate the main parameters which determine the quality of a low-pressure discharge: plasma density, electron temperature, Debye length and plasma frequency. Eventually, a model is introduced which was originally derived by Langmuir and MottSmith and which is now known as the Global Model.
3.1.1 Phenomenology One of the most prominent appearances of a glow discharge is the glow itself which is created by relaxation processes of atoms which have been excited by inelastic electronic collisions. The zones which emit light at very different intensities reveal that these relaxation processes do not happen with the same intensity across the discharge (Fig. 3.1). Dark areas can be identified adjacent to the electrodes. Their extension differs with pressure, power input and type of gas, and they are called “dark space”. Both cathodic dark spaces belong to the darkest zones of all: Aston’s and Crooke’s dark space, respectively. When Langmuir denoted the different zones in 1928, he applied the word “sheath” for this zone [12]. However, it has become common to use the word “dark space” for DC sheaths solely. The anodic dark space is significantly lighter than the cathodic glow which appears to be light only because it contrasts with the confining dark spaces. The negative glow with its gas specific color (Table 3.1) exhibits its highest intensity a little way behind the very sharp boundary to the cathodic dark spaces and is smeared out to Faraday’s dark space. All these three zones exhibit nearly the same length. This is approximately the length which has to be covered by the electron to gain enough energy of motion to at least equal the ionization energy or ionization potential. Therefore, this ionization length is proportional to the inverted pressure of the discharge as well as the potential difference between G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 3,
41
42
3 The plasma
Table 3.1. Characteristic colors in the negative glow and in the positive column gas hydrogen nitrogen oxygen helium neon argon chlorine bromine iodine HCl CCl4 sodium potassium rubidium cesium mercury
negative glow faint blue red yellowish blue green orange blue purple blue yellowish brownish green faint green whitish faint blue blue light green green
positive column rosa blue faint blue pink brick red dark red greenish reddish violet rosa whitish green yellow green light red yellow brown greenish
the electrodes.1 Faraday’s dark space is again relatively dark compared with its adjoining zones, in particular the positive column between this dark space and the anode. Its color is characteristic for the ambient, but differs not only in brightness but also in wavelength from the light radiated by the negative glow (cf. Table 3.1), in most cases, it is shifted to longer wavelengths. Its brightness is often uniform but sometimes exhibits several zones of different glow intensity, in particular in molecular gases, so called “striations”, which are either spatially stationary or can oscillate.2 In the positive column, the gas is in the ideal plasma state: The number of carriers equal each other, and there is only a very weak longitudinal field. Reducing the distance between the electrodes or decreasing the pressure will affect the nature of the discharge dramatically. First, the positive column will vanish, and afterwards also Faraday’s dark space. When the electrodes are so close that the anode dips into the negative glow, the discharge is quenched, if the voltage is not simultaneously increased. From this behavior, we can conclude 1 As we have seen in Chap. 2, the cross section of ionization increases exponentially just after the ionization threshold. Hence, this ionization length depends on the energy distribution of the electrons. 2 The positive column is used to generate light with countless variations. The gas filling consists of an inert gas and a metal vapor: mercury in fluorescent lamps, sodium in sodium vapor lamps. These exhibit the highest optical efficiency; which exceeds the light output of a normal incandescent bulb by approximately a factor of 10 at 100 W input power.
3.1 Direct current glow discharge
43
cathodic layers negative glow
anodic glow
_
+
positive column Faraday
Aston Crooke
dark spaces
glowing intensity
anode
I
dc potential
Vc
V
electric field
n+ nj- j +
Ex
space charge density j- + j+ current density
Fig. 3.1. Crooke’s glow discharge tube, approximately 50 cm in length at a pressure of about 133 Pa (1 Torr). The glowing areas are shaded. Below, the spatial dependence c J. Wiley of some properties which are characteristic for the discharge is shown [82] ( & Sons, Inc.).
that the most important zones of a discharge are Crooke’s dark space and the negative glow (Fig. 3.1). The potential does not vary linearly across the distance between the electrodes. Due to the different mobility of the positively and negatively charged carriers, space charges are built up which change the spatial shape of the voltage in a characteristic way: A steep decrease across the cathodic dark space in the so-called cathode fall is followed by a region of constant potential and vanishing field in the negative glow. In the positive column, the potential slowly grows — in the form of a staircase in a sandwich column with striations. In the anodic zone, the anodic sheath exhibits a potential increase which is a little steeper than in the positive column, which is of the order of the ionization potential of the gas in the tube. The carrier distribution is intimately correlated with the course of the potential and the electric field via the Poisson equation. Entering the dark space from the quasineutral plasma, both charge densities decline steeply, with a steeper decrease of the electron density to vanish far in front of the cathode (vice versa for the anode). Due to the different mobilities of the two carrier types, not
44
3 The plasma
Vd I V0
R S
_
+
Fig. 3.2. Circuit to generate a glowing discharge with a well defined source voltage V0 which must exceed a certain voltage Vb (b for breakdown). The (overall) voltage of the discharge will be determined according to Eq. (3.1).
only the sheath thicknesses but also the physics in the sheaths are different. For example, the thin anodic sheath is the only region with an excess of negative charge; the ion density dominates in all other parts of the discharge, in particular the region of the cathode fall. The electric field vanishes in the negative glow and the electric carriers move randomly and mutually screen each other (cf. Sect. 3.3), in the positive column, they migrate with medium velocity according to their polarity to their mutual sheaths. To maintain this process, a relatively low electric field is sufficient. Capture of positive ions at the anode would cause a depletion of them, if there were not a sufficient high field to generate additional ionization. 3.1.2 V-I characteristic As a first experiment, we evaluate the (overall) characteristics of the voltage between the electrodes as a function of electric current with external means, in other words, the V-I characteristic or I(V) characteristic of the discharge as a single phenomenon. For this end, we use the discharge tube of Fig. 3.1 and a circuit with voltage source V0 and adjustable resistance R (Fig. 3.2). After having passed a certain threshold, a current I will flow through the tube, and the voltage of the discharge can be simply calculated according to Vd = V0 − IR.
(3.1)
This characteristic depends on a large number of parameters, not only gaseous density and type of the gas, but also the electrode material. The principal characteristic is plotted in Figs. 3.3. At very low current densities of 1μA cm−2 or less, this discharge is denoted as Townsend or dark discharge because very little light or no light is generated. The main feature of this discharge is the linear rise of the potential between the electrodes which is closely connected with a remarkably linear electric field. In the region of about 1 μA cm−2 , the first space charges begin growing. The zone of potential drop is now confined to the cathode fall. The glow becomes gradually lighter with growth of current, but covers only part of the cathode. When the current is further increased, the cross section of
3.2 Temperature distribution in a plasma
45
Vb Townsend's discharge
normal abnormal glow discharge
abnormal
V [V]
potential
Vb
Vn
normal Townsend
Vn 10-9
10-6
10-3
cathode
I [A]
anode distance
Fig. 3.3. V (I) characteristic of a self-sustained discharge without positive column, and the schematic spatial behavior of its potential (Vb is the potential required for c Academic Press). breakdown) [83], [84] (
the glow continues to grow thereby maintaining the current density constant. This is the range of the normal glow discharge or the range of normal cathode fall. After having extended across the whole electrode, the cathodic current density can only rise with increasing current, which, in turn, causes a growth of the cathodic voltage: abnormal glow discharge or abnormal cathode fall. Further growth of current leads to a maximum of the voltage and an eventual breakdown as an arc [85]. Now, the thermal emission of electrons begins to play a decisive part. The main difference between an arc discharge and a glow discharge is the relatively low cathode fall of the former type, which is connected with a negative I(V ) characteristic because of the steep thermal increase in electrons. Due to the occurrence of space charges und the simultaneous generation of different potentials within the discharge at growing discharge current, recording of the spatial resolved I(V ) characteristic has been a challenge since the 1920s. These potentials are measured with a Langmuir probe [86, 87]. In principle, it consists of a small metallic electrode (in the case of a cylindrical system simply a wire) which is immersed in the plasma and connected to a voltage source by which its potential can be changed in relation to the plasma. Thus the relatively small disturbance of the plasma due to the measuring current will be neglected, at least to first order. This is justified for plasma densities larger than 1010 cm−3 for DC discharges [88] (but cf. [89] and Sect. 9.1 for RF discharges).
3.2 Temperature distribution in a plasma As mentioned in Chap. 1, the plasma state is a dynamic equilibrium of generation and loss of electric carriers. The negatively charged carriers are normally
46
3 The plasma
electrons, the positively charged carriers ions of the ambient gas. The energy which is coupled into the plasma, i. e. the work which has been performed by the electric field (W = F x), can be calculated according to W = e0 Ex with x = 1/2 at2 . Since the acceleration a = e0 E/m, we obtain W =
(e0 Et)2 ; 2m
(3.2)
the energy of the external field will be predominantly transferred to the electrons. The efficiency of the transfer of kinetic energy between the plasma constituents depends on • the collision rate between the particles, and • the mean energy transferred during every collision. As we have seen in Chap. 2, the collision rate itself depends on • the cross section of the specific type of collision, • the mean energy of the projectiles and • the density of the molecules. The process most often considered is the elastic collision between electrons of mean energy < ve > and neutrals of density nn , for which we have the cross section of momentum transfer σm : νm = σm nn < ve > .
(3.3)
In low-temperature/low-pressure plasmas, number densities are in the range 1012 − 1014 /cm3 , and typical values of the cross sections for momentum transfer are in the ˚ A2 range, the electrons exhibit kinetic energies of several electronvolts (2 − 8 eV or 20 000 − 80 000 K; 1 eV equals 11 600 K), which leads to values for νm between 104 and 108 sec−1 . Due to the very large ratio of the masses of electrons and other plasma constituents (me mi ), the transfer of kinetic enery via elastic collisions is nearly neglible (the efficiency is approximately 10−5 , since it is proportional to the ratio of the masses me /mi , see Sect. 2.1). Hence, ions and neutrals remain at relatively low temperatures (≤ 0.1 eV; at room temperature, kB T is about 1/40 eV). This means: • Significant deviations from the conditions for thermodynamic equilibrium: Pressure and temperature have to be the same in the whole system— we must distinguish between the temperature of the electrons and the neutrals/ions, and
3.2 Temperature distribution in a plasma
47
Table 3.2. Mean velocities of electrons and argon ions Energy [eV] 0.1 1,0 10 100 1000
v [cm sec−1 ] of electrons argon 1.9 × 107 6.9 × 104 5.9 × 107 2.1 × 105 1.9 × 108 6.9 × 105 5.9 × 108 2.1 × 106 1.9 × 109 6.9 × 106
• Very high mean energy of the electrons, which amounts to approximately 1 % of the speed of light (Table 3.2).3 Provided the electrons are distributed following a Maxwellian distribution function, their mean energy is
< ve > =
8kB Te , πme
(3.4)
which leads to a current density of the electrons (j = 1/4 ρ v with ρ the density of the carriers) a factor of 103 higher than that of the Ar ions. Because of the low number of collision processes with heavy particles, the energy distribution of the electrons remains different from that of the ions and neutrals which remain at a very low level, typically in the upper range of 102 K (Sect. 9.4.2). With growing pressure the number of collisions between electrons and heavy particles will increase dramatically (this number is proportional to the pressure squared) which leads to an effective equalization of the different temperatures (cf. Sects. 3.5 and 14.1 for a critical understanding of distribution functions and the phenomenon of temperature). This is the main difference between low pressure and high pressure plasmas, which start at about 1 bar and are used in quartz burners, for example. And we understand why a low-pressure plasma serves as an efficient light source: The electric energy is mainly transferred to the electrons which effectively excite, in turn, the optical levels of mercury atoms. Therefore, a low pressure plasma is also a cold plasma. The plasma density of a cold plasma is confined to a range between 108 and 1012 cm−3 and is determined by physical limits: At too low a charge density, the electrostatic interaction is very low, which leads to some isolated charges which are separated across a significant distance; when the plasma density exceeds a certain threshold, the gas will heat up significantly by collisions between electrons and neutrals. 3 We realize a resemblance to the free electron Fermi gas in metals. Here, the Fermi energy EF reaches some eV (e. g. Cu: 7.00 and Au: 5.51 eV); as a first consequence, the Fermi velocity, which formally assigns the velocity of the electrons to their kinetic energy reaches very high values.
48
3 The plasma
Direct measurement of the temperature of the heavy particles (ions and neutrals) of the plasma bulk are impossible. One method to determine the speed of the particles, and hence their temperature, is measurement of the width of a spectral line, which is predominantly determined by the Doppler shift [besides radiation damping [90] and lifetime broadening (so called fourth uncertainty relation ΔE · Δt ≥ h ¯ [91])]: An object radiating with a wavelength which moves away from a resting watcher with velocity v, is recognized by him as having the apparent wavelength (1 + v/c)λ, an approaching object, however, will have apparent wavelength (1 − v/c)λ (cf. Chap. 9). Another possibility is the recording of a rotational vibration spectrum. In absorption, the distribution of intensities in a Bjerrum double band is largely determined by the temperature-dependent population of the rotational levels of the vibration ground state [92], cf. Sect. 9.4.
3.3 Neutralization of charges in an undisturbed plasma The discharge potential between the electrodes is mainly characterized by a steep decline across both the dark spaces, whereas the negative glow remains almost free of fields (constant potential). In the positive column, we note a small rise of the potential with constant electric field (Fig. 3.1). The plasma is effectively screened against disturbing fields, which are caused in the simplest case by ions. This is achieved by the formation of electron clouds which are arranged around the ions. By this, the range of the Coulomb potential is reduced to very small values. This shielding effect was first described by Debye and H¨ uckel in their theory on strong electrolytes in aqueous solutions [93]. Due to a mutual orientation of the ions, clouds of the lighter ions will be formed around the more inert ions. The ionic density around a certain ion in a solution is then given for thermal equilibrium by
e0 Φ(r) n(r) = n0 exp − kB T
(3.5.1)
with n0 the ion density in the undisturbed solution, where n+ + n− = n0 , the charge density ρ(r), however, vanishes at large distances from the ion because of the different polarities of the ions (z the valence of the ion) ±zi e0 (n+ + n− ): ρ(r) =
i
e0 Φ(r) zi e0 n0 exp − kB T
(3.5.2)
For small values of the orientating electrostatic potential, Eq. (3.5.2) may be expanded4 4 This linearization is necessary to apply the fundamental principle of superposition. Accounting for members of higher order would violate this principle.
3.3 Neutralization of charges in an undisturbed plasma
ρ(r) ≈
zi e0 n0
i
49
e0 Φ(r) 1− . kB T
(3.5.3)
We demand electroneutrality for the whole solution, hence, the first sum van ishes ( i zi e0 = 0), and we obtain for one sort of carrier (zi = 1)
ρ(r) = e0 n(r) ≈ e0 n0 −
e0 Φ(r) kB T
(3.6)
and likewise for the potential: Φ(r) ≈ −
kB T n(r) . e0 n0
(3.7)
Inserting Eq. (3.7) into the Poisson equation d2 Φ(r) n(r)e0 = dr2 εε0 eventually leads to the Poisson-Boltzmann equation −
d2 Φ(r) e20 n0 = Φ(r). dr2 εε0 kB T
(3.8)
(3.9)
A solution is5
Φ(r) = Φ0 exp −
r d
(3.10)
with
d = λD =
εε0 kB T , n0 e20
(3.11)
which is the Debye length. The equilibrium between thermal and electrostatic energy determines the value of the screening length: Increasing the charge density reduces the screening length. The ion cloud would collapse if there were no thermal movement of the ions. At its edge, where e0 Φ ≥ kB T , e. g. in close vicinity of the electrodes, the ion cloud is severely disturbed. In a similar way, we can argue for a plasma because highly mobile carriers exist as well. However, they differ significantly in their energy [Eq. (3.2)] which has to be taken into account in a comprehensive analysis (cf. Sects. 3.4 and 14.2). For immobile ions, we obtain eq. (3.11), replacing n0 by ne . In fact, there is a slight thermal movement of the ions as well.6 In a low-pressure discharge, however, they are far away from thermal equilibrium. Even small dragging forces 5
For positive values of the exponent, the equation diverges. Provided also the ions are in thermal equilibrium (Te Ti ), the equation for the reduced temperature for both types would read 6
potential energy [a. u.]
50
3 The plasma
Fig. 3.4. The long-range tail of the Coulomb field 1/r is screened by the formation of an ion cloud with the screening length λD (λD is normalized to unity [94]).
screened Coulomb potential Coulomb potential
distance [a. u.]
cause the ion drift velocity to rise beyond its thermal velocity (a potential of already 0.05 V would equal the thermal velocity of Ar+ ions exhibiting a temperature of 600 K). λD ≈ kTe /ρ has a length of about 100 μm for kB Te = 2 eV and ne = 1010 cm−3 . Across a distance of λD , the electrostatic interaction drops to 1/e (37 %), and in 3 λD , to values of less than 1 % (cf. Fig. 3.4). Numerically, we get approximately (for T in K and ne in electrons/cm3 ) for λD in cm:
λD = 6.91
Te [cm], ne
(3.12.1)
if we prefer to express the electron temperature in eV which is not exact but common:
λD = 743
Te [cm]. ne
(3.12.2)
λD decreases with increasing electron density, and increases with rising electron temperature. Only electron temperature and electron density are inserted into the equation to determine the Debye length, because it is only the electrons which cause the screening effect of the ions due to their tremendously high mobility, no matter if a surplus of negative charge is to be equalized by draining or if a positive excess charge has to be compensated by a flow of electrons into this drain. From an external viewpoint, we find this volume containing mobile charges to behave quasineutral since ne = ni . 1 1 1 1 = + ≈ , T Te Ti Ti and λD would be solely determined by the ions.
3.3 Neutralization of charges in an undisturbed plasma
51
The numerical values of the Debye length cover some orders of magnitude; e. g. in the photosphere of our sun with T = 5 600 K and a plasma density of 1012 cm−3 , λD is about 5 μm, in the Orion nebula with T = 10 000 K and a plasma density of 100 cm−3 , λD is approximately 70 cm (Fig. 1.1). In dense and cold plasmas, the quasineutrality can only be disturbed in the interior of relatively small volumes, in a thin and hot plasma, however, the Debye length can exceed the dimensions of the plasma container. In this case, ions and electrons move independently, and there is no correlation mechanism which would compensate or equalize high potentials within this agglomeration of charges. By means of the Debye length, we can sharply distinguish between gases which are ionized, i. e. which contain but some charged carriers (ne is very small), and plasmas. For those “plasmas”, we calculate very large values for the Debye length that are eventually in the order of the dimensions of the reactor. Evidently such a system cannot be termed quasineutral any more. λD has to be small against the spatial dimensions of the plasma (Langmuir’s definition of a plasma). As a second requirement, we demand a large number N of screening electrons. N can be calculated as the product of the electron density of the undisturbed plasma, ne , and the spherical volume with the Debye length as radius: N=
T 3/2 4π ne λ3D ∝ √ ∨ λD > ne−1/3 , 3 ne
(3.13)
N must be very large (> 100). The importance of this parameter is elucidated from its simple linguistic usage as the plasma parameter: The quasineutrality can be violated only within a sphere of radius λD . Only within this volume can the kinetic energy of the electrons exceed the directing effect of electrostatic interaction.7 To balance out deviations of equilibrium, it does not take longer than λD = τ=√ < v2 >
ε0 m e ne e20
(3.14)
with
the mean squared velocity (= 3kB Te /me ). This length is the border between the motion of individual particles and the motion of an ensemble. Mathematically speaking, the long-range tail of the Coulomb potential with its singularity at x = 0 is replaced by the Thomas-Fermi potential (Fig. 3.4). Compared with the mean free path of the electrons λe , which is in the order of cm, λD is between 0.1 and 1 % of λe . This means that the properties of a plasma are not, or only very slightly, disturbed by collisions between the electrons and heavy particles. 7 In the case this condition cannot be met, the plasma behaves rather as a liquid than as a gas.
52
3 The plasma
Vp
potential [a. u.]
0 Vf
floating grounded electrode
Vc
Fig. 3.5. Potential distribution of a discharge which lacks a positive column and Faraday’s dark space; the positive electrode is immersed in the negative glow, and the adjoining sheath of the negatively charged electrode separates it from the negative glow.
distance [a. u.]
3.4 Potential variation in the plasma
As we have seen from Eq. (3.4), the velocity of the electrons exceeds that of the ions by some orders of magnitude. Hence, they can easily evade the glowing plasma and are captured by the walls, leaving behind a positive residual charge. Due to the effective Debye screening, this excess charge cannot easily be distributed over the plasma bulk. On the contrary, this potential difference will be built up across a distance of only several Debye lengths. Across this layer (dark space), the carrier density drops to very small values, and therefore, intense fields can occur without generating large currents, which is impossible in the negative glow where the plasma density exceeds the upper limit of a dark Langmuir plasma (np ≥ 107 cm−3 ). The developing potential is called the wall potential [95] or floating potential, Vf , since it is set up at all isolating walls which are in contact with the plasma. This floating potential can be readily measured against ground and can even be calculated in simple cases. The layer itself is denoted sheath; in particular, in DC discharges this layer is commonly referred to as dark space (cf. Sect. 3.1). One of the physical functions of this sheath is the formation of a potential barrier to bound the electrons electrostatically. It adapts to values to ensure equality of the electron and ion currents. The height of this sheath is relatively low, but sheaths in front of electrodes can very significantly exceed kB Te /e0 . Looking from the plasma bulk, it has charged up to a positive value, the plasma potential, Φp , with respect to all the walls confining it, and the potential distribution can be sketched according to Fig. 3.5.
3.4 Potential variation in the plasma
53
3.4.1 The low-voltage plasma sheath To make this quantitative, we investigate an isolated electrode being immersed in a plasma. In thermal equilibrium, for both types of electric carriers, we take the Maxwellian distribution for granted within the negative glow. In this region, n0 = ne = ni [96]. Due to their considerably different mass, the current densities for both carrier types incident on the substrate are initially different: 1 e0 n e je = e0 ne < ve >= 4 4
8kB Te kB Te = e0 ne πme 2 πme
(3.15.1)
and 1 e0 n i ji = e0 ni < vi >= 4 4
8kB Ti kB Ti = e0 ne , πmi 2 πmi
(3.15.2)
so we obtain for their ratio je = ji
Te mi , Ti me
(3.16.1)
and with ne = ni = n0 < ve > = < vi >
Te mi . Ti me
(3.16.2)
The initial current densities of ions and electrons differ by a factor which exceeds the ratio of their velocities! This gives immediately rise to a surplus of negative charge on the substrate, the potential drops from the initial zero to the floating potential Φf , and in the vicinity of the device, a positive space charge surrounds it due to the deficiency of captured negative electrons. Hence, by the creation of such a potential 1. the electronic current will be reduced to reach a level such that 2. the ionic current will equal the electronic current. 3. Looking outward to the undisturbed plasma, the electric field which is connected to the sheath is rapidly damped out. Therefore, the ionic flux incident on the electrode is by no means enlarged, since the ions entering the sheath have left the quasineutral plasma with very low energy. Only those ions which have reached the sheath will be accelerated. 4. The net result is a depletion of charges in the sheath.
54
3 The plasma
3.4.1.1 Approximation of first order. Although we are predominantly interested in the plasma potential Φp , the potential that is readily measured is the floating potential Φf . To calculate its order of magnitude, we use condition (2), the evenness of the currents incident on the floating electrode. The correct electron current density is the mean current density times a retarding Boltzmann factor, which takes into account its drop from the initial large value due to the creation of the retarding potential Φf < 0 (uphill for the electrons); for the ions, there does not exist any barrier which they have to overcome (downhill for the ions):
e0 (−Φf ) . < vi >=< ve > exp − kB Te
(3.17)
Combining Eqs. (3.16.2) + (3.17), we get
e0 (−Φf ) Te mi = exp − Ti me kB Te
⇒ −Φf =
kB Te Te mi ln , 2e0 Ti me
(3.18)
from which follows the floating potential:
Φf =
kB Te Ti me ln . 2e0 Te mi
(3.19)
Φf is negative in all cases. 3.4.1.2 Approximation of second order. In Sect. 14.2, the ion current density at the boundary presheath/sheath is shown to be8
ji = n 0
kB Te mi
(3.20)
from which we derive the Bohm velocity at the sheath edge to be
vB =
kB Te . mi
(3.21)
Solving for the kinetic energy of the ions, we obtain the Bohm potential according to 1 1 kB Te e0 ΦB = mi vB2 ⇒ ΦB = , 2 2 e0
(3.22)
8 The ions are accelerated across the presheath to a velocity that balances the electronic velocity component along the electric field, i. e. 1/3 of their thermal energy: 2 /2 mi vB = 1/3 3/2 kB Te
1
from which Eqs. (3.20) and (3.21) will follow.
3.4 Potential variation in the plasma
55
which represents the plasma potential Φp at the boundary between the presheath and the sheath. The electron current density is the mean current density times a retarding Boltzmann factor which gives rise to its drop from the initial large value, and dividing by n0 , we get
je 1 e0 (−Φf ) = < ve > exp − . n0 4 kB Te Equating Eqs. (3.20) and (3.23), and using < ve >=
Φf =
(3.23)
8kB Te , πme
we obtain
kB Te πme ln , 2e0 2 mi
(3.24)
which is negative in all cases. In argon, the floating potential would reach 16 V for an electron temperature of 3 eV and 11 V for a Te of 2 eV. 3.4.1.3 Approximation of third order. In the vicinity of the sheath and in the sheath itself, the electrostatic energy is much greater than the thermal energy. To calculate the flux of electric carriers incident on the electrode, the Poisson equation has to be supplemented to arrive at the Poisson-Boltzmann equation (cf. Sect. 3.3). d2 Φ = βΦ, dx2
(3.25)
(with Φ = rφ, φ is a potential which must not be linearized because its value is large compared with kB T, cf. Sect. 14.2). Now, we also take account of the fact that, lacking a reference potential, we can measure but the potential difference, and as the result of a tedious calculation, we find that the fraction of electrons with energies greater than e0 (Φp − Φf ) with Φf the floating potential, that can penetrate the sheath entirely to eventually reach the wall, drops to
e0 (Φp − Φf ) , ne (x) = n0 exp − kB Te
(3.26)
with Te , the electron temperature. Their velocities deviate from the mean electron velocity according to Eq. (3.4) to a large extent. At the sheath boundary, where Φp = ΦB , the electron density equals ne = n0 still. Therefore, the electronic current incident on the negatively charged substrate will be je (x) =
1 ne (x)e0 < ve >, 4
(3.27)
the current density of the positive ions, however [Eq. (14.43)]
1 ji = n0 e0 vB exp − 2
= n0 e0
kB Te 1 exp − mi 2
(3.28)
56
3 The plasma
i. e. the condition for equilibrium ji + je = 0 ∨ −je (x) = ji can be written in this approximation [97]: Φp − Φf =
kB Te 2π me kB Te me ln ⇒ Φp − Φf = ln 2.31 , 2e0 emi 2e0 mi
(3.29)
which is the potential of an isolated floating surface in contact with the plasma for which the net current must be zero. Since the logarithm is negative, the floating potential Φf is always numerically larger and opposite in sign to the plasma potential Φp . This difference is the barrier potential which has to be overcome by the electrons to cross the sheath of an isolated electrode, and it equals approximately the logarithmic ratio of the masses between ions and electrons. The ions are accelerated towards the wall by the initial voltage Φf , since they reach the presheath with a “random” motion leaving the plasma bulk with vanishing electric field. Hence, their kinetic energy is thermal and very small. When entering the presheath, they are accelerated by an electric field to enter the sheath with the directed velocity vB . The calculated values for Φf generally have an upward trend. Cox et al. pointed out that these deviations can often be attributed to the wrong energy distribution of the electrons, which is supposed to be Maxwellian [98] (cf. Sect. 14.1). This assumption is questionable, at least in low-density plasmas. A second assumption refers to the temperature of the ions (room temperature assumed). The detailed discussion of high-voltage plasma sheaths are deferred to Chap. 4 for DC discharges and Chap. 6 for RF discharges. On a qualitative level, the potential of the cathode sheath is set to a level of approximately Φc = Φp − Φcathode , and the potential of a grounded electrode is about Φp . The range of potential drop between uniform plasma and the electrode itself, the sheath, can be divided into two subparts. Hence, above an electrode, we can distinguish at least three zones (Fig. 3.6): • Undisturbed plasma, far away from an electrode, no field, the ions move randomly, and the densities of positively and negatively charged carriers equal each other. Disturbances which occur randomly, and deviations from equilibrium are equalized within the Debye length. • Slightly disturbed plasma, the ions will be accelerated by a weak electric field (transition zone). The densities of positive and negative carriers mutually equal still, its thickness is of the order of the ionic mean free path. This directed velocity is due to a potential drop of 1/2 Te which happens to occur over a long distance between electrode sheath and undisturbed plasma (i. e. a very tiny electric field, so called “presheath”). The ions are accelerated by the Bohm potential ΦB , and the Bohmic presheath terminates when the ions have gained the (directed) velocity Bohm velocity vB ; here, their density has been approximately diminished to
3.4 Potential variation in the plasma
potential energy [a. u.]
Vp VB
57
neutral plasma no field Bohmic presheath weak field
Fig. 3.6. Undisturbed neutral plasma, the quasi-neutral transition zone (Bohmic presheath) and the cathode fall within the positively charged sheath adjacent to the negatively charged surface.
positively charged cathode sheath
Vc
distance [a. u.]
1
/2 mi vB2 ni = n0 exp − kB Te
= n0 exp −
1 2
n0 =√ . e
(3.30)
For a stable sheath with a surplus of positive charges, it is mandatory that the ionic density across the sheath always exceeds the electronic density. Only then ∀x > 0 :
d2 φ <0 dx2
(3.31)
2
is valid across the whole sheath. If ddxφ2 became positive, oscillatory motions would result in instabilities of the sheath. • Severely disturbed plasma, so-called electrode fall or sheath adjacent to the electrode, strong acceleration, ΔΦ kTe /e0 , at every point, already at the boundary between presheath and sheath, the ion density is larger than the electron density. In RF discharges, the electron density can eventually vanish leading to a sheath which consists only of a small number of positive ions, whereas in DC discharges, a small, but permanent conduction current is incident on the electrode. Its thickness is very thin (of the order of some Debye lengths) for isolated and grounded electrodes but can grow to considerable thicknesses (several millimeters) for sheaths of powered electrodes. From Table 3.3, we can see the ion current density as a function of plasma density and electron temperature. In every case, this leads to a tremendous loss of dissipated energy. If we take a low-density plasma (5 × 109 /cm3 ), sheath voltages are of the order of some hundreds volts, say 400 V, the power density will be 84 mW/cm2 , for a high density plasma with 1 × 1011 /cm3 and a sheath
58
3 The plasma
voltage of 20 V, we will find 100 mW/cm2 , almost the same loss. On a wafer 4 inches in diameter, that means a loss of 10 W, on a wafer 8 inches in diameter, a loss of 40 W. Table 3.3. Current density of Ar ions incident on the walls of a plasma reactor, electron temperature: 3 eV plasma density [cm−3 ] 109 1010 1011 1012
current density [mA/cm2 ] 0.042 0.42 4.2 42
Within the sheath with very low electric conductivity, intense electric fields can be present without having triggered high fluxes of carriers. This zone screens the plasma from high potentials. Therefore, its function is comparable with that of the surface of a metallic conductor.
3.5 Temperature and density of the electrons In the preceding two sections, we have seen that a cold plasma is far away from thermodynamic equilibrium (so-called athermal plasma), and we distinguish between the temperature of the neutrals and ions, which are normally very similar and do not exceed 1 000 K, and the temperature of the electrons. Every plasma is outwardly neutral because the number of positively charged carriers exactly equal the number of negatively charged carriers; in a so-called electropositive plasma (for example argon), the negatively charged carriers are but electrons. Following Langmuir and Mott-Smith who established a very simple steady-state and volume-averaged model in the early 1920s which is known as the Tonks-Langmuir model [99], we want to give a qualitative answer to the question which external parameters determine the very important plasma properties electron temperature Te and electron density ne , which we denote primary plasma parameters in contrast to other (derived) properties, e. g. the collision frequency of electrons with neutrals (cf. Chaps. 2 and 9). This model was revitalized by Lieberman in the 1980s and is often referred to as the “Global Model” [100]. For a stable discharge, power has to be transferred continuously into the plasma to balance the losses in energy and highly excited neutrals and charged particles (electrons and ions), either by wall reactions (diffusion) or by recombination reactions. Electrons absorb energy from the electric field, and this kinetic energy is lavishly dissipated in the plasma by elastic collisions. In a stable plasma, a sufficient
3.5 Temperature and density of the electrons
59
number of electrons collide with neutrals inelastically to sustain the discharge. By this electronic impact, neutrals are ionized or excited to optical transitions (cf. Chap. 2). The energy losses of the electrons can be mainly attributed to the processes of • elastic scattering, • excitation to rotations, oscillations, UV/VIS transitions and • ionization. 3.5.1 Elastic scattering and thermalization Elastic scattering leads to thermalization, and for electrons which are distributed Maxwellian, its mean energy is closely connected to their well-defined temperature. For the state of equilibrium, we describe the energy distribution of the neutrals by a Maxwellian distribution which is assumed to be valid also for the charged particles (cf. Sect. 14.1) and can be written in the one-dimensional case as
1 v f (v) = √ vw exp − π vw
2
,
(3.32)
with vw the most probable velocity: vw = 2kB Te /me . With this assumption (equipartition of energy), it is possible to define a temperature of electrons, Te (to avoid confusion with the intensity of the electric field, in this section the symbol for energy is ε): 1 2 1 3 me < ve2 > . < ε > = kB Te = me < v 2 >⇒ kB Te = 2 2 3 2
(3.33)
The mean squared velocity < v 2 > and the square of the most probable velocity vw2 show a mutual ratio of 3/2 [101]: 2 <ε> . (3.34) 3 The plasma is outwardly screened by the sheath; hence, the interior of the plasma, at least the negative glow, becomes almost free of electric fields. As a direct consequence, the thermal velocity of the electrons far surpasses their drift velocity, which is additionally reduced by collisions with neutrals, v = μE with μ the mobility and E the electric field. Hence, the influence of the electric field on the energy of the electrons (and at second order also the energy of the molecules and ions at the border plasma/sheath, the Bohm edge) is small—and this holds true also for the processes of ionization and diffusion. kB Te =
60
3 The plasma
3.5.2 Energy loss When thermalized electrons hit the wall during their chaotic course, they are annihilated, and all their energy gets lost as well. As it is shown in the kinetic theory of gases, the electron flux incident on the wall with positive direction +x is given by 1 (3.35) Γx = ne < ve >, 4 and the energy flow of n electrons with average energy 1/2 me < v 2 > incident on the wall in positive direction +x with mean velocity < vz > yields to Sx = 2kB Te Γx ;
(3.36)
hence the energy loss is 2kB Te . As we have seen in Chap. 2, the processes of excitation are characterized by low efficiency. On the average, the energy which has to be piled up by an electron must exceed the excitation threshold significantly, and after excitation, the abundant energy will mainly get lost in elastic collisions. The lower threshold of this amount of energy can be generally given by Eexc , in particular for ionization, by Eion . Ions which have traversed the Bohmic presheath have gained a certain amount of directed energy (EB = 1/2 kB Te ) and are about to get lost. Just before they hit a surface, they are further accelerated across the sheath, either in front of the wall of the reactor, or in front of the electrode. In either case, they absorb an amount of energy which is large compared with EB and which, in turn, causes a heavy power demand. In total, the energy consumed by lost ions is 1 (3.37) Ei = kB Te + e0 Φs . 2 It is these three contributions which define the lower limit of energy loss per time and area: 1 E ≥ 2kB Te + Eion + kB Te + e0 Φs , 2 and the energy flow into the plasma must exceed
(3.38)
Pabs (3.39) ⇒ Sabs ≥ nB vB ε O where O denotes the surface of the plasma bulk which terminates at the Bohm edge. In equilibrium, the carrier densities are balanced out, and the temperatures of molecules and electrons differ significantly but are separately constant in time. This is the case provided that Sx =
• the rate of formation for charged carriers equals the rate of loss, and if
3.5 Temperature and density of the electrons
61
• the energy losses of the electrons are balanced out by their energy gain. We will first estimate the temperature of the electrons and then take a closer look to their density. 3.5.3 Electron temperature Applying the law of mass action, we can get an idea of some important properties of electropositive tenuous plasmas in which the plasma density np (= ne ) is small compared with the density of neutrals n0 : np n0 . • How large is the plasma density np ? • How large is the rate of charge generation? We assume a single-step ionization by collisions of neutrals and electrons: A + e− −→ A+ + 2 e− and only monovalent ions, the losses should be due to diffusion to the reactor walls, and we disregard all other loss mechanisms such as recombination processes. Since from a neutral A, two charges are generated, one ion (i) and one electron (e), the bimolecular reaction of formation can be formulated A + e− −→ A+ + 2 e− ,
(3.40)
ni = n e ∧ nA = n 0 − ne ≈ n 0 .
(3.41)
and therefore
According to the law of mass action, we can this express as
εion ni n2e = exp − n0 ne kB Te ne =
√
n0 exp −
⇒
(3.42)
εion , 2kB Te
(3.43)
(simplified Saha equation without consideration of potential degeneration of states). The formation rate of the ions is with kion the bimolecular rate constant for the ionization from A to A+ , ∂nA+ (3.44) = kion ne nA ; ∂t it depends linearly on the electron density, and kion depends only on Te . Hence, in the cylindrical volume πr2 L, the formation rate of the ions amounts to ∂NA+ = kion ne nA πr2 L. ∂t
(3.45)
62
3 The plasma
This is countered with losses by diffusion to the walls of the reactor (velocity B Te ; and at sufficiently low pressures, the ionic mean at the Bohm edge vB = km i free path λi exceeds the dimensions of the reactor (λi R ∧ λi L) yielding approximate solutions for the densities at the radial and axial boundaries. The density of charges at the radial surface is slightly different from the axial one [102, 103], but is neglected during this derivation; np is the plasma density in the center of the reactor:
∂NA+ = nA+ vB 2 πr2 + 2πrL . (3.46) ∂t For a stable system, this loss equals the generation according to a continuity equation
(3.47) kion nA+ nA πr2 L = nA+ vB 2 πr2 + 2πrL ⇒
−
vB nA vB = ⇒ kion = Op,spec kion Op,spec nA
(3.48)
with 2(r + L) , (3.49) rL a factor which reflects the geometric ratio volume over surface. The index “spec” denotes the spectroscopically determined electron temperature. Inspecting Eq. (3.48), we note that the quotient on the left-hand side implicitly includes the electron temperature (i. e. the mean electron energy): kion depends strongly, vB weakly on Te (see also Figs. 2.12 and 9.34).9 In particular, the rate coefficient for ionization must decrease with increasing neutral density (growing pressure), hence, the electron temperature Te must go down [eq. (3.48)]. This effect is more pronounced at lower pressures. We must compare the rate coefficient for ionization, which is closely connected with the cross section for ionization, via Op,spec =
kion = < ve > σion
(3.50)
and which can be obtained by scattering experiments with data obtained by plasma diagnostics. The generative and diffusive parameters are analyzed in Figs. 3.7. For a given geometry of the reactor volume (here: electrode gap: 5 cm, reactor diameter: 43 cm), in the range of low pressures (two orders of magnitude), the plasma property nO does not show any tendency to rise sharply beyond a certain pressure (border at about 100 mTorr). According to this simple model, the electron temperature Te rises 9 In particular, this holds true for energies close to the ionization threshold where the characteristic jump in the inelastic cross section by several orders of magnitude is observed (cf. Sect. 2.5).
3.5 Temperature and density of the electrons
1013
1014
n [cm -3]
1015
20
1015
10
1014
0
1
6
1016
nO
10
Te [eV]
electrode distance: 5 cm electrode diameter: 30 cm O
nO [cm-2]
O [cm]
8 1017
40 30
63
4
2 0
2
10
100
4
10 10 vBohm/k ion [1015/cm2]
p [mTorr]
6
10
10 8
Ar gap 5 cm diameter 43 cm
Te [eV]
6 4 2 0 0
25
50 p [mTorr]
75
100
Fig. 3.7. For a given geometry, the parameters for electron generation and electron loss are correlated with electron temperature. The rate coefficient is modeled applying the (energy dependent) ionization cross section and assuming a Maxwellian distribution of electrons [104, 105] (Sect. 9.4). The smaller the range O, the larger the loss rate by diffusion. To sustain a stable plasma, the electron temperature must consequently rise. The electron temperature is calculated considering different ion densities at the boundary sheath/presheath in axial and radial direction according to the approach by Godyak and Lieberman [102, 103].
• with falling particle density nA (for constant gas temperature, this is the discharge pressure), • when the reactor volume gets smaller, and • when the confining walls around the plasma become larger. In particular, the temperature of the electrons, Te , is determined by the density of the neutral molecules (i. e. for constant gas temperature: the discharge pressure), and via the geometry factor by their gain (by collisional impact) and loss (by diffusion), but it definitely does not depend on the plasma density, np and the power (incoming or absorbed). In this model, the thermal velocity of the electrons scales inversely with ln nA [cf. Eq. (3.43)]. This simple dependence
64
3 The plasma
is confirmed by approaches which take the electron energy distribution function (EEDF) into account (cf. Sect. 14.1). 3.5.4 Electron density The temperature of the electrons is determined by the dynamic equilibrium between their generation and loss. According to Ohm’s law, the power absorbed by a current of ions j in an applied field of intensity E is given by dPabs = js Ed3 x ∨ dPabs = σ E 2 d2 x dx.
(3.51)
The current density js has to be measured at the boundary between Bohmic presheath and sheath and is the product of the density of √ the ions, ρA+ = e0 nA+ = e0 NA+ /V , the Bohm velocity vB and the factor 1/ e [Eqs. (14.43/44)]: e0 NA+ vB , js = √ e V
(3.52)
whereas the (averaged) electric field is given by the drop of the electric potential between electrode level and plasma boundary, so we get dPabs =
e0 NA+ vB d2 x √ E dx. e V
(3.53.1)
From this consideration, we see that dPabs =
e0 NA+ vB d2 x √ Edx e V
(3.53.2)
2
where the ratio dV x describes again the ratio between volume and its confining surface which can be associated again with Op,spec introduced by Eq. (3.49), the specific geometry factor. The absorbed power scales with ion density density and field intensity, and, to a weaker extent, with electron temperature (via vB ). Furthermore, we can infer the number of generated ions, NA+ , to depend directly on the absorbed power, but inversely on the effective geometry, and the intensity of the electric field: N A+ ∝
dPabs , vB Op,spec Edx
(3.54)
and comparing this equation with the rough estimation of eq. (3.39), we see that nB × O must equal NA+ × Op,spec . To achieve high plasma densities, the specific area, the electric field and the thickness of the sheaths (electrode sheath and Bohmic presheath) should be kept as low as possible. On the other hand, failing to achieve a high plasma density can be caused by these fundamental parameters.
3.6 Plasma oscillations
65
Furthermore, the dependence of vB on Te determines the ion impact energy, and the rate coefficient of ionization is a single function of Te , which primarily depends on the gas density (at the sheath edge, the ions have gained a directed energy of order 1/2 kB Te [106], cf. Chap. 9 for the difference between generated and absorbed power). Even in the simplest approximation of a DC discharge, Peff scales with the squared electric field (cf. Chap. 5 for the effective electric field in an HF discharge). This derivation is based on some simple assumptions, primarily on the law of mass action for generation and the diffusion as mechanism for loss and should generally be valid. In particular, this model does not depend on the method of plasma excitation, irrespective of whether they are DC based or HF methods.
3.6 Plasma oscillations We encounter two different types of electrical carriers in an electropositive plasma, as in argon, namely ions and electrons. However, the collision frequency is very high even at modest discharge pressures. Hence the mean free path of electrons, λe , and that of the ions, λi , respectively, is sufficiently short to ensure electroneutrality, even if electric currents are flowing under the influence of an external field. At low frequencies or large wavelengths, we can regard the plasma as a system consisting only of one component (cf. Sect. 3.3). With increasing frequency, this model breaks down. Electrons and ions move mutually independently, causing a separation of charges, and accompanying electric fields are set up which generate electrostatic oscillations. Further rising frequency eventually highlights the inertia of the ions since they cannot follow the fluctuations of the field any longer. They become a background of positive discharge to ensure electroneutrality, and the electrons move against this background as in a viscous liquid (jelly). This plasma is termed Lorentz plasma. This disturbation can be described with the Poisson equation, and for a linear field we obtain ρx ρ dE ⇒E= . = dx ε0 ε0
(3.55)
The force which is applied to the charge density ρ and causes its acceleration, is F = −ε0 e0 E = −e20 nx ⇒
d2 x ne0 x + = 0, dt2 ε0 m 0
(3.56)
which is the well-known equation of a harmonic oscillation with eigenfrequency
ωp =
ne20 , ε0 m 0
(3.57)
66
3 The plasma
so so-called plasma frequency (for a more exact treatment of this problem cf. Sect. 14.3). Since Eq. (3.57) contains only constants, except for n, ωp can be written for electrons (m0 = me ) as ωp = 5.64 × 104 10
√
n.
(3.58)
−3
As an example, for n = 10 cm , we obtain ωp = 5.64 GHz. This would be the angular frequency of the plasma electrons around their position of equilibrium without any damping. However, the oscillations are damped by collisions with neutrals and electrically charged carriers, irrespective of whether they are electrons or heavy ions. To excite such oscillations, the collision frequency νm must be small compared to the plasma frequency, the reciprocal of the time τ which we defined in Eq. (3.14): ωp > νm .
(3.59)
As we can see later on (Chap. 9 and Sect. 14.6), this is a very good approximation, especially in high-density plasmas. But already in low-density plasmas, as in capacitively coupled discharges, νm amounts to only a few percent of the value for ωp . The product of Debye length and plasma frequency results in ε0 kB Te ne20 kB Te 1 = = √ vw 2
ne0
ε0 m e
me
2
(3.60)
with vw the most probable velocity [maximum of the Maxwellian distribution, cf. Eq. (3.34)]. That means: In a plasma, carriers are displaced only to an extent which is of the order of magnitude of the Debye length. Hence, the paramount importance of the Debye length for the description of plasmas will be evident from the fact that also dynamically generated deviations from equilibrium are not stable across a range which is larger than this length. The Debye length separates the plasma into two parts of interaction with waves: For disturbances k kD (with kD the inverted Debye length, the Debye wavevector), the plasma behaves as a continuum; it oscillates as a single ensemble (“cooperative effect” [107]); for disturbances k kD , however, Eq. (3.25) is still valid, which describes the behavior as individual particles. We must distinguish between transverse and longitudinal optical oscillations which are not mutually coupled in the case of no external fields. Transverse optical oscillations are excited by electromagnetic waves in the range ω > ωp . For ω < ωp , the dielectric constant becomes negative; which makes the propagation impossible beyond the skin depth, but opens up the possibility to measure the highest charge density by reflection of microwaves (cf. Sects. 14.6 and 14.7) [108]. Longitudinal optical oscillations, however, are excited by inelastic scattering of electrons with energies greater than their mean energy. The plasma waves exhibit an energy of typically 10 eV and more. For wavevectors k kD , we can approximate the phase velocity according to
3.6 Plasma oscillations
67
vph ≈ ωp /k.
(3.61)
The oscillations will propagate (i. e. the local oscillation will turn into waves) by thermal motions of the electrons. These waves can propagate almost undamped since a particle moving slowly against the wave feels almost the same field as a resting particle. For further growth in k, a large fraction of the electrons exhibit thermal velocities which are comparable with vph . This opens the window for a very effective, collisionless energy transfer which becomes so efficient at k ≈ kD that it is useless to speak of organized oscillations (Landau damping [109], cf. Sect. 14.3). Now, at latest, we have entered the range which emphasizes the interaction between mutual pairs rather than with ensembles and between them. Ions oscillate in a plasma as well. According to Eq. (3.57), however, they will move at considerably lower velocities and lower frequencies. For argon with a plasma density of 1010 cm−3 , we find ωp,i = ωp,e me /mAr = 1/300 ωp,e ≈ 20 MHz. Some authors make these oscillations responsible for the striations in the positive column (cf. Sects. 3.1, 4.5 and 7.5).
4 DC discharges
The behavior of a DC discharge is first described in a phenomenological sense, which is followed by the avalanche theory of carrier generation. Next, the issues of energy gain in the cathodic zone and charge generation in the negative glow are dealt with. For breakdown, these processes of charge generation must equal the carrier losses which are the focus of the next section. The features of the positive column, in particular diffusive processes, and the anode zone are topics of the following section. Some special DC discharges are briefly mentioned, and eventually, the similarity rules are discussed.
4.1 Introduction Let’s start with secondary electrons which have been released by γ-processes at the cathode. These secondary electrons leave the cathode with very low energies of only several electronvolts, but are considerably excited across the adjacent electric field of the dark spaces. Within a very short distance (Aston’s dark space), they have accumulated enough energy to excite atoms, which causes the sharply defined cathode layer. As expected, the lines of lower exitation energy are closest to the cathode. Farther away from the cathode, the electrons have already gained sufficient energy that they can easily perform multiple ionizations, which causes an avalanche of ionization in this part of the dark space called Crooke’s dark space. Simultaneously, the electric field diminishes steeply until it has almost entirely vanished when the electrons enter the negative glow. The main physical feature of the adjacent positive column is the diffusive loss of the charged particles, therefore, these issues are discussed in this section. Eventually, with a small increase in potential, the anode zone is reached. In the following section, the possibilities of ionization at the cathode and in the cathodic dark space are discussed, applying Townsend’s approximation. According to this theory, at the surface of the cathode electrons cannot be generated by electrons. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 4,
69
70
4 DC discharges
4.2 Gaseous generation of carriers 4.2.1 Townsend’s equation There remain the two possibilities that • electrons between cathode and anode can generate secondary electrons in the gas phase (so-called α-electrons) and that • ions incident on the cathode can generate γ-electrons by bombardement of the cathode. With the simplifying assumptions that • the electric field drops linearly between cathode and anode, that • the ion flux is mobility-limited and that • secondary electrons generated on the surface of the cathode do not exhibit any energy distribution but show beam-like character, we can apply the ionization theory developed by Townsend [110, 111]. We denote α and γ the first or primary and the second or secondary Townsend’s ionization coefficient, respectively, and Γe (x) the electron flux. Thus, the generated electron flux yields dΓe (x) = αΓe (x)dx ⇒ d ln Γe (x) = αdx ≈ e0 Φn /Eion ,
(4.1)
Γe (x) = Γe (0) eαx ;
(4.2)
hence with Γe (0) the flux of secondary electrons at the cathode, Eion the ionization energy, and Φn the so-called normal cathode fall. At each successful electron impact, a pair of charges, electrons and ions, is generated with current density je (0)(eαx − 1), which are attracted by the cathode where they release γΓe (0)(eαx − 1) electrons of the second generation ad infinitum (Fig. 4.1).1 At the anode, the current of electrons has grown to (with d the distance between the two electrodes, j = e0 Γ): je (d) = je (0)eαd + γje (0)(eαd − 1)eαd + γ 2 je (0)(eαd − 1)2 eαd + · · · ,
(4.3.1)
or 1 This derivation neglects collisions of the ions. In fact, the ions incident on the cathode experience several collisions, since at pressures between 5 and 100 mTorr (1 and 13 Pa), the ionic mean free path λi amounts to values between 1 mm and 5 cm. Although we neglect ionizations by ionic impact in the gas phase (so-called β-processes), the mean energy of the ions is reduced by these collisions, and this reduces the impact at the electrode.
4.2 Gaseous generation of carriers
-
+
71
Fig. 4.1. Schematic representation of the generation of an avalanche of charged carriers by electron impact. LHS: cathode, RHS: anode.
je (d) = je (0)eαd (1 + γ(eαd − 1) + γ 2 (eαd − 1)2 + · · ·),
(4.3.2)
which represents an infinite geometric series, with sum for γ(eαd − 1) < 1, je (d) =
je (0)eαd je (0) =M 1 − γ(eαd − 1) 1 − γ(M − 1)
(4.4)
with M = eαd .2 This is the common expression for the current growth in a Townsend’s discharge, provided the current density is small (≈ 10−10 A cm−2 ) and the diffusion losses remain negligible [112]. For increasing voltage between the electrodes, both the coefficients α and γ will increase, the denominator in eq. (4.4) approaches zero and can even become negative. At any rate, the current rises sharply, and the striking of the plasma is connected with gas breakdown, or simply breakdown. In fact, ignition of the plasma happens at lower voltages since a positively charged layer is generated in front of the cathode, which leads to a steep increase of the electric field. The characteristics of the discharge changes fundamentally: A feature of the Townsend’s discharge is a constant electric field or a linear rise in potential between cathode and anode. Now, we observe the formation of a positively charged layer in front of the cathode. Due to this field, secondary electrons are accelerated away from the cathode, and the electron density falls to very low values but remains finite (Fig. 3.1). After ignition, the production of secondary electrons at the cathode is sufficient to sustain the discharge: The loss of ions is balanced by released secondary electrons which, on their course to the anode, generate this very number of ions. Hence, the current has become independent from its external cause, and we can write down the condition for stationarity of a self-sustained normal discharge in 2 The electron current will be amplified by M times and would alone not satisfy the continuity equation. However, every ionization impact in the plasma creates the same amount of electrons and their counterparts, and therefore, the continuity equation is valid for the total current. Due to their small mobility, the ions arrive the cathode with a certain time lag.
72
4 DC discharges
Townsend’s approximation [vanishing denominator in Eq. (4.4)], the so-called condition for stationarity:
1 αd = ln 1 + . γ
(4.5)
γeαd = γM = 1 :
(4.6)
Since γ 1, this yields
αd
αd
e positive ions release γe secondary electrons, or that number of ions + 1 ion, which are generated by one electron. With Eq. (4.1), we obtain for the normal cathode fall: Eion Φn = e0
1 1+ . γ
(4.7)
4.2.2 The primary ionization coefficient Provided that no pressure dependent phenomena play any role in the ionization, the ratio between electric field and pressure, E/p (which scales at constant gas temperature with the ratio between electric field and number density, E/n), determines the energy gain of the electrons between two collisions. Since the mean free path of the electrons, λe , is inversely proportional to the number density n, we find as an additional condition (in the following, we use ε instead of E for the energy to avoid confusion with the electric field):
< ε >= f
e0 E p
=f
e0 E n
= f (e0 Eλ).
(4.8)
The dependence on α, the number of ionizations per cm in the direction of the field, must be the same; in addition, α depends on the collision rate per cm (at constant temperature, p ∝ n): α α = pf (e0 Eλ) ⇒ = f (e0 Eλ) = f p
e0 E . p
(4.9)
According to Eq. (4.6), the product of γ and the multiplication factor M is constant. Inserting α using Eq. (4.9) yields
γ exp pd f
E p
= γ exp p df
Vb pd
= 1.
(4.10)
This requires for fixed pd the breakdown voltage Vb to remain constant as well: Vb is a single function of the product between the discharge pressure p and the electrode distance d (for Tgas = const : p ∝ n, Paschen’s law). For large values of E/p, the electrons reach the anode without having collided, i. e. without any chance to ionize, whereas for small values of E/p, the
4.2 Gaseous generation of carriers
73
2000 1600
Vb [V]
1200
hydrogen
800 air
400 0 0
Fig. 4.2. Paschen curves for hydrogen and air after [113]. 5
10 15 pd [Torr mm]
20
25
30
distance between two collisions is too short to gain that amount of kinetic energy which is sufficient for ionization. These counteracting dependencies lead to a relatively broad minimum which can be located at d ≈ λe , the so-called Paschen minimum (Fig. 4.2). α is often empirically approximated by
Bp α Ap (4.11) = · exp − E E E with A and B properties which are characteristic for the gas under consideration (Fig. 4.3). η=
ln a/E
A/eB
B
Fig. 4.3. Characteristic of α/E vs. E/p in double-logarithmic scale. The maximum can be found for the argument B and the value A/Be for the condition α/E = Ap/E exp(−Bp/E) (after [114]).
ln E/p
The decrease in α/E for small arguments is closely related with the excitation of higher electronic states (instead of “successful” ionizations); for large values of E/p, however, the increase in kinetic energy leads to a significant reduction of the ionization cross section. Deviations from this conduct can often be referred to as the Penning effect. To make this visible, it is common to plot η = (nσ)/E vs. E/p (Fig. 4.4). Since the electric field decreases almost entirely across the cathode fall, d in Eq. (4.5) equals the thickness of the cathodic dark
74
4 DC discharges
space, and it is assumed to begin where the field vanishes when the electrons enter the negative glow, neglecting the thin Bohmic presheath [115].
Ne+10-2 % Ar
104 a/E [10-6 V]
Ar
Ne+10-3 % Ar
Ne+ 10-4 % Ar
Fig. 4.4. Enlargement of α by collisional impact with metastables: Penning ionization Ne∗ +Ar → Ne+Ar+ +e− c [70] ( Review Modern Physics).
103 Ne
102
Ar
1
10
102
103
E/p [V/Torr cm]
4.3 The normal cathode fall 4.3.1 The secondary ionization coefficient Attraction of the ions by the cathode and mutual repulsive forces of the ions determine the thickness of the cathodic dark space. Therefore, the dark space is expected to shrink with rising pressure (n ∝ p). Expressed with the condition for stationarity, the thickness of a normal cathode fall, dn , increases for rising gaseous ionization or α-ionization or lessening surface ionization or γ-ionization. From Eq. (4.7), we can see that the height of the cathode fall, Φn , increases with rising ionization potential of the plasma-constituting ambient. Furthermore, γ plays an appreciable role in this dynamic equilibrium by which the height of the cathode fall is eventually fixed. Some qualitative approaches exist regarding Φn as a function of the work function of the electrode material for the same plasma ambient, since electrodes with the lowest work function Wa exhibit the lowest cathode falls (which is similar to gases with the lowest ionization potential or largest cross section for ionization by electrons). However, it seems quite likely that Φn cannot show a simple dependence on Wa because Φn is also determined by γ, which depends on the material of the cathode, its geometry and its surface quality and from the gas ions. Equations (4.1) and (4.7) seem at last to supply us with a satisfactory relation for the thickness of the normal cathode fall, which confirms the preceding assumption:
1 1 . · ln 1 + pdn = αp γ
(4.12)
4.3 The normal cathode fall
75
The thickness of the dark space increases in size for rising α-ionization or falling γ-ionization. The breakdown condition opens an experimental window to crosscheck γ. For this end, we take the logarithm of the condition for stationarity γeαd = 1, and express the exponent by αd = α/E Ed or α/E Vb which yields ln
1 1 = ln Vb − ln . γ η
(4.13)
Plotting the difference of the logarithms of Vb and 1/η vs. the logarithm of E/p yields the reciprocal secondary coefficient γ (Fig. 4.5).
air, brass 5
10
4
F [V]
10
1/h
Ne, Fe
Vb
103
Vb
2
10
Fig. 4.5. Values for the breakdown voltage Vb and 1/η for a parallel-plate capacitor for air and neon after [116].
1/h
1
10 0 10
101
102 E/p [V/Torr cm]
103
4.3.2 V-I characteristic After ignition, the initial linear potential drop between the electrodes will be distorted by the occurence of space charges, and the field cannot be regarded as homogeneous. Therefore, the current density between the electrodes cannot be determined with the simple equations
j = ρv = e0 nv ∧ v =
2e0 2e0 ⇒ j = e0 n , m m
(4.14)
because it has become spatially dependent. To describe these new features of the discharge, in particular the cathodic dark space, which happens to become the layer with the steepest potential drop, besides the condition for stationarity [eq. (4.5)], a relation between • current density of the ions • height of the cathode fall, and
76
4 DC discharges • the thickness of the dark space
is required. 4.3.2.1 Matrix sheath. To get an idea what will happen if we neglect the continuity equation, we consider the ion density to remain constant across the dark space. According to the (one-dimensional) Poisson equation, we find e0 n0 x e0 n0 dE ⇒ E(x) = = dx ε0 ε0 and the potential Φ turns out to depend on the distance squared:
(4.15.1)
e0 n 0 2 x, (4.15.2) ε0 which yields a square-root behavior between the thickness of the dark space, dc , and the potential drop Φ: Φ(x) = −
dc =
2ε0 Φ, e0 n0
(4.16.1)
2e0 √ Φ: kB Te
(4.16.2)
or, in terms of the Debye length,
dc = λD
The thickness of this matrix sheath would exhibit an extension of many Debye lengths. 4.3.2.2 Child-Langmuir sheath. To meet the requirements for continuity, the current density of the ions must drop with increasing velocity.3 This can be caused either by space charges in the high-vacuum regime (case 1) or by friction in the high-pressure regime (case 2), depending on the ratio of the mean free path of the ions over the thickness of the dark space: λi < 1 : mobility limited dc λi > 1 : space charge limited dc The relation between these three properties was first derived by Child in the year 1911. Other derivations are due to Langmuir (1912) and Schottky (1914). 3 From the sheath criterion, the electron density should drop even more, with the remaining electrons accelerated to 0.1×c or more. Both facts are responsible for the low level of emissivity. Few but rapidly moving electrons will lose their energy by exciting neutrals to levels which are far beyond the optical levels.
4.3 The normal cathode fall
77
The ion current leaving the quasineutral negative glow, which exhibits a vanishing electric field (dE/dx)x=0 = 0) and is incident on the cathode, is given by j = ρv, where v is calculated via the electrode potential e0 Φ = 1/2m0 v 2 . The Poisson equation can be written as d2 Φ C = −√ dx2 Φ
(4.17.1)
with C=
ε0
j , 2e0 /m0
(4.17.2)
and C < 0 since we move from zero potential downward to the cathode potential. If the first derivative of Φ is regarded as a function of Φ (in many textbooks, this is called multiplication by dΦ/dx): d2 Φ dF dΦ dF dΦ = F [Φ(x)] ⇒ =F , = dx dx2 dΦ dx dΦ this equation can easily be separated yielding
F
C dF =√ ⇒ dΦ Φ
F dF =
√ C √ dΦ ⇒ 1/2 F 2 |Φ 0 = 2C Φ + A. Φ
(4.18)
(4.19)
Assuming dΦ/dx to vanish in the negative glow it follows that A = 0, and repeated integration yields √ √ dΦ 4 3 d = 2 Cdx ⇒ Φ /4 |Φ 0 = 2 Cx |0 , 3 Φ1/4
(4.20)
and finally for Φ and j:
3
9 4 j 2e0 Φ /2 Φ /2 = d2 ∧ j = ε0 . 4ε0 2e0 /m0 9 m0 d2 3
(4.21.1)
Following Bohm, the maximum ion current is given by eqs. (14.43/44). Inserting this relation into Eq. (4.21.1), we get finally for the sheath thickness (with n the plasma density): 2 d= 3
ε0 n
4
3 2e Φ /4 . e0 kB Te
(4.21.2)
This is the high vacuum version of Child’s current density equation (Fig. 4.6). As an example, for an electron energy of 4 eV, a plasma density of 1010 cm−3 and a potential Φ of 100 eV we obtain a sheath thickness of d = 2.06 mm. For the same plasma parameters, we get a Debye length of only 0.16 mm. For higher pressures, the current leaving the negative glow is mobility limited: j = ρu = ρμE, and the mobility is supposed to be independent of the electric field (with u the drift velocity).
78
4 DC discharges 300
300 0.1 mA/cm2 1 mA/cm 2 10 mA/cm2
0 mA/cm 2 10 mA/cm2
200 F[V]
F[V]
200
100
100
0 0.00
0.25
0.50
0.75
1.00
0 0.00
0.25
0.50
0.75
1.00
distance [cm]
distance [cm]
Fig. 4.6. Space charge limited current between two plates, one of which acts as electron source. LHS: various current densities, RHS: attenuation of the potential by ions in the sheath adjacent to the electrode in comparison to a linear potential drop.
j d2 Φ ρ j =− . =− =− dx2 ε0 ε0 u ε0 μE The separation is easily performed with E = − dΦ dx and the product rule
(4.22) d dx
dΦ dx
2
=
2 2 ddxΦ2 dΦ dx :
dΦ 1 d 2 dx
2
dΦ j dx ⇒ = = ε0 μ dx
2j √ d, ε0 μ
(4.23)
and finally for Φ and j: 2 Φ= 3
2j 3/2 9 Φ2 d ∧ j = ε0 μ 3 . ε0 μ 8 d
(4.24.1)
This is the mobility limited version, which is known in the solid state (with finite ε) as Mott-Guerney equation. Inserting Eq. (14.44) into Eq. (4.24.1), we get finally for the sheath thickness in the mobility-limited case
d=
9 3 8 ε0 μ
n0 e0
1 / 6 e 2 mi
kB Te
2
Φ /3 .
(4.24.2)
• In the high vacuum version, the potential of the space charge scales with 4 1 3 x /3 , the field with x /3 , and the current density with Φ /2 . • In the high pressure version (drift velocity is limited by collisions) the 3 1 potential is proportional to x /2 , the field proportional to x /2 , and the 2 current density proportional to Φ .
4.3 The normal cathode fall
79
1.00 linear rise in potential space charge limited current mobility limited current
1.00
0.75 0.75
V/Vc
jnorm
0.50
0.25
0.00 0.00
space charge limited mobility limited homogeneous
0.50
0.25
0.25
0.50
0.75
0.00 0.00
1.00
0.25
0.50
0.75
1.00
Fnorm
d1/dc
Fig. 4.7. LHS: Spatial dependence of the potential for some different distributions of the space charge: The difference is very minute between a - linear potential drop without any screening and a - potential drop which is attenuated by a current which is either - space charge limited or - mobility limited. RHS: The same holds true for the cathodic ion current density: Compared to the 1 limit of the homogeneous field which assumes a vanishing charge density (j ∝ Φ /2 ), the differences between the two cases are too minute to be detected, here pictured for normalized properties. The Ohmic law simply scales with Φ1 .
In both cases, the dependence between current density and potential considerably differs from the case of a homogeneous field with a very low charge density, where we find the current density j to depend on the square root of the potential Φ. In particular, the charge density (and consequently the current density) are spatially independent. However, it was Ingold [117], who 3 pointed out that the assumed differences between these possibilities (Φ ∝ d /2 4 for mobility limited current, Φ ∝ d /3 for space charge limited current would be far too small to be distinguished experimentally; even a linear field (Φ ∝ d1 ) would be difficult to confirm [118], Fig. 4.7). Furthermore, the mean free path of the ions, λi , is comparable to the thickness of the cathodic dark space, dc , and we eventually conclude that neither of these equations is correct (at least in principle). In the case of the mobility limited current density, the current density at the surface of the cathode (u = μE) would yield j = ε0
9μ Φ2n 8 d3n
(4.25)
with the plasma regarded as emitting electrode and the cathode as collector electrode with vanishing electric field at the boundary of the cathode fall.
80
4 DC discharges
For the thickness of the dark space at an isolating wall which exhibits a floating potential of typically Φf = −10 V, we can calculate for a collisionless dark space dc to be about 0.2 mm: The dark space is invisible! Becoming more precise, we can express the thickness of the dark space in terms of the Debye length with eqs. (3.15) and (4.24.1) for a one-dimensional Maxwellian distribution of the ions (ion current incident on the electrode). We use Child’s equation to calculate the positive ion current neglecting the electron space charge within the sheath:
3
2e0 Φ /2 4 ε0 = e0 ni vB . 9 m0 d2c
(4.26)
Considering the velocity of the ions at the Bohm edge [cf. Eq. (14.30)] this yields √ d2c =
√ 3 3 32 e0 Φ /2 2 2e0 Φ /4 × λ2D ⇒ dc = × λD , 9 kB Te 3 kB Te
(4.27.1)
or
3
Φ /2 λD : (4.27.2) Φf The cathodic sheath thickness must be considerably larger than the Debye length. Furthermore, we draw the following conclusions from this discussion: dc ≈
• For vanishing je (0) also je (d) becomes zero: To trigger an electron avalanche some free electrons are required. To facilitate the ignition, plasma reactors are often equipped with a filament which emits electrons thermally. • For a self-sustaining discharge, a certain initial electron density must be available. Once the discharge is ignited, the ion current drops by diminishing the discharge voltage, and the required ion current density is provided by radial shrinking of the negative glow. The minimum voltage required to release a sufficient amount of γ-electrons is denoted normal cathode potential Φn . In these discharges, current density and height of the cathode fall remain constant even when the current rises by some orders of magnitude. That is accomplished by enlarging the cross section of the current path until the whole electrode is covered. Now, further rising current leads to an avalanche of increasing current density which will trigger a higher efficiency of γ-processes which, in turn, is connected with a further rise in potential. It is these abnormal cathode falls which are used extensively for sputter deposition and plasma etching. Since the thickness of the sheath depends on flux and potential drop, the characteristics of an etching process or a deposition process can gradually or
4.4 The abnormal cathode fall
81
Table 4.1. Calculated values of properties which characterize an abnormal discharge through argon [122]. energy [eV] 15.8 16 20 30
pd1 [mTorr cm] 319 224 61 27
pd2 [mTorr cm] 323 233 75 53
pdc [mTorr cm] 1545 1103 353 218
α/p [cm−1 mTorr−1 ] 34.8 49.2 157 258
Φc [V] 127 128 147 195
suddenly change when we alter one of these determining properties. Although derived for a high-voltage DC sheath, this model is appropriate for employment in the treatment of the RF sheath. This has been extensively shown by Riemann and coworkers during the late 1980s who also incorporated an energy distribution of the ions for his derivation [119, 120]. For an arbitrary ratio between dc and λi , they obtain a formula which unifies the approaches of a space-charge limited current with a mobility-limited current according to
9 12π dc j d2 × 1 + Φ /2 = 4ε0 2e0 /m0 125 λi 3
2
.
(4.28)
4.4 The abnormal cathode fall As we have previously noted, the current density and the thickness of the dark space depend on the height of the cathode fall in this type of discharge. For ionization in the dark space, we require the mean free path of the electrons, to be short compared with its thickness. By this scattering process, the beam-like, monoenergetic characteristic of the γ-electrons is terminated; the electrons exhibit a certain energy distribution, hence, α is not constant any more. Furthermore, conditions are required to calculate the balance for particles, momentum and energy which depend on the number of collisions between electrons and neutrals. As invariant properties, j/p2 and pd exhibit the following dependencies on the abnormal cathode fall [121]: ⎫
j/p2 ∝ Φc3/2 : space charge limited current; ⎪ ⎪ ⎪
j/p2 ∝ Φ2c : mobility limited current; pd ∝
1 αd 1−
Φ Φc
ln 1 . γΦc
⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭
(4.29)
Using these equations, the numbers in Table 4.1 are calculated for a collision frequency νm of 6.6 × 109 sec−1 and a γ of 0.1.
82
4 DC discharges
After having fallen across the (reduced) length pd1 , the electrons are accelerated to energies which equal the ionization potential or even larger. This length is insignificantly shorter than pd2 , which defines the upper limit of enery accumulation due to elastic collisions (mobility limit). On the other hand, pd2 is considerably shorter than pdc , about a factor of 5 just beyond the ionization threshold to drop to about 1/10 at twice the ionization potential; all the length which characterizes the dark space declines with rising potential, in particular, the thickness of the dark space [123]. Its thickness roughly equals the ionization length for the γ-electrons to trigger an ionization. This model gives a rather good description of the experimental data up to the Paschen minimum as communicated by v. Engel [124]. Deviations are mainly caused by the approximation for α (as just mentioned) and its energy dependence (in particular for rapidly moving electrons), but also by the heating of the gas, by which the density is reduced; hence, pd (pd ∝ nd) is no longer an invariant property. Furthermore, Φc begins to rise in the negative glow; therefore, the thickness of the dark space is difficult to fix although the borderline for high cathode falls becomes sharper and better defined. Its length varies between 4 and 10 ×λe [125]. Hence, we can clearly distinguish electrodes which sustain a DC discharge from additional surfaces with adjustable potential. The thickness of the dark space of these electrodes can be calculated using the sheath equation (cf. Sect. 14.2) or by Child’s equation. This difference is mainly caused by the release of γ-electrons which have emerged as a constitutive part of a self-sustained DC discharge. Irrespective of whether the electric field does depend on the thickness of the dark space, for rising γ or increasing current density and cathode fall, the sheath thickness becomes thinner, which leads to a striking increase of the field strength (Fig. 4.8).
10
-1
0.5 j/p
2
0.3
-2
10
-3
dp
0.2 0.1 0.0
10
250
500
750 1000 VC [V]
1250
1500
j/p 2 [A cm -2 Torr -2]
dp [cm Torr]
0.4
Fig. 4.8. Reduced thickness of the dark space pd and reduced current density j/p2 as function of the cathode fall in nitrogen (iron cathode). The values for high voltages have not been corrected by the falling c Oxford gas density [126] ( University Press).
4.4 The abnormal cathode fall
Plotting the product pd which follow
83 j/p2 vs. Φc yields the straight lines in Fig. 4.9,
d j = A + B (Φ − Φ0 ),
(4.30)
an equation which was reported for the first time by Aston [127].
d j 1/2 [10-4 A 1/2 cm -1]
5 H2
theoretical experimental
4
He
3 N2
2 He
H2
Kr
N2
1 CO
0
CO
200
400
Kr
600 800 Fc [V]
1000 1200
Fig. 4.9. Reduced thickness of √ the dark space p d j as function of the cathode fall for several gases. j is calculated according to the space charge limited Child’s equation [127] c Springer-Verlag). (
4.4.1 Discussion of Townsend’s approximation Townsend’s theory postulates the existence of two electron sources: By electron impact, the gaseous constituents are ionized (α-electrons), by ionic impact, electrons are released from the surface a solid (γ-electrons). It should be mentioned again that Townsend’s equation has been derived but for a dark discharge. Effects in other discharges may come to be better understood, but this model cannot be strictly applied to other discharges. This is shown by an example by Davis and Vanderslice [128]: Example 4.1 For a DC discharge through argon, DC voltage: 600 V, at a pressure of 60 mTorr (8 Pa), which means a density of the neutrals of 2 × 1015 cm−3 , the ionization cross section at 100 eV is for electron impact 3 ˚ A2 , for ion impact 0.5 ˚ A2 , 2 ˚ and the total cross section equals 30 A , we calculate an electronic mean free path λe of 1.67 mm, and a thickness of the dark space dc of 1.3 cm. This means for the two tracks of ionization: 1. Ionization by electron impact: The upper limit of ionization is then about 1.9, i. e. at a yield for secondary electrons γ ≈ 0.2, the yield of generation equals 0.2 × (1.9 − 1) = 0.18 because it is a two-step process. 2. Ionization by ion impact: yield of 1.12 ⇒ a maximum rate of 1.12 − 1 = 0.12.
84
4 DC discharges
Albeit σi is smaller than σe by approximately one order of magnitude, the yields are almost equal because of the low density of the secondary electrons. With the supposed thickness of 1.3 cm, every electron would be able to ionize just one neutral species, far too low to sustain the discharge.
• Therefore, the thickness of the dark space should be considerably larger since the yield for secondary electrons does not depend strongly on energy. Also the cross section for ionization is considered to have reached its maximum. No doubt that both the mechanisms just discussed are important, however, they are not sufficient to sustain a discharge. • Moreover, the current density in a normal discharge is between 10−5 and 10−2 A cm−2 , which increases to 10−2 A cm−2 to peak at about 0.5 A cm−2 in an abnormal discharge. Thus, there are considerable carrier losses due to ambipolar diffusion (cf. Sect. 4.7). Eventually, the fraction of electrons released by the photoelectric effect is assumed to be of comparable size with that due to ion bombardement [129]: γi ≈ γhν . • At the anode, an additional ionization must take place to maintain the ion current. Hence, an effective ion source is required either in the negative glow or in the anodic dark space. The height of the anode fall remains in the order of the ionization potential—far too low for further discussion concerning this topic. It remains the negative glow that has been the subject of the scientific discussion for decades. For example, Little and v. Engel could not suppose the negative glow to be the region of any carrier generation [129]. This assumption was grounded on the fact that the only collisions which change the velocity of the ions incident on the cathode are due to resonant charge transfer: + A+ rapid + Aslow −→ Aslow + Arapid .
(4.31)
In most cases, elastic collision cause only small-angle scattering, by which the velocity of the ions is not significantly changed. According to a model developed by Scherzer, all electrons are generated in the negative glow [130]. As Druyvesteyn and Penning had pointed out, the equation of stationarity [Eq. (4.5)] had to be extended to
αdc = ln
1 1+ γ
(1 − β)
(4.32)
with β the quotient between the current densities of the ions and the electrons, respectively ji /je [131]. If ji were a considerable fraction of the total current, α could become significantly smaller. They distinguish between a normal and an abnormal discharge. In the latter discharge, most of the ions are generated in the negative glow, whereas in the first discharge, the main area of ion production is shifted into the region of
4.5 Negative glow and positive column
85
energy transfer, i. e. the cathodic dark space. This is mainly caused by its shrinking width compared with the mean free path of the electrons, λe , but it is also due to the smaller ionization cross section at larger energies (cf. Sect. 2.2). Above all, Townsend’s approximation has been derived for a homogeneous field. This is definitely not the case for the cathode fall which, on the contrary, is extremely inhomogeneous [here, values for E/p of up to 105 V/(Torr cm) across 100 μm in length can occur]. Therefore, α does not depend solely on E/p, but also on the thickness of the dark space: E = E(d).
4.5 Negative glow and positive column Although the negative glow, regarded as a whole, is quasineutral, it is by no means uniform; this is mainly caused by the electron bombardement from the cathodic dark space. Traditionally, three classes of electrons are distinguished, in a DC discharge, their ratio amounts to about 1 : 10 : 1 000 [132]: • Primary electrons of high energy (in fact, these are the γ-electrons ejected from the cathode). • Secondary electrons which are generated by α-processes (electrons → neutrals) or β-processes (ions →neutrals) with energies which are considerably lower. • Thermalized electrons which are decelerated down to the plasma temperature, so-called last electrons, they form the greatest part of these three fractions. When entering the negative glow, the first two groups will be considerably slowed down due to multiple excitations (optical levels and ionizations) which causes the intense glowing at the head of this zone. At these high energies (some hundreds or even thousands of eV), however, the cross section for both these excitations has dropped to very low levels. Hence, the intensity of the glow does not peak at the border to Crooke’s dark space but is somewhat shifted into the negative glow, until the kinetic energy has been decreased to values of about 100 − 150 eV. That is why some primary electrons can pass the negative glow and reach the anode (cf. Sect. 10.2). All the qualities which are characteristic for a beam are lost in the negative glow; a small negative space charge is generated at the anode which causes the formation of a weak electric field which, in turn, drags the electrons into Faraday’s dark space. After having been accelerated across a sufficient distance, they have gained enough kinetic energy to excite neutrals to optical levels or to ions once again which is manifested by the head of the positive column. Since the beam qualities were already lost in the negative glow, this head appears washed out. The cathodic dark space and Faraday’s dark space differ mainly
86
4 DC discharges
by the mean energy of the electrons. In the first, the mean energy is far too high to excite molecules to optical levels, in the latter, the mean energy is insufficient for this excitation. The field of the positive column adapts to that value that ensures a constant density of carriers, i. e. the rate of generation equals the rate of loss. Often, zones of different glowing intensity are observed in the positive column, so-called striations which many authors consider repetitions of the ionization processes in the head of the positive column. From their distance, the mean free path for optical excitation can be inferred. Morgan, however, interprets these striations as nodes and crests of ionic oscillations [133]. Both of these zones are scenes of prototypic processes. In the negative glow, most of the ionization will happen, whereas the positive column is best suited to explain diffusion losses.
4.6 Ionization Due to their high kinetic energy, primary electrons can perform multiple ionizations. But even if the cross section for ionization is supposed to be at maximum (for argon about 29 ˚ A2 at 100 eV) and we extend the negative glow to an unusual length of 5 cm, a rough estimation reveals that only 5 % of charged carriers are generated which are required to sustain the discharge [134]. This is due mainly to the low energy of the electrons which are generated by the ionization impact because they are hardly accelerated by the weak electric field in the negative glow by which the avalanche is readily terminated. Since the exponential term is lacking, the ionization is solely determined by the product je × σ × nAr . Assuming a Maxwellian thermalization of the electrons around the temperature Te = 2 eV/kB and the condition for reaching the electrodes • cathode: E ≥ e0 (Φp + Φcathode ), • anode: E ≥ e0 Φp , the probabilty for a thermalized electron to reach the cathode becomes zero whereas the anode will be reached with a probability of 1%; i. e. thermalized electrons are almost completely trapped in the negative glow, unless they do not get lost by wall reactions. 4.6.1 Ionization in the negative glow This energy between 2 and 8 eV is definitely insufficient to ionize argon, for its first ionization potential is 15.7 eV. For a Maxwellian distribution of electrons, we can calculate the fraction of electrons with energies larger than 15.7 to 0.1 % at a mean electron energy of 2 eV, which rises to 28 % for a mean energy of 8 eV. The generation rate GR of ions (in fact the ionization frequency νion )
4.6 Ionization
87
dnA+ = GRion = kion nA dt is proportional to these quantities:
(4.33)
• The ionization cross section σion for electronic impact. • The energy E of the electrons. • The fraction of electrons with energies f (E) larger than 15.7 eV. • The density of the neutrals nA : GRion =
∞ Eion
nA σion (E) E f (E) dE
(4.34)
with f (E) the normalized EEDF. The rate coefficient kion is obtained by dividing GRion by the density of the neutrals (Sect. 2.3.5): kion =
∞ Eion
σion (E) E f (E) dE.
(4.35)
Provided that the cross section for ionization by electron impact can be linearized close beyond the threshold Eion , σion (v) = a(Φ − Φion ),
(4.36)
with a = 4.05/cm mTorr K for argon [135], which is a very good approximation for the negative glow, the generation rate can be calculated. When we apply analytic expressions for f (v), the integration is made possible in closed form (Sect. 14.1). For a plasma density of 1010 cm−3 and a typical pressure of 50 mTorr (7 Pa) we can calculate the yield for electrons as compiled in Table 4.2. Example 4.2 For a cathodic current density of 1mA cm−2 and a γ of 0.2, the electron current at the edge of the negative glow can be calculated to 0.2 mA cm−2 or 3.5×1015 electrons per cm2 sec. For a length of the negative glow of 3 cm, 1015 electrons per cm2 sec have to be generated. This task can be accomplished by Maxwellian electrons exhibiting a mean energy of 2 eV. Electrons with a temperature slightly higher can also compensate the losses which are caused by diffusion and recombination. {A similar result was obtained by Winters et al. for the generation of ions in a discharge through CF4 (ionization potential of CF4 : 16 eV) [136]}.
Although numerous experiments have shown that the electrons in the negative glow are not distributed according to a Maxwellian distribution, inspection of Table 4.2 clearly shows that a pure Druyvesteynian distribution does not describe the facts as well. This is known as Langmuir paradoxon and has been explained by electron trapping and generation of plasma oscillations (Sects. 14.1 and 14.3). Others supposed these differences to be induced by the measuring technique (artifacts of the Langmuir probe technique) [137]. This
88
4 DC discharges
Table 4.2. Ionization of argon by electrons with varying mean energy < E > as obtained by application of two different distribution functions (Maxwell and Druyvesteyn). E [eV]
Eion / < E >
8 6 5.3 4 3 2.5 2 1.5 1
2 2.6 3 4 5.3 6 8 10 15
GR [cm3 sec−1 ] Maxwell Druyvesteyn 2.72 × 1017 9.8 × 1015 17 1.41 × 10 2.0 × 1014 16 9.96 × 10 5.0 × 1013 3.64 × 1016 1.4 × 1010 1.13 × 1016 4.30 × 1015 1.04 × 1015 1.01 × 1014 1.05 × 1012
could be definitely excluded by comparison of data obtained by the Langmuir probe with data gained with the non-invasive optical emission spectroscopy [98]. The main zone of carrier generation is the negative glow, the main zone of energy gain is the cathodic dark space. The energy is shared among ions and electrons. By ionic impact on the cathode, particles are ejected, by γ-processes, electrons will be released and some other processes will occur (cf. Chap. 10). The generated γ-electrons are accelerated by the cathode fall and gain energy until they enter the negative glow.
4.7 Loss of carriers These processes of generation of carriers are exceeded by losses (still no plasma) or just evened out by losses (breakdown or ignition). The main mechanisms for loss are recombination of charged carriers, and diffusion. Although these mechanisms are important, we have already seen that breakdown, for example the Paschen minimum, can be explained without having an idea of the explicit knowledge of how the electrons are lost. Although both mechanisms are of importance, at low pressures, diffusion dominates, and for pressures higher than about 10 Torr (1 333 Pa), recombination takes over [138]. The pure mechanisms can easily be distinguished since recombination scales with carrier density squared, which simply explains its range of validity in the high-pressure range, whereas in the diffusion regime, losses are exponentially proportional to the carrier density. Recombination takes place in the plasma bulk, diffusion means leaving the plasma bulk and reaction with the wall (surface) of the reactor
4.7 Loss of carriers
89
(heterogeneous reaction). In this section, we limit our interest to diffusion with vanishing field. 4.7.1 Free diffusion We consider an electropositive plasma with ni = ne = n. It is only the electrons which can act as ionizing species. As we have seen in Sect. 4.2, ionization by electron impact is an avalanche process, and the number of ionizations, i. e. the number of charges of the next generation n , is proportional to the number of parent charges n, the density of the gas nn , and a bimolecular rate constant k2 which is connected with the frequency of ionization via the generation rate GR GR = k2 ne nn ⇒ k2 =
νion , nn
(4.37.1)
and it is shown here for electrons: dne = νion ne . (4.37.2) dt Losses in the plasma bulk are recombinations and are neglected for the pressure range considered here. For slow changes, we can start with the extended continuity equation [Eqs. (4.38/39)] and Fick’s first law [Eq. (4.40)]: ne = ne eνion t ⇒
∂n ∂n (4.38) + ∇ · Γ = 0 −→ + ∇ · Γ = νion n, ∂t ∂t and for the steady case, all charges that are generated in the volume are lost by diffusion processes through the (imaginary) walls, which confine the plasma, so we can drop the first term on the LHS to obtain ∇ · Γ = νion n.
(4.39)
Γ = −D∇n
(4.40)
Remembering that
we obtain the three-dimensional Helmholtz equation [139] νion n=0: (4.41) D The amount of generated carriers equals the amount of lost carriers, and the generation of new carriers is proportional to their number. We have introduced the three-dimensional diffusion coefficient, D, which is well-known from kinetic gas theory where we have derived a relationship between the mean square distance < x2 >, mean velocity < v >, the mean free path λ, and the reciprocal time between two collisions (1/τ = νm , random walk) ∇2 n +
90
4 DC discharges
1 < x2 > 1 = λ . (4.42) 3 τ 3 It differs for electrons and ions by some orders of magnitude. In a plasma, it is regarded as spatially constant. For a rectangular box, this boundary-value problem yields D=
πx πy πz cos cos , (4.43) I J K with maximum density at the origin [ne (0) = ne,0 ] and vanishing at the walls, i. e. for x: ne = ne,0 cos
I ∂ne = 0 at x = 0 ⇒ ne = 0 at x = ± , ∂x 2 and the diffusion length D 1 ∨ = νion Λ which can be written in an equivalent form
Λ2 =
νion , D
2
π 1 1 1 1 1 = 2 + 2+ 2∨ 2 = Λ2 Λx Λy Λz Λ I
(4.44)
+
(4.45)
π J
2
+
π K
2
.
(4.46)
Further simplifying for a cube with I = J = K, we obtain 1 νion = 2 =3 D Λ
2 π
I
,
(4.47)
and 3D D ∧ Λ2 = . (4.48) νion νion We note that Λ is independent of pressure because both its constituents, D and νion , scale inversely with pressure. Keeping in mind that the net current at the boundaries will vanish, the same consideration for the ions yields an important relationship between the densities of the two different types of charges Λ2x =
De ne . (4.49) Di The high electron flux directed outward leads to a depletion of the negative carriers and to an excess of positive ions, and no stable plasma can be formed. ni =
4.7.2 Ambipolar diffusion coefficient Since the diffusion coefficients for electrons and ions differ by some orders of magnitude, this also holds true for their fluxes. Above all, the few charges move independently and undisturbed. However, after ignition, in plasmas which meet
4.7 Loss of carriers
91
the requirement for quasineutrality, i. e. their dimensions are large compared with the Debye length (Λ λD , cf. Sect. 3.3), the charge density has tremendously increased. Space charges generate electric fields which exert deflecting forces on the charges which happen to come across. In particular, the motions of electrons and ions will be linked together, and the diffusion rates of electrons and ions equalize to one value in the stationary state. This is achieved by the build-up of an electric field which holds the electrons back in the plasma bulk but accelerates the ions out of it. The total fluxes of the two carriers will become equal [validity of Ohm’s law provided and μ the mobility of the carriers (j = ρ v = e0 n v = e0 n μE)] which are composed of a drift term (due to external forces) and a diffusion term (due to internal forces: random walk): j+ = e0 Γi = e0 μi ni − e0 Di ∇ni ∧ j− = −e0 Γe = −e0 μe ne − e0 De ∇ne . (4.50) Assuming that the deviations from equilibrium remain very small, we demand that the densities of ions and electrons be equal (condition of electroneutrality). We obtain a new value for the electric field E=
Di − De ∇n , μe + μi n
(4.51)
and in this case, the densities in the drift terms vanish, and we arrive at a new diffusion coefficient Da =
μ i De + μ e Di . μi + μe
(4.52)
This coupled motion (diffusion and drift) can be regarded as a modified diffusion which is termed ambipolar diffusion. Remembering that μe μi , this expression can be further simplified by neglecting the ion mobility in the denominator, yielding Da ≈
μi De + D i , μe
(4.53)
and applying the Einsteinian relation D = kB T μ
(4.54)
we arrive at
D a = Di 1 +
Te Ti
≈ Di
Te , Ti
(4.55)
and we can write for the second Fick’s law for vanishing drift dn = e0 Da Δn. dt
(4.56)
92
4 DC discharges
The diffusion of the ions will be accelerated, but that of the electrons is slowed down, and the total diffusion velocity is determined mainly by the mobility of the slower species, but enhanced by the ratio of the mean energies of the mutually coupled species. Hence, ambipolar diffusion coefficients are considerably smaller than that for free electrons. As a first consequence, the electron density falls more slowly, and the electric field which is necessary to sustain the discharge decreases from the topmost value required for ignition or breakdown to values which are considerably lower [23]. Example 4.3 From thermodynamics, we remember typical values for D to be around 10−1 cm2 /sec at 1 bar (DO2 = 1.8 × 10−1 cm2 sec at 0 ◦ C), and since D scales with the mean free path (D = 1/3 λ < v >), it will become larger by four orders of magnitude at 100 mTorr (end of the viscous flow regime), say 750 cm2 /sec. Let’s consider an electron temperature of 3 eV and an ion temperature of 600 K, we find for the ratio Te /Ti a value of around 60 which leads to coefficients for the ambipolar diffusivity of 45 × 103 cm2 sec at 100 mTorr. A parallel-plate reactor 20" (50 cm) in diameter and a gap of 7.5 cm between its plates exhibits a volume of 15 l, and for a gas flow of 100 sccm, this leads to a residence time τ of 2.4 sec. Remembering the equation for the random walk [eq. (4.38)], we eventually obtain a value for the diffusion length Λ of 570 cm, large in size compared with all the dimensions of the reactor.
Both carrier types diffuse very rapidly, which has two consequences: • All the problems concerning density gradients can be linearized. This has the advantage that we can treat issues concerning plasma properties separately from those associated with transport behavior. • Of the two mechanism for equalizing density gradients, diffusion is predominant over convection, i. e. real transport of gas volumes different in density (caused by heating). In particular at discharge pressures of some tens of mTorr, which we encounter in most of the RF reactors which are driven capacitively or inductively, the charged carriers diffuse rapidly thereby causing the plasma to extend readily in the reactor. Raising the pressure, however, confines the plasma to the path between the electrodes. The diffusion becomes ambipolar for plasma densities larger than 108 cm−3 . In low-pressure discharges in which loss is dominated by wall reactions, the diffusion time, which is the averaged time to cover the distance from the origin to the wall, is approximately given by the ratio Q/I: τ=
Q ρV = , I jA
(4.57)
which reads for the cylindrical symmetry of a parallel-plate reactor [l: cylinder length, r: cylinder radius, O: surface: 2πr(r + l)], τ=
n0 e0 πr2 l . vI O
(4.58)
4.7 Loss of carriers
93
Since the carriers must pass a dark space (sheath) we have to insert for ji the current density at the Bohm edge [eq. (14.44)] and we obtain for the typical geometry (r l):
τ ≈l
mi . kB Te
(4.59)
For an argon plasma with an electron energy of 2 eV confined within a tube of 10 cm in length, τ becomes ≈ 45 μsec. In fact, the values are higher by a factor of 2 which can be referred to the nonlinear distribution of charged carriers within the dark space of a DC discharge or the sheath of a HF discharge. The conditions become more complex in discharges of electronegative gases with rising pressure. Now, the density of the negatively charged carriers is considerably influenced by the negative ions, which can even lead to a clear predominance in discharges through SF6 .
4.7.3 Modified boundary Reducing Eq. (4.43) to one dimension and considering (4.46), we find
ne = ne,0 cos
x , Λ
(4.60)
and inserting it into Fick’s first law, we obtain for the flux
Γ=D
ne,0 x sin Λ Λ
(4.61)
which propagates with the diffusion velocity of
u=
Γ x D : = tan n Λ Λ
(4.62)
To meet the the conservation laws which are expressed in the continuity equation (4.38), for vanishing number density a steeply rising diffusion velocity is mandatory, which would become singular at the wall! To circumvent this singularity, we must exclude this hard boundary condition and we demand for real systems a finite velocity and a finite residual density. For high discharge pressures operated at some Torr this is no issue at all since D scales with inverted pressure, and we can set n = 0. However, DC discharges as well as RF discharges are driven at significantly lower pressures, and for them, this boundary condition has to be modified by replacing it by a finite density at some distance in front of the wall. This boundary is the Bohm edge (Sects. 14.2 and 14.4) which takes into account the inertia of the ions.
94
4 DC discharges
4.7.4 Diffusion processes in the positive column We have seen in the last section that the positive column is a relatively placid zone. Its field adapts to that value that ensures a constant density of carriers, i. e. the rate of generation equals the rate of loss. According to Schottky’s assumption that the mean free path of the electrons is small compared with the radius of the column, some collisions will take place. Carrier loss is dominated by ambipolar diffusion in a radially outward direction and subsequent capture by the walls of the reactor. Hence, the charge density is at its maximum in the center, declining gradually when going outward and vanishing eventually at the reactor walls. The potential of the column is more positive than the wall potential due to the larger mobility of the electrons compared with the other charges. The current in the positive column is maintained predominantly by the electrons which are delivered from the adjacent Faraday’s dark space. Provided the length of the positive column is large compared with its diameter, its properties can be regarded as radially symmetric. Furthermore, the diffusion laws can be applied when the radial dimensions exceed the mean free path of the electrons, and we assume the densities of both the carrier types to be equal [cf. derivation with Eqs. (4.50) and (4.51), ni = ne = n0 ], which holds then also be true for their radial gradients dn/dr. In the state of equilibrium, positively and negatively charged carriers migrate outward with the same ambipolar diffusion coefficient. A two-dimensional toroidal element with inner diameter r and outer diameter r + dr can absorb
dn dt
= 2πrDa r
dn dr
(4.63) r
carriers, but will lose
dn dt
= 2π(r + dr)Da r+dr
dn dr
(4.64) r+dr
carriers. Thus, its balance can be calculated according to
dn dt
= −2πDa r+dr
dn r dr
dn −r dr r+dr
dn + dr dr r
,
(4.65.1)
r+dr
or
dn dt
= −2πDa r+dr
d2 n dn r dr + dr , dr2 dr
(4.65.2)
provided no recombinations occur. For the stationary state, dn/dt will vanish:
4.8 Anodic region
95
1 d dn νion r + n = 0, r dr dr Da
(4.66)
which is solved by the Bessel function of the first kind and zero order
n(r) = n0 J0 r
νion Da
(4.67)
with n0 the central concentration at r = 0. The Bessel function of first kind and zero order is comparable to a damped cosine function with shifted roots; n(r) = n0 J0 (2.405r/R) with R = rmax (Sect. 14.7.2, Figs. 14.35), and in three dimensions
ne = n 0 J 0 r
νion πz cos Da K
(4.68)
and for the diffusion length
2.405 1 = Λ2 R
2
+
π K
2
.
(4.69)
4.8 Anodic region The description of the various zones is completed with the anodic zone. Here, the current is drawn from the glow discharge to the external circuit. The properties of the anodic zone depend mainly on the position of the anode with respect to • the quasineutral plasma of the positive column or to • the slightly negative space charge of Faraday’s dark space or to • the positively charged negative glow. Is the anode located within the positive column, the density of ions vanishes at the anode, and a small negative space charge is created in front of the anode, the anodic dark space with the anode fall; the anode is considerably more positive than the potential of the negative glow. To sustain equilibrium in the dark space, the electrons which establish the negative space must balance the number of ions entering the boundary between the positive column and Faraday’s dark space. That is why the energy of the electrons (and the potential, resp.) rises to values which amount approximately the ionization potential of the ambient. Would the anode fall rise to higher values, the number of ionizations would be enlarged until the condition of state is met again. Hence, the anode fall cannot significantly surpass the ionization potential [140] (as in the case of the cathode, Fig. 4.10). Neglecting all the secondary processes, v. Engel developed a simple theory for the anode fall, which can be applied at low pressures below 100 mTorr (15
96
4 DC discharges
Pa) and for low current densities (j < 10−3 A/cm2 ) [141]. A parabolic variation of the field assumed, the anode fall yields V (x) = 3Va
2 x
d
1−
2x 3d
(4.70)
with Va the anode fall and d the thickness of the anodic dark space. 15
f(x) [a. u.]
Fig. 4.10. Anode fall in argon after the theory of von Engel, which dives into the positive column in which we observe a constant rise of potential (and a constant electric field). Va is somewhat smaller than the ionization energy of argon [141].
potential electric field
10 5 0 -5 -3
-2
-1
0 x [mm]
1
2
3
This limitation does not apply for very small anodes and is also not valid for discharges through electronegative gases which form negative ions. By this electron capture, electrons are consumed, and the resulting negative ions are significantly slower than electrons. This gives rise to a steep increase of the anode fall, which can reach 1 000 V in discharges through halogenes [142]. From this dicussion it is obvious that the thickness of the anodic dark space is determined chiefly by the condition of a stable space charge in front of the anode, but secondly by the condition of ionization. The ion current density is considerably smaller than the electron current density; it is approximately ji = −
Di je De
(4.71)
with De , Di the diffusion coefficients. To meet the condition for ionization, the thickness of the dark space must be sufficiently large to generate the ion current according to Eq. (4.71), i. e. to accelerate the electrons to energies which are somewhat larger than the ionization potential Ekin ≥ EIon : da
ji =
0
α je dx = −
Di je . De
(4.72)
Since the electron current density is almost constant, we obtain da Di = α dx. De 0
(4.73)
4.9 Hollow cathode discharge
97
According to this condition, we can calculate a value for pd1 (invariant in similar discharges) of ≈ 10−4 mbar cm (0.1 mTorr cm) in helium, which is considerably smaller than the thickness of the anodic dark space; da has been calculated by Ingold to 3 Torr cm employing the mobility-limited Child’s equation [143]. Further reduction of the electrode distance at constant discharge current will reduce the height of the anode fall, which can eventually become negative if the anode reaches the negative glow, because ions and electrons diffuse to the anode meeting the requirement for neutrality of charges [144, 145]. In this case, the potential of the negative glow is the topmost value of the discharge—a normal case in plasma systems for technological applications (Fig. 3.5). Further reduction to values smaller than the thickness of cathodic dark space gives rise to a steep increase of the potential because ionization at the same current density is evidently made more difficult. This is a further hint at the importance of the negative glow to sustain the glow discharge [a similar phenomenon can be obtained by reducing the discharge pressure (similarity rules)]. We speak of an obstructed glow discharge [130].
4.9 Hollow cathode discharge Other obstructed discharges are the spray discharge and the hollow cathode discharge, which can increasingly be found not only in ion plating systems but also in ion beam systems [146]. The electrons which are ejected from the cathode exhibit a beam-like character in directions normal to the cathode. Sharp bending of a cathode that either end will migrate close to its counterpart gives rise to a melting of their glows (the anode can degenerate to a ring of larger diameter). If the acute angle has reached a certain lower limit, the electrons which are emitted from the first cathode into the mutually glowing zone can be reflected from the second cathode (pendulum effect). Ideally, this angle should reach 180◦ . For constant potential, current density and intensity of the glow will increase from this threshold (pa ≈ 4/3 Torr cm) [147]. For small values for pa (with a the distance between the electrodes), the current density in hollow cathode discharges can far surpass the value observed in normal discharges up to factor of 1 000 (Fig. 4.11); the functional relationship can be approximated to j/p2 ∝ 1/(pa)5/2 in the range a ≈ d [129]. This increase in current density is referred to the dramatic reduction of diffusion to the walls with subsequent loss for charged particles (electrons and ions) and excited particles. Shortening the distance between the two electrodes gives rise to a decrease of the thicknesses of the dark space, which leads, in turn, to a rising field strength at constant cathode fall Φc and an increasing ion density. Furthermore, since the radiation of the discharge will become more intense, the electron yield will rise due to the photoeffect. Instead of the application of two
98
4 DC discharges
VC =
1000 He 250 V
He 400 V
100 j/jn
H2 300 V
10
Ar 300 V
1 10-1
Fig. 4.11. Hollow cathode discharge after [148]. The reduced current density j/jn is plotted vs. the reduced distance between the two electrodes for several gases and cathode falls. The material of the cathode c Oxford University was iron ( Press).
1 pa [Torr cm]
plane-parallel plates, in most technological cases a cylindrical hollow cathode is used. A rise in glowing intensity can be observed in normal reactors at small diameter welded flanges; we speak of a special hollow cathode effect only if the reduced length p a exceeds values of 1 Pa m. At a pressure of 375 mTorr (50 Pa), this means a gap between the electrodes of 2 cm.
4.10 Similarity laws As has become clear from the preceding sections, the exact description of a discharge is difficult and extensive because of the mutual dependencies of the plasma properties. The oldest systematization was presented by de la Rue and M¨ uller [149] who first pointed out that the breakdown voltage Vb does not change significantly if the product pd is held constant (with p the pressure and d the gap between two plane-parallel plates, cf. Sect. 4.2). Since number density n scales with pressure p for constant temperature, but the mean free path λe scales with inverted pressure, we find that pd is proportional to the number density between the electrodes. However, the energy which is gained by an electron between two collisions is proportional to 1/pd; hence, the generation rate will remain constant: Vb ∝ pd, which is the Paschen law. This has been investigated most thoroughly by Steenbeck and can be formulated as follows [150]: Discharges are denoted as similar when potential and current are equal at similar positions, and all the linear dimensions will only vary by a factor of a. Assume two discharges of the same gas (same electrode material), in two reactors which are distinguished—or similar—by a factor of a only in their linear dimensions. Provided that
4.10 Similarity laws
99 T1 = T2 ; V1 = V2 ; u1 = u2 ,
(4.74)
in the case of a Maxwellian distribution, we further have < Ekin,1 > =< Ekin,2 >
(4.75)
d1 = a d2 ∧ r1 = a r2
(4.76)
and
which yields λ1 = a λ 2 ⇒ p 1 =
p2 E2 E2 E1 = , ; E1 = ⇒ a a p1 p2
(4.77)
i. e. E/p is an invariant property. Moreover, from E1 = E2 /a, we can derive with Gauß’s law (E = σ/ε0 ): σ1 = σ2 /a ∧ ρ1 = ρ2 /a2 .
(4.78)
Simplified, we can further state I1 = ρ1 vA ⇒ I2 = ρ2 /a2 v a2 A2 = ρ2 vA2 : I1 = I2 4
(4.79)
Inserting this in Eqs. (4.77), j/p2 evolves as an invariant property as well as E/p: j1 = I/A1 ∧ j2 = I/A2 ∧ p1 = p2 /a ∧ p21 = p22 /a2 ⇒ I Ia2 j1 I j2 j1 = = ∧ 2 = = 2. 2 2 2 2 2 p1 A1 p1 a A2 p2 p1 A2 p2 p2
(4.80)
As we have seen in this chapter, it has become common to plot functional relationships against the properties E/p, E/d or j/p2 which are invariant against alterations of the reactor geometry. Deviations from the similarity laws or rules hint at complicated processes. For normal discharges, j/p2 is constant over a wide range. For abnormal discharges, j and Vc will rise which can be approximated most simply by j/p2 = aV b . The most important reason for this behavior is the heating of the cathode by which the gas is heated as well. If the pressure is not corrected by this increase in temperature, the gas density will decline. If this effect is not taken into account, we calculate values for j/p2 which are far too low. A similar argument holds true for the invariant pd. For example, Druyvesteyn and Penning could show that in a discharge through hydrogen a variation of dc by a factor of 100 lead to a variation of pdc by a factor of only 2 [151]. In abnormal discharges, V and j/p2 are connected via a complicated function [152]; the ionization via metastable atoms can also be detected by deviations from this rule. 4
More precise is j = ρi vi + ρe ve .
100
4 DC discharges
8
v [106 cm/sec]
6 hydrogen
4 2 0 0.0
0.5
1.0
1.5
Fig. 4.12. In discharges through hydrogen, the drift velocity of the electrons linearly increases with the c J. Wiley ratio E/p [153] ( & Sons, Inc.).
E/p [mV/(Torr cm)]
For some gases, the similarity rules are followed with an astonishing precision. A linear functional relationship exists between the mean molecular velocity of hydrogen, helium and a mixture between helium (He) and mercury (Hg), socalled Heg gas, and E/p (Fig. 4.12, cf. Sect. 2.3). The physical meaning behind this similarity rule is the fact that E/p determines the mean energy of the electrons. The mean energy < εe > which is gained by the electron between two collisions is < εe >= e0 Eλ,
(4.81)
during a collision, the mean energy Δε = L (< εe > − < εg >)
(4.82)
with L the Langevin’s energy loss parameter L=
2me mi + me
(4.83)
and < εe > and < εg > the mean enery of the electrons and the gas molecules, respectively, will be transferred. Equating (4.81) and (4.82) yields e0 Eλ = L (< εe > − < εg >)
(4.84)
or < εe > − < εg >≈< εe >=
e0 Eλ , L
(4.85)
E . pσ
(4.86)
eventually, we obtain with σ the cross section < εe >∝ Eλ∨ < εe >∝
4.11 Conclusion
101
Within certain limits, the energy which is transferred to the electron scales with E/p. In a similar sense, j/p2 behaves invariant as well, j = σE with σ the electric conductivity which is, according to the model of free electrons, proportional to the electron density which, in turn, scales with the number density n; the free parameter is τ , the time between two collisions.
4.11 Conclusion • In a DC discharge, a DC current flows across the electrodes through the plasma, electrons are accelerated to the anode, ions to the cathode, respectively. • It is therefore impossible to apply isolating substrates as electrodes. • The discharge is ignited after having reached a certain ratio of electric field to number density. The required voltage is denoted breakthrough voltage. • Once the discharge is stable, currents flow through the discharge, and an avalanche is generated which serves for a self-sustained discharge, provided the operating voltage is maintained. This voltage is significantly lower than the voltage for breakthrough because the mobilities of the carriers (in an electropositive discharge: Positive ions and negative electrons) are mutually coupled. • The region for power transfer is the cathodic dark space, the region of carrier generation is the negative glow. • For maintenance of the discharge, the processes at the cathode are of extreme importance (during γ-processes, ions incident on the electrode with high a kinetic energy, often some keV, liberate γ-electrons from the electrode which take part of the avalanche for carrier generation). • Due to the high mobility of the carriers, the potentials of the electrodes are screened by charges of unlike type forming a space charge around the electrodes. The existence of this space charge suppresses a continuous growth of the elecric field: The zone of steep increase is confined to a thickness of several centimeters at maximum (space charge limited current).
5 High-frequency discharges I
The phenomenological difference in breakdown behavior between DC discharges and HF driven plasmas is used to discuss the processes of carrier gain, predominantly Ohmic heating, and carrier loss (diffusion) which are required for the understanding of HF breakdown. The conditions for equilibrium of gain and loss in the plasma bulk are extensively discussed. Next, power coupling is investigated from an electrical viewpoint on two levels. The chapter is finished with a discussion of matching networks and waveguides for the different methods of coupling.
5.1 Phenomenological introduction In a DC discharge, an electrode which is covered by an electrically isolating plate will be charged to the weakly negative floating potential Φf ; the fluxes of both types of carriers (electrons and ions) become equal in magnitude, irrespective of whether the potential is applied at the rear side of the insulator (cf. Sect. 3.4). At its surface, ions and electrons will be terminated or will recombine, and there is no need to draw off an electric current (which would be impossible). At plasma densities of 1010 cm−3 , a voltage of 10 − 20 V develops across the sheath. The sheaths represent capacitancies which can pile up charges. Capacitance is defined as C = Q/V ; since Q ∝ V , and it takes a certain time to charge up the capacitancies (Q = Idt), the voltage does not alter instantaneously, i. e. without any time of retardation. After ignition, both sides of the insulator drop to the (negative) cathode potential. During bombardment with positively charged ions, the potential becomes more positive (less negative) because electrons are consumed to neutralize the ions: The potential at the surface facing the plasma gradually reaches Φf , which is sufficient to generate an ion bombardement which can clear off the surface from weakly bounded contaminations. However, the energy of the ions incident on the surface is too low to make substantial sputtering feasible [154]. To solve this problem, we can use alternating current (AC). During the positive part of the cycle, the negative charging of the electrode will be removed by ion bombardement. The frequency which is required for successful neutralization can be roughly estimated as follows, provided the electrode current is assumed to remain constant (in fact, it will diminish): C = Q/V = I t/V, ⇒ t = C V /I. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 5,
103
104
5 High-frequency discharges I
p [mTorr]
1
Fig. 5.1. For rising operating frequency, the pressure which is required to ignite a plasma steeply drops up to some hundreds of kHz to settle down for even higher frequencies [155] c Elsevier). (
10-1
10-2
100
1000 f [kHz]
5 4 S [W/cm2]
14 MHz 7.5 MHz
3 2
3.7 MHz
Fig. 5.2. For rising operating frequency, the efficiency of power coupling increases [157].
1 0 1
10
100
1000
p [mTorr]
For a quartz 3 mm in thickness, its specific capacitance amounts to about 1 pF cm−2 ; for V = 1 000 V and j≈ ≈ 1 mA cm−2 (the current density is estimated from DC measurements and sputtering rates) we calculate a rise time of about 1 μsec. Since the electrode does not charge up instantaneously because the current does not remain constant but drops with falling potential, AC discharges with isolating cathodes can already be operated at frequencies beyond some ten kHz. Koenig and Maissel were the first to demonstrate in 1970 that plasmas can be ignited with rising frequency [156]. Norstrøm extended these investigations [157] (Fig. 5.1). It turned out that the coupling of RF energy close to the ignition voltage is almost independent of pressure but rises steeply for larger pressures. The pressure dependence of power coupling is larger for higher frequencies which indicates further ionizations or even the onset of additional ionization mechanisms (Fig. 5.2). However, the slope of the I(V ) curves is almost the same
5.2 Generation of carriers
105
from which we infer similar electron temperatures (ln je = const − e0 V /kB Te ). This effect can be detected up to some MHz, from which we further conclude that the discharge is sustained by electrons which are not generated by electrode processes (for more details see Sects. 6.1 and 14.4)!
5.2 Generation of carriers To explain this fact, we consider the movement of a free, undamped electron which is accelerated by an oscillating electric field. According to Eqs. (5.1) and (5.2) (a property which is multiplied by “i” exhibits a phase shift of π/2 : eiπ/2 = cos π/2 + i sin π/2 = i), the velocity exhibits a shift of π/2, and the amplitude a shift of π against the exerting force: me
due = −e0 E 0 eiωt dt
(5.1)
e0 1 e0 E 0 eiωt . (5.2) E 0 eiωt ∧ xe = m iω mω 2 On the time average, the power absorption is determined by the integral over 2π iωt 0 e dt = 0: Neglecting the losses by radiation damping, the electron cannot absorb any energy! Hence, another mechanism must be responsible for power absorption. This is suggested also by the fact that the maximum amount of energy that can be absorbed by a carrier within one half of a microwave cycle (2.45 GHz), is less than 0.1 eV at the measured field for breakdown of 100 V/cm. This is more than two orders of magnitude less than the ionization potential of argon [Eion (Ar) = 15.76 eV, Table 5.1 and Fig. 5.3]. ⇒ ue = −
Table 5.1. Velocity, kinetic energy, and maximum amplitude of an electron which oscillates freely in an HF field at the operating frequencies of 13.56 MHz and 2.45 GHz. ν 13.56 MHz
2.45 GHz
E [V/cm] 1 10 100 10 100 1000
v [cm/sec] 2.07 × 107 2.07 × 108 2.07 × 109 1.14 × 106 1.14 × 107 1.14 × 108
Ekin [eV] 0.12 12.2 1217 3.6 × 10−4 3.6 × 10−2 3.6
ψ0 [cm] 2.42 24.2 242 7.4 × 10−5 7.4 × 10−4 7.4 × 10−3
We have seen that an electric field exerts an accelerating force on an electron as long as it collides with a second particle. During the impact, the field drift is destroyed and kinetic energy is transferred. This amount is negligible for an
106
5 High-frequency discharges I
elastic collision with a heavy gas constituent. After this incident, the electron will be accelerated again, but on a higher energy level. By this repeatable process, the electron can accumulate a tremendous amount of energy until it is by a wall collision. Hence, the motion of the electron consists of a large random component and a small drift component. As we have extensively discussed (similarity rules, Sect. 4.10), the energy transferred to the electron scales with E/p with E the electric field and p the discharge pressure. Although the operating frequency in an RF discharge is typically lower by a factor of 10 to 50 compared to a microwave discharge, the pressure is lower by about the same factor. Hence, the picture is similar: We observe many collisions during one oscillation period [at a pressure of 100 mTorr (13 Pa), νm = 9 × 108 sec−1 for 12 eV electrons] and a so-called random phase movement. The elastic collisions between electrons and neutral molecules are mandatory for this mechanism of energy transfer to occur. The (directed) energy which is piled up by the electrons during the drift motion exerted by the electric field is transformed into a random phase motion. Albeit in the two half-waves the electron is accelerated to and fro the electrode, the electron can gain energy in both parts of the AC cycle. In a mathematical sense, the absorbed energy scales with the field squared, and it becomes independent of the sign: Ohmic heating, [Eqs. (5.4) and (5.9)]. For the absorption of power and the increase in conductivity, the ratio ω/νm with ω the operating angular frequency and νm the frequency of elastic collisions and σm the cross section of elastic scattering of electrons with neutrals (m for momentum) or the cross section of momentum transfer νm = σm ue ne
(5.3)
is of paramount importance. This ratio expresses how rapidly a concerted electron movement will be damped by elastic collisions with neutrals. The energy gain of an electron amounts to 1 (5.4) Pabs = − e0 E 0 · ue 2 with ue the drift velocity of the electron, which can be derived from the second Newtonian axiom for harmonic distortion:
me
due + νm ue = me (iωue + νm ue ) = −e0 E 0 eiωt dt
(5.5)
yielding e0 1 E 0 eiωt . (5.6) m iω + νm We define the AC mobility as connecting factor between drift velocity and electric field ue = −
5.2 Generation of carriers
107
e0 1 , (5.7) me iω + νm which differs from the DC mobility by the imaginary summand iω in the denominator by which electron inertia a/(due /dt) is accounted for. For sufficiently low frequencies, this term can be neglected against the damping term mνm ue by which the dissipative losses by elastic collisions are taken into consideration; in this range, μ is real and constant.1 Hence, the AC mobility is a complex scalar function, and its real part describes the energy transfer. This can be readily understood by splitting the function into its real and imaginary part, respectively: μAC = −
e0 iω νm μ=− − 2 ; (5.8.1) 2 2 2 m e ω + νm ω + νm e0 ω νm e0 ; (μ) = − ; (5.8.2) (μ) = − 2 2 2 2 m e ω + νm m e ω + νm and inserting into Eq. (5.4). For the power absorption of N electrons in the volume V (n = N/V ), we obtain for one HF cycle 1T P j · E dt = V T 0 e 1 1 P = ne0 ue · E ⇒ ne0 μAC E 2 , V 2 2 which yields, taking the real part of μ [Eq. (5.8)]:
(5.9.1) (5.9.2)
P νm 2 E02 νm 2 E02 ne20 = = σDC 2 , (5.9.3) 2 2 V mνm νm + ω 2 νm + ω 2 2 with σDC the DC conductivity, and further provided that νm is independent of the electron velocity (this holds true only for hydrogen and √ helium and approximately for the high-energy tail of the EEDF), E = E0 / 2 denotes the RMS field. From Eq. (5.9.3), we see that the ratio ω/νm , which is in fact the ratio of the imaginary part over the real part (μ)/(μ), mainly determines the efficiency of power coupling into the discharge. • For small values of ω/νm (low operating frequency ω and/or high pressure causing high νm ), the mean free path of the electrons decreases, and the power gain per one mean free path decreases with the mean free path squared. Vice versa, 1 Free electrons would exhibit a purely imaginary mobility (and because σ = e0 nμ a purely imaginary conductivity as well) due to the phase shift of 90◦ = 1/2 π against the exciting electric field.—To compare with metals: Even against microwaves, the conductivity of metals remains almost completely real, i. e. field and conduction current are in phase. For example, for copper, n = 8 × 1022 electrons/cm3 ; σ = 5 × 1017 sec−1 = 5, 5 × 107 Ω−1 cm−1 and νm —it is denoted as damping constant g or γ—about 3 × 1013 Hz. However, appreciable deviations are observed in the IR or VIS range which occur in the tenuous plasmas treated here some orders of magnitude lower.
108
5 High-frequency discharges I
• with rising operating frequency f = ω/2π, the amplitude of the electric field must be increased to keep the power input constant (cf. Sect. 6.5). Hence, with constant power input and fixed electron density ne , the energy transferred to the electrons would decrease with rising operating frequency −1 rather than ω −1 (Fig. [158], since the time of electron acceleration is νm 5.3). • Dropping the pressure, νm ω, causes the mean free path to grow, and the absorbed power can be approximated by ne20 P ≈ V 2me νm
νm ω
2
E02 :
(5.10)
Higher plasma densities or higher electric fields of some kV are required to maintain the discharge, and below a certain pressure (number density) threshold, the discharge will extinguish. For vanishing collision frequency νm , the absorbed power equals zero.
0 0.8
10
20
p [mTorr] 30 40
50
60
0 0.8
p [mTorr] 2000 3000
4000
5000
0.6
0.4
Eeff /E 0
0.6 Eeff/E 0
1000
hydrogen ½
w = nm
0.2
E 0/2 13.56 MHz 27.12 MHz
0.4
w= nm
hydrogen
0.2
E 0/2½ 2.45 GHz
0.0
0.0 0
50
100
150 200 250 nm [10 6 sec-1]
300
0
350
5
10
15
20
25
30
nm [10 9 sec-1]
Fig. 5.3. In the case of RF discharges through hydrogen, driven at f = 13.56 MHz, the collision frequency νm equals the angular frequency at 85 MHz, in the case of MW discharges, νm equals the angular operating frequency at 15 GHz. In both cases, the effective field is considerably smaller than the RMS field. For vanishing collision frequency, the effective field becomes zero: No power transfer takes place.
These effects are often taken into account by defining an effective field by 2 = Eeff
2 νm E02 ; 2 + ω2 2 νm
(5.11.1)
this field would transfer the same amount of energy as a steady field E0 , and the denominator effectively averages over these two frequencies. Writing
5.2 Generation of carriers
109 2 Eeff =
1 E02 , ω2 2 1 + ν2
(5.11.2)
m
we readily see that the more efficient coupling with rising operating frequency is attained at the expense of the intensity of the oscillating electric field which will in all cases be smaller than the steady field E0 . For νm ω, i. e. for high discharge pressures, the effective field will √eventually turn into the RMS field √ E0 / 2, for νm ω, we obtain Eeff ≈ E0 / 2 νm /ω (Figs. 5.3 and 5.4 showing two different presentations). The evenness ω = νm marks the lower limit for efficient power transfer. In RF driven discharges, the effective field is reduced to only a few percent at 10 mTorr in argon, which increases to some ten percent at 100 mTorr. In microwave discharges, the pressure should exceed 1 000 mTorr for an effective discharge. Striking the discharge becomes difficult below a certain pressure threshold. 100
100
10 Torr 100 mTorr
10-2
argon
Eeff /E0
Eeff/E 0
10-1
13.56 MHz 40.68 MHz
1000 mTorr
10-2
10 mTorr
100 mTorr
argon 2.45 GHz
10-3 0
2
4
6 Ekin [eV]
8
10
10-4 0
2
4
6 Ekin [eV]
8
10
Fig. 5.4. In the case of discharges through argon, driven at two RF frequencies at 13.56 and 40.68 MHz (LHS) and at the MW freqency of 2.45 GHz (RHS), the relative effective field is plotted vs. the mean energy of the electrons. At 10 mTorr, the effective field is reduced to only a few percent, which makes ignition more difficult and eventually prevents striking the discharge; and also MW discharges should be operated at pressures above 1 Torr.
By this derivation, it has been shown that the power transfer per HF cycle depends on the ratio (μ)/(μ). For fixed pressure (fixed number density), the efficiency of energy input should decline with rising frequency. However, this approach does not take into account an important fact: With rising frequency, the efficiency of ionization steeply increases, and this effect more than makes up for the decrease in power transfer. The mechanism of Ohmic heating is the result of interaction between an electron, the force which is exerted by an electric field, and neutrals which collide with the accelerated electron. This mechanism is mainly active in the plasma bulk provided there is a considerable electromagnetic field. But this can happen
110
5 High-frequency discharges I
only at low plasma densities or at operating frequencies which exceed the plasma frequency of the electrons, which is the case in low-density microwave-driven discharges. These discharges are driven “electrodeless” (they can be operated through a so-called “microwave window”, mainly made of quartz) and thus have very low plasma potential. This, in turn, causes the formation of very thin plasma sheaths. In the theory of continua which is extensively dealt with in Sect. 14.6, we see that this absorption of power can be expressed by a dielectric constant which is larger than zero but less than unity [eq. (14.171)]. For rising plasma density, however, we enter the regime of evanescence, which is characterized by a negative dielectric constant (ωp,e > ω = 2πf , this will happen for an operating frequency of 2.45 GHz at 1.54×1010 Hz or a plasma density of 7.45×1010 cm−3 ). In the high-pressure regime (νm > ω), the plasma impedance shows a complex behavior between inductive and capacitive characteristics, in the low-pressure regime (νm < ω), the impedance is inductive (positive imaginary). As main phenomenological result, the electric field of the incoming wave is damped to 1/e within the skin depth, but this volume is not static as in the DC case but instead, it is vigorously moving. We speak of the creation of electron-deficient breathing sheaths, and Ohmic heating in the plasma becomes less effective. These sheaths are responsible for three effects, displacement current heating, Ohmic heating, and stochastic heating. As they are mainly associated with capacitively coupled plasmas, they are discussed in Chap. 6.
5.3 Operating frequency and the EEDF At the end of the last section, we pointed out that this approach does not take into account the frequency dependence of the different excitations. What we are interested in is high an ionization efficiency but not a waste of energy. It was shown by the groups around Moisan and Wertheimer that in fact it is the efficiency of ionization which increases with rising frequency, albeit the power input declines. This can happen because the electron energy distribution function EEDF will change its slope with increasing frequency (see also Sect. 14.1). In particular, the convex shaped EEDF exhibiting a typical Druyvesteynian characteristic with an extremely low portion of high-energy electrons beyond 15 eV which is characteristic for the DC case gradually reinforces this part of the EEDF which is so important for ionization (cf. Sect. 4.6). This is shown in Fig. 5.5 for same mean electron energy of 3.5 eV and rising operating frequency from DC (A) to microwaves (D) [159, 160]. This adjustment of the EEDF, however, is self-limiting, since for high plasma densities, Coulombic collisions between electrons become more probable and eventually become dominant, and every distribution function transforms to a Maxwellian distribution. The distribu√ tion functions are normalized to 0∞ f (E) E dE = 1.
5.4 Loss mechanisms
111
1
f(E)
10-1
DC
DC
10-2
0 MB
1.25
10-3 10-4
2 DC
0
5 10 electron energy [eV]
Fig. 5.5. Keeping the total energy constant, the portion of higher-energetic electrons increases with rising operating frequency. Mean energy < E > = 3.5 eV; various νm /ω-ratios: A: → ∞ (DC); B: 2; C: 1.25; D: 0 (microwave plasma). (pa = 0.15 Torr cm (20 Pa cm), after [159, 160].
15
Since the total absorbed power at constant discharge pressure (number density, which means constant νm ) decreases with rising operating frequency, especially in microwave plasmas, the plasma density is larger than the other types operated at DC or RF for the same absorbed power, Moisan and Wertheimer concluded that less energy is spent in atomic excitation. But this means: Since the generation of carriers depends on the operating frequency, also within the same band (e. g. the RF band between 1 and 100 MHz), the plasma density can be increased by raising the operating frequency. This was first pointed out by Surendra and Graves [161] and is the basic principle of dual-frequency operation (Sect. 6.7).
5.4 Loss mechanisms We have already identified two mechanisms of electron loss, diffusion and attachment. 5.4.1 Diffusion The loss mechanism of diffusion is of paramount importance in the low-pressure regime (cf. Sect. 4.7). However, this mechanism is also modified in HF discharges and has to be analyzed precisely. We have noted that the diffusion coefficients of electrons and ions are mutually coupled after ignition of the discharge. As a first result, the electric field required for maintenance is significantly reduced compared to the breakdown field. In discharges of electropositive gases, the ambipolar diffusion coefficient Da happens to be smaller by orders of magnitude compared with that of the electrons (Fig. 5.6). The solutions of the Helmholtz equation [139]
112
5 High-frequency discharges I 7
10
-1
nion/p [s Torr ]
106
-1
105
Fig. 5.6. Field dependence of the diffusion coefficients in an RF discharge (upper value: breakdown, lower value: sustaining) after [58].
104 3
10
2
10
0
10
20 30 40 E/p [V/Torr cm]
50
60
d2 n νion + n = 0, dx2 Da
(5.12)
simplified for a one-dimensional problem, are set by the boundary condition of vanishing plasma density at the wall, which is justified by the fact that the diffusion constant scales inversely with pressure. Since no odd function can be a solution, Eq. (5.12) is simply solved by the cosine function:
n = n0 cos
νion x Da
(5.13)
with the boundary condition n = 0 at x = ±1/2 L the gap between the electrodes. This represents the solution for the dominant (lowermost) diffusion mode. Higher diffusion modes are solved by the higher harmonics of the cosine function. Moreover, this yields for the ionization frequency and the number density:
νion =
π 2 Da νion π ∧ = 2 L Da L
2
=
1 , Λ2
(5.14)
x . (5.15) Λ with Λ the diffusion length of the electrons. Inspecting Eqs. (5.13) − (5.15), we remark that the constant in the spatial-dependent part, 1/Λ, equals the ratio of the two complicated functions D and νion . Whereas the former depends on gaseous pressure (in fact: on number density and gas temperature), the latter depends exponentially on the temperature of the electrons. To obtain a constant result, the two dependencies must cancel exactly. Equation (5.12) reads, in cylindrical coordinates taking the boundary conditions into account (electron density and its derivations vanish at the walls and at the electrodes): n = n0 eνion t cos
5.4 Loss mechanisms
113
d2 n 1 dn νion +n + =0 2 dr r dr Da
(5.16)
which yields
n = n0 e
νion t
J0
νion r. Da
(5.17)
To have n vanish at the wall rmax = R, we demand that R νion /Da = 2.405, which opens the possibility to determine the ionization frequency νion . This represents the solution for the dominant (lowermost) diffusion mode. Higher diffusion modes with several maxima will be described by Bessel function of higher order [162]. Inserting Eq. (5.17) into the diffusion equation yields for a sustaining discharge
νion 2.405 1 = = Λ2 Da R
2
+
π L
2
(5.18)
with R the radius of the cylinder and L the distance between the electrodes. The diffusion to the walls is described by the first part of the RHS, the diffusion to the electrodes is described by the second part of the RHS of Eq. (5.18). For a uniform electric field, breakdown will occur when the losses by diffusion are at least counterbalanced by ionizations within the plasma, i. e. when νion = Da /Λ2 or Λ2 =
Δn . n
(5.19)
5.4.2 Recombination Whereas diffusion losses are heterogeneous reactions which happen at surfaces, other loss mechanisms take place in the gaseous phase of the plasma. In either case (recombination or attachment), a bimolecular reaction is required. With rising pressure, recombination processes become more likely to occur, and the condition for loss has to be extended by a term which takes the diminishing of carriers due to recombination into account. Since this must be a reaction of second order, Eq. (5.12) has to be extended by a term which is of second order in the ion density: d2 n νion + n − kr n2i = 0 : dx2 Da
(5.20)
The negative sign is because of carrier loss, ni squared because we assume an electropositive plasma with electron density and ion density equal in magnitude. From this equation, we see that recombination processes are likely to occur only at high densities where they readily become the dominating loss mechanism.
114
5 High-frequency discharges I
5.4.3 Attachment in electronegative gases In plasmas with electronegative molecules, we observe strong deviations from this conduct. A simple mass balance which takes into account generation (ionization) and loss (electron attachment), both considered as bimolecular reactions under general consideration of losses by diffusion [Eqs. (5.15) and (5.18)], can be described as follows: Da ne = kion ne nn − ka ne nn (5.21) Λ2 with kion and ka the rate coefficients of ionization and electron detachment, respectively, ne and nn the densities of electrons and neutrals, Da the ambipolar diffusion coefficient and Λ the diffusion length (approximate distance between the electrodes). This equation can be simplified to kion − ka =
Da , nn Λ2
(5.22)
provided the mobilities of positive and negative ions are considered equal and are small compared with that of the electrons. Mobility and diffusion coefficient are joint via the Einsteinian relation 2 D = < εk > μ 3
(5.23.1)
where < εk > denotes the averaged electron energy which is known in this connection as characteristic energy [58] and can be expressed for a Maxwellian distribution as D = kB Te μ
(5.23.2)
σDC . ne0
(5.24)
1 + 2α De 1 + 2α + μe /μIon
(5.25)
with μ= Da can be expressed by [163]: Da =
with α = nn /ne (nn : density of the negative ions). Of course, α is zero for argon, but becomes about 100 for SF6 : Hence, the nature of discharges through electronegative gases (i. e. Da ) can be determined predominantly by the density of the negative ions which can far surpass the density of the electrons appreciably, sometimes, by some orders of magnitude [56]. In discharges through electropositive or inert gases, we observe equilibrium between gain (ionization) and loss (ambipolar diffusion). Therefore, the electron
5.4 Loss mechanisms
115
density can be described with cosine functions or Bessel functions (maximum in the middle of the reactor, zero at the walls). In discharges through strongly electronegative gases, equilibrium is achieved by electron detachment, and diffusion can be neglected, and to first order, the electron density becomes spatially independent. From equation kion − ka =
Da , nn Λ2
(5.26)
we can further extract that reducing the discharge pressure (i. e. diminishing the particle density nn ) will lead to a rise in Da since μ scales inversely with the density of the neutrals. • In discharges of electropositive molecules (kion ka ), this means an increase of kion , which can be only realized for rising mean energy of the electrons (electron temperature, cf. Sect. 3.5). • In discharges of electronegative gases, ka is of comparable magnitude of kion ; kion and ka will change in the same sense, and the electron temperature will change considerably less. Additionally, in this equation, neither the densities of the charged carriers neither their energies (temperatures) will directly appear (the densities are hidden in the ambipolar diffusion coefficient); and to first order, these properties are independent of the coupled power. Within certain limits, Te depends only on gas pressure and on the ratio Te /Ti . 5.4.4 Decay The generation of carriers by electronic impact can be described by the ionization rate νion which depends exponentially on the electron temperature. When the power is turned off, the decay rate of the carriers is determined by the ratio of the Bohm velocity over the critical dimensions of the reactor (real volume over effective surface). The Bohm velocity scales with the square root of Te , and the time-dependent part of the continuity equation becomes ∂ ln ne (5.27) = νloss . ∂t The electron density is expected to decay exponentially after having switched off the external power (“afterglow”, Fig. 5.7), which is the main advantage of pulsed plasmas: The mutual coupling of the diffusion coefficients will be lost, the electrons evade the plasma bulk which, in turn, causes an enrichment of positive ions. When the next HF cycle starts by plasma ignition, the reactions at the substrate surface can be tremendously different from the behavior associated during “normal” operation (Chaps. 11 + 12). −
116
5 High-frequency discharges I
n e [10 9cm -3]
3
Fig. 5.7. The electron density in a cavity resonator exponentially decays after having switched off the HF power c J. Wiley & Sons, [164] ( Inc.).
1
2
4
6 8 t [msec]
10
12
5.5 Breakdown Having discussed extensively the mechanisms for charge gain and charge loss, we want to tackle the question of breakdown in HF discharges, when the electric field is still uniform throughout the plasma volume, for example for the operating frequency f = 13.56 MHz (λ ≈ 22 m). A discharge can be ignited when the electron losses by diffusion, recombination or electron attachment can be compensated by ionization mechanisms. If the losses are solely due to diffusion (cf. Sects. 4.7 and 5.4), the diffusion current can be determined by ∂ρ (5.28) ≥ ne0 νion − ∇ · j ∂t with νion the frequency of ionization, and ρ the charge density. Typical for a mechanism which is controlled by diffusion is the dependence of the breakdown voltages on pressure with distinct minima. We can readily understand this conduct by deriving Eq. (5.9.3) yielding 2 ne2 E 2 ν 2 + ω 2 − 2νm ∂P = 0 0 m 2 ∂νm me 2 (νm + ω 2 )2
(5.29)
which peaks at ω = νm (Fig. 5.8). We can readily identify two limiting cases. 2 ω 2 (many collisions during one oscilla1. High discharge pressure νm tion period): The energy which is transferred from the field to the electron will be dissipated into the plasma by elastic collisions between electrons and neutral molecules, the energy loss per collision Δε is defined via Langevin’s energy loss parameter 2me /(mi + me ), approximately 2me /mi :
power coupling efficiency [a.u.]
5.5 Breakdown
117
w = nm
f = 13.56 MHz
0
50
100 nm [106 sec-1]
150
Δε ≈
200
Fig. 5.8. The efficiency of power coupling peaks when the angular operating frequency matches the collision frequency νm .
2 1 e20 Eeff 2e0 me = < εe > | νm | m e ν m mi
(5.30)
with < εe > the mean energy of the electrons [165], which gives for the electric field
Eeff = νm
2me 2 < εe >, e0 mi
(5.31)
the effective field must be linearly increased with rising pressure (νm ∝ p). Therefore, the dependence is similar to the DC case (energy which is piled up is proportional to E/p with E the electric field, cf. Fig. 4.12). 2 ω 2 (many oscillations per collision): Since in this 2. Low gas pressure νm range, λe will increase, the probability of gaining energy from the field will diminish, which will lead to a rise by the amount of the effective field. Simplifying facts by the assumption that all inelastic collisions will cause an ionization,2 the absorbed power becomes Pabs = νion < εion >, which means for the frequency of ionization:
νion =
2 νm Pabs e20 E02 = , 2 + ω2 εion me εIon νm νm
(5.32.1)
Pabs e2 E 2 νm = 0 0 2. εion me εIon ω
(5.32.2)
simplified for νm ω: νion =
2 Again, the model gas is helium with mercury vapor (Heg gas); He exhibits a metastable level at 19.8 eV which has a lifetime of some msec: Almost every collision of a metastable He atom, He∗ , with an Hg atom will lead to an ionization; the effective ionization potential amounts to about 19.8 eV; νm (Hg) then becomes 2.37 × 109 sec−1 (p in Torr).
118
5 High-frequency discharges I The condition for breakdown is given by νion = D/Λ2 . We note that Λ, the diffusion length, solely depends on the geometry of the reactor, and D can be evaluated from kinetic gas theory according to D = λ √ < v > /3 (with λ the mean free path). More precisely, we have to write < v 2 > instead of < v >, this yields νion =
√ λ < v2 > . 3Λ2
(5.32.3)
√ √ Since < v 2 > can be written as < v 2 > = λνm , we eventually obtain for the ratio between the frequencies of collision and ionization: νion λ2 = νm 3Λ2
(5.33.1)
or νion νm =
< v2 > . 3Λ2
(5.33.2)
With < εe >= 1/2 me < ve2 >, the field eventually becomes ω E0 = e0 Λνm
2 εIon < εe >. 3
(5.34)
The minimum of the field can be located where the frequency for momentum transfer νm equals the operating angular frequency ω. The electric field for breakdown scales with • the inverse frequency of elastic collisions νm [and approximately with the inverted discharge pressure (cf. Sect. 14.1)], and with • the operating angular frequency ω (Fig. 5.9). Furthermore, this dependence holds also true for maintenance of the discharge. This causes the time-independent EEDF to vary with the operating frequency [160], and the efficiency of power input becomes dependent on pressure. For example, for microwave discharges through helium (σ ∝ 1/v) the range for largest power input can be confined between 4.5 and 9 Torr (600 and 1 200 Pa), and the maximum can be located at about 6 Torr (900 Pa). The breakdown at high frequencies is solely determined by the (primary) α-ionization. Knowing the ionization coefficient, it is feasible to determine the strength of the electric field required for breakdown. To put in another way, the ionization coefficient α can be evaluated from this breakdown experiment. From the spatial dependent diffusion equation (5.12), it is evident that νion /D is equivalent to the first Townsend’s coefficient α squared (dependent on the gas, its pressure, the electric field, and, additionally, the operating
5.5 Breakdown
119
E [V/cm]
104
103
2 L = 0.159 cm
2 L = 0.475 cm
102
10 -1 10
1
10 p [Torr]
102
103
Fig. 5.9. Microwave breakdown in He/Hg (Heg gas), L: distance between the electrodes after [166]. Raising the pressure and the collision frequency weakens the effective electric field. Hence, the amount of E which is required for breakdown must be increased.
frequency). α describes the growth of the electron current n/n0 as function of the cathodic distance: n = n0 exp(αx), i. e. the number of ionizations per cm which can also expressed by νion < ue > with νion the frequency of ionization and < ue > the drift velocity (< ue > = μE) or
(5.35)
νion . (5.36) μE α, however, refers to an ionization which is caused by a drift in the DC field, whereas νion /D describes a motion in an oscillating HF field, which is significantly smaller than the motion caused by a steady DC field. Alternatively, this can be expressed by an ionization (creation of an electron-ion pair) by an electron that falls across a voltage of 1 V (instead across a distance of 1 cm). Both the coefficients are related by α=
η=
νIon α ⇒ νion = ημE 2 ; [η] = 1/V, [νion /D] = 1/cm2 . = E μE 2
(5.37)
By definition of ζ ζ=
νion 1 , 2 D Eeff
(5.38.1)
the coefficients are related via ζ=η
μ 1 1 = 2 2 . D Λ Eeff
(5.38.2)
120
5 High-frequency discharges I
From the Einsteinian relation, we see that the ratio D/μ measures the mean energy of the electrons and depends on E/p. Therefore, it is possible to determine η from measurements of ζ and vice versa (Fig. 5.10). However, the mean energy of the electrons can be calculated in a different way for the AC case and the DC case, which can lead to difficulties when the mean energies, i. e. the electron temperatures, are compared.
h[ionizations/V]
10-2
Ayers (DC)
Hale (DC)
10-3 Varnerin and Brown (AC)
10-4 10
100 Eeff/p [V Torr-1 cm-1]
Fig. 5.10. Comparison of the coefficients for ionization α and ζ in a discharge through hydrogen after [167].
The increase of the breakdown field with rising pressure occurs for the same reason as in the DC case: λe drops, and the kinetic energy which can be piled up by the electron between two collisions, decreases as well. Therefore, the field must scale inversely to λe , i. e. direct proportional to the discharge pressure. For low pressures, the mechanisms become different. The efficiency of the energy transfer deteriorates, thereby causing an increase of the breakdown field. In the limit of collisionless plasma, no energy can be absorbed at all. Hence, this minimum is no Paschen minimum [168]. In contrast to DC discharges, widening the distance between the electrodes will cause a reduction of the breakdown field: The equilibrium between electron loss and electron generation is achieved more readily (Fig. 5.11). As in DC discharges, the minimum is the result of two competing mechanisms. 5.5.1 Microwave discharges: model for breakdown To shed light on this breakdown problem from another side and to discuss the limits of energy gain, Brown and Bell investigated the influence of all the parameters which are involved in microwave breakdown [58, 169]. Since it is of high heuristic value and conventional and highly sophisticated microwave discharges have regained much interest as plasma sources for ion beam processing (SLAN), it will be presented here.
5.5 Breakdown
121
60
-1
E/p [V Torr cm ]
50
-1
40
Fig. 5.11. Reducing the gap between the electrodes (reduction of Λ) causes higher losses by diffusion, and the effective field which is required for breakdown must rise (H2 , after [58]).
30 20 10 0 0
1
2
3 4 pL [Torr cm]
5
6
7
We will use the property pd (p: gas pressure, d: distance between the electrodes) which behaves as almost invariant against alterations of the dimensions of the reactor [similarity rules (Sec. 4.10)]. As far as the influence of the HF field is concerned, pλ (with λ the wavelength of the operating field) is the wellmatched parameter. The energy is denoted by ε instead of E to avoid confusion with the electric field, and for the mean free path λ, we use l to avoid confusion with the wavelength λ. For all parameters, the influence on the ratio Λ/λ has to be considered. 5.5.1.1 Frequency. As we have stated in the introductory remarks, a breakdown will occur at the state of equilibrium between generation by ionization and loss by diffusion. At low frequencies, d λ: The field across the dimensions of the reactor is uniform, this is the border of uniform field. For high frequencies, however, a threshold is set by the condition d = πΛ ≥ λ/2
(5.39)
with Λ the diffusion length, yielding pλ . (5.40) 2π We note that this restriction comes into play only for microwave discharges where λ ≈ Λ. pλ = 2π(pΛ) ⇒ pΛ =
5.5.1.2 Pressure. Next, we find that the pressure will confine the range of validity of the kinetic gas theory, that is, when the mean free path le becomes equal to the diffusion length Λ. The probability for collision Pm is inversely proportional to the discharge pressure p, and le = Λe , yielding
122
5 High-frequency discharges I pΛ =
1 , Pm
(5.41)
this property does depend on the (mean) energy of the electrons, Te . This defines the border of the mean free path. 5.5.1.3 Oscillation amplitude. If the amplitude of the HF oscillation becomes sufficiently large, the electrons can reach the electrodes at the peak of every HF cycle and are instantaneously annihilated. The amplitude can be inferred from the second Newtonian law: e0 dx d2 x + νm = − E0 eiωt dt2 dt m
(5.42)
yielding x=−
e0 e0 E0 E0 eiωt = eiωt . m iνm ω − ω 2 m −iω(iω + νm )
(5.43)
The amplitude ψ0 of this oscillation is the amount of the complex number e0 E0 mω
−ω iνm − 2 2 2 ω 2 + νm ω + νm
ψ0 =
:
e0 E0 1 , 2 2 mω ω + νm
which can be written with Eqs. (5.11) for the effective field as √ 2e0 Eeff ψ0 = . mωνm
(5.44) (5.45.1)
(5.45.2)
The amplitude cannot exceed half the distance between the electrodes: √ L 2e0 Eeff . (5.46) = 2 mωνm Substituting 1 2πc v ∧ω = ∧ νm = , ple λ le
(5.47)
√ mπc pL Pm v ∨ pλ = 2 . e0 Eeff /p
(5.48)
Pm = we obtain pλ =
Inserting the constant yields (with L = πΛ): pλ = 106
pΛ , Eeff /p
(5.49)
5.5 Breakdown
123
which can be solved for known breakdown field. This is the border of the oscillation amplitude. When entering this regime, all the electrons are lost within one half of a HF cycle by recombination with the wall (electrodes), and the field must be increased tremendously to compensate for these losses (Fig. 5.12). 350 300 400 mTorr
250 E [V/cm]
300 mTorr
200 150 500 mTorr
Fig. 5.12. At the border of oscillation limit, the breakdown field must be increased tremendously (after [170]).
100 50 0 0
50
100 l [cm]
150
200
5.5.1.4 Low gas pressure. Next, there exists the transition from high pressure to low pressure (many collisions per oscillation → many oscillations between two collisions). The borderline can evidently be drawn at √ < v >2 . (5.50) ω = νm = le For hydrogen [νm = 5.9×109 p0 (p in Torr) or 7.89×1011 p0 (p in Pa)], pλ = 4246. This is the border of the transition of the collision frequency. 5.5.1.5 Breakdown. Eventually, we find the optimum border of the breakdown [Eq. (5.34)] with ω = 2πc/λ yielding 2πc E= e0 Λλνm Inserting the values for hydrogen:
2 εIon < εe >. 3
(5.51)
• νm = 7.89 × 1011 p0 [in Pa] • εion = 15.76 eV, assuming < ε >= εion , we obtain pλ = This borderline is drawn in Fig. 5.13.
0.56 . pΛ
(5.52)
124
5 High-frequency discharges I
104
Fig. 5.13. Meeting the borders of stability in a HF discharge through hydrogen, (1) border of the uniform field, (2) border of the mean free path, (3) border of the oscillation amplitude, (4) border of low gaseous pressure, (5) optimum border of breakdown (after [171]).
pL [Pa cm]
5
10
3
10
2
3 1 4
10
2
1 10-1 10
102
103 104 105 pl [Pa cm]
106
107
108
5.5.1.6 Conclusion. For an HF discharge, there exists an island of stability, which is completely confined by diffusion processes with respect to every (primary) parameter. These boundaries encircle this island of stability perfectly for discharges driven by microwaves (300 MHz or higher), whereas for RF discharges, some conditions become obsolete.
5.6 Maintenance 5.6.1 EEDF and the electric field We first note that νion = nkion = nσion < ve >
(5.53)
using σion , the cross section for ionization by electron impact, and < ve >, the mean electron velocity. For athermal plasmas, the mean energy of the electrons is considerably greater than that of the ions. Balancing generation and loss, and remembering that the diffusion coefficients of both carriers are mutually coupled by the ambipolar diffusion coefficient, Da can be expressed approximately as Da ≈ μ i ε k
(5.54)
with μi the ion mobility and εk the so-called “characteristic electron energy” which is obtained via the Einsteinian relation [Eqs. (5.23)] εk =
De . μe
(5.55)
Approximating Da with the diffusion coefficient defined by Eq. (5.54), we eventually obtain
5.6 Maintenance
125
nkion ≈
μi εe kion μi → ≈ : 2 Λ εk n Λ2
(5.56)
Although both quantities on the LHS (n and kion ) depend on the EEDF, its ratio can be expressed as a function of specific constants of the plasma-constituting gas (which determine μi ), its pressure (which mainly determines n) and the reactor geometry (by which Λ is fixed). In particular, this ratio does not depend on the operating frequency. The independent properties which determine the shape of the EEDF adapt to those values to ensure the ratio kion /εk being kept constant. Following Moisan and Wertheimer [160], we can find two limiting cases: • For low plasma densities, the Coulombic collisions between electrons (i. e. the electron density itself) and the stepwise ionization can be neglected in shaping the EEDF, both kion and < εk > remain unaffected and become sole functions of E/n and E/ω. – In particular, the averaged electron energy in ultrahigh frequency plasmas (some hundreds of MHz or microwave) should always by smaller than that in an RF plasma, even for higher intensity of the electric field. – RF discharges follow neither the E/n-dependence (DC discharges) nor the E/ω (microwave) but show a somewhat mixed behavior. Thus Eq. (5.31) defines the magnitude of the discharge maintenance field (which differs from the externally applied field with operating angular frequency ω), and • for sufficiently high plasma densities, the EEDF becomes Maxwellian, the electron temperature is well-defined and is given by kB Te = 2/3 < εe >. For a given operating frequency ω at a certain number density n, the electron temperature Te will determine the electric field required for maintenance. 5.6.2 Collision frequency In a high-frequency discharge, the frequency of elastic collisions between electrons and neutrals, can be obtained by measuring the complex conductivity (which is connected to the mobility via σ = e0 nμ) (Fig. 5.14): νm σr = . σi ω However, two objectives make matters complicated:
(5.57)
126
5 High-frequency discharges I
• In all other gases except hydrogen and helium, the frequency for elastic collisions νm varies with electron energy and consequently, the concept of the effective field becomes questionable, if we are not interested in the high-energy tail of the EEDF. • The EEDF depends complexly on the operating frequency [172]. The power transfer exhibits a maximum at νm = ω which is a manifestation of the frequency dependence of the (time-dependent) EEDF.
10 -1
s [W -1 cm -1]
10 -2
sr sr si si
1 eV
0,05 eV
10 -3
Fig. 5.14. Comparison of the real part and the imaginary part of the electric conductivc ity of a plasma after [173] ( J. Wiley & Sons, Inc.).
10 -4
10 -5
107
10 9
10 11
n e [cm -3]
Therefore, it is worthwhile to get an idea of the deviations from the simple theory. For chlorine, νm rises from 6.3 × 106 sec−1 at about 1 eV to 135 × 106 sec−1 at about 15 eV. The first value is small even against the angular FCC frequency (13.56 MHz) of 85 MHz, and so we expect the characteristic electron energy to depend solely on the square of E0 /ω, whereas the averaged energy of the tail electrons should show some pressure dependence (Sect. 14.1).
5.7 High-frequency coupling: qualitative approach Considering the generation of carriers, the operating frequency should be 1. lower than the plasma frequency to enter the cutoff regime or regime of evanescence: ω < ωp , and should be 2. matched to the discharge pressure: ω = νm . Also to use dielectric electrodes, high operating frequencies would be advantageous. However, inductivities are then severely damped. It is for this reason that the upper operating limit is given by severe grounding problems which are difficult to overcome. Furthermore, below 100 MHz, the reactor can be designed very flexibly; electrodes and coils can be shaped into almost all geometries. The
5.7 High-frequency coupling: qualitative approach
127
most serious disadvantage is the lack of condition (2). That is why the range between 10 and 20 MHz is a good compromise; and in the 1950s, 13.56 MHz was the only frequency in the RF band which was permitted by the Federal Communications Commission (FCC) to avoid interference with the networks used for communication [174]. Non-linear effects, however, cause the generation of numerous strong higher harmonics: The sixth one is located in the VHF band, the seventh and eighth are located in the band used for airspace surveillance. Moreover, there are several frequency-dependent effects which have to be discussed in the following. As a first summary, we have found that the high cathode voltage which is required in a DC discharge to generate secondary electrons can be considerably reduced at the same power input in an HF discharge: • γ-electrons are not necessary any more to sustain the discharge (so-called α-regime [175]). • The discharge pressure can be significantly lowered. As main result, we can get rid of the electrodes as constitutive elements within the discharge, and this discharge is spoken of an electrodeless discharge. The coupling occurs through the dielectric wall of a bell-jar or a window by means of condensor plates (capacitive coupling) or coils (inductive coupling) (Figs. 5.15, cf. Sect. 5.8).
Fig. 5.15. The coupling can happen capacitively, placing the electrodes within the discharge (LHS) or externally (M); alternatively, coupling can occur in an inductive way (electrode always located externally, RHS), here shown for a barrel reactor [176].
Hence, high-frequency discharges can be divided into the following three classes which are characterized by the interaction between the electromagnetic field and the plasma: 1. E-type: Capacitive or voltage coupling between the RF electrodes which are placed within or out of the discharge reactor, respectively: High a capacitance is required between electrode and plasma, and the sheaths resemble the dielectric layer of a capacitor between the two plates, the electrode
128
5 High-frequency discharges I and the plasma. That the electrodes can be placed outside the plasma reactor is a clear manifestation of the fact that emission of γ-electrons is not required for the maintenance of the plasma. The orientation of the exciting electric field is normal to the surfaces of the electrodes, the reactor walls and their sheaths. The degree of ionization depends on the amplitude of the RF voltage. This is the vast area of parallel-plate reactors (Chaps. 6, 10 and 11, Figs. 5.16). With increasing degree of ionization, i. e. larger plasma density, plasma voltage and plasma resistance decrease dramatically, and we observe an abrupt change to the
2. H-type: The time-varying current induces a time-varying magnetic field in the plasma zone which, in turn, generates a second time-varying electric field which is oriented in a parallel direction with respect to the reactor walls and which sustains the plasma. This excitation requires high conductance between the electrode and the plasma. This regime is denoted inductive or current coupling with large RF currents. The electric field loops will close within the plasma which allows an electrodeless operation. Very high plasma densities (≈ 10%) can be achieved. This is the area of the ICP reactors which are now in common use (Chaps. 7, 10 and 11). For types (1) and (2), the wavelength exceeds the dimensions of the reactor appreciably. For example, the corresponding vacuum wavelength of an RF wave with f = 13.56 MHz (ω = 85.20 MHz) amounts to 22.12 m. Hence, we speak of “quasistationary” discharges or steady-state discharges. Albeit the plasma density achieved with the method of inductive coupling is at least higher by one order of magnitude than that obtained with capacitive coupling, it suffers from spatial inhomogenities; therefore, the plasma is applied in the “downstream” mode [95]. RF excitation
RF excitation target + dark space shield
gas shower head
dss
gas shower head
plasma
plasma
plasma
electrode with substrates
electrode with substrates
electrode with substrates
dss
dark space shield HF, RF
vacuum system
vacuum system
vacuum system
Fig. 5.16. In a countless number of modified parallel-plate reactors, capacitively coupled plasmas serve to modify surfaces either by diode sputtering (LHS), ion etching (middle) or by plasma etching and deposition (RHS). dss denotes dark space shield.
5.7 High-frequency coupling: qualitative approach
129
3. With further rise of the operating frequency, the wavelengths of the operating field become comparable or even smaller than the dimensions of the plasma apparatus. For example, the corresponding vacuum wavelength for the common microwave frequency of 2.450 GHz is only 12.25 cm. Following our considerations in Sect. 5.5, the topmost efficiency is reached for pressures of about 1500 mTorr (200 Pa) in discharges through argon, but microwave discharges can also be operated at elevated pressures, sometimes even at atmospheric pressure, however, at the expense of spatial homogenity. According to Eqs. (14.170), waves with frequencies exceeding the plasma frequency will propagate through the plasma almost undisturbed, the plasma acts as a low-loss dielectric, and the amount of dissipated power is very small still; for a driving frequency of 2.45 GHz, this will happen at plasma densities below 8 × 1010 cm−3 . Beyond this threshold (cutoff: ω = ωp ), the evanescent or cutoff regime is entered, and the penetration depth of the wave is restricted to a zone which is determined by the skin depth with a thickness a several centimeters [eq. (14.166)], the heating zone. According to this equation, the skin effect is far more pronounced in microwave-driven discharges (cf. also Figs. 14.28). In this frequency regime, plasma generation appyling capacitive or inductive coupling is almost impractible, and instead, waveguides, coaxial cables and antennae are used (cf. Sect. 5.10). In principle, widely spread, simple microwave reactors consist of four parts: the power supply with the waveguiding system (for low powers: coaxial cables, for high powers: waveguides), an applicator for power matching and a circulator to annihilate the reflected power, and the plasma reactor, a geometrically simple device with either cylindrical or rectangular symmetry, constructed of metallic or dielectric walls (Fig. 5.17). magnetron
TE10
dielectric tube with plasma
rectangular waveguide
Fig. 5.17. Simplified schematic of a microwave discharge in a reactor (here: dielectric tube) which is transparent for electromagnetic waves which are generated in a magnetron and guided via a rectangular waveguide. For better coupling, the reactor can be constructed as resonator with a sliding short. The main sources for power loss are waveguide losses, impedance matching, and radiation.
• The power supply itself consists of a microwave source (magnetron or clystron) and an attached waveguide or resonant cavity to allow easy striking of the discharge. Waveguide losses can be minimized
130
5 High-frequency discharges I by highly-reflecting coatings (silver or gold), the impedance matching can be significantly improved by terminating the waveguide with a horn, gradually flared over several wavelengths. This design also enhances the directivity of the radiation pattern. • The applicator, in its simplest form, consists of the microwave window by which the vacuum recipient is sealed against atmosphere. Application of an antenna improves the characteristics of the radiation pattern, however, at the expense of contamination. If the window is fitted with a converging lens, the refraction index of the plasma which is less than unity has to be taken into account (Sect. 14.6). Therefore, they have to be concave to focus. • To prevent local heating due to the above mentioned skin effect behind the microwave window which consists in most cases of quartz, the plasma vessel has to be designed to act as a resonant cavity. Now, a standing wave is generated in the reactor, and the maximum intensity of the field has moved away from the microwave window and can be found somewhere inside the reactor. A combination of both of them has been realized by Engemann et al. with their slotted antenna (SLAN) [177]. The microwave field is coupled into an annular structure via a tuning element (rod antenna) which is located within a conventional rectangular waveguide with perpendicular orientation. To match the varying conditions of the capacitive plasma, not only can this antenna be varied in length but also the rectangular waveguide can be adapted by a sliding short. The microwave power is selectively radiated into the interior of the reactor through slots which are located at the nodes of the electric field. • The circulator serves to annihilate the reflected power by a dummy load [178]. The problems associated with the skin depth and poor performance at low discharge pressures can also be circumvented by application of an intense static magnetic field which guides the wave (plasmas generated by whistler waves, cf. Chap. 7 and Sects. 14.6 + 14.7).
In these high-frequency discharges, the plasma density is considerably higher than in DC discharges for the same power input. For convenience, we compile some basic properties of the most common plasmas in Table 5.2.
5.8 High-frequency coupling: quantitative approach To ensure a comprehensive understanding of the two most important coupling modes (capacitive and inductive), we will focus on some of the electrical considerations. We will begin with the limiting cases of the two resonant circuits,
5.8 High-frequency coupling: quantitative approach
131
Table 5.2. Plasma sources and the most important parameters discharge pressure, plasma density, and electron temperature along with their most common application. type DC glow
p [mTorr] 1 − 102
Te [eV] 0.1 − 10
np [cm−3 ] 108 − 1011
RF CCP, low p RF CCP, high p RF ICP RF helicon MW convent. MW SLAN MW ECR E beam
1 − 102 102 − 103 1 − 103 0.1 − 10 10 − 103 100 − 103 0.1 − 5 0.01 − 0.1
1 − 10 1−5 1−5 <1−3 <1−3 2−5 3 − 10 <1
108 − 1011 108 − 1010 1010 − 1012 1010 − 1013 108 − 1011 109 − 1011 1010 − 1013 108 − 1012
application sputtering, etching, chem. syntheses, radiation sputtering, ion etching deposition, plasma etching real ion etching, chemistry real ion etching chemistry, depos., etching chemistry, depos., etching real ion etching, chemistry sputtering, ion beam etch.
either in series or in parallel, which will be followed by coupled parallel circuits and the dual wiring of the approximated capacitively coupled plasma. 5.8.1 Absorption circuit According to Kirchhoff’s second law, the magnitude of the voltage which is drawn from the voltage source is the sum of the voltage drops within the circuit: V˜ = V˜L + V˜C + V˜R
(5.58)
with V˜L = iωLI˜
(5.59.1)
iI˜ V˜C = ωC
(5.59.2)
and
which yields for V˜
1 V˜ = I˜ R + i ωL − ωC The amounts of current and voltage are then I= R2
V
+ ωL −
1 ωC
2
˜ = Z I.
∧ V = I R2 + ωL −
From Eq. (5.60), it will become evident that Z is real for
(5.60)
1 ωC
2
.
(5.61)
132
5 High-frequency discharges I
ωL =
1 : ωC
(5.62)
Resonance: I˜ and V˜ are in phase, since the reactance Zs = iXs vanishes; the voltages V˜L and V˜C can exceed the source voltage V˜ by the quality factor; and we observe a voltage resonance. According to Thomson, the resonance frequency of the oscillation circuit is given by ωr,series = √
1 . LC
(5.63)
˜ we can evaluate the damping of the Plotting the characteristic ω → I, resonance. For R = 0, I → ∞, for R → ∞, a maximum cannot be identified because the initial sharp peak will broaden (cf. Figs. 5.18). 1.0
100 1 mH, 0.01mF
0.8
80
1 mH, 0.1mF
0.6 Z [W]
I [A]
60 10 mH, 0.01mF
0.4
10 mH, 0.1mF
0.2 0.0 0.0
10 mH, 0.1mF 10 mH, 0.01mF
40
1 mH, 0.01mF 1 mH, 0.1mF
20
2.5
5.0
7.5 w [MHz]
10.0
12.5
0 0.0
2.5
5.0
7.5 w [MHz]
10.0
12.5
Fig. 5.18. LHS: resonance curves of a series circuit for various values of L and C at constant R = 1 Ω. The current show a maximum at resonance, but is strongly damped. This is due to the variation in impedance (RHS).
5.8.2 Eliminator circuit According to Kirchhoff’s first law, the total current which flows through the circuit equals the (geometrical) sum of the partial currents: I˜ = I˜L + I˜C + I˜R
(5.64)
with the coil current V˜ V˜ I˜L = = −i , iωL ωL and the capacitance current
(5.65.1)
5.8 High-frequency coupling: quantitative approach
133
I˜C = V˜ iωC :
(5.65.2)
the inductance causes the current intensity to lag behind the voltage (in an ideal case by 90◦ ), whereas the capacitance causes the current intensity to take the lead (in an ideal case 90◦ against the voltage). V˜ yields again 1.00
10
1 mH, 0.01mF
1 mH, 1mF
1/Z [W -1]
0.75 I [mA]
50 mH, 0.5mF
10 mH, 0.1mF
1 0.0
10mH, 0.1mF
0.50
50 mH, 0.5mF
0.25 1 mH, 1mF
1 mH, 0.01mF
2.5
5.0
7.5 w [MHz]
10.0
0.00 0.0
12.5
2.5
5.0
7.5 w [MHz]
10.0
12.5
Fig. 5.19. LHS: In a parallel resonant circuit, the current skims a minimum for vanishing first derivative of the I(V) curve; here displayed for the inverse impedance [several values for L and C at constant R (1 Ω)]. RHS: For minimum damping, the ratio C/L should be kept as low as possible.
1 V˜ = I˜ R + i ωL − ωC and the amount of the current is given by
I=V
1 R
2
+ ωC −
˜ = IZ
1 ωL
(5.66)
2
.
(5.67)
From Eq. (5.66), we see that Z becomes real for 1 . (5.68) ωC The condition for resonance in a parallel circuit is identical with that in a series circuit. Current and voltage are in phase; however, the current maximum is replaced by a minimum in current at resonance (Figs. 5.19). ωL =
5.8.3 Coupled parallel circuits After these preliminaries, we will deal with 1. inductive voltage coupling;
134
5 High-frequency discharges I
C1 L 1
U2
C2
U1 R1
Fig. 5.20. Two oscillation circuits which are inductively coupled with primary series feed (transformer coupling).
L2 R2
2. inductive current coupling; 3. capacitive voltage coupling; 4. capacitive current coupling; 5. transformer coupling (transducer). For our considerations, the odd numbers are of importance. These three cases can be depicted using the same equivalent circuit (Fig. 5.20). As model, we will focus on transformer coupling, which we shall meet again when we deal with inductive coupling; it is easily adapted to the two other cases. 5.8.3.1 Transformer coupling. Applying Kirchhoff’s second law, the oscillator circuit (index 1) and resonance circuit (index 2) follow the equations V˜ = I˜1 Z1 + I˜2 iωM 0 = I˜2 Z2 + I˜1 iωM
(5.69)
with M the coefficient of mutual induction
M = k L1 L2 ,
(5.70)
with k the coupling factor. In the oscillator circuit, the source voltage V˜ equals the voltage drop across the impedance Z1 + the voltage which is transmitted back from the resonance circuit to that circuit; in the resonance circuit, the transmitted voltage equals that voltage dropped across impedance Z2 . For the two impedances, it it generally valid that
Z1,2 = R1,2 + i ωL1,2 −
1 ωC1,2
(5.71)
From Eq. (5.69.2), we see that iωM I˜2 = −I˜1 Z2
(5.72)
and for the other properties
ω2M 2 V˜1 = I˜1 Z1 + Z22
(5.73)
5.8 High-frequency coupling: quantitative approach
I˜1 = I˜2 = −
135
V˜1 2 2 Z1 + ω ZM2
(5.74)
iωM V˜1 . Z1 Z2 + ω 2 M 2
(5.75)
The voltage, which can be measured at the capacitance C2 , is I˜2 V˜2 = iωC2
(5.76)
which yields V˜1 M C2 (Z1 Z2 + ω 2 M 2 )
V˜2 =
(5.77)
together with Eq. (5.75). Regarding both circuits as equal in size, R1 = R2 = R; C1 = C2 = C; L1 = L2 = L; M = kL,
(5.78)
we find furthermore R 1 ω2 (5.79) ∧ x=1− 2 ∨ x = 1 − 02 ωL ω LC ω with d the damping. This yields for the impedance Z of both the circuits, the current I˜2 which flows in the oscillator circuit, and the voltage V˜2 which is induced into the oscillator circuit: d=
Z =r 1+i V˜1 I˜2 = −i R 1+
x , d
(5.80)
k d
−
k2 d2
V˜1 V˜2 = − ωCR 1 +
x2 d2
+ 2i xd
,
k d k2 d2
−
x2 d2
+ 2i xd
(5.81)
;
(5.82)
from which we obtain for the amounts for V2 and I2 : V2 = kV1
1 (d2
+
k2
− x2 )2 + 4x2 d2
V1 k I2 = R d 1 +
1 k2 d2
−
x2 d2
2
2
+ 4 xd2
,
.
Deriving Eq. (5.83.1), we find that the extrema are located at
(5.83.1)
(5.83.2)
136
5 High-frequency discharges I x(k 2 − d2 − x2 ) = 0
(5.84)
with the roots and Eq. (5.79) for x √ x1 = 0 ∧ x23 = ± k 2 − d2 , ω1 = ω0 ∧ ω23 =
ω0 . √ 1 ± k 2 − d2
(5.85)
(5.86)
Besides the resonance frequency at ω0 , two more maxima in current will occur which are due to the feedback of the resonator circuit to the oscillator circuit, and they are defined by the coupling factor k: • Loose or undercritical coupling: k < d. • Critical coupling: shallow maximum with three roots (k = d). • Tight or overcritical coupling: k > d. • Very tight coupling: k d ⇒ k 2 d2 , by which Eq. (5.86) is simplified to
ω23 = √
ω0 . 1±k
(5.87)
Using Eq. (5.83.2), the currents which flow in the resonance circuit can be written as I2,1 =
V1 kd , R d2 + k 2
(5.88.1)
V1 ; 2R
(5.88.2)
and I2,23 =
in the case of the critical coupling (k = d), eventually I2,1 =
V1 2R
(5.88.3)
√ as well. For the detunings k = d, k = d, k = d2 and k = d3 , the resonant characteristics are displayed in Fig. 5.21.
5.8 High-frequency coupling: quantitative approach k=d
k = d3
2
k=d
I2
0.50
137
0.25
2
k=d
Fig. 5.21. Resonance circuit: resonant characterics of the flowing current I2 for for different detunings of k/d.
1/2
k=d
0.00 -10
-5
0 x
5
10
5.8.4 Capacitive and inductive coupling The transformer coupling is a special case of voltage coupling which can be derived from Eqs. (5.69) for the two remaining cases by replacing iωM for 1 or iωL for inductive coupling: capacitive coupling by iωC V˜ = I˜1 Z1 + 0 = I˜2 Z2 +
I˜2 iωC I˜1 . iωC
(5.89)
Then M −→ − C 1ω2 , which yields for capacitive coupling k 1 −B 1 −→ − = with B = d ωCR R ωC and for inductive coupling k ωL B −→ = with B = ωL, d R R and we find for the resonance circuit: U˜1 I˜2 = ± · i R 1+
B R B2 R2
U˜1 ·i U˜2 = ± iωCR 1 +
−
x2 d2
+ 2i xd
B2 R2
−
x2 d2
(5.90.2)
,
B R
+ 2i xd
(5.90.1)
(5.91.1) .
(5.91.2)
The ⊕ sign is valid for inductive coupling, the sign holds for capacitive coupling. 5.8.5 Dual circuit of the capacitively coupled plasma The equivalent circuit of a glowing plasma bulk exhibiting a complex impedance which is separated from the confining reactor walls by capacitive sheaths is
138
5 High-frequency discharges I
Cs,1
sheath 1
Fig. 5.22. Equivalent circuit of a capacitively coupled discharge with a complex impedance of the plasma bulk which is isolated by two capacitive sheaths.
Lp
Rp
plasma bulk
sheath 2
Cp
Cs,2
shown in Fig. 5.22. We see that two resonances will occur, one series resonance and one parallel resonance. Neglecting the real part of the plasma impedance Zp , the parallel resonance is simply 2 ωparallel =
1 . Lp Cp
(5.92)
The resistance of the series resonance can be calculated according to ˜ = −i + R ωCs,1
1 Rp
1
+ i ωCp −
1 ωLp
−
i . ωCs,2
(5.93)
5.8.5.1 First approximation (symmetric discharge). Neglecting the plasma resistance Rp and assuming that the two sheath capacitancies are equal in size (Cs,1 = Cs,2 = C) yields at resonance: −
1 2i + ωC ωCp −
1 ωLp
= 0,
(5.94)
and the two resonance frequencies are connected via 2 2 ωseries = ωparallel
2Cp . 2Cp + C
(5.95.1)
2d . dp + 2d
(5.95.2)
Considering further C = ε0 Ad , this equals 2 2 ωseries = ωparallel
5.8.5.2 Second approximation (asymmetric discharge). Neglecting the plasma resistance Rp and assuming that the two sheath capacitancies are different in size (Cs,2 = Cs,2 ) yields at resonance: − which yields
1 i + ωCs,1 ωCp −
1 ωLp
−
i = 0, ωCs,2
(5.96)
5.9 Matching networks
139
2 2 ωseries = ωparallel
(Cs,1 + Cs,2 )Cp . Cs,1 Cs,2 + Cp (Cs,1 + Cs,2 )
(5.97)
For the limiting case Cs,1 Cs,2 , both the resonance frequencies become equal: ωparallel = ωseries .
(5.98)
5.9 Matching networks For DC discharges, the maximum of power which can be transferred into the plasma is limited by the source resistance and by the character of the load (V : source voltage, Rs : source resistance, Rp : plasma resistance) P = V I = I 2 R; I =
V Rp ;P = V 2 . Rs + Rp (Rs + Rp )2
(5.99)
Hence, the power depends on the ratio of the resistances and exhibits a maximum for Rs = Rp . To generate a given power for decreasing plasma load, the current must rise to the current limit (P = I 2 R ⇒ I = P/R), for increasing plasma load, √ the voltage has to be increased to the voltage limit (P = V 2 /R ⇒ V = P R). In all these cases, no matter whether we reach the current limit or the voltage limit at maximum power output, the voltage along the line is uniform. The same consideration is valid for the AC case. However, we have to take into account the complex character of the load impedance. In nearly all cases, the load impedance does not match the characteristic impedance of the generator (usually 50 Ω), and a standing wave will be set up along the line (Sect. 5.10). Even the load without plasma is complex; in this simple case, it is purely capacitive. Its value can be evaluated according to the simple formula for a parallel-plate capacitor according to C = ε Ad with ε the dielectric constant of the gas within the reactor, A the surface of the electrodes, and d their distance apart. Since the plasma impedance depends on several parameters, it is customary to match the source impedance with the plasma impedance by inserting a network in the line to meet the requirement R ± ix = Rs . The source impedance Rs is usually terminated with 50 Ω which is less by orders of magnitude than the plasma impedance Rp (between 5 and 15 kΩ, inductive plasma impedance, capacitive plasma impedance, capacitive impedance of the sheaths, a typical value at 13.56 MHz: 0.1 pF is a capacitive load of 0.12 MΩ). The RF generator should not only be perfectly matched to the discharge during plasma operation but it should also provide a substantial contribution to the ignition voltage. This is best accomplished by coupling the circuit to the AC current resonance.
140
5 High-frequency discharges I
According to the discussion in Sect. 5.8.5, the capacitive discharge can be regarded as a dual circuit: plasma as a circuit connected in parallel, with parallel resonance according to Eq. (5.92), where the voltage peaks, and the second one, series resonance or geometric resonance according to Eqs. (5.95.1) and (5.97) where the current peaks (Fig. 5.22). From the theory of networks, we know that the output impedance of a well-matched network equals the conjugate-complex of the connected system, i. e shifted by −π and equal in size. This is solved practically either by installation of an • L-matching network, consisting of two variable capacitancies (one of them connected in series with the capacitive plasma load) and one coil connected in series, or of a • π-matching network, consisting of two variable capacitancies (both of them connected in parallel with the capacitive plasma load) and again one coil connected in series between generator output and the powered electrode [89] (Fig. 5.23). The advantage of the π-network is a large tuning range. Since the L-network consists mainly of a blocking capacitor connected in series to the plasma, which easily facilitates the built-up of a so-called DC bias (see Sect. 6.1).3 Lmatch
C (var.) C (var.)
RF generator
C (var.)
capacitively coupled discharge
RF generator
Lmatch
inductively coupled discharge
C (var.)
Fig. 5.23. LHS: a capacitively coupled discharge with a matched L-network which is connected between the generator output and the powered electrode. RHS: an inductively coupled discharge with a matched π-network which is connected between the generator output and the powered electrode.
With this means, the system can be tuned to resonance. A double-tuned transducer-coupled output is very useful since it can be coupled slightly overcritical which results in a resonance characteristic with a broad and flat peak, and the power output remains nearly constant. A matching network will be required only for the frequency range beyond approximately 1 MHz (more exactly: beyond the angular plasma frequency of the ions, ωp,i ), for lower frequencies, a (coupling) transformer is sufficient. 3 In the case of metal electrodes, a blocking capacitor has to be inserted between generator and electrode to suppress the DC current.
5.10 Transmission line
141
5.10 Transmission line The HF source is connected to the matching network via the transmission line which exhibits the impedance ZL =
Ev E Er = = = H Hv Hr
μ0 μr μr = Z0 ε0 εr εr
(5.100)
with Z0 = 377 Ω,
(5.101)
the characteristic wave impedance of free space and in air (this is the characteristic wave impedance of the far field). Since μr in cables is close to unity, but εr significantly exceeds unity, the impedance of the transmission line is Z0 ZL = √ < 377 Ω. εr
(5.102)
Because c=
l ω =√ k LC
(5.103)
√ and the phase velocity vph in the line with refractive index n = εR with εR the real part of the dielectric constant (see Sect. 14.6 for an extensive discussion) vph =
c c =√ , εR n
(5.104)
the connection between the characteristic impedance of the transmission line ZL and the capacitance per unit length Cl and the inductance per unit length L , respectively, is given by l √ l εR (5.105.1) ZL = cC and Lc ZL = √ : l εR
(5.105.2)
The characteristic impedance ZL is inversely proportional to the capacitance per unit length Cl , but scales with the inductance per unit length Ll . In a transmission line which is terminated with the characteristic impedance of the generator (usually 50 Ω) and lacks any imaginary fraction,
Z0 =
L0 , C0
(5.106)
142
5 High-frequency discharges I
there will be no reflexions, since the load will completely absorb the delivered power. In all other cases, the load impedance will not match the source impedance, a reflected wave will result and a standing wave will be set up along the line. In contrast to the normal process of transmittance, on a non-matched line, steady peaks of voltage can cause arcing, and steady peaks of current can result in overheating. Voltage and current are sampled by a directional coupler which is inserted into the transmission line. They are converted to two quantities which are denoted as forward and reflected voltage Vf and Vr , respectively: V˜ + Z˜ · I˜ V˜ − Z˜ · I˜ ∧ Vr = . (5.107) 2 2 The topmost and lowermost voltage, respectively, is the sum or the difference of the maximum voltages of the forward and reflected wave. At the topmost voltage, the impedance exhibits its maximum and the current its minimum; the extremes of voltage and current are separated apart by half a wavelength, the extremes of impedance by a quarter of a wavelength, respectively. From Eqs. (5.107), the two powers Vf =
| Vf,r |2 (5.108) Z can be obtained. We see from Eqs. (5.107/5.108) that Pr will vanish for ˜ but Pr takes large values for extremes in the load impedance (voltage V˜ = Z˜ × I, limit or current limit). It is these considerations which will limit the capabilities of an RF generator to handle but a certain amount of reflected power. The power which is transmitted into the complex load has to be corrected by the power which is consumed by the transmission line. The reflected power, however, is not absorbed by the generator which can be easily seen from the extreme case, termination by an open circuit. In this case, all the currents will vanish, and no power can be transmitted. But there is a standing wave on the line whose components are mutually extinguished by the equal amplitudes of forward wave and reflected wave, assuming no loss in the transmission line. In fact, we simply measure but the forward voltage and the reflected voltage, and on the dials of the directional coupler, power is displayed which would be dissipated in a 50 Ω resistor [P = V 2 /(R = 50 Ω)].4 In fact, conductive losses and dielectric losses occur in the load which will throw the phases of voltage and current out of step. Conductive losses are Ohmic resistance and frequency dependent resistance which is described by the skin effect. Dielectric losses are caused by the directing effect on molecules by the Pf,r =
4 HiFi enthusiasts labor under a similar misapprehension when they connect their brandnew super amplifier to real critical impedances, e. g. electrostatic speakers, which will become −i , and due to the acoustic short-circuit, very low-ohmic in the pitch range because of R = ωC at the very first trumpet signal, blue smoke will soar from the amplifier . . .
5.10 Transmission line
143
electric field; they are difficult to calculate. In total, they are proportional to the excitation frequency, whereas conductive losses scale with the square root of the frequency: At low frequencies, the conductive losses will dominate, at high frequencies, the dielectric losses will take over. In dielectric media, timely varying fields (E and H) are orientated perpendicular with respect to the direction of propagation. In the limiting case ω → 0, the fields are longitudinal: They can be derived from scalar potentials and are orientated in parallel fashion with respect to the direction of propagation. For non-vanishing conductivity, we expect a mixed behavior, i. e. modes with either a longitudinal component of the electric field E or the magnetic field H, respectively. The mode with the lowest cutoff frequency will dominate this propagation. Below this frequency, propagation cannot occur because the wave will be damped to 1/3 of its initial intensity within the skin depth. The TEM mode does not exhibit a cutoff, but its tangential components Et and Ht must simultaneously vanish at the equipotential surface of the conducting medium. But vanishing field at the equipotential surface entails vanishing everywhere: Propagation of TEM waves within this waveguide is impossible and can be achieved only in coaxial cables and waveguides, respectively.
5.10.1 Coaxial cable The transmission is made feasible by applying coaxial cables. In these devices, the TEM mode is the dominant mode, E and H are mutually perpendicular, the current in the walls is normal with respect to the magnetic field. The currents are considered to flow along the outside surface of the inner conductor and to flow back along the inside surface of the outer conductor or shield which is larger in size. They can be either rigid, using a central rod as inner conductor and a solid metallic tubing for the outer conductor, or they can be flexible with one or more layers of conducting braid for the outer conductor (coaxial cable). In the latter case, the inner conductor can be supported by an elastomeric polymer, or by spaced dielectric beads. Since the dielectric loss limits the power that can be transmitted, the second construction is superior to a complete filling by a continuous dielectric between the two conducting lines. The impedance of these coaxial cables strongly depends on the operating frequency; in particular, standing waves can occur not only at the design frequency, which leads to considerable loss especially at the dielectric beads. In solid coaxial cables, the dielectric loss can be further reduced by stubs, a section of a coaxial line 1/4 λ long and short-circuited at the inner conductor. Since the input admittance of this device vanishes at the design frequency of the coaxial cable, this stub does not influence the operation of the transmission line. In its active version, it can be used to vary the characteristic impedance of the cable. In particular, a mismatch which is caused by standing waves can be
144
5 High-frequency discharges I
eliminated, either only at the design frequency, or, equipped with a broadband stub, over a broader frequency range. 5.10.1.1 Characteristic impedance. To calculate the velocity of phase propagation of a transmission line, we should know the capacitance per unit length C , and the inductance per unit length Ll according to Eqs. (5.105). The magl netic energy which is stored in a coaxial cable can be simply calculated to (ε for energy, E for electric field strength) ε0 c2 2 3 1 B d x. (5.109) ε = LI 2 = 2 2 For a cylindrical volume element of length l and thickness r d3 x = l2πrdr, this `re’s law B = 1/2πε0 c2 I/r yields with Ampe ε=
2 ε0 c2 D I l 2πrdr 2 d 2ε0 c2 πr
(5.110.1)
I 2l D ln 2 4πε0 c d
(5.110.2)
or ε=
(with D the inner diameter of the outer conductor and d the diameter of the inner conductor), and we obtain for L l D ln 2πε0 c2 d and the inductance per unit length L0 L=
(5.111)
1 D ln . (5.112) 2πε0 c2 d Calculating the charge which has been stored upon the surface of the cylinder with area 2πrl L0 =
Q = 2πε0 rlE
(5.113.1)
or Q = −2πε0 rl
dV , dr
(5.113.2)
we obtain Q=
2πε0 l (V1 − V2 ). ln D/d
(5.113.3)
For the capacitance per unit length, this yields C0 =
Q 2πε0 = , Ul ln D/d
(5.114)
5.10 Transmission line
145
and eventually, for the characteristic impedance, Z0 =
ln D/d . 2πε0 c
(5.115)
Since the geometric factor does depend on the dimensions of the cable only logarithmically, and 1/(2πε0 ) amounts to approximately 60 Ω, the characteristic impedance can be found to range between 50 and several hundreds of ohms. 5.10.2 Waveguide Removing the central conductor from a coaxial cable leads to the second type of transmission line, the waveguide. This is made use of in the coaxial to waveguide couplings. For a given power, the electric field is less intense in a waveguide compared with a coaxial cable. Therefore, the transmission of higher powers is feasible until breakdown occurs. Compared with a coaxial cable, the losses, especially at higher frequencies, are considerably lower: Dielectric loss scales with frequency, Ohmic loss, however, only with the square root of frequency. Additionally, the current densities across the inner conductor are very large, and since the loss scales with the current density squared, the lower current densities in a waveguide are of advantage once more. To reduce these Ohmic losses further, its inner surfaces are often coated with silver or gold. Again, the characteristic impedance can be varied with broadband stubs, especially for coupling microwaves using a three-stub tuner. For practical reasons, the transmission through waveguides is feasible only in the wavelength range between 1 and 30 cm; but within this interval, the FCC frequency of 2.45 GHz is located which is radiated by the widely used microwave magnetrons (λ = 12.25 cm). 5.10.3 Mode patterns in transmission lines 5.10.3.1 Coaxial cable. The dominant mode is always the mode with the lowest cutoff frequency. Below this frequency, the transmitted energy is rapidly attenuated (Figs. 14.25/26). The first cylindrical mode of higher order is the TE11 , which can propagate if λ = (D − d)/(2n) ≈ πD (Fig. 5.24). The first subscript denotes the number of nodes of the radial component, the second subscript denotes the number of nodes of the azimuthal component. These modes degenerate to similar modes for a cylindrical waveguide for vanishing diameter of the inner conductor (Fig. 5.24). The frequency separation of the higher modes is a function of D/d. 5.10.3.2 Waveguide. For rectangular waveguides, the subscripts denote the number of maxima: the first index for the wide or a dimension, the second one for the narrow or b dimension. The dominant mode is the TE10 with only one
146
5 High-frequency discharges I
E H
Fig. 5.24. The TE11 mode in a coaxial cable (LHS) will transform to the TE11 mode of a cyclindrical waveguide for vanishing diameter of the inner conductor.
maximum and the longest cutoff wavelength λc . The modes next to follow are the TE20 or TE02 ; but in the vast range between cutoff and double cutoff, the TE10 dominates (Fig. 5.25).
E E
Fig. 5.25. The TE10 mode in a rectangular waveguide is the dominant mode in the vast range between λc = 2a and λc = a.
5.10.4 Electrode The transmission line is terminated by the electrode (in the case of ion etching). In the case of sputtering, this is identical with the target. The impedance should be as low as possible to avoid stray capacitancies which reduce the efficiency of power input. For a target, this is most easily accomplished for a large ratio area over thickness—a tradeoff between long-term stability, mechanical stability and electrical properties. Smooth, rounded edges reduce high field intensities; but one of the most important issues is a perfect metallization of the backside of the target since the power loss per volume is defined by P = ε0 εω E 2 tan δ, (5.116) V with E the intensity of the electric field across the thickness of the target and tan δ the energy dissipation factor of the electrode, which depends on the quality
5.11 Shielding
147 Faraday shield
interference circuit
CS
interfered circuit
interference circuit
interfered circuit
Fig. 5.26. The capacitive coupling of an interference circuit is effectively prevented by installation of a Faraday shield between interference circuit and and interfered circuit.
of the target material (grain size, purity, . . . ) [179]. The decisive parameter is E.
5.11 Shielding The devices within a plasma are subject to at least three disturbing influences: • Capacitive coupling (straying of electric fields). • Inductive coupling (straying of magnetic fields). • HF fields (straying of electromagnetic waves). Capacitive coupling by the presence of disturbing electric fields which are generated by the storage of electric charges in capacitors will cause oscillations of a voltage signal when the electric flux is altered.5 By wrapping the circuit to be protected in a metallic shield, the penetration of electric fields is effectively prevented. This electrostatic shield is the well-known Faraday shield. In most cases, this shield is merely grounded for mechanical reasons, and stray capacitancies between circuit and shield are eliminated rather than acting as sources for crosstalk (Fig. 5.26). In the case of suspecting inductive coupling, the magnetic flux must be prevented from penetrating into the circuit. According to Faraday’s law, the magnitude of the induced surge depends on the penetrated area of the circuit to be protected, the strategies are either drilling of the wires (f < 1 MHz) or the use of coaxial cables (f < 1 GHz). Since the near field of HF fields [characteristic wave impedance significantly different from the value of free space (377 Ω)] amounts to about 1/6 λ, RF sources with an operating frequency of 13.56 MHZ threaten a double hazard capacitive 5 Among disturbing subjects, the experimenting scientist with rubber-soled shoes standing on a floor made of highly non-conductive, organic polymers should not be forgotten.
148
5 High-frequency discharges I
and inductive in nature within a distance of about 6 m. This field is neutralized most elegantly by induced eddy currents in a Faraday shield, i. e. aimed inductive coupling in a sacrificial circuit. On the other hand, the skin depth of an electric field scales inversely with its frequency and the conductivity of the substrate; the remaining field will not be absorbed but will be reflected: aimed capacitive coupling in a sacrificial circuit. Harmonic generation considered, the power density of the losses caused by eddy currents with their vortex diameters in the order of magnitude of the thickness of the shield d can be approximated by [180] dB 1 , (5.117) jE = σE 2 ≈ σd2 ω 2 4 dt this is proportional to the thickness and the operating frequency, both squared, but scales only linearly with the relative permeability (B = μ0 μH) (Fig. 5.27).
absorption loss [dB]
150
100
steel d = 3 mm
Fig. 5.27. The losses by eddy currents depend on the thickness of the metallic sheets, their conductivity and the relative permeability, but above all on the frequency of the penetrating wave field [181].
copper d = 3 mm
50
copper d = 0.06 mm
0 10
102
103
104
105
106
107
n [Hz]
The skin depth of the electric field in good metallic conductors is given by [182] d= √
1 , πμ0 μωσ
(5.118)
which scales at high frequencies with 3
jE ∝ d ω /2 .
(5.119)
This makes readily comprehensible the fact that a thin, highly conducting sheet is more than sufficient for effective capacitive coupling in the sacrificial circuit; for effective inductive coupling, high frequencies, good conductivities and relative permeabilities are advantageous. The last two properties show opposite behavior; however, steel is often favored over copper, especially at frequencies below 100 kHz, which displays its advantage beyond 1 MHz (Fig. 5.27). The generation of eddy currents is much more efficient in bubble-free metals; therefore, the shielding metal used against magnetic fields should be chosen very carefully.
5.11 Shielding
149
In conclusion, it should be emphasized that the coupling of HF power is considerably more difficult than the generation of DC plasmas. To obtain reproducible results, ground loops should be avoided and a well-matched network should respond to slight but inevitable variations of primary process parameters.
6 High-frequency discharges II
After the excursion into transmission techniques, we want now to discuss the electrode processes for capacitively coupled plasmas. Reactors with this coupling technique are widely used for plasma enhanced vapor deposition (PECVD) and for plasma and ion etching. They are introduced in Chaps. 10 and 11. The treatise will be on three different levels: in Sect. 6.1 on a qualitative level, in Sects. 6.2 − 6.6 on a semi-quantitative level, and in Sect. 14.4 on a comprehensive quantitative level which considers the most recent developments. Section 6.7 is devoted to dual-frequency discharges, the most recent development of this type of plasma generation, and Sect. 6.8 deals with scattering of ions within the sheath, a big issue in capacitively coupled discharges exhibiting a thick RF sheath, and which requires knowledge gained in all the preceding sections.
6.1 Introduction When we described the processes of the time-dependent discharge of a target in Sect. 5.1, we neglected the tremendous difference in masses between ions and electrons. Since the acceleration is inversely proportional to the inert mass (a = F/m), we see that the lighter the charge, • the larger the current density (atρ) for a given field, or • the weaker the field that will be required to generate a given current (j = σE = ρ v = ρ at = ρ t F/m). Let’s start with very low frequencies. From the glow lamp, it is well known that the glow occurs in-phase with the net frequency at either electrode — the discharge is ignited and extinguished again and again because the rate of generation of carriers is less than its rate of loss. In this frequency range, the plasma behaves as a DC discharge with fast changing polarity, which could be proven with stroboscopic methods. Great sheath potentials accelerate the electrons into the plasma where they further ionize neutral atoms or molecules and simultaneously attract ions to fall into the deep negative potential thereby gaining more than enough energy to generate γ-electrons during the collision G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 6,
151
152
6 High-frequency discharges II
with the electrode. Sheaths and glowing zones will form at either electrode. The frequency threshold for continuous glowing is several kHz (Sect. 5.1). With growing frequency, we observe a drop in the breakdown voltage. Continous increase lets the frequencies of excitation and the plasma frequency of the ions become similar. For argon ions with a density of 1010 cm−3 we find ω = 18 MHz and ω/2π = 3 MHz [eq. (3.57)], whereas the excitation frequency lies far below the electronic plasma frequency ωp,e and the frequency of elastic collisions between electrons and argon atoms νm [ωp,e = 5.6 GHz, νm ≈ 650 × 106 sec−1 for nAr = 3.54 × 1015 cm−3 (100 mTorr, 13 Pa) and σe = 14 × 10−16 cm2 ]. That means that at the customary excitation frequency of 13.56 MHz, the excitation frequency exceeds the plasma frequency of the ions considerably. But above their plasma frequency, due to their inertia, the ions are no longer able to follow the oscillations of the time-dependent electric field. Hence, they cannot absorb energy from the RF field any longer! In the first half of the RF cycle, the inertialess electrons flow to the electrode. In the second half, the ions are attracted; however, their current is lower by three orders of magnitude [see eqs. (3.15) + (3.16)].1 Hence, we measure a DC current in the circuit through the discharge (so-called “DC coupling”). If the electrode is insulated or we have inserted a high-pass filter (“blocking capacitor”), blocking the electrode will charge it up to a certain negative, time-independent potential. As we can learn from Eq. (3.23), the fraction of electrons which can overcome a potential barrier decreases exponentially. As result, the electron current will drop until it will equal the low ionic current, and the net discharge current will vanish (so-called “capacitive coupling”). A sinusoidal voltage does not oscillate around zero but around a somewhat negative value (Figs. 6.1/6.2)2 (the so-called “DC offset”). In a microscopic picture, the inertless electrons follow the instantaneous electric field within a cloud of stationary positively charged ions. The acceleration of the electrons and their extinction on the surface of the electrode in the positive half wave lead to the formation of a positive space charge above the negatively charged electrode, combined with high an electric field which accelerates the inert ions towards the electrode. At the sheath boundary, the Bohm edge, the ions have already been accelerated to sound velocity:
vB =
kB Te mi
(6.1)
and they have a drift component incident to the electrode of 1/2 vB . Hence, the electrode will be continuously bombarded by ions during the whole cycle, 1 A typical value for np = 1010 /cm3 is je between 30 and 60 mA/cm2 , ji about 0.02 mA/cm2 . 2 In general, the sheath potential at the excited electrode is negative, provided the plasma does not contain a high concentration of negative ions. For this case, the plasma potential can become more negative than the electrode potential (cf. Sect. 9.8)!
6.1 Introduction
153
electrode potential
0
VDC
0
360
720 1080 phase angle [°]
1440
current density [a. u.]
potential [a. u.]
ion current
0 electron current
VDC
0
360
720 1080 phase angle [°]
1440
Fig. 6.1. As a primary result of the different mobilities of the charged carriers, the sheath develops a negative DC bias potential for capacitive coupling (LHS). As a secondary result, the electrons succeed in reaching the electrode only when its potential becomes positive (rigorously taken, this holds only for Te = 0). In contrast, the inert ions experience only the DC potential, which leads to a nearly time-independent ionic current density (for didactical purposes, somewhat exaggerated shown here).
however, with a relatively low current density. The positive current density has become independent of the phase and constant in time. Due to the inertia of the heavy ions, an anode sheath cannot occur any more, and the cathode sheath exists over the whole oscillation period, that is across every surface which is exposed to the plasma. The electronic current density instantaneously follows the potential but is almost zero over the whole cycle because of the retarding negative DC bias developing at the electrode and the reactor walls. It has a sharp peak only when the potential attains its most positive value. Only during the two moments when the pulsating cloud of electrons has just reached the electrode and the potential has been entirely broken down, can the electrons evade the cloud. During most of the cycle, the plasma potential remains positive against electrode and reactor walls. In particular, the electrons of the plasma bulk are confined by the positively charged sheaths. As a result, we obtain a diode-like characteristic for the I(V ) dependence. The generation of this DC offset is intimately connected to the absence of a DC current path between the electrodes, which can easily be fabricated by the implantation of a blocking capacitor into the circuit. Hence, in RF plasmas the discharge can be reduced to the (cathodic) sheaths and the plasma bulk. In the former area, the carriers are accelerated, in the latter one, ionizations take place. In contrast to a passive sheath (without any flow of currents) which exhibits a typical thickness of several Debye lengths and a potential drop of 10 − 15 eV (the floating potential), the application of high RF voltages causes a steep rise of its thickness and its DC component. This can
6 High-frequency discharges II
Ie [a. u.]
Ie [a. u.]
154
net current
net current dc offset rf voltage
rf voltage
Ve [a. u.]
Ve [a. u.]
Fig. 6.2. In their famous 1962 paper, Garscadden and Emeleus first pointed out that application of an RF voltage will develop a negative DC offset and a positive ion sheath. The I(V ) characteristic becomes extremely nonlinear [183].
be inferred from Child’s equation and the Bohmic equation for maximum ion current at the Bohm edge [Eqs. (4.21) (space charge limited ion current) and (14.43)]: 3
4 V2 d2 = ε0 9 j
2e0 mi
∧ jmax = n0 e0
d2 =
4√ e0 V 2e 9 kB Te
kB Te emi
(6.2.1)
3/2
λ2D .
(6.2.2)
For a sheath lying on a potential V = 3 kB Te /e0 , its thickness for the collisionless case is about 2.5 λD . This behavior is directly due to rectification across the sheath which is triggered by nonlinearities of the sheath impedance, its Ohmic and capacitive portion. Concordance with DC sheaths: In sheaths of abnormal discharges, ionization does occur to only a little extent, the potential of the plasma bulk is almost constant, the RF voltage mostly appears across the sheaths. Difference from DC sheaths: The potential of the plasma bulk is far more positive than in DC discharges because the strong screening of the electrode potentials is weakened by the high-frequency oscillations. Hence, the electrons are instantenously sucked out of the plasma bulk which raises its potential. Moreover, the small conduction current across the sheaths is almost entirely due to the ions, which is synonymous with the statement: The current across the sheaths is almost pure displacement current, and the RF current density is significantly larger than the current density of the ions. The lower limit has been estimated by Zarowin to be at about 0.01 n0 (with n0 the electron density in the undisturbed plasma bulk) [184]). Since the density of positively charged
6.1 Introduction
155
carriers across the sheath is constant over time, its visible outer boundary is as sharply defined as in a DC discharge, although the electronic boundary oscillates (cf. Sect. 14.4). An additional collisionless process plays a significant role at very low discharge pressures (< 100 mTorr or about < 10 Pa), when the mean free path of the electrons becomes larger than the electrode gap. The electron sheaths oscillate with the excitation frequency (more or less in-phase, cf. Sect. 14.4). A sheath 1 cm in thickness and growing and shrinking more than ten million times per second must move with a velocity of more than 107 cm/sec, which is equal to electron thermal velocities (cf. Table 5.1). We can discriminate between a spatial and an energetic oscillation. A spatial oscillation occurs during an oscillation period because the boundary of the electron sheath between Bohm edge and surface of the electrode moves back and forth. An energetic oscillation takes place since the sheath potential changes periodically. In either case, the local plasma potential is subject to change as well: During sheath retreat, the plasma potential will rise and electrons will drift to the electrode. When the sheath expands in the second half of the cycle, the plasma potential decreases. This potential instantaneously acts on the inertless electrons which are accelerated out of the sheath into the plasma — the electric field is still more intense than in the plasma bulk: Es > Ep ! The energy gain or loss is approximately the difference between the plasma potential and the Bohm potential at the end of the Bohmic presheath because it is this very place where the plasma electrons are hit by the breathing sheath. As result, we observe a high-energetic tail in the EEDF at the Bohm edge. The collision will happen at the sheath boundary and scales with the square of the frequency. This is the so-called stochastic heating, which was named after the stochastic randomness of the collisions between sheath and electrons (cf. Sect. 6.5). The magnitude of the potential of the plasma bulk will be positively biased against either electrode to such an extent that most of the plasma electrons are effectively trapped in this zone. For DC discharges, the thickness of the dark space is roughly proportional to the inverse pressure: dc ∝ 1/p. In RF discharges, however, the sheath thickness will become constant for discharges pressures below 150 mTorr (20 Pa, Fig. 6.3). This behavior is mirrored in the parallel dependence of the sheath potential (Fig. 6.4). Since γ-electrons have become unimportant for the maintenance of the discharge, a description of the properties which determine the nature of the sheath is possible applying the sheath equation (or even better approximations, so-called α-mode) [186]. For even higher frequencies, ωRF and ωp,e and νm , respectively, will become comparable. This has at least these consequences: • The wavelengths are comparable to the plasma dimensions (diameter of the reactor). For ω/2π = 2.45 GHz, λ is 12.25 cm, and the radial field becomes inhomogeneous.
6 High-frequency discharges II
100
10 powered
impedance [W]
sheath thickness [mm]
156
electrode grounded
10 1 50
100 discharge pressure [mTorr]
Fig. 6.3. Sheath thickness in oxygen as a function of discharge pressure. Circles: excited electrode, triangles: grounded electrode. On the right ordinate, the RF c impedance is plotted [185] ( The American Institute of Physics).
500
• If the excitation frequency exceeds the plasma frequency of the electrons, the discharge behaves as a low-loss dielectric and the sheath potential drops to the floating potential. In the evanescent regime, the plasma is opaque, and it will become more or less transparent when we pass the threshold of the electron plasma frequency ωp,e . Furthermore, resonance effects can occur (cf. Chap. 7).
bias potential [V]
1000
RF electrode, measured RF electrode, calculated grounded electrode, measured grounded electrode, calculated
100
10 40
100 discharge pressure [Pa]
Fig. 6.4. Since the sheath voltage depends on the discharge pressure to almost the same extent as the sheath thickness, the electric field across the sheath does not exhibit a strong pressure dependence (completely capacitive behavior assumed) [185] c The American Institute of ( Physics).
500
We distinguish the E-types by their coupling (DC or capacitive) or by the reactor type (hexode reactor or parallel-plate reactor, respectively). The capacitive system is discriminated between symmetric and asymmetric reactors. In this chapter, we focus on capacitively coupled discharges but will not refrain from mentioning some of the features of the other types.
6.2 Electric fields across the sheaths
157
6.2 Electric fields across the sheaths In this section, we will focus on the plasma behavior at the border of the plasma frequency of the ions, ωp,i , whereas frequency effects far beyond ωp,i are deferred to Sect. 6.7. From the preceding section, we can draw the conclusion that the DC conductivity in the plasma bulk is determined by the electrons, and in the sheaths, however, by the ions. In these zones we must discriminate between two borderline cases: • Below the plasma frequency of the ions, ωp,i , the ions are not inert, but mobile, i. e. we observe conduction current in the plasma and in the sheath (low-frequency range). • Above ωp,i , we find capacitive displacement current (except a very small ionic current which is caused by the drift in the electric field), in particular at the common frequency at 13.56 MHz (high-frequency range). That means that the total current density can be calculated according to j = jc + jd = (σi + σe + ε0 iω) E
(6.3)
with jc the density of the conduction current, jd the density of the displacement current, σi the ionic conductivity, σe the electronic conductivity and ω the angular operating frequency. Using the free-electron model, the so-called DC conductivity can be calculated according to σDC =
e20 n τ e2 n = 0 , m mνm
(6.4)
with τ the time between two collisions and 1/τ = νm ω.3 τ can be estimated by 1/τ = n0 σm < v >
(6.5)
with n0 the density of the neutrals and σm the cross section for momentum transfer. • Within the plasma, σi σe and σe → σDC for νm ω in the pressure range down to 50 mTorr (in argon), for the low-pressure region, which is used for reactive ion etching, νm ω (Sect. 2.2), and σe → iε0 ω ε˜ [eq. (14.154)], • but in the sheath because of ni ne : σi σe . 3 Because n = n(x), σ exhibits a spatial dependence (cf. Chap. 7). σ is connected with the mobility μ via σ = ρμ = e0 nμ.
158
6 High-frequency discharges II
Hence, the current density in the plasma can be approximated by jp ≈ (σe + ε0 iω) Ep ,
(6.6.1)
which further simplifies to ⎫ ⎬
νm ω : jp ≈ σe Ep ;
(6.6.2)
νm ω : jp ≈ ε0 iωEp . ⎭
In both cases, the plasma frequency of the electrons exceeds the typical excitation frequency by one or two orders of magnitude. On the other hand, we find in the sheaths σi ε0 ω, which means ⎫
high frequency range js ≈ ε0 ωEs ; ⎬ low frequency range js ≈ σi Es .
(6.6.3)
⎭
Since σi depends on the ionic density which strongly decreases across the sheath in the direction of the electrode due to the enhancement of the ionic drift velocity, the electric field must vary as well, but in an inverted dependence because of the constancy of js . For high frequencies, however, the electric field across the sheaths will be constant (only displacement current). Due to the principle of continuity, the current in the plasma must equal the currents across the sheaths, i. e. Ip = Is or jp Ap = js As , and we get for the currents in the two limiting cases ⎫
High frequency range σe Ep Ap = ε0 ωEs As ; ⎬ ⎭
Low frequency range σe Ep Ap = σi Es As .
(6.7)
The differential resistance in the plasma bulk can be calculated with the resistivity, dp its length and Ap its cross section dRp = ·
ddp ddp = , Ap σe Ap
(6.8)
and we get for the Ohmic heating within the plasma bulk4 dPp = kΩ
Ip2 ddp j 2 Ap j 2 Ap = kΩ p ddp ⇒ Pp ≈ kΩ p dp σe Ap σ σ
(6.9)
kΩ is a constant which contains the time-averaged sheath thickness and the Debye length [187]. Eventually, we obtain the electric fields for the highfrequency and low-frequency range, respectively (ε0 Es,1 As,1 = ε0 Es,2 As,2 ):
High frequency range Es =
1 ε0 ωAs
Low frequency range Es =
1 σ I As
P p σ e Ap ; kΩ dp
P p σ e Ap kΩ dp
⎫ ⎪ ⎪ ⎬
⎪ ⎭ : ⎪
(6.10)
4 Additionally, the electron density is axially and radially dependent, but only to a weak extent.
6.2 Electric fields across the sheaths
159
the electric field across the sheath is inversely proportional to the electrode surface and directly proportional to the plasma current and the square root of the power coupled into the plasma bulk, respectively. The electric fields in the plasma and across the sheaths can be obtained from σe Es ≈ Ep σi
(6.11.1)
for the low-frequency range, and for the high-frequency range from Es iωε ≈ . Ep iωε0
(6.11.2)
According to Eq. (14.171), the collisionless dielectric constant remains negative for ω < ωp,e and ε ε0 : The electric field across the plasma bulk is directed in the opposite direction to the fields across the sheaths and is much smaller. We find the important results: 1. The RF voltage will drop across the sheaths generating an intense electric field, and only a small electric field is set up across the plasma bulk. 2. The electric field in the LF case is much higher than in the HF case.
Example 6.1 With < v >= kB T /mAr and a discharge pressure of 100 mTorr (13 Pa), a plasma density of 109 /cm3 and the values for the temperatures T , the cross sections σm , the collision frequencies νm , the specific conductivity σ and the resulting electric resistivity for the • low-frequency case with Es /Ep = σe /σi = 67; • high-frequency case with Es /Ep = σe /(2π × 13.56 MHz) = 4.7. For the high-frequency case, the sheath field has dropped to about 1/10 of that for the low-frequency case!
Table 6.1. Calculated values for the electric conductivity of electrons and argon ions according to the free electron model.
T [eV] electrons 3 ions 0.1
σm νm σ [10−16 cm2 ] [10−6 Ω−1 cm−1 ] 20 0.7 GHz 400 25 350 kHz 6
[kΩ cm] 2.5 167
Since the sheath field determines the ionic drift across the sheath, and for the high-frequency case, this field is significantly smaller than for the low-frequency case—all other conditions kept constant—this difference must be taken into consideration for the construction of reactors for ion or plasma etching (cf. Sects. 11.2). In the HF case, the relative strength of the electric field is weakened by
160
6 High-frequency discharges II
• rising excitation frequency (Fig. 6.5) and • increasing collision frequency (i. e. growing particle density or rising discharge pressure).
bias potential [V]
400
300
Fig. 6.5. Bias potential as a function of RF power coupled into the plasma bulk at 13.56 MHz and the second harmonic at 27.12 MHz (high frequency case above ωp,i ) after [188]. p = 600 mTorr (80 Pa); flux ratio GeH4 :H2 = 10:90; area ratio of the electrodes (“hot” to “cold” including the quartz cylinder: 0.65.
RF electrode, 13 MHz RF electrode, 27 MHz grounded electrode, 13 MHz grounded electrode, 27 MHz
200
100
0
0
25
50 RF power [W]
75
100
At lower frequencies, the higher electric field leads to further growth of the sheath thickness. This, in turn, leads to a rise in electron temperature [Eqs. (3.48) and (3.49)], since the bulk volume shrinks but the relative plasma surface remains the same. ε0 ∇E s = e0 (ni −ne ) we obtain for the directionz normal to the electrode ε0 dE ≈ e0 ni dz. Integrating and considering that E will vanish at the (with E = E0 at the electrode surface). If we sheath boundary yields ds = ε0eE(0) 0 ni suppose the potential gradient to be linear (or a potential which depends on d2 ), this remains valid also for the voltages (Vs = Es ds ; and with the dependencies discussed in the preceding section with a slightly higher dependence on As ): ε0 Es As =
ε0 Vs As Ip Ip ds ⇒ ε0 Vs = = ds ω ωAs
(6.12)
with ε0 As /ds the sheath capacity [189].
6.3 Current-voltage characteristic at one electrode In Figs. 6.2 we have seen that the I(V ) characteristic of the sheath is similar to that of a rectifying diode. Its Ohmic part has been outlined in Sect. 3.4; the nonlinearities are controlled by the sheath thickness and the (potential dependent) ion distribution across the sheath. Due to the progressive depletion of electrons across the sheath, which starts at equilibrium values at the Bohm edge but quickly falls to zero, at RF frequencies above the plasma frequency of the ions, ωp,i , e. g. at 13.56 MHz
6.3 Current-voltage characteristic at one electrode
161
(ωp,i < ω ωp,e ), the capacitive displacement current across the sheath nearly equals the conduction current of the electrons within the plasma bulk.5 However, not only the electron density is inhomogeneous but also the ion density must drop to smaller values because the ions are accelerated to the RF driven electrode by the negative DC potential. Even if the discharge is excited by a harmonic voltage which is responded by a harmonic displacement current, the dropping charge density across the sheath leads to a nonlinear response of all the other properties. In terms of the equation of continuity, we can express the condition at the migrating Bohm edge, the last point for which evenness between ni and ne is given [with ω the operating frequency and a one-dimensional problem in x-direction with ˜j = j0 cos ϕ(x) with ϕ the spatial-dependent phase angle which peaks at the sheath boundary with ϕ = 0] dϕ ∂ e0 ni = 0 ⇒ j0 sin ϕ = e0 ωni . (6.13) ∂t dx Evidently, the nonlinear response is caused by the spatial dependence of the phase angle. Appropriate models for a symmetric discharge have been established by Godyak and Sternberg [190, 191] and Lieberman [192]; a model for a capacitive/resistive sheath has been introduced by Klick [193, 194]. Provided that the quantities are known which define the character of the capacitively coupled plasma, the plasma density np , ji , ion current density to the electrodes, and ds , the thickness of the sheaths adjacent to the electrodes, it is possible to calculate the efficiency of power coupling. It has turned out that an uncritical application of equations by which the DC discharge can be described (Bohm’s theory, Child’s law and Langmuir theory), is very questionable [191]. Due to the extreme complexity of the exact models describing the sheath dynamics, in this section, we restrict ourselves to a qualitative treatment of the properties which control the capacitive coupling. However, it is possible to understand the development of the high negative potentials at the RF driven electrode with a one-dimensional model. In this approximation, electrons and ions both exhibit zero temperature. The DC fraction of the sheath voltage is caused by the different mobility of the charged carriers which leads to initial current densities which differ by about three orders of magnitude. Equilibrium at the electrode is reached after several RF cycles when the electron current density is reduced to the ion current level, and the condition for equilibrium reads ∇ · j˜ +
1 2π [ji − je (t)]d(ωt) = 0. (6.14) 2π 0 The electron current density, je , instantaneously follows the electrode potential, i. e. the charge density of the electrons is time-dependent, and it drops from the equilibrium value at the Bohm edge to zero above the electrode (cf. Fig. 5 In more exact treatments, the positive conduction current across the sheath must not be neglected.
162
6 High-frequency discharges II
6.6). The current density of the positive carriers, ji , is independent of time and phase above their plasma frequency due to the inertia of them. Across the whole sheath between x = 0 and x = ds , an excess of positive carriers is required for its stability (cf. Sect. 14.2).6 For harmonic excitation, the extension of the “breathing” sheath is given by
electrode
plasma _ ne
Fig. 6.6. Principal structure of the RF sheath with quasi-stationary ions after Godyak and Lieberman [191, 192].
ni
ne(t)
x
se(t)
dS - se(t) dS
0
se (t) = se cos ωt,
(6.15.1)
ds = s0 + se cos ωt
(6.15.2)
which leads to
as the total extension of the sheath. In the direction of the electric field, the (linear) charge density which is piled up in the sheath amounts to ds −se cos ωt 0
e0 nB dx = e0 nB (ds − se cos ωt),
(6.15.3)
with nB the density of the charged particles at the Bohm edge by which the developing electric field [Es Ep (Sect. 6.2)] is given according to Poisson’s law to Es ≈
e0 nB (ds − se cos ωt) , ε0
(6.16)
and the potential Vs = VDC VDC = −
VDC = −
e0 nB [ds − se cos ωt]2 ⇒ 2ε0
(6.17.1)
e0 nB 2 s2 ds − 2se ds cos ωt + e (1 + cos 2ωt) . 2ε0 2
(6.17.2)
If the potential of the DC sheath becomes positive, it will be opened, and in this short part of the RF cycle the same amount of electrons will flow to 6 This assumption is not entirely correct, but simplifies the picture of two telescoped sheaths.
6.4 Sheath potentials
163
the electrode to equalize the ion current over nearly the whole part of the RF cycle. Furthermore, we see that the response leads to the generation of a second harmonic and a constant average value.
6.4 Sheath potentials 6.4.1 Symmetric system A typical symmetric system in a parallel-plate reactor consists of two electrodes of equal size; one of them is grounded, the other one connected to the high-frequency source via a blocking capacitor which acts as a high-pass filter suppressing the DC current.7 In the state of equilibrium, the electron current to the excitation electrode must equal the positive current because the capacitor blocks the conduction current. Since the plasma bulk must lose an equal number of ions and electrons, this very condition applies also to the grounded electrode. Due to its high electric conductivity, the plasma bulk is an equipotential zone, and the high-frequency generator couples to the plasma across both the electron-deficient dark spaces with same geometry, i. e. same thickness d, same electrode area A and same capacitity C (Fig. 6.7).
electrode sheath
Rh
RP
plasma potential VP(t)
Rw wall sheath
Ch
Cw
Fig. 6.7. Electric model of an RF discharge with an excitation frequency ω/2π above the plasma frequency of the ions ωpi . The plasma is regarded as an electric conductor with highly mobile electrons, whereas in the sheaths of the electrodes, the electron density (and hence the electric conductivity) drops to zero which let them act as capacitancies for the RF current. The diodes in the circuit describe an electron flux across the sheaths for the situation Φp < Ve . By this, the uppermost value of the c The American Institute of Physics). sheath potential is fixed at zero [195] ( 7 The area of the grounded electrode encompasses all the surfaces which are not connected with the RF source.
164
6 High-frequency discharges II
To even out the currents of electrons and ions, the excitation electrode must develop a negative bias voltage, thereby creating an uphill potential for the electrons and a downhill potential for the ions. Hence a DC component is added to the (symmetric RF) sheath potential. At the RF driven electrode, the difference between the applied and (self-biased) electrode potential Ve (t) + VDC and the plasma potential Vp (t) yields the sheath potential Vs (t). At the grounded electrode, this difference between zero and the plasma potential Vp (t) equals the sheath potential Vs (t) (Figs. 6.8).
Fdc [a.u.]
p/2, 3p/2
p
F [a.u.]
0 0
Vdc
0
electrode distance [a. u.]
electrode distance [a. u.]
Fig. 6.8. LHS: the spatial dependence of the DC potential across the discharge in a symmetric discharge. The plasma potential exhibits a significant DC component and equals the sheath potential at the grounded electrode. RHS: the electrode potentials at the RF driven electrode for four equidistant moments of the RF cycle.
The lowermost value of the plasma potential is controlled by the high mobility of the electrons. It must not be less than the lowermost potential of any surface in contact with the plasma bulk, the grounded counterelectrode. This condition defines the instantaneous plasma potential according to 1 (6.18) Vp (t) = VRF (1 + sin ωt) . 2 Both electrodes are almost equivalent and are equally bombarded by plasma particles (Figs. 6.8). The counterphase shift with respect to the other electrode is reflected by an equation similar to eq. (6.17): VDC =
e0 nB (ds + se cos ωt)2 2ε0
(6.19)
which yields for the entire voltage drop VΣ across both sheaths [sum of eqs. (6.17) + (6.19)]: VΣ =
2e0 nB ds se cos ωt. ε0
(6.20)
6.4 Sheath potentials
165
The voltage drop depends only on the fundamental (ω = 2πf ) of the operating voltage.
6.4.2 Asymmetric system 6.4.2.1 Sheath potential theorem. In a typical asymmetric system, we choose the magnitude of the blocking capacitor to be much larger than the capacities of the sheaths. Because of the series connection of the capacities, it does not affect the properties of the plasma bulk and the sheaths. In this configuration, the DC component of the sheath potential of the grounded electrode can differ significantly from its counterpart at the RF driven electrode. In the model of a parallel-plate capacitor, the sheaths are assumed to play the role of the capacitive medium between the two electrodes. For a fixed charge, the area of the electrodes mainly determines the capacitance and the potential drop. This was the begin of a long-lasting debate. Koenig and Maissel pointed out that the assumption of a space-charge limited current would lead to an extremely strong dependence of the voltage ratio across the sheaths on the area ratio [156]:
A1 V1 = V2 A2
4
.
(6.21)
The sheath theorem has been subject for numerous investigations. For its validity, the ion current is supposed to be space-charge limited which does not hold true for pressures above 1 mTorr (0.1 Pa). This means in practice, that we have to face a hybrid between a space-charge limited and a mobility-limited discharge, and the exponent drops down to values between 1 and 2 [95, 196]. Evidently, the discharge will contract at higher discharge pressures, and a lower exponent is just pretended [185] (cf. Sec. 11.4). On the contrary, the exponent can be fixed at 2 and a ratio of “active” surfaces can be defined [188]. Eventually, it could be shown that it is the capacitance of the sheaths rather than the area of the electrodes that controls the ratio of voltages. The voltage which is applied to the RF driven electrode will be divided between the two electrodes and the plasma bulk: VRF sin ωt = (Vh − Vc ) sin ωt + E(t) [l − dt (h) − dc (t)] .
(6.22)
l the distance between the electrodes; dh , dc is the sheath thickness of the RF driven electrode (hot) and grounded electrode (cold), resp., E(t) ist the electric field across the plasma bulk. Provided that the voltage drop across the plasma bulk can be neglected, i. e. that the definition of a plasma potential Vp does really make sense and that the applied RF voltage is not severely distorted by electrical nonlinearities, we can calculate the sheath potentials.
166
6 High-frequency discharges II
6.4.2.2 Calculation. For vanishing DC conduction current in the sheaths, i. e. for Rh ∧ Rc → ∞, the sheaths represent two capacitors connected in series which are charged up to charge Q by VRF : V RF = V h + V c = Q ·
1 1 + , Ch Cc
(6.23)
with V h and V c the potential differences between plasma and target and between plasma and wall, respectively—both voltages are difficult to measure—and Ch and Cc the capacities of the two sheaths. Provided that the DC current through the external circuit vanishes which is ensured by the blocking capacitor, the current incident on the RF driven electrode must equal that incident on the grounded electrode Ih = Ic ⇒ QT = QW ⇒ Ch (Vh − Vp ) = Cc (Vc − Vp ),
(6.24)
which defines the capacitive voltage division: Ch Vh − Vp = . Cc Vc − Vp
(6.25)
In the approximation to first order, the magnitude of the capacitance is controlled by the area ratio of the RF driven electrode and the other surfaces exposed to the plasma, morever, it is determined by the voltage drop across the sheaths. This voltage drop, in turn, influences the sheath thickness via Child’s law: C = C(A, V, d), leading to a thick high-voltage sheath across the RF driven electrode, and a thin low-voltage sheath across the other surfaces. This implies that the capacitive impedance of the larger electrode will have a larger capacitance per unit area and rises by a factor which is larger than the pure area ratio. In the following paragraph, we deal with a completely capacitive coupling of the plasma bulk with plasma potential Vp against the RF driven electrode. As has become evident from previous sections, we must distinguish between a DC part and an RF fraction [195]. The electrode potential of the RF driven electrode can be written as Ve (t) = VDC + VRF sin ωt.
(6.26)
Provided that the sheaths act purely capacitively, the instantaneous plasma potential oscillates around a DC value, V p with amplitude aVRF : Vp (t) = V p + aVRF sin ωt,
(6.27)
with a the coupling factor a=
Ch (V ) . Ch (V ) + Cc (V )
(6.28)
6.4 Sheath potentials
167
The DC component of the plasma potential, V p , is controlled by the ratio of the mobilities electron/ion—as in a DC discharge. Vp (t) cannot be less than the potential of any surface exposed to the plasma: • The potential of the grounded electrode, V = 0. • The potential of the RF driven electrode, V (t) = VDC + VRF sin ωt. The maximum of the plasma potential is V p + aVRF ≥ VDC + VRF ,
(6.29.1)
V p − aVRF ≥ 0.
(6.29.2)
and its minimum
Since we discuss a capacitively coupled plasma with a vanishing external current, both types of carriers must succeed in reaching the electrode during a comparatively long part of the RF cycle (ions) or a sufficiently short part (electrons). Hence, its minimum turns out to be zero, and its maximum to be VDC + VRF , i. e. both the inequalities (6.29) become equations: 1 Ch − Cc . (6.30) (VRF + VDC ) ; VDC = VRF 2 Ch + Cc The sheath potential across the RF driven electrode is the difference between the electrode potential (sum of the applied RF voltage and the DC bias, VDC = Vh − Vc ), and the plasma potential, and can be written approximately as Vp =
Vs (t) = VRF
Cc 1 (sin ωt − 1) = VRF (sin ωt − 1) , Ch + Cc 1 + CChc
(6.31)
which can be written with Eqs. (6.28) − (6.30), (VDC ≤ 0): VRF − VDC · (sin ωt − 1) . (6.32) 2 The sheath potential of the grounded electrode is equal in magnitude but opposite in sign from that of the plasma potential (Fig. 6.9) (VDC ≤ 0) : Vs (t) =
Vp (t) =
VRF VRF + VDC · (1 + sin ωt) = · (1 + sin ωt) . 2 1 + CChc
(6.33)
We distinguish between these borderline cases: • Symmetric system (Ch = Cc )VDC = 0, Vp (t) = 1/2 VRF sin ωt. • Asymmetric system (smaller electrode is RF driven): VDC < 0; Vp (t) < 1 /2 VRF sin ωt; Vp,min = 0, Vp,max = VRF .
168
6 High-frequency discharges II
Vp(t)
V [a. u.]
0 Vs(t)
Ve(t)
VDC
0
90
180
270
360
Fig. 6.9. Comparison of the instantaneous potentials of the plasma bulk, the RF driven electrode and its sheath for a capacitively coupled plasma at the RF driven electrode (ωRF > ωp,i , sheath: Vs , RF driven electrode: Ve , plasma potential: Vp ); phase shifts are neglected. Electrons can flow only for Ve > 0.
phase angle [°]
• Asymmetric system (larger electrode is RF driven): VDC > 0; Vp (t) > 1 /2 VRF sin ωt; Vp,min = 0, Vp,max = VRF . Besides the rectification across the sheaths, we observe in asymmetric systems the additional effect of voltage division between the electrodes. The plasma potential oscillates around a positive DC value, and its lowermost value during the negative half of the RF cycle is just zero. On the other hand, the electrode potentials oscillate around a negative DC value, and their uppermost value during the positive half of the RF cycle equals zero as well. The negative DC value of either electrode develops according to the law for voltage division for pure capacitive coupling. Using laboratory slang, this DC value is denoted “self-bias” voltage or simply “DC bias” or “DC voltage”. Hence, the voltage of the sheath Vs (t) is composed of several components: • A time-dependent RF voltage around a negative DC component for the electrode. • A time-dependent RF voltage around a positive DC component in the plasma bulk. They are split according to the ratio of capacitive voltage division between the electrodes (Fig. 6.10). In the case Ch ≈ 1/3 Cc , the DC bias becomes half the peak-to-peak voltage (VDC ≈ −1/2 VRF and V p ≈ 1/4 VRF ). Typical values for parallel-plate reactors are less than 0.05 [195], and VDC → −VRF . These functional relationships are compiled in Fig. 6.11 along with the grounded case, a configuration with several severe disadvantages. To begin with, no DC bias can be developed. If the small electrode is RF driven, the plasma potential will never come close to ground. The energy of the ions incident on the electrode, however, is the sheath potential, a quantity which is composed of several components. Among them, the plasma potential and the electrode potential are the two most important
6.4 Sheath potentials
169 Fig. 6.10. Spatial dependence of the DC component of the potential across an asymmetric discharge. The sheath potential drop across the grounded electrode differs substantially from the sheath voltage at the RF driven electrode. The plasma potential exhibits but a small DC component.
Fdc [a.u.]
0
electrode distance [a. u.]
potentials. Both of them exhibit a DC and an RF component. Therefore, it is a disastrous misapprehension to conclude etch rates or sputter rates from a knowledge of only the DC bias which have been established in different plasma reactors. For the long-term process control in the same reactor, however, this datum is of high heuristic value.8 For this derivation we did neglect the floating potential, which is calculated according to Eqs. (3.24). In particular for small capacitancies, the absolute value of VDC can become approximately VRF , which almost causes the plasma potential to vanish. Hence, small errors in measuring VDC and VRF strongly influence the value of the plasma potential (for further details see Sect. 14.4.2).
6.4.3 Resistive Coupling The main reason for the difference between the measured plasma potential and its derived value is the assumption of a totally capacitively coupled discharge. To put it shortly, the resistivities of the sheaths, Rh and Rc , which have been neglected so far, have to be taken into account. There is not only a displacement current, IRF , across the sheaths, but we must consider also the small positive ion current Ii [197]—this is simply required for a clear definition of the plasma potential. This effect becomes effective at higher plasma densities and leads to a drop of the plasma potential, since the sheath begins leaking (Fig. 6.12; again, we neglect Vf ), and this results in an additional inductive/resistive component. Let us compare the capacitive coupling with the inductive/resistive coupling as borderline cases: 8 To check the asymmetric behavior of the reactor, the etch rate can be checked with substrates which have been mounted to both electrodes.
170
6 High-frequency discharges II capacitive coupling
DC coupling V(t)
V(t)
V(t)
VP(t) asymmetric small electrode powered
0 VDC
0
VP VS 0 VDC
VP VS 0
V(t)
VP(t) symmetric
0
0
VP VS 0
VP VS 0
V(t)
VP(t) asymmetric large electrode powered
VDC
0
0 VP VS 0 VDC
VP VS
0
90
180 270 360 Phase angle [°] Phasenwinkel [°]
450
540
0
90
180 270 360 Phase angle [°] Phasenwinkel [°]
450
540
Fig. 6.11. Schematic representation of the time-dependent variation of the plasma potential Vp (t) (dashed) and the electrode potential Ve (solid) for three different system geometries, totally capacitive conduct of the sheaths for DC coupled and capacitively coupled electrode, respectively. System geometries: (1) sputtering and ion c The American etching, (2) PECVD and plasma etching, (3) hollow cathode [195] ( Institute of Physics).
• Capacitive coupling: Rh , Rc → ∞, Ch , Cc : finite. • Inductive/resistive coupling: Ch , Cc → 0, Rh , Rc : finite. Ions can only respond to the time-averaged DC potential, i. e. for capacitive coupling, there remains a small but continously flowing current, whereas for inductive/resistive coupling, parts of the wave are omitted by clipping (rectifying behavior of the sheath, Figs. 6.1/6.2 + 6.7). Harmonic excitation V (t) = VRF cos ωt provided, the sheath potential crosses zero for
6.4 Sheath potentials
171
Vp(t) for IRF >> Ii
Fig. 6.12. Comparison of the plasma potentials Vp (t) for purely capacitive (IRF Ii ) and purely resistive (IRF Ii ) cases and a case lying in between. Vf has been neglected, i. e. VRF + VDC Vf [195] c The American Institute of ( Physics).
potential [a. u.]
Vp(t) for IRF ~ I i
0
Vp(t) for IRF << Ii
V(t)
VDC
V(t)
phase angle [a. u.]
1 αi VDC = , arccos (6.34) ω VRF ω the first roots can be found at α/ω and 2π/(ω − α/ω). Therefore, the timeaveraged mean value of the sheath potential—and the ions respond only to this value due to their inertia—is (VDC ≤ 0) : ti =
1 t2 (VRF cos ωt + VDC )dt. T t1
(6.35)
1 (VRF (sin(2π − α) − sin α) + VDC (2π − 2α)) 2π
(6.36)
Vs = For resistive coupling: Vs =
sin α α , (6.37) + VDC 1 − π π for capacitive coupling (with t1 = 0, t2 = 2π) the phase factors can be omitted: V s = −VRF
Cc Vh − Vp = ; C Vc − Vp h
(6.38.1)
Cc VRF − VDC . (6.38.2) =− Cc + Ch 2 For a given ratio VDC /VRF , the sheath potential in the capacitive case is at least equal to or greater than the sheath potential in the inductive/resistive case. As a matter of fact, reality lies in between, and both types of coupling will become comparable: Logan et al. proved that the ratio between the capacitive load and the resistive load is about a factor of 6 for a typical argon discharge at 13.56 MHz between 7.5 and 20 mTorr (1 to 3 Pa) [198] (Fig. 6.13). Detailed investigations for the case 1 to 100 mTorr (0.1 to 13 Pa), which is the working window for reactive ion etching and RF diode sputtering, have been carried out by Horwitz [199]. He pointed out a change of mechanism in Vh − Vp = V s = −VRF
172
6 High-frequency discharges II
1.00
I
C-coupling
Vt
0.75
R/L-coupling
0.50
0.25
0.00 0.00
0.25
0.50 VDC /VRF
0.75
1.00
Fig. 6.13. Time-averaged electrode potential for a capacitively coupled and an inductive/resistively coupled plasma as function of the ratio VDC /VRF .
the lowermost pressure range (≤10 mTorr or 1 Pa). The lowering of the DC bias voltage is interpreted as a manifestation of the growing importance of electrode processes (generation of γ-electrons). Due to their larger λe , the electrons are instantaneously sucked off by the electrodes and consequently are not available to generate new carriers in the plasma bulk. Since these electrons have to be delivered by the smaller electrode, its (negative) bias voltage will decrease. This conduct is emphasized by the wall material. Because the yield of γ-electrons is far higher in dielectrics, the discharge can be operated at significantly lower pressures in reactors with anodized surfaces than in reactors with metallic walls. For a low-frequent operation with a small excitation voltage, the resistive model can be applied. The higher the operating frequency, the better fits the capacitive model in with reality. This holds particularly true for high operating voltages since the plasma potential and possible deviations from the sinusoidal behavior play an ever-decreasing role. Significant voltages can occur at the grounded electrode. Hence sputtering is possible from substrates which are placed on this electrode. To avoid sputtering from the walls, their potential must not exceed 20 V which can be controlled via the area ratio powered/grounded electrode. In the DC case, an insulating substrate can be charged only to very low voltages. In RF driven discharges, however, the sheath voltage of an insulator can reach disastrous voltages which leads to sputtering and subsequent crosscontamination.
6.5 Power input In the last section, we have extensively dealt with capacitive coupling and the development of a self-bias voltage across the RF driven electrode which is caused by the different mobilities of ions and electrons. This effect can be tremendously
6.5 Power input
173
intensified by capacitive voltage division. We have seen that the accuracy of this model is restricted to high frequencies well above the plasma frequency of the ions ωp,i and to low power input with small electron conduction current. However, this model is purely static and does not take into account the dynamics of the processes in the sheath. We can describe the equivalent circuit of an asymmetric discharge using Fig. 6.14 when we consider the resistance of the plasma bulk to be purely resistive. The Ohmic load of the sheath Rs should consist of two parts: the resistance caused by the Ohmic heating, RΩ , and the part resulting from stochastic heating, Rst : Rs = RΩ + Rst . These two contributions have to be completed by a capacitive part, C0 , and an inductive part, L0 , which can be referred to the electron inertia in the HF field. The sheaths are assumed to be composed of two parts which are connected in parallel: • The Ohmic resistance, Rs , which is caused by the acceleration of the charged electric carriers by the rectified sheath voltage which leads to energy dissipation. • The capacitive part of the displacement current, Cs . This capacitance can be calculated for two sheath resistances of area A and thickness ds connected in series according to
Lp
Rp
Cs,1
sheath 1
Fig. 6.14. Equivalent circuit of a capacitively coupled discharge with L-network which is considered symmetric for Cs,1 = Cs,2 .
plasma bulk Cp
sheath 2 Cs,2
1 1 1 = + Cs Cs,1 Cs,2
(6.39.1)
with Cs,1 = Cs,2 = ε0 ε
A , ds
(6.39.2)
i. e. Cs = Cs,1 /2 we obtain for Rs Rs = 1/ωCs =
2ds . ε0 εωA
(6.39.3)
174
6 High-frequency discharges II
Since the density of accelerated carriers drops to zero in the sheath, we can set ε = 1. Provided that the resistances of sheath and plasma bulk, respectively, are connected in series in the simplest model: R = Rp + RΩ + Rst = Rp + Rs ,
(6.40)
then Ip = Is , and we find: V = I(Rp + Rs ) = Vp + Vs , 9
(6.41)
and the absorbed power is approximately 1 Pabs = IRF (Vp + IRF Rs ) 2
(6.42)
with Vp the plasma potential, the subscript “RF” denotes the amplitude of the RF field. From Eqs. (6.40/6.41), it becomes evident that the RF component of the sheath voltage equals the discharge voltage only for small Rp . This holds true for highly excited plasmas. The DC component of the sheath voltage behaves similarly (cf. Sect. 14.4). For low sheath voltages it approaches Vf which drops at every sheath. But even then, Vs remains large compared with kB Te . For frequencies which exceed ωp,i , the displacement current dominates the I(V ) characteristic in the sheath, whereas in the plasma bulk, almost all current is electronic conduction current (see Sect. 9.1 for a detailed discussion). Furthermore, we find for the potential drop across the sheath with Eq. (6.41) Vs =
V 2 − I 2 (Rp2 + 2Rp Rs ).
(6.43)
In the glowing plasma bulk, the potential drop is almost independent of the discharge current due to the equilibrium of ionization. Borderline cases are: • For small currents I, the absorbed power depends linearly on I;since the constant is Vp,RF , the I(V ) characteristic follows an Ohmic law. • For large currents, the coupled power is expected to depend on the square of the discharge current with a small linear part of the first summand from Eq. (6.41): P ∝ I 2 . In a vivid picture, this change manifests the transition from the heating of electrons to the heating of ions. Furthermore, two other mechanisms are of importance, especially at low discharge pressures the displacement current heating in the breathing sheath and stochastic heating. 9
This sum is the real part of the resistance; the capacitive part is not included!
6.5 Power input
175
6.5.1 Sheath heating 6.5.1.1 Displacement current heating. Due to this breathing (beyond the plasma frequency of the ions ωp,i ), during each HF cycle the voltage moves back and forth through the plasma, and since the sheath region is deficient in electrons, the current is mainly displacement current. Its magnitude is governed by ∂E , (6.44) ∂t and during its flow, it causes a voltage drop due to the sheath impedance Zs = ˙ = ωCs VRF 1/(ωCs ) where the sheath capacitance Cs is given by I = Q˙ = Cs VRF (here, we just take into account the first harmonic). This leads to heat dissipation according to P = V × I. The energy flux S which is dissipated into the volume V of length l can be calculated roughly according to j = ε0
S= V
Zj 2 dx ⇒ S ∝
2 Zω 2 VRF l . σDC
(6.45)
Since the current is proportional to frequency, and the power scales with the squared current, the amount of power dissipated is proportional to the frequency squared. Due to reasons of charge continuity, this displacement current must equal the electron current at the Bohmic edge. 6.5.1.2 Ohmic heating. For this effect, we note that the density and the field will vary across the sheath. Assuming a sinusoidal response of the current density to the sinusoidal excitation (this is a rough approximation; in fact, the processes within the sheath are highly nonlinear) and assuming a constant electric field, we get for the real part 1 2 (6.46) j dx σ with x the instantaneous position of the sheath, which depends itself on the phase dP = IdV ⇒ dP = jAEdx ⇒ dS =
x = x[φ(t)].
(6.47)
The power gained in one RF cycle from φ = 0 to φ = 2π or from x = 0 to x = s is then dS =
m e νm 2 2 j sin (ωt)dx e20 ne (x) 0
(6.48)
with dx = ωdt. Likewise, applying the differential Ohmic law (j = σDC E), Eq. (6.48) can be written
176
6 High-frequency discharges II P j2 = RF V 2 σDC
(6.49.1)
with 2 = jRF
j02 sin2 (ωt). 2
(6.49.2)
In the case of finite electron density, Ohmic heating occurs [kΩ a constant (Sect. 14.4.1), nB the electron density at the Bohm edge and n0 the electron density in the plasma bulk, Ie,s the electronic sheath current, Rs the Ohmic part of the sheath resistance, σ its DC conductivity, As its area and ds its thickness]: 1 dPs = kΩ j · E dx, As Ss = kΩ
ds 2 je,s 0
σ
dds ⇒ Ss = kΩ
(6.50.1) 2 je,s ds . σ
(6.50.2)
In particular for high sheath potentials and large sheath thicknesses, Ohmic heating is comparable or can even exceed the contribution of the plasma bulk. 6.5.1.3 Stochastic heating. From Eqs. (5.11) which consider the effective field, as well as from Eqs. (6.45) and (6.49), we see that all these heating mechanisms depend on νm which drops with falling pressure. However, stable operation even at pressures where the discharge is expected to be already extinguished (depending on the gas, between 30 and 100 mTorr), added experimental evidence to the suggestion that a collisionless mechanism would take over at low pressures which might be overshadowed at higher pressures but is mainly responsible for the opening of a working window of low-pressure operation in HF discharges, the interaction of plasma electrons with the pulsating sheath [200, 201]. To get an idea of the impact, we consider a sheath 1 cm in width which moves back and forth 107 times per second. Its velocity of 107 cm/sec is comparable to thermal electron velocities. As result of this collision (the sheath is supposed to play the role of a rigid barrier with its mass assumed to be infinite compared with the electrons), the electron velocity will rise from v0 , the initial velocity before the collision, to v0 + 2vs with vs the velocity of the oscillating sheath boundary (stochastic heating); this mechanism becomes important in discharges of inert gases below 100 mTorr (13 Pa) [202].10 Since the electrons move instantaneously in response to the field to which they are subjected, a net gain for an ensemble of electrons can only be ensured for collisions which happen randomly or in a stochastic manner, as in the case of Ohmic heating (Sect. 5.2). 10 Recently, its significance has been demonstrated in capacitively coupled discharges of reactive gases as well [203].
6.5 Power input
177
Since this picture is adopted from the conservation laws of momentum, Lieberman coined this behavior the hard wall model [204].11 It is abbreviated either HWM or HWA (for approximation). Due to its enormous complexity, there is still no really comprehensive and mature theory [207]. Stochastic heating across the sheath. At zero order, Ohmic heating is characterized by the collision frequency of electrons with neutrals which reads νm = nn σ < ve > (nn : density of the neutrals, σ: elastic cross section, < v e >: electron mean velocity) which must be rewritten in the case of multiple gases, taking into account the dependence of the cross section on < v e >, and with the discharge pressure p as the controllable parameter νm =
p pi nn σi (ve ) < ve >, k B Tn i p
(6.51)
where i denotes the index and pi the partial pressure of species i, Tn the temperature of the neutrals in K.12 Especially at low discharge pressures, when the mean free path approaches the characteristic length of the reactor, the transfer of energy from the electric field to the electrons cannot happen by Ohmic heating. Due to the relatively small number of collisions, a collisionless mechanism dominates the heating of electrons. This process is caused by the “breathing” RF sheath. This movement pushes the electrons back and forth and transfers energy to them. This possibility of such a collision-free mechanism for RF discharges, based on the Fermi acceleration [208] was first introduced by Godyak [209, 210] (Fig. 6.15). 9
10
8
-1
neff [s ]
10
experiment theory
107
106 -4 10
10-3
10-2
10-1
Fig. 6.15. The effective electron collision frequency in a RF discharge through mercury at 40.8 MHz (3 × 13.56 MHz), length of the tube: 6 cm, diameter: 10 cm) [210].
discharge pressure [Torr]
11 As experimentally shown by Popov und Godyak, this mechanism is (at least in discharges through metal vapors) not sufficient to explain the high degree of ionization [205]. They concluded that a collisionless mechanism is reponsible for this effect (surfing or wave riding), at least at pressures below 10 mTorr (1 Pa) [206]. 12 In several cases (H2 and He) the dependence of the cross sections of the neutrals σ on < ve > cancels approximately out and then the collision frequency is merely a function of particle density (cf. Sect. 2.3.4).
178
6 High-frequency discharges II
Historical approach. According to kinetic gas theory, we can evaluate the energy gain of the electrons with initial velocity v incident on a breathing sheath with velocity vw . Due to the sheath’s velocity, only electrons with velocities larger than vw can collide with the sheath in its retreating phase: The number of collisions is reduced, whereas in its expansion phase, the number of collisions is enlarged. We consider the electrons at the sheath edge (density n) to be distributed according to n=
∞ −∞
fw (v, t) dv dt,
(6.52)
where fw (v) denotes the electron energy distribution function EEDF at the sheath boundary, and the electrons which will collide with the sheath (expressed as current density) is dj = (v − vw ) dn = (v − vw )fw (v, t) dv dt.
(6.53)
For v ≥ v w : collision, but reduced intensity, for v ≤ v w : no collision, for v s = 0: immobile wall, we obtain the same result as in kinetic gas theory, for v w < 0: collisions, more intense than against an immobile wall: je (v) =
∞ v−vw
(v − vw )fw (v, t) dv dt
(6.54)
and the force per unit area exerted on the electrons can be expressed with vr = −v + 2vw as dp vw = 2m(vw − v) (v − vw )fw (v, t) dv dt dt dx which yields for the transferred power F =
(6.55)
dP = F dv = 2m(vw − v)(v − vw )vw fw (v, t) dv.
(6.56)
In order to determine fw , we consider a Maxwellian distribution in the undisturbed plasma bulk denoted g(v ) and connect it with the distribution within the sheath via v = v − vs with vs the drift velocity of the electrons at the sheath edge [192] to obtain [211] P = −2m
∞
∞
vw −vs
vw −vs
(vw − vs )(v + vs − vw )2 g(v ) dv+
(6.57)
vs (v + vs − vw )2 g(v ) dv .
Conservation of current requires vw = vs at the sheath edge which lets the first integral in Eq. (6.57) vanish: Reflection from the breathing sheath does not change the distribution of thermal electron velocities. It was the second integral which was supposed to be responsible for stochastic heating [192, 212]. However, Gozadinos et al. pointed out that the second integral of Eq. (6.57) describes the maintenance of the electron drift in the field and its time-averaged integral vanishes as well [211].
6.5 Power input
179
Pressure heating. Gozadinos et al. used an extended fluid approach leading to pressure heating [207, 211]. Despite the pressure heating appears to be much more consistent, both approaches represent different points of view to the same mechanism. The usual fluid approach based on the first-order moments of the Boltzmann equation provides for the permittivity ωe2 =1+ , 0 iω (iω + νm )
(6.58)
2 , a series expansion and for the dissipated RF bulk power assuming ω 2 νm 2 considering all terms of the order of ω or lower
PΩ =
m e νm l e20 ne
1+2
ωk2 1 2 j , ωe2 k 2 k
(6.59)
taking into account higher harmonics k > 1 of the discharge current (density) e2 n 2 j. The term containing the plasma eigenfrequency ωp,e = m0e e0 in Eq. (6.59) reflects approaching local plasma oscillations which also cause an increase of RF power dissipation into the plasma bulk due to an increase of the RF electric field through a low electron density. The first term is the ohmic (real) part of the resistivity and l denotes the effective length of the plasma (electrode gap minus the sheath lengths). Strictly speaking, calculation of the RF current within the chamber and a subsequent averaging were required, but for an approximation of first order, it is sufficient to note that the effect has to be taken into account also when the electron density decreases only in the outer regions of the plasma bulk, which can occur in particular in electronegative gases. The collisionless heating is also expected to depend on j 2 , so that it can be also characterized by a “sheath” collision rate νse , and an effective collision rate can be defined as
νeff = νm 1 + 2
ωk2 2 ωp,e
+ νse
(6.60)
replacing now νm and the first-order plasma oscillation term in Eq. (6.59). Combined model. The models known from the literature describe the collisionless heating as stochastic or pressure heating and rely on the questionable assumption of a sinusoidal discharge current. They provide, after some algebra with the assumption of a constant ion density within the sheath (matrix sheath) [206], 2v e , (6.61.1) se with self-consistent modeling of collisionless movement of ions in the sheath but assuming the minimum sheath thickness to be zero [192], νse =
180
6 High-frequency discharges II
1 e0 V , νse = se πme
(6.61.2)
and, using Lieberman’s collisionless sheath model [192] with a new, more realistic and consistent pressure heating approach [207, 211]:
νse =
⎛ 7 e0 V ⎜ ⎝1 +
18se
πme
⎞−1 1 2e0 V ⎟ ⎠ ,
45π
kB Te
(6.61.3)
with V the mean sheath voltage. The stochastic heating approach results in enhanced heating in case of a decrease of the ion density within the sheath. This is based on a higher electron drift velocity in the quasi-neutral sheath region. The pressure heating depends entirely on the gradient of the ion density within the sheath only and disappears for a homogeneous ion density within the (matrix) sheath [211]. But all Eqs. (6.61) indicate the heating to be independent of gaseous pressure and electron density in the case of a collisionless sheath. However, for the pressure range under consideration this is really true only at very low pressures (below 1 Pa, 7.5 mTorr). Compared to the collisionless case, the gradient of the ion density within the sheath is decreased with growing pressure by the lower drift velocity due to energy loss by collisions with neutrals. This can also be concluded from Gozadinos’s pressure heating model [211] where the core model does not depend on the sheath model but depends strongly on the ratio of the electron density at the boundary to the undisturbed plasma bulk and the sheath edge to the “real” sheath nse /nB . This ratio in “quasi-neutral” range of the sheath equals the ratio of the corresponding ion densities. We can therefore expect a pressure-independent νse in the tiny range between the minimum pressure and slightly higher pressure where the free mean path is still well above the sheath thickness. For a medium pressure above 1 Pa (7.5 mTorr), where the (stochastic or pressure) sheath heating is still dominating, increasing pressure results in a lower νse and νeff . Using the general pressure heating approach [211] provides approximately π ve νse ≈ 8 se
'
ns ln e nB
2 (
.
(6.62)
The Bohm criterion, the conservation of mass and energy and estimating the average in Eq. (6.62) provide √ π e0 V νse ≈ , 8 se me
(6.63)
the square root dependence on V in Eqs. (6.61.2) and (6.61.3) is also caused by the dependence of the density of the ions on their energy. At sufficiently low
6.5 Power input
181
pressures, the ion energy is determined by the sheath potential V . For arbitrary pressures, this provides finally the estimation νse ≈
νse |p=0 1+
σ
p
se knB+Tn
,
(6.64)
where se and σn+ denote the mean sheath thickness and the ion-neutral collision cross section, respectively. Thus the pressure dependence described above can be reduced to the ratio of the sheath thickness and the ionic free mean path, i. e. both collisionless and collisional ion motion will contribute to the heating mechanism. With further growing pressure, Ohmic heating increases and will become the important heating mechanism at pressures well above 75 mTorr (10 Pa) [205] which also leads to a significant change of the EEDF [204, 191]. Substituting the term for the frequency according to Eq. (6.60) leads to a slight modification of the formula for stochastic heating as published by Misium et al. [187]:
3
se /2 me νeff 2 se je A, (6.65) Pst = kst λD e20 nB which takes into account the inhomogeneous ion density and the instantaneous electron density across the sheath. The occurence of the sheath thickness in Eq. (6.65) stresses the importance of this heating mechanism especially at low pressures. 6.5.2 Two regimes of power transfer Capacitively coupled plasmas are generated in the evanescent regime in which the operating frequency f falls short of the plasma frequency of the electrons but exceeds the plasma frequency of the ions. As conseqence, the penetrating electric field is reduced by a factor of 3 within 60 cm at topmost plasma densities of 1010 cm−3 (collisional plasma), which is shortened to 30 cm in collisionless plasmas; reducing the plasmas density lengthens these values by a factor of 3 for every decade (cf. Sect. 14.6). No parallel-plate reactor is known to exceed this dimension nor is any likely to be built. But is should be kept in mind that the mode of excitation will gradually change with increasing plasma density. For low-density plasmas, the intensity of the electric field within the plasma bulk Ep is relatively high, Ep =
j0 me νm , e20 np
(6.66)
and the heating of electrons occurs mostly by Ohmic heating in this area, which is large compared with the sheaths. The mean energy of the electrons is relatively low, and therefore, these electrons remain trapped in the plasma bulk which masters the processes in the discharge.
182
6 High-frequency discharges II
For plasmas with higher density, sheath heating becomes predominant. Since the density of the sheath electrons is dramatically lower than their plasma counterpart, we find Es =
j0 m e ν m Ep , e20 ns
(6.67)
and due to this acceleration, their velocity significantly exceeds that of the electrons in the plasma bulk. The intensity of the electric field Es is determined by the balancing of the ionization processes (which determine the ion density in the sheath). Here, the sheaths will dominate the processes in the discharge.
6.6 Spatial distribution of charged carriers In weakly ionized plasmas, the spatial distribution of the charged carriers is determined by the equilibrium between carrier gain and carrier loss. The latter mainly depends on ambipolar diffusion. At pressures of several Torrs which are customary in microwave discharges, the axial diffusion can be approximated by a cosine function, and the radial diffusion by a Bessel-function of the first kind (Schottky profile):
n(x, r) = n0 cos
πx r , J0 2.405 2L R
(6.68)
with L = a − ds half the distance a between the electrodes (plasma half-length), reduced by the sheath thicknesses ds (Sect. 5.4). We have seen in Sect. 4.7.3 that the zero boundary condition which are applied to solve the diffusion equation leads to a singularity in the ambipolar drift velocity. This is no issue in microwave-driven discharges since they are operated at comparably very high pressures for which the diffusion coefficient Da drops to very small values. However, in RF driven discharges, the discharge pressure is lower by at least two orders of magnitude, and in this pressure range, the drift velocity of the ions exceeds their thermal velocity considerably, and the carrier profile is not very well fitted by Schottky’s approach. This problem has been circumvented by Person who took the inertia of the ions into account assuming that the plasma density at the boundaries remains finite [213]. To make this happen, the ambipolar drift velocity must be equal to the Bohm velocity, i. e. the velocity of the ions is on the brink of supersonic behavior. Based on this assumption, Self and Ewald presented a one-dimensional distribution which contained the two cases at the lower and upper pressure limit model λi L and λi L (Langmuir profile vs. Schottky profile). For the lower limit, the ion transport is not mobility-limited any longer but determined by resonant charge transfer, and the drift velocity has to be replaced from ui = μi E to ui = 2Ekin,i /pi which exceeds the thermal ion velocity considerably. During their course across the Bohmic presheath, the ions have been accelerated to the Bohm velocity, which
6.6 Spatial distribution of charged carriers
183
is always large compared with their thermal velocity in the plasma bulk. But this means that the frequency of the ion-atom collisions becomes independent on the ambipolar drift velocity which prevents a smooth joining of the two borderline cases. For a better approximation, a non-linear transport equation has to be solved, which has been worked out for the first time by Valeri Godyak [214] and has been further refined by Lieberman and Lichtenberg [103]. As in the previous model introduced by Self and Ewald, it is closely correlated with the ionic mean free path, λi . Following Godyak, the plasma density at the edges for x = X and r = R amounts to [214]: ns,X 0.86 ns,r 0.80 ≈ ∧ ≈ . X n0 n0 3 + λi 4 + λRi
(6.69)
Both the distribution profiles (radial and axial) resemble a Langmuir profile with a plasma density on the boundary which amounts 1/2 to 1/3 of that along the discharge axis. At the Langmuir limit, the axial value saturates at 0.5 n0 , and its radial counterpart at 0.4 n0 (Fig. 6.16). λi is defined according to the 0.5 0.4
n a,r/n0
0.3 0.2 0.1 0.0 0.00
axial shape radial shape
0.25
0.50
0.75
1.00
Fig. 6.16. Cylindrical plasma: The plasma density at the axial and radial boundary has dropped to less than half its value along the axis after [102].
l i/L,R
kinetic theory of gases 1 ; (6.70) nσi for σi , we have to insert the sum of the cross sections for elastic scattering and the resonant charge transfer, which are in the order of 10−14 cm2 , and we obtain a λi of about 3 mm in argon at 10 mTorr (1.3 Pa). λi =
The integration of the nonlinear differential equation is sketched in [214] and in great detail in [215]. They start with the equation for ion continuity (ρi = e0 ni ) ∇ · ρi + νi ρi = 0, a Maxwellian distribution of the electrons
(6.71.1)
184
6 High-frequency discharges II
ne = n0 exp
e0 Φ , kB Te
(6.71.2)
and a certain drift velocity of the ions
ui =
2λi e0 . πmi μi
(6.71.3)
After having introduced reduced variables which define a variable α, a normalized ionization rate which describes the balance of ionization and loss at the boundary, and which is the product of the reduced ionization length c=
νi d vB
(6.72.1)
and the ratio between the axial dimensions of the reactor and the ion mean free path
b=
πd , 2λi
(6.72.2)
α = cb
(6.72.3)
they consider slab geometry, and inserting (6.72.3) into (6.72.1) yields the nonlinear differential equation with ξ = x/d, y = n/n0
vB
2λi d π dx
−n
dn = νi n dx
(6.73)
which yields a value for the parameter α and for the relative distribution y(ξ) for the boundary conditions v0 = 0 ∧ vi = vB : ⎡
2 α /3 ξ
1
⎤
1 1 1 π 2(y 3 − 1) /3 − 1 ⎦ √ = ln[(1 − y 3 ) /3 + y] + √ arctan ⎣ + √ . 2 3 3 6 3
(6.74)
For λi /d → 0, α reaches its maximum value at
α0 =
2π √ 3 3
3
=
4 3
(6.75)
which can be seen from Fig. 6.17.
From Eq. 6.74, we obtain the general dependence of y. The relative density 2 n/n0 solely depends on the parameter α /3 ξ with α determined by Eq. (6.72.3) and shown in Fig. 6.18. In this figure, also the Schottky profile and the circular equation y 2 + ξ 2 = 1 are shown. For the low-pressure case, the plasma density within the plasma bulk is much flatter (almost constant) than in the highpressure case but drops significantly steeper than we could expect from eq. (6.68).
6.7 Dual-frequency discharges
185
1.4 a0 = 1.33
a
1.3
1.2
Fig. 6.17. Normalized ionization rate as function of the relative ionic mean free path after [102].
1.1
1.0 1E-3
0.01
0.1 l/2d
1.00
Y
0.75
0.50
y(x) x 2 + y2 = 1 cos(x)
0.25
0.00 0.00
0.25
0.50 X
0.75
1.00
Fig. 6.18. The relative plasma distribution between the electrodes for a planar geometry in axial direction after [102].
6.7 Dual-frequency discharges 6.7.1 Frequency dependence of plasma density In Sect. 5.2, we dealt with spatially homogeneous discharges, and found that raising the operating frequency at equal power input leads to an increase in effective field and high-frequency current. This may be explained by electron inertia which will come into play (although at a plasma density of 1010 /cm3 , the angular plasma frequency is 5.64 GHz, i. e. higher by a factor of 66 still than the operating frequency of 13.56 MHz). Since the instantaneous oscillations of the high-frequency field are not exactly followed as for lower frequencies, higher electric fields are required to obey Poisson’s law or to ensure the effective control of space-charge even in small parts of the RF cycle. As a result, electron inertia counteracts the frequency effects discussed in Sects. 6.2 and 6.5. According to Surendra and Graves who were the first to present MonteCarlo simulations, the sheath thickness scales almost inversely with frequency, whereas the plasma density rises with the square of the applied frequency [161]:
186
6 High-frequency discharges II
• The rising sheath field which is mainly due to a decreasing sheath thickness leads to enhanced ionization rate ⇒ • The plasma density in the plasma bulk will be increased (Fig. 6.19). • Since the electrode sheaths shrink in thickness, the plasma bulk will extend in size [216] (Fig. 6.20). This leads to a reduction of the ratio of plasma volume over surfaces confining the plasma [(Eqs. (3.43 and 3.44)]: • Although the plasma density increases with rising operating frequency, the electron mean energy (electron temperature) remains almost unaffected (Fig. 6.19).
100
jRF
10
1010
0.1
<je>
109 50
100
150 200 p [mTorr]
250
150 200 p [mTorr]
P [mW/cm2]
10
1 <Ee>
1 50
100
150 200 p [mTorr]
250
0.1 300
Fig. 6.19. The plasma density scales with the squared operating frequency, whereas the sheath thickness scales nearly inversely with ωRF according to Monte-Carlo simulations by Surendra and Graves. [161]
electrons
100
Ei dsh
0.01 300
ions
50
E [eV]
np
sheath thickness [mm]
e0 Fp
1
j [A/cm2 ]
np [cm -3]
1011
250
6.7.2 Mutual influence of two electrodes Operating at two different frequencies opens the window to steer ion flux and ion bombardement energy almost independently. Ideally, we want to control the plasma density by the high-frequency excitation [Eq. (6.45)] S=
2 Zω 2 VRF P dx. = A σDC d
(6.76)
6.7 Dual-frequency discharges
187
200
<Ez> [V/cm]
100 0 13.56 MHz 27.12 MHz 40.68 MHz 60 MHz
-100 -200 -300 0.0
0.5
1.0
1.5 2.0 z [cm]
2.5
3.0
Fig. 6.20. The plasma bulk extends its volume at the expense of the shrinking sheaths with rising frequency [216]. Shown here is the axial electric field for a discharge through hydrogen at VRF = 100 V and p = 300 mTorr (40 Pa), reactor dimension: diameter: 12.8 cm, electrode distance: 3.0 cm.
Since the power is proportional to the square of the sheath voltage, the amount of the dissipated power scales as the square of the frequency, the fundamental equation of this effect. 6.7.2.1 Narrow gap. When we turn on the low-frequency electrode, a highvoltage sheath will form. Its extension from the surface of the electrode up to the Bohm edge where the plasma condition of time-averaged quasi-neutrality is violated) is governed by the Poisson equation along with Gauss’s law: V
∇ · E d3 x =
A
E · dA =
A
ρ ρ ρ dA ∧ ∇Φ = − ⇒ Φ ∝ d. ε0 ε0 ε0
(6.77)
To ensure the stability of the sheath for rising voltage, the sheath must thicken. This, in turn, leads to a reduction of the plasma volume. Since the effective boundaries have not been changed significantly, the electron temperature must rise according to Eqs. (3.43 + 3.44) which causes an increase in plasma potential [cf. Eq. (9.2)]. Comparing the modeled sheath thickness, the plasma potential and the mean bombardment ion energy 1 (6.78) Ei = VDC + kB Te 2 with 1/2 kB Te the (directed) ion energy at the Bohm edge and considering a collisionless sheath, Boyle et al. found an almost perfectly congruent behavior [217] (Fig. 6.21). The steeper slope of the plasma potential compared with the mean ion energy is due to the enhanced number of collisions when traversing the thickening sheath. 6.7.2.2 Wide gap. Since the electron temperature remains almost constant after having added the low-frequency electrode to the discharge, we must con-
188
6 High-frequency discharges II
6
80
60 F [V]
ds [mm]
5
4
40
3 0
50
100 150 RF voltage [V]
200
20
250
0
50
100 150 RF voltage [V]
200
250
0
mean ion energy [eV]
-10
DC bias [V]
-20 -30 -40 -50 -60
0
50
100 150 RF voltage [V]
200
250
30
powered electrode
20 grounded electrode
10 0
50
100 150 RF voltage [V]
200
250
Fig. 6.21. Using a Monto-Carlo method, Boyle et al. simulated the sheath width, the plasma potential, the self-bias voltage, and the mean bombardment ion energy in a dual-frequency discharge as functions of the low-frequency voltage and found almost congruent behavior [217]. The power at the high-frequency electrode has been kept constant at 1 200 W.
clude that the processes triggered by the two electrodes have become almost mutually independent. Only in the high-energy regime, there remains an influence of the electrode driven with lower frequency [218]. It is these electrons which are important for the operation of the discharge. The characteristic of the EEDF has now almost entirely transformed to a Maxwellian shape, a straight-line behavior in contrast to a convexly shaped slope for a Druyvesteynian distribution (Fig. 6.22). It is noteworthy to mention the resemblance of this transformation from a Druyvesteynian behavior to a Maxwellian conduct when lowering the discharge pressure, as was first noted by Godyak and Piejak [219]. They reduced the operating pressure in a discharge of argon from 3 000 mTorr (400 Pa) to 70 mTorr (10 Pa). The EEDF changed in shape from a convex shaped Druyvesteynian behavior to a straight-line Maxwellian behavior, but with two different temperatures; i. e. straight lines but different slopes (Fig. 6.23).
189
1000
1000
100
100 EEDF [a. u.]
EEDF [a. u.]
6.7 Dual-frequency discharges
10 1 0.1 0.01 0
10
1
Vlf = 0 V Vlf = -280 V
0.1
Vlf = 0 V Vlf = -200 V
5
10
15
20
25
0.01 0
30
5
10
15
20
25
30
electron energy [eV]
electron energy [eV]
Fig. 6.22. Influence of the electrode gap. LHS: Small gap (large a part of the gap consists of sheaths), the convexly shaped Druyvesteynian distribution transforms to a Maxwellian distribution with two temperatures when the voltage at the low-frequency (LF) electrode rises from 0 V to 280 V. RHS: Large gap (sheaths are only a small fraction of the gap), the shape of the EEDF is nearly Maxwellian [218].
3.00
1010
2.00 109
[T or r]
pr es s
0.30 0.20 0.10
ur e
0.50 107
ch ar ge
f [eV-3/2 cm-3 ]
1.00 108
di s
106 105
0
5
10
15
0.07
electron energy [eV]
Fig. 6.23. Lowering the discharge pressure leads to a transformation of the EEDF. At large pressures, a Druyvesteynian behavior is observed which gradually changes into a Maxwellian distribution with two temperatures [219].
190
6 High-frequency discharges II
This behavior was repeated by Abdel-Fattah and Sugai when they enhanced the ωHF /ωLF -ratio (HF means high(er) frequency, LF stands for low(er) frequency) from unity to 4 (from 13.56 MHz to 54.24 MHz) [220] (Fig. 6.24). This transformation is explained by the increased number of collisions between electrons by which every Druyvesteynian distribution will eventually turn into a Maxwellian distribution (cf. Sect. 14.1).
Tbulk Ttail
f [eV -3/2 cm-3]
109
108
50 44 37
107
27 106
0
5
10
2p w/
[M
]
Hz
13
E [eV]
Fig. 6.24. Raising the discharge frequency by a factor of 4 from 13.56 MHz to 54.24 MHz entirely alters the characteristics of the EEDF from a Druyvesteynian distribution into a Maxwellian distribution, but with two temperatures. This behavior indicates the transition to a second heating mechanism from collisional Ohmic heating to collisionless sheath heating [220].
6.8 Collisional sheaths The maximum kinetic energy of the bombarding ions which have traversed the sheath can be calculated as (Fig. 6.9): e0 Vs = e0 · (V p − Ve );
(6.79)
this is the kinetic energy of an ion incident on the electrode which had not suffered any collisions when traversing the sheath. Relatively early, it became obvious that several scattering mechanisms must exist which cause significant broadening of the ion energy distribution function
6.8 Collisional sheaths
191
(IEDF) and the ion angle distribution function (IADF) which leads to a lowering of the mean energy of the ions. Eventually, a state of equilibrium between the heavy constituents of the plasma is achieved [221]. Several questions arise: • How long is the mean free path λ for an ion in a sheath of thickness ds , or how large is the ratio λ/ds , or how many collisions will happen when an ion traverses the sheath? • Is there an influence of ion inertia on the ion velocity, or does there exist an influence of the phase of the RF cycle on the ion velocity, and if yes: Can this amount be quantified? • Does there exist a correlation between the kinetic energy of the ions and its broadening in the spatial (IADF) and energetic domain (momentum, IEDF)? In general, the ions which enter the pulsating sheath need a time which equals some periods of the RF cycle before they have traversed the sheath. In a typical capacitively coupled plasma, the maximum velocity of an Ar+ -ion in a sheath field of 500 V/cm would be 5 × 106 cm/sec; the time required for traversing the sheath would then be 0.2 μsec. The duration τ of a complete RF cycle is 1/13.56 × 10−6 = 0.074 μsec. We infer from Fig. 6.9 an accelerating force on the ions in the negative half-cycle when the sheath extends and the instantaneous plasma potential decreases; in the positive half-cycle, this force inverts its sign. Therefore, ds (t) and V (t) exhibit a mutual phase shift of π. This means that it takes at least three oscillation periods until the argon ion has crossed the sheath and consequently, it will hit the electrode with the time-averaged sheath potential. As a result, we will see but a small width in the ion energy distribution function IEDF around the mean potential. For excitation frequencies below the plasma frequency of the ions, ωRF < ωp,i , however, ion energies can vary from very small values (when the sheath thickness is thin) to energies which equal the peak RF voltage (when the sheath thickness is at its maximum). The sheath thickness itself can be split into two parts, an RF component and a DC component: ds (t) = ds,DC + ds,RF sin ωt.
(6.80)
Kushner estimated the ratio ds,DC /ds,RF to about 10 in argon for perfectly capacitively coupled sheaths, i. e. for ωRF ωp,i [222]. In this case (typically ωp,i /2π ≈ 1 MHz), the ion motion can be regarded as the response to a mean RF field. • For a sheath thickness which significantly exceeds the mean free path of the ions, λi , the mean sheath voltage is large compared with the
192
6 High-frequency discharges II kinetic energy of the ions incident on the electrode (divided by their charge).
• For comparable values of ds and λi , the mean ion energy is expected to equal the mean sheath voltage. A precise approach to solve this problem would require the solution of Boltzmann’s transport equation for ions, electrons and neutral molecules which are coupled with the Poisson equation for a self-consistent field. The solution of the coupled integral-differential equations with correct boundary conditions would yield the corresponding distribution functions [223]. Another track is the Monte-Carlo method or particle-in-cell method (PIC) [189, 221]. A third track has been found by Economou et al. 1988 [224]. Since it is lucid and transparent, this model will be presented here in detail. Without temperature gradients and magnetic fields, we can consider a plasma consisting of neutrals and singly ionized ions which can be described by three mutually coupled differential equations: 1. The continuity equation [thereby the sheath is expected to lack any source (i. e. vanishing divergence)], which is a rather good approximation due to the very low electron density, which eventually vanishes. 2. The equation of motion of ions traversing the sheath with Newtonian friction (in principle, this equation should take into account the density gradient ∂ni /∂dx). 3. The Poisson equation: ∂ni ∂ni ui + = 0; ∂t ∂x
(6.81.1)
∂ui qE Fi ∂ui + ui = + ; ∂t ∂x m m
(6.81.2)
∂E qni . = ∂x ε0
(6.81.3)
To estimate the force of friction—and every scattering can be regarded as friction—all possible collisions of a test ion with an ensemble of neutrals which are Maxwellian distributed are averaged. The result is a quadratic dependence in ui : Fi = −γi u2i with γi the so-called resistance term. For that end, we consider the relative density ni of particles with mass mi after scattering into the solid angle element dΩ = 2π sin ϑdϑ. Considering conservation of energy and momentum yields for the geometric factor 1 − cos ϑ (cf. Sect. 2.2) which promotes large-angle scattering. For a hard sphere potential 2 , the resistance term can be written as σ(ϑ) = πrhs
6.8 Collisional sheaths
193
2 γi = mi ni πrhs .
(6.82)
γ scales with discharge pressure. The resonant charge transfer consists in an (almost) entire exchange of momentum between a fast moving ion and a neutral atom with relatively low velocity. The idealized cross section is then approximated with a δ-function according to 2 σct (η) = πrct
δ(π − η) , sin η
(6.83)
which yields for the total cross section 2 2 σtot = π(rhs + rct ).
(6.84)
The boundary conditions are given by the Bohm criterion (cf. Sect. 14.2). Albeit derived for DC sheaths, we can apply it to the quasi-stationary RF sheath. First of all, the three coupled differential equations can be simplified with the time-averaged E-field with E = −∇V. d(ni ui ) = 0; dx ui
dui q dV γi u2i = − ; dx m dx m d2 V qni =− . 2 dx ε0
(6.85.1) (6.85.2) (6.85.3)
vi = kB Te /mi denotes the (maximum) initial velocity at the Bohmic edge. In fact, it is somewhat smaller, since the ions suffer friction losses in the Bohmic presheath, which exhibits a thickness of about one a Debye length. At the boundary presheath/sheath (x = 0), the boundary conditions are approximate ni (0) = n0 ;
ui (0) =
dV dx
kTe ; m i + γi λ D
= x=0
kB Te . qλD
(6.86.1)
(6.86.2) (6.86.3)
Inserting the Bohm velocity into the formula for the collision number per second, which is well-known from kinetic gas theory, Z =< u > /λ =< u > nσ (with the mean free path λ and the mean velocity < u >: < u >→ vB ) yields a lower value for the collision number between the ions and the neutrals in the sheath. For an electron temperature of 2 eV in a discharge through argon (σtot : 26 × 10−16 cm2 ), we obtain the following values:
194
6 High-frequency discharges II
• p = 10 mTorr (1.5 Pa), ds = 10 mm: one collision, • p = 50 mTorr (7 Pa), ds = 5 mm: 3.5 collisions, and • p = 500 mTorr (66 Pa), ds = 4 mm: 30 collisions. Already at relatively low discharge pressures, we find equilibrium between the electrostatic force and friction [225]. 6.8.1 Experiments In the following paragraph, we deal with • scattering by elastic collisions, • scattering by elastic collisions and resonant charge transfer, and • effects of the pulsating RF sheath (RF modulation). It is common to apply a series of vertically arranged consecutive grids which are driven at different voltages (Sect. 2.5) [226]. The resolution depends on their distance, their open area (wire diameter) and the aperture A (Fig. 6.25) [227, 228]. Using various grids with different open surface (1−3), it is possible to vary the disc’s section C which is exposed to the beam: between 0.4 and 5 cm2 , i. e. more than order of magnitude. By application of accelerating or retarding potentials (from −125 to +125 V), it is possible to repel one type of carriers completely (so-called retarding field analyzer). ion beam
Fig. 6.25. Retarding field analyzer: Applying various apertures A and grid distances with various open areas (1 − 3) it is possible to vary the current which will caught by the disc C and eventually measured by the detector. By continous sweeping of the DC potential, the IEDF is determined [229].
The aperture A and the first grid are grounded to generate an area free of fields between the neutralization grid of the ion source and the analyzer for the ion energy. To suppress the influence of the electrons on the current signal, the potential of the downmost grid was fixed at −80 V. This is sufficient to
6.8 Collisional sheaths
195
block the electrons from the downstream region and secondary electrons from the collector electrode from penetrating the beam. By deriving the curve for ion current with respect to voltage, the ion energy distribution function IEDF is determined (Fig. 6.26). 0.003 ion current density
0.100 ion energy distribution
0.075 0.050
0.002
0.001
dj/dV [a.u.]
ion current density [mA cm-2 ]
0.125
0.025 0.000
0.000 0
50
100
150
200
Fig. 6.26. Ion current density and ion energy distribution function (dI/dV ) of an argon discharge at 10 mTorr, 150 W, boundary voltage: 50 V [229].
retarding potential [V]
6.8.1.1 Elastic scattering and resonant charge-transfer. The resonant charge transfer is the most significant scattering mechanism for an ion in its parent gas, e. g. Ar+ in Ar (so-called “symmetric” charge transfer); its cross section is remarkably large (cf. Sect. 2.5). On the contrary, the “asymmetric” cross section is significantly smaller; this has been shown for discharges through − − SF6 (SF6 → SF+ 5 + F ; also ions as SF5 are detectable in high concentrations which determine the electronegative character of SF6 -discharges) by Brandt and Jungblut [230]. In most cases, these competing dissociations in several fragments reduce the density of the M+ -ion noticeably, and as a consequence, the resonant charge transfer does not play this paramount role in discharges through molecular gases than in ones through inert gases, however, it cannot be simply neglected. According to analyses with the mass spectrometer, the M+ peak does not disappear until the electron energy has passed 50 eV or higher [231]. But the ionization processes mainly occur in the plasma bulk where the mean electron energy amounts to only several electronvolts! DC discharges. The energy spectrum of ions incident on the electrode was measured for the first time by Davis and Vanderslice in a DC discharge for the three lightest inert gases [128]. Their experiment consisted of an electrode with a drilled hole and attached mass spectrometer; the (optical) sheath width could be measured with an accuracy of ±5 %. Provided that • the field declines linearly across the dark space, • all ions are generated in the negative glow,
196
6 High-frequency discharges II
• the only interaction between ions and neutrals is a symmetric charge transfer and • that the cross section is constant within the regarded energy interval, the energy distribution of the ions incident on the electrode IEDF could be described consistently. As can be seen from Fig. 6.27, in argon the IEDF at 60 mTorr (8 Pa) is very broad: Only a vanishingly small fraction of the Ar+ -ions traverses the dark space collisionless (λ/dC = 0.06 with λAr+ in Ar = 0.4 mm).
relative intensity
1.00 Ar+ in Ar VC: 600 V dC: 1.3 cm p: 60 mTorr dC/l = 15 for -16 2 s = 53 * 10 cm
0.75
0.50
Fig. 6.27. Energy distribution of argon ions in a DC discharge c The American Phys[128] ( ical Society).
0.25
0.00 0.00
0.25
0.50
V/VC
0.75
For constant discharge voltage, the IEDF is relatively weakly influenced by the discharge pressure which is a manifestation of the similarity rules (see Sec. 4.10, the product of pressure and sheath thickness is relatively constant). The same holds true for the collision number in the sheath. The functional relationsship can be written as ⎧ ⎨
⎛
⎞⎫
1 d ⎝ V /2 ⎠⎬ 1− 1− f (E) ∝ 1 · exp ⎩− ⎭ λct Vc (1 − VVc ) /2
1
(6.87)
with λct the mean free path for resonant charge transfer. RF discharges. In a capacitively coupled RF discharge, Ingram and Braithwaite measured the EEDF and IEDF of electrons and ions incident on the grounded electrode as function of pressure with a modified Langmuir probe [232]. The glass tube which serves as a shield for the tungsten rod is sealed by grid with fine mesh on ground potential. Since its width is significantly smaller than the Debye length, we obtain an equipotential surface in front of the electrode. A second grid exhibits a retardation potential to let through only one sort of carrier which are incident on a collector with sweeping bias potential; the collector current can be measured with an amplifier which simultaneously generates the first and second derivatives dI/dV and d2 I/dV 2 .
6.8 Collisional sheaths
197
By biasing the second grid to sufficiently negative values, the electron current is suppressed, and all the collector current is pure ion current which is subject to an energy-dispersive analysis, i. e. we record the IEDF. The uppermost value of the bias potential with vanishing probe current marks the highest ion velocity. In Fig. 6.28, this strong dependence of the IEDF on pressure or the mean free path, respectively, is displayed. A sharp, almost monochromatic beam of argon ions traversing through argon has become a diluted distribution by raising the pressure by two orders of magnitude. The onset of thermalization has communicated to about 40 mTorr (5 Pa). Of paramount importance, however, is the fact of the exponential decrease of f (v) for declining v which is an important part of f (v)dv.
1.0
normalized IEDF [a. u.]
150 mTorr 105 mTorr 60 mTorr
0.8 0.6
30 mTorr 15 mTorr
0.4 7.5 mTorr
0.2 0.0 0
1.5 mTorr
10
20 Ekin [eV]
30
Fig. 6.28. Measured ion energy distribution functions (IEDF) in argon for various pressures. With rising pressure, the maximum shifts to lower energies and diminishes in intensity. The maximum energy corresponds to the free fall from the plasma potenc IOP Publishing tial to the potential of the Langmuir probe Φp − ΦLp after [232] ( Ltd).
A2 Example 6.2 The cross section for argon at 30 eV, σ(30 eV Ar+ ) equaling 38 ˚ 13 −3 yields at a discharge pressure of 20 mTorr (3 Pa, n = 70 × 10 cm ) a mean free path λ of 5.8 mm, at 60 mTorr (8 Pa), λ = 2 mm. For a sheath thickness of 12 mm, this means two collisions at 20 mTorr and six collisions at 60 mTorr, or stated another way: At 20 mTorr, a fraction of e−d/λ = 12.5% traverse the sheath without collisions, which drops to 0.2 % at 60 mTorr.
198
6 High-frequency discharges II
Both the mechanisms of ion energy broadening can be separated only by energy-dispersive mass spectrometry. In fact, even at discharge pressures of 55 mTorr (7 Pa), the plasma potential Φp at 23 eV, measured at full width at half maximum is broadened by only 2 eV when the Ar+ 2 -ion is used for its determination [195, 196]. For Ar+ 2 , Davis and Vanderslice find a sharply-peaked distribution when V /Vc equals unity, i. e. maximum sheath voltage, which resembles Fig. 6.27 at A2 at 500 eV) [128]. The integral below the peak low pressures (σscatt is only 7 ˚ had shrunk to only 12 % of the total area! These measurements clearly indicate that the resonant charge transfer dominates the scattering processes in the discharges of inert gases [201]. 6.8.1.2 Elastic scattering. This assumption is confirmed by measurements of B.E. Thompson et al. in discharges through molecular gases such as SF6 , CH3 Cl and CH3 Br [189]. Their experimental setup was similar to that of Ingram and Braithwaite, however, they investigated the high-pressure range between 190 mTorr and 1 000 mTorr (25 to 130 Pa) [232]. Since in these gases, the resonant charge transfer does not play any role because most of the ions are only fragments of the neutral species, the mean free path λ is significantly increased. For example, at 190 mTorr (25 Pa) λ amounts to about 350 μm at a sheath thickness ds of about 1 mm, and about 1 /4 of the ions traverse the sheath without any collision. For argon, λi of an ion beam with an energy of 500 eV amounts to about 27 cm to 13 m within a pressure range between 70 and 1 mPa, respectively. At the upper pressure limit, we find a significant number of fast neutrals after only a few cm. Since current densities are normally measured with a Faraday cage, calculations for the sputtering yield have to be corrected by these fast neutrals. For an exponential decay of the ion beam flux with respect to distance, resonant charge transfer can be identified as dominating scattering process. 6.8.1.3 Modulation. But even in collisionless sheaths, a broadening of the IEDF is observed which is due to RF modulation effects. This phenomenon is more distinct for lighter ions and is more easily visible at lower excitation frequencies ωRF . Its occurrence is closely related to the ratio τRF /τi with τi the transit time of an ion across the sheath and τRF the time of an RF cycle. When τi falls short of τRF , then the ions become sensitive for the instantaneous sheath potential. In the simplest case, we investigate a collisionless sheath. Provided the spatial variation of the sheath potential can be calculated according to Child’s law, and the ion velocity is monoenergetic (vx = 2e0 Vs /mi ), then [cf. Eqs. (4.24)]
6.8 Collisional sheaths
199
4
Vs (x) = A x /3 ∧ A =
9ji 4ε0
2/3
1 mi /3 × , 2e0
(6.88)
with the constant which does not depend on x (x = 0 means wall). It takes the ion (s the time-averaged sheath thickness) τi =
s dx 0
v(x)
⇒ τi = 3s
mi 2e0 Vs
(6.89)
to fall across the sheath, and for the ratio τi /τRF we obtain 3sω τi = τRF 2π
mi . 2e0 Vs
(6.90)
For τi τRF , i. e. in the low-frequency regime, the ions can respond to the instantaneous sheath potential which finally results in a bimodal IEDF with ΔEi the difference between minimum and maximum sheath drop which scales with the ratio τRF /τi . For more realistic sheaths, the modulation effect has been evaluated by Benoit-Cattin and Bernard for the first time [233] (Fig. 6.29):
f(E)
8e0 VRF ΔE = 3ωds
e0Vs
2e0 VDC , mi
(6.91)
Fig. 6.29. Benoit-Cattin and Bernard derived the bimodal IEDF for a monoenergetic initial ion velocity. The two singular peaks are symmetric about e0 Vs pictured here for two different masses [233].
ion energy [a. u.]
with ds the sheath thickness. Their calculation yielded a bimodal IEDF with two peaks which are also symmetric about e0 Vs , and the energy difference ΔE between them is proportional to ωi /ωRF : Rising ωRF and/or enlarging mi will decrease ΔE, and the two peaks will approach each other and will eventually merge to one single peak. Since the IEDF is assumed to be initially monoenergetic, both the peaks are singular.
200
6 High-frequency discharges II
Four years later, Coburn and Kay performed an experiment in the sheath region of a grounded electrode [196]. In fact, they observed a severe broadening in the ion energy distribution, in particular in light gases (Fig. 6.30).
Eu+ +
Fig. 6.30. Energy distribution of several ions very different in mass in an RF discharge at 13.56 MHz through argon, 75 mTorr, electrode distance: 50 mm, measured at the grounded electrode [196] c The American Institute of ( Physics).
IEDF [a. u.]
H2O
H3+ H2O+ DE
50
H3+
75 100 125 ion energy [eV]
150
175
√ Since the spread in energy ΔE/E scales inversely to mi , this means for water (M = 18) at 100 eV a ΔE of 20 eV, but for Eu (M = 151) only 4 eV. Since the quantity that is responsible for this effect is the ion transit frequency which has to be compared with the operating frequency, we obtain nearly the same spread when we investigate same masses at various operating frequencies. Again, we find the maximum more sharply peaked for E = e0 V s , when we have left the regime for almost instantaneous field response (cf. Eq. (11.3) and Fig. 6.31). 104 +
Cl2 ; 13.56 MHz +
Cl2 ; 100 kHz Cl+; 100 kHz Cl+; 13.56 MHz
IEDF [a. u.]
103
102
Fig. 6.31. Energy distribution of Cl+ - and Cl+ 2 -ions at two different operating frequencies (100 kHz and 13.56 MHz) [234] c The American Institute of ( Physics).
101
100
0
100
200 300 ion energy [eV]
400
500
Hence, ion inertia causes three RF modulation regimes:
6.8 Collisional sheaths
201
• (Very) light masses (H+ ): The ion transit time is comparable or even less than the operating frequency. Low DC bias values are accompanied with a distinct bimodal energy spectrum with a typical saddle profile around e0 (VDC ± Vs ). • Medium masses (H2 O+ , Ar+ ): The width of the saddle profile shrinks, it is still remarkable that the profile is asymmetric around the value of VDC and the height of the lower peak of the IEDF always exceeds that of the high-energy peak. Coburn and Kay put this down to the resonant charge transfer which cannot be separated from the RF modulation. For example, the ratio ΔE/E for the Ar+ 2 -ion (no resonant charge transfer possible with the mother molecule) is similar in sharpness to that for Eu+ , although the mass ratios amount to mEu /mAr2 = 152/80 ≈ 2. • Heavy masses (Xe+ ): The ions can only respond to the time-averaged potential V s (t), ΔE/E becames progessively sharper for rising mass or operating frequency. 6.8.2 Computer simulations Simultaneously, numerous Monte-Carlo simulations were performed which took account of collisions within the sheath. Solving the equations of motions in the electric field across the sheath yields the trajectories of the ions between collisions. Applying the Monte-Carlo method randomizes the moment of collision. The cross section itself can be expressed as the sum of the three contributions σtot = σelast + σinelast + σct .
(6.92)
In the case of great uncertainties —this is the case for very low energies —model potentials an be applied (model of hard spheres for elastic collisions: σelast → σhs , Lennard-Jones for the inelastic collision between ions and neutral molecules, σinelast → σLJ ). After every collision, the velocity components, i. e. momentum and kinetic energy, are calculated again. Since there are large differences in velocity between ions and thermal molecules (< E > ≈ 1/40 eV) the calculation can be simplified by the assumption of resting molecules, and collisions between ions can be neglected due to their low density. Next, the oscillating sheath has to be taken into account for the specific moments of inertia of ions and electrons which can respond to the instantaneous electric field up to the GHz range (see Sect. 14.4). It is evident that the time interval for the computation should be small compared • to the time between two collisions: Δt < τ ≈
1 ν
=< v > /λ and
• to the reciprocal value of the operating frequency: Δt < 2π/ω,
202
6 High-frequency discharges II
otherwise, we would take into consideration only the time-averaged E-field and a Maxwellian distribution. On its course, the single ion trajectory is calculated until the final impact on the surface of the electrode. This calculation is then repeated for many ions. Under the assumption that the time average equals the ensemble average, this yields simulated IEDFs and IADFs, often displayed as histograms. The accuracy of this simulation depends on the choice of • the initial conditions: The ions enter the sheath with Bohm velocity vB , • the correct sheath potential and its dependence on the thickness, i. e. the electric field across the sheath, • the accuracy of the differential cross section, • the ratio of sheath thickness over the ionic mean free path ds /λi , and • the size of the ensemble (N = 20 000 is required for substantial statistics, whereas trends are already recognizable for N ≈ 6 000 [235]). Kushner calculated the pressure dependence of the IEDF in discharges through argon [221] and fixed the threshold of thermalization at 40 mTorr (5 Pa) which is in accordance with the results of Ingram and Braithwaite. However, he took account only of the resonant charge transfer. Therefore, B.E. Thompson et al. repeated the calculations and added elastic collisions [189] which does not only modifies the IEDF but also broadens the IADF, i. e. the fragmentation and scattering of the ion beam (the IADF cannot be changed by the resonant charge transfer since the transfer of momentum is supposed to happen in only one collision). Their results can be compiled as follows: • If the ions are supposed to act as hard spheres, in the uniform field of a sheath exhibiting a thickness of 3λi , about 80 % of the ions are thermalized. To have a significant fraction of ions traversed without any collision across the sheath, the sheath must significantly shrink (Fig. 6.32, E ∗ denotes the gain of kinetic energy by the time-averaged field). • For fields exhibiting a spatial dependence as strong as across the sheaths, this value can be doubled to reach about 6λi (Fig. 6.33). But even for ds /λi = 1, i. e. after only one collision, a siginificant fraction of the ions exhibit a direction which deviates from normal incidence.13 • Considering only charge transfer yields an exponential decrease of ion energy, as has been described by Davis and Vanderslice [128]. The mean energy has been cut down to half that value obtained by elastic scattering, since the amount of energy transferred in the latter case is 13 The ratio ds /λi does not exactly equal the number of collisions, since by simple occurence of collisions, the effective sheath thickness grows, and therefore, the transit time as well.
6.8 Collisional sheaths
203
1.5 5.5
0.06 0.23
1.0
1
IADF
IEDF
0.04 1
3
0.5
3
0.02 5
5
0.0 0
7
7
2
4 * E/E
6
8
0.00 0
20 40 60 angle of incidence [°]
80
Fig. 6.32. Applying a Monte-Carlo method, B.E. Thompson et al. calculated the IEDF and IADF for several ratios of ds /λi (scales approximately with the collision number) during its course across the sheath for hard spheres and a uniform DC field. For better guiding of the eye, the vertical axis is shifted, E ∗ denotes the gain of kinetic c The American Institute of Physics). energy by the time-averaged field [189] (
significantly less (Pex = 0 for elastic scattering). The case of Pex = 0.5 lies between total elastic scattering and total charge transfer (Fig. 6.34). • For higher energies, the IADF becomes more sharply peaked: A larger fraction of ions traverses the sheath without any collision, since the time for interaction declines (the cross section for elastic scattering decreases with increasing energy beyond the maximum at 100 − 150 eV). • Raising the discharge pressure broadens both the functions IEDF and IADF. • Ions with large a deviation from normal incidence exhibit a lower kinetic energy according to tendency. Therefore, for simulations which model etch profiles, the dependence of the ion flux on energy and angle of incidence has to be taken into account. In addition, B.E. Thompson et al. simulated RF effects with parameter M = ωt, the number of oscillations between the collisions. In general, in the upper limit M → ∞, the distribution should be solely determined by the DC field, and in the lower limit M → 0 by the AC field, respectively. For λi /ds ≈ 1, they found a variation in energy of about 30 % and in angle of about 20 %, at λi /ds ≈ 1/3 , however, these values have been reduced to about 5 and 2 %, respectively. In conclusion, for large values of λi /ds , the IEDF and IADF are tremendously affected by the operating frequency, but for values of λ/dS ≤ 5, the influence is negligible.
204
6 High-frequency discharges II
0.15
0.10
2 IADF
IEDF
3
2
1
2 4
0.05
4 6
6
8 12
0 0
2
4 * E/E
8
6
12
0.00 0
8
20 40 60 angle of incidence [°]
80
Fig. 6.33. IEDF and IADF, calculated with the Monte-Carlo method, for several number of collisions of hard spheres on their course across the sheath supposing a linear field. The vertical scales have been shifted to distinguish the various curves. E ∗ c The American denotes the gain of kinetic energy by the time-averaged field [189] ( Institute of Physics).
1.0
0.06
d(Pex=1.0) = 1.00 d(Pex=0.5) = 0.537
0.8
0.04 IADF
IEDF
0.6 Pex = 0.0 Pex = 0.5 Pex = 1.0
0.4
Pex = 0.0
0.02
0.2
Pex = 0.5
0.0
0.00 0
1
2
3 *
E/E
4
5
0
20 40 angle of incidence [°]
60
Fig. 6.34. Applying a Monte-Carlo method, the IEDF (LHS) and IADF (RHS) are calculated for a scattering mixed in character (collisions between hard spheres and symmetric charge transfer) for a linear DC sheath field. Pex denotes the probability for a symmetric charge transfer which does not cause a change of the IADF (δ functions peaking at ϑ = 0◦ with height 1.00 and 0.58), but does influence the IEDF tremendously. E ∗ denotes the gain of kinetic energy by the time-averaged field [189] c The American Institute of Physics). (
6.8 Collisional sheaths
205
6.8.3 Hybrid sheath model Brinkmann et al. have developed a (one-dimensional) Monte-Carlo model that couples the equations of motion with the Poisson equation for both types of carriers [cf. Eqs. (6.81)] [236]: ∂ne,i ∂ne,i ui + = 0; ∂t ∂x
(6.93.1)
qE ∂ui Te,i ∂nr,i ∂ue,i + ue,i = − νm(e,i) ue,i ; + ∂t ∂x me,i me,i ne,i ∂x ε0
∂E qi,j ni,j + e0 ne = 0. = ∂x j
0.025
(6.93.3)
0.07 50 mTorr (6.7 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa)
0.015 0.01
IEDF [a. u.]
0.06
0.02
IEDF [a.u.]
(6.93.2)
109/cm3 10 3 10 /cm 11 3 10 /cm
0.05 0.04 0.03 0.02
0.005
0.01 0
0
50
100 150 200 energy [eV]
250 300
0
0
50
100 150 200 energy [eV]
250
300
Fig. 6.35. IEDF as functions of discharge pressure p (LHS) and plasma density np (RHS). Since the sheath thickness scales inversely with pressure and plasma density, already at np = 1010 /cm3 the energy of the ions incident on the electrode is only marginally influenced by collisions, and it is no longer detectable at 1011 /cm3 [237].
The term for friction in Eq. (6.93.2) depends on Langevin’s energy loss parameter and on the cross sections [as in Eqs. (6.81) and (6.82) − (6.84), the cross section of elastic scattering is denoted by σelast , and that of the resonant charge transfer is σct ]. As emphasized in Sect. 2.5, in particular in discharges of inert gases, the resonant charge transfer dominates the scattering reactions; by this process, fast neutrals are generated which influence the character of the discharge itself but also take a significant part of secondary reactions (surface reactions: sputtering or etching). Hence, in the study of Sabisch et al., a large part is devoted to the contribution of neutral molecules and their distributions in energy (N EDF ) and angle (N ADF ), respectively [237].
206
6 High-frequency discharges II 0.3 0.4 50 mTorr (6.7 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa)
0.2 0.15
9
IADF [a. u.]
IADF [a. u.]
0.25
normal incidence: 0°
0.1
0
3
10 /cm 10
3
11
3
10 /cm 10 /cm
0.2 0.1
0.05 0
0.3
2
4 6 angle deviation [°]
8
0 0
10
6 4 2 angle deviation [°]
8
10
Fig. 6.36. IADF as function of discharge pressure (LHS) and the plasma density np (RHS). The beam melts away with rising pressure and rising plasma density, but high a plasma density (1011 /cm3 ) guarantees an almost perfect forward direction [237].
0.07
0.12
0.06
0.10
0.05 0.04 0.03
50 mTorr (6.7 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa)
0.02
NADF [a. u.]
NEDF [a. u.]
6.8.3.1 Ions. For pressures beyond 45 mTorr (6 Pa), no bimodal distributions are calculated any more (cf. Figs. 6.35−6.37), and the mean energy declines with rising pressure; simultaneously, the beam incident on the electrode broadens in energy and angle (normal incidence: 0◦ ).
0.01 0 10 20 30 40 50 60 70 80 90 100 energy [eV]
0.08 0.06 0.04
50 mTorr (6.6 Pa) 85 mTorr (11.3 Pa) 120 mTorr (16 Pa) normal incidence: 0°
0.02 0 0
5 10 15 20 25 30 35 40 45 50 angle deviation [°]
Fig. 6.37. N EDF and N ADF as functions of discharge pressure. The kinetic energy can reach 100 eV, and higher pressures lead to a reduction in mean energy. The angle distribution sharply peaks at normal incidence, but exhibits a second, less pronounced maximum around 20 − 30◦ [237].
6.8.4 Measurements and modellings 6.8.4.1 IEDF in the sheath. These measurements were extended by Liu et al. applying a time-of-flight method in discharges through argon at 13.56 MHz
6.8 Collisional sheaths
207
with pressure as parameter (Fig. 6.38) and compared with own Monte-Carlo calculations (Fig. 6.39) [238]. Although the pressure dependence of the double peak which is created by the harmonic excitation of the sheath could not be reproduced, the calculations resulted in fully developed distribution functions at 500 mTorr (67 Pa) of energy (IEDF) and angle of incidence: After having experienced more than thirty collisions during traverse across the sheath, the shape of the IEDF has changed to Maxwellian behavior. 0.10
Fig. 6.38. Measured dependence of the IEDF on pressure in a capacitively coupled discharge through argon at 13.56 MHz. The double peak at very low pressures is well resolved; at 500 mTorr (67 Pa), the IEDF is fully developed exhibiting Maxwellian behavior (after [238]).
10 mTorr (1.3 Pa) 50 mTorr (6.7 Pa) 500 mTorr (67 Pa)
0.08
IEDF
0.06 0.04 0.02 0.00 0
10
20
30
40
ion enenergy [eV]
The maximum has retreated to only several electronvolts and has become independent of DC bias. Furthermore, the IADF exhibits isotropic behavior: The ion energy is independent of the angle of incidence. Albeit care has to be taken when interpreting this result —in particular, the large discrepancy between the primary beam and its simulation is striking—this marks the upper limit of an anisotropic angular ionic distribution (Fig. 6.39).14 The occurence of a bimodal IEDF can be observed only at low pressures with low collision numbers. Only in this case can ions traverse the sheath without any collision responding to the instantaneous sheath potential. Hence, we are far away from equilibrium. For higher pressures (and correspondingly higher collision numbers) the ion distribution within the sheath turns to a Maxwellian distribution beyond a collision number which typically exceeds 6, which allows us to define an ion temperature within the sheath (cf. Sect. 14.1). 6.8.4.2 IEDF in the sheath of the powered electrode. By means of an energy dispersive mass spectrometer, Becker et al. investigated the sheath of the powered electrode in discharges of SF6 which exhibits a strongly electronegative behavior even at very low pressures. The ions enter the spectrometer through an aperture with radius 50 μm. Since the floating system is shielded against RF 14 As has been pointed out in Chap. 2, the differential cross section at very low energies is difficult to measure and still remains unknown for most of the reactive gases; the collision frequency, however, does depend on the cross section.
208
6 High-frequency discharges II 0.20
0.10
0.15
IADF
IADF
10 mTorr (1.3 Pa) 50 mTorr (6.7 Pa) 500 mTorr (67 Pa)
0.05
10 mTorr (1.3 Pa) 50 mTorr (6.7 Pa) 500 mTorr (67 pa)
0.10
0.05
0.00
0.00 0
10
20
30
0
40
incidence angle [°]
10
20
30
40
incidence angle [°]
Fig. 6.39. IADFs of a discharge through argon at 13.56 MHz and various pressures, LHS: measured, RHS: Monte-Carlo simulation (after [238]).
fields and against the electrode, the potential difference between spectrometer and DC bias can be adjusted to zero. In particular, the aperture is free of DC fields (but is still exposed to the RF field of the powered electrode). In Fig. 6.40, the IEDF for the SF+ 5 ion is depicted at two different pressures. The simulated spcetrum has been calculated according to Wild’s model [240].
count rate [a. u.]
10 mTorr (1.5 Pa)
calculated experimental 1 mTorr (0.15 Pa)
0
50
100
150
200
250
Fig. 6.40. Measured and calculated IEDF of SF+ 5 at 3 and 10 mTorr (0.5 and 1.5 Pa) at VDC = −200 V. At the low pressure, the saddle-shaped profile is very well resolved. The peaks at low energies are caused by additional ionizations in the sheath of the powered electrode [239].
ion energy [eV]
They calculated the sheath thickness to be 3.4 mm at 3 mTorr (0.5 Pa) and 10 mm at 10 mTorr (1.5 Pa). In the range of the saddle profile, the agreement between measured and calculated distribution is relatively poor which is attributed to additional ionizations in the sheath. According to Biehler’s model, these ionizations are also made responsible for the peaks at significantly lower energies [241]. In conclusion, the elastic scattering changes the kinetic energy of the ions with a strongly forward direction into one with a substantial component of the random walk. Since scattering scales with particle density, a rise in pressure
6.9 DC discharges and capacitively coupled RF plasmas
209
lowers the mean energy of the carriers. In discharges of inert gases, resonant charge transfer is the main scattering mechanism.
6.9 DC discharges and capacitively coupled RF plasmas The character of capacitively coupled discharges is dominated by the sheaths in front of the electrodes. Without the RF field, there exists a boundary between the quasi-neutral plasma and the grounded surface, however, this sheath is very thin and the sheath potential drops to several kB Te /e0 [Eq. (3.24)]. In RF discharges, the sheath potential at the grounded electrode can rise to some hundreds of volts, which is joint with an increase in sheath thickness to several tens of a Debye length because of the rectifying response of the plasma to the RF field. This effect is mainly caused by nonlinearities of the sheath impedance (Sects. 14.4.1/2). Across the sheath, the electron density drops and eventually vanishes which causes the displacement current in the sheath to become comparable with the conduction current in the plasma bulk. For a collisionless sheath, both become equal at the plasma frequency of the ions, ωp,i , in the plasma bulk, however, at the plasma frequency of the electrons, ωp,e , with ωp,e ωRF ωp,i (ωRF denotes the operating angular frequency). The RF discharge exhibits the characteristics of two different types: properties of the negative glow and that of a hollow cathode. • Negative glow: Dissipation of electric power into the plasma by accelerated electrons which collide with neutrals elastically and inelastically. This process dominates in the plasma bulk for x > ds . In the plasma bulk, the electric field induced by the electromagnetic field is small, but finite. For example, at a power density of 0.4 W/cm2 , the electric field
E2 =
1 μ0 /2 S, ε0
(6.94)
amounts to about 12.5 V/cm. • Hollow cathode: In this type of discharge, power is transferred by beamlike γ-electrons which are emitted by the electrode. This process can become dominant in RF sheaths (for plasma electrons, the sheath is uphill, seen from the valley; for sheath electrons, the sheath is downhill, seen from the hill). This is the so-called γ-regime [175]. An additional—collisionless—process with high efficiency, stochastic heating by the pulsating sheath, takes over at about 75 mTorr (10 Pa) when the
210
6 High-frequency discharges II
electron mean free path λe becomes larger than the distance between the electrodes. Especially in the high-energy tail of the EEDF, this heating mechanism dominates all other processes. The HF ionization mechanisms are far more effective than in DC discharges (α-ionization); the breakdown voltages are lower at the same pressure, and electrode processes become of secondary importance. Eventually, the electrodes can be located out of the vacuum reactor (α-regime). AC discharges with an operating frequency significantly lower than the ion plasma frequency (ωRF ωp,i ) behave as DC discharges with changing polarity; but it is only in the RF range that these changes become so fast that charging of insulators is effectively suppressed. With rising operating frequency, the sheath impedance declines; and the center of ionization shifts further into the center of the plasma bulk. Albeit in both types of discharges, the ionization processes take place predominantly in the plasma bulk, but to sustain the discharge, electrode processes are of paramount importance in DC discharges. This is quantitatively summarized in a model which has been developed by Graves and Jensen [242]. Provided that the electron mean free path is large compared to the Debye length: λe λD , the electron temperature Te soars across the cathode fall, since freshly generated γ-electrons are accelerated along the electric field across the dark space. With weaker electric field, but still very low electron density, Te begins declining. Simultaneously, the ionization rate goes up (cf. Sect. 3.5). Since the ionization rate is a reaction of second order, it linearly depends on ne , but exponentially on Te according to the simplest approach of Arrhenius):
d[A+ ] Eion . = k2 [e− ][A] ∧ k2 = k0 exp − dt kB Te
(6.95)
Due to the opposite direction of these effects, a maximum in the ionization rate should occur. After having cooled off, the electrons remain at almost the same temperature (energy) across the distance between the electrodes (Fig. 6.41). 4
8 6
ionization rate
2 4 2 0 0.0
electron temperature
0.2
0.4 0.6 relative distance
0.8
0 1.0
ionization rate [a. u.]
electron temperature [eV]
10
Fig. 6.41. Electron temperature and ionization rate in a DC discharge as a function of the relative electrode distance from anode (0.0) to cathode c IEEE). (1.0) [242] (
6.9 DC discharges and capacitively coupled RF plasmas
211
electron temperature [eV]
4
normalized ionization rate
5
0° 90° 180° 270°
3 2
90°
1
270°
0 0.00
0.25
0.50 0.75 relative distance
0.50
0.25 0° 90° 180° 270°
0.00 0.00
1.00
0.25
0.50 0.75 relative distance
1.00
Fig. 6.42. Electron temperature (LHS) and ionization rate (RHS) in an RF driven c discharge as a function of relative electrode distance within one RF cycle [242] ( IEEE).
Whereas in RF discharges the electron temperature exhibits a similar behavior, the ionization rate has shifted farther into the plasma bulk. Although the role of the sheath for ionization cannot be neglected, it is highly modulated and can become almost zero (Fig. 6.42). Compilation of excitation mechanisms: • Excitation of γ-electrons (predominant in DC discharges). • Ohmic heating (very efficient in RF discharges). • Stochastic heating (only in CCP RF discharges).
electron temperature [eV]
10
1 DC RF
0.1 -3 10
Fig. 6.43. Electron temperatures in DC and RF discharges through mercury vapor [243]. 10-2
10-1 100 pd [Torr cm]
101
Compilation of ionization mechanisms:
212
6 High-frequency discharges II
• γ-electrons generate an avalanche of charged carriers during their fall across the dark space (sheath); • photoionization (negligible in RF discharges); • α-ionization in the negative glow (plasma bulk); • symmetric charge transfer. In RF discharges, frequency-dependent scattering processes become of great importance; among them, the frequency of elastic collision, νm , the angular operating frequency (ωRF or simply ω), and the plasma frequency of the ions, ωp,i , play a significant role (RF modulation). For both types of excitation, in the pressure regime below 1000 mTorr (130 Pa) the carrier generation is dominated by collisions, the carrier loss by ambipolar diffusion. Hence, the electron temperature, Te , remains almost constant and depends on the product p a with p the gaseous pressure and a the dimensions of the reactor a = 1/(1/R + 1/L) with R the radius and L the length of the plasma, provided the drift velocity is small compared to the thermal velocity of the electrons [243] (Fig. 6.43).
6.10 Summary • In capacitive RF plasmas of electropositive atoms, the discharge impedance is dominated by the sheath capacitancies, and the plasma resistivity can be neglected. • All the applied voltage is dropped across the two sheaths, and the sheaths act as a capacitive divider for the applied voltage. The large differences in electron and ion mobilities cause the plasma to act as a rectifier. • Model: Due to the high electron mobility, the instantaneous plasma potential cannot drop below the instantaneous potential of either electrode. • Model: For symmetric capacitively coupled systems, this leads to a sinusoidal plasma potential with lowermost values of zero, whereas the topmost values of the sheath potentials, which are equal in size, just reach zero. The potential drop across the electrode sheaths which is caused by the flowing displacement current is symmetric in size, and this negative selfbias amounts to about 1/2 VRF . The symmetric system develops a relatively high plasma potential and a very low self-bias voltage. • Reality: During a very short time of the RF cycle, the electrodes become positive with respect to the plasma, and a strong electron conduction current flows.
6.10 Summary
213
• In asymmetric capacitively coupled systems, the negative self-bias at the smaller electrode is larger in size to maintain the continuity condition for current. Simultaneously, the plasma potential adjusts to lower values, to keep the electrons on ground potential for a short time of the RF cycle. To check whether an intended asymmetric system acts properly, an easy, highly recommended procedure is etching of wafers, covered for example with SiO2 , at either electrode. If the etch rates are comparable or, even worse, are almost equal, then the system is symmetric. • The most severe limitation is the impossibility of enlarging the plasma density with increasing power input. Parasitic processes at low pressures are increased ion bombardment which is mainly due to a steep rise of the plasma potential, and generation of hot electrons at high pressures by the heating of plasma constituents (neutrals and ions) because of the strong increase of elastic collisions between electrons and heavy particles.
7 High-frequency discharges III
We start with a comparison of advantages and disadvantages of capacitively coupled discharges, by which it becomes obvious why and how the semiconductor industry has triggered the demand for high-density plasmas. In the first half, this reflection is devoted to inductively coupled plasmas, in the second half to resonant excitation methods. To focus in this chapter on fundamental properties and interrelations, important, but tedious derivations are deferred in Chap. 14 (Sects. 14.5 − 14.7).
7.1 Introduction The steep potential decrease in the sheaths and an extended zone of constant, relatively high charge density, which is closely connected with a uniform, constant potential, is the main advantage of capacitive discharges. This further simplifies the description of the discharge, since it can be confined to only one dimension. In a parallel-plate reactor, we have two adjacent electrodes whose diameter 2r is large compared to their distance d, in its hexagonal counterpart, we have a coaxial configuration of the electrodes (r/d 1). The excitation frequency f = ω/2π, typically 13 or 27 MHz) is far below ωp,e , but significantly above ωp,i which leads to • a quasi-stationary character (λ/r 1), • and a separation into the glowing, quasi-neutral plasma with thermal electric carriers and sheaths across the electrodes. Here, the plasma density decreases by orders of magnitude, whereas the potential drops to very low values. As a direct consequence, the ions will hit the surface of the negatively biased electrode with high a specific kinetic energy of 1/2 VRF in symmetric discharges, and VRF in highly asymmetric discharges. With the wisdom of hindsight, we now see that despite their advantages compared with DC discharges, the capacitively coupled plasmas suffer from the following inherent disadvantages: • The degree of coupling is very low and deteriorates with decreasing pressure which is required for good anisotropy and radial uniformity of etching processes (cf. Chaps. 11 and 12). G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 7,
215
216
7 High-frequency discharges III
• The energy which is transferred to the electrons scales with the ratio of electric field intensity and discharge pressure E/p (cf. Sects. 4.10 and 5.2). Ion flux and accelerating field cannot be varied independently but will be altered in the same direction. For high ionization which leads to a heavy ion flux incident on the hot electrode, intense electric fields are required inherently, for good radial uniformity, the discharge should be kept low. Hence, for an etching process with good performance, the ratio E/p must reach large values. However, with increasing E/p, we have to face severe setbacks: Not only crystal damage, but also the fidelity of pattern transfer will deteriorate, and the Line Edge Roughness, LER, increases [244, 245]. • From first principles, it is the electrons which gain energy from the RF field. However, an amount sufficient for ionization can only be piled up by collisions with heavy particles (conservative field, ∇ · E = 0). Especially at high power input, this leads to significant parasitic heating of the ions which finally defines the upper limit of capacitive coupling.
To overcome the limits of capacitively coupled plasmas, several high-density plasma sources have been invented in the last two decades. To identify the parameters which are responsible for an enlargement of the plasma density, we observe an ion current nB emanating from the Bohmic edge incident on the electrode with initial Bohm velocity vB . The energy of the electromagnetic wave is transferred to the electrons which dissipate the energy via collisions with the heavy plasma constituents. Hence, the absorbed power Pabs will be Pabs ∝ e0 nB vB V
εi
(7.1)
i
with εi the energies of the various excitations for electrons and ions, plasma density nB at the Bohmic edge, which delivers for nB : nB ∝
Pabs . vB V i εi
(7.2)
Hence, nB can be enlarged by these measures: 1. Increasing the power input Pabs . 2. Decreasing vB {only limited application since vB depends only on the square root of the electron temperature (and from the ionic mass) [cf. Eq. (14.30)]}. 3. Reduction of the surface of the plasma by reduction of the volume of the plasma reactor by which carrier losses at the walls are diminished.
7.2 Inductively coupled plasma
217
4. Selectivity of the energy transfer, i. e. excitation of only the electrons and avoidance of ionic excitation by reducing the sheath potential. One possible improvement is the application of a triodic system in which a second carrier electrode serves to separate the processes of plasma generation (first electrode) and control of the ionic flux and ionic energy (second electrode). This is used extensively in so-called dual-frequency systems which have been en vogue since the mid-1990s. Here, a frequency higher than 13.56 MHz is capacitively coupled to the “hot” electrode (the plasma density scales with the operating frequency following a relationship with a slope less than unity) and the sample is placed above the second electrode, which is driven with 13.56 MHz or lower frequencies. Another method is Magnetically Enhanced Reactive Ion Etching, MERIE: By application of a magnetic field, the efficiency of power transfer and plasma confinement are improved (item 2) [246]. For a given cylindrical reactor with radius R and length L, by application of a strong axial magnetic field, the area for possible losses is reduced from 2πR(R + L) to 2πR2 ; and in directions rectangular to the magnetic field, the diffusion length increases (cf. Sect. 14.5). Disadvantages are E×B drifts which cause radial and azimuthal inhomogenities. Therefore, • the magnetic field should be separated from the process chamber or • should be entirely avoided. The first end is pursued in plasmas which are excited by guided whistler waves, the second one in inductively-coupled discharges (cf. Figs. 7.1). In the first mentioned discharges, the coupling of wave fields is enforced through a dielectric window (ECR discharges) or by a λ/4 or U-shaped antenna (helicon discharges).1 For these discharges, an additional static magnetic field is required. The plasma is penetrated by slow whistler waves.2 In these plasmas, the plasma source is spatially separated from the reaction chamber, and we can separately control the density of the ions and their energy and flux. Therefore, we speak of “downstream” or “remote” plasma sources. By renouncing the magnets, the antenna will mutate to a coil, and we will generate an inductively coupled plasma.
7.2 Inductively coupled plasma For inductive coupling, the antenna is wound around the vacuum recipient and is denoted as coil. This is the helical or solenoidal configuration where the turns 1 The application of a conventional RF design with a π-network is advantageous: The antenna itself serves as inductivity L. 2 “Slow” means: The group velocity of these waves is comparable with or equals the thermal velocity of the electrons: vg2 ≈< ve2 > which allows Landau damping (Sect. 14.3).
218
7 High-frequency discharges III RF ICP coil
RF
ICP coil
CCP electrode
CCP electrode
pumps
pumps
microwave n n
dielectric window
solenoids n
solenoids
n
n
RF
n
antenna
CCP electrode CCP electrode
pumps pumps
Fig. 7.1. Various types of reactors for high-density plasmas. Top: two different coils (wound helically or spirally around the reactor) for inductive coupling, bottom: reactor with magnetically guided whistler waves. LHS: helicon wave, RHS: ECR (lower solenoid not necessary).
have constant diameter. In another widely used arrangement, the coil is placed above the cylindrical reactor in a spiral, planar or stovepipe-type configuration. For example, the GEC reactor has this configuration. Both configurations have their merits and disadvantages; one of the most remarkable features of the GEC reactor is the small gap (several centimeters) between the upper dielectric window and the substrate electrode which can be capacitively coupled to the plasma [247, 248]. The principle of excitation is explained using the helical arrangement, the self-screening at high plasma densities and the transitional jump from the E-mode to the H-mode is shown using the planar configuration. The
7.2 Inductively coupled plasma
219
fields are transmitted through a dielectric window which also serves as a sealant against the atmosphere. To reduce capacitive coupling, a metal sheet or, better, a mesh is placed upon the dielectric window, because the losses caused by eddy currents are smaller in the mesh (Fig. 7.2).
RF generator 13.56 MHz
CH4
p-network
BCl3
ICP coil Faraday shield gas ring
SEERS Langmuir OES
substrate electrode with He-backside cooling
Cl2 H2 O2 Ar
Fig. 7.2. Principal setup of an inductively coupled plasma source with Faraday shields and helical arrangement of the RF coil: turns wound around a quartz cylinder to get rotational symmetry with respect to the axis of the cylinder. The frequencies for plasma generation in the ICP source and for the capacitively coupled substrate holder are often different. The diagnostic tools are installed in the downstream region.
RF generator 2 - 13.56 MHz
L-network
7.2.1 Transformer Model As is evident from Fig. 7.3, the RF current in a coil with n turns and length l induces an azimuthal magnetic field which is orientated normal with respect to the coil current and radially outward from the middle axis. This H-field, in turn, induces a second azimuthal electric field by which a plasma current is generated that is directed antiparallel with respect to the coil current (Lenz’s law). Due to their very high carrier density, the plasmas effectively screen themselves against the field across a skin length δ. Since both the “coils” are mutually telescoped and the inner “coil” consists of only one turn, the induced current density in the plasma is extremely high (principle of the induction oven). Inductively Coupled Plasmas (ICPs) are often denoted as Transformer Coupled Plasmas (TCPs). The power is mainly transferred by Ohmic heating from the toroidal induced electric field; due to the small thickness of the sheath, stochastic heating is negligible. In most cases, the plasma potential is significantly lower than 20 V. As can be seen from Fig. 7.4, the plasma flows to a substrate holder which can be capacitively coupled to the plasma. Since it can be steered independently from the plasma source, plasma density and ion energy can be controlled without
220
7 High-frequency discharges III
mutual influence. In contrast to a capacitively coupled sheath with its electric field directed normally to the electrode, this induced electric field is aligned in parallel fashion with respect to this electrode. The advantages of inductively coupled sources are:
.
B
d
H
E
IRF
r R
l
plasma
IP
Fig. 7.3. LHS: The time varying magnetic field generates a time-varying azimuthal electric field which induces a circumferential current in the plasma which is aligned in parallel fashion with respect to the reactor wall. RHS: Principal sketch of the currents in a plasma source (coil and plasma) driven by inductive coupling (helical configuration, turns wound around a quartz cylinder to get rotational symmetry to the axis of the cylinder, electrons are negatively charged).
• Simple configuration of reactor and plasma source. • Spatial separation of ion generation (by Ohmic heating in the inductively coupled plasma source) and ion energy (in the sheath of the capacitively coupled electrode). • No static magnetic fields, but only RF fields. • Compared with capacitively coupled plasmas, a significantly higher plasma density (1 to 11/2 orders of magnitude higher); compared with resonant plasma excitation methods (ECR or helicon), the plasma density is lower by about one order of magnitude. • No resonant coupling, these discharges rather “forgive” deviating primary plasma parameters (pressure, RF power, gas composition) than discharges which rely on resonant energy transfer. • The magnetic field is orientated parallel to the reactor walls, which remain free of fields and currents; hence, the plasma potential Φp drops only in the sheath of the reactor wall and possible sputtering effects remain negligible.
7.2 Inductively coupled plasma
221
In the plasma source itself, we detect spatial inhomogenities in density and temperature of the electrons due to the small skin depth, which is transferred to the heavy ions via the ambipolar plasma properties [249] (Figs. 7.5, 7.12 and 7.14). These inhomogenities become even more pronounced in the downstream direction, by which compensating currents are triggered. The magnitude of this diffusion potential can be calculated using the Einstein-Smoluchowski equation (mobility μ = e0 D/kB T with D the diffusion coefficient, np : plasma density in the plasma source, n: plasma density at the capacitively coupled electrode):
V =
n kB Te ln . e0 np
(7.3)
RF generator 13,56 MHz 2 ERF
p-network 1 ERF
1 ERF
BRF afterglow
EDC
Fig. 7.4. Principal sketch of the experimental set-up and fields and currents in a plasma source driven by inductive coupling (helical configuration, turns wound around a quartz cylinder to give rotational symmetry to the axis of the cylinder as combined from Figs. 7.2 and 7.3).
RF generator 2 - 13.56 MHz
L-network
As a final result, the most important plasma parameters (plasma density, electron temperature) are nearly equalized downstream at wafer level (common distance: 10−15 cm). As prominent evidence, the supreme uniformity in etchrate for most materials and ambient gases has already been mentioned. Today, radial uniformities within ±1% are demanded across diameters between 8 and 12". For too small a gap between window and substrate, these inhomogenities are likely not to be levelled out. Hopwood et. al. swept a Langmuir probe across the plasma at a constant distance z below the window and could clearly judge a distance of 5.7 cm as being too short to have the inhomogenities levelled off [250]. They could significantly improve the radial uniformity by applying a magnetic multipole arrangement (Fig. 7.5).
222
7 High-frequency discharges III
1.00
normalized Is
0.75
Fig. 7.5. Normalized ion saturation current measured across the diagonal of an ICP source with and without magnetic multipole confinement [250].
0.50 0.25 0.00
measured at z = 5.7 cm magnetic confinement no confinement
-20
-10
0 r [cm]
10
20
7.2.2 Power input for inductive coupling In this section, we apply basic electrodynamic principles to calculate the efficiency of power input for inductive coupling (helical coil) [251, 252]. A cylindrical plasma volume V in a cylindrical glass cylinder with inner radius r, wall thickness R − r and length l, is surrounded by a coil with radius R with n turns (Figs. 7.2 − 7.4). In this volume V between r − δp and r across a length l, the absorbed Ohmic power per turn can be evaluated considering harmonic excitation E = E 0 sin ωt and neglecting terms of second order: IRF
LRF
Ip
Lp
Fig. 7.6. Equivalent circuit of an inductively coupled discharge.
VRF RRF
Rp
Pabs =
V
Sohm d3 x = − r
Pabs =
r−δp
1 r ˜ ˜ j · E l 2πrdr, 2 r−δP
j2 j(r)2 lπrdr ≈ lπrδp σ σ
(7.4.1)
(7.4.2)
with ˜j the current density which is induced in the plasma by the RF field, thereby opposed to the direction of the complex coil current density and E˜ , the complex-conjugated field intensity in the sheath.3 3
The energy flow SOhm is given by 1/2 ˜j · E˜ ; Eq. (5.4) is the special case for an electron.
7.2 Inductively coupled plasma
223
7.2.2.1 Plasma resistance and plasma impedance. The complete induced eddy current within the plasma amounts to IP = jlδp ,
(7.5)
1L . σA
(7.6)
with the plasma resistance Rp =
Using L the length of the current path (L = 2πr) and A the penetrated area, or A = lδP , we further obtain 2πr σlδp
(7.7.1)
e20 n . m e νm
(7.7.2)
Rp = with σ the DC conductivity σ=
The inductivity of the plasma, Lp , can easily be calculated via the magnetic flux Φ = Lp Ip = μ0 Ilp πr2 , which is generated by the current within the skin layer of thickness δp (the induced magnetic field H is directed perpendicular to the induced current density)4 Lp =
μ0 πr2 . l
(7.8)
7.2.2.2 Coupling between coil and plasma. For n turns with radius R at a total length l of the coil, the equations for the transformer apply (cf. Sect. 5.8, induction matrix): VRF = iωL11 IRF + iωL12 Ip Vp = iωL21 IRF + iωL22 Ip ,
(7.9)
with the self inductances L11 in the coil, and L22 in the plasma and the mutual inductances L12 = L21 L11 = L22 = Lp = L12 = L21 = 4
μ0 πn2 R2 , l μ0 πr 2 , l μ0 πnr 2 . l
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
See Sect. 14.6 for the skin layer, in particular Eq. (14.166).
(7.10.1)
224
7 High-frequency discharges III
7.2.2.3 Primary circuit. How large is the impedance measured across the whole coil? Considering the inverted direction of the current within the plasma (Vp = −Rp Ip ), and inserting (7.9.2) into (7.9.1) gives −Rp Ip = iωL21 IRF + iωL22 Ip ⇒ Ip (iωL22 + Rp ) = −iωL21 IRF , and we obtain for the total impedance of the source ZS =
(7.10.2)
VRF IRF
VRF = iωL11 −
iωL21 IRF iωL21 iωL22 + Rp
(7.10.3)
ZS = iωL11 −
iωL21 iωL21 iωL22 + Rp
(7.10.4)
ZS = RS + iωL11 = iωL11 +
ω 2 L221 . iωL22 + Rp
(7.11)
For δp r, it follows that Rp iωL p , and we
can expand the denominator of Rp 1 1 ⇒ 1 − , which yields Eq. (7.11) to obtain Rp +iωL iωLp iωLp p
Zs = iωL11 −
iωL221 Rp 1− Lp iωLp
= iωL11 − iω
L212 Lp
+ Rp
L12 Lp
2
, (7.12.1)
shortened to Zs = iω(L11 − nL12 ) + n2 Rp
(7.12.2)
with Rs = n2 Rp and Ls = L11 − nL12 . 7.2.3 Limits of power input From Eqns. (7.4.2) (P ∝ ) and (7.6 and 7.7.1) (Rp ∝ 1/(σ δp ), the Eqns. (7.7.2) for the DC conductivity and (14.166) for the collisional skin depth which e20 n decreases with increasing plasma density (σ = me νm ∧δp = 2/μωσ), it becomes
evident that the power input for fixed current scales with n1p . The RF field penetrates deeper in thinner plasmas, and the electric field does not fade within the skin depth δp but can be transmitted across the whole plasma column for sufficiently low damping (δp ≥ r). In this case, the excited current density is proportional j∝
IRF IRF = = IRF (n0 r), A V /r
(7.13)
and for this case, Eqs. (7.4) must be integrated from r = 0 (center of the plasma filled cylinder) to r = r (reactor wall):
power input [a. u.]
7.2 Inductively coupled plasma
225 Fig. 7.7. Inductive coupling, power input: As competition between linear increase at √ lower densities and 1/ conduct at high densities, we observe a maximum of power input at approximately r = δp with δp the screening length (skin depth) [253].
1/SQRT behavior product linear increase
nP [a. u.]
Pabs =
r j(r)2 0
σ
πrLdr ≈
j2 2 πr L : σ
(7.14)
For fixed current, the power transferred by inductive coupling evidently scales with plasma density: The efficiency at very low plasma densities is poor since there is no means to absorb the energy. The efficiency increases with rising plasma density but eventually saturates at some topmost level because the plasma screens itself against penetrating electric fields by Debye shielding [253]. Since the reasons for low power input at low and high plasma density are entirely different, it is not possible to evaluate the maximum analytically. By graphical analysis, the maximum for power input can be found at r = δp (Fig. 7.7). This means: Inductive discharges only become stable beyond a certain threshold of the induced current. Hence, the ignition process itself and the low-current operation take place via capacitive coupling. Without a plasma, only Ohmic losses occur in the coil, and a large voltage is set up across the coil. Eventually, the breakdown voltage is reached, and the plasma is ignited by capacitive coupling. At lower current values, coupling will remain capacitive across the high potential difference of the terminal turns of the coils. Immediately after ignition, the plasma is always in this mode. With growing power input and higher coil current/voltage, the degree of ionization will increase which, in turn, leads to a decrease of the resistivity (increase of the conductivity) of the plasma column, and we can find a positive slope for the characteristic plasma current vs. coil voltage. Simultaneously, with growing coil current, the magnetic flux density increases and, again, the induced voltage. Hence, the absorbed power will be increased by these two effects which are mutually contingent. Eventually, the change from capacitive to inductive coupling happens out of the sudden: After having passed the threshold, coupling will take place inductively via the induced electric field within each turn of the coil. Passing the upper limit, due to the high plasma density which causes an electronic plasma frequency ωp,e compa-
226
7 High-frequency discharges III
rable to the angular operating excitation frequency 2π f , the plasma screens itself against disturbing HF fields. To avoid excessive shielding, the operating frequency f can be reduced. In most of the reactors which are commercially available, f is fixed to values below 2 MHz by which the skin depth is enlarged by more than a factor of 4 (cf. Sect. 14.6). 7.2.4 Top coil configuration This evanescent behavior has been studied in detail by Hopwood et al. applying a spirally wound coil atop the cylinder [254] who actually found an exponential decay of the azimuthal B field in the axial direction (Fig. 7.8), measured with a magnetic induction probe [255]. Since the electron plasma frequency is higher than 13.56 MHz the electromagnetic wave is cutoff within several centimeters (i. e. within a few percent of its vacuum wavelength) inside the plasma. The curves in Fig. 7.8 are calculated with δp between 2.1 cm for 900 W and 3.7 cm for 300 W. The radial and axial variation of the radial component of B is shown in Fig. 7.9. Its ring-shaped field geometry is the result of the spiral form of the coil in creation of a double-peak design, which is mirrored in the annular form of the induced E field; its magnitude in the z-direction, however, is severely damped in the overdense plasma due to strong absorption (Fig. 7.9). The magnitude of the B field is typically a few Gauss at the window leading to B(r, z) = B(r)e−z/δ .
(7.15)
6
Br(z) [Gauss]
5 4
900 W 700 W 500 W 300 W
3 2
Fig. 7.8. Br (ω) decreases exponentially with distance from the window [254].
1 0
0
2
4
6
distance below window, z [cm]
7.2.4.1 E-mode and H-mode. In the spiral configuration of the coil, this transition from the E-type to the H-type can be excellently found as jump in the plasma potential [256] (Figs. 7.10/7.11).
7.2 Inductively coupled plasma
227
2.50
B(r)
B D F
Fig. 7.9. ICP discharge of argon, 500 W ICP power, 5 mTorr: radial variation of Br (ω) at three distances below the window [254].
1.25
0.00 -20
-10
0
10
20
Titel X-Achse
30
25
plasma potential
1010
20
electron density
plasma potential [V]
electron density [cm-3 ]
1011
Fig. 7.10. E-mode → H-mode I: sudden change in the characteristics for continuous rise of RF power in an ICP discharge of oxygen/argon (80/20 %) at 20 mTorr (2 2/3 Pa) after [256].
15
109 0
50
100 150 ICP power [W]
200
250
This is caused by both the modes: In E-mode, the potential of the coil is topmost at its center. Hence, an ICP discharge is ignited between the center of the coil and the lower electrode—as in a CCP mode. With rising coil current a sharp transition can be observed because the induced electric field is most intense between center and outer diameter of the coil, and the induced current flows in a ring-shaped region between inner and outer reagion of the coil (Fig. 7.10). This behavior is even more pronounced for pulsed plasmas. As can be seen from Fig. 7.11 (electron and ion density vs. time of a complete RF cycle (2000 μsec, RF off time 100 μsec), the densities for both the carrier types decrease after having extinguished the plasma, but they continue falling after the RF power is restored. The electron density limps behind the ion density to explode after 1000 μsec by about two orders of magnitude. Now, the H-mode is brought back into effect, and the densities of ions and electrons remain stable during the rest of the RF cycle.
228
7 High-frequency discharges III Fig. 7.11. E-mode → H-mode II: sudden change in the density of ions (ni ) and electrons (ne ) in a pulse period characteristics for (RF off time: 100 μsec) [256].
100 msec
H mode
-3
ne, ni [cm ]
1011
E mode
10
10
ions 9
10
electrons
0
500
1000
1500
2000
t [msec]
In a helical arrangement, however, the plasmas are generated along the chamber wall in both E-mode and H-mode. Therefore, the transition occurs smoothly, and the mode transition becomes dim. 7.2.5 Modeling of ICP discharges In the interior of the plasma source, inhomogenities in plasma density and electron temperature are caused by the small skin depth: The excited RF field quickly fades within a distance which is small compared to the spatial dimensions of the plasma source. This nonuniformity creates a diffusion potential which serves to equalize this gradient by ambipolar diffusion of the electric carriers. This equalization leads to a very homogeneous radial distribution of plasma density and electron temperature. This can be easily proven by the superior radial uniformity of the etchrate, provided the wafer is placed at a certain distance in the downstream zone of the reactor. We present two modeling studies. Numerical calculations of some primary properties which characterize the plasma: Plasma density, electron temperature and plasma potential have been carried out by Stewart et al. for argon and can be summarized as follows [249]: • For low pressures (5 mTorr, < 1Pa), the radial profile of the carrier density nP does not depend on the coil configuration (top coil or cylindrical coil). • The electron temperture Te peaks at the edge of the cylindrical plasma reactor, exactly at the locus of power input where the electrons gain energy from the RF field. But even in the center of the cylinder, Te decreases by only 10 % (Fig. 7.12.1). np peaks on-axis; it approximately meets the Cosine-Bessel profile (Schottky profile) which is required from eq. (6.46) and which results from the isothermal, ambipolar diffusion equation (Fig. 7.12.2). The plasma potential remains very shallow and
7.2 Inductively coupled plasma
229 1.00
1.00
0.50
0.75
5 mTorr (2/3 Pa) R/L =1
normal. ne
normal. ne
R/L =1 5 mTorr (2/3 Pa)
R/L =2,5 20 mTorr 2 2/3 Pa
0.75
5 mTorr (2/3 Pa) R/L =2.5
0.25
5
10
0.50
R/L =2.5 5 mTorr (2/3 Pa)
0.25
20 mTorr 2 2/3 Pa R/L =1
0.00 0
20 mTorr 2 2/3 Pa R/L =1
R/L =2.5 20 mTorr 2 2/3 Pa
0.00 0
15
5
10
15
radius [cm]
radius [cm]
3.5
3.5 R/L =2.5
R/L=2.5
5 mTorr (2/3 Pa)
3.0
3.0 Te [eV]
Te [eV]
R/L =1
2.5 20 mTorr 2 2/3 Pa
2.0 0
5 mTorr (2/3 Pa) R/L =1
2.5
R/L=2.5
R/L =2.5 20 mTorr (2 2/3 Pa)
R/L =1
5
R/L =1
10
2.0 0
15
5
10
15
radius [cm]
radius [cm]
6 1.0
5 kIon [10-10 cm3/sec]
0.8 normal. ne
non-uniform Te
0.6
uniform Te + substrate (h = 2 cm)
0.4 uniform Te + substrate (h = 1 cm)
0.2
4
R/L = 2.5 5 mTorr (2/3 Pa)
3 2
R/L = 1 5 mTorr (2/3 Pa)
1
R/L = 2.5, 20 mTorr (2 2/3 Pa)
R/L = 1; 20 mTorr (2 2/3 Pa)
uniform Te
0.0 0
5
10 radius [cm]
15
0 0
5
10
15
radius [cm]
Fig. 7.12. Two-dimensional model of the radial distribution of charged carriers in an inductively coupled discharge for a top coil configuration and a laterally fixed coil after [249] (1: top LHS, 2: top RHS, . . . , 6: bottom RHS).
peaks at about 3 1/2 kB Te in the center of the cylinder. For discharge pressures beyond 20 mTorr (3 Pa), the maximum of np can be shifted radially outward, since the diffusion coefficients become smaller (Fig. 7.12.6).
230
7 High-frequency discharges III
• A non-uniform electron temperature is a necessary but not sufficient condition for a maximum of np which does not peak on-axis of the plasma source (Figs. 7.12.2/7.12.4). • For a coil which surrounds the cylinder in a helical configuration (Figs. 7.12.2, 7.12.4, 7.12.5, 7.12.6), Te rises radially outward which enhances the ionization rate and likewise np in the toroidally shaped border region (skin effect). • The modeling can be entirely changed by installation of a substrate holder (Fig. 7.12.6). This change in geometry has consequences for nearly all dependencies. This equalizing behavior has been also modelled by Panagopoulos et al. for an ICP reactor with an asymmetric pumping port (Fig. 7.13) with chlorine as an etchant [257]. In contrast to the normal design, an annular downstream pumping port, the pumping port is located at one lower quadrant of the reactor, thus generating an azimuthal asymmetry. RF ICP coil inlet 1 pumping port
inlet 3
inlet 2 focus ring CCP electrode
Fig. 7.13. Sketch of a cylindrical ICP reactor with top– coil configuration (15 cm from wafer level) and asymmetric pumping port (90◦ 2.6 cm from wafer level). There are four gas inlets, symmetrically located around the periphery of the reactor (12 cm from wafer level) [257].
We expect the plasma density, and in particular the density of Cl+ ions, which are mainly responsible for the significant increase of the etchrate in ICP discharges (cf. Chap. 12), to follow this azimuthally asymmetric design of the reactor. This should be manifested in the most sensitive property, the etchrate, which should exhibit the same shape. However, it was shown by Panagopoulos et al. applying a modified three-dimensional finite element fluid model that the asymmetry of the reactor design is by no means reflected in the etchrate (Fig. 7.14). Furthermore, it was be demonstrated that a focus ring could play an important role in alleviating these azimuthal nonuniformities. Although the power density at the dielectric window radially varies by a factor of 5 from the
231 7
7.2 Inductively coupled plasma
3
7
5
7
79
3 3
3
5
5 7
3
3
3
1
5 3
3
3
3 7
75
7
3
7
8
7 2
7 5
6
5
5
8
9
6
7
9
5
7
8
5
2
10
9
7
9
8
9
12
9
7
6
1
Fig. 7.14. Because of the inhomogeneous distribution of the charged carriers within the plasma source, equalization processes are triggered which lead to a radially more uniform plasma density and electron temperature at the locus of the substrate. This is shown here for a planar coil atop the bell-jar. In particular, the contour lines denote: - Top left: The radial power gradient is steepest at the outmost turn where also the current peaks [about a factor of 3 larger, between 3.44 × 105 (corresponds to 3) and 10.78 × 105 W/cm2 (corresponds to 9), locus: 14 cm above the substrate position. - Top middle: The density of atomic Cl fluctuates only by ±10 % [1.40 × 1014 (corresponds to 1) and 1.67 × 1014 /cm3 (corresponds to 7)], locus: 14 cm above wafer level. - Top right: The density of Cl+ fluctuates stronger [1.60 × 1011 (corresponds to 3) and 2.80 × 1011 /cm3 (corresponds to 7)], locus: 14 cm above wafer level. - Bottom left: On wafer level, the density of atomic Cl fluctuates by about ±15 % [8.33 × 1013 (corresponds to 2) and 11.00 × 1013 /cm3 (corresponds to 10)]. - Bottom middle: On wafer level, the Cl+ -density fluctuates by a factor of 1.4 [2.00 × 1010 (corresponds to 1) and 5.56 × 1010 /cm3 (corresponds to 9)]. - Bottom right: The etchrate itself, however, is but weakly influenced [fluctuation about ±5 %, 0.51 μm/min (corresponds to 7) and 0.57 μm/min (corresponds to 9)] [257].
center to the edge, which is reflected by a poor radial uniformity of both the
232
7 High-frequency discharges III
chlorine radicals, Cl·, and the Cl+ ions, at wafer level these nonuniformities have levelled off (deviation less than 5 %) [257].5 7.2.6 Conclusion Both types of coils, the planar type and its helical counterpart, generate azimuthal fields, in the case of the planar type an additional vertical (usual z) dependence. Small azimuthal irregularities will be generated at the ground end of the coil, where the current is highest, at pumping ports and other prominent loci. In principle, energy from the RF field to the electrons can be transferred without collisions over a whole period, i. e. the time-averaged amount of energy does not vanish, since the magnetic field is not conservative [254]. Therefore, capacitive and inductive coupling are fundamentally different excitation methods.
7.3 Generation of plasmas supported by magnetic fields 7.3.1 R´ esum´ e of the properties of HF discharges Up to now, we considered methods of electronic energy gain which required elastic collisions between electrons and neutrals. After a consecutive series of these collisions, the electrons have piled up a sufficient amount of energy to ionize the neutral constituents of the plasma, and furthermore, this is a very efficient means to dissipate energy into the plasma. In capacitively coupled RF discharges, the plasma potential and the sheath potential can rise to very high values (hundreds of volts). For different electrode areas, the sheath potential at the smaller electrode can additionally exhibit a large DC component which can skim the peak value of the amplitude of the applied RF voltage. The sheath potential of the electrode is the difference between the potential of the electrode and the plasma potential. At the grounded electrode, the sheath potential equals the negative plasma potential, at the RF driven electrode, it is equal to the difference of the sum of applied RF potential and DC self bias, V˜RF + VDC , and the plasma potential V˜p on the other hand. A further small part appears in the plasma bulk as longitudinal electric field. Since the density of the highly reactive species which are generated by collisional impact is relatively low, for high etchrates high sheath potentials are required. The inherent disadvantage of this type of excitation consists in the nature of coupling: The DC bias which develops at the RF driven electrode depends on pressure and RF power, furthermore, it is directed normal to the surface of the 5 Since the etchrate exhibit a similar dependence, the rate-limiting step is supposed an attack of Cl+ ions, cf. Chap. 12.
7.3 Generation of plasmas supported by magnetic fields
233
electrode (on which the wafer is to be placed). This enhances the anisotropic component of the process of removal, but inevitably causes severe damage to the remaining topmost layers. Therefore, the strategy for high, anisotropic etchrates with combined low damage is obvious: • High plasma density. • Low electron temperature. • But above all: spatial separation of plasma generation and ion acceleration. The first two steps have been made in changing from DC excitation to RF excitation by capacitive coupling. In fact, higher plasma densities are associated with lower bias values at the RF driven electrode. To further enhance the energy transfer, magnetron-supported methods were applied. Among them, MERIE, Magnetron Enhanced Reactive Ion Etching, has become most popular. As was shown by M¨ uller et al., the application of a static magnetic field (130 Gauss or 13 mT) which was orientated in parallel fashion to the surface of the electrode (and the wafer), could enhance the etchrate of silicon by a factor of 7 and the etchrate of Si3 N4 by a factor of 2.5. Simultaneously, the DC bias dropped by a factor of 6 for Si and 5 for Si3 N4 , respectively, all the other parameters left unchanged [246]. All progress considered, the excitation of the plasma remains a capacitively coupled method with the disadvantages just mentioned. Inductively coupled plasmas clearly avoid these setbacks, but, due to the skin effect, they are limited to plasma densities to upper values between 1012 cm−3 in the plasma source, but by about one order of magnitude at wafer level. A qualitative leap forward is achieved by resonant coupling methods. The penetration depth of the wave fields is tremendously enlarged, elegantly avoiding all the problems and setbacks associated with the limitations caused by the skin effect (cf. Sects. 14.6 and 14.7). These experiments with whistler waves can be performed either in the microwave range (ICC frequency: 2.450 GHz); socalled ECR excitation or ECR heating, or in the RF range at typically 10 − 100 MHz (for technological purposes: still at 13.56 MHz): so-called regime of helicon waves. Two effects are benefitted from: the resonance with transverse electromagnetic waves and the enhancement of the diffusion coefficient by the magnetic field. Hence, comprehensive knowledge of the following topics is demanded: • Which waves propagate in the plasma, which will be absorbed? • What is the polarization of these waves? • ⇒ In short: What is the quality of the dispersion in a magnetized plasma?
234
7 High-frequency discharges III
First, we will recapitulate some characteristic phenomena of RF discharges. The AC field is weaker than the DC field at same amplitude. In the plasma bulk, the current is almost completely electronic conduction current, in the sheaths, it is ionic conduction current (small contribution) but mainly displacement current. j p ≈ (σe + iε0 ω)E p ;
(7.16.1)
j s ≈ (σi + iε0 ω)E s ;
(7.16.2)
whereas the proportionality factor between j and E is given by the conductivity (σ = e0 np μ) and can be calculated after Eqs. (5.7) and (6.4) to yield ε0 ωp2 1 np e20 ⇒σ= . (7.17) me iω + νm iω + νm np = np (x) is a spatial dependent function: Starting from topmost values in the center of the discharge, it falls down in outward direction to vanish at the walls. In a cylindrical plasma, this radial decline follows a Bessel function. For a capacitively coupled plasma, the degree of ionization exhibits a typical value between 100 ppm to 0.1 %, that means plasma densities of up to some 1010 /cm3 , and the angular plasma frequency ωp,e exceeds the operating angular frequency by more than one order of magnitude (5.64 GHz vs. 85 MHz), and the RF power can be readily absorbed by the plasma. For further rising plasma density, for example in inductively coupled plasmas, a sharp cutoff will occur in a cold plasma (νm = 0) at ωp,e = ω, and the RF wave will be reflected (Sect. 14.6). At finite temperature with the possibility of elastic collisions, this sharp transition is softened. σ=
7.3.2 Whistler waves 7.3.2.1 Phenomenology. Below the plasma frequency, waves exhibit an evanescent propagation behavior: Within the skin depth, the wave is damped to about one third of its initial value which leads to the phenomenons of cutoff and reflection. However, by application of a strong magnetic field, the sharp border between dielectric matter and metall is softened (cf. Sect. 14.7): Transmission becomes possible in a frequency band below the plasma frequency which is otherwise “forbidden.” The propagation behavior is determined by the dispersion relations and confines the upper boundary to ωc,e : ωc,e ≥ ω. As result, we observe slow-propagating [phase velocity vph is smaller than √ or equals to about 1 % of the TEM value (c = 1/ ε0 μ0 )] whistler or helicon waves. The name “helicon” was first used by Agrain for waves propagating in a high-conductive metal at very low temperatures [258]. “Helicon” is derived from the helical guiding lines of force which cause the electrons to follow. The inherent property of these waves is their almost exclusive coupling of their energy
7.3 Generation of plasmas supported by magnetic fields
235
density to the magnetic field [cf. Sect. 14.6, Eq. (14.164)]. In the gaseous plasma, these waves are normally denoted “whistler waves”; their propagation band is located between the cyclotron resonance frequencies of ions and electrons: ωc,i ≤ ω ≤ ωc,e with ωc = e0 B/m. It was Barkhausen who was the first to note the sound of atmospheric whistler waves [259]: Since it is a sensation which lies within the audible range, sometimes we hear a descending glissando starting at very high pitches when we listen to the radio in the short-wave band. It is caused by lightnings which generate a broad band wave signal. Among them are the whistler waves which propagate along the lines of√the earth’s magnetic field; and since their dispersion behavior follows ∂ω/∂k ∝ ω for ω < 1/2 ωc,e , the higher pitches arrive in front of the lower pitches which explains the audible sensation (cf. Sec. 14.7.1, Fig. 7.15). However, these whistler waves can not only be observed at ground level but also from spacecraft [260]. Vice versa, these waves can generate a plasma. Fig. 7.15. Spectrogram of a whistler recorded in northern Nevada, 1999. The tone, starting from the highest pitches of the piano, descends to the pitch of high a female speech voice, c exhibiting a steep glissando [261] www.auroralchorus.com.
7.3.2.2 Dispersion and absorption. The magnetic field causes the electromagnetic wave to propagate with anisotropic behavior. When viewed along the static magnetic field, we can distinguish between a right-hand circularly polarized wave and a left-hand circularly polarized wave. The first are denoted R-waves, the latter are called L-waves. First, the threshold frequency for propagation is different for both the types. For the R-waves —and it is but this type of waves which can transfer resonant energy to the electrons —this interaction causes a rise in the resonance frequency from ωp,e to ωr , and ωr is a function of both frequencies, ωp,e and ωc,e (cf. Table 14.2). In addition, a band of propagation at low frequencies will arise (n > 1): the regime of slow-propagating whistler waves. Its upper boundary is set by ωc,e . At its lower limit, it is operated with radio frequency, at its upper limit with microwaves. We can chose the intensity of the static magnetic field to match the electron cyclotron frequency with the operating frequency yielding a resonant excitation: so-called “ECR heating”. This holds true for the frequency range6 6 Waves with larger wavelength (lower frequency) are below the cutoff; they exhibit evanescent behavior and will be reflected at the sheath. This mechanism is comparable to the conduct of short waves at the Heavyside layer of the atmosphere.
236
7 High-frequency discharges III ωl ω ≤ ωc,e ,
(7.18)
whereas the plasma frequency of the electrons, ωp,e , should be high against the product of operating frequency ω and cyclotron resonance frequency of the electrons ωc,e : 2 ωp,e ωωc,e
(7.19)
with ωl the lower hybrid resonance, which is defined by 1 1 1 = 2 + . 2 ωl ωp,i ωc,i ωc,e
(7.20)
In this equation, the subscript “i” refers to the ions and their response to the fields which becomes important at very low frequencies. For the conditions (7.18/19), we can neglect ionic movements and electronic gyromotions, and following Stix, we denote this excitation coupled resonance [262]. The range for ECR operation is confined by ω = ωc,e < ωp,e . Besides simple Ohmic heating, another heating mechanism is supposed to become important in this case: Landau damping by helicons, low-frequent whistler waves which slowly move in a narrow tube in a certain angle with respect to the static magnetic field B 0 . Fast electrons can extract energy from the wave field with high an efficiency when their thermal velocity equals the phase velocity of the helicon: vph =
ω , k
(7.21)
which happens to be even lower than the thermal velocity of the electrons in the vicinity of resonance: The phase velocity vph and the wavelength drop to very small values (ideally, they would vanish), and the wavevector k becomes infinite. In contrast to the R-wave, which can push the electrons exclusively, this damping can also happen to L-waves when they interact with electrons of the high-energy tail of the EEDF. It was Frank Chen who suspected Landau damping as the main absorption mechanism in discharges excited by helicons. Besides stochastic heating, this is the second collision-free mechanism for energy transfer. However, the excitation of electrons by Landau damping is restricted to a very narrow area adjacent to vph . At a ratio of ve /vph = 0.1 the efficiency of energy transfer ξ almost vanishes (ξ ≈ 10−51 ), whereas ξ has risen to 0.17 at ve /vph = 0.5 (cf. Sect. 14.3). 7 Landau damping would lead to high a portion of electrons in the tail of the EEDF which could not be proven [264]. On the contrary, the damping of helicon 7 That this mechanism must play a significant role has made obvious by experiments with double Langmuir probes [263]. As Popov has found, the absorption of R-waves increases in the vicinity of the resonance (as it is expected since their phase velocity should decrease to very low values), but he succeeded in separating the two effects by measuring two electron temperatures far away from the ECR layer in the vicinity of the vacuum window.
7.4 Helicons in a bounded plasma
237
waves could be explained by Coulomb collisions [265] since at these high plasma densities of almost 100 %, every EEDF, no matter which is its initial character, turns into a Maxwellian distribution simply by Coulomb collisions. Although the fate of the L-wave during the process of absorption is still unclear, this explanation could shed light onto this open question. In particular, in plasma-filled cylinders at such frequencies far below the Larmor frequency, also the L-wave should contribute to plasma heating by evanescent absorption leading to an enhanced movement, albeit chaotic in character. It is but the R-wave which can solely rise the electron energy. After a long discussion over more than two decades, an additional heating, namely the strong damping of the waves named after Trivelpiece and Gould has evolved the main mechanism of absorption (cf. Sec. 14.7.2). In free space, these are the aforementioned whistlers.
7.4 Helicons in a bounded plasma 7.4.1 Introduction Which waves can propagate in a bounded plasma? As shown in Sect. 14.7.1, the regime of the R-waves encircles two frequency ranges, which are separated by ωc,e . Whereas the upper band exhibits the normal evanescent behavior, the lower band with ω < ωc,e is denoted the range of whistler waves which can propagate through the plasma, albeit with severe damping. Since cylindrical symmetry is provided, the waves can be described with Bessel functions, and the solutions for the wave fields can be obtained in rising accuracy: • Cold plasma: neglect of pressure gradients, • Lossless plasma: neglect of the ηj term (cf. [266] − [269] and Sect. 14.7.2). The propagation of these waves in a hollow wave guide is described by means of the Maxwell equations along with the dispersion relation. In contrast to the free-space behavior, the confinement restricts the propagation, and only certain modes can be transmitted through the plasma. Propagation of waves in free space can be modeled by a transverse wave with E- and B-vectors, respectively, orientated perpendicular to the direction of propagation, which is denoted by k, and all ω(k) values are possible for positive index of refraction n. In contrast to this simple conduct, the modes propagating through a waveguide exhibit field components which are orientated in parallel fashion to the wave vector k. It is possible to have either the complete electric field or the complete magnetic field orientated perpendicular to the k vector, but not both of them. In the former case, we speak of TE-modes, in the latter case, we denote these modes as TM-modes. These modes are caused by currents flowing in the walls of
238
7 High-frequency discharges III
the wave guides which generate a magnetic field in the direction of propagation k. The mode patterns are displayed in Fig. 7.16 for the m = 1 mode and in Sect. 14.7.2 also for the m = 0 mode. It has to be kept in mind, however, that the modes in a plasma-filled cylinder only resemble the modes obtained by application of classical electrodynamics. In contrast to the propagation of cylindric modes in a hollow waveguide which are characterized by vanishing divergence, we must take into account the conductive medium (occurrence of free charges) which causes the electric field to possess a finite divergence (open E-field lines), and the anisotropy which is caused by the static magnetic field. 7.4.2 Dispersion and wave fields The dispersion relation for helicons and whistler waves is derived in Sects. 14.7.1/2. For whistler waves, we can easily calculate the dispersion relation when we neglect electron inertia via Eq. (14.223) according to Eq. (7.22). For bounded helicons, we adopt this approach [Eq. (14.239.1)], and the equation to determine the axial wave vector yields kk =
μ0 e0 ωnp , B0
(7.22)
2 which can be solved along with k 2 = k2 + k⊥ . In combination with the dispersion relation (the wave vector consists of two components which are orientated in parallel and normal directions with respect to the static magnetic field), we obtain solutions of coupled differential equations of second order which further depend on the conditions of the reactor (perfectly isolating or ideally conducting walls and conditions for the equation of motion, see Sect. 14.7.2). As result, the fields of the helicons exhibit hybrid character between pure electrostatic (∇ × E = 0) and pure electromagnetic (∇ · E = 0) behavior. For example, the pattern for m = 1 in Fig. 7.16 resembles that of the TM11 mode for a circular waveguide, and especially the phase-sensitive patterns for the m = 0 exhibit a close relationship between the borderline cases (Figs. 14.39, 14.43 and 14.44).
7.4.3 Antenna coupling To couple energy into a bounded magnetoplasma and to transfer energy to the plasma constituents, the plasma modes should be excited which have their maximum on-axis or between edge and center. Ideally, they should not exhibit any azimuthal dependence to ensure a maximum of radial and azimuthal uniformity. In Figs. 7.17, the radial variations of the first two azimuthal modes with m = 0 and m = 1 for radial wave number n = 1 are displayed along with the squared current density for the first three modes. Since only the first two modes
7.4 Helicons in a bounded plasma
239
Fig. 7.16. Simplified pattern of the electric field (solid) and magnetic field (dashed) in a helicon wave (m = 1) after [270]. Note the open field lines for E which are caused by the free charges in the plasma– filled waveguide.
peak in the center, we are restricted to apply one of them for plasma generation. The specific feature of the mode with m = 0 is its axial symmetry, therefore, this wave should be radiated from a coaxial antenna in the center of the plasma. Due to reasons of feasibility and contamination, in most cases the wave with m = 1 is chosen and this mode is radiated from an externally wound antenna. The whistler waves are tuned to resonance by variation of • the length of the antenna La , • magnetic induction B0 and • excitation frequency ω; their axial wavelength (λ = 2π/k ) is calculated according to Eq. (14.223) (cos Θ = 1) 2πc λ= ωp
ωc ε0 B0 B0 = 2πc ≈ 5 × 109 [cm] ω ne e0 ω ne ω/2π
(7.23)
with B0 in Gauss, n in cm−3 and ω in Hz. The wavelength is proportional to the square root of the ratio of the static magnetic field over the electron density; this resonance condition is good for weak magnetic fields ≤100 Gauss (10 mT) and = 5 × 1011 cm−3 plasma densities ≤ 1011 cm−3 : for B0 = 100 Gauss (10 mT), n 8 and ω/2π =13.56 MHz we obtain λ ≈ 60 cm. λ scales with 1/np , and vice versa. This resonance condition imposes restrictive conditions on the excitation frequency and the antenna wavelength which are limited through the dispersion 8 We obtain the same result when inserting for the dielectric constant ε = n2 (Maxwell’s relation). Using Eqs. (14.208/209), we find an equation between k, np , and B0 : μ0 e0 np ω B0 2π 5 ≈ 5.6 × 10 [m] k≈ ⇒λ= B0 k np ω
with B0 in T, np in cm−3 and ω in Hz.
240
7 High-frequency discharges III 1.0
1.0 Bz
Br
0.5
Bq
B [r. u.]
B [r. u.]
0.5
0.0 Br
m=1 b=2
0.0 Bz
-0.5
-0.5 0.00
Bq
0.25
0.50 r/a
0.75
1.00
-1.0 0.00
0.25
0.50 r/a
0.75
1.00
1.00 m=0 m=1 m=2
I2
0.75 0.50 0.25 0.00 0.00
0.25
0.50 r/a
0.75
1.00
Fig. 7.17. Radial dependence of the magnetic wave fields and the current density squared (bottom) for the azimuthal modes m = 0 (LHS) and m = 1 (RHS) after [271]. k /k = 1/3 , radial wave number: n = 1.
relation: In a bounded system, λ and k = 2π/λ are free parameters no longer. It turns out that there is a set of eigenvalues of permitted angles which are determined by the radial modes. Hence, only a set of wavelengths N λ /2 fit the length of antenna La , beginning with N = 1 leading to k1, = π/La [270]. According to Eq. (7.22), at fixed ω, k or vph = ω/k , the phase velocity along the tube, the plasma density np should be proportional to the intensity of the magnetic field B0 or vice versa, np is expected to rise linearly with increasing B 0 . However, density jumps will occur when the static magnetic field is enlarged, which leads to the feared mode jumping (see Fig. 7.18): The wavelength does not remain constant along the axis. For the m = 0 mode, k⊥ is fixed to 3.83/a with a the tube radius (Fig. 7.19) [272]. Remembering that k ≈ π/La with La the length of the antenna, we see that for a given radius a, the density np scales with B0 , or vice versa, for a given np , the required field intensity B0 should vary linearly with respect to the tube radius a. To heat electrons specifically by Landau damping, antennas have been constructed according to the condition
7.4 Helicons in a bounded plasma
241
plasma density [1011 /cm3]
15
Fig. 7.18. Mode jumping occurs when the intensity of the static magnetic field is smoothly increased. The thick line rising from the origin is the theoretically expected dispersion for a helicon wave excited with a wavelength that doubles the length of the antenna [271].
10
5
0 0
500 Bz [Gauss]
1000
antenna length [cm]
60
Fig. 7.19. The axial wavelength as function of np /B [Eq. (7.23)] for an m = 0 mode after [272] for two different tube radii in an approximation neglecting electron inertia yielding typically 2/3 of the wave length applying full theory.
a = 5 cm
40
20
0
a = 10 cm
0
1
2 B0/np [10
3 -11
4
5
3
Gauss cm ]
k =
2π π ≈N λ La
(7.24)
with La the length of the antenna to obtain [273] me E= 2
ω k
2
(7.25)
with E the electron energy. Another effect which is encountered with changes in plasma density or magnetic field, is the sudden change from resonance (propagating and counterpropagating wave act to form a standing wave) to propagation, also in various directions [272]. These observations shed light on the heating mechanism which is explained for the radial component: To begin with low magnetic field intensities, the radial electric field is inductively coupled, i. e. the amplitude evanescently drops during penetration into the plasma, and power input is limited by the conventional skin effect. Hence, the maximum of the electric field is located
242
7 High-frequency discharges III
close to the coils, its local minimum is on-axis (center of the tube). But intensifying the magnetic field results in further penetration of the radial electric field. Eventually, we reach the first onset of a standing wave in radial direction. Simultaneously, the axial plasma density increases due to the improved radial penetration, and the phase velocity of the azimuthal electric field begins to drop. When the azimuthal field either encounters a boundary or can interfere with a counterpropagating wave, a standing wave of the azimuthal field in axial direction begins to form, and the characteristics of propagation changes from radially dominated to axially dominated with an increase of wavelength for both the radial and axial directions according to Eq. (7.22). As result, we find: • The power input through radial electric fields rises with increasing radial mode number, or the other way round: • Standing wave patterns with higher order will appear as eigenfunctions for the radial electric field at higher magnetic fields (Fig. 7.20).
amplitude [a. u.]
30
20
Fig. 7.20. The axial variation of Br and Bz for B0 ≈ 100 Gauss clearly shows a standing wave pattern: At each minimum of the wave fields, there is a phase change of π [271].
Br
Br
10
Bz
Bz
length of antenna
0
20
40 60 axial distance [cm]
80
To ensure a significant energy transfer to the electrons which should be in the order of the ionization maximum of argon (approximately 50 eV), the antenna has to meet the condition La a, a typical value would be about 120 cm. Following Eq. (7.23), however, a short antenna is required with La R. Plasmas in thin, long pipes are difficult to be operated because the electron temperature rises sharply with decreasing cross section of the reactor (cf. Sect. 3.5, also [272]). Therefore, an increase of the static magnetic field is required to enclose the plasma. Provided that Landau damping plays the significant role in exciting these helicons, Chen has compiled the achievable plasma densities along with the conditions for a ratio of L/λ = 7 and an electron temperature of 4 eV (Table 7.1) [274]. These values have to be slightly modified for a different choice of parameters (Te , L/λ and a). For more precise calculations, the dependence of the wave vector k on temperature and the resistance term should be taken into account.
7.4 Helicons in a bounded plasma
243
Table 7.1. Resonant tuning of the helicon waves to experimental parameters [a: diameter of the cylinder with perfectly conducting walls, λ: 2π/(k), L: 1/(k)], Lp : plasma length. n [cm−3 ] 1011 1012 1013 1014 1015 1016
a [cm] 25.1 7.9 2.5 0.79 0.25 0.08
B [T] 0.004 0.013 0.040 0.126 0.40 1.26
ω [MHz] 0.288 0.912 2.88 9.12 28.8 91.2
λ [cm] 500 159 50 15.9 5.0 1.6
Lp [cm] 3500 1100 350 111 35 11
7.4.4 Operation The discharge is operated in the radiofrequency range (5 − 50 MHz) with the excitation frequency of 13.56 MHz for technical purposes. The antenna is placed upon a plasma cylinder whose walls consist either of a conducting or isolating material. Across its length La , the antenna radiates a time-dependent B-field that couples with the transversal magnetic field of the helicon mode. Furthermore, an electric current is induced directly beneath the antenna wires which is opposite in direction to the antenna current. This current, in turn, excites a transverse electric field across the antenna which can couple with the transverse electric field of the helicon mode (cf. Fig. 7.21) [270]. This leads to a rotation of the wave fields in space and time. A significant assumption of this model is a very high plasma density which justifies the neglect of displacement currents within the plasma, and we regard the discharge as a plasma filled circular hollow waveguide with either isolating (glass) or perfectly conducting walls (metal). The plasma densities are among the highest which ever have been achieved in low-temperature plasmas (up to 100 % which is only possible with high an amount of doubly-ionized species) in conjunction with very low plasma potentials Vp (typically 10 − 15 V). The magnetic fields which are required for this performance are remarkably low (50 − 150 Gauss or 5 − 15 mT, Fig. 7.22). As real challenge, however, remains the matching. In the first matching networks, the π-matching consisted of two variable capacitors and the antenna serves as inductive device. In retrospect—the inventor Boswell himself investigated the matching network for more than two years—it was mainly this difficulty which prevented a commercial breakthrough [273]. Boswell eventually succeeded with his Alcatel reactor mainly constructed by himself [278] (Fig. 7.23).
244
7 High-frequency discharges III
Fig. 7.21. Power coupling of helicon waves (m = 1) via a Boswell-type antenna after [275]. The antenna matches the condition that the currents should alternate in direction every half wavelength. In fact, the solenoid consists of a series of small magnets, and the glass tube is very tall. The ratio height over diameter often exceeds 20 [271, 276].
antenna
RF
downstream applications
30
20 Vp
20
15
np
10 15
5 0
10
150
300
450
11
25
25
plasma density [10
plasma potential [V]
-3
cm ]
30
Fig. 7.22. The sharp drop in the plasma potential at an antenna power of about 350 W indicates the onset of helicon generation. This is reflected in the simultaneous steep increase in plasma density. Below that threshold, the discharge remains in the capacitively coupled modus [277].
600
antenna power [W]
7.4.5 Experiments Boswell described the first experiment in argon at a pressure of 1.5 mTorr (0.2 Pa) in 1970 [273]. The experimental setup consisted of two coaxial cylinders: a source made of glass, and a reactor of glass or metal. The solenoid at the source chamber is required to couple the RF power into the plasma, and the solenoid at the processing chamber confines the plasma. An axial static magnetic field with an intensity of 1.5 kGauss (150 mT) surrounds the tube. The axially orientated antenna generates a standing helicon wave by coupling the transverse antenna fields to the radial plasma fields (Ez = 0). With a double Langmuir probe, Boswell et al. measured plasma densities between 1.5×1011 and 3.8 × 1012 cm−3 , i. e. a degree of ionization between 0.4 and 9 %, dependent on B0 . This causes an index of refraction of up to 1012 by which the classification as resonance phenomenon is justified. Boswell coined the acronym RIPE for etching in these discharges (Resonant Inductive Ion Etching) [278].
7.4 Helicons in a bounded plasma
245
matching network
RF
antenna
source solenoid
substrate table (isolated or ccp) reactor solenoid
vacuum system
substrate cooling matching network
Fig. 7.23. The first reactor which was commercially available was equipped with an antenna of the Boswell-type c Societe (Alcatel) [278] ( Fran¸caise du Vide).
RF
Later on, Boswell et al. reported on a degree of ionization almost 100 %, measured with a microwave interferometer at electron energies of about 3 eV [275].9 Even when we take into account that highly ionized plasmas consist of a large fraction of double-ionized species which distort the degree of ionization to higher values, this result is very impressive.10 At low static magnetic fields, the color of an argon plasma is white reddish which is due to the lines of atomic argon [Ar(I)]. Increasing B0 to values of 500 Gauss (50 mT), the color suddenly turns to blue (Ar lines at 480.6 and 488 nm); the plasma density has increased by a factor of 5. Additionally, a sharper confinement of the radial profile is combined with the doubling of the axial frequency. Even further rise of the magnetic field to 75 mT causes another turn-over which is associated with a doubling of the plasma density [271]. Boswell himself conducted some RIPE etching runs in the silicon system applying the reactive gases SF6 and Cl2 [278, 279]. At a DC bias of −100 V and a radiated RF power of 500 W (13.56 MHz), very high etchrates could be found [0.4 μm/min at 0.5 mTorr (0.07 Pa) and 1.5 μm/min at 3.75 mTorr (0.5 Pa)]; 9 In most cases, the cutoff will be measured, i. e. the frequency for which the measuring frequency equals the plasma frequency (ω = ωp ), and the wave will be reflected. 10 From the two branches of the Langmuir curve, we obtain values for ni und ne . For single-ionized plasmas holds ne = ni , from the deviations, we can conclude the fraction of multiple ionization with reserve.
246
7 High-frequency discharges III
the anisotropy, defined as A = tan α = lv /lh with lh the horizontal and lv the vertical etchrate, however, dropped by nearly 50 % from 1.0 to 0.6. The systems with coupled resonance are distinguished by an excellent radial uniformity (±1 % across 4"), which can be attributed to the high spatial plasma homogenity, combined with extremely low bias potentials. Furthermore, the competion between etching and deposition in carbon-containing gases can be decided in favor of etching even at very low bias values [280]. 7.4.6 Summary Helicons are superpositions of bounded low-frequency whistler waves which propagate at a common but fixed angle with respect to an axial static magnetic field. From their dispersive behavior, they are mixtures of electromagnetic (∇ · E = 0) and electrostatic (∇ × E = 0) fields. Their field pattern can be described by E ∝ exp [i(ωt − kz − mΘ)]
(7.26)
with Θ the angle between the wave vector and the static magnetic field, and m the number of the the azimuthal mode. In reactors for helicon wave excitation, an RF driven antenna is mounted directly on the reactor wall. Combined with relatively low static magnetic fields, excitation to very high plasma degrees is made feasible. The waves transmit their energy by collisions or a mechanism which is closely connected to Landau damping, interaction with the Trivelpiece-Gould mode. Due to high plasma densities, the sheaths are considerably thinn and completely collisionless. For vanishing magnetic field, a smooth transition to inductively coupled plasmas will happen.
7.5 Electron cyclotron resonance 7.5.1 The electric field and the diffusion length The introduction of plasma sources which were driven with electron cyclotron resonance marked a significant progress in low pressure plasma technology, a spin-off process of projects of developments of nuclear fusion and thrusters.11 Energy of a microwave field is coupled into a plasma by means of a static divergent or radially inhomogeneous magnetic field (quadrupole, octopole or even higher order) at a certain frequency, the Electron Cyclotron Resonance frequency (ECR). As a simple example, we start with the plasma-filled cylindrical tube which is penetrated by an axialsymmetric, static magnetic field. Enforced by the 11 A “spin-off” process denotes the transfer of production processes, materials and know-how from the military/governmental sector into the civil/private sector.
7.5 Electron cyclotron resonance
247
Lorentz force, the electrons will gyrate on orbits and helices with the Larmor radius perpendicular to the lines of this field (Fig. 7.24).
B
F v
Fig. 7.24. Charges are forced to gyrate around the lines of a magnetic field. Due to its small mass, the radius of the electron is smaller by orders of magnitude compared to the ion radius. Changes in the electron velocity (electron energy) alter the radius of the orbit but not the angular frequency. The electrons all rotate with the same Larmor frequency.
At sufficiently low pressures the electron mean free path becomes farther than at least one orbit. If this tube is exposed to a microwave field in the axial direction, a synchronization of these movements can happen at ωc,e = ωop : The electron cylotron frequency equals the angular frequency of the microwave, and the electrons can pile up energy from the wave field which causes the orbits and helices to degenerate to spirals. This resonant enlargement of the energy transfer from the wavefield to the electrons is denoted as ECR heating. For convenience, the incoming microwaves are supposed to be linearly polarized, and they can easily be decomposed into two wave components which are polarized circularly, and L-wave and an R-wave (Figs. 7.25). +E
CCW: R
CW: L
Fig. 7.25. A linearly polarized wave can be decomposed into two circularly polarized components with vanishing phase angle. For positive z, the CW rotating component is the L-wave, the CCW rotating one is the R-wave.
-E
Due to the negative charge of the electron, it is only the R-wave which can enhance the tangential component of the orbital motion of the gyrating electron (Figs. 7.26). Neutral atoms and molecules are excited and ionized by inelastic collisions with the heated electrons. Both the types of electric carriers which are coupled by electric forces experience a slow drift out of the divergent or radially inhomogeneous magnetic field. As a matter of fact, a beam of electron ion pairs leaves the plasma source. This movement is caused by a conversion process of transverse kinetic energy to longitudinal kinetic energy (cf. Sect. 14.5). By steering the degree of divergence, the longitudinal drift velocity component can be
248
7 High-frequency discharges III
L
B
F v
R 0°
90°
180°
270°
Fig. 7.26. From the two circularly polarized waves, it is only the electric field of the R-wave which can continuously transfer energy to the gyrating electron, shown her for four distinct phase angles. For the same phase angles for the L-wave, only two of them happen to push the electron (0◦ and 180◦ ), the remaining two will retard the orbital component, so there is not net result any.
adjusted over a large range. Provided ions moving downstream do not suffer collisions, their trajectories should remain in parallel direction with respect to the static magnetic field, and they eventually approach the substrate with normal incidence and small velocity. Hence, the crystal damage can be kept low — at the expense of a low etchrate. By the spatial separation of plasma source (generation of carriers) and substrate immersed by the plasma (definition of the kinetic energy of the impinging particles), it was possible for the first time to control density and energy of the ions independently. The absorption of energy of the electrons as a whole is shown in the appendix (Sec. 14.7.1). Here, we want to investigate the behavior of a single electron in an electromagnetic field which is perpendicularly penetrated by a static magnetic ˜ field B 0 , which should be large against the fluctuating magnetic wave field B: ˜ ∧ B0 z k B0 B
(7.27)
with k the wave vector of the incident wave, we start with e0 (7.28) B 0 ∧ E = E 0 eiωt ∧ v = v G eiωt m for the equation of motion of the gyration center G [Drude or Langevin equation, Eq. (14.172)] ω=−
m
dv = −e0 E(t) − e0 v × B 0 − νm v, dt
(7.29.1)
7.5 Electron cyclotron resonance
249
which yields in components ⎫
(iω + νm )vx + e0mB0 vy = − em0 Ex ⎪ ⎬ − e0mB0 vx + (iω + νm )vy = − em0 Ey ⎪ (iω + νm )vz = − em0 Ez ⎭
(7.29.2)
or as matrix with ωp2 = ne20 /mε0 ⎛
1 ⎜ ⎝ ε0 ωp2
⎞⎛
⎞
⎛
⎞
iω + νm ωc 0 jx Ex ⎟⎜ ⎟ ⎜ ⎟ −ωc iω + νm 0 j = ⎠⎝ y ⎠ ⎝ Ey ⎠ . jz Ex 0 0 iω + νm
(7.29.3)
This is the matrix equation σ −1 ij · j = E;
(7.29.4)
but we are interested in the solution j = σ ij E ⇒ ρ v = ρ μij E ⇒ v = μij · E.
(7.30)
For this end, we must invert Eq. (7.29.3) and obtain the doubly inverted matrix −1 μ−1 with ij ) A=
1 nε0 ωp2
iω + νm [(iω + νm )2 + ωc2 ]
:
(7.31.1)
⎛
⎞
(iω + νm )2 −ωc (iω + νm ) 0 ⎜ ⎟ (iω + νm )2 0 A × ⎝ ωc (iω + νm ) ⎠. 2 2 0 0 (iω + νm ) + ωc
(7.31.2)
μij can then be expressed as ⎛
⎞
μ⊥ iμ× 0 0 ⎟ μij = ⎜ ⎝ −iμ× μ⊥ ⎠, 0 0 μ
(7.32)
which is denoted as usual [281]: μ⊥ : μtransverse : μ× : μnormal : μ :
μparallel :
e20 iω+νm ; m (iω+νm )2 +ωc2 e20 −ωc ; m (iω+νm )2 +ωc2 e20 1 . m iω+νm
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(7.33)
If the operating frequency equals the cyclotron frequency, a singularity will occur. The electron velocity perpendicular to the lines of the static magnetic field diverges, and Eqs. (7.29.2/3) are not valid any longer: At resonance, the
250
7 High-frequency discharges III
spherical movement degenerates into a spiral movement. The resonance is very sharp and only effective within a frequency interval of about ±5 %. As a typical resonance phenomenon, a strong energy absorption occurs even for small fields at ω = ωc , which is but weakly damped at low discharge pressures. For pressures between 10 and 75 mTorr (1 to 10 Pa), the breakdown field is significantly smaller than without a magnetic field (cf. Fig. 7.27. For 2.450 GHz, the magnetic induction for resonance amounts to 875 Gauss or 87.5 mT). The magnetron radiates unpolarized wave fields. Hence, the radial component which is normal with respect to direction of propagation can be split into a left and right circularly polarized fraction, resp. Helicon waves of both polarizations cannot propagate in the plasma-filled cylinder if the refraction index sinks below unity, i. e. at a certain diameter less than λcutoff , for magnetic fields B < Bc,e , the waves will be reflected. However, R-waves can propagate in that part of the plasma column where B > Bc,e . It is but this wave that can push the gyrating electrons (cf. Sect. 14.7.1) [263]. Walking on the resonance ridge (n → ∞, vph → 0), it should be considered that this condition for strong absorption is only valid for the R-wave and can be narrowly missed. The L-wave is likely to be reflected when the cutoff is about to be reached (n → 0, vph → ∞), and perhaps mode hopping is caused at this threshold [282]. Although the fate of the L-wave during the process of absorption still remains unclear, Popov’s proposal of Landau damping of the L-wave by highvelocity electrons could shed light on this open question [263].
EB [V/cm]
100
10 0.0
30 Torr (4 kPa) 18 Torr (2.4 kPa) 8 Torr (1.05 kPa) 4 Torr (530 Pa) 2 Torr (270 Pa) 1 Torr (130 Pa)
0.1
B [T]
0.2
Fig. 7.27. Already in the 1950s, the dependence of the breakdown voltage on the magnetic flux density was subc ject to research [283] ( The American Institute of Physics).
0.3
The resonance is definitely not caused by collisions of electrons, on the contrary: The energy absorption will be limited by collisions of the electrons with heavy particles. Lower damping will increase the Q-factor of the resonance (Fig. 7.28. Hence, the lower condition for resonance should be formulated as
7.5 Electron cyclotron resonance
251 ω ≥ νm ,
(7.34)
which means discharge pressures in argon of significantly less than 10 mTorr (less than 1 Pa) for the typical excitation frequency of 2.45 GHz. The effective electric field for a motion perpendicular with respect to the ˜ ⊥ B 0 ) will change as well. To show this, we take the static magnetic field (E real part of Eq. (7.29.3), take the square, average over time and obtain 2 Eeff
ν2 = m 4
1 1 + 2 E02 , 2 2 νm + (ω − ωc ) νm + (ω + ωc )2
(7.35)
2 (ω + ωc,e )2 ). which turns into the RMS field at high discharge pressures (νm At the singularity, the second summand vanishes, and the shape of the effective field will become Lorentzian. For this case, the FWHM equals the frequency of elastic collisions:
νm =
√
Δω,
(7.36)
which facilitates a direct measurement of νm . The absorbed kinetic energy is given by 1 1 2 Ekin = mv 2 = ma2 /νm , 2 2
(7.37)
or
Ekin
1 1 e2 E 2 1 e20 1 = m 0 2 eff2 = + 2 E02 . · 2 + (ω − ω )2 2 m νm 4m νm ν + (ω + ωc,e )2 c,e m
(7.38)
For high discharge pressures, the kinetic energy is approximately (ω νm ): Ekin ≈
e20 E02 . 2 2mνm
(7.39)
This equation is very important for hydrogen and helium, since in these gases, νm does not depend on the particle’s velocity. Only in this case can the absorbed energy be described as a function of frequency and magnetic field. For lower discharge pressures, the resonance at ω = ωc,e will become sharper and the power absorption can be approximated by P (r) ≈
ne (r)e20 2me νm
νm ω − ωc
2
E(r)2 .
(7.40)
Additionally, the diffusion length enlarges, and in cylindrical geometry we obtain (B 0 L, R: radius, h: height of the cylinder): 2 νm 1 = 2 2 Λ νm + (ω + ωc,e )2
2
2.405 π + R h
.
(7.41)
252
7 High-frequency discharges III
nm = 2 wc nm = wc n m = w c/2 n m = w c/4
Eeff/E0
0,6
0,4
Fig. 7.28. For lower discharge pressures, the Q-factor of the c resonance increases [283] ( The American Institute of Physics).
0,2
0,0
0
1
w/w c
2
3
Just at very low pressures the energy transfer is extremely efficient. The magnetic field causes an effective enlargement of the plasma volume in all those directions which are perpendicular with respect to the magnetic field, by the factor 2 νm + (ωc,e + ω)2 . 2 νm
(7.42)
For too low a pressure, the mean free path λe and the closely connected energy absorption will steeply increase, but the ionization drops due to • the density reduction of neutral molecules, and • the gradual decrease of the energy dependent cross section of ionization with increasing energy. Since by increasing the pressure, the efficiency of power absorption will again deteriorate, the working window for ECR is relatively narrow, and the discharge pressures should be kept below a few mTorr [284]. Further increase of the discharge pressure leads to the condition νm ω. When entering this regime, the mean power absorption per collision will decrease, the voltages for breakdown will grow and become proportional to the number density (Fig. 7.27). These competing mechanisms directly lead to a maximum of ionization in the pressure range between 1 and 0.1 mTorr [0.1 and 0.01 Pa, Eq. (7.34)] [285]. Measurements with Langmuir probes in argon revealed a monotonous drop of the electron temperature with rising pressure which has been referred to the increasing collision frequency (cf. Sect. 14.1) [286]. These competing processes cause a maximum in the saturation current measured by a Langmuir probes at a pressure of about 0.075 mTorr (10 mPa): To lower pressures, the number of collision partners decreases, to higher pressures, the electron temperature goes down. Furthermore, due to the low collision number number between the
7.5 Electron cyclotron resonance
253
charged carriers and the resonant excitation, it is questionable to speak of an electron temperature at all. At least Asmussen reported on strong deviations from a Maxwellian behavior [284]. In the resonance zone, the losses, predominantly by diffusion, are exactly balanced by carrier generation. Albeit the solenoidal magnetic field minimizes these radial diffusion losses, the very high “electron temperature” effectively counteracts this radial confinement of carriers.12 Temperatures of up to 8 eV have been measured at pressures below 1 mTorr [287]. On the other hand, electrons are guided by the axial field into the downstream region, thereby conserving the structure of the radial electron temperature distribution function and density, and, via ambipolar coupling, also the ionic parameters as well [288]. The electrons that are accelerated in the source volume will ionize the gas molecules to a comparable extent as on the low-frequency border of the band of whistler waves (ω ωc,e ), depending on discharge pressure, flow rates and the microwave power.13 7.5.2 Coupling of microwaves The models which describe the coupling of microwaves assume the microwaves to propagate in the plasma bulk against the magnetic beach of the ECR layer where they are resonantly absorbed. Therefore, the microwave window and the region adjacent to it are the zones which will decisively influence the quality of coupling. Stevens et al. demonstrated with conventional transmission line calculations that for a given plasma density, the optical thickness (product of geometrical thickness and refraction index) will reduce the reflection of the microwave at the surface of the window, thereby stabilizing the preferred mode [290] (cf. Figs. 7.31 + 7.34). 7.5.3 Electron cyclotron resonance heating The electric field of a linearly polarized wave (sum of a circularly right-hand polarized (RHP) wave and a circularly left-hand polarized (LHP) wave of same amplitude), which may propagate in z-direction, is given by (i: unit vector in x-direction) 12 It should be kept in mind that the statistic definition of temperature requires processes of equalization, most simply realized by collisions between like particles (electrons) or collisions between unlike particles (electrons and neutrals). At these low discharge pressures, both processes are not very likely to happen, despite the high degree of ionization. 13 In retrospect, this type of discharge was used to generate the first high-density plasmas. Since this discharge can be operated without metallic electrodes, a contamination with easily sputtered electrode material could be avoided. For the first time, discharges with reactive gases were operated over longer periods (several hundreds of hours [289]) under stable conditions which led to their application as plasma source for reactive ion beam etching (RIBE, cf. Chap. 8).
Intensität [a. u.]
254
7 High-frequency discharges III
Fig. 7.29. The wave should enter the plasma at the highest value of the magnetic field and will transmit its energy to the electrons within the ECR layer that is sharply defined at low discharge pressures.
RHS-Welle in der Resonanzzone Restwelle nach Absorption Begrenzung der Resonanzzone
x [a. u.]
E(r, t) = [iEx (r)]eiωt .
(7.43)
The electric field of the RHP wave rotates around B 0 in counterclockwise direction with frequency ω and transfers energy to the electrons. Secondly, the magnetic field B 0 exerts the Lorentz force −e0 v × B on the electrons which are forced to gyrate with the Larmor frequency ωc,e around B 0 . By these two interlinked processes, a continuous transfer of energy from the wave field to the electrons is ensured.14
Pabs =
V
Sohm d3 x = n0 e0 (x − iy)ERHP
(7.44)
taking account of the radial inhomogenity of the magnetic field which we assume as locally linear, which allows an expansion according to
B(z) = B0 +
∂B ∂z
z = B0 + B z = B0 (1 + αz),
(7.45)
ω=ωc,e
with z the distance between the zone of exact resonance (z = 0), and for the cyclotron frequence, we obtain in an analogous manner: z = ωres (1 + αz), ωc,e (z) = ωres + ωres
(7.46)
which gives the gain in kinetic energy: 1 e2 E 2 t2 Ekin = me v 2 = 0 RHP . 2 2me
(7.47)
For t, we insert the time of gyration in the resonance layer. We start with a thermal electron which will pile up energy in this zone, will further diffuse and 14
On the other hand, the ions are almost not deflected by the LHP wave.
7.5 Electron cyclotron resonance
255
1.0
intensity
0.5
0.0
Fig. 7.30. The absorption of the RHS wave by the plasma can be described by an Airy function.
-0.5
-1.0 0.0
0.1
0.2
0.3
0.4
0.5
B/Bc
will be reflected at the peak of the static field. On its course back, it will pass the ECR layer for a second time (Figs. 7.29/7.30). Its initial velocity is 2 v02 = v2 + v⊥
(7.48)
with v the component orientated in parallel fashion and v⊥ the component directed in perpendicular fashion with respect to the static magnetic field B 0 . The equations for the velocity components orientated in parallel and perpendicular direction with respect to B 0 are dv dt dv⊥ dt
≈ −
2 αv⊥,0 , 2 ∼
+ iω(1 + αz)v⊥ ≈ − em0 Ee ,
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
(7.49)
with the approximative oscillatory solution for t [291] t=
8 2 π sin x3/2 + , παv ω 3 4
(7.50)
with
2ω x=− αv⊥,0
2/3
αΔz.
(7.51)
The oscillation of t comes about because the resonance zone is passed twice (Fig. 7.30). From Eq. (7.51), we can further see: • t scales inversely with v⊥,0 : the higher the transverse energy of the electron, the less effective the ECR heating. • Additionally, t scales inversely to v : the slower the velocity directed in parallel fashion to the static magnetic field, the longer the residence time
256
7 High-frequency discharges III within the ECR layer, the higher the amount of energy which can be piled up.
7.5.4 Electron cyclotron resonance reactors In the past, different types of ECR reactors have been offered on the market (now, it is cut back to the Hitachi/NTT variant). They mainly differed by the method of microwave coupling into the zone of resonance (Fig. 7.1). The technical realization is displayed in Figs. 7.31 and 7.34/7.35, respectively, for the two prototypes: waveguide or cavity applicator. The microwave energy is coupled into the recipient via a coaxial input probe (cavity applicator) or via a waveguide (waveguide applicator). In this case, the static magnetic field can be oriented in parallel or perpendicular fashion with respect to the direction of the wave with wave vector k (helicon waves and TG waves): • k B 0 or • k ⊥ B 0 15 In both cases, the metal housing confines the discharge, however, often a quartz tube is placed into the interior of the waveguide to avoid severe deterioration of the waveguide’s surface. The microwave field (so-called TE-wave) is generated in a microwave oscillator at constant frequency but variable cw power output (magnetron) and is guided into a tightly flanged rectangular waveguide (waveguide applicator) or a coaxial input probe (cavity applicator). Just at its entrance, the power is rectified and measured by a detector diode. The power coupled into the plasma column is the difference between the forward power and the reflected power. An orthogonal circulator serves to protect the magnetron by guiding the reflected ground wave into a water-cooled dummy load. 7.5.4.1 Waveguide applicator. The microwave energy is coupled into the recipient via a hollow waveguide. This waveguide can come either as radial waveguide (Figs. 7.31 + 7.34) or cavity resonator (Fig. 7.35) with the case as outward boundary (either made of metal or protected by a cylinder of quartz to avoid contamination what has consequences for the dispersion of the helicons, Sect. 14.7.2). To optimize the transmittance, the waveguide is equipped with a three-stub tuner. It consists of three stubs, mostly made of brass, which serve to match the wave to the highly inductive load of the plasma-filled hollow waveguide. With these stubs (section of coaxial line 1/4 λ long and short-circuited at one end), dielectric losses can be fought. 15 This case is realized in so-called distributed ECR systems (DECR). Here, the X-wave with a resonance at the upper hybrid frequency [Eq. (14.207) and Table 14.2] takes over the role of the R-wave.
7.5 Electron cyclotron resonance
257
Because of the cylindrical symmetry, their solutions are Bessel functions. Only whistler waves can propagate which are guided by the static magnetic field B 0 (q B 0 , for details see Sects. 14.7.1/2). dummy load three-stub tuner
orthogonal circulator magnetron
coupler
detector diode
l/4-window solenoides
ECR plasma
ion beam grid optics
Fig. 7.31. Schematic view of a divergent field ECR waveguide applicator with opc Oxford Instruments 1993). tional ion beam grid optics used for etching purposes (
By guiding the microwaves through waveguides, their “freedom of movement” is severely confined, and the waves answer by generation of rectangular and cylindrical waves in the waveguide and by cylindrical whistler waves in the plasma-filled tube. In vacuum, the electric and magnetic fields are always rectangular to the wave vector k. For guided waves, however, this holds only true for TE waves (for the electric field vector), or TM waves (for the magnetic field vector). The amplitudes of these waves exhibit a radial and azimuthal dependence and vary also in time which can be described by the general wave equation ψ1,2 = A1,2 Jm (γ1,2 r)eikz eimϑ e−iωt
(7.52)
with Ai amplitude constants and Jm the Bessel function of order m. Via a mirror tilted by 45◦ or a taper, the wave is deflected by 90◦ . Just in front of the dielectric window made of quartz or sapphire, a waveguide transformer transforms the rectangular TE wave into a cylindrical TM wave. The simplest mode converter were a cuboid that generated a circular TE11 wave from a rectangular TE10 wave (Fig. 7.32 M, λcutoff = 6.12 cm). At a frequency of 2.45 GHz, this mode required a plasma tube radius of 3.59 cm (λcutoff = 7.19 cm) [292] (Fig. 7.32 LHS). Albeit for this mode, the electric field peaks on-axis, it is azimuthally asymmetric. This could lead to etch profiles which are non-radially symmetric. Hence, a cylindrical converter is applied
258
7 High-frequency discharges III
which generates the TM01 cylindrical wave. To propagate, this mode requires a minimum radius of 4.67 cm [293] (Fig. 7.32 RHS, λcutoff = 9.34 cm). Its profile is concentric with outward-bound growing or fading magnetic field.
H
E
E
E
H
Fig. 7.32. LHS: The TE11 wave is the dominant wave with a cutoff-wavelength of 3.59 cm at 2.45 GHz. Albeit this mode peaks on-axis, it exhibits an azimuthal dependence. Therefore, the TM01 wave is generated applying a cylindrical converter (RHS) [294]. M: the rectangular TE10 wave.
By application of two magnets, the resonance condition can be met at three positions (Fig. 7.33). At the upper magnet [coming from higher a magnetic field than required for resonance (875 Gauss)] and above and below the lower magnet, so that the residue of the helicon wave can be absorbed. But even for lower magnetic field intensities at the lower magnet, the electrons will be mirrored between the two magnets (“magnetic bottle”, cf. Sect. 14.5). At a certain lower threshold, the magnetic bottle will leak, and electrons will flow downstream. Although the Larmor radius of the ions is larger by orders of magnitude, so they should be not affected, Coulombic interaction will force the ions to a collimated, equal flow of electron ion pairs out of the magnetic bottle. Provided they do not suffer collisions, their trajectories should remain parallel to the static magnetic field, and they eventually impinge on the substrate with normal incidence (so-called “plasma stream” mode). A third magnet at wafer level is scarcely used. The static magnetic field is divergent in radial and axial directions: • Radial direction: The intensity of the magnetic field rises in outward direction from the center to the edge of the discharge. As result, carriers drift to the discharge center increasing the charge density. This counteracts the steep gradient which has been created by the strong ionization in the annular, very narrow shaped ECR layer. By varying the intensity of the magnetic field, the zone of resonant energy absorption can be radially shifted. • Axial direction: The topmost value of the magnetic field should be found at the microwave window. The field strength must exceed the value which
7.5 Electron cyclotron resonance
259
1250
B [Gauss]
1000
125 A 875 Gauss
120 A 110 A
750
90 A 70 A
500
substrate level
50 A
250
30 A 10 A
0 0
15
30 d [cm]
45
60
Fig. 7.33. Principal sketch (“brontosaurus” shape) of a typical waveguide system with its magnetic field gradient for two magnets [295]. The efficiency of ionization is higher for the two-magnet system since the magnetic bottle mirrors the electrons back and forth, however, also the diffusion losses are higher than in a one-magnet system.
is necessary for resonance (cf. Fig. 7.33, cutoff), otherwise, the wave will be reflected rather than absorbed. Since the intensity of the magnetic field gradually decreases in the downstream direction, a magnetic bottle can be formed which leaks a little in the downstream direction. At the microwave window, a strong magnetic field serves to have the axial component of the electron velocity vanish, at the other end (downstream), B 0 should exhibit its lowest value which exerts a force downstream on the plasma (cf. Sect. 14.5.1). In thermodynamic equilibrium, a potential difference ΔΦ is generated which causes a flow of charged particles: ΔΦ = −
μ ΔB , e0 kB Te
(7.53)
and by this variation in plasma density, an additional potential difference is generated: (np : plasma density in the plasma source, n: plasma density at the capacitively driven RF electrode):
kB Te n V = ln . e0 np
(7.54)
For an excellent performance of the ECR source, this divergent magnetic field has evolved to be extremely important. Its tasks are twofold: • The ECR layer is confined to a very narrow annular zone where the condition for resonance is met. By means of the divergent magnetic field, the
260
7 High-frequency discharges III
microwave magnetron inert gas feed symmetric plasma coupler
rectangular waveguide
mirror
l/4 quartz window magnet 1
monochromator photomultiplier or photodiode array
magnet 2 reactive gas feed
mass spectrometer for RGA
wafer HR grid
RF
substrate electrode RF generator + matching network
turbo pump mechanical pump computer computer
Fig. 7.34. Besides the application of two magnets to effectively screen electrons from evading the plasma source, the separation into inert and reactive gases is strongly recommended. In particular, film-generating gases will belong to this group. Some of c Oxford Instruments them polymerize in the plasma rather than on the surface ( 1993).
zone of equal ion density is extended a little bit, above all, the density will be reduced. • As main result, a small ambipolar field is generated which exerts a net force which is directed outward, and the ions are forced to leave the weakening field. 7.5.4.2 Cavity applicator. Alternatively, the resonant excitation of a plasma volume can be achieved by coupling microwave power into a system which consists of a tunable microwave cavity and a radially inhomogeneous static magnetic field (Fig. 7.35). Only a fraction of the cavity volume is occupied by the plasma source which is sealed “upstream” by a quartz window. The plasma source can be optionally confined by a screen or grid in downstream direction to operate as an ion beam source (“ion beam mode”) as well. Typically, it consists of N pairs of permanent magnets with a pole-face maximum field of about 3000 Gauss, assembled as an octagon with eight magnets [284] or as a circle with twelve magnets [296] (two of them are shown in Fig. 7.35) to form a cylindrical resonant zone in the plasma source which exhibits a diameter of 12.5 cm to ensure the condition of resonance, approximately 1 cm inside the discharge zone. The process gas is introduced via pinholes (not shown) in the annular ring below the magnets [284]. The microwave power is coupled into the cavity via a coaxial
7.5 Electron cyclotron resonance
261
input port, and the cavity is tuned to resonance via the sliding short [297]. The volume of the resonant cavity is defined by the
2r sliding short
coaxial
input probe
h
viewing port
input port quartz window solenoid
12.5 cm ECR plasma to process chamber
solenoid
Fig. 7.35. Cross sectional view of a cavity multicusp ECR applicator after [284].
• diameter of the cylinder (2R), and the • height (h) of the remaining cylinder between sliding short (upper level), and the end plate (lower level) for excitation, and for operation by the volume denoted as “plasma”. Hence, various single modes can be excited (common are the TE111 or TE211 ). The modes of the fields vary in axial and radial directions, the modes with numbers greater than zero also in the azimuthal direction. For example, the TM-mode for a cylinder with inner radius R and height h has the solution ψ(r, θ) = Jm (γmn r)e±imθ
(7.55)
with γmn = xmn /R with xmn the nth roots of the Bessel functions Jm (x) = 0 (m is the azimuthal mode number), and the electric field Eθ must vanish at r = R. Hence, the resonance frequencies of the TM-mode are given by
ωmnp
c =√ εμ
x2mn p2 π 2 c 1 p2 π 2 R 2 + 2 ⇒ ωmnp = √ x2mn + 2 R h R εμ h2
(7.56)
with m, p = 0, 1, 2 . . . , but n = 1, 2, 3 . . . (Table 7.2 and Figs. 14.36). The resonance frequency of the lowermost mode is given by
262
7 High-frequency discharges III
2.405 c , ω010 = √ εμ R
(7.57)
Table 7.2. Roots of the Bessel functions which are important for the design of cavity resonators. order 0
1
2
Jm x01 x02 x03 x11 x12 x13 x21 x22 x23
= 2.405 = 5.520 = 8.654 = 3.832 = 7.016 = 10.174 = 5.136 = 8.417 = 11.620
Jm x01 x02 x03 x11 x12 x13 x21 x22 x23
= 3.832 = 7.016 = 10.174 = 1.814 = 5.331 = 8.536 = 3.054 = 6.706 = 9.970
and is independent of h! Hence, simple tuning by changing the geometry of the reactor is impossible [298]. The relations for the fields are obtained by Ez = E0 J0
2.405r −iωt ; e R
(7.58.1)
2.405r −iωt √ Bθ = −i εμJ1 . (7.58.2) e R For TE-modes, Eq. (7.55) is valid in principle, but due to the boundary condition for Bz
∂ψ ∂r
= 0,
(7.59)
r=R
we have the condition γmn = xmn /R where xmn equals the nth root of the first derivative of the Bessel function Jm (x) = 0, and we obtain for the resonance frequencies
ωmnp
c =√ εμ
2 p2 π 2 c 1 p2 π 2 R 2 xmn 2 + xmn + 2 ⇒ ωmnp = √ 2 R h R εμ h2
(7.60)
with m = 0, 1, 2 . . . , but n, p = 1, 2, 3 . . . The lowermost TE-mode with m = n = p = 1 exhibits a resonance frequency of
ω111
1.841 c R2 = √ 1 + 2.912 2 . εμ R h
(7.61)
7.5 Electron cyclotron resonance
263
For h > 2.03 R, this mode dominates. Since the frequency depends on the ratio h/R, a simple matching is made possible by variation of the length of the cavity resonator. Since the Q-factor just for this mode is extremely low, in most cases the TE011 -mode is used for matching. After ignition, the waves will be absorbed and the E-field breaks down, the Q-factor reaches unity, and no intensity pattern can be detected any more. The lowermost TM-mode which will be solved by a Bessel function of zero order exhibits a resonance frequency which is independent of the axial distance. Therefore, no tuning with this mode is possible, and the lowermost TE-mode is used (TE111 ). For h > 2.03 R, its resonance frequency is below the lowermost TM-mode, and the TE111 -mode will become the fundamental mode; its eigenfrequency does depend on the h/R ratio and the process of tuning is carried out with ease. The efficiency of coupling is mainly set by the damping of the whistler waves which is expressed by its inverted property, the Q-factor. Besides the conductivity of the plasma, the external parameters which influence the Q-factor are • the operating frequency [the ratio between the plasma frequency (ωp ∝ n2e ) and operating frequency determines the regime of propagation and the skin depth δ, Sect. 14.6], and • the dimensions of the cavity resonator which both determine the • special mode which matches these conditions and will dominate the oscillating behavior of the plasma. The Q-factors peak around unity, and the resonance curves are found to be relatively broad. Referring the skin depth δ to the wavelength of the operating mode which scales with the Q-factor and plotting this ratio vs. the geometric ratio 2R/h for a cylindrical cavity resonator, Q for TE-modes peaks for 2R = h (Fig. 7.36). After having struck the plasma which causes the electric resistance to decline precipitously, the electric field collapses. The advantages over the waveguide applicator are: • No external matching stubs are required; this is perfectly accomplished by the sliding short. • A standing wave is impressed against the plasma source sealed by the quartz window without the difficulties caused by the occurence of reflected powers. • It can easily be operated in the two modes: plasma stream mode and ion beam mode.
264
7 High-frequency discharges III
0.8 TE012
Q [d/l]
0.6
TE011 TE112
0.4
TM012 TE111
TE211 TM011
0.2
TM010
0.0
0
1
2
3
Fig. 7.36. Q as function of the dimension (cross section/height) of a cylindrical cavity resonator for different modes after [299]. Albeit the Q-factors peak at 2r/h = 1, this maximum is not very distinct.
2R/h
7.5.4.3 Conclusion. In ECR discharges, the electron temperatures Te are typically between 2 and 6 eV, but peaking at 8 eV at discharge pressures below 1 mTorr, and the plasma densities are in the range between 1011 cm−3 and 1012 cm−3 , i. e. higher by up to three orders of magnitude, compared with capacitively coupled discharges, and about one order of magnitude higher than inductively coupled discharges [282]. This leads to significantly shorter Debye lengths [λD ≈ 6.91× nTPe with np in cm−3 and Te in K, Eq. (3.11)]. The pressures should be less than 2 mTorr to avoid severe damping of the resonance (collisional damping), but higher than 0.5 mTorr to ensure an effective ionization: a classic example of a ridge walk. Due to the resonant power transfer which happens at pressures which are significantly lower than in other discharge types, the degree of ionization is higher than for most of the other types, but at the expense of higher electron temperatures [300]: A larger fraction of them can overcome the retarding potential of an unbiased substrate [282]. Evidently, this stress would lead to higher substrate temperatures (and eventually to damage) than in comparable capacitively coupled discharges but this constructive feature can be effectively fought by installation of a second magnet and the configuration of a magnetic bottle. Applying a certain substrate bias causes the etchrate to rise and improves the degree of anisotropy. But most important is the fact that ion generation in the plasma source is uncoupled from ion acceleration at the substrate. To enlarge the plasma density, high substrate potentials are no longer mandatory. This uncoupling is carried out by electric means and offers the possibility to separate these two processes also spatially apart (distance typically 5 − 30 cm), the plasma processes at the substrate happen “downstream”, but the discharge pressure remains still the same. This is the plasma beam mode. Installing a grid optics which has been optimized according to the laws of Paschen and Child the ECR source is turned into an electrodeless broad beam source (“ion beam mode”, Chap. 8, Fig. 7.31).
7.5 Electron cyclotron resonance
265
The essential parts of an ECR source are: • A stable microwave source. • The application of a circulator, to transmit the reflected power in a dummy load where it is annihilated. • A rectifying detector diode to measure forward and reflected power. • A three-stub tuner to match the source with the inductive load of the plasma including the λ/4-window. • A mode converter that transforms the rectangular wave into a circular wave. • Finally, the correct dimensioning of the divergent magnetic field: topmost value at the microwave window, lowermost value at the plasma exit (Sects. 14.5 and 14.7.1). ECR plasmas offer the opportunity to study the term and the conditions for quasineutrality quite easily. The neutral gas dissociates into charges carriers with very different mobilities: 5 eV-electrons at 875 Gauss exhibit a Larmor radius of only 86 μm, whereas this radius amounts to 2.15 mm for thermal (500 K) Ar ions (ωc,i is 2.1 × 105 Hz at 875 Gauss). By the order of magnitude, it reaches the dimensions of the reactor. Therefore, the IEDF is not expected to show any dependence on the intensity of the magnetic field. The strong magnetization, however, causes an ambipolar electric field in both directions parallel and normal with respect to the magnetic field which influences and guides the ions as well! The electrons which have gained velocity (are accelerated) in the ECR layer immediately experience a longitudinal force due to the diverging magnetic field. Thus its transverse velocity component is converted into a longitudinal velocity component. This causes a drain of electrons in the excitation zone which is instantaneously counterbalanced by Coulombic forces between electrons and ions. As result, also the ions are longitudinally guided (i. e. accelerated) by the magnetic field, and a neutralized beam of electron-ion pairs leaves the diverging magnetic field and impinges on the substrate [284, 301]. The plasma flickering which can also be observed in discharges excited by helicon waves at 13.56 MHz can be caused by several phenomena [302]: • Oscillations which are fed back between the discharge and the matching network. • Mode hopping. • Instabilities within the discharge caused by the collisions of gas flows with strongly different temperatures (hot gas current leaving the plasma source, cold gas flow which comes from the nozzles of the gas ring just above the RF electrode).
266
7 High-frequency discharges III
7.6 Comparison of high-density discharges For all the methods introduced, plasma generation is spatially separated from its application, and this is called downstream configuration. Two of the methods discussed are distinguished by resonant absorption of whistler waves; and collisionless mechanisms for power transfer are effective. In ECR discharges, the RHP whistler waves are resonant at the cyclotron frequency; another possibility for power transfer is the absorption by Landau damping [263]; this would include also the absorption of LHP whistler waves. To match the condition of resonance, a certain intensity of the magnetic field is mandatory for a given frequency: For 2.45 GHz, the intensity of the magnetic field is 875 Gauss. The microwaves penetrate the plasma via a λ/4 window; for reasons of process stability, its optical quality must remain constant. Employing more than one magnet, the condition of resonance can be met at different positions within the plasma source which opens the possibility to construct a magnetic bottle with two magnetic mirrors which are more or less transparent, causing the electrons to evade the plasma with a relatively strong longitudinal component. Since the electrons are partly mirrored back into the plasma source, they can trigger further ionizations, and these systems are characterized by an even higher plasma density despite the inevitable diffusion losses which are caused by the geometrical construction of this source. However, the lower the pressure the sharper the resonance; and high-efficiency processes are often connected to razor-edge situations and results—not very desirable for a production-line process. In the case of helicon-driven discharges, the transverse modes of an RF driven external antenna couple with a standing internal helicon wave. The πmatching network can be configured with the antenna itself as inductance, and the plasma densities can exceed 100 %, neglecting multiple ionizations. Its big disadvantage, however, is the phenomenon of mode hopping which is caused by the quasi-resonant coupling method. To avoid mode hopping, the design of the reactor has to be shaped very carefully. The magnetic fields exceed the intensity of the earth’s magnetic field (0.2 Gauss) by a factor of 50 − 100 but are less intense than in ECR-driven discharges (875 Gauss for 2.45 GHz). Interfering E×B drift which would cause radial and azimuthal inhomogenities are considerably lower (they can be observed for oblique mutual orientation of the two fields). For vanishing magnetic field, the helicon-driven discharges gradually becomes a simple inductively coupled discharge without any resonant coupling behavior. Although the plasma density is lower by one order of magnitude compared to the resonant methods which declines further at wafer level, it is more than sufficient for most applications. In contrast to parallel-plate reactors which can be considered as “fat” systems with a ratio radius/length much larger than unity, all these high-density plasma systems exhibit geometric configurations with a ratio radius/length approaching unity. This causes radial inhomogenities in the plasma source which
7.6 Comparison of high-density discharges
267
have not yet been treated mathematically. The very high plasma densities (between 1011 and 1013 cm−3 in the plasma source) are leveled down by more than one order of magnitude at wafer level, which is closely connected with a radial homogenization. Due to the resonant excitation, the electron temperatures in ECR discharges exceed values of 5 eV with ease to peak at values of 10 eV for very low pressures. Furthermore, high plasma densities cause very thin, collisionless sheaths. The resonant coupling leads to very high plasma densities and the coupling itself has been subject to many experimental investigations and theoretical derivations, but for production plants, it has been the inductively coupled discharges which have won the competition for a high-quality, most reproducible tool. Even in these discharges, the etchrates remain very high, and most frequently the additional application of moderating gases is recommended.
8 Ion beam systems
8.1 Introduction The reactors considered so far consist of a plasma source which is either spatially connected with the processing chamber (DC reactors and CCP reactors) or partially separated as in the high-density reactors, which are spoken of “downstream”reactors. Spatial separation between plasma source and processing chamber leads to ion beam processing which was initially pushed by NASA at the end of the 1950s when they dreamt of interplanetary missions which chould be made feasible with ion thrusters [303, 304]. For optimized thrust, the fuels should combine high atomic weight with low ionization potential to facilitate high ion densities and high a momentum at the same power input. Simultaneously, corrosion should be kept at low a level. Therefore, most experiments were performed with mercury (instead of cesium). This research remained but a dream, but as a typical spin-off process, these systems were identified as plasma sources for inert gas processing. Systems which were progressively improved are now known under the names ion milling system or microetch system [305]. These systems consist of an ion beam source and a spatially separated processing chamber, i. e. a vacuum reactor with numerous manipulating systems (Fig. 8.1). The plasma source consists of the plasma generation system itself which is confined against the other part of the reactor by the grid optics which serves in a first step to extract ions out of the plasma source and in a consecutive step to accelerate them. Plasmas can be generated by all possible methods. We can distinguish between DC excitation using a Kaufman source or a hollow cathode and HF excitation: RF or ECR. But we can also oppose the Kaufman source against filamentless sources which allow the generation of an ion beam free of contamination—at least in principle (Fig. 8.2). Although the extracted ion beam is denoted broad beam, it consists of a bundle of single beamlets (Fig. 8.3). The optics itself consists of a system of grids (1 − 4) which are aligned in “eclipsed” position and all exhibit the same diameter of the holes, at least to first order. In fact, they are delicately constructed, not only as far as their slightly different diameter is concerned (radially outward on the same grid and between different grids), but also the G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 8,
269
270
8 Ion beam systems inlet for reactive gases (CAIBE) ion beam source
ion current aperture (pivotable)
inlet for reactive gases (RIBE) load-lock valve process chamber substrate table (rotating, water cooled,tiltable)
ion beam
argon
anode
acceleration grid
turbomolecular pump
substrates current probe (pivotable)
process chamber in opened position
roughing pump
Fig. 8.1. Sketch of a reactive ion beam etching system (RIBE system with CAIBE option, equipped with a Kaufman source). By means of a rake containing a chain of Faraday cups, the radial characteristic of the ion beam can be measured (IEDF and IADF).
ratio hole diameter/thickness, distance of the grids and shape (dished or plane grids). Since the grids are always set to different potentials, this optics is denoted acceleration deceleration system. The core of the processing chamber is the substrate holder. This system is a machine by itself, since it can be tilted against the beam, it can be cooled or heated and to cap it all, it can even be rotated. On top of this substrate holder, the sample is placed. In the case of a wafer, it must be fixed mechanically with cramps when a coolant will flow below the wafer (gaseous or liquid). From this sample, material can be removed (ion milling), but it is also possible to fix this system with respect to a target from which material is sputtered away by the ion beam. This sputtered material will eventually deposit on the substrate. Then, this system acts in the mode of ion beam deposition or secondary ion beam deposition [306], although the atoms which will deposit onto the substrate are uncharged. These two chambers are totally separated in an electrical sense, but not as far as the vacuum is concerned. To meet technological challenges, i. e. high current density along with high radial uniformity of the beam, the latter must be guaranteed at the entrance of the grid optics, whereas the first can be influenced by the plasma source and by the grid optics. By means of the grid system, the vacuum conductivity is severely deteriorated, which leads to a pressure rise of
8.2 Plasma sources
271
about one order of magnitude above the vacuum in the processing chamber. By this configuration, very low pressures are possible (typically ≤ 10−4 Torr or 0.1 Pa) in the plasma source at pressures in the processing chamber of 10−5 Torr (10 mPa) or even less. Therefore, of all the plasma processing methods available, cross contamination is kept at lowest levels.
8.2 Plasma sources 8.2.1 Kaufman source In Fig. 8.2, a short historical sketch of ion beam technology is presented. The oldest plasma source is the Kaufman source [307] (see also Fig. 8.1). It consists of grounded or isolated cylindrical walls and a glowing wire which emits electrons thermally (cathode). Adjacent to the reactor wall, the cylindrical anode is installed which is set to the topmost potential within the discharge. The discharge voltage is defined as the difference between cathode potential and anode potential. By this configuration, the ions are accelerated towards the grid optics, whereas the electrons remain confined in the plasma source. The working gas, most frequently argon, is introduced into the plasma source through a port in the grounded wall. Electrons are emitted from the glowing wire and on their course to the anode, they strike the neutrals and ionize by collisional impact. Since the mean free path of the electrons becomes very long at these low pressures, which would increase the required volume for a stable discharge up to several m3 , the electrons must be contained by magnetic fields (electron confinement by magnetrons). By this configuration, not only the pressure which is required to sustain the discharge can be lowered, but also the discharge voltage between cathode and anode can be fixed at very low values. Whereas a low pressure also contributes to a lean beam due to the fact that collisions between the ions are suppressed, the latter effect serves to keep the plasma potential at low levels. Since the second ionization potential of argon is 27.6 eV (two-step process) or 43.4 eV (one-step process: 15.8 eV + 27.6 eV), only very few double-charged ions are generated, which leads not only to poor performance of the ion beam due to inevitable effects in the acceleration system of the grid optics but also to lattice damage. Its main advantages are the ease of use and simplicity, however, it suffers from a limited cathode lifetime (cathode wear by evaporation processes) which is further reduced when reactive gases are applied. Hence, a good performance with ever-increasing demands for radial uniformities becomes more and more difficult to meet.
272
8 Ion beam systems
RF (13.56 MHz)
1995
PBN coil
MW 2.45 GHz
PBN
1985
ECR SLAN ion optics
1975
permanent magnets filament DC
neutralization filament
Fig. 8.2. The progress in the ion beam technology mainly consists in the development of new plasma sources [from the Kaufman source to MW sources (ECR and SLAN) and RF sources], and in the application of refined neutralizing elements (from simple tungsten wires in the oldest Kaufman sources to so-called plasma-bridge neutralizers). In contrast to the Kaufman source, with the newly developed sources, almost c Oxford Instruments 1993). no contamination is detectable (
8.2.2 HF sources The experiments with HF sources can be traced back to the 1960s, mainly to the group of H.W. Loeb at Justus-Liebig University in Gießen, which favored RF sources, mainly coupled inductively. Others experimented with capacitive coupling and ECR sources [308, 309]; comprehensive treatments have been reported by Jongen and Lyneis and Wolf [310, 311]. In all cases, the use of a dielectric window is required to bring the RF energy into the chamber, or RF electrodes have to be applied within the chamber which itself is isolated from ground. If capacitively coupled, they are subject to ion bombardement (and source of contamination), if a dielectric window for inductive coupling is the favored solution, this requires maintenance. But no matter which method has been applied, the plasma density is significantly higher than in DC discharges, although the upper plasma density in ICP discharges is limited to about 1011 cm−3 due to the
8.2 Plasma sources
273
screening which is denoted the skin effect (cf. Sect. 14.6). But most technicians voted for ICP instead of ECR, simply because of practical considerations (too bulky and expensive, Fig. 8.2). But it should be kept in mind that due to the strong magnetic field in ECR discharges, the electrons are effectively confined within the plasma bulk and therefore very seldom hit the walls, by which, in turn, also ion bombardement is avoided. But also in ICP discharges, the plasma potential is kept at low voltages, and since the walls are on floating potential, the ions approach wall and grid optics by ambipolar diffusion. ICP driven ion beam sources dominate the market, therefore, we want to focus on this method of plasma generation.1 8.2.2.1 Design of a grid optics with RF source. Excitation by inductive ˜ induces an electric field coupling means that an oscillating magnetic field B ˜ field. Hence, we have ˜ ind which is orientated perpendicular to the exciting B E two possibilities to wind a coil around the vacuum cylinder which has to be made of quartz or alumina (so-called axial arrangement or radial arrangement). Mounting the coil in a planar arrangement at the top of the plasma chamber is the third possibility. This configuration comes very close to the standard of the GEC cell. Remembering the demands for low density of contaminants and a homogeneous ion beam with high ion flux, we see that only the radial arrangement meets these requirements. • Only when the axes of the plasma source and the coil coincide, is the in˜ ind which guides electrons and ions orientated parallel duced electric field E to the chamber walls, so that they do not intersect them, which avoids ion bombardement. This is only possible with the radial arrangement. • The ion beam density ji is the product of ion density ni and their (mean) velocity < v >. The ion density peaks at the axial center and decays in outward direction due to diffusive loss. Fortunately, the mean ion energy drops from its topmost values at the wall to lower values in the axial center, but only for the radial configuration: These two counteracting effects can effectively cancel out, and the radial uniformity of a high-density beam is ensured. Again, this is only possible for the radial arrangement. To make this quantitative, we find that the RF coil generates an axial magnetic field BRF : n (8.1) jRF,0 sin ωt, l (n: number of turns, l: length of the coil), which is inhomogeneous for l r, i. e. in the axial center, the radial intensity B(r) is lower than at the chamber ˜ induces an azimuthally orientated electric vortex field wall. This fluctuating B BRF = μ0
1
The SLAN source has been introduced in Sect. 5.6.
274
8 Ion beam systems
1 (8.2) rω BRF,0 cos ωt. 2 Because of the skin effect2 and the inhomogenity of BRF , this is no more than a rough estimation.3 This radial rise of electron temperature Te which causes an increase in the velocity of the ions via the Bohm velocity vB is characteristic for RF sources and cannot be found in Kaufman sources. Following Bohm, the extractable current density depends on the electron temperature Te according to [Eqs. (14.43/44)] Eind =
ji = 0.6065 e0 n0
kB Te . mi
(8.3)
Because of the counteracting effects which determine ni and vi , we obtain the flat response for the extractable current density ji = ji (r), and we can simply calculate the total extractable current emerging of N aperture holes with diameter ρ according to Ii = Aji = N πρ2 ji .
(8.4)
The radial dependence of the ion flux Γi (r) which is measured just below the grid optics of a 14" source exhibits an almost flat response within the inner 10", a slight increase which is caused by the rising field which is followed by the steep drop at the edge. The densities in the center and at the maximum (1" from the edge) differ by no more than a factor of 2 [313].
The voltage peaks between the terminals of the coil, and we have found that an inductively coupled discharge is ignited in the CCP mode (cf. Sect. 7.2.3). Capacitive coupling can be minimized by lowering the operating frequency. If it is lowered between the expected plasma frequency, no capacitive coupling can occur. In this case, problems to strike the discharge are likely to occur. And since we are involved with inductive coupling, a π-network must be applied to match the plasma impedance to the DC output of the RF generator (50 Ω). 8.2.2.2 Boundary voltage. Installation of a circuit which would push the ions towards the grid optics and would simultaneously retain the electrons in the ion source would improve the process of beam generation. For this purpose, Ranjan et al. installed an accelerator circuit, consisting of a coil and a tuneable capacitor in series, which taps a fraction of the power from the plasma to the RF accelerator electrode and generates a boundary voltage Vb (t) (Fig. 8.3) [229]. This voltage is the sum of a DC offset VDC and RF voltage with a peak-to-peak voltage Vpp (cf. Sect. 6.4): 2 At 13 MHz, the skin depth can be calculated to 7 mm in a cold (collisionless) plasma at a plasma density of 5 × 1011 /cm3 [cf. Eq. (14.167)]. This value is enlarged to several centimeters at lower operating frequencies. 3 Due to the cylindrical geometry, E(r) rises as a Bessel function of first order [312].
8.3 Grid optics cooling water
to oscilloscope
275 gas inlet 6” flange
beam acceleration electrode inductive coil
RF power supply
quartz tube
Fig. 8.3. Sketch of an ion beam source with accelerator electrode [229].
neutralizer grid
Vb (t) = VDC + Vpp sin(ωt).
(8.5)
The acceleration electrode is connected over a high voltage connector to a high-frequency digitizing oscilloscope. The shape of the boundary voltage can be explained by the capacitive coupling of the electrode to the plasma. Because the areas of both acceleration electrode and walls with ground potential are different, a DC potential can be formed. The discharge is asymmetric and VDC ≈ Vpp . Vpp scales with VDC and both can not be tuned independently. In this case, VDC was always set and Vpp followed. Because of their mass, the ions in the plasma can not follow the sinusoidal potential Vpp (typically at the radiofrequency of 13.56 MHz), but they are repelled by VDC by the acceleration electrode and are accelerated towards the neutralizer grid. The distance between acceleration electrode and grid optics is several centimeters and much longer than the M F P , so the ions can thermalize.
8.3 Grid optics 8.3.1 Configuration and potential adjustment The ions which are generated by collisional impact in the plasma source are now extracted and subsequently accelerated by the grid optics in order to generate a collimated beam which should simultaneously exhibit high intensity (ion flux) and high radial uniformity. To realize these opposing requirements, at least partly, highly sophisticated extraction devices have been fabricated which consist of at least one grid but which have recently been extended to a maximum of four grids. Each of them consists of hundreds of holes which are drilled into a wafer of molybdenum or graphite in a specially calculated design. Eventually, they are carefully aligned in the eclipsed configuration. Out of each hole, a beamlet will be radiated which is divergent in shape, but its angle of divergence can be adjusted by several parameters (angle of divergence α (Fig. 8.4, rB : radius of the broad beam at the target, rG : radius of the broad beam at the grid optics, s: distance between grid optics and target):
276
8 Ion beam systems
a rB
rG
s Fig. 8.4. The broad beam consists of a bundle of single beams, so-called beamlets, which are extracted from the plasma source by means of a grid optics (ion optics is also in common use). Depending on the variation of the potential, we obtain a focused (α < 0◦ ), collimated (α = 0◦ ) or defocused (α > 0◦ ) beam. Under rG , we understand the radius of the broad beam at the grid optics level, rB is the radius at target level which are separated by the distance s which determines the angle of divergence [eq. (8.6)]. The target itself can consist of a substrate to be etched, or a target from which material is sputtered on a substrate, or a rake containing a chain of Faraday cups for purposes of optimization (beam current, ion angle distribution function IADF, ion energy distribution function IEDF).
r B − rG . (8.6) s For a beamlet which is principally divergent in nature (although the broad beam itself can be parallel, convergent or divergent following this definition), we use the similar equation to define the angle of microdivergence as the truncated cone angle which encloses 95 % of the beamlet current [314], Fig. 8.5. arctan α =
ρB − ρ G . (8.7) s However, ρB and ρG are now the radii of the beamlet at grid optics level and target level, respectively, and s the distance between lowest grid and target. For optimization of the grid optics, measurements are made of the voltages which are applied to the grids and the currents which are picked up by them, the ion-beam current itself and its radial dependence by application of a chain of Faraday cups. The grid currents increase for poor focusing of the beamlets and have to be tuned to zero for optimized focusing.4 arctan α =
4 Since the ion beam current scales with the emission current in the Kaufman source and the RF current which is transmitted by the RF generator in the RF case, respectively, only these currents are subject to measurement when the ion beam source is operated in the “normal” mode. Therefore, the parameter of second order (etch rate or sputter rate), will depend on the state of the individual machine.
8.3 Grid optics
277 a
2rg
s
2rb
Fig. 8.5. For a diameter of a beamlet of 2ρg = 2 mm at the ion optics, a divergence of only 5◦ will cause a widening of the beamlet cone 2ρb to more than double its original value on its course of about 300 mm in length. The angle of divergence of a beamlet will follow the equation arctan α = (ρb − ρg )/s.
8.3.2 Screen grid 8.3.2.1 Kaufman source. The task of the inner grid is not only to suppress evading of electrons out of the plasma chamber but also to focus the ions into small beamlets, and is is denoted screen grid or electron suppressor. This grid is usually set at cathode potential, i. e. a little less than the plasma potential Φp , and the discharge voltage is the difference between cathode and anode (inner skin of the reactor or separate electrode). This additional discharge voltage is small compared to the potential of the inner grid; and it is often referred to as plasma potential, and its absolute value is set to the topmost value within the discharge which leads to an effective confinement of the electrons but repels the ions. Even further containment of these primary electrons within the plasma chamber is accomplished by means of an axial magnetic field which additionally serves to prolong their lifetime. It must be at least as intense as to prevent a primary electron from reaching the anode without have experienced a collisional impact. On their enlarged spiral course, they can additionally contribute to a higher degree of ionization by impact. Albeit the ions are influenced to but significantly lower extent, the counteracting electrostatic force which is exerted by the electrons will also keep the ions confined, and the inner skin of the plasma source is almost completely prevented from ion bombardement since the plasma potential which determines the sputtering power is only slightly higher by some volts than the discharge voltage. 8.3.2.2 RF source. The ions pass the screen grid which is set to high a posiB Te , which is considerably higher tive potential with the Bohm velocity vB = km i than the mean molecular velocity; the other walls of the plasma source are isolated from ground. Therefore, the walls of the plasma source are only subject to the bombardement of ions which are accelerated to no more than the Bohm velocity [Eqs. (3.21/3.22)]. Despite the low plasma density (a few percent at maximum), the flux of the ion current incident on the screen grid is dramatically higher than • the flux of the neutral molecules incident on the grid optics and
278
8 Ion beam systems
• the flux of the ions to the other surfaces which confine the plasma source since the fluctuating magnetic field reduces diffusive losses. This behavior justifies neglect of other losses and taking into account only the losses which happen at the screen grid—and which are intended.
8.3.3 Accelerator grid
potential [a. u.]
The potential of the outer grid or accelerator grid is fixed to negative values with respect to the potential of the screen grid. Typically, its absolute value is set to 1/10 up to 1/4 of the plasma potential which generates the required electric field to extract the ions out of the plasma source. Furthermore, the electrons are prevented from flowing back into the plasma source, which can be misinterpreted as an increasing ion current. The ions are generated at Φp , are accelerated to Φa − Φp = Vt and decelerated to ground potential within the ion beam, and their final energy is then e0 (Vp − Vg ) = e0 Vn (Fig. 8.6).
Vn Vt
ground aperture potential
screen grid
grid potential acceleration grid
Fig. 8.6. Variation of potential in the acceleration system of a two-grid optics. The plasma potential is set approximately to the anode potential, the potential of the beam which has passed the neutralization system is set to ground potential. The focusing is mainly determined by c R = Vn /Vt [315] (The American Institute of Physics).
distance [a. u.]
8.4 Qualitative treatment of beam extraction The ion trajectories are mainly determined by the potential distribution in the system of the ion optics which consists of two or three grids in customary designs (cf. Figs. 8.5 and 8.6, also for the used nomenclature); in simple extraction systems, also single-grid optics are in use, and in the most advanced system, a four-grid optics serves to optimize the beam performance.
8.4 Qualitative treatment of beam extraction
279
8.4.1 Extraction without a grid In principle, it is possible to extract ions by means of a divergent magnetic field. This principle is realized in the mirror system of ECR reactors where the magnetic bottle is adjusted to leak at one mirror into the processing chamber (see Sects. 7.5 and 14.5). In thermodynamic equilibrium, a potential difference ΔΦ is generated which causes a flow of charged particles: μ ΔB . (8.8) e0 kB Te Despite its ease of application, the main severe setback is the relatively high discharge pressure which is required to strike the discharge, but then also to sustain it. Since the mean free path scales inversely with pressure, the beam readily diverges at higher pressures, and furthermore, the energy of the ions is difficult to change. Since a grid exhibits a poor conductance, its application within one reactor wich is evacuated with one pumping system makes a mode of operation possible, in which we simultaneously observe two different pressures in two compartements. We get benefit from the low conductance of the grid: high vacuum in the processing chamber, poor vacuum (sometimes more than an order of magnitude less) in the plasma source. ΔΦ = −
8.4.2 Extraction with one grid A plasma sheath will be formed around each hole (aperture) which serves to screen the grid potential against the presheath of the plasma. As we have seen in Sect. 14.2, this presheath is necessary to build a stable sheath. In this presheath, the ions are accelerated to the Bohm velocity, in the sheath, the ions are accelerated further and focused to form a single beamlet. The dimensions of the aperture are solely determined by the mechanical stability and manufacturability. Therefore, they are fabricated of molybdenum or graphite. This sheath consists of a space-charge limited layer of positive ions; its thickness scales inversely with the square root of the plasma density. Using a single grid, we easily see that the hole diameter must be small against the sheath thickness as calculated using Eq. (4.21.2). Otherwise, the electrons would not be influenced by the grid and its electric field would flow unaffected through the (intended) apertures. For example, for an ion density of ni = 1011 cm−3 , an electron energy of 5 eV, and an extraction voltage of 25 V, the diameter of the holes must not exceed 0.2 mm! From this condition, we can also deduce that the thickness of the grid also must shrink; otherwise, we would face large losses in beam intensity due to the deflection of the ions which would impinge the grid within the acceleration “tunnel”, and so they get lost. Since the single-grid extraction lacks an effective protection against the ionic attack, it is subject to intense erosion.
280
8 Ion beam systems
8.4.3 Extraction with two or more grids The screen grid determines the absolute magnitude of the plasma potential Φp , by the potential of the acceleration grid, which is always negatively biased against the screen grid, the electric field is set up which exerts a force on the ion beam flux by which the beamlet is focused. En route this distance, the velocity of the ions is dramatically enhanced, and the current density increases. According to continuity of the current, the penetrated area must shrink. The narrowest diameter of the beam is the meniscus. The third grid is set to ground potential and retards the beam effectively to the beam potential. Moreover, it effectively prevents the sputtering of the acceleration grid by ions which would be directed to its downstream side. Poor screening of the middle grid mainly leads to contamination. The wear of the grid is almost unmeasurable. But above all, it improves the performance: better ion beam divergence at higher ratios of R = Vn /Vt [313].
8.5 Quantitative treatment of beam extraction In the approximation to first order, the ions arrive the sheath at the screen grid at the Bohm velocity; they are accelerated by the strong field between the two grids which causes a focusing of the beam. In the case of correct focusing, it will pass the aperture of the acceleration grid.
8.5.1 Current density 8.5.1.1 Derivations from Child’s law. Since the ion beam is space-charge limited, it can be determined applying Child’s law: 3 / 2e0 Vt 2 4 , ji = ε 0 9 mi le2
(8.9)
with
le =
lg2 +
d2s . 4
(8.10)
Although derived for the one-dimensional case, this equation can be applied to calculate the extractable current density of a broad-beam ion source. Vt denotes the potential difference between the two grids, and for l, the diagonal le is inserted l instead of lg , the distance between the two grids. Hence, we have to consider the thicknesses of the two grids, their distance and the acceleration
8.5 Quantitative treatment of beam extraction
281
voltage between them, and eventually the radius of screen holes (cf. Fig. 8.7) and the ratio mass to charge.5 screen grid
accelerator grid
discharge chamber plasma
Fig. 8.7. Beam divergence in a two-grid system as function of the normalized perveance. The ions are extracted from the volume with diameter le [cf. Eq. (8.9)] [315] c The American Institute of ( Physics).
le ds
da a/2
ts
lg
ta
3 • I varies with V /2 . This is why the ratio of these properties, the so-called perveance, is regarded rather than currents for a given voltage [316];
4 2e0 Ii ; P = 3 = ε0 9 mi V /2
(8.11)
it is constant for a given optics and gas composition; it rises for a given extraction voltage when the ion current is increased or when the extraction voltage is reduced at a given current density. • From Eq. (8.11), it becomes also evident that the upper limit of the perveance depends only on the ionic mass mi . • Furthermore, we infer from Eqs. (8.9) and (8.10) that the current density scales inversely with l2 which represents the extraction area. This yields, for the ion current itself that can be drawn through an aperture with diameter d:
2e0 3/2 4 V : Ii = ji le2 = ε0 9 mi
(8.12)
5 This derivation is based on the assumption that the ratio of the hole diameter of the accelerator grid, da , and that of the screen grid, ds equals unity. For different hole diameters, in particular in the case da < ds the extractable current is smaller than the maximum current calculated with Eq. (8.9). Most grid designs are constructed this way as it has turned out that the best collimation is obtained at current densities which are just a little short of the possible maximum.
282
8 Ion beam systems it is independent on the aperture da itself because le and da both scale similarly, and their variations cancel out.
Assuming that the source area of ion emission for one beamlet equals the area of one hole screen grid, 1/4 πd2s , we define the normalized perveance per hole Π: ji Π= 3 V /2
le ds
2
4ε0 = 9
2e0 . mi
(8.13)
The upper limit for current drawing is given by the breakdown field which is roughly 20 kV/cm. But above all, with increasing acceleration voltage Vt , the divergence of the ion beam eventually exceeds any tolerable mark. 8.5.1.2 Grid transparency. The extractable current can also be enhanced by raising the grid transparency η. For a hexagonal configuration of spherical apertures, the well-known formula for hexagonal close-packed structures yields η=
1 πrs2 2 √ 3 2 d 4
(8.14)
with d the distance between the centers of the adjacent holes (d = rs +x, x being the width of the bridge between adjacent holes with radius rs ). The transparency cannot be raised beyond η = 0.907, since this is the limiting value for vanishing bridges between the holes. By raising the hole diameter from 1.5 to 3.5 mm, the transparency increases by a factor of 5.5, and due to the reduced surface of the grids, issues of contamination also become less serious.6 The focusing of the ion beamlets, however, deteriorates with rising perveance. Furthermore, issues of fabrication and stabilization, respectively, limit the lower length of the aperture tunnel: Grids which are thinner than 250 μm are hardly stable across a larger diameter of roughly 8". Since the sheaths extend into the plasma, the ions are focused when they pass the apertures of the acceleration grid. Therefore, the diameters of the accelerator holes can be significantly smaller than those of the screen holes without sacrificying much of the beam current. Values down to 2/3 for da /ds have been reported. In particular, this is advantageous for larger ion beam sources, since this feature reduces the conductance of the grid optics by which the consumption of the process gases will go down. For smaller sources or for maximum current, however, the diameters of the apertures of all grids should be equal. For an optics with a two-grid design with a screen hole diameter ds of 2 mm and a grid spacing lg of 1 mm, at a total voltage of 500 V a single aperture should 6 It has to be mentioned that the extractable current density drops below the value calculated by eq. (E.9) when the apertures of the screen grid are smaller than 2 mm in diameter. The reason for this behavior is unclear [303].
8.5 Quantitative treatment of beam extraction
283
extract a maximum of 0.1 mA of singly ionized Ar ions. A typical hexagonal aperture configuration exhibits then a density of 18.5 holes/cm2 with a center-to-center hole spacing of 2.5 mm. The average current density is then 1.85 mA/cm2 . This dependence on Vt should hold up to the breakdown limit of roughly 20 kV/cm across the gap lg [317].
For hole diameters of less than 2 mm, the plasma is almost shielded from the grid optics. If the ratio hole diameter/grid spacing exceeds unity, aberrations begin deteriorating the beam collimation.
8.5.2 Focusing and divergence From first principles, it is evident that a grid with holes that first accelerates the ions and then decelerates them can never generate a collimated beam, a beam without any divergence, since acceleration means focusing and deceleration is synonymous with defocusing. In fact, the broad beam consists of hundreds of beamlets [cf. Eq. (8.7) and Figs. 8.4 and 8.7]. We find the lowermost angle of defocusing to be no less than 5◦ . In contrast to the method to calculate and optimize the current density via Child’s law, the methods to determine the parameters which influence the collimation of the beam are still empirical. The ion beam divergence depends most strongly on the ratio R = Vn /Vt , the distance between all the grids, and the normalized perveance per hole. The focusing deteriorates with increasing perveance (rising current) which is caused by the growing constriction of the ion beam current density. Simultaneously, the electron current also increases which shallows the constriction of the ion current at the outer boundary of the accelerator grid (Fig. 8.7). Further increasing perveance enlarges the diameter of the beamlet by electrostatic repulsion: The divergence and the grid current increase once more which is mainly caused by the thinning of the sheaths. For excessive extraction, the outer part of the ion-beam current impinges the edges of the aperture tunnel. These ions contribute to the grid current (which is, in fact, loss current) and cause substantial sputtering. These effects determine the upper limit of the extractable ion current. The lower limit is determined by the backstreaming of the electrons which are necessary for the neutralization of the ion beam. To suppress this flow, a sufficiently negative potential is required (see Sect. 8.6). We insert Vt = Vs + Va into Eq. (8.9) and see that the absolute value of Vt can be influenced by both grid potentials. A highly negative value for Va is responsible for a high current density, whereas the ion energy is determined by the value for Vs . On the other hand, the electric field of this grid also prevents neutralization of the space charge below this grid, which causes a widening of the beam (deterioration of the collimation). Therefore, this effect can only by circumvented by lowering the absolute value of Va , which mainly determines
284
8 Ion beam systems
Vt . But this enhances the ratio R = Vn /Vt and, in turn, lowers the extractable ion current. This beam widening, the performance of the ion current can be evaluated by plotting the angle of divergence vs. the perveance (Fig. 8.8) [315].
relative beam expansion
1.00
0.75 Vn /Vt = 0.5
0.50
0.9
0.25
0.00 0.00
Fig. 8.8. Beam divergence in a two-grid system as function of the normalized perveance c The American Insti[315] ( tute of Physics).
0.7
0.25
0.50 0.75 normalized perveance
1.00
Although this sketch is the shape of a broad beam, we can use this pattern to describe the divergence of a beamlet as a function of the I(V ) characteristic. With rising normalized perveance, the positive space charge of the beamlet exerts a stronger electrostatic force on the diffuse electrons. With deeper penetration of the electron cloud, however, the deflection of the beamlet becomes shallower which causes, in turn, a smaller angle of divergence. Further increase of the normalized perveance, however, gradually deteriorates the collimation of the beamlet: First of all, a rising ion current will lead to enhanced charge transfer between ions and neutrals (cf. Sect. 8.7.3), and although the slow ions are readily captured by the accelerator grid, the generated rapidly moving neutrals are not affected by the potential at all. Furthermore, ions get increasingly lost due to impingement at the orifice of the accelerator grid (the aperture of the accelerator grid is smaller by typically 20 % compared with that of the screen grid). A good collimation is considerably influenced by a low ratio for R. This ratio can be varied over a limited range only and has to be kept above at least 0.5 to prevent backstreaming of the electrons. Example 8.1 To enhance the focusing, the ratio lg /ds can be increased. Consider doubling this value from 0.5 to 1.0 at a value of R of 0.7. This change in the grid optics reduces the angle of divergence α from 26◦ to about 20◦ (the half-angle α1/2 is reduced from 13◦ to 10◦ , respectively) [317]. Since R can be enhanced to 0.9 without fearing the danger of electron backstreaming, the divergence can be further improved to only 12◦ .
The divergence is mainly dependent on the normalized perveance, the distance between the two grids and the ratio R. The first parameter most strongly depends on the ratio screen hole diameter over grid thickness of the accelerator grid, but relatively weakly [318].
8.5 Quantitative treatment of beam extraction
285
8.5.3 Conclusion The requirements for collimation and current density are a classical example for mutually opposing targets: • High collimation requires high R. • High beam current density requires low R. That is why in Kaufman sources, this ratio can be varied between 0.5 and 0.9, or to put it another way: Within certain limits, ion energy and ion flux are mutually independent. At relatively high ion energies of 0.5 − 3 keV which are necessary for etching and secondary ion beam deposition, the ion beam can be sufficiently collimated (several ten degrees at a FWHM of 10 eV). For surfaces which should be kept free of lattice damage, the ion energy must not exceed 100 eV. For an R of 0.5, this would mean an acceleration voltage Vt of approximately 200 V, which causes a steep decrease of the ion flux (0.5 mA cm−2 at 1 000 eV represents about 0.015 mA cm−2 or a bombardment of less than a monolayer in 10 sec at 100 V [319]). To solve this dilemma, we have several possibilities: • The single electrode [319]: When using only one electrode, we can get rid off the space-charge limitation. This allows a significantly higher ion beam current density because the course of ion acceleration is now determined by the sheath thickness of the single apertures. By application of micromesh grids with mesh diameters of 100 − 300 μm which are a little smaller than the sheath thickness or just equal it, the ion current density can be considerably enlarged [> 10 mA/cm2 at very low energy (10 − 200 eV)] [314]. But operating with ions of very low energy is mandatory to avoid severe grid erosion; otherwise, the lifetime is restricted to hours or days. • The configuration with three grids by which the divergence is reduced at low values for R; at high values for R, the improvement is vanishingly small (Figs. 8.9 and 8.10).
8.5.4 Three-grid ion beam source The higher expenditure for three-grid design is justified only by a performance which is significantly improved compared with two-grid optics (Fig. 8.9). In particular, the collimation is better, especially at low values for R (cf. Fig. 8.10). In a two-grid optics, the ions are focused to the smallest beam diameter when they have just passed the holes of the accelerator grid. At this boundary, they leave the zone of well-defined potential and emerge into a region with poorly defined, but rising potential (cf. Fig. 8.6) where they are decelerated
286
8 Ion beam systems screen grid
accelerator grid
decelerator grid
discharge chamber plasma
screen grid
accelerator grid
decelerator grid
discharge chamber plasma
neutralization surface
neutralization surface
le ds
da
dd a/2
ts
lg
ta
ld
td
Fig. 8.9. The beam divergence can be effectively reduced by adding a third grid: c The American Institute of Physics). three-grid optics [315] (
and simultaneously defocused according to basic laws (Poisson equation). The zone of retardation and defocusing can be clearly defined by application of a third grid which is set to ground potential and positioned downstream of the accelerator grid, and which is called the decelerator grid. Fixing the surface of neutralization at this grid results in flatter equipotential surfaces between this grid and the accelerator grid. And flatter equipotential surfaces cause less off-axis deflection of the emerging ions [313].
beam divergence [°]
30
electron backstreaming limit two-grid optics
20
three-grid optics
10
0 0.0
0.2
0.4
0.6 R
0.8
1.0
Fig. 8.10. Reducing the beam divergence by installation of a third grid downstream of the accelerator grid and set to ground potential, here shown as function of R = Vn /Vt which is called the decelerator grid; data are from [313] and [320]. Triangles down: ld /ds = 1.0, triangles up: ld /ds = 0.123.
A reduction of the hole diameter in the decelerating grid (referring to the hole diameter in the screen grid) serves to further improve the collimation of the ion beam. A value of 0.83 for dd /ds was found as the result of an optimization procedure [316]. With similar parameters (hole diameter, grid distances),
8.5 Quantitative treatment of beam extraction
287
potential, energy [a.u.]
potential, energy [a.u.]
a three-grid optics can deliver an ion beam of double the intensity of a two-grid optics [321]. The principal shape of the ion energy distribution function (IEDF) is sketched in Figs. 8.11.
potential ion energy
0
potential ion energy
0
potential, energy [a.u.]
x [a. u.]
x [a. u.]
potential ion energy
0
x [a. u.]
Fig. 8.11. Ion optics: variation of the potential for a grid optics with one, two and three grids (solid) and its influence on the ion energy (dotted): (1) One grid: large divergence but high current density, operation possible only at low energies due to severe grid erosion. (2) Two grids: low divergence and large ion-beam current density. (3) Three grids: lowest divergence at medium ion-beam current density, but complic Oxford Instruments 1993). cated set-up (
Especially at low ion energies which should be favored to avoid lattice damage, this ratio can be extended by a factor of 100 [319, 322]. Studies have shown that the three-grid optics is superior to the two-grid optics, especially what the divergence of the ion beam is concerned. For best results, the three grids should be spaced as closely as possible. However, d, the grid distance, is mainly limited by • the intensity of electric fields, which is mainly determined by the acceleration voltage which can cause arcing, but in almost all cases by the
288
8 Ion beam systems
• (failing) mechanical stability of the grids due to thermal stress and their mutual displacement by thermal expansion. That is why most grids are made of molybdenum or graphite (another reason is their low sputtering yield which reduces the risk of contamination).
8.5.5 Four-grid ion beam source Focusing is achieved by subjecting the ion beam to high potential differences which limits the extractable current by excessive ion impingement to the grid optics. To combat this upper limit, an additional grid could serve to split this voltage into two separated regions. The fourth grid then electrostatically focuses the ions after they are extracted from the plasma and are accelerated through a set of focusing and defocusing grids (Fig. 8.12) [323]. As has been shown by screen grid
accelerator grid
focus grid
decelerator grid
neutralization surface
discharge chamber plasma
Fig. 8.12. By installing a fourth grid, the beam divergence can be further reduced [323, 324]. 250 V
-470 V
-1650 V
0V
Hayes et al., the collimation of the beam can be effectively improved by adding a fourth grid (Figs. 8.13). Over a wide range of ion energies, the divergence can be suppressed to values smaller than 4◦ .
8.6 Neutralization 8.6.1 Principle of operation A beam of ions, all positively charged, would readily disrupt due to mutual repulsion. What is really achieved within the grid optics is a tremendous transfer of energy from the electrons (which carry the predominant part of the energy in all non-equilibrium plasmas) to the ions. Not only is the charge of the entire ion beam balanced by a stationary electron cloud but in each small volume in the ion beam, the number of negative charges equals the number of positive charges. At operating energies of some hundreds of eV, however, the cross sections of the
8.6 Neutralization
289
10 8 6 4
beam divergence angle [degs]
beam divergence angle [degs]
250 V Vt: 470 V
3-grid, 98 % 3-grid, 95 % 4-grid, 98 % 4-grid, 95 %
2 0 0
1
2
3
4
5
8 750 V Vt: 1050 V
6 3-grid, 98 % 3-grid, 95 % 4-grid, 98 % 4-grid, 95 %
4 2 0 0.5
P [10-9 A/V3/2]
1.0
1.5
2.0
2.5
P [10 -9 A/V3/2]
Fig. 8.13. The angle of divergence can be further reduced for the broad beam by application of a fourth grid (LHS: 250 V beam energy, RHS: 750 V) [324].
ions for electron attachment have already dropped to very low values; therefore, the ions still remain “alive”. This type of neutralization is called charge neutralization. One main source for charge neutralizing electrons is a neutralizer, which serves to compensate the charge of the ion beam completely. In the case of its malfunctioning, mainly because the coupling between ion beam and neutralizer is not working properly (either distance too far or adjustment problems between ion beam and neutralizer), the required negative charge is drawn from every grounded surface, which can be observed as arcing. If the charge neutralizing electrons travel with the same average velocity towards the target as the ions, then the beam itself is neutralized: current neutralization. The origin of these electrons is the target surface (cf. Chap. 2), where the electrons are generated by • γ-process: generation of secondary electrons due to collisional impact by ions, • δ-process: generation of secondary electrons due to collisional impact by electrons, • γhν -process: generation of photoelectrons, which all together play a significant role at voltages of some hundreds of eV. When the ion beam impinges on the grounded surface of a metal, ions will recombine with electrons at the surface, and the electric current that will flow from the DC source to the target can easily be measured and is used to adapt the conditions at the neutralization element dynamically, because it equals the instantaneous ion current. Problems can arise at the surface of insulators or floating surfaces of a metallic conductor since the lack of neutralizing electrons
290
8 Ion beam systems
will reduce the velocity of the ion beam incident on the surface by which the efficiency of the electron-generating processes severely deteriorates. A deficient beam neutralization manifests again by arcing and dropping the etchrate (sputtering rate).
8.6.2 Neutralization elements To avoid this defocusing, we can use a simple filament which has to be located a short distance downstream with respect to the ion optics to guarantee effective coupling to the extracted ion beam. The potential of the hot filament is set to values between 20 and 100 V below the potential of the beam plasma,indexbeam plasma by which the filament current density becomes space-charge limited: The emitted current cannot be increased by either raising the potential (referred to the potential of the beam plasma) or by raising the temperature of the filament but only by increasing the length of the rod which is immersed in the ion beam plasma. This leads into an insoluble dilemma: To increase the lifetime of the filament to tolerable values, this must exhibit a certain thickness which requires a relatively high operating voltage. On the other hand, the potential with respect to the potential of the beam plasma should be kept as similar as possible to avoid sputtering. No matter how this optimization problem is solved, it has become evident that contamination by the filament is the main concern of this technique. This problem cannot be circumvented by other geometries. Because of the space-charge limitation of the electron current, the efficiency of a toroidal element which encircles only the outer regions of the ion beam is considerably lower [325]. Applying this method, it is possible to sputter from isolating targets that will not become charged during operation. Widening of the beam to angles of up to 30◦ , however, shows that the neutralization often does not work properly. In most of these cases, it is recommanded to set the emission current of the neutralizer slightly above the ion-beam current. To avoid these problems of contamination and short lifetime, filamentless sources have become popular. They make use of small discharges which are driven by RF, mostly in the ICP mode; but also DC hollow cathode discharges are applied. In both cases, the main difference to open sources consist in the presence of an aperture in the keeper electrode by which effective penetration of the ion beam is ensured at low voltage difference and large electron current density, thereby providing a conductive plasma bridge between ion beam and neutralizer whose current density is controlled by means of the bias voltage (Fig. 8.14) [326, 327]. To avoid electron backstreaming into the ion optics (which could be confused with the same amount of ions leaving the ion optics), the ion optics have to be protected by a highly retarding potential. Hence the effective neutralization can be achieved by
8.7 Process optimization
291
keeper electrode
bias voltage
keeper voltage
Fig. 8.14. One of the niches of hollow-cathode discharges are neutralizers in ion beam systems which are driven separately from the main source and electrically connected to the ion beam by means of a plasma bridge.
• adjusting the filament current until the emission current evens up the ion beam current measured at the surface of the grounded target, i. e. until the net current to the grounded target will vanish, or by • adjusting the filament current until the electron current which is incident on the floating target can even up the potential of the floating target to ground. At zero voltage, • adjusting the power of the plasma source in order to bring the coupling voltage between ion beam and plasma source down to zero.
8.7 Process optimization A process should be optimized taking at least these parameters into account 1. acceleration voltage and substrate damage, 2. plasma potential and substrate damage, 3. ion-beam current density and the resulting loss current at the accelerator grid, and 4. electron backstreaming. 8.7.1 Maximum power and substrate damage As we have derived in Sect. 8.5, the maximum ion-beam current density can be calculated according to the space-charge limited Child equation [eq. (8.9)]. Since substrate damage is mainly caused by high energies of the projectiles incident on the surface, and this energy is determined by Φa , the potential of the accelerator grid, Φa cannot rise beyond −750 V, and the current is in fact
292
8 Ion beam systems
limited by the source volume lg .7 It turns out that an upper value of 0.5 W cm−2 for a substrate with good thermal contact at its backside (He backside cooling preferred) must not be exceeded; at 500 V acceleration voltage8 this means an upper current density limit of 1 mA cm−2 .
8.7.2 Discharge voltage and substrate damage The density of double-ionized species steeply rises with increasing plasma potential Vp which is closely connected to the discharge voltage. For example, Kaufman estimated the fraction of double-ionized argon ions at 20 % at discharge voltages Vd between 60 and 70 V, whereas it is negligibly small at 35V ≤ Vd ≤ 40 V [328]. But double-ionized species generate severe substrate damage due to the larger penetration depth. Since all these processes are connected with damage of chemical bonds, or jumping to crystal sites etc., they exhibit a certain threshold (activation energy). After having passed this threshold, the rate constant for this process rises exponentially, and since the dose for the generation of lattice damage is relatively low (1 − 2 mA sec/cm2 ), the actual fraction of double-ionized species is almost insignificant.
8.7.3 Power and grid current Evidently, excessive beam currents cause rapid wear of the grid optics which causes, in turn, contamination problems. But if the beam currents drop below a certain threshold, process times become intolerably long. The loss current which can be measured at the acceleration grid consists of two parts: the direct ion current generated in the plasma source and the charge transfer current (Sect. 2.5.2). On their course to the ion optics, rapidly moving ions will hit slowly moving neutrals and will mutually change their velocity (or their charge). These rapidly moving neutrals will cause the same collisional impact than an ion would do. But a slowly moving ion happens to be captured more readily by the grid than a rapidly moving ion. Therefore, the grid current predominantly consists of these slow ions; and they will impinge on the upstream side of the grid optics. However, at higher acceleration voltages, two other effects will come into play: 1. Rapidly moving ions from the downstream side of the grid optics will take over the importance of the just mentioned effect on the upstream side of the grid optics. 7 Substrate damage can be caused by physical and chemical effects, from the degradation of photoresists to the complete amorphization (“atomization”) of the surface. 8 At this acceleration voltage, the Ar+ ions exhibit velocities of 50 km/sec or 180 000 km/h (the second cosmic velocity to leave the gravitational field of the earth is about 40 000 km/h).
8.7 Process optimization screen grid
293 Fig. 8.15. The characteristics of the ion trajectories will change for rising ion-beam current. More and more ions will impinge the grid during their course through the grid optics which increases the loss current. This current is measured at the accelerator grid. Schematically shown here are trajectories for normal current (dotted), maximum current (broken), and excessive current (solid, after [328]). This also causes contamination problems.
accelerator grid
discharge chamber plasma
accelerator current [a.u.]
2. The number of collisions caused by ions which are on course to or within the aperture tunnel will increase steeply (Fig. 8.15). Beyond a certain threshold, this current dominates all other contributions and can be recognized by plotting grid current versus beam current. Deviations from linearity which are due to the charge transfer current indicate the optimized operating point (Fig. 8.16).
impingement current maximum beam current
charge exchange current
Fig. 8.16. At larger beam currents, the onset of direct impingement of the ions on course through the grid optics is indicated by a steep rise in the loss current measured at the accelerator grid. It will become the main mechanism for ion loss (after [328]).
beam current [a.u.]
8.7.4 Electron backstreaming In the case of keeping the accelerator voltage at too low an (absolute) value, electrons will flow back into the plasma source. This movement cannot be elec-
294
8 Ion beam systems
trically distinguished by a forward motion of the ions in the outward direction. This effect can effectively be suppressed by tuning the acceleration voltage to sufficiently negative values (larger numbers). Plotting the I(V ) curve, the onset of electron backstreaming is indicated by a steep increase at very low values of Va . Backstreaming of the electrons into the plasma source will be effectively prevented for Va beyond this point (Fig. 8.17).
ion-beam current [a.u.]
electron backstreaming current
ion-beam current
Fig. 8.17. For low accelerator voltages, electrons are likely to stream back into the plasma source, thereby simulating an increasing ion-beam current (after [328]).
accelerator voltage [a.u.]
8.7.5 Current density of the ion beam For sputtering, damage is of no concern, but the ion flux should be as high as possible to shorten the process time. For etching, issues of beam divergence and the voltage of the screen grid, Vn , come to the fore, since a high-quality collimation allows larger distances between the ion optics and the substrate which reduces contamination problems. The beamlets superimpose eventually to form a homogeneous broad beam with a relatively flat beam profile. We can influence the flux of this broad beam at a certain distance by focusing or defocusing. This can be most readily achieved by a systematic displacement of the holes in one grid with respect to the holes of the other grid. This can be further improved with dished grids (Figs. 8.18). 8.7.6 Uniformity Even provided the divergence of the beamlet is no larger than 5◦ , this means a widening to more than double its initial diameter over a distance s of 300 mm (Fig. 8.5). For a one-dimensional array of holes 2 mm in diameter at a distance of 7 mm, the pattern of the grid optics remains clearly visible for a distance between
8.7 Process optimization
295
Fig. 8.18. By measuring the ion-beam current density across the target, we obtain evidence of its radial uniformity. If the broad beam appears focused at target level, it is spoken of as a convergent beam (top), if it is defocused, the beam is denoted divergent (bottom). The radial uniformity can be controlled not only by a systematic mutual displacement of the grids but also by shaping into the third dimension (dishing, RHS). A high ion flux is advantageous for sputter processes, whereas for etching processes, c Oxford Instruments 1993). the broad beam should be radially uniform (
grid optics and substrate of 300 mm and an angle of divergence of 5◦ . At a ratio of 1 : 1.2 (diameter/distance of the centers of adjacent holes), however, the pattern is already blurred (Figs. 8.19). In reality, these are two-dimensional grids with a ratio between aperture diameter and distance of the centers of considerably larger than unity in a hexagonal configuration. Inspecting the radial uniformity of the broad beam measured with a rake containing a chain of Faraday cups, this is sufficiently homogeneous (Fig. 8.20). However, this measurement does not allow a statement concerning the collimation of the beamlets. On the other hand, we can qualify the radial uniformity of the broad beam with respect to the current density from this measurement. The homogenity does slightly deteriorate with sinking current density (at 70 mA/cm2 , the radial uniformity deviates by ±1.8 % and is improved to ±1.04 % at 80 mA/cm2 ). This is advantageous for ion beam coating processes (cf. Sect. 10.9), but has some disadvantages as far as etching is concerned; the IADF is always uncertain (cf. Sect. 11.6). From this, it is evident that homogenity of the beam consisting of hundreds of overlapping beamlets, measured in mA/cm2 , is not at all an issue. Therefore, no beam pattern is visible at the target. The distribution of the atoms sputtered from the target mainly follows a cosine distribution, since at these low pressures of 0.1 mTorr or less, there is no scattering of the sputtered atoms flying to the
296
8 Ion beam systems 150
current density [a. u.]
current density [a. u.]
150
100
50
0
0
25
50
-250 a. u.
100
50
0
75
d [mm]
0
25
50
75
d [mm]
Fig. 8.19. LHS: For a diameter of a single beamlet at the accelerator orifice of 2 mm, the pattern of the grid optics (linear chain of holes 7 mm in diameter) remains clearly visible for a distance between grid optics and substrate of 300 mm and an angle of divergence of 5◦ Lower: two adjacent beamlets, upper: resulting pattern of the current density. RHS: With the same diameter of the aperture hole, but at the very close distance of only 2.4 mm between centers, the beamlets smear out to a homogeneous linear beam profile. Lower: six adjacent beamlets, upper: resulting profile of the linear beam, which is nearly homogeneous (±5% between mean value and extreme values. In reality, the grids show a two-dimensional, hexagonal configuration with a ratio between aperture diameter and distance of the centers of considerably larger than unity.
relative beam intensity
1.00 0.95
Fig. 8.20. The radial uniformity of the curent density of the ion beam, measured with a rake containing a chain of Faraday cups at target level, is prerequisite for a homogec Veeco Instr. neous coating 2002.
0.90 0.85 0.80 0.75 0.70 -15
70 mA/cm 2
-10
-5
75 mA/cm 2
0
80 mA/cm 2
5
10
15
position [cm]
substrate by ambient molecules (as is the case for RF diode sputtering). This is shown as a conclusion in Chap. 10. What is really interesting, however, is the deviation from normal incidence. Widening the beam by only 5◦ has dramatic consequences for the etching angle (Chap. 11). This etching angle is a function of the angle of incidence of the ions
8.7 Process optimization
297
and the distribution function of their velocity (denoted as ion angle distribution function IEDF and ion angle distribution function IADF) [329].
9 Plasma diagnostics
This chapter is divided into two parts: In the first part, several techniques are introduced and discussed from the very beginning to a level of being able to read the original literature. The second part is devoted to some results which are presented for high-frequency plasmas (CCPs, ICPs and whistler wave generated plasmas), especially discharges through electronegative gases.
The various plasmas can be analyzed in several ways to obtain the parameters which determine their qualities: • Plasma density; more precise carrier density (ni ). • Electron temperature. • Temperature of molecules and ions. • Frequency of elastic collision between electrons and neutrals. • Chemical and electrical composition of the plasma. • Chemical and energy resolved composition of the plasma sheaths. • Absolute determination of the densities of reactive species. • Determination of electrical data. Since the methods are different, the results are different as well, even when we consider the same property. Prominent examples are the electron temperature, measured with optical emission spectroscopy (OES) or with a Langmuir probe, or the plasma density (measured with electron spectroscopy (SEERS), microwave interferometry, or with a Langmuir probe, cf. Table 9.1). In Chap. 6, we have already discussed plasma analysis, applying energy resolved mass spectrometry to determine the IEDF. Since no method can give answers to all the questions, some methods are introduced and critically compared. It must be admitted that a definitive method for determining the plasma density is still lacking, which explains the numerous different approaches. G. Franz, Low Pressure Plasmas and Microstructuring Technology, c Springer-Verlag Berlin Heidelberg 2009 DOI 10.1007/978-3-540-85849-2 9,
299
300
9 Plasma diagnostics
Table 9.1. Advanced methods to determine important plasma parameters. AA: Advanced Actinometry; SEERS: Self-Excited Electron Resonance Spectroscopy; Z: analysis of the impedance Z; μ: microwave interferometry. Physical property VP nP Te Tbulk PRF,eff νm cos ϕ Vrms Irms
Method of determination Langmuir AA SEERS Z √ √ √ √ √ √ √ √ ? √ √ √ √
μ √
9.1 Langmuir probe 9.1.1 Introduction A Langmuir probe consists of a spherical or cylindrical electrode which is immersed in the plasma. By applying a potential difference which is referred to a system immersed in the plasma as well, the so-called reference electrode, we can record a current voltage characteristic (Fig. 9.1). ¿From its shape (Fig. 9.2), important plasma parameters can be deduced. The theory was established by Langmuir and Mott-Smith in a series of papers beginning in the year 1926 for DC discharges through argon [330]. If not explicitely stated, the following remarks refer to such an electropositive gas without any negative ions. We first start with an inductive approach of the phenomenology of the I(V ) characteristic, will continue with a detailed explanation of grounding problems and the definition of the reference potential, and the later sections are devoted to certain details of the catching of carriers by the probe (absorption of electrons and ions).
9.1.2 Conditions for performance Planar probe theory applies for the following conditions: • The dimensions of the plasma should be large compared to the Debye length.
9.1 Langmuir probe
301
aluminum wings KF 40 flange vacuum feedthrough tungsten rod aluminum tube
clamping ring glass tube
Fig. 9.1. Setup of a Langmuir probe. Vacuum part with a tube made of aluminum to protect the glas capillary which contains the probe rod made of refractory metal with large-area stainless steel wings which serve as high frequency compensating electrode and a vacuum flange includc [331]. ing electric feedthroughs This system is attached to a linear translator which allows spatial measurements along the axis of the reactor.
• The temperature of the electrons Te should be much higher than the temperature of the ions Ti . • The discharge pressure should be so low that the mean free paths of both carriers should be large compared to the sheath thickness of the probe: collision free sheath. • The sheath thickness rs should be small compared to the radius rp of the tip of the Langmuir probe.1 In these cases, the area of the sheath edge is comparable (only slightly larger) than the area of the surface of the rod). Otherwise, the effective area of the probe is difficult to define. The length of the rod lp which is not protected by the glass sleeve and protrudes into the plasma should be large compared to its diameter to meet the requirements for a cylindrical probe. • The material of the electrode should be chemically inert, exhibit a low sputtering yield and a high work function: Its conduct should be neutral. This can be best fulfilled using refractory metals as tungsten, rhenium or rhodium. • The Debye length of the plasma should be much smaller than the radius of the probe to keep the disturbance of the charges as small as possible [250]. The area of the reference electrode should be large. Especially in reactors with electrically insulated walls, or when their electric state will change during a process [e. g. processes with coating plasmas (CF4 , CH4 1
See Chap. 4 for the relationship sheath thickness and Debye length.
302
9 Plasma diagnostics etc.)], its application is recommended [332]. Moreover, its additional advantage is the straightforward determination of the floating potential Vf (cf. Sects. 9.1.5/6).
• The inevitable disturbance of the charge distribution should not alter the properties of the plasma significantly. • In the (weak) field of the probe, there should be no temporary variation of the spatial distribution of the carriers. • The probe should be an absolute drain for the carriers. • The geometry of the probe should be spherical or cylindrical; in the following, we confine to the cylindrical type. ¿From these conditions, it is evident that a Langmuir probe must not be used in zones with low electron density and hot electron temperature, especially not in the sheaths. For this end, Gottscho, Zarowin and Czarnetzki have proposed (and brought to perfection) the application of the Stark effect, i. e. the splitting of originally degenerate lines in the strong electric field of a sheath [333] − [335]. With this method, a spatial resolution of about 0.1 mm3 in volume has been demonstrated. However, these measurements require a homogeneous field to avoid Stark broadening, which does not show any temperature dependence — in contrast to e. g. the Doppler effect [336]. As result, with this unique method the determination of the electric field of a sheath has been made possible — and by this, the independent control of the sheath thickness which is electrically active (in contrast to the sheath thickness which is determined by optical investigations). A biased electrode will collect electrically charged carriers which are accelerated by the electric field according to Coulomb’s law. This loss disturbs the quasi-neutrality of the plasma which responds by equalization processes: a flow of electrons into the (positively charged) drain or a flow of electrons out of the (negatively charged) source. This process leads to the build-up of a selfconsistent sheath in front of the electrode. To act as plasma probe, its electric field must not exceed the threshold of electron generation by secondary processes (thermal excitation or γ-processes). Likewise, the electrode current must be kept below a disturbing level (cf. Sect. 9.1.12 for determining the plasma potential). • During operation, the potential distribution is disturbed within the Debye length. • This leads to a flow of charged particles and, furthermore, to the built-up of space charges. Due to the strong asymmetry of the I(V ) characteristic, which is caused by the large mass ratio between electrons and ions, the initial electron current far surpasses the ion current by some orders of magnitude (cf. Sect. 3.4).
9.1 Langmuir probe
303
ji ∝ je
me . mi
(9.1)
Hence, the negatively charged space charge is much more important. Since the ion current is very small in magnitude, its measurement is very delicate, and the determination or at least estimation of the two properties electron density ne and electron temperature Te constitutes a significant part of the theory of the Langmuir probe. 9.1.3 Characteristic of the Langmuir probe
electron-saturation current
3
ln IS [mA]
VP
2
retarding-field regime ln j = const - e0 /kTe * V
1
Fig. 9.2. Scheme of an idealized I(V ) characteristic of a Langmuir probe to evaluate the most important plasma parameters electron density, ion density, electron temperature, and plasma potential [337].
VP (d2j/dV2 ) = 0 VP = VS
VF
0 ion-saturation current B A -20
0
20
40
DF[V]
9.1.3.1 Principle of the measuring technique. We regard the plasma bulk as a homogeneous zone with constant plasma density which lacks an electric field. It is confined by sheaths which are terminated by conducting reactor walls to ensure a proper circuit. In this case, a current from the Langmuir probe to the reactor walls will flow through plasma and sheaths if an electric potential is applied to the Langmuir probe. The principle can be seen from Fig. 9.2 [337]. The potential of the cylindrical rod is compared with that of the anode or with ground potential. Applying highly negative potentials will retard the electrons but accelerate ions which create a positively charged space layer around this negative potential. Therefore, this ion current quickly saturates (so-called space charge limited current, points left from A). With growing potential (i. e. less negative values), fast electrons can be absorbed by which the net current of of positive ions is diminished (condition: 1/2me ve2 ≥ e0 V , Point B). At a voltage which is still negative, 0 > V = Vf , both the fluxes are equal in magnitude, and there is no net current. This potential is
304
9 Plasma diagnostics
denoted the floating potential, because it is the potential of a conductive system immersed in the plasma but not connected to a reference potential, in particular not to ground. Further increase of the probe potential leads to weakening of the ion current, but to a steep growth of the negative current (which consists of electrons). This region is called the electron collection region, transition region or retarding-field region. Eventually, the potential of the plasma equals the probe potential. Only at this point will an electric field not repel or accelerate either carrier type, and they move to the probe according to their thermal velocities. According to Poisson, this is synonymous with a lack of space charges. Here, at the plasma potential, the retarding-field regime turns into the electron saturation regime, more or less sharply. To determine Φp exactly, we plot the logarithm of the probe current density vs. probe voltage. Due to the asymmetrical mobilities of the electric carriers, the V (I) characteristic is asymmetrical to the origin and increases steeply in the retarding-field region. Thanks to the logarithmic suppression, it seems to be a straight line in most cases. In principle, this steep increase is terminated at the plasma potential, and the slope of the characteristic should become significantly shallower and can be approximated by a second straight line. Following this definition, the plasma potential is determined by the point of inflection of the I(V ) characteristic (d2 I/dV 2 = 0). Another common definition is the intersection of both of the extended lines of electron saturation current and retarding-field region (Fig. 9.2). The concordance between these two approaches will become better for steeper slopes of the I(V ) characteristic. Only electrons which have invaded the sheath can traverse the sheath and can be eventually collected as probe current. If the thickness of the sheath (and hence its volume) were independent of the applied potential, the electron current could not grow even with further increasing potential but saturates at a real constant value. In fact, not only the area of the probe is subject to change but also the thickness of the negative space charge. Since this sheath consists of higly mobile electrons, it is significantly more extended than the ionic sheath in the regimes of the ion saturation current and the retarding-field regime, respectively. This is the reason why the electron current further increases even beyond the plasma potential, however, with a smaller slope. 9.1.3.2 Extraction of plasma parameters. According to the equation 1 (9.2) me ve2 ≥ e0 (Φp − ΦLp ) = e0 VLp 2 with Φp the plasma potential, ΦLp the probe potential and VLp the probe voltage, respectively, the energy distribution of the electrons EEDF can be determined by variation of the electron current with respect to the potential of the Langmuir probe ΦLp . The mean value of the energy is straightforwardly obtained by application of a normalized distribution function f on the measured energy values according to the relation
9.1 Langmuir probe
305
<E > =
∞
E f (E) dE.
(9.3)
0
For a Maxwellian distribution of the electrons, the fraction of the electrons which have a higher energy than e0 VLp and are able to overcome the sheath, is given by
Ne = N0 exp −
e0 VLp , kB Te
(9.4.1)
which yields for the current density
e0 Ne < v > e0 VLp je = exp − V 4 kB Te
⇒ je = e0 n
e0 VLp kB Te exp − , (9.4.2) 2πme kB Te
from which follows ln je = const −
e0 VLp . kB Te
(9.5)
¿From this equation, two properties can be derived: • Equation (9.5) is a straight line with a slope of e0 /kB Te (since d/dVLp ln je = 1/je × dje /dVLp = −e0 /kB Te ). Provided the effective area of the probe is exactly known, we can extract the electron temperature.2 • For VLp = 0, i. e. at Φp = ΦLp , the electron density n0 is given from the current density:
je = e0 n0
kB Te . 2πme
(9.6)
¿From a linear increase of the logarithmic current density in the fieldretarding regime, we can conclude that the electrons are distributed Maxwellian. In all other cases, the energy distribution of the electrons must be described by other functions. A very popular function is the Druyvesteynian distribution function [339], which has been derived for distributions in electric fields. Its application is superior for higher electron energies and low(er) electron densities (cf. Sect. 14.1).3 2 Numerous trials are reported to apply this equation to the ions which led to ridiculously high values for Ti . The reason is a change in the energy distribution of the ions IEDF caused by the sheath formation [338]. 3 This description is valid only for the ideal case. As a matter of fact, in most cases, no sharp bend in the I(V ) characteristic is found. Then two or even more slopes must be determined — with a necessary spread in Φp (i. e. Maxwellian distribution of the electrons with two temperatures).
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9 Plasma diagnostics
9.1.4 Plasma potential We have already realized that “plasma potential” is not only difficult to measure but also complicated to define. The potential of the Langmuir probe or the potential of the reference electrode can be absolutely determined. The plasma potential Φp is defined as that potential of the probe where no space charges (and electric fields) around the probe are present. Hence, the carriers move to the probe solely with their thermal velocity, experiencing no drift in an accelerating or retarding field. This voltage can only be evaluated from the I(V ) characteristic, and it is the value of its abscissa of the point of inflection, which coincides with the straight lines which approximate the electron saturation current and the electron retarding zone. The point which can be directly extracted from the I(V ) characteristic, however, is the floating potential Φf , which exhibits the additional advantage that it can be theoretically determined (Eq. 2.21) when the current of positive carriers is neglected, e. g. in discharges of a electropositive gas (inert gas or oxygen at low pressures). In the context with the electron temperature Te , we have to discuss (Fig. 9.3): • The plasma potential Φp , referred to the potential of the reference electrode. • The potential of the probe ΦLp , referred to the plasma potential, i. e. the sheath potential of the Langmuir probe. • VLp , the potential difference measured with the Langmuir probe.
potential
Fs
Fp Vp
Fig. 9.3. Going from the plasma potential Φp to more negative values for the voltage, the probe is surrounded by a cloud of positive ions creating a positively charged sheath potential which has to be corrected for.
distance
We measure VLp which deviates from Φp by the sheath potential Φs . Only when the potential ΦLp equals the plasma potential Φp , is no sheath present, and only here it holds true that
9.1 Langmuir probe
307 Fd probe 2
probe 1
relative potential
Fb
Fg
Fa
Fig. 9.4. Sketch of the potentials across two electrodes which are immersed in the plasma: Steep sheaths and a voltage difference between the electrodes which is rather caused by space charges than by Ohmic resistance. Φγ − Φα is the external voltage after [340].
Fa distance
ΦLp = Φp .
(9.7)
To evaluate the I(V ) characteristic in order to obtain the significant plasma parameters, ΦLp is the highest necessary abscissa value. For even more positive potentials, the probe is surrounded by a cloud of negative carriers, i. e. electrons; consequently the measured value for Φp has to be corrected by a negative value. For even more negative potentials, however, the probe is surrounded by a cloud of positive carriers, i. e. ions, consequently the measured value for VLp has to be corrected by a positive value (Fig. 9.3): Φs = ΦLp − Φp .
(9.8)
In the following sections, we will focus on this principal problem of determining the plasma potential and the floating potential. 9.1.5 Principle of the double probe We have dealt with the potential of the Langmuir probe in an absolute way insofar as we considered the floating potential as easily measurable (in a floating system, the net current drawn from the plasma is zero). However, it is not that easy. In Fig. 9.4, we have sketched the voltages between the potentials Φ of points β and α and δ and γ, respectively. These denote the sheath voltages, and the voltage between the potentials Φβ and Φδ is the potential drop in the neutral, undisturbed Ar+ plasma, consisting only of (fast) electrons and (slow) Ar+ ions. This potential difference is caused by differences of space charges in the plasma rather than by the existence of an Ohmic resistance and is small compared to Vβ,α and Vδ,γ , further simplied: This potential difference vanishes.
308
9 Plasma diagnostics
Provided that the sheath voltages do not depend on the external voltage, we can start with the condition of current continuity in this circuit and the definition of the potentials [340] (Ip denotes the current in the circuit of the two electrodes defining the Langmuir probe): Ip = I1− + I1+ ∧ Ip = −(I2− + I2+ );
(9.9.1)
(Φβ − Φα) + (Φδ − Φβ ) = (Φγ − Φα) + (Φδ − Φγ )∨ Vβα + Vδβ = Vγα + Vδγ .
(9.9.2)
Since the currents of ions and electrons both depend on the potential difference between plasma and the probes, both the sheath potential drops will adjust to values that condition (9.9.1) is fulfilled. For equal or comparable probe areas, we denote this setup a double probe system. For a floating probe system, which does not draw any net current from the plasma, it is symmetric with reference to the floating potential Φf . In its simplest form, this system consists of two identical probes which are connected to a voltage source. The current flowing in the circuit can be measured as a function of the voltage between these two probes. The inherent advantage over a simple Langmuir probe is twofold: • The electrodes are on floating potential Φf and are not connected to the electrodes of the discharge tube. Fluctuations of the plasma potential Φp will cause a similar fluctuation of the floating potential Φf which leads to constancy of the sheath potential Φs : Φs = Φp − Φf . • The probe current can be kept at very low values and hence the (negative) influence on the plasma should be small [341]. Discharges which are lacking in clearly defined electrode potentials, in particular electrodeless high-frequency discharges, are the main candidates for their application. The symmetrical setup leads to a I(V ) characteristic which is symmetrical to the origin for vanishing potential difference in the plasma (i. e. for constant plasma potential). Otherwise, the I(V ) characteristic is shifted by this difference (Fig. 9.5). The main advantage of the double probe system is the smaller disturbance of the plasma compared with a single probe measurement [341]: For equal areas of both the electrodes, the electron current which flows to the positively biased electrode is limited by the flow of positive ions to the other electrode, the socalled reference elecetrode, and not by reaching the plasma potential. Hence, the assumption that the sheath potentials are not influenced by the externally applied voltage, is justified to an extent.
9.1 Langmuir probe
309
1.0
Is [mA]
0.5
II,1 Rs = 1/(D Is/DV)
0.0 II,2
-0.5
-1.0 -20
-10
0 DF [V]
10
20
Fig. 9.5. I(V ) of an idealized double probe with a characteristic symmetrical to the origin (no difference in plasma potential for both the electrodes) [341].
However, this system lacks the possibility of measuring either a plasmapotential or a floating potential. But for a Maxwellian distribution of the electrons, the two common parameters, i. e. electron temperature and plasma density can be evaluated from the I(V ) characteristic. To avoid this dilemma, at least two possibilities do exist: The application of a large-area electrode or the addition of a third electrode whose potential with reference to the plasma is kept constant under all circumstances. 9.1.6 Principle of the asymmetrical double probe By choosing a strong asymmetrical ratio, we can heighten the ion current at the large area electrode (1) to values which are large compared to the electron current at the small area electrode (2) even at the plasma potential (Vδγ = 0): I2− I1+ .
(9.10.1)
I1− = −I+,1 ∧ Ip ≈ 0,
(9.10.2)
Then eq. (9.9.1) simplifies to
because of I1+ I2+ ∧ I1− I2− : The resulting small current only consists of an ionic and electronic contribution at the large electrode which mutually cancel out. This can happen only at high a negative potential. Now, even strong variations of the external voltage Vγα at the small area electrode influence the current at the large area electrode only to a very small extent. Thus, we can vary the probe potential by variation of Vγα without affecting the voltage between the large area electrode and the plasma. Henceforth, we refer to the large area electrode as “reference electrode”, exhibiting a strong negative potential with respect to the plasma potential and to the smaller probe. Remembering our
310
9 Plasma diagnostics
definition of the floating potential Φf , we realize the accordance of the reference potential with the floating potential [342]). Decreasing of the external potential Vγα from Φp to smaller values at the smaller electrode (going left in the I(V ) characteristic), leads to strong variations of the electron current with the ion current remaining nearly constant: free fall in the electron retarding regime. This effect is accompanied by relatively small variations of either currents to the reference electrode, and it can be made more distinct by increasing the ratio of the probe areas: the transformation ratio.4 By increasing the potential to values beyond the plasma potential Φp , however, only little changes are expected because the range of saturation of the electron current is quickly reached. Setting the potential of the reference electrode to zero leads to a positive shift for the I(V ) characteristic. In particular, the value for Φf shifts to positive values. If the electrons are assumed to be Maxwellian distributed, we can connect the probe current with the voltage between Langmuir probe and reference electrode by ln Ip = ln Ip,s −
e0 (Φp − Vb ) kB Te
(9.11)
with Ip the probe current, Ip,s the probe current in the saturation regime, Φp the plasma potential, referred to the reference electrode, and Vb the voltage between probe and reference electrode. As result, we have obtained the characteristic of a single probe, but with clearly defined reference potential. 9.1.7 Determination of potentials in high-frequency discharges If a Langmuir probe lacks a reference electrode with constant potential, we are faced with severe difficulties of interpretation. Additionally, plasma fluctuations have a strong influence on the characteristic of a Langmuir probe [183]. Boschi and Magistrelli were the first to point out the difficulties in interpreting the Langmuir probe data “polluted” with RF noise [343], and afterwards, the HF modulation of the EEDF and various plasma properties, e. g. the electron density, respectively, has been extensively discussed by Winkler et al. [344, 345], Godyak [346] and Wiesemann [347], just to mention a few. As could be shown by Flender et al. [347], the second derivation of the Langmuir characteristic which serves to evaluate the EEDF, very sensitively reacts to an insufficient RF compensation, whereas the characteristic itself remains almost unaffected (Fig. 9.6). Hence, the plasma density np and the plasma potential Φp can be evaluated from the Langmuir characteristic, but the EEDF can be not. 4 In the best case, the inner surface of the plasma reactor serves as reference electrode. To ensure constant surface conductivity, reactive gases must not be processed which could contaminate the reactor walls.
9.1 Langmuir probe
311 15
0.7
0.5
I [mA]
0.4
B D
10 5 I [mA]
0.6
B D V(t) H
0.3 0.2
0
0.1
-5
0.0 -0.1 15
20
25
30
35
-10 15
40
20
25
30
35
40
V [V]
V [V]
Fig. 9.6. I(V ) characteristic and its second derivation, as measured with good RF compensation (solid) and the original signal convoluted with a sinusoidal voltage of V0 = 5V (dashed) after [347].
Winkler et al. showed that in capacitively coupled plasmas driven at 13 MHz, i. e. significantly beyond ωp,i , a threshold of approximately 75 mTorr (10 Pa) can be drawn which has to be exceeded to find the derived plasma properties significantly frequency-modulated. Hence, at discharge pressures which are common for reactive ion etching (RIE), no HF modulation is detectable: Only the HF component of the plasma potential has to be considered because of an additional potential drop across the forming sheath, the RF self-bias [183]. Hence, we expect the probe voltage to be modulated by a DC component VDC and an AC component V˜ , respectively [348]. For sinusoidal variation, the voltage in Eq. (9.5) is modified from5 Vp −→ VDC + V˜ sin ωt.
(9.12)
Averaging over an RF cycle of duration T , i. e. e0 V˜ 1T exp[iV˜ sin(ωt)] dt = J0 − T 0 kB Te
(9.13)
yields the Bessel function of zero order with pure imaginary argument [349], and we obtain for the current density ln je = je,0 −
e0 (VDC + ΔV ) kB Te
(9.14)
with [343]
kB Te e0 V˜ ΔV = − ln J0 − e0 kB Te
:
(9.15)
5 Being more precise, there is an additional voltage drop across the sheath caused by the electron current of the order of R A je (t) with A the effective area of the tip.
312
9 Plasma diagnostics
rounded "knee" I (sat) e
3
VP
ln Ip [mA]
2 D Vdc
1 VF
VP (d 2I/dV2) = 0
VF
0 Ii (sat)
-1 -40
-20
0
20
Fig. 9.7. High frequency discharge: The characteristics is shifted by the self biasing process, the electron current across the sheath distorts its upper part [rounded “knee”, (after [348])].
V [V]
Due to the balancing of electron and ion current, the ln I(V ) characteristic of the probe is shifted by ΔV with respect to the applied bias. For small ratios of e0 V˜ /kB Te which are smaller than or equal unity, only a negative shift is imposed on the ln I(V ) characteristic, for large ratios, however, a significant rounding will occur at the “knee” induced by the Ohmic term R A je (t) (cf. Fig. 9.7) [337]. Careful attention has to be paid to filter the signal from high-frequency noise. This can be accomplished by a transmission line beginning with an inductor which terminates the tungsten rod of the Langmuir probe. This transmission line contains a series of circuits with resonance frequencies which equal the fundamental and its overtones of the excitation frequency. To avoid stray capacitancies of the probe wire to the outer ground, the mechanics of the Langmuir probe often comes along in triaxial technique. For this end, an additional inner shielding is used which is capacitively coupled to the plasma by means of large-area compensating electrodes, for example metal wings, which are attached to the Langmuir probe (cf. Fig. 9.1). Since the probe impedance is mainly determined by the capacitance between Langmuir probe and plasma, an additional shunting capacitance which is exposed to the same plasma potential Φp , will reduce the impedance of the Langmuir probe to low values, which would be unachievable by other means. Now, a requirement established by Godyak et al. can be readily met [346]: 1 Te Zs ≤ Zext 2 VRF
(9.16)
with Zs and Zext the impedances of the Langmuir probe (and plasma) and the external probe circuit, mainly the DC resistance of the RF driven electrode sheath (which are comparable without this shunting impedance). It should be kept in mind that this requirement has to be followed by all the Fourier
9.1 Langmuir probe
313
components which are generated by the rectification of the sinusoidal voltage applied to the RF driven electrode. Hence, it can follow, at least in principle, the high-frequency amplitude of the plasma potential without any phase shift. In this inner shielding, no currents would flow except the inevitable capacitive coupling to the outer shielding. Another possibility is the application of a double probe or a triple probe, respectively [350]. With this probe, it is possible to create a reference potential even in electrodeless discharges [351]. On the other hand, the oscillations can be investigated with a carefully designed circuit. Wood et al. connected the rod to a series of parallel LC elements within the probe shaft [352]. If they were tuned to resonate at 13.56 MHz they would follow the oscillations of the plasma potential, and the typical parameters (ne , Te ) could be extracted as if they were taken in a DC discharge. They could verify the self-consistent model with a highly nonlinear sheath motion which was established by Lieberman in 1988 [192] (cf. Sect. 14.4). This model improved the picture of a sinusoidal sheath oscillation which had been introduced by Godyak ten years earlier [206]. Later on, Lieberman’s model was brilliantly confirmed by Vender and Boswell with particle-in-cell (PIC) simulations in 1990 [353], and by Gozadinos et al. with their considerations for stochastic heating [211]. 9.1.8 Details We will now focus on some details of the collecting process of carriers which arrive at the Bohmic presheath of the Langmuir probe. We will briefly discuss the limits of the theory for a thin and a thick sheath, respectively, and will finally focus on operating a Langmuir probe in a magnetic field. 9.1.9 Probe radius The geometry of the probe and the density of the plasma (not only the density of electrically charged carriers but also the number density) mutually influence the regimes in which an adequate evaluation of the probe measurements is possible without a strong simplification. In principle, we distinguish between these regimes: 1. A regime for which the radius of the probe, rLp , is large compared to the thickness of the sheath, rs − rLp , around the probe (Fig. 9.8) [354]. This is a layer of positive ions which begins on the surface of the probe and extends to a plane where the finite electron density must be considered. 2. A regime for which the radius of the probe, rLp , is very small or comparable to the Debye length and to the thickness of the sheath, rs − rLp , around the probe (cf. Sect. 3.3, Fig. 9.8).
314
9 Plasma diagnostics
3. Eventually, also the mean free path of the ions and its relation to the Debye length plays an important role in discharges at higher pressures → (1). In the first case, the surface of the cylinder with radius rLp is comparable to the surface of the cylinder with radius rLp , whereas in the latter case, the surface of the rod is small compared to the radius of the sheath. Since the radius of the sheath edge is a property which evolves from a model, the area of this cylinder depends on its accuracy. In particular, the current density becomes a questionable quantity.
rs rp li rs rp Fig. 9.8. Top, simplest assumption: Probe radius rLp and radius of the sheath rs are both comparable, but small against the mean free path of the ions λi : collisionless sheath. For rLp rs , every electron which has arrived at the Bohm edge will be collected. Bottom, next assumption: Probe radius rLp is small compared to the sheath radius rs . Carriers are influenced (from slight deflection to collection) up to the sheath edge.
Since the metallic rod represents an equipotential surface, the upper limit for the probe radius should be the Debye length. As outlined in Chap. 4, the sheath can be estimated to be several Debye lengths in thickness.6 For a ratio rLp ≤ 1, (9.17) λD the motions of the carriers can and must be taken into consideration. This is possible with an analytical solution (OML). In low-pressure plasmas and in 6
This assumption is questionable for large negative values of the bias voltage.
9.1 Langmuir probe
315
high-density plasmas, respectively, this condition can easily be fulfilled. In these discharges, the mean free path of both carrier types is large compared to the sheath thickness rs (collisionless sheath); at higher discharge pressures, however, the probe itself alters the gradient of carrier distribution. In this pressure regime, further approximations are required, which were first numerical in nature [355], but recently, some parametrized solutions have been developed [356, 357]. The electron density around a spherical probe deviates from the value for infinite distance from the probe by [358]
n = n∞ 1 −
rs rLp
(9.18)
with rs the radius of the negative sheath whose thickness rs − rLp equals several Debye lengths [359]. At large number densities, the number of collisions for ions and electrons within the space charge becomes finite. This leads to a weakening of the steep increase of the electron current, and is very complicated to treat. Therefore, we limit our discussion to the collisionless sheath with the condition λe,i (rLp , rs ). 9.1.10 Thin sheath: space charge limited current 9.1.10.1 Positive ions. For strongly negative probe potentials, a positively charged space charge will form around the electrode which outwardly screens B Te . Although this voltage is small comthe probe potential by the fraction k2e 0 pared with the total plasma potential, Φp , the leaking field associated with this potential leads to a considerable increase of the ion flux, and the ions have attained a component of (directed) kinetic energy which vastly exceeds their thermal energy. At the boundary between Bohmic presheath and probe sheath, they are accelerated to the Bohm velocity vB . In the sheath, the electron density drops sharply to very low values and eventually vanishes (Fig. 9.9), and we get for the ion flux and the ion current density [Sect. 14.2, Eqs. (14.43/44)]:
ji,max = e0 Γi,max =
kB Te n0 e0 = 0.606 n0 e0 emi
kB Te , mi
(9.19)
it is considerably enlarged (the square root is the sound velocity of the ions!): The ions are accelerated from the very low thermal energy Ei ≈ 0.1 eV to approximately the mean electron energy (several electronvolts)! It has turned out in many cases that the probe current may be calculated applying the area of the rod of the Langmuir probe [360]. 9.1.10.2 Electrons. The region of the linear increase of the probe current for which not only the name retarding zone is customary but which is also denoted the electron repelling current, starts at negative values of the probe potential. In this regime, a change of the characteristic of the electron current will occur.
316
potential energy [a. u.]
Vp VB
9 Plasma diagnostics
neutral plasma no field Bohmic presheath weak field positively charged cathode sheath
Fig. 9.9. The three zones in front of an electrode: undisturbed quasi-neutral plasma, quasi-neutral Bohmic presheath, positively charged sheath with cathode fall.
Vc
distance [a. u.]
At its beginning, it is determined by a retarding potential which will be replaced by a diffusion-limited behavior at less negative potentials. The current density can be calculated [with ΦLp the probe potential (Sect. 3.4, Eqs. (3.20) − (3.23))]
1 1 e0 ΦLp kB Te je = ne (x)e0 < ve >= n0 e0 · exp − . 4 4 2πme kB Te
(9.20)
Plotting the logarithmic probe current versus probe voltage should result in a straight line, and from its slope, we can evaluate the electron temperature Te , provided the energy of the ensemble is distributed Maxwellian. For rs −rLp rLp , all electrons will reach the probe. In that case, the current for both carrier types can be calculated according to I± = j± As
(9.21)
with As the surface of the sheath. In principle, the plasma density can be extracted from the ion current. Since it is smaller by orders of magnitude than the electron current, the errors are always very large [361]. A theory for the case of very small probe radii and collisionless sheaths has been established by Allen, Boyd and Reynolds [362], the limiting case of a finite sheath was thoroughly treated by Bernstein and Rabinowitz [363] and Laframboise [355] also for higher number densities with a collisional sheath. 9.1.10.3 Finite electron temperature. For finite electron temperature, the electrons can leave the surface of the probe without an extracting field by which a retarding potential is caused (V < 0 V) which suppresses any further emission. Hence, the V (x) characteristic drops across its whole course: For small distances, the potential values are below zero, and also at the anode, we do not reach the values which we are accustomed to when we apply the Child-Langmuir
9.1 Langmuir probe
317
equation (Sect. 4.3, Fig. 9.10). Here, Vm denotes a potential which has been established by Langmuir [86]. For a value of 3 Vm2 , the V (x) characteristic exhibits its minimum. 20
300
200
0 mA/cm 10 mA/cm 2 100 mA/cm2
10 F[V+Vm]
h = 0.1
F[V]
Vm: 0 V Vm: -15 V Vm: -30 V Vm: -50 V
15
2
100
5 0 -5 -10
0 0.00
-15 0.25
0.50 d [cm]
0.75
0.00
1.00
0.05
0.10
0.15
0.20
d [cm]
Fig. 9.10. For finite electron temperature, the potential in the vicinity of the probe drops to values below 0 V, and the minimum of the potential dependence is located in front of the electrode; this is shown here for a ratio η = keB0 VTpe of 0.1 for a sheath thickness of 400 μm and various values for Vm and an η = 0.1 and electron current densities je of 10 mA/cm2 . Note the larger scale of the abscissa at the RHS figure.
For thin sheaths (rs rLp ), the current is space-charge limited and can be expressed by Eq. (9.22). Its value depends on potential via the the changing thickness of the sheath. Provided that ΦLp kB Te /e0 , we can apply the diode equation (4.21), and for a planar probe with a one-dimensional sheath, the current is given by
Ie = je ALp
3/2
ΦLp −4 2e0 = ALp . ε0 9 me (rs − rLp )2
(9.22)
¿From the I(V ) characteristic, we can eventually calculate the sheath thickness rs . For finite electron temperature and/or small Langmuir probe potential ΦLp , Eq. (9.22) has to be extended by a correction term [364], and we define xs = rs − rLp :
