Reactive Sputter Deposition (Springer Series in Materials Science)

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Reactive Sputter Deposition (Springer Series in Materials Science)

Springer Series in materials science 109 Springer Series in materials science Editors: R. Hull R. M. Osgood, Jr.

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Pages 589 Page size 346.46 x 592.43 pts Year 2008

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Springer Series in

materials science

109

Springer Series in

materials science Editors: R. Hull

R. M. Osgood, Jr.

J. Parisi

H. Warlimont

The Springer Series in Materials Science covers the complete spectrum of materials physics, including fundamental principles, physical properties, materials theory and design. Recognizing the increasing importance of materials science in future device technologies, the book titles in this series ref lect the state-of-the-art in understanding and controlling the structure and properties of all important classes of materials. 99 Self-Organized Morphology in Nanostructured Materials Editors: K. Al-Shamery and J. Parisi

105 Dilute III-V Nitride Semiconductors and Material Systems Physics and Technology Editor: A. Erol

100 Self Healing Materials An Alternative Approach to 20 Centuries of Materials Science Editor: S. van der Zwaag

106 Into The Nano Era Moore’s Law Beyond Planar Silicon CMOS Editor: H.R. Huff

101 New Organic Nanostructures for Next Generation Devices Editors: K. Al-Shamery, H.-G. Rubahn, and H. Sitter

107 Organic Semiconductors in Sensor Applications Editors: D.A. Bernards, R.M. Ownes, and G.G. Malliaras

102 Photonic Crystal Fibers Properties and Applications By F. Poli, A. Cucinotta, and S. Selleri

108 Evolution of Thin-Film Morphology Modeling and Simulations By M. Pelliccione and T.-M. Lu

103 Polarons in Advanced Materials Editor: A.S. Alexandrov 104 Transparent Conductive Zinc Oxide Basics and Applications in Thin Film Solar Cells Editors: K. Ellmer, A. Klein, and B. Rech

109 Reactive Sputter Deposition Editors: D. Depla amd S. Mahieu 110 The Physics of Organic Superconductors and Conductors Editor: A. Lebed

Volumes 50–98 are listed at the end of the book.

D. Depla

S. Mahieu

Editors

Reactive Sputter Deposition With 341 Figures

123

Professor Dr. Diederik Depla Dr. Stijn Mahieu Gent University, Department of Solid-State Sciences 281-S1 Krijgslaan, 9000 Gent, Belgium E-mail: [email protected], [email protected]

Series Editors:

Professor Robert Hull

Professor Jürgen Parisi

University of Virginia Dept. of Materials Science and Engineering Thornton Hall Charlottesville, VA 22903-2442, USA

Universit¨at Oldenburg, Fachbereich Physik Abt. Energie- und Halbleiterforschung Carl-von-Ossietzky-Strasse 9–11 26129 Oldenburg, Germany

Professor R. M. Osgood, Jr.

Professor Hans Warlimont

Microelectronics Science Laboratory Department of Electrical Engineering Columbia University Seeley W. Mudd Building New York, NY 10027, USA

Institut f¨ur Festk¨orperund Werkstofforschung, Helmholtzstrasse 20 01069 Dresden, Germany

ISSN 0933-033X ISBN 978-3-540-76662-9 Springer Berlin Heidelberg New York Library of Congress Control Number: 2007938637 All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without the written permission of Kodansha Ltd. (except in the case of brief quotation for criticism or review.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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. Springer is a part of Springer Science+Business Media. springer.com © Springer-Verlag Berlin Heidelberg 2008 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data prepared by SPi using a Springer LATEX macro package Cover concept: eStudio Calamar Steinen Cover production: WMX Design GmbH, Heidelberg Printed on acid-free paper

SPIN: 12067657

57/3180/SPi

543210

Preface

The most straightforward method to change the surface properties of a material is to deposit a thin film or coating on it. Hence, it is not surprising that an overwhelming amount of scientific and technical papers is published each year on this topic. Sputter deposition is one of the many so-called physical vapour deposition (PVD) techniques. In most cases, sputter deposition uses a magnetically enhanced glow discharge or magnetron discharge to produce the ions which bombard and sputter the cathode material. In the first chapter of this book (Chap. 1), the details of the sputter process are discussed. Essential to sustain the discharge is the electron emission during ion bombardment. Indeed, the emitted electrons are accelerated from the target and can ionize gas atoms. The formed ions bombard again the target completing the sustaining process. A complete chapter is assigned to this process to highlight its importance (Chap. 2). Although the sustaining process can be described quite straightforward, a complete understanding of the magnetron discharge and the influence of different parameters on the discharge characteristics is only possible by modelling (see Chap. 3). With these three chapters, the reader should be able to form an idea of the target and plasma processes occurring during a DC magnetron discharge. When a reactive gas is added to the discharge, it becomes possible to deposit compound materials. This process is called reactive sputter deposition. The changes of the deposition process as a function of the reactive gas addition are discussed in Chaps. 4 and 5. The former (Chap. 4) describes the well-known “Berg” model, which enables a better understanding of the general aspects of reactive magnetron sputtering. The next chapter (Chap. 5) summarizes several experimental results and shows the need to more detailed models to describe the reactive sputtering process. One of the major problems of the reactive sputter process is its complexity and several fundamental aspects of the process have not been elucidated yet. Only by understanding all its details, it is possible to understand the properties of the obtained thin film as a function of the deposition conditions. In this respect, it is necessary to describe the deposition flux towards

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the substrate. This can be achieved by modelling the transport of sputtered particles towards the substrate (Chap. 6). However, not only the metallic flux but also the energy flux towards the substrate is important. This latter forms the subject of the following chapter (Chap. 7). More details of the reactive magnetron sputter process can be achieved by a good knowledge of the available plasma diagnostic tools. Hence, they form the content of the next two chapters (Chaps. 8 and Chap. 9). A nice illustration of the implementation of these tools is shown in the Chap. 10 which demonstrates their use in the study of the cross-corner and cross-magnetron effect (Chap. 10). Finally, the book ends with some interesting examples of materials deposited with reactive sputter deposition. Indeed, complex materials such as solid electrolytes (Chap. 11), complex oxides (Chap. 12) and electrochromic thin films (Chap. 13). In the last chapter, the reader can learn more about the simulation of the growth of thin film deposited by magnetron sputtering (Chap. 14). Although the technique is easy to use, it conceals enough challenges to remain scientific interesting. This explains its popularity in the academic world. Also in the industrial world, reactive magnetron sputtering remains an interesting and often used technique, due to its flexibility and scalability and has gained in this way a strong position for large-area deposition of thin films. As for most thin film deposition techniques, sputter deposition was once considered a black art and only in the last two decades, there has been a vast increase in the range of material types which can be deposited, the complexity of thin films which are possible, the ability to deposit precisely controlled heterostructures and the reproducibility of film deposition. The origin of this change is the trend to analyse in more detail all relevant processes during the thin film deposition to maximize the level of control. This forms exactly the goal of this book, i.e. to give the reader on overview of the important processes during sputter deposition and of the aim to describe them by modelling and to use them to deposit complex materials such as perovskite, solid electrolytes and electrochromic thin films. A good understanding of the reactive sputtering process is essential when tailoring the thin film properties. This reasoning formed also the guideline for the table of content which mimics a virtual journey from target towards substrate. Together with the authors, we hope that the different topics discussed in the book will help the novice and experienced scientist to solve some of the problems encountered during the use of this interesting deposition technique. Gent, January 2008

Diederik Depla Stijn Mahieu www.draft.ugent.be

Contents

1 Simulation of the Sputtering Process T. Ono, T. Kenmotsu, and T. Muramoto . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Computer Simulation Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Total Sputtering Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Incident-Energy Dependence of Sputtering Yield . . . . . . . . . . 1.3.2 Incident-Angle Dependence of Sputtering Yield . . . . . . . . . . . 1.4 Differential Sputtering Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Energy Spectrum of Sputtered Atoms . . . . . . . . . . . . . . . . . . . . 1.4.2 Angular Distribution of Sputtered Atoms . . . . . . . . . . . . . . . . . 1.5 Sputtering from Rough Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Sputtering of Compound Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Electron Emission from Surfaces Induced by Slow Ions and Atoms R.A. Baragiola and P. Riccardi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Physical Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Excitation Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Separation of PEE and KEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Electron Transport and Escape into Vacuum . . . . . . . . . . . . . . 2.3 Electron Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Dependence of the Electron Yields on Ion Velocity . . . . . . . . . 2.3.2 Electron Energy and Angular Distributions . . . . . . . . . . . . . . . 2.3.3 Electron Emission from Contaminant Surface Layers . . . . . . . 2.4 The Role of Ion-Induced Electron Emission in Glow Discharges . . . 2.4.1 Effect of Electron Recapture at the Cathode . . . . . . . . . . . . . . 2.4.2 Effect of Changes in the Chemical Composition of the Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 5 9 16 16 21 28 30 37 39

43 43 44 44 46 46 46 47 48 48 50 53 56

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2.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3 Modeling of the Magnetron Discharge A. Bogaerts, I. Kolev, and G. Buyle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1.1 The Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.1.2 The Magnetron Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.3 The Particle–Target Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.4 Particle Transport in the Gas Phase . . . . . . . . . . . . . . . . . . . . . 64 3.1.5 Film Growth on the Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Overview of Different Modeling Approaches for Magnetron Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.1 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.2 Fluid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.3 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.4 Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.5 Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.6 PIC-MCC Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 Challenges Related to Magnetron Modeling . . . . . . . . . . . . . . . . . . . . . 74 3.3.1 Secondary Electron Emission Yield (γ) . . . . . . . . . . . . . . . . . . . 75 3.3.2 Recapture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.3 Electron Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.3.4 Modeling “Industrially Relevant” Magnetron Discharges . . . . 76 3.4 Two-Dimensional Semi-Analytical Model for a DC Planar Magnetron Discharge . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.4.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.2 Examples of Calculation Results . . . . . . . . . . . . . . . . . . . . . . . . 81 3.5 PIC-MCC Model for a DC Planar Magnetron Discharge . . . . . . . . . 83 3.5.1 Particle-In-Cell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.5.2 Monte Carlo Collision Method . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.5.3 Methods for Speeding Up the Calculations . . . . . . . . . . . . . . . . 99 3.5.4 Examples of Calculation Results . . . . . . . . . . . . . . . . . . . . . . . . 103 3.6 Extension of the PIC-MCC Model: To Include Sputtering and Gas Heating . . . . . . . . . . . . . . . . . . . . . . . . 115 3.6.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.6.2 Examples of Calculation Results . . . . . . . . . . . . . . . . . . . . . . . . 119 3.7 Conclusions and Outlook for Future Work . . . . . . . . . . . . . . . . . . . . . . 123 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4 Modelling of Reactive Sputtering Processes S. Berg, T. Nyberg, and T. Kubart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.2 Basic Model for the Reactive Sputtering Process . . . . . . . . . . . . . . . . 133 4.3 Steady State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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4.4

Influence of Material Properties and Processing Conditions . . . . . . . 141 4.4.1 Reactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4.2 Sputtering Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.4.3 Pumping Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.4.4 Target Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.4.5 Mixed Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.4.6 Two Reactive Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.4.7 Reactive Co-Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.4.8 Comment on Pulsed DC Reactive Sputtering . . . . . . . . . . . . . 150 4.4.9 Secondary Electron Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.4.10 Ion Implantation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5 Depositing Aluminium Oxide: A Case Study of Reactive Magnetron Sputtering D. Depla, S. Mahieu, and R. De Gryse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.2 Some Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.2.1 A First Series of Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.2.2 A Second Series of Experiments: Oxygen Exposure and Plasma Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.2.3 Stability Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.2.4 First Conclusions Based on the Experiments . . . . . . . . . . . . . . 166 5.3 An Extended Model for Reactive Magnetron Sputtering . . . . . . . . . . 167 5.3.1 The Description of Reactive Ion Implantation During Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.3.2 The Influence of Chemisorption and Knock-On Effects . . . . . 171 5.3.3 Calculation of the Reactive Gas Partial Pressure as a Function of the Reactive Gas Flow . . . . . . . . . . . . . . . . . . 173 5.4 Confrontation Between Experiment and Model . . . . . . . . . . . . . . . . . . 175 5.4.1 Simultaneous Reactive Ion Implantation and Sputtering Without Chemical Reaction . . . . . . . . . . . . . . . 175 5.4.2 Simultaneous Reactive Ion Implantation and Sputtering with Chemical Reaction . . . . . . . . . . . . . . . . . . 176 5.5 Towards a More Complete Model for Reactive Magnetron Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.5.1 Plasma-Related Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.5.2 Deposition Profiles and Erosion Profiles . . . . . . . . . . . . . . . . . . 187 5.5.3 Rotating Magnetrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

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6 Transport of Sputtered Particles Through the Gas Phase S. Mahieu, K. Van Aeken, and D. Depla . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 6.2 Radial Distribution Where Sputtered Particles Leave the Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.3 Energy and Angular Distribution of Sputtered Particles Leaving the Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.3.1 Sigmund–Thompson Theory for the Linear Cascade Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.3.2 Other Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.3.3 Comparison to Experimental Results . . . . . . . . . . . . . . . . . . . . . 206 6.3.4 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.4 Describing the Collision with the Gas Particle . . . . . . . . . . . . . . . . . . 208 6.4.1 The Mean Free Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6.4.2 The Scattering Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.4.3 The Interaction Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.5 Gas Rarefaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.6 Typical Results of a Binary Collision Monte Carlo Code . . . . . . . . . 218 6.7 Specific Example: In-Plane Alignment of Biaxially Aligned Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7 Energy Deposition at the Substrate in a Magnetron Sputtering System S.D. Ekpe and S.K. Dew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.2 Energy Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.3 Factors Affecting Energy Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.3.1 Magnetron Power and Pressure . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.3.2 Substrate-Target Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 7.3.3 Electrical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.4 Total Energy per Deposited Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.5 Energy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.5.1 Sputtered Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 7.5.2 Reflected Neutrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.5.3 Plasma Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 7.5.4 Charge Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 7.5.5 Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 7.5.6 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

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8 Process Diagnostics J.W. Bradley and T. Welzel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.2 Electrical Probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 8.2.1 Probe Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 8.2.2 Use of Electrical Probes in Pulsed Reactive Sputtering . . . . . 258 8.3 Mass Spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 8.3.1 Mass Spectrometry Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 8.3.2 Application of Mass Spectrometry to Reactive Sputtering . . 271 8.4 Optical Emission Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 8.4.1 Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 8.4.2 Plasma Emission Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8.4.3 Time-Resolved OES in Pulsed Reactive Sputtering . . . . . . . . 288 8.5 Optical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 8.6 Laser-Induced Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 9 Optical Plasma Diagnostics During Reactive Magnetron Sputtering S. Konstantinidis, F. Gaboriau, M. Gaillard, M. Hecq, and A. Ricard . . . 301 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 9.2 Emission Spectroscopy of Magnetron Plasmas . . . . . . . . . . . . . . . . . . . 302 9.3 Resonant Absorption Spectroscopy of Magnetron Plasmas . . . . . . . . 311 9.4 Laser Spectroscopy of Magnetron Plasmas . . . . . . . . . . . . . . . . . . . . . . 320 9.4.1 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 9.4.2 Principle and Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 9.4.3 Application to Magnetron Discharges . . . . . . . . . . . . . . . . . . . . 326 9.5 Optical Diagnostic of High-Power Impulse Magnetron Sputtering Discharges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 9.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10 Reactive Magnetron Sputtering of Indium Tin Oxide Thin Films: The Cross-Corner and Cross-Magnetron Effect H. Kupfer and F. Richter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 10.2 The CCE and CME as an Inhomogeneous Target Erosion . . . . . . . . 338 10.3 Evidence of the CCE and CME From In Situ Measurements . . . . . . 346 10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 10.3.2 The CCE and the Plasma Properties of Unipolar-Pulsed Single Magnetron Discharges . . . . . . . . . . 349 10.3.3 The CME and the Plasma Properties of Bipolar-Pulsed Dual Magnetron Discharges . . . . . . . . . . . . . 350 10.3.4 The CME and the Thermal Load of the Substrates . . . . . . . . 353

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10.4 CCE, CME and Film Property Distribution: ITO as an Example . . 354 10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 10.4.2 Unipolar DC-Pulsed Single Magnetron: CCE and ITO Film Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 10.4.3 Bipolar-Pulsed Dual Magnetron: CME and ITO Film Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 10.5 CCE, CME and the Role of the Atomic Oxygen in the Process Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 11 Reactively Sputter-Deposited Solid Electrolytes and Their Applications P. Briois, F. Lapostolle, and A. Billard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 11.2 Crystallographic Basis of the Solid Electrolyte . . . . . . . . . . . . . . . . . . 369 11.2.1 O2− Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 11.2.2 H + Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 11.2.3 N a+ Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 11.2.4 Li+ Carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 11.2.5 Mixed Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 11.3 Application of Solid Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 11.3.1 Solid Oxide Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 11.3.2 Microbatteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 11.3.3 Smart Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 12 Reactive Sputtered Wide-Bandgap p-Type Semiconducting Spinel AB2 O4 and Delafossite ABO2 Thin Films for “Transparent Electronics” A.N. Banerjee and K.K. Chattopadhyay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 12.2 Spinel and Delafossite Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 12.2.1 Spinel Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 12.2.2 Delafossite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 12.3 p-Type Transparent Conducting Oxides Based on Spinel and Delafossite Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 12.3.1 Introduction to Transparent Conducting Oxides . . . . . . . . . . . 417 12.3.2 Transparent Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 12.3.3 p-TCO with Spinel Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 420 12.3.4 p-TCO with Delafossite Structure . . . . . . . . . . . . . . . . . . . . . . . 422 12.3.5 Other Deposition Techniques: PLD, RF Sputter Deposition, Magnetron Sputtering with RTA, and Ion Exchange Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 12.3.6 Reactive Sputtered p-TCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

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12.4 Transparent Junctions Based on Spinel and Delafossite Oxides . . . . 441 12.5 Origin of p-Type Conductivity in Wide-Bandgap Spinel and Delafossite Oxide Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448 12.6 Reactive DC Sputter Deposition of Delafossite p-CuAlO2+x Thin Film . . . . . . . . . . . . . . . . . . . . . . . . . . 455 12.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 12.6.2 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 12.6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 12.7 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 12.7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 12.7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 13 Oxide-Based Electrochromic Materials and Devices Prepared by Magnetron Sputtering C.G. Granqvist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 13.2 Energy Efficiency of Chromogenic Building Skins . . . . . . . . . . . . . . . . 486 13.3 Electrochromic Device Design and Materials . . . . . . . . . . . . . . . . . . . . 487 13.4 Properties and Applications of Electrochromic Foil . . . . . . . . . . . . . . 490 13.5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 14 Atomic Assembly of Magnetoresistive Multilayers H. Wadley, X. Zhou, and W.H. Butler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 14.1.1 Giant Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 14.1.2 Atomic Scale Structure Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 501 14.1.3 Deposition and Growth Processes . . . . . . . . . . . . . . . . . . . . . . . 504 14.2 Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 14.2.1 Interatomic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 14.2.2 MD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 14.3 Growth of Metal Multilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 14.4 Ion-Assisted Growth of Metal Multilayers . . . . . . . . . . . . . . . . . . . . . . 532 14.5 Dielectric Layer Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 14.6 Ion-Assisted Reactive Growth of Dielectric Layers . . . . . . . . . . . . . . . 553 14.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

Contributors

Arghya N. Banerjee Nevada Nanotechnology Center Department Electrical Engineering and Computer Science University Nevada Las Vegas, NV 89154, USA banerjee [email protected]

Annemie Bogaerts Research Group PLASMANT Department of Chemistry University of Antwerp Universiteitsplein 1 2610 Wilrijk-Antwerp, Belgium [email protected]

Ra´ ul A. Baragiola Laboratory for Atomic and Surface Physics University of Virginia Engineering Physics Charlottesville, VA 22904, USA [email protected]

James W. Bradley Department of Electrical Engineering and Electronics The University of Liverpool Brownlow Hill Liverpool L69 3GJ, UK [email protected]

S¨ oren Berg The ˚ Angstr¨ om Laboratory Uppsala University Box 534 751 21 Uppsala, Sweden [email protected]

Pascal Briois Laboratoire d’Etudes et de recherches sur les Mat´eriaux les Proc´ed´es et les Surfaces (LERPMS-UTBM) Site de Montb´eliard 90010 Belfort cedex, France [email protected]

Alain Billard Laboratoire d’Etudes et de recherches sur les Mat´eriaux les Proc´ed´es et les Surfaces (LERPMS-UTBM) Site de Montb´eliard 90010 Belfort cedex, France [email protected]

William H. Butler Department of Physics and Astronomy University of Alabama Tuscaloosa, Alabama, USA [email protected]

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Contributors

Guy Buyle Department of Solid State Sciences Gent University Krijgslaan 281/S1 9000 Gent, Belgium [email protected] Kalyan K. Chattopadhyay Thin Film and Nanoscience Laboratory Department of Physics Jadavpur University Jadavpur, Kolkata 700032, India kalyan [email protected] Roger De Gryse Gent University Department of Solid State Sciences Krijgslaan 281 (S1) 9000 Gent, Belgium [email protected]

F. Gaboriau Laplace, University of Paul Sabatier 118 route de Narbonne 31062 Toulouse, France [email protected] M. Gaillard CAPST, SungKyunKwan University 300 Chun-Chun-Dong Jangan-gu, Suwon 440-746, Korea [email protected] C.G. Granqvist Department of Engineering Sciences The ˚ Angstr¨ om Laboratory Uppsala University Box 534 75121 Uppsala, Sweden Claes-Goran.Granqvist@ Angstrom.uu.se

Diederik Depla Gent University Department of Solid State Sciences Krijgslaan 281 (S1) 9000 Gent, Belgium [email protected]

M. Hecq Laboratoire de Chimie Inorganique et Analytique Universit´e de Mons-Hainaut Avenue Copernic 1 7000 Mons, Belgium [email protected]

Steven K. Dew Department of Electrical and Computer Engineering University of Alberta Edmonton, Alberta T6G 2V4, Canada [email protected]

Takahiro Kenmotsu Department of Environmental Risk Management Kibi International University 8 Iga-cho, Takahashi-shi Okayama 716-8508, Japan [email protected]

Samuel D. Ekpe Department of Electrical and Computer Engineering University of Alberta Edmonton, Alberta T6G 2V4, Canada [email protected]

Ivan Kolev Research Group PLASMANT Department of Chemistry University of Antwerp Universiteitsplein 1 2610 Wilrijk-Antwerp, Belgium [email protected]

Contributors

Stephanos Konstantinidis Laboratoire de Chimie Inorganique et Analytique Universit´e de Mons-Hainaut Avenue Copernic 1 7000 Mons, Belgium Stephanos.Konstantinidis@ umh.ac.be Tomas Kubart The ˚ Angstr¨ om Laboratory Uppsala University Box 534 751 21 Uppsala, Sweden [email protected] H. Kupfer University of Technology Chemnitz Institute of Physics 09107 Chemnitz, Germany [email protected] Fr´ ed´ eric Lapostolle Laboratoire d’Etudes et de recherches sur les Mat´eriaux les Proc´ed´es et les Surfaces (LERPMS-UTBM) Site de Montb´eliard 90010 Belfort cedex, France [email protected] Stijn Mahieu Department of Solid State Sciences Gent University Krijgslaan 281 (S1) 9000 Gent, Belgium [email protected] Tetsuya Muramoto Department of Computer Simulation Okayama University of Science 1-1 Ridai-cho, Okayama 700-0005, Japan [email protected]

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Tomas Nyberg The ˚ Angstr¨ om Laboratory Uppsala University Box 534 751 21 Uppsala, Sweden [email protected] Tadayoshi Ono Department of Computer Simulation Okayama University of Science 1-1 Ridai-cho, Okayama 700-0005, Japan [email protected] A. Ricard Laplace, University of Paul Sabatier 118 route de Narbonne 31062 Toulouse, France [email protected] Pierfrancesco Riccardi Laboratorio IIS INFN gruppo collegato di Cosenza Dipartimento di Fisica Universit` a della Calabria 87036 Arcavacata di Rende (CS) Italy [email protected] F. Richter University of Technology Chemnitz Institute of Physics 09107 Chemnitz, Germany [email protected] .de Koen Van Aeken Department of Solid State Sciences Ghent University Krijgslaan 281/S1 9000 Ghent, Belgium [email protected] Haydn Wadley Department of Materials Science and Engineering University of Virginia Charlottesville, Virginia, USA [email protected]

XVIII Contributors

Thomas Welzel Chemnitz University of Technology Institute of Physics 09107 Chemnitz, Germany [email protected]

Xiaowang Zhou Department of Mechanics of Materials Sandia National Laboratories Livermore, California, USA [email protected]

1 Simulation of the Sputtering Process T. Ono, T. Kenmotsu, and T. Muramoto

1.1 Introduction In this chapter, we deal with sputtering of target materials bombarded with energetic particles. In this process, target atoms are removed from the surface by collisions between a projectile and/or recoil atoms produced and the atoms in the near-surface layers of the target material. Sputtering is utilized widely and positively as a useful technique to produce thin films, to make trace impurity analyses of materials of all sorts (e.g., Secondary Ion Mass Spectroscopy), for surface treatment and surface processing, and also for a variety of many other technological applications. However, sputtering plays an undesired role in some cases, on the other hand. The first wall and the divertor of a thermonuclear fusion device, for example, are eroded mainly by impinging plasma ions, which causes the core plasma to be contaminated if sputtered atoms enter it, resulting in insufficient heating of fuel. Therefore, quantitative knowledge on sputtering is required to be determined accurately and compiled for these scientific researches or technological development. Computer simulations of sputtering have contributed greatly to such purposes and to elucidate the pertinent processes. In what follows, we give a quantitative account of and highlight the results obtained mainly by us from computer simulations of sputtering. In Sect. 1.2 we introduce our computer codes (Monte Carlo (MC) binary code, dynamical Monte Carlo code, and molecular dynamics (MD) code) used for sputtering calculations and the outlines of the codes necessary to discuss the results are described. Section 1.3 is concerned with incident-energy dependence and angular dependence of total sputtering yield. The sputtering mechanisms of a target material bombarded with heavy and light ions are outlined. Then, semiempirical formulae for energy dependence are compared with experimental and calculated data. The sputtering mechanism for light-ion bombardment from oblique incidence is mentioned in terms of the knockout process. New semi-empirical formulae to reproduce experimental and calculated data of

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incident-angle dependence of total sputtering yield are introduced. We also present incident-angle dependence of tungsten self-sputtering yield calculated with our computer codes for very low incident energy. The large difference found at glancing angles of incidence between the MC and the MD data is accounted for using the contour maps of deposited energy density on the surface calculated in these simulations. Section 1.4 describes differential sputtering yields with respect to energy and ejection angle of sputtered atoms. In particular, we indicate that the energy spectrum of sputtered atoms from heavy target materials with low-energy light ions can be reproduced well by semi-empirical formulae proposed recently by us. The energy spectra of sputtered atoms calculated with the MC and the MD codes are compared with each other. The components of sputtered atoms with very low energy obtained with the MD are discussed. We present typical angular distributions of sputtered atoms obtained with experiments and calculations. The difference in the angular distributions for heavy- and light-ion irradiation is described. A new semi-empirical formula that can reproduce even the heart-shaped distribution corresponding to heavy- and moderately heavy ions with very low-energy bombardment is presented. In Sect. 1.5, we mention the effect of surface roughness on sputtering with low- and high-energy ions. Low-energy ions incident on a rough surface is equivalent to randomizing incident angle, and sputtering with high-energy ions is affected by the averaged low-density of a rough surface. Section 1.6 goes into sputtering of compound materials. Sputtering of multicomponent targets is complex compared with that of monoatomic targets. In sputtering of multicomponent targets, compositional change near a surface is unavoidable under ion bombardment and depends on target systems. Kinetic and thermal processes occurring in a material result in the compositional change. Ion-fluence dependence of the compositional change for a Cu–Ni alloy was calculated with the dynamic Monte Carlo simulation code which includes both kinetic and thermal processes.

1.2 Computer Simulation Codes Comprehensive reviews on computer simulation codes used widely for sputtering have been done [1–4]. According to how atomic collisions occurring in a solid are modeled, these codes are classified basically into two groups, i.e., MC codes which treat atomic collisions with a MC method in a binary collision approximation (BCA), and classical MD codes which assume that incident projectiles collide with a system of many atoms involved and solve time evolutionally the classical dynamics of the system from knowledge of the interaction forces between particles. Generally, MC codes are suited for simulations in much larger space- and timescales than MD ones. However, as incident energy of ions reaches a low-energy range, e.g., around 100 eV,

1 Simulation of the Sputtering Process

3

the effective interactive region that a moving atom feels widens. Accordingly, BCA breaks down, and MD is required, instead. Since ACAT [5] code, ACAT-DIFFUSE [6] code, DYACAT [7], and a MD code [8] have been described in detail, we describe here only the main features of these codes necessary to discuss the results obtained with these codes. The ACAT and the TRIM.SP [2, 9] codes are of the MC type. While the TRIM.SP code pursues atomic collisions by using a mean free path like many other MC codes, the ACAT code assumes an amorphous target by employing the so-called “cell model,” i.e., an amorphous target is composed of simple cubic cells, in each of which the site of a target atom is chosen stochastically with a lattice constant λ = Ω 1/3 , where the atomic volume Ω = 1/N, N the number density of the target material, and the surface is atomically rough in scale of λ/2. Moreover, one can prepare desired surface roughness by adding more cells on the original surface, as indicated in Fig. 1.1. The DYACAT code is a dynamical version of the ACAT code. In this code, the binary collision events are arranged in order of time they occur. The code can consider collisions between two moving atoms as well as a non-Markov process, which is very important for polyatomic ion bombardment on a solid. The non-Markov process is a random process whose future probabilities are determined by its most recent and all the past values. In the code, the process corresponds with collisions between a moving atom and point defects produced from a simultaneously ongoing collision cascade. Since distant collisions become important in a low-energy range as mentioned above, the code considers many-body collisions among a moving atom and several target atoms located in an effective interaction range by a limited-MD calculation. Therefore, the DYACAT code can be regarded as a MD code-like unique BCA code.

Fig. 1.1. Cell model of a rough surface employed in the ACAT code. Open circles stand for atoms located randomly in cubic cells

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Fig. 1.2. Schematic diagram of the work flow of the ACAT-DIFFUSE routines

The ACAT-DIFFUSE code can calculate sputtering of an amorphous monoatomic and composite materials being irradiated by ions. The code is based on the ACAT code and the DIFFUSE [10] code and its basic concept is schematically shown in Fig. 1.2. The ACAT part of the code calculates the slowing-down processes of implanted projectiles and recoil atoms produced, together with the associated vacancy and range distributions. A large total dose Φ is divided into smaller dose increment ΔΦ during which the target composition is not varied appreciably by incident ions. The bunch of ions corresponding to ΔΦ is assumed to hit the target material simultaneously and to be slowed down instantaneously. Slowdown of ions, together with the associated vacancies and range distributions are calculated in the ACAT part. Mobile atoms, not trapped in trapping sites, diffuse during the time interval of ΔΦ/J (J being the ion flux). The DIFFUSE part copes with the thermal processes of preimplanted and currently implanted ions and recoil atoms produced by solving the diffusion equations numerically. The two routines are iteratively repeated n times, where n = Φ/ΔΦ. As mentioned above, MD determines the dynamics of a system of many atoms by solving Newton’s laws of motion for each particle of the system. A well-known numerical calculation method is Verlet algorithm [11]. The most important factor in MD calculations is atomic interaction force. For a fewbody system, the force can be calculated by quantum dynamics. However, for a large-scale system, the force is described by semi-empirical many-body potentials in view of calculation speed. If a constrained MD simulation on

1 Simulation of the Sputtering Process

5

temperature is required, the Langevin MD (LMD) method [12] may be employed to combine the system with an external heat bath. In sputtering MD simulations, since one can prepare only a finite target, energy dissipation to the external heat bath has to be considered using a LMD method. A large number of computer codes have been developed in the fields of ionimplantation and ion-surface interactions, as discussed partly in this chapter. For convenience, those codes widely used are categorized as to atomic process and type of atomic collisions occurring in solids they employ, as depicted in Table 1.1.

1.3 Total Sputtering Yield 1.3.1 Incident-Energy Dependence of Sputtering Yield A measure of the erosion due to ion irradiation is sputtering yield, which is defined as a ratio of the number of sputtered atoms to the number of incident projectiles. A large amount of experimental and calculated data on incidentenergy dependence of sputtering yield of monoatomic and multicomponent solids for normal incidence have been produced and accumulated [13–19]. The sputtering mechanism can be classified into two categories as schematically indicated in Fig. 1.3. When relatively heavy ions hit a solid surface, they deposit their energy near the surface and a collision cascade develops, resulting in the ejection of target atoms from the surface (mechanism I) [20]. On the other hand, incoming light ions such as H+ and D+ can not produce a collision cascade near the surface because the energy they transfer to target atoms in collisions is not enough to generate a collision cascade. Instead, those ions are reflected from inside the target material and hit near-surface atoms, causing those recoil atoms to leave from the surface if they receive sufficient energy to overcome the surface barrier (mechanism II) [21–23]. As the mass of incident ions becomes lighter, the sputtering mechanism shifts gradually from mechanisms I to II. For ions with intermediate mass, such as Ar+ , both mechanisms I and II contribute to the actual total sputtering yield. Considering the above sputtering mechanism, Yamamura et al. [17] revised their old semi-empirical formulae [24, 25] by interpolating them to propose a new one (hereafter called Yamamura formula) for the energy dependence of sputtering yield Y (E) of monoatomic solids for normal incidence of projectiles, as given by s   Q(Z2)α∗ (M2 /M1 ) Sn (E) Eth Y (E) = 0.042 1− , (1.1) Us 1 + Γ kε0.3 E where E is projectile energy and M1 , M2 are the masses of a projectile and a −2 target atom in a.m.u., respectively, and the numerical factor in units of ˚ A . The factor Γ has the form

Slowing Down Process Thermal Process Binary Collision Approximation Molecular Dynamics Local Mixing Simple MC Dynamical MC MDACOCT24 MD-TOPS26 Model ACAT1 ACOCT5 TRIDYN10 EVOLVE12 PARASOL25 SPUT2SI29 DIFFUSE16 2 6 11 27 28 TRIM MARLOWE T-DYN dynamicMODYSEM SPUT3 PIDAT17 TRIM.SP3 Crystal-TRIM7 SASAMAL13 MOLDYCASK30 QDYN32 SASAMAL4 COSIPO8 MOLDY31 REED33 IMSIL9 DYACAT14 DYACOCT15 ACAT-DIFFUSE19 ACAT-DIFFUSE-GAS20 EDDY23 TMAP418 TRIDYN+PIDAT21 TRIDYN+DIFFUSEDC+YCEHM22 (1) Y. Yamamura and Y. Mizuno, Research report of Institute of Plasma (19) Y. Yamamura, Nucl. Instr. Meth. B 28 (1987) 17. Physics, Nagoya University IPPJ-AM-40 (1985). (20) M. Ishida, Y. Yamaguchi and Y. Yamamura, Thin Solid Films 334 (2) J.P. Biersack and L.G. Haggmark, Nucl. Instr. Meth. 174 (1980) 257. (1998) 225. (3) J.P. Biersack and W. Eckstein, Appl. Phy. A34 (1984) 73. (21) W. Eckstein, V.I. Shulga, J. Roth, Nucl. Instr. Meth. B 153 (4) Y. Miyagawa and S. Miyagawa, J. Appl. Phys. 54 (1983) 7124. (1999) 415. (5) Y. Yamamura and W. Takeuchi, Nucl. Instrum. Methods B29 (22) K. Schmid and J. Roth, J. Nucl. Mater. 313–316 (2003) 302. (1987) 461. (23) K. Ohya and R. Kawakami, Jpn. J. Appl. Phys. 40 (2001) 5424. (6) M.T. Robinson and I.M. Torrens, Phys. Rev. B9 (1974) 5008. (24) K. Yorizane, T. Muramoto and Y. Yamamura, Nucl. Instr. Meth. (7) M. Posselt, Radiat. Eff. Def. Solid. 130/131 (1994) 87. B153 (1999) 292. (8) M. Hautala, Phys. Rev. B30 (1984) 5010. (25) G. Betz, R. Kirchner, W. Husinsky, F. Rudenauer and H.M. (9) G. Hobler, H. Potzl, L. Gong and H. Ryssel, in: Simulation of Urbassek, Radiation Effects and Defects in Solids 130/131 (1994) 251. Semicondoctor Devices and Process, eds. W. Fichtner and D. Aemmer, (26) Javier Dom´ınguez-V´ azquez, E. Pablo Andribet, A. Mari Carmen Vol. 4 (Hartung-Gorre, Konstanz, 1991) p.389. P´ erez-Mart´ın, Jos´ e J. Jim´ enez-Rodr´ıguez, Radiation Effects and (10) W. M¨ oller and W. Eckstein, Nucl. Instr. Meth. B 2 (1984) 814. Defects in Solids 142 (1997) 115. (11) J.P. Biersack, S. Berg and C. Nender, Nucl. Instr. Meth. B59–60 (27) V. Konoplev and A. Gras-marti, Philosophical magazine A71 (1995) (1991) 21. 1265. (12) M.L. Roush, T.S. Andreadis and O.F. Goktepe, Radiat. Eff. 55 (28) M.H. Shapiro and T.A. Tombrello, Nucl. Instr. Meth. B84 (1994) 453. (1981) 119. (29) M.H. Shapiro, T.A. Tombrello and D.E. Harrison, Jr., Nucl. Instr. (13) Y. Miyagawa, M. Ikeyama, K. Saitoh, G. Massouras, S. Miyagawa, Meth. B30 (1988) 152. J. Appl. Phys. 70 (1991) 7289. (30) T. Diaz de la Rubia and M.W. Guinan, J. Nucl. Mater. 174 (1990) (14) Y. Yamamura, Nucl. Instrum. Methods B33 (1988) 493. 151. (15) Y. Yamamura, I. Yamada and T. Takagi, Nucl. Instr. Meth. B37–38 (31) B.L. Holian, The MOLDY program is filed in mass storage at the Los (1989) 902. Alamos National Laboratory (1975). (16) K.L. Wilson and M.I. Baskes, J. Nucl. Mater. 76/77 (1978) 291. (32) D.E. Harrison, Jr. and M.M. Jakas, Radiat. Eff. 99 (1986) 153. (17) W. M¨ oller, Max-Plank-Institute f¨ ur Plasmaphysik, Report IPP (33) Jeong-Won Kang, E.S. Kang, M.S. Son, and H.J. Hwang, Journal 9/44 (1983). of Vacuum Science & Technology B18 (2000) 458–461. (18) G.R. Longhurst, D.F. Holland, J.L. Jones, B.J. Merrill, TMAP4 User’s Manual, EGG-FS-10315, Idaho National Engineering and Environmental Laboratory, 1992.

Table 1.1. Computer codes used in the fields of ion-implantation and ion-surface interactions

6 T. Ono et al.

1 Simulation of the Sputtering Process

7

Fig. 1.3. Schematic diagram of the sputtering mechanisms. The case (I) indicates the mechanism for relatively heavy-ion bombardment, and the (II) that for light-ion bombardment

Γ =

W (Z2 ) . 1 + (M1 /7)3

(1.2)

The surface binding energy of a target material Us , the best-fit values of the dimensionless parameters W (Z2 ), Q(Z2 ), and s are tabulated in [17]. The best-fit values of α∗ are described as a function of a mass ratio M2 /M1 in the following manner:  0.249(M2/M1 )0.56 + 0.0035(M2/M1 )1.5 , M1 ≤ M2 , ∗ (1.3) α = 0.0875(M2/M1 )−0.15 + 0.165(M2/M1 ), M1 ≥ M2 . Eth is sputtering threshold energy and is expressed by the following best-fit functional relation 6.7 Eth , = Us γ 1 + 5.7(M1 /M2 ) , = γ

M1 ≥ M2 , M1 ≤ M2 ,

(1.4)

where γ is energy transfer factor in an elastic collision defined as γ = 4M1 M2 /(M1 + M2 )2 . ke is the Lindhard electronic stopping coefficient [26], and Sn is the nuclear stopping cross section expressed by Sn (E) = 

84.78Z1Z2 M1 TF 1/2 M + M sn (ε), 2/3 2/3 1 2 Z1 + Z2

(1.5)

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2 in units of eV ˚ A per atom. Note that the numerical factor of 8.478 in (21) in Ref. [17] should be corrected to 84.78 as indicated in (1.5). ε is reduced LSS energy [26], and sTF n (ε) is reduced nuclear stopping cross section and is approximated by an analytic fit to the Thomas–Fermi potential [13]. Yamamura formula was applied successfully to the cases of multicomponent target materials as described below. There are other semi-empirical expressions for Y (E), called the “Bohdansky formula” [27] and the “revised Bohdansky formula”, which were used to represent a large number of experimental and calculated data [14, 15, 28]. The discrepancy of the sputtering yields calculated with the TRIM.SP and the ACAT codes is generally small. The above-mentioned revised Bohdansky and Yamamura formulae can represent experimental results within the errors of better than 30% in many cases, but more than 50% in special cases such as H+ and D+ ions onto C for low E in which chemical erosion might take place [14]. The deviations are seen in some cases for incident energies of high keV and in the threshold energy region. Sputtering yield of Cu target materials by 5 keV Ar+ impact calculated with the ACAT and the MD codes is tabulated in Table 1.2. In this MD simulation, the Cu–Cu interactions are described by TB-SMA potential [29] and the Moliere potential [30] for an equilibrium range and a short range, respectively. The Ar–Cu interaction is calculated by the Lennard– Jones [31] and Moliere potentials which are smoothly connected in a similar manner to TB-SMA + Moliere potential. We use a local electronic energy loss model based on the Firsov model [32], which is corrected by the Lindhard– Scharff formula [26]. In a microscopic view, a polycrystalline target looks like a randomly oriented crystal. Therefore, in this MD simulation, we used a random surface model to reproduce the sputtering of a polycrystalline target. In this model, the target crystal is produced by a set of primitive translation vectors, which is oriented randomly for each incidence. It is shown the current simulations could reproduce almost the experimental sputtering yield [33, 34] of polycrystalline and crystal Cu target materials. The energy dependence of sputtering yield for Ar+ ions incident on Cu obtained with experiments, MD calculations and Yamamura formula (1.1) are illustrated in Fig. 1.4. It is clear from the figure that the simulation results and the formula are in good agreement with the experimental values.

Table 1.2. Calculated and experimental sputtering yield for 5 keV Ar+ → Cu at θ = 0◦ Poly. Experiment ACAT MD a Reference [33] b Reference [34]

a

5 5.2 6.8

(111) 9b 9.6

1 Simulation of the Sputtering Process

9

Fig. 1.4. Energy dependence of Ar+ ions incident on Cu obtained with experiments, ACAT and MD calculations and Yamamura formula (1.1)

Yamamura formula (1.1) can represent sputtering yield data of multicomponent materials by replacing Z2 , M2 , and Us in (1.1) withtheir average  values such as Z2  = ci Z2,i , M2  = ci M2,i , and Us  = ci Us,i where ci is the atomic fraction of ith element [35,36]. For threshold energy, however, the maximum value of {Eth,i } is chosen except for oxygen ion bombardment for which the threshold energy is a fitting parameter, where Eth,i is the threshold energy of the monoatomic solid of ith element for the projectile. The experimental and calculated sputtering yields of oxide targets displayed in Fig. 1.5 are represented well by (1.1) with the above replacements. 1.3.2 Incident-Angle Dependence of Sputtering Yield A large quantity of experimental and calculated data on incident-angle dependence of sputtering yield of monoatomic and composite solids have been produced and accumulated [14, 18, 19, 37–43]. Generally, while the incident angle, measured from the surface normal, of ions is small, the yield increases with increasing incident angle because of a cascade developed more close to the surface; after passing the maximum, it decreases with increasing incident angle for large incident angles since the screening effect of neighboring surface atoms begins to prevent incident ions

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Fig. 1.5. Energy dependence of sputtering yields of Si, Ta, SiO2 , and Ta2 O5 for argon and oxygen bombardment, where solid lines correspond to (1.1), and the experimental data of monoatomic solids are from [17] and those of oxide from [36]

from entering the surface; finally it decreases sharply since incident ions are almost all reflected without giving energy to the solid. Taking into account the probability that incident ions can enter the surface of a solid to add a corresponding factor to a formula given by Sigmund [20], Yamamura [43] proposed a semi-empirical formula for incident-angle dependence of sputtering yield expressed by Y (E, θ) = X f exp[−Σ(X − 1)], Y (E, 0)

(1.6)

where X = 1/ cos θ, and the term X f was proposed by Sigmund [20]. Y (E, θ) is sputtering yield for incident angle θ measured from the surface normal. Σ is a physical quantity that is proportional to scattering cross section, and f and Σ are fitting parameters to be determined by adjusting the formula to experimental or calculated data. It was shown that (1.6) can generally reproduce well experimental and calculated data on the incident-angle dependence of sputtering yield with light ions [14, 43]. The validity of (1.6) was checked to indicate that it is acceptable for not-too-low energies, i.e., above 1 keV, though it is not correct for self-sputtering and for heavy projectiles with energies near the threshold energy [28]. However, the second conclusion seems to be not worth special mention, since heavy projectiles with such low energies can not fulfill the necessary condition for developing a genuine collision

1 Simulation of the Sputtering Process

11

cascade near a solid surface that (1.6) assumes as a premise for heavy-ion bombardment [43]. To account for the binding energies of incoming ions, for example, selfbombardment, where large angles of incidence cannot be reached, (1.6) has been improved, resulting in a new formula expressed as c −f c 



θ π θ π , exp b 1 − 1/ cos Y (E, θ) = Y (E, 0) cos θ0 2 θ0 2 (1.7)  π 1 (1.8) θ0 = π − arccos ≥ , 1 + E/Esp 2 with the fit parameters f, b, c [44]. Esp , the binding energy of a projectile, has to be provided. For self-bombardment, it is equal to the surface binding energy of target atoms; for noble gas projectiles Esp = 0; for hydrogen isotopes Esp = 1 eV is assumed. With the use of large datasets [18], new fits threshold energies for different ion-target combinations and for incident angle at several mass ratios are given [18]. The angular dependence of sputtering yield obtained with the experiment and (1.6) for 1 and 8 keV H+ incident on Ni is shown in Fig. 1.6a, and that for 30 eV W+ onto W from (1.6) and (1.7) in Fig. 1.6b [44, 45]. Modeling of surfaces is a very important factor for sputtering simulations. For oblique incidence, a collision cascade tends to be developed more closely to the surface. Thus, the surface-model dependence is expected to be clearly shown up for grazing incidence. Hence, we performed simulations of low-energy ion sputtering to examine it. Again, we make a comparison of incident-angle dependence of sputtering in a low energy range calculated with both ACAT and TRIM.SP codes. Figure 1.7 shows simulation results of the TRIM.SP and the ACAT codes. The TRIM.SP result for grazing incidence approaches a finite yield. To understand this feature, let us mention the surface treatment of the TRIM.SP code. Target atoms are searched in an inner cylinder and coaxial outer ring cylinders with volume Ω and length λ along the pass of a moving atom, where the atomic volume Ω = 1/N, λ = Ω 1/3 and N is the number density of a target material. The bottom faces of the cylinders are called “target disks.” A target atom is set randomly on each target disk. A target atom found outside the surface is disregarded. If several target atoms are found, the one on the inner disk is selected. Thus, the code contains more short-range collisions than the ACAT code. Open squares in Fig. 1.7 indicate sputtering yield derived with a modified ACAT code that forces the projectiles to collide first in the cell at an origin, and that contains more short-range collisions. For grazing incidence, the TRIM.SP result comes somewhere between the ACAT and the modified

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Fig. 1.6. Angular dependence of sputtering yield. (a) Obtained with the experiment and (1.6) for 1 and 8 keV H+ incident on Ni and (b) that for 30 eV W+ onto W from (1.6) and (1.7) [44]

ACAT results. Thus, the large energy deposited by short-range collisions increases sputtering yield in the TRIM.SP case. The knockout process of a surface target atom executed by an incident light ion becomes dominant at large angles [46, 47]. The knockout process at large angles is divided roughly into direct and indirect ones, where a “direct” one means the direct knockoff of a surface atom by an incoming ion and an “indirect” one the knockoff of a surface atom by an incident ion which is scattered just before near the surface by the other target atoms, as illustrated in Fig. 1.8. While only the indirect one works for the not-too-oblique incidence, the direct one plays a major role at grazing angles of incidence. Equation (1.6) does not include the contribution from the direct knockout process to the sputtering yield. Later, Yamamura et al. [47] presented a new formula for light ions where that process was also taken into account, as given by

1 Simulation of the Sputtering Process

13

Fig. 1.7. Incident-angle dependence of sputtering yield for 100 eV W → W. Closed diamonds are the results of TRIM.SP [2, 9], the open squares are the results of the modified ACAT code that contains more short-range collisions

Y (E, θ) = T f exp[−Σ(X − 1)], Y (E, 0)

(1.9)

where T = (1 + A sin θ)/ cos θ, X = 1/ cos θ. The term sin θ included in T reflects the contribution of the direct knockout process. f, Σ, and A are parameters and estimated by adjusting the formula to experimental or calculated data. Most recently three parameters were estimated for the ACAT data of D+ ions incident obliquely on C, Fe, and W materials in the energy ranges from tens eV to 10 keV [48]. Then, the parameters were expressed as a function of incident energy. As displayed in Fig. 1.9, the dependence of normalized physical sputtering yield on incident-angle derived from (1.9) using the functions has been compared with the ACAT data for 200 eV and 1 keV D+ ions incident obliquely on C and with those from (1.9) with not using the functions and from (1.6). We found that the three formulae all agree well with the ACAT data, except for 20–40% difference between the ACAT data and (1.6) at angles of not-too-oblique incidence for 1 keV ions. Figure 1.10 indicates incident-angle dependence of tungsten self-sputtering for incident energy of 100 eV, where the solid squares, triangles, and circles represent ACAT, DYACAT, and MD results. The MD simulation assumed a random surface. The yields for normal incidence obtained with the DYACAT and MD codes are almost equal to 0.013 atoms/ion which was derived by extrapolating the experimental database of sputtering in [17] using (1.9). For grazing incidence (θ ≥ 80◦ ), the ACAT yield approaches zero since the surface normal component of the incident velocity also becomes close to zero. As a

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T. Ono et al.

βout

θ β

δ α (a)

θ

B PA

PB Δθ

A

(b ) Fig. 1.8. Schematic diagram of knockout processes by light ions for oblique incidence. (a) Direct knockout process. (b) Indirect knockout process

result, impulse and deposited energy onto the surface decrease. However the DYACAT and MD yields approach a finite value (≈0.07). Figure 1.11 illustrates a density distribution of deposited energy at θ = 80◦ on the surface, where incident atoms travel from the right side to the left side, and the grid width is 5 ˚ A. A low-deposited energy density for the DYACAT and MD results is seen to expand in the left area. This is caused by complex trajectory development under many-body collisions, resulting in an increase in the deposited energy, which is given by the integration of the deposited energy density. In fact, the deposited energies are 27, 50, and 37 eV for the ACAT, DYACAT, and MD cases. Therefore, enhanced energy deposition by manybody collisions increases sputtering yield. The maximum of the deposited −2 energy density is greater than 1.6 eV ˚ A for the MD case, while it is less −2 than 1.6 eV ˚ A for the ACAT and the DYACAT cases. In the MD simulation, incident energy of a projectile temporarily changes to potential energy among the surface atoms at an impact point because the atoms behave like a cluster under cohesive potential. This is not considered in the limited-MD calculation by the DYACAT code. As a result, the deposited energy density in the MD case is higher than that in the DYACAT case. Since the highly deposited

102 D+ Æ C 200eV 101

100

10−1 ACAT Eq.(8) with functions Eq.(8) Eq.(7)

10−2

0

30 60 Incident angle θ(degree)

90

Normalized Physical Sputtering Yield

Normalized Physical Sputtering Yield

1 Simulation of the Sputtering Process

15

102 D+ Æ C 1keV 101

100

10−1 ACAT Eq.(8) with functions Eq.(8) Eq.(7)

10−2

0

30 60 Incident angle θ(degree)

90

Fig. 1.9. Comparison of the normalized physical sputtering yield vs. incident angle. Left: The legend indicates the curves calculated with the three formulae. The closed circles are the data obtained with the ACAT code for 200 eV D+ onto C. Right: The caption is the same as in (left) except the incident energy is 1 keV. Refer to (right) for legend

Fig. 1.10. Incident-angle dependence of sputtering yield for 100 eV W+ → W. The solid squares, triangles, and circles indicate the results obtained with the ACAT, DYACAT, and MD codes. The curve shows (1.9) with A = 1.32, f = 3.24, and Σ = 1.87

energy density enhances sputtering yield, the MD and DYACAT results are equivalent to grazing incidence, although the deposited energy in the MD case is less than that in the DYACAT case. As indicated in the figure 1.10, this fact influences the yield for incident angles in the range 10◦ ∼ 60◦ .

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Fig. 1.11. Contour map of deposited energy density on a surface for (a) ACAT, (b) DYACAT, and (c) MD simulations of 100 eV W+ → W at θ = 80◦ . Incident atoms travel from the right side to the left side. The grid width is 5 ˚ A. The contour lines in eight colors from dark grey to light grey indicate the densities of 0.003, 0.006, 0.016, −2 0.04, 0.1, 0.3, 0.6, and 1.6 eV ˚ A

1.4 Differential Sputtering Yield While total sputtering yield is required in some cases, differential sputtering yields with respect to energy and ejection angle of sputtered atoms are of importance in other cases. We discuss differential sputtering yields below. 1.4.1 Energy Spectrum of Sputtered Atoms Quite a large quantity of experimental and calculated data on the energy spectrum of sputtered atoms from solids has been produced and summarized [2, 49–59]. It has been well established experimentally, theoretically, and by computer simulations that an energy spectrum of sputtered atoms coming from a welldeveloped collision cascade can be well reproduced by Thompson formula [60].

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17

Such a developed cascade is generated by high-energy medium-heavy and heavy ions. The formula is expressed in terms of differential sputtering yield Y (E) of atoms sputtered with ejected energy E for incoming ions of normal incidence as    1 − (Us + E) /γE0 dE, (1.10) Y (E)dE ∝ 3 E 2 (1 + Us /E) where E0 and E are the energies of incident ions and sputtered atoms and US is the surface binding energy of a target material, γ ≡ 4M1 M2 /(M1 + M2 )2 , where M1 and M2 are the masses of an incoming ion and a target atom. When γE0 Us , E, (1.10) is approximated by Y (E)dE ∝

E dE. (E + Us )3

(1.11)

Normalized Sputtering Yield (Arb. units / ion / eV)

As an example, we show in Fig. 1.12 an energy spectrum of sputtered Fe atoms due to 5 keV Ar+ ion bombardment of normal incidence calculated with the ACAT code, together with the Thompson formula [(1.11)]. It is clear from the figure that the formula reproduces the calculated data quite well. Falcone et al. [61] also derived a similar formula. Equation (1.11) indicates that the energy spectrum does not depend on incident energy and angle, and ejection angle. However, more rigorously, it is expected to be dependent on incident energy and angle. The dependence seems to be more prominent for a cohesive material. To be able to discuss the nature of the energy distributions quantitatively, (1.11) should be generalized to have a form,

0.08 5 keV Ar+

Fe

0.06

0.04 ACAT Thompson

0.02

0

0

10 20 30 40 Energy of sputtered atoms (eV)

50

Fig. 1.12. Energy spectrum of Fe atoms due to 5 keV Ar+ ion bombardment of normal incidence

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Fig. 1.13. Best-fit values of k in (1.12) applied to the energy distribution of sputtered atoms from Nb irradiated normally by Ar+ ions with a variety of incident energies

Y (E)dE ∝

E dE. (E + Us )k

(1.12)

The value of k depends on the interaction potential for elastic collisions. Equation (1.12) peaks at Us /(k − 1). By computer simulations, we obtained an energy distribution at each ejection angle of sputtered atoms from cohesive Nb (Us = 7.75 eV) irradiated normally by Ar+ ions with a variety of incident energies. The best-fit value of k was derived by fitting (1.12) with the energy spectra, and potted in Fig. 1.13 as a function of the ejection angle. It is clear from Fig. 1.13 that a variation in k is around k = 3.0 with a width ±0.2. In addition, for less cohesive target materials like Cu, the dependence of k on ejection angle is estimated to be weaker (see for more discussion [62]). Thus, the Thompson formula can almost represent energy distributions of sputtered atoms from target materials. Deviations from the Thompson formula have been observed in measurements for incident energy below 1 keV, particularly, for light ions [52–54, 59, 62–66]. The peak of an energy spectrum tends to shift to lower energy and the width of the spectrum becomes narrower. As described in Sect. 1.3.1, the sputtering mechanism II becomes dominant in such a low energy range. To explain the energy spectrum of sputtered atoms from heavy target materials by low-energy light-ion bombardment, by assuming that sputtered atoms are only primary recoils that undergo only elastic collisions, Falcone [67] derived a formula Y (E0 , E)dE ∝ dE

E γE0 ln . 5/2 E + Us (E + Us )

(1.13)

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19

Assuming likewise that light-ion sputtering is mainly due to the primary knock-on atoms, i.e., recoils, produced by ions backscattered from inside the material, Kenmotsu et al. [68] obtained a formula on the basis of the transport theory.

2 Tmax E ln , (1.14) Y (E0 , E)dE ∝ dE E + Us (E + Us )8/5 where Tmax = γ(1 − γ)E0 . It is notable that (1.10), (1.13), and (1.14) depend on incident energy of a projectile. Figure 1.14 illustrates the energy spectra of sputtered atoms calculated with the ACAT code for a Fe target material bombarded normally by 50, 100, and 500 eV D+ ions, together with (1.11), (1.13), and (1.14). Each spectrum was normalized to unity at the maximum value. Equation (1.14) predicts quite well the energy spectra for 50 and 100 eV D+ ion bombardment as compared with (1.11) and (1.13). Equation (1.14) differs from the ACAT results for the 500 eV case. The reason for this difference is considered to be due to the neglect

Fig. 1.14. Energy spectra of sputtered atoms calculated with the ACAT for 50, 100, and 500 eV D+ ions incident normally on a Fe target, together with (1.11), (1.13), and (1.14)

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Table 1.3. Peak energies of spectra obtained by fitting (1.11), (1.13) and (1.14) with the ACAT results Incident energy (eV) 50 100 500

Present work (eV)

Falcone (eV)

Thompson (eV)

0.43 1.29 2.77

0.83 1.58 2.22

2.14 2.14 2.14

of an inelastic energy loss in the model. An inelastic energy loss is dominant for 500 eV, because this energy loss increases in proportion to the projectile’s velocity. Therefore, (1.14) is valid for the incident energy of light ions below a few hundred eV. From these figures, the peak position of the ACAT results shows a slight shift toward higher energy as incident energy increases, as pointed out above. Table 1.3 lists peak energies obtained by fitting (1.11), (1.13), and (1.14) with the ACAT results. Both (1.13) and (1.14) indicate that the peak position of the energy spectrum is an increasing function of incident energy. Equation (1.11) is not a function of incident energy. The peak energy of (1.13) is larger than that of (1.14) and the ACAT data for the 50 and 100 eV D+ cases. Most recently, Ono et al. [69] extended (1.14) by taking into account an inelastic and elastic energy loss while keeping the same sputtering mechanism as assumed in [68]. In addition, they assumed that primary knock-on atoms produced by backscattered ions do not lose energy while penetrating to the material up to the surface, instead of the energy loss model employed in [68]. The extended formula is expressed in terms of normalized energy-distribution function and fits well with the data calculated with the ACAT code for 50 eV, 100 eV, and 1 keV D+ ions impinging on a Fe target. Figure 1.15 shows the energy spectra of sputtered particles for 5 keV Ar+ ions incident normally on a random surface. The ACAT result is plotted for comparison. The ACAT result is reproduced successfully by a Thompson distribution [60]. In the high-energy range, the MD result agrees with the ACAT result, which suggests that sputtered particles come from a collision cascade. But, sputtered particles with low-energy are considered to come from a thermal spike, and are comprised of many Cu clusters. The latter fact indicates the main difference between the MD and the ACAT results. It is well known that the flux of particles sputtered from a solid surface bombarded with energetic ions is composed not only of atoms but of molecules and clusters [70]. Compared to the understanding of the properties of monomer emission in sputtering, the understanding of the formation and emission of clusters in energetic ion irradiation is much less incomplete. Both computer simulations and experiments have indicated the importance of fragmentation processes and detailed atomistic mechanism which imparts energy to an aggregation of atoms at the surface and causes its ejection [70–72].

1 Simulation of the Sputtering Process

21

Fig. 1.15. Energy spectra of particles sputtered from a Cu random surface for 5 keV Ar impact. The circles and squares are the MD and ACAT results. The open circles show the monomer yield in the MD result. The solid curve is from the approximate Thompson [60] with surface binding energy Us = 3.1 eV

1.4.2 Angular Distribution of Sputtered Atoms The collision cascade theory predicts that an isotropic distribution of recoil atoms is formed from a well-developed cascade in a target material bombarded normally by energetic projectiles, which results in a cosine-type distribution for the angular distribution of sputtered atoms [20,73]. Experimental data and simulation results indicate that the angular distribution depends on incident energy [74–79]. For low incident energy, a collision cascade is formed, but it is not well developed. As a result, the angular distribution of recoil atoms in the cascade is no longer isotropic, resulting in the angular distribution of ejected atoms being of the under-cosine or heart-shaped type [80]. For high incident energy, on the other hand, an angular distribution tends to become an over-cosine type [75], being different from the cosine shape expected from the collision cascade theory. A qualitative explanation for the over-cosine ejection is that the contribution to sputtering from recoils produced in deeper layers that reach the surface without suffering collisions or that create new recoils (multigenerated) to be sputtered increases as incident energy becomes higher; in this case, the event that the recoil atoms will travel along the direction of the surface normal from the bulk to the surface is highly probable, resulting in an over-cosine distribution [80]. Figure 1.16 shows the types of angular distribution of sputtered atoms, cosine, over-cosine, under-cosine, and heart-shape. The heart-shaped type will be explained later. The angular distributions of sputtered atoms due to high-energy projectiles can be represented by cosn θ, where θ is ejection angle and n is a fitting parameter [75]. Yamamura and Muraoka [78] calculated, with the ACAT code, angular distributions of sputtered recoils produced in the first three atomic

22

T. Ono et al. cosine over - cosine under - cosine heart - shape

0o 0.15

−30o

30o

0.1

60o

0.05 0

90o Yield (atoms / ion / sterad., arb. units)

Fig. 1.16. Types of angular distribution of sputtered atom. Thick solid line, cosine type; doted line, over-cosine type; dashed-dot line, under-cosine type; thin solid line, heart-shaped type Table 1.4. Best-fit values of n obtained by fitting cosn θ with experimental data and the calculated angular distributions of sputtered atoms for normal incidence with the ACAT code Ion Ar Ar Ar Ar Ar Ar Ar

Target Fe Fe Fe Fe Fe Fe Fe

Energy (keV) 0.3 0.5 1.0 2.0 3.0 4.0 5.0

Beat-fit n-values

Exp.

First

Second

Third

Total

0.64 0.87 1.10 1.21 1.28 1.31 1.33

2.06 2.07 2.58 2.60 2.68 2.63 2.65

4.23 4.01 4.44 4.33 4.01 4.54 4.44

0.67 0.97 1.28 1.45 1.53 1.57 1.60

– 0.88 1.0 1.35 1.45 – –

layers of a Fe target irradiated normally by 0.3–5 keV Ar+ ions and derived best-fit values of n for them, as listed in Table 1.4. The best-fit values of n from the first atomic layer derived from the ACAT results are in good agreement with those of experiments, and the degree of the over-cosine distribution, n, is an increasing function of ion energy E. This energy dependence of n is mainly due to the sputtered atoms from the first layer. The angular distribution of sputtered recoils produced at the deeper layers has nearly the same degree of the over-cosine distribution as a function of ion energy. For E > 1 keV, the angular distribution from the first layer shows an over-cosine distribution. Figure 1.17 shows the angular distributions of sputtered atoms for 5 keV Ar+ ions incident normally on Cu (100), (111), (110) and random surfaces. For the former first three surfaces, the angular distribution has some characteristic peak. From Fig. 1.18, one can see that the spot pattern corresponds to the closed-pack direction of the fcc crystal. The focusing in the

1 Simulation of the Sputtering Process

23

Fig. 1.17. Angular distribution of sputtered atoms for 5 keV Ar+ impact on a Cu random surface. The circles, squares, triangles, and diamonds correspond to the results from the random, (111), (110), and (100) surfaces. The solid curve is an over-cosine fit (cos θ)1.8 . The ejection angle is measured from the surface normal

Fig. 1.18. Spot patterns of sputtered Cu atoms for 5 keV Ar+ impact on Cu (100), (110), and (111) surfaces. Radial and rotational axes correspond to the polar and azimuth angles, respectively. Lines in nine colors from dark grey to light grey indicate contour lines from 10 to 90% with increment of 10%, of the maximum differential sputtering yield

closed-pack direction causes these peaks. For the random surface, on the other hand, the angular distribution has no characteristic peak and turns out to be of the over-cosine type of cos1.8 θ. This means the effect of the crystal structure disappeared for a random surface. The following formula can also represent an angular distribution for normal incidence [25, 81, 82]: Y (E, θ) ∝ cos θ(1 + B cos2 θ),

(1.15)

24

T. Ono et al. Kr+

ACAT B = −0.62

0.04

−30o

0o

Cu 100 eV

30o

0.03

60o

0.02 0.01 0

90o

0.5 0.4 0.3 0.2 0.1 0

Yield (atoms / ion / sterad., arb. units) Kr+

ACAT B = 2.0

3 2

−30o

0o

Kr+

ACAT B=0

−30o

0o

Cu 500 eV

30o 60o

90o Yield (atoms / ion / sterad., arb. units)

Cu 5 keV

Kr+

ACAT B = 2.0

30o

4 60o

1

3

−30o

0o

Cu 20 keV

30o 60o

2 1

0

90o Yield (atoms / ion / sterad., arb. units)

90o

0 Yield (atoms / ion / sterad., arb. units)

Fig. 1.19. Angular distributions calculated with ACAT for 100 eV, 500 eV, 5 keV Kr+ ions incident on a Cu target. Also shown are the results of (1.15)

where B is a fitting parameter. The cosine distribution corresponds to B = 0 in (1.15). B > 0 and B < 0 accord with angular distributions of over-, undercosine, and heart-shaped types, respectively. The fitting function, (1.15), is useful compared with the function cosn θ, since (1.15) can describe even a heart-shaped distribution (smaller negative value of B) corresponding to very low incident energy. Figure 1.19 displays angular distributions of sputtered atoms calculated with the ACAT code for 100 eV, 500 eV, 5 keV, and 20 keV Kr+ ions incident normally on a Cu material. Also shown are the results of the fitting formula [(1.15)]. The best-fit values are B = −0.62 for 100 eV, B = 0 for 500 eV, B = 2.0 for 5 and 20 keV. It is quite clear from the figure that above formula [(1.15)] fits very well with these calculated distributions. The value of B increases with increasing incident energy, except for 5 and 20 keV for which it is the same. Thus, it is not a linear increasing function of incident energy. The behavior of B for incident energies from 5 to 20 keV is understood as: for such high incident energy, since main cascades are formed at great depth from the surface, the recoil density created which is responsible for sputtering becomes nearly constant near the surface, resulting in tending to an isotropic distribution. The sputtering mechanism for normally incident light ions is described in Sect. 1.3.1, and that for obliquely incident light ions is also presented in Sect. 1.3.2. While an angular distribution formed by the direct knockout pro-

1 Simulation of the Sputtering Process

25

: few collison part : random part

Fig. 1.20. Schematic representation of an angular distribution of sputtered atoms due to light ions of oblique incidence

cess is expected to be narrow, that by the indirect knockout process becomes broad, because in the latter case light ions at nearly normal incidence are backscattered nearly isotropic from inside a material. This corresponds to the fact that the angular distribution due to light ions does not lead to the under-cosine type, i.e., cosine or over-cosine types [73]. Therefore, as shown in Fig. 1.20, the angular distribution by light ions of grazing incidence consists of a random and a few collision parts. Figure 1.21 illustrates the normalized angular distributions of sputtered atoms from a Ni material bombarded by 450 eV and 1 keV H+ ions at several angles of incidence [39, 75, 76]. The simulation results calculated with ACAT are in good agreement with the experiments except for the case of 450 eV H+ ions for incident angle 80◦ . This exception is considered to come from the effect of the surface roughness effect examined and the low incident energy. A comparison of the distribution for normal incidence with that for 60◦ indicates that, as incident angle becomes larger, a distribution tends to be sensitive to surface roughness, resulting in being broad. Figure 1.22 indicates that the angular distribution by 100 eV H+ ions incident normally on Cu is the over-cosine type (B = 0.5). In contrast, that by 100 eV Kr+ ions incident on Cu becomes the under-cosine type, as illustrated already in Fig. 1.19. Thus, it is understood that a few collision process characteristic of light-ion bombardment leads to the over-cosine type even in low incident energy. Assuming sputtering due to the direct knockout process between an incident ion with energy E and a target atom, and planar surface potential Us , the relation between a recoil angle δ and an ejection angle βout (see Fig. 1.8) is given as [83] cos2 δ sin2 (θ + δ) , (1.16) cos2 βout = cos2 δ − q 2

26

T. Ono et al.

ACAT Experiment [39, 75]

450eV

−30º



−30º

30º

−60º −90º

60º

90º 1

60º

0.5

−30º

0

0.5

1

90º

−30º

0

0.5

1

90º 0

0.5

1

0

0.5

1

0º 30º 60º

90º 0.5

−30º 60º

0.5

−90º 1

30º

−90º 1

90º 0.5

−60º



80º −60º

30º 60º

−30º

−90º 0.5

−90º 1

30º 60º



−60º



−60º

1

1keV

H→Ni

0

0.5

1

0º 30º

−60º

60º

−90º

90º 1

0.5

0

0.5

1

Fig. 1.21. Normalized angular distributions of experimental and ACAT data for 450 eV and 1 keV H+ ions incident for a Ni material for 0◦ , 60◦ , and 80◦

 where θ is an angle of incidence, q = Us /γE, γ = 4M1 M2 /(M1 + M2 )2 , where M1 and M2 are the masses of a projectile and a target atom, respectively. By taking the derivative of (1.16) with respect to δ and set equal to zero, the flowing transcendental equation is obtained: cos3 δ cos(δ + θ) − q 2 cos(2δ + θ) = 0.

(1.17)

In addition, preferential ejection angle βp due to the direct knockout process under q 1 is roughly given as [79, 83]

1 Simulation of the Sputtering Process

H+

ACAT B = 0.5

27

Cu

100 eV 0o

0.0006

30o

−30o

0.0004

60o

0.0002 0

90o Yield (atoms / ion / sterad., arb. units)

Fig. 1.22. Angular distributions calculated with the ACAT code for 100 eV H+ ions incident on a Ni target. Also shown are the results of (1.15) Table 1.5. Comparison with the experimental data and analytical results calculated with (1.18) on the preferential angle βp Ion Target Energy (keV)

Ar Ar Ar Ar Xe Ar Ar Ar Ar Xe H H H H H He

Cu Cu Cu Cu Cu Zr Zr Zr Zr Zr Ni Ni Ni Ni Ni Ni

500 500 150 900 200 500 500 900 150 200 0.45 1 1 4 4 4

Incident angle (degree) 80 85 85 85 85 80 85 85 85 85 80 70 80 80 80 80

βp (degree) Theory (1.18) Exp. 10.2 5.2 5.4 5.2 5.4 10.3 5.3 5.3 5.6 5.5 46.6 45.9 33.1 32.1 21.2 15.9◦

βp = cos−1 {(cos θ + q)(cos θ + 2q)}1/2 .

∼10.0 ∼5 ∼5 ∼5 ∼5 ∼11 ∼9 ∼8 ∼9 ∼5 49.3 43.0 45.0 33.7 21.0 15.0

(1.18)

Table 1.5 [79] lists the experimental data and analytical results calculated with (1.18) for βp . Equation (1.18) is in good agreement with the experimental data for the extensive energy range (0.45–500 keV). Figure 1.23 illustrates the differential sputtering yield of tungsten selfsputtering with incident energy of 100 eV and at θ = 30◦ , 60◦ , and 80◦ . Anisotropy was seen for βp , and βp increases slightly with increasing inci-

28

T. Ono et al.

Fig. 1.23. Contour map of differential sputtering yield for 100 eV W → W at θ = 30◦ , 60◦ , and 80◦ . The radial and rotational axes correspond to the polar and azimuth angle, respectively. Lines of nine colors from dark grey to light grey indicate couture lines from 10 to 90% with inclement of 10%, of the maximum differential sputtering yield

dent angle. This result corresponds qualitatively with the feature that (1.18) indicates.

1.5 Sputtering from Rough Surface It is known that the surface in the BCA code is atomically rough. The random surface in the MD code is also atomically rough. To observe the effect of a surface roughness that is greater than the atomic roughness, we also performed the MD simulations using a fractal surface, which is constructed by the Fourier filtering method [84]. The height z at a horizontal position r = xi+yj is given by the two-dimensional discrete inverse Fourier transform as   z(r ) = {A(k ) cos(k · r ) + B(k ) sin(k · r )}, (1.19) kx =0 ky =0

where k = kx i + ky j is the wave vector, the spectral density is S(k ) = A2 (k ) + B 2 (k ) ∝ (kx 2 + ky 2 )D−4 , and D is the fractal dimension. Figure 1.24 shows the incident-angle dependence of tungsten self-sputtering yield with incident energy of 1 keV calculated with the MD code. Open circles indicate sputtering yields for the fractal surface with D = 2.15 and the RMS roughness of about 2λ, where λ is defined in Sect. 1.2. We found that the rough surface reduces incident-angle dependence because the local surface normal vectors are randomized for the rough surface which is shown in Fig. 1.25a. This is equivalent to randomizing incident angle. The randomization is derived from low wave number components of (1.19) because we should consider the size of the local surface that includes a collision cascade. In Fig. 1.26, we show, with the ACAT code, the incident-angle dependence of Cu self-sputtering yield with incident energy of 1 keV. From Fig. 1.26, one can find that ions normally incident on rough surfaces (D = 2.2, 2.5) gives

1 Simulation of the Sputtering Process

29

Fig. 1.24. Incident-angle dependence of sputtering yield for 100 eV W → W. The open circles correspond to the MD results on a rough surface

Fig. 1.25. Schemata of rough surface sputtering. (a) Low-energy ions incident on a rough surface is equivalent to randomizing incident angle. (b) High-energy ions incident on a rough surface is equivalent to a low-density surface

30

T. Ono et al.

Fig. 1.26. Incident-angle dependence of sputtering yield for 1 keV Cu → Cu with D = 2.0, 2.2, and 2.5, respectively

rise to lower sputtering yield than that on a smooth surface (D = 2.0). Since a high-energy collision cascade develops in a wide area of the surface, highenergy sputtering is affected by the averaged low-density surface which is schematized in Fig. 1.25b, which reconfirms the results obtained by Kenmotsu et al. [85] about low-density effect of the fractal surface on sputtering. This effect was not found in low-energy sputtering as shown in Fig. 1.24. When a low-energy projectile collides with surface atoms, it is influenced by them, and feels the direction of the local surface normal. Thus, randomization of a surface normal is important for low-energy sputtering.

1.6 Sputtering of Compound Targets When a multicomponent material is bombarded strongly with an ion beam, near-surface compositional alteration is an unavoidable phenomenon which results from a combination of several kinetic (preferential sputtering, collision mixing, etc.) and thermal processes (radiation-induced diffusion, Gibbsian segregation and radiation-induced segregation, etc.). The driving forces of surface segregation are strain energy and difference in the chemical potentials between the first and the second layers. The former is radiation-induced segregation, and the latter known as Gibbsian segregation. The relative importance of each process depends on an alloy system, the elements, energy and flux of bombarding ions, and irradiation temperature [86]. To analyze the surface change of multicomponent targets under ion bombardment, Auger electron spectroscopy (AES), ion scattering spectroscopy (ISS), Rutherford backscattering spectroscopy (RBS), secondary ion mass spectroscopy (SIMS), and X-ray photoelectron spectroscopy (XPS) are used.

1 Simulation of the Sputtering Process

31

Kimura et al. [87] made an experiment of 6 keV As2 + ion implantation on Si wafers and measured the depth profile up to 10 nm with high-resolution RBS. These analytical techniques are useful to estimate surface concentration and the compositional change near a surface, but the real surface concentrations of constituents are possibly different from those measured by respective technique, because of the different resolutions of these techniques. Therefore, it is difficult to estimate the compositional change near the surface under ion bombardment for multicomponent targets experimentally. At temperatures lower than room temperature, the surface compositional change of most alloys can be explained mainly by kinetic processes, because the time constants of the processes such as the recession speed due to sputtering are much faster than those of thermal processes. At high temperatures, since the time constants of the latter processes are comparable to or longer than those of the former processes, the phenomenon becomes more complex. The first study of bombardment-induced alterations of surface composition at high temperature was carried out on a Cu−Ni alloy by Shimizu et al. [88]. After this work, there were a lot of measurements on surface and/or surface compositional changes in binary alloys by the AES and ISS techniques [89, 90]. Lam et al. [91] measured changes on the surface and surface compositions of Ni–40 at.% Cu alloys at temperatures between 25 and 700◦C, using the ISS technique. They found that the steady-state surface composition was noticeably temperature-dependent above 400◦ C, which was interpreted in terms of the significant contribution of the second atomic layer to the sputtering flux. There are two typical theoretical approaches to understand alterations of a surface composition. One is an analytical approach which is based on a diffusion equation [92–95] or on a kinetic balance equation [96]. The other approach is Monte Carlo simulation [97–99]. Ho [92] introduced radiation enhanced diffusion into the model proposed by Patterson and Shirm [93]. Wiedersich et al. [94] presented a theory for radiation-induced segregation by introducing a concept of the preferential migration of vacancies and interstitial of a certain atom. Kelly [95] suggested that surface segregation might be important for the interpretation of compositional changes at the surface. Sigmund and his coworker [96] investigated multicomponent sputtering theoretically on the basis of the kinetic balance equation. For multiple-component targets, total sputtering yields, Y , are defined with its partial sputtering yields of constituents Yi as follows [86]:  Yi (1.20) Y = i

At sufficiently low temperature, where thermal processes are not important, the ratio of sputtering yields for a binary target is expressed by  Y1 c2 c1 , (1.21) = Y2 c2 c1

32

T. Ono et al.

where Y1 and Y2 are partial sputtering yields of the two constituents, and c1 , c2 , c1 and c2 are surface concentrations of each element before the ion bombardment and in the steady state. If a change in the surface concentration for a binary target due to sputtering does not occur, the ratio Y1 /Y2 is equal to unity. The ratio Y1 /Y2 for a binary target is usually determined from (1.21) by measuring the surface concentrations of constituents before the ion bombardment and in steady state. The measured sputtering yield ratio of alloys is listed in Table 1.6 [100]. A theory of sputtering on multiple component targets was proposed by Sigmund [73,96,101]. Partial sputtering yield of the i-component for a binary infinite medium is given as: Yi = ci Λi FD (E, θ, 0), where Λi

= Λi

(U )i Ui

1−2mi

i = 1, 2,

(1.22)

1 Gi . ci (G)i

Table 1.6. Sputtering yield ratio of alloys. Also listed are atomic weight ratio and ratio of sublimation energies of constituent atoms of pure metals [100] System (A−B)

Mass ratio (MA /MB )−1

Sputtering yield ratio (YA /YB )

Ration of sublimation energy in eV (UA /UB )

Ion

Method and references

Al−Cu Al2 −Au Al−Au2 Si−Ni Si−Pt Si−Pt2 Au−Cr Cu−Ni Cu−Ni Ni−Pt Pd−Ni Cu−Pd Cu−Pt Cu−Au Cu3 −Au Cu3 −Au

0.42 0.14 0.14 0.48 0.14 0.14 3.3 1.1 1.1 0.3 1.7 0.63 0.32 0.32 0.32 0.32

2.1 1.9 1.3 1.6 2.1 1.6 1.4–2.0 1.7 1 1.5–1.9 1.5–1.7 1.5–1.6 1.6–3.0 1 1.1 2.3–2.8

3.36/3.52 3.36/3.80 3.36/3.80 4.70/4.46 4.70/5.86 4.70/5.86 3.80/4.12 3.52/4.46 3.52/4.46 4.46/5.86 3.91/4.46 3.52/3.91 3.52/5.86 3.52/3.80 3.52/3.80 3.52/3.80

Xe (1 keV) Ar (40 keV) Ar (40 keV) Ar (40 keV) Ar (40 keV) Ar (40 keV) Ar (0.5–2 keV) Ar (0.5–2 keV) Ar (3 keV) Ar (2 keV) Ar (0.5–2 keV) Ar (2 keV) Ar (2 keV) Ar (1 keV) Ar (40 keV) Ar (0.5–5 keV)

Ag−Pd Au−Pd Ag−Au Ag−Au Ag−Au

1.0 2.0 0.56 0.56 0.56

1.5–1.6 1.0–1.4 1.7–1.8 1.2 1

2.97/3.91 3.80/3.91 2.97/3.80 2.97/3.80 2.97/3.80

Ar (2 keV) Ar (2 keV) Ar (1 keV) Ar (40 keV) Ne (1.5 keV)

RBS [99] RBS [100] RBS [100] RBS [100] RBS [100] RBS [100] AES [101] AES [102] ISS [103] AES [104] AES [105] AES [106] AES [107] AES [107] RBS [100] Electron diffraction [108] AES [109] AES [109] AES [107] RBS [100] ISS [110]

1 Simulation of the Sputtering Process

33

Here ci is atomic concentration of the binary alloy. The relation between c1 and c2 is expressed by c1 + c2 = 1 for the binary alloy. The quantity F (E, θ, 0) is the deposited energy per unit depth in the alloy surface, mi exponent in the inter-atomic power law potential for an elemental the i-target, Λi a material constant for an elemental i-target, (U )i surface potential in an elemental itarget, Ui and surface potential for an i-atom of the alloy. The factor Gi /(G)i is ratio of the flux functions between i-atoms in the alloy and elemental itarget. Gi and (G)i are densities of moving target atoms in a collision cascade generated by incident ions or recoil atoms for the alloy and an elemental target, respectively. For roughly equal masses M1 , M2 of the constituents, one may approximate Gi /(G)i ≈ ci . Using (1.22) and assuming m1 = m2 , the ratio of sputtering yields for the binary alloy is expressed by c1 Y1 = Y2 c2



M2 M1

2m

(U )2 (U )1

1−2m .

(1.23)

Heat of sublimation is generally taken as an approximate value for surface binding energy of elemental targets [73]. It is difficult to estimate surface binding energy of each element of multiple-component targets, because the energy depends on the surface concentration of the elements and surface coordination numbers. For a random binary alloy the surface binding energy U1 is derived as a function of surface concentration of the constituents and the nearest-neighboring bond strengths [95, 102]: U1 = −Zs [c1 U11 + c2 U12 ],

(1.24)

where ZS is surface coordination number, and U11 and U12 are bond strengths. To estimate the ion-fluence dependence of depth profiles and compositional changes during ion bombardment, there are several Monte Carlo simulation codes that are based on the BCA. The EVOLVE code was developed by Roush et al. [97]. The TRIDYN was made by M¨oller and Eckstein [98], and the ACAT-DIFFUSE was proposed by Yamamura [6]. The ACAT-DIFFUSE code calculated compositional changes near a surface and depth profiles in Ni–40at. % Cu alloys at various temperatures (25–500◦ C) when sputtering is taking place with normal incident-beam of 3 keV Ne+ ions [103]. The compositional change near surface calculated with the ACAT-DIFFUSE code is shown below. As Gibbsian segregation, the ACAT-DIFFUSE code uses the Darken description [104]. The application of the modified Darken equation leads to the following coupled rate equations [103–105]:

34

T. Ono et al.

  ⎫⎤ ⎧ φ B1 ⎨ ⎬ X 1 − X 1 1 ⎣ ΔG1 + ln   ⎦, ⎩ X φ 1 − X B1 ⎭ RT 1 1  ⎫   ⎫⎤ ⎧ ⎡ ⎧ B B ⎨ X1 2 1 − X1 1 ⎬ a 2 ⎨ X1B1 1 − X1φ ⎬ 2 ⎣ln   −   ⎦, ln ⎩ X φ 1 − X B1 ⎭ ⎩ X B1 1 − X B2 ⎭ a1 ⎡

∂X1φ ∂t

=

Ξ1 X1B1 a21

Ξ1 X1B2 ∂X1B1 = ∂t a22

B

Ξ1 X1 i+1 ∂X1Bi = ∂t a2Bi+1

1

1

.. .

⎫ ⎧ B  2 ⎨ X1 i+1 1 − X1Bi ⎬ a Bi+1 ⎣ln   − ⎩ X Bi 1 − X Bi+1 ⎭ aBi 1 1   ⎧ ⎫⎤ ⎨ X1Bi 1 − X1Bi−1 ⎬   ⎦, ln ⎩ X Bi−1 1 − X Bi ⎭ 1 1 ⎡

1

1

(1.25)

.. . where X1φ and X1Bi are fractional concentrations of species 1 in the topmost and the ith layers, respectively. In deriving above equations we use chemical potential terms for an ideal binary alloy, i.e., μ = μ0 + RT ln X, where μ0 is standard chemical potential, R universal gas constant, and T is temperature. The coefficient Ξ1 corresponds to a diffusion constant given by Ξ1 = M1 RT . In this model segregation takes place from the bulk B to the surface Φ, because 0φ 0B 0φ 0B 0B μ0φ 1 = μ1 . Segregation energy is given by ΔG = μ1 − μ1 −2 +μ2 . Abovecoupled equations lead to the Langmuir–McLean relation in a stationary phase [91]

X1B ΔG X1φ . (1.26) = exp kT 1 − X1B 1 − X1φ Lam et al. [91] measured the surface compositional change of Ni–40 at.% Cu alloy bombarded normally with flux of 3.75×1013 ions cm−2 s−1 of 3 keV Ne+ ions. The simulations were done under the same conditions as the experiments. When the specimen is heated to a high temperature, a thermodynamic driving force gives rise to Cu enrichment at the alloy surface even in the absence of sputtering. Lam et al. gave the following expression for segregation energy ΔG of a Ni–40 at.% Cu ally at high temperature (500◦ C): ΔG = −2.6kT + 0.42.

(1.27)

From (1.26) and (1.28), we have the reasonable estimation of the surface atoms fraction Cuφ and Niφ at the topmost layer before ion bombardment which is denoted by [Cux Ni1−x ] in the ensuring discussions. In the calculations with ACAT-DIFFUSE we considered only one trap site which is vacancy and the basic diffusion equation is for interstitial diffusion. The respective activation

1 Simulation of the Sputtering Process

35

Fig. 1.27. Time evolution of the Cuφ /Niφ ratio in the first layer during 3 keV Ne+ ion sputtering of a Ni–40 at.% Cu alloy at 400◦ C, where the ion flux is 3.75 × 1013 ions cm−2 s−1 . Solid symbols mean the ACAT-DIFFUSE data and open symbol Lam’s experimental data [91]. The parameters of cases A, B, C and D are listed in Table 1.7

energies of Ni and Cu are assumed to be 0.93 and 0.88 eV for the interstitial diffusion, and the vacancy diffusion is considered indirectly by solving the rate equation of trapped atoms concentration, where the trapping energies of Ni and Cu are set to be 2.03 and 1.76 eV, respectively. We use smaller detrapping energies than the diffusion activation energies in the material with no irradiation, because trapped atoms may be in an excited state compared with a lattice atom. This leads to the radiation-induced diffusion. Figure 1.27 shows the calculated Cuφ /Niφ ratio in the topmost layer at 400◦ C as a function of bombarding time. We made simulations for four cases in which the physical parameters used are listed in Table 1.7, where ΔG0 is the segregation energy before bombardment. Since (1.28) can not be applied to T < 500◦C, both ΔG0 and [Cux Ni1−x ] are determined from the extrapolation of the experimental data [91]. The ion-fluence dependence of Cuφ /Niφ ratio of case A is in agreement with the experimental data only at the early stage (t < 10 s), and the parameters of case A give the large steady-state Cuφ /Niφ ratio compared with the experiment. The segregation speed of case B is very small and it gives good agreement with the experimental steady-state Cuφ /Niφ ratio, but the ratio drops more rapidly than the experiment. There are experimental evidences [106, 107] that segregation energy under irradiation is smaller than that before bombardment. Referring to ion-fluence dependence of the calculated Cuφ /Niφ ratios for case A and B, we adopt the

36

T. Ono et al.

Table 1.7. Initial surface atoms fractions Ξ, ΔG0 , ΔGS and decay constant τ for cases A, B, C and D Parameters Initial surface atoms fractions Ξ (cm2 s) ΔG0 (eV) ΔGst (eV) τ (ions cm−2 )

Case A

Case B

Case C

Case D

[Cu0.99 Ni0.01 ]

[Cu0.99 Ni0.01 ]

[Cu0.99 Ni0.01 ]

[Cu0.4 Ni0.6 ]

1.0 × 10−15 0.175 0.175 –

3.5 × 10−17 0.175 0.175 –

1.0 × 10−15 0.175 0.050 2.0 × 1016

1.0 × 10−15 0.175 0.050 2.0 × 1016

Fig. 1.28. Time evolution of the Cuφ /Niφ ratio at first layer during 3 keV Ne+ ion sputtering of a Ni–40 at.% Cu alloy at various temperatures, where the ion flux is 3.75 × 1013 ions cm−2 s−1 . Solid symbols denote the ACAT-DIFFUSE data and open symbols denote Lam’s experimental data [91]. The parameters used in the calculation are listed in Table 1.7. The cross marks denote the ACAT-DIFFUSE simulation of T = 100◦ C with the initial atom fraction [Cu1.0 Ni0.0 ]

following ansatz: ΔG = max[ΔG0 exp(−Φ/τ ), ΔGst ],

(1.28)

where ΔGst is segregation energy in a steady state. As shown in Fig. 1.27, the ion-fluence dependence of the calculated Cuφ /Niφ ratio for case C is in good agreement with the experiment in the whole range considered. To show the dependence of the initial surface atoms fraction on Cuφ /Niφ ratio, we use the bulk atoms fraction as an initial surface atoms fraction in case D. Before 50 s, Cuφ /Niφ ratio is strongly dependent on the initial surface atoms fraction, but, after 50 s, the Cuφ /Niφ ratio shows the same dependence as that of case C.

1 Simulation of the Sputtering Process

37

Table 1.8. Best-fit parameters of the initial surface atoms fractions Ξ, ΔG0 , ΔGS and decay constant τ for various temperatures Parameters

Temperature 100◦ C

200◦ C

300◦ C

400◦ C

550◦ C

Initial sur- [Cu0.4 Ni0.6 ] [Cu0.79 Ni0.21 ] [Cu0.89 Ni0.11 ] [Cu0.99 Ni0.01 ] [Cu0.92 Ni0.08 ] face atoms fractions −1 6.0 × 10−16 1.0 × 10−15 1.0 × 10−14 Ξ (cm2 s ) 2.0 × 10−17 1.0 × 10−16 ΔG0 (eV) 0.003 0.006 0.120 0.175 0.200 0.003 0.006 0.045 0.050 0.080 ΔGst (eV) 2.0 × 1016 8.0 × 1016 τ (ions – – 2.0 × 1016 −2 cm )

The Cuφ /Niφ ratio calculated with the ACAT-DIFFUSE code at various temperatures is displayed in Fig. 1.28, where solid symbols mean the ACAT-DIFFUSE data and open symbols Lam’s experimental data [91]. The parameters used in the calculation are listed in Table 1.8, where ΔG0 and the initial surface atoms fraction [Cux Ni1−x ] are also determined from the extrapolation of experimental data. In low temperature ( 2U ). For most singly charged ions, PEE does not occur at surfaces that have been exposed to the atmosphere, since the relation ε > 2U is not generally satisfied for adsorbed molecules. In kinetic electron emission (KEE), electrons are excited as a consequence of time-varying Coulomb interactions between the nuclei and electrons of the incident particle and the surface, within the constraints of the Pauli exclusion. Here, several mechanisms are possible, depending on the type of surface. In metals, projectiles can interact with loosely bound (“conduction”) electrons and shallow core levels (such as the d-band in light transition metals). Deeper core levels are not accessible at the energies of interest. Binary collisions of “free” valence electrons of the targets with the screened Coulomb field of the projectile are the dominant excitation mechanism for light projectiles (H, He, Li) on metals. A collision with a nearly free electron cannot result in a large energy transfer because the mass mismatch between the heavy particle and the electron prevents transferring sufficient momentum. The maximum energy transfer to an electron by a much heavier atomic particle occurs in head on collisions, where the electron gains twice the projectile velocity after a single scattering. The resulting energy transfer is T = 2mv(v + ve ), where v and ve are the velocity of the projectile and the target electron before the collision, respectively, and m the electron mass. Therefore, there is a minimum impact velocity that will result in a free electron excited to a state with energy above the vacuum level, when T = W , given by [12]   vF 2W vth = 1+ −1 , (2.1) 2 mvF2 where W is the work function, vF the Fermi velocity, and m the electron mass. This corresponds to 1.5–3 × 107 cm s−1 (117–470 eV amu−1 ) for most metals, consistent with extrapolation of the experimental results for light projectiles [12]. Deviations from (2.1) contain information on the velocity distribution of electrons at the surface [13]. For heavy projectiles, the observed threshold velocities are an order of magnitude lower than (2.1). This is due to an additional electron excitation mechanism. This is the electron–electron interactions during the interpenetration of the electron clouds of the projectile and one of the target atoms which promote electronic levels directly into the ionization continuum or through autoionizing states [14].

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A direct demonstration of the importance of excitations in such binary atom–atom collisions below the free electron threshold of (2.1) was made by Rabalais et al. [15] who studied the impact parameter dependence of KEE in collisions of 4 keV Ar+ with Ni(110) and found that electron emission required a minimum impact parameter in a collision below 0.3 ˚ A. In binary collisions between projectile and a target atom, the minimum possible threshold for KEE allowed by energy conservation is when the center-of-mass energy equals U . However, quite above this threshold the electron yield is usually undetectable above the background of other secondary processes. 2.2.2 Separation of PEE and KEE PEE and KEE are separable in two limiting cases. When Ecm < U , and ε > 2U , only PEE can occur. If, on the other hand, ε < 2U and Ecm > U , electrons can only be ejected by the kinetic mechanism. In KEE, the threshold condition is when the maximum energy transfer is equal to U . Energy transfer center-of-mass energy Ecm has to exceed U . PEE does not require the motion of the projectile but can be affected by it, whereas KEE can be affected by the degree of excitation of the projectile. Electron emission statistics have been used to separate PEE and KEE for singly charged ions, based on the fact that PEE generally results in at most one electron being emitted [16]. 2.2.3 Electron Transport and Escape into Vacuum At low projectile impact energies, many of the electrons are emitted from the surface layer and it is difficult or incorrect to separate scattering inside the solid and transmission through the surface barrier. Inside the solid, excited electrons undergo a cascade of collisions until their energy is degraded into heat or stored in long-lived excited states. Those excited electrons which are directed outward may cross the surface of the solid and escape before being thermalized, giving rise to electron emission. Therefore, the energy distribution of emitted electrons does not represent an equilibrium situation. An important characteristic of the low-energy ion impact case discussed here is that the vast majority of the initially excited electrons have very small kinetic energies, insufficient to excite additional electrons above the vacuum level. That is, electron multiplication [17] is typically unimportant.

2.3 Electron Yields Experiments involve measuring the dependence of electron emission on different factors. These include the projectile and target properties and the bombardment geometry. Important properties of the projectile are its excitation and ionization state, its mass, and whether it comes as a single atom,

2 Electron Emission from Surfaces Induced by Slow Ions and Atoms

47

a molecule, or a cluster of atoms. Relevant target properties are the elemental composition and atomic and electronic structure in the surface region, the magnitude of surface electric fields, the temperature, and, for magnetic materials, the degree of magnetization. The relevant geometrical factors in experiments include the angle of incidence with respect to the surface and to the possible preferred crystallographic planes. Electrons can absorb a much larger energy transfer (even a substantial fraction of the center-of-mass energy in the collision) if they are not free, but bound to atoms. In this case, the treatment of the collision is similar to that of ionization in gas-phase collisions, where electronic excitations occur due to electron–electron interactions which promote electronic levels directly into the ionization continuum or through autoionizing states [5]. Atom–atom collisions can excite electrons from the projectile or the target and are thought to be the main mechanism ejecting electrons during impacts on gas-covered surfaces [18]. The atomic collisions considered here can occur between a projectile atom and a target atom, between fast target recoils and other atoms, or even between atoms in the projectile in the collision spike formed on impact.

2.3.1 Dependence of the Electron Yields on Ion Velocity Early measurements using atomic ions have shown that, at low velocities, the KEE yields have the velocity dependence γ = γo (v − vth ),

(2.2)

for ν not too close to νth . The extrapolated value of νth is ∼4.5 × 106 cm s−1 (10.6 eV amu−1 ) for gas-covered surfaces [19, 20], roughly independent of the type of bombarding ions. The implicit acceptance of (2.2) has influenced the way data have been plotted in the literature. One can ask whether the extrapolated value νth is affected by a limited experimental sensitivity or the way the data are plotted. KEE yields induced by Xe+ ions on atomically clean gold [21, 22] depart from the linear behavior of (2.2) and do not have a definite threshold down to ν = 1.2 × 106 cm s−1 (100 eV). KEE down to 1.5 × 106 cm s−1 (160 eV) has previously been observed by Waters [23] for Cs on W who also found that gas adsorption increases the electron yields by three orders of magnitude at these low velocities. We will return to this last point later, because of its significance for discharges. The model of KEE resulting from binary low velocity atomic collisions suggests that the projectile velocity be measured in terms of the characteristic velocity aΔE vc = , (2.3)  which follows from Massey’s [24] adiabatic criterion, where a is the range of the interaction causing the transition with energy defect ΔE. At low velocities,

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this suggests a similar dependence that has been proposed for ionization collisions in the gas phase, using a straight-path approximation [25, 26], γ = γo exp(−vc /v),

(2.4)

or a sum of exponentials if several processes are present with different values of νc . Breaks in the plot of log γ vs. 1/ν could then be used to identify different excitation channels. In the limit of very low velocities, (2.4) should break down for at least two reasons (1) the inability to reach interatomic distances where the coupling to the continuum is sufficiently strong and (2) the existence of an absolute binary threshold when the center-of-mass energy equals U . Situations close to this absolute binary threshold have been identified in the impact of relatively light ions on gas-covered surfaces [18], from comparisons with gas-phase ionization cross sections. Beyond the binary collision approximation, the minimum possible threshold results from energy conservation, when the projectile energy equals U . For this to occur, the lattice has to absorb all the momentum of the projectile and all the available energy has to go into the excitation of a single electron. This is very unlikely for heavy particle impact, since there is a large number of alternative pathways for the dissipation of the incident energy. 2.3.2 Electron Energy and Angular Distributions Ferr´on et al. [27] have measured electron energy distributions for 4 keV Ar+ on clean and oxidized Al and Mo surfaces. The peak of the distributions occurs at a few eV and there are relatively fewer high-energy electrons from oxidized surfaces. Other collision systems are characterized by electron energy distributions with sharp or broad peaks characteristic of specific processes, such as autoionization, plasmon decay, or the decay of shallow core levels [5]. Recently, the method of factor analysis of energy distribution curves has been used to identify different mechanisms of ion-induced electron emission [28]. The angular distribution of ejected electrons has, in general, a cosine dependence on angle with respect to the surface normal. Structure is superimposed on this dependence in the case of experiments on single crystals, due to the diffraction of the electrons [29]. 2.3.3 Electron Emission from Contaminant Surface Layers Figure 2.1 depicts the results of analytical expressions given by Phelps and Petrovic [30] for electron yields γi (γa ) from metal surfaces under the impact of Ar+ ions (Ar neutrals) both for clean and contaminated (dirty) surfaces, based on a compilation of experimental data. This graph serves to illustrate several points. For Ar+ on clean surfaces, the electron yields are constant at very low energies, indicating the predominance of PEE. In contrast, Ar neutrals,

2 Electron Emission from Surfaces Induced by Slow Ions and Atoms

49

Electron emission yields

101

100

10−1

Ar+ 10−2

Ar0 10−3 10

100

1000

10000

Ion Energy (eV)

Fig. 2.1. Electron emission yields for argon ion and atom bombardment for clean (dashed curves) and dirty (continuous curves) metal surfaces (adapted from [30])

which carry no potential energy, do not produce PEE. The electron yields for neutrals are exclusively due to KEE and increase rapidly with energy above a threshold which is below 40 eV. The KEE component of Ar+ is essentially the same as that of Ar, due to the spatial separation of the PEE and KEE mechanisms. PEE occurs outside the surface by Auger neutralization producing neutral Ar which then enters the solid producing KEE. For dirty surfaces, the PEE yield by Ar+ is suppressed, a general property of PEE caused by the stronger bonding of electrons in adsorbates compared to metals. Strikingly, the KEE yield, seen clearly for neutral Ar, is much larger for dirty than for clean surfaces and the ratio decreases with increasing projectile energy, as reported earlier [31]. This behavior of dirty surfaces is similar to that of insulators which also have larger KEE yields than for clean metals or semiconductors. In most of the literature, this is attributed to a larger mean electron escape depth due to the reduced electron scattering for electron energies below the band gap. However, this effect is not important at the low energies appropriate to electron discharges, where ion penetration is very shallow. A more important factor is the easier electron escape from insulators due to a reduced surface barrier. Comparative experiments on the adsorption of oxygen on clean Al and Mo surfaces showed very large effects on the yields [27] with γ increasing for Al and decreasing for Mo. Since one can expect a similar electronic configuration in the oxygen in both cases, the reason for the large differences in the two materials is not likely to result from differences in direct excitation. Rather, it was argued that yields were mainly affected by the change in the surface barrier but not necessarily related to changes in the work function.

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2.4 The Role of Ion-Induced Electron Emission in Glow Discharges The pivotal role of electron emission yields in electrical discharges has become better understood in the last decades. Recent complex numerical models [32–39], categorized as fluid, kinetic (Monte Carlo), and hybrid (combination), have shown that the modeled discharge characteristics are seriously influenced by the accuracy of the input data. Models require realistic cross sections for electron, ion, atom and photon collisions with the gas, as well as realistic probabilities for electron, ion, atom and photon interactions with electrode surfaces. Whereas the data for modeling the relevant gas-phase processes are relatively well known [30, 39, 40], those describing electrode processes are often uncertain, such as the electron emission yields from cathode materials under actual gas discharge conditions. Therefore, there has been significant effort spent lately in understanding issues associated with secondary electron emission processes in modeling the characteristics of gas discharges [30,40–45]. A comparison with measured voltage–current characteristics is generally the first way to check the results of model calculations. An important parameter is the ratio of the electron to ion current densities at the cathode surface, γ  = Je,tot /Ji,tot , called the apparent or effective secondary electron emission yield per ion. The derivation of this quantity is not trivial, since several particles (positive ions, metastable atoms, fast neutrals, and photons) contribute to electron emission from the cathode and their relative importance changes with the operating parameters and conditions of the discharge. Furthermore, as compared with ion beam measurements of electron emission yields of heavily sputtered samples in ultrahigh vacuum, cathode materials operated under discharge conditions can have significantly different secondary emission yields [30]. Nevertheless, until recently, it was common practice to assume a value of γ  independent of discharge conditions. An example of the effect of the choice of γ  is shown in Fig. 2.2, taken from [39], depicting the voltage–current density curve of the helium discharge for pL = 3 Torr cm, where p is the pressure and L the electrode separation. The simulation is based on a 1D hybrid model, using different values of γ  . As can be seen from Fig. 2.2, the particular choice of the value of γ  results in pronounced differences in the calculated curves, none of which are consistent with experimental data. We note that the curve obtained assuming γ = 0.3, a value that corresponds to typical electron yields for He+ on clean metal surfaces, disagrees strongly with the experimental data. Figure 2.2 reports also the characteristic obtained from a fluid model calculation, using γ  = 0.16. It agrees quite well with experiment, suggesting that cathode was not cleaned by the low-sputtering yield of He since contaminated surfaces have low PEE yields. However, as discussed in [39], the fluid model fails to reproduce other characteristics of the discharge, such as the spatial distributions of ion and electron densities.

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51

Fig. 2.2. Electrical characteristics of helium glow discharges. The 1D hybrid modeling results (filled symbols) obtained for pL = 3 Torr cm are shown for apparent secondary electron yield values γ = 0.1, 0.13, 0.16, and 0.3. kTe = 0.3 eV is assumed in the model. The dashed line is the result of a fluid calculation with γ = 0.16 and kTe = 1 eV, while asterisk denotes experimental data obtained at pL = 3.38 Torr cm (from [39])

Phelps and Petrovic [30] reviewed the processes responsible for the production of the initial secondary electrons required for the growth of current at electrical breakdown and for the maintenance of cold-cathode discharge in argon, a commonly used as a benchmark gas for discharge studies. These electrons are produced in collisions of Ar ions, fast Ar atoms, metastable atoms, or photons with the cathode or in ionizing collisions of fast atoms or ions with the neutral Ar atoms in the gas phase. Since electron emission yields for ions, fast atoms, metastable atoms, and photons vary greatly with particle energy, surface conditions and discharge conditions, the effective yield γ  has to include the effect of all the relevant mechanisms as accurately as possible. The authors assembled a large amount of data for photoelectric yields and secondary electron emission yields by argon ions and fast argon atoms, as a function of ion and atom energy, for a large number of different metal surfaces, both clean and dirty. Since data for different metal surfaces are rather close to each other, they made analytical fits [30, 46] for electron emission yields as a function of impact energy for both clean and dirty surfaces, as shown in Fig. 2.1. The fits should be best for KEE yields from contaminated surfaces

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R.A. Baragiola and P. Riccardi

because of the insensitivity to the target material. For clean surfaces, it is better to use the PEE yield from the fit to a large number of materials [47] γp = 0.032(0.78Ei − 2φ).

(2.5)

Here Ei is the neutralization energy of the ion and φ the work function of the surface. For contaminated surfaces, the PEE is quite smaller, as shown in Fig. 2.1 for Ar+ (Ar0 only produces KEE). In this case, there is no equivalent expression to (2.5) since yields depend not only on φ but also on the type of contaminant. The analytical expressions provided by Phelps and Petrovic stimulated several studies that improved the description of discharge processes in argon [30, 39, 41–45]. Given the energy-dependent electron yields for each relevant process, the apparent yield can be calculated self-consistently from the simulation. As an example of the procedure, we mention here the work by Bogaerts and Gijbels [41] who developed a hybrid model to describe the discharge conditions typically used for analytical glow discharge mass spectrometry. In this technique, the material under analysis is used as the cathode of the glow discharge, and the particles sputtered by energetic plasma species are identified by mass analysis. Under the typical conditions of such a discharge, only Ar+ and fast Ar play a significant role in electron emission from the cathode. Hence the apparent yield can be calculated from the simulation as γ =

Je,tot = Ji,tot

Ei,max 0

Fi (E)γi (E)dE + Ei,max 0

Ea,max 0

Fa (E)γa (E)dE

,

(2.6)

Fi (E)dE

where γi (E) and γa (E) are the yields induced by ions and atoms shown in Fig. 2.1 while Fi (E) and Fa (E) are the calculated ion and atom flux energy distribution at the cathode. The energy distributions Fi (E) and Fa (E) depend on discharge conditions and may significantly affect the value of the effective yield. For example, the model of Phelps and Petrovic calculates the effective yield describing the fluxes of electrons, ions, atoms, and photons in a spatially uniform electric field. Attempts have been made to use this effective electron yield, calculated for a uniform electric field, to model also the cathode fall of abnormal glow discharge [40, 46]. The failure of these attempts led to intense debate [39, 40, 44, 46, 48], recently clarified by Donk´ o and coworkers [39, 44]. Using a hybrid model, they showed that the different electric field distributions result in dissimilar energy distributions of species bombarding the cathode, thereby modifying the effective electron yield. This points to the important fact that the electron emission yields induced by ions and neutrals (γi (E) and γa (E)) can be used in the simulations, but the effective yield γ  has to be calculated under the actual discharge conditions. The analytical expressions for γi [30, 46] were recently used in modeling the breakdown behavior in radiofrequency argon discharges [49]. The dependence of γi on the incidence angle θ was included and taken to be ∝ 1/ cos θ.

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53

It was found that, at low gas pressures, the breakdown voltage is a multivalued function of pressure, i.e., a single pressure corresponds to several breakdown voltages [50]. The multivalued branch of the breakdown curve could be reproduced only by taking into account the energy dependence of the ion-induced secondary electron emission yield. The inclusion of the angular dependence of the yield improved the agreement with experimental data. It is important to remark that many of the above-mentioned studies used the yields γi and γa for “dirty” metal surfaces. However, it is known that, for untreated metal surfaces, the KEE yields decrease with successive sputtering cycles as contaminants are removed [47] while, at energies below a few hundred eV, the PEE yields increase upon sputter cleaning. Furthermore, changes in the electron emission yields, due to spatial and temporal modification of the surfaces of electrodes during the discharge, can produce plasma nonuniformities and instabilities in discharge parameters [51], which are major concerns in plasma-based processing of materials, such as film growth and etching. We now consider the magnetically assisted magnetron glow discharges, where secondary electrons are trapped in the region near the cathode by an external magnetic field. The electron confinement allows operation at low pressures of a few millitorrs with typical applied voltages between 200 and 500 V, making the magnetron discharge very convenient for sputter deposition. The magnetic field strength is adjusted to avoid affecting the ions, which are extracted from the plasma and hit the cathode, where they give rise to sputtering and electron emission. Models of the magnetron discharge [35, 42], discussed in the following section of the book, show that sputtering is predominantly determined by ions of the buffer gas, with a much smaller contribution of fast neutral and a marginal contribution of sputtered target particles. Therefore, electron emission from the cathode is primarily determined by ion impact. The dependence of the magnetron discharge characteristics on secondary electron yields has been investigated by self-consistent calculations. As an example, Kondo and Nanbu [52] found with a Monte Carlo model a substantial decrease in both the electron density and ionization rate when decreasing γ from 0.15 to 0.12. Shidoji et al. [53] discussed the influence of the electron yield on the current–voltage characteristic of the Ar discharge using a hybrid model, with γ = 0.0005 Va, where Va is the applied voltage (in the range 240–400 V). The authors found that the Va in the simulated J–V characteristic was lower than the experimental value obtained with a Cu cathode, and an agreement required halving the value of γ, which they state would make it too small compared to experimental values. The authors attributed the discrepancy to diffusion loss of electrons. 2.4.1 Effect of Electron Recapture at the Cathode Inelastic collisions and a fraction of elastic collisions prevent the ejected electrons from returning to the cathode. However, since the electron mean free

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path is inversely proportional to pressure, at the low gas pressures of a magnetron discharge [42,54,55], a significant number of these electrons can return to the cathode and be recaptured without producing ionizations. The effect of electron recapture on the argon discharge characteristics has been discussed in a simplified [54] and a detailed model of the dc planar magnetron [42]. Since electron emission from the cathode is primarily determined by ion impact, the effective electron emission yield is given by [42] γ =

Nej − Nrec Nrec  = γion − , Nion Nion

(2.7)

where Nej is the rate of electrons produced by ion impact, Nrec the rate of electrons recaptured at the cathode, and Nion the rate of ion impacts on the cathode. The effective electron yield without recapture, i.e., only due to ion  = Nej /Nion . The authors calculated impact at the cathode, is given by γion  γion using the energy-dependent expression for γi given for clean metal surfaces [30]. For these types of surfaces, Fig. 2.1 shows that at the Ar+ energies typical of a magnetron discharge electron emission is determined by PEE with a nearly constant yield. The effective yield γ  is a function of the pressure and the electron reflection coefficient RC, which in turn depends on the cathode material, including contaminant layers. Kolev and Bogaerts [42] calculated that γ  is significantly reduced by electron recapture and exactly one half of γ for the intermediate value of RC = 0.5. The influence of different discharge parameters on the discharge voltage during magnetron sputtering has been recently discussed by Depla et al. [55] who compiled a list of electron emission yields for several metals from experimental data and empirical formulas found in literature [47, 56, 57]. As seen in Fig. 2.3, the discharge voltage for constant current operation of the magnetron resulted inversely proportional to the PEE yield of the target material [58]. This finding can be understood on the basis of the well-known equation proposed by Thornton [59] for Vmin , the minimum discharge voltage to sustain a magnetron discharge, W0 Vmin =  , (2.8) γ εi εe where W0 is the effective ionization energy (≈30 eV for Ar+ ), εi the ion collection efficiency, and εe the fraction of the maximum number of ions Vmin /W0 that can be made on average by a primary electron before it is lost from the system. Expressing the effective yield γ  as the product of the ion-induced electron emission yield of the target material γi and the effective gas ionization probability E(p) [56], we get Vmin =

W0 . E(p)γi εi εe

(2.9)

Assuming a material-independent recapture probability and a constant pressure leads to the inverse proportionality between the discharge voltage and γi , reported in Fig. 2.3.

Inverse of the discharge voltage (x10−31 / V)

2 Electron Emission from Surfaces Induced by Slow Ions and Atoms

55

Ag Al Au Ce Cr Cu Mg Nb Pt Re Ta Ti Y Zr

4.0

3.5

3.0

2.5

2.0

1.5 0.00

0.05

0.10

0.15

0.20

ISEE coefficient Fig. 2.3. The inverse of the measured discharge voltage VAr as a function of the electron emission yields shown in the horizontal axis, for different metallic targets, measured at a constant current of 0.3 A and argon pressure of 0.3 Pa. The error bars are based on the minimum and maximum values reported in the literature (from [55]) 60

β(x105V/A2)

50

Al Ag Au Cu Cr Mg Nb Re Ta Ti Y

40 30 20 10 0 0.05

0.10

0.15

0.20

Fig. 2.4. The influence of the electron emission yields (horizontal axis) on the coefficient β (from [56])

The electron emission properties of the cathode material influence also the I–V characteristics of the discharge. The measured I–V curves [56] have been reproduced by the equation I = β(V − Vmin )2 [60], where β is a constant. As shown in Fig. 2.4, the value of β increases with increasing γi . As discussed

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R.A. Baragiola and P. Riccardi

in [56] this is because, for a fixed discharge voltage, a higher γi produces a higher plasma density [52], which means a lower plasma impedance and, therefore, a higher value of β. 2.4.2 Effect of Changes in the Chemical Composition of the Cathode It is noteworthy that, for some target materials (Al, Cr, Mg, Nb, Re, Ti) at low discharge current, the discharge voltage decreases with increasing current [55], possibly due to target contamination by residual gases during sputtering. Indeed, since water is the main species in the residual gas, its chemisorption on target surfaces cannot be ruled out, especially at low discharge current. Chemisorption is critical in reactive magnetron sputtering of a metallic target, where a reactive gas is added to the discharge to deposit a compound material. The addition of the reactive gas results in the formation of a compound material not only on the substrate (anode) but also on the target (cathode) where its formation is balanced by the sputtering process. This changes the electron emission properties of the cathode which, together with the modification of the plasma composition, changes the plasma impedance. At a constant discharge current, this change of the plasma impedance results in a change of the discharge voltage. Whether the voltage increases or decreases with respect to the discharge voltage with metallic targets depends on the target material [61–63]. It has been found [60] that the discharge voltage during reactive sputtering of aluminum decreases when adding oxygen or nitrogen, in contrast to the case of other metallic targets. The difficulty in the understanding of these contrasting behaviors is that both the target condition and the plasma composition change when adding the reactive gas. A series of interesting experiments [58, 64] have been recently performed to study the behavior of the discharge voltage, at constant current, during reactive sputtering of oxides. In these experiments, the metal target is first sputtered in pure Ar until a constant discharge voltage, VAr , is established. Then Ar was replaced by pure oxygen and the new discharge voltage, VO2 , was measured. In a third step, O2 was replaced with Ar and the discharge voltage with the oxidized target, Vox,Ar , was measured. In this way Vox,Ar can be directly compared to VAr , since both are measured under identical experimental conditions and plasma composition. As shown in Fig. 2.3, the measured discharge voltage for metallic targets VAr is inversely proportional to the ion-induced electron emission yield γi of the target. A straight line can be fitted to the data in Fig. 2.3 1 = A + Bγi . VAr

(2.10)

Given the fitting coefficient A and B, which are only valid for a given series of experiments because they depend on the experimental conditions (discharge

2 Electron Emission from Surfaces Induced by Slow Ions and Atoms

57

current, pressure, and magnetic field), the authors attempted to extract the electron emission yield of the oxidized target as 1 1 −A . (2.11) γi,ox = B Vox,Ar The literature data for ion-induced emission yield of oxides are scarce. For magnesium oxide, the authors found a value of 0.4, which is of the same order magnitude of the value of 0.2, measured by ion beam experiments [65,66]. We note here that the yield obtained by this procedure should be more properly  as defined in (2.7), compared with the effective yield without recapture, γion and may not be directly compared with ion beam-induced emission yields γi . In fact, (2.9) and (2.10) are based on a compilation of PEE yields for clean metal surfaces, i.e., constant yield γi in the ion energy range of interest for  magnetron discharge for which γion = γi (see, e.g., (2.6)). For dirty or oxidized  , surfaces and in general for KEE, there will be differences between γi and γion this last being determined by the dependence of γi on ion energy and by the ion energy distribution at the applied discharge voltage. The extent of this difference will depend on the narrowness of the ion energy distribution, but it should be carefully taken into account as it can be a source of discrepancy when comparing electron emission yields obtained by different experimental techniques and procedures. Another source of discrepancy may be the assumption of a constant probability of recapture of electrons at the cathode. Chemisorption affects the electron recapture at the cathode and hence the effective ionization probability. As shown by Babout et al. [67] for copper, the electron reflection coefficient decreases with oxygen exposure, i.e., recapture becomes more important. A remarkable finding of these experiments is that two groups of materials could be distinguished from the analysis discussed above: one where the electron yield increases on oxidation (Al, Ce, Li, Mg, Y) and one where it decreases (Cr, Nb, Re, Ta, Ti). This finding is in apparent contrast with the generally accepted idea that oxide materials have larger yields than metals, although it is not inconsistent with the different behavior of Al and Mo upon oxidation [27], mentioned above. A suggestion for the uneven behavior of different materials upon oxidation has been proposed based on a study by Wittmaack [68] of electron emission from n-type silicon bombarded with O+ and O+ 2 ions. A conclusion of this work is that the enhanced electron yield generated by oxygen implantation in silicon is directly proportional to the fractional coverage of the surface with SiO2 islands growing at the bombarded surface. Isolated oxygen atoms embedded in Si or suboxides (SiOx , x < 2) apparently produce only a negligible change in the electron yield. This interpretation, however, has not been accompanied by arguments based on electron excitation, transport, and escape mechanisms. Based on Wittmaack’s work, Depla et al. [58] concluded that the discharge voltage behavior during reactive sputtering of metal oxides originates in the formation of an oxide layer of the order of 1–2 nm, as estimated from the sum of the ion range and the straggle

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of the oxygen implanted during the plasma oxidation. The formation of this oxide layer changes the electron yield of the target; depending on the type of oxide formed during plasma oxidation and subsequent sputtering in argon, the yield is high (oxides) or low (suboxides). The low yield of suboxides is a very interesting feature that has not yet been elucidated and deserves further investigations with the use of in situ surface analysis techniques.

2.5 Outlook The description of the understanding of physical mechanism and its application to some types of gaseous discharges points to directions for research. The most important variable in electron emission seems to be the state of the surface, which changes dynamically during the operation of the electric discharge. This is because of the competition of sputtering which removes native surface contamination and chemical alterations due to ion implantation (in reactive discharges) and recondensation of impurities from surfaces in the vacuum system. Possibly the most promising first step would be the study of electron emission yields using reactive gas ions over a range of targets, incident energies, and accumulated ion fluences. Fundamental studies need to relate observed changes in electron yields to measured changes in surface composition, as determined, e.g., by X-ray photoelectron spectroscopy. This is particularly important to understand the materials dependence of the changes of electron yield upon oxidation.

References 1. J. Schou, in Physical Processes of the Interaction of Fusion Plasmas with Solids, Chap. 5, ed. by W.O. Hofer and J. Roth (Academic, New York, 1996) 2. K. T¨ oglhofer, F. Aumayr, H.P. Winter, Surf. Sci. 281, 1430 (1993) 3. D. Hasselkamp, H. Rothard, K.-O. Groeneveld, J. Kemmler, P. Varga, H. Winter, Particle Induced Electron Emission II (Springer, Berlin Heidelberg New York, 1992) 4. R.A. Baragiola (ed.), Ionization of Solids by Heavy Particles (Plenum, New York, 1993) 5. R.A. Baragiola, in Low Energy Ion-Surface Interactions, Chap. 4, ed. by J.W. Rabalais (Wiley, New York, 1994) 6. H. Kudo, Ion-Induced Electron Emission from Crystalline Solids (Springer, Berlin Heidelberg New York, 2001) 7. H. Winter, H. Burgd¨ orfer (eds.), Slow Heavy-Particle Induced Electron Emission from Solid Surfaces (Springer, Berlin Heidelberg New York, 2007) 8. B.A. Brusilovsky, Appl. Phys. A 50, 111 (1990) 9. S. Valeri, Surf. Sci. Rep. 17, 85 (1993) 10. R.A. Baragiola, Radiat. Eff. 61, 47 (1982) 11. H.D. Hagstrum, in Chemistry and Physics of Solid Surfaces VII, ed. by R. Vanselow, R.F. Howe (Springer, Berlin Heidelberg New York, 1988), p. 341

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3 Modeling of the Magnetron Discharge A. Bogaerts, I. Kolev, and G. Buyle

3.1 Introduction Because of the industrial importance of sputter deposition magnetrons, there is a strong drive to simulate the entire magnetron deposition process, to replace trial-and-error experiments. This can lead to serious cost reduction, both for the manufacturers and the users of magnetron sputter equipment. The reasons are straightforward. For a typical coating plant, one of the main costs is the installation cost. Hence, it is a major advantage to have equipment that can realize a large throughput which fits specifications like required deposition speed, uniformity, reproducibility, or target lifetime. This means that magnetron manufacturers strive to a minimum “setup time,” which requires from them the ability to predict whether the proposed design will work or not before the machine is actually built. This could be achieved by simulating the machines’ characteristics. Of course, also the possibility of optimizing the deposition parameters without actually performing any real world experiment is attractive. The ideal magnetron sputter deposition simulation would use as input the desired coating characteristics (e.g., electrical resistance, adhesion, refractive index, etc.) and process requirements (e.g., deposition speed, price per square meter, etc.). It would yield as output the necessary process parameters, e.g., sputter mode (DC, RF, pulsed, etc.), gas pressure, magnetic field strength, electrical power input, etc. In reality, we are still far away from such a model. The model becomes more realistic when the input and output are switched (see Fig. 3.1). In this way, the parameters defining the deposition process are the input; the deposited film properties form the output. Such model is referred to as a “virtual sputter magnetron.” Efforts are made to develop such a simulation tool, e.g., [1–4]. Now, we will consider the virtual sputter magnetron of Fig. 3.1 in more detail. Basically, it consists of the following modules: – Magnetic field modeling – Magnetron discharge modeling

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System: geometry, B - field, electrical power,…

Magnetic field

Magnetron discharge

Particle Target interaction

Film properties: Physical: adhesion, conductance, hardness,… Economical: cost per m2,…

Transport in gas phase

Film growth

Fig. 3.1. Sketch of a virtual sputter magnetron, a simulation tool that would allow simulating the entire magnetron sputter-deposition process. The basic parts needed in such a tool are also shown

– Particle–target interaction, sputtering – Transport of the sputtered particles through the gas phase – Deposition and film growth at the substrate From a scientific view-point, these modules cover a wide range of disciplines (plasma physics, surface physics, materials science, etc.). Some of the modules can be considered as known physics, others as challenging research topics. An example of the latter is the relation between the particles arriving at the substrate and the properties of the deposited film. An accurate self-consistent virtual sputter magnetron that can operate over a wide range of parameters requires careful treatment of each of these modules. Like in a chain, the weakest link will determine the total strength: no matter how good the other modules, if there is one in the global model that is not accurate, the outcome of the whole model will be affected. In the following, the different parts will be very briefly discussed and references to some relevant literature will be given. 3.1.1 The Magnetic Field Essential for a magnetron discharge is of course the magnetic field. Hence, it is necessary for any simulation to have accurate values for the magnetic field. This can be achieved for the most complex magnetic configurations with high accuracy by using finite element models that are available as commercial packages or as free- or shareware. Packages reported in the literature regarding magnetron sputtering simulation are, e.g., POISSON [5, 6], OPERA [7], FEMME [8] or ELF/MAGIC [9]. A disadvantage of this method is that a high accuracy is only reached for a very dense mesh, which makes the magnetic field calculation time consuming. Very high accuracy is needed, however. Indeed, the length scale over which the electrons move during a time step when retracing their orbits is very small, in the order of 0.1 mm or less. For an accurate

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simulation of the electron orbits, the magnetic field must vary smoothly over this scale length. Another possibility is to calculate the magnetic field analytically. This has the major advantage that the field strength can be determined almost instantaneously at any point in space, which guarantees the smooth variation of the magnetic field along an electron orbit. Drawback is that only relatively simple magnetic configurations can be modeled and that the effect of magnetic shunts or complex magnet shapes cannot be dealt with. The last possibility is to start from a measured magnetic field and interpolating or fitting it by an analytical expression. This also leads to a smooth spatial variation of the magnetic field. Of course, in such an approach the principle of virtual coater is violated: it does not allow simulating different magnetic field configurations, unless they are built in reality and measured accurately. 3.1.2 The Magnetron Discharge The simulation of the magnetron discharge, based on the magnetic field, the gas pressure, and the electrical power input, is the next step. The underlying basic physics is at microscopic level “only” the motion of charged particles in a region subjected to an electric and magnetic field. This, combined with the necessary cross sections and electron yields, is in principle sufficient to describe the magnetron discharge. However, the emerging behavior of the plasma as a whole can be very complicated and turns this module of the virtual coater into one of the most difficult hurdles to take. Indeed, the small timescales required for the electron motion (10−11 –10−12 s) combined with the large timescales required for reaching equilibrium (10−5 s) make the computational load of numerical modeling extremely, and for certain configurations even unrealistically, high (see also Sect. 3.2 below). As the magnetron discharge is in principle the heart of the process, it is surprising that the more advanced simulation of the sputter process presented in [2] can completely bypass this and the previous module. Instead, the simulation starts with a profile characterizing the sputtered particles. This module of the virtual sputter magnetron will be dealt with more extensively in Sect 3.2, where the different modeling approaches and their specific (dis)advantages will be discussed. Also the rest of the chapter will be devoted specifically to the magnetron discharge modeling. 3.1.3 The Particle–Target Interaction The sputtering process is based on the removal of target atoms by ion bombardment. The simulation of ion bombardment on a solid is well developed. These packages do not only allow determination of the sputter yield, i.e., the average number of atoms removed per incoming ion, but are also able to reproduce the angular and energy distribution of the atoms that leave the

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surface. Very well known are Monte Carlo (MC) codes, such as TRIM [10,11], SRIM [12], and packages based on these. These packages have as main shortcoming that they are not dynamical, i.e., changes in the solid because of the incoming ion flux are not taken into account. Packages that take this into account are for instance TRIDYN [13, 14] and KALYPSO [15]. Another MC simulation package on sputtering is ACAT [16]. All the mentioned packages are numerical simulations, implying that the computational load is heavy. This is especially true for the dynamical codes. Hence, efforts are made to determine properties like the sputter yield and the angular and energy distributions of the sputtered particles by analytical expressions. These analytical expressions can be purely empirical or can be based on a simplified model. For the sputter yield, an example of an empirical expression is the well-known formula of Matsunami [17], whereas an example of a simplified model is the work reported in [18]. Also for the energy and angular distribution of the sputtered particles analytical models exist. A nice overview of this issue can be found in [19]. 3.1.4 Particle Transport in the Gas Phase Once the particles are sputtered from the target, they start spreading out through the vacuum chamber. The collision dynamics of these rather low energy (typically some tens of eV or below) particles are known. However, discussion exists about which interatomic potential is to be used [20]. This is not an academic discussion as the choice of the interatomic potential influences the simulated energy and angular distribution of the sputtered particles at the substrate [21]. As such, the exact potential is required for realistic thin film growth models. Ideally, the sputtered particle transport is described by combining a MC based model (for nonthermalized particles) and a diffusion model (for thermalized particles) [22]. In [23] the transport is described by a MC approach (SIMSPUD) and by a diffusion approach. Also in [24] a MC model is described. In [25] the film thickness distribution is simulated using a MC model for different target materials and for different target-substrate distances at different pressures. The results agree well with the experimental results, even for conditions where the ratio of the substrate and target diameter is large. Of course, also for this aspect of the magnetron sputter process simplified models have been developed, e.g., [26]. More information on simulations for particle–target interaction and particle transport in the gas phase can be found in the chapter “Transport of sputtered particles through the gas phase” by S. Mahieu et al. 3.1.5 Film Growth on the Substrate For this part, the input consists of the energy and angular distribution of the particles arriving at the substrate and their arrival rate. With “particles” is not only meant the sputtered particles but also the electrons and discharge

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gas particles (neutrals and ions) as they also contribute to the energy deposition at the substrate. The relation between this input and the resulting film properties is one of the more difficult parts when modeling the magnetron sputter process. A first step is the simulation of the film thickness distribution and/or its deposition rate, as this follows rather straightforwardly from the information about the incoming particles and the sticking coefficients. In practice, in a lot of cases only this aspect is considered. A more advanced form of this is “feature scale modeling.” In microelectronics, the deposited coating might be required to fill structures like a thin trench or via. Feature modeling tries to simulate the exact coating thickness on all surfaces of such a feature [27–31]. One step further is to actually simulate the microstructure of the deposited material. The most accurate simulations are based on molecular dynamics, but this is extremely computationally intensive. Consequently, (empirical) approximations are needed. A sound discussion of this problem together with some examples of predicted structures can be found in [32]. In [33] the deposition on grain boundaries was simulated for Ti. Examples of the microstructural evolution during film growth (and the kinetic MC simulation thereof) can be found in [34]. Of course, also simplified models are developed, e.g., for the biaxial alignment in yttria-stabilized zirconia layers [35].

3.2 Overview of Different Modeling Approaches for Magnetron Discharges The term numerical modeling refers to the process of finding approximate numerical solutions to a system of proper physical equations that adequately describe the system of interest. In a slightly broader aspect, also analytical models fall in this category. The latter are based on simplifications of globally valid, but complicated physical equations, rather than on numerical solutions of these globally valid equations. The choice of the proper set of equations is normally based on the given operating conditions. In other words, certain physical approximations are assumed already prior to the process of numerical solving, which in its turn introduces mathematical (numerical) approximations. The overall goal is always to obtain the most useful results at the lowest possible computational price. A complete model of a magnetron discharge must include the magnetron plasma and its interactions with (some of) the solid surfaces that surround it. There are two surfaces that cannot be omitted in a model. These are the cathode (also known as target) and the substrate where the film is deposited. In many cases, the substrate is grounded, and therefore, acts as an anode. An intrinsic part of the operation of the magnetron is the effective electric field that results from the external power source and the movement of the plasma charged particles. Therefore, a procedure that calculates this field

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should also be a part of the model. When the calculation of the resulting electric field is included in the model, the model is called self-consistent. It should be mentioned that for real self-consistency, some external electrical circuit should be coupled to the magnetron plasma [36–39]. However in the literature, it is common to use the term self-consistent even when no external circuit is included. The existing models of magnetrons can be classified in several ways. The first division is between analytical and numerical models. The numerical models can further be split into fluid or kinetic models. The latter (i.e., kinetic) models can be based on numerical solutions of the Boltzmann equation, or on Monte Carlo (MC) simulations, also called particle simulations. These models are, however, not self-consistent, and need to be combined with a solution of the Poisson equation, resulting in either hybrid models or PIC-MCC simulations. In following sections, a literature overview of these different modeling approaches will be given. 3.2.1 Analytical Models Analytical models (e.g., [40–57]) usually approximate the magnetron chamber with a one-dimensional (or sometimes even zero-dimensional) domain in the coordinate space, where the electric and magnetic forces are perpendicular to each other and the electron transport is considered in the direction parallel to the electric field and perpendicular to the magnetic field. In addition, the magnetic field is considered constant. Such a picture results, however, in a particularly simple form of the electron transport coefficients. Indeed, only the diagonal elements of the tensors of mobility and diffusion are nonzero and can be regarded as permutations of the mobility and diffusion referring to the nonmagnetized case [58]. In such an approach, classical diffusion is considered. Lieberman and coworkers [5, 41–43] have done many efforts for developing analytical models for magnetron discharges. Wendt and Lieberman [5, 42, 43] developed a two-dimensional analytical model. The discharge area was split up into arch-shaped areas, with the shape determined by the magnetic field lines. In this way, the authors were able to relate the width of the sputter erosion profile to the discharge voltage and current, and the magnetic field strength, through the Larmor radius [5]. Also the light emission could be described with this model [41]. In [43], two different models for the movement of the electrons in a cylindrical postmagnetron have been proposed and compared with experimental data. In [46–48], a particularly interesting result has been the prediction of the appearance of a negative space charge region due to the highly restricted electron mobility at strong magnetic fields and low pressures. This result has been later on validated by more realistic particle simulations for both cylindrical [36] and planar magnetrons [37]. Recently, an analytical model has also been proposed for an RF sputtering system [49].

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Analytical models have also been used intensively by Bradley and coworkers. In a series of papers, they have investigated different parts of the magnetron discharge, such as the cathode fall [50], the presheath [51], and the bulk plasma [52]. All these studies are based on an earlier work by the same author [59], which is a one-dimensional fluid description. In [50, 51] the classical diffusion has been replaced by Bohm diffusion and the results have indicated that the later is a more realistic mechanism for the electron transport across the magnetic force lines. Recently, M¨ oller [53] proposed an analytical model for a reactive magnetron plasma (argon – nitrogen gas mixture), based on a balance between ion generation and transport. Ar+ , as well as N2 + and N+ , ions were considered in this model, and it was assumed that all ions, created in the plasma by electron-impact ionization, arrive at the target in a free-fall transport regime. Beside the different ion fluxes, also the fluxes of reactive N atoms and N2 molecules to the target were estimated. These data formed the input for surface processes, such as sputtering and implantation for the various ions, and adsorption for the neutral species [53]. Finally, Buyle et al. have recently developed a self-consistent two-dimensional semi-analytical model for a DC planar magnetron, to reproduce the magnetron dependence on external parameters over a wide range [54–57]. This model will be explained in more detail in Sect. 3.4 below. A general disadvantage of analytical models is that they are unable to describe quantitatively the complex electron motion in the multidimensional electric and magnetic fields crossed at arbitrary angles, which is the situation in real sputter magnetrons, especially in the planar ones. However, analytical models have proven to be an important step in the understanding of magnetron discharges, with several advantages compared to complicated numerical simulations. Their main advantages are the relative simplicity, the ability to produce fast results, and the fact that the results are intuitively easy for understanding. They are particularly suitable in situations where the main purpose of the model is to define whether a given phenomenon is important or negligible. Such example is the electron recapture at the cathode, which has been studied by the semi-analytical model described by Buyle et al. (see also Sect. 3.4 below) [54–57]. The model has shown unambiguously the major importance of electron recapture at the cathode, and these results have later been confirmed by particle simulations [37]. 3.2.2 Fluid Models Fluid models are based on the continuity, the momentum and mean energy conservation equations of the plasma species. By coupling these equations to the Poisson equation, the electric field distribution can be calculated in a self-consistent way. Fluid methods find extensive use, for instance, in the field of aerodynamics. However, they are not so commonly used to describe magnetron

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discharges [48, 59–64]. Indeed, magnetron discharges cannot be so easily considered as a fluid. The main reason is that they are operated at relatively low pressure (typically several millitorr). At such low pressure, the main assumptions of the fluid theory are not necessarily valid. An additional difficulty arises from the complicated forms of the equations of the magnetohydrodynamics, which describe a magnetized fluid. If these equations are applied to arbitrary magnetic and electric fields, they require elaborated discretization, resulting in long computation times. Another issue is the validity of the classical diffusion at high ratios of the magnetic field and the gas pressure. At relatively weak magnetic fields and relatively high gas pressure, fluid models with simplified transport coefficients can be employed. This has been done in [61], where the bulk plasma in an RF cylindrical postmagnetron has been simulated and the results have been compared to a particle simulation. A fluid description has been used also to develop a theory of the cathode sheath in magnetrons [62]. A two-dimensional fluid model using the drift-diffusion theory has been reported in [63]. It is, however, applied to conditions, which are different from typical sputtered magnetron conditions, i.e., a pressure of several torr and a magnetic field weaker than 100 G. An attempt to overcome the complexity of the full magnetohydrodynamics equations of the electron transport [65] was recently proposed in [64]. In this paper, the presence of the magnetic field is included as a perturbation to the flux equations describing the nonmagnetized case and assuming a classical transport. However, it can be argued that the electrons are so strongly magnetized that this aspect cannot be represented in their transport by a perturbation. Hence, the condition for applying the perturbation formalism is thus violated.

3.2.3 The Boltzmann Equation The maximum information with minimum assumptions about the discharge can be obtained by kinetic models. As mentioned above, they can be divided into two groups: either solutions of the Boltzmann equation or particle simulations. The direct solution of the Boltzmann equation has been a popular tool for plasma modeling for many years. For magnetron discharges, the Lorentz force needs to be included. Adding the Lorentz force term, however, complicates significantly the solver and makes it impractical in two and three dimensions and for arbitrary magnetic fields. There is one situation where it can, and has been, successfully used in magnetrons. This is the simulation of cylindrical postmagnetrons, where in a large part of the discharge the magnetic field is one-dimensional and constant, which simplifies the Boltzmann equation. This is illustrated by the work of Porokhova et al. [66–73]. To find a self-consistent solution for a given input of magnetic field strength, gas pressure and discharge current, the Boltzmann equation was

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supplemented by some other requirements. In [67], self-consistency was reached by adding an equation, describing the ion motion in the collisional regime (i.e., a fluid model for the ions) and the Poisson equation. In [69] the radial distribution of the electric field was determined by fixing the discharge voltage to the experimentally measured one and by demanding that the radial potential distribution provides a radial electron density distribution that is similar to the experimentally measured one. This latter approach seems “less self-consistent” because both the discharge current and voltage are used as input parameters. In [67] the model was one-dimensional, whereas in [69] the model was extended to two dimensions, so that also the axial inhomogeneities generated by the shields at the ends of the cylindrical postmagnetron can be modeled. A major advantage of the method is the small computational load: a typical calculation requires about 2–10 min, depending on the number of grid points [74]. The presented results in the above-mentioned references show a very good agreement with experimental data. However, the results were obtained at relatively high gas pressure (above 3 Pa) and relatively weak magnetic field strengths (maximum 400 G). It would be interesting to see whether this modeling approach can deal with low pressures and strong magnetic fields, as these are the working conditions where anomalous electron transport can be expected. Furthermore, it is also not evident to adapt this technique to a twodimensional model for the planar magnetron discharge, due to the nonuniform magnetic field [74].

3.2.4 Monte Carlo Simulations In MC simulations, computational test particles that represent a large number of real plasma particles are followed. Their movement is subject to the applied forces and the collisions of the particles are included by using probabilities and random numbers. This technique is very easy to implement and fast to compute. Its main disadvantage is that the forces must be an input, i.e., the simulation is not self-consistent. There are many MC simulations, often called direct Monte Carlo simulations (DMCS), of different types of magnetrons reported in the literature. Among them are [75–83], which have contributed largely to the understanding of the transport and collision properties of the charged particles in magnetron discharges. The group of Sheridan–Goree et al., e.g., [75–79], was able to simulate the ionization distribution using a two-dimensional MC model and compared it with optical emission measurements. The basis of their model was a simple retracing of the high energy electron orbits by numerically integrating the Lorentz equation and combining this with a MC approach for the collision events. Similar work is reported in [84,85]. The group Sheridan–Goree also simulated the ion motion [78] and the influence of the magnetic field [79] using the same Monte Carlo model.

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Ido et al. used a MC model to simulate the erosion profile in planar magnetrons. First, the erosion profile in a cylindrical and a rectangular magnetron was simulated for various pressures and magnetic field configurations [85–87]. Then, the erosion profile using ferromagnetic targets was simulated. They report how the eroded region can be influenced by placing a ferromagnetic ring on top of the target [88] and by the outer part of the yoke [89]. For a ferromagnetic target, the eroded region strongly influences the magnetic field strength. In [90] it is reported how this effect is taken into account, which leads to a very accurate erosion profile simulation. The main advantages of the MC method are that it is easy to implement, it can handle the low gas pressures encountered in magnetron discharges, and it is much faster than self-consistent methods, such as PIC-MCC models. Therefore, it can be used on (large) three-dimensional geometries, see e.g., [91– 96]. In all these references the MC model is basically used to investigate the cross corner effect. This term is used to denote the enhanced ionization and erosion that occurs at opposite sides of a (long) rectangular target, and which is attributed to a disturbed E × B drift of the electrons after they drifted through the end region of the racetrack. Here, the MC method has an advantage over the PIC-MCC method: In [97] a three-dimensional magnetron was simulated but the variation of the ionization along the racetrack was too large to notice the cross corner effect. Although the PIC-MCC method should undoubtedly be able to reproduce this effect, it would cost an enormous computational effort. Another example illustrating the beneficial use of a MC model is the explanation of the so-called plasma emission redistribution (PER) effect [98]. This effect occurs during pulsed magnetron sputtering. When the plasma emission of a single pulse is observed, it was noticed that at the very beginning of a pulse the emission came predominantly from the turnaround regions of the racetrack, whereas in steady state, the emission came predominantly from the central straight sections of the racetrack. Using a MC model and adapting the settings for the situation at the very beginning of a single pulse, this effect could be reproduced and the origin of the effect could be retraced, as explained in [98]. Finally, the last application mentioned here for MC simulations of magnetron discharges is the simulation of the sputtered atoms transport, as was also briefly discussed in Sect. 3.1.4. These are normally non-self-consistent trajectory calculations of the sputtered particles with included collisions with the background gas atoms [23, 24, 89, 91, 99–105]. In this way the spatial distribution of the sputtered atoms and their energy and angular distribution at the substrate are calculated. This allows accurate predictions for the quality of the deposited film. Possible effects of the spatial location of the target and the substrate can be easily studied. Input is needed for the starting positions of the sputtered atoms at the cathode. This is usually taken either from an experimentally measured erosion profile or from a calculated erosion profile by a particle model.

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3.2.5 Hybrid Models While electrons are heavily magnetized, as discussed above, the ions can be considered practically nonmagnetized, due to their large masses. Therefore, in contrast to the electrons, they can be well described by a classical fluid approximation. This assumption forms the foundation of the so-called hybrid models. Hybrid models are usually a combination of fluid and particle models. In the simulations of nonmagnetized glow discharges (e.g., [106]), the ions and the bulk electrons are regarded as a fluid, while the energetic (beam-like) electrons, emitted from the cathode or generated in the sheath, are simulated as particles, i.e., fully kinetically. This approach combines the precision of the kinetic models with the relatively higher speed of the fluid codes and successfully overcomes the low pressure restrictions. Magnetron simulations based on the hybrid model are explained in [9, 107–115]. Shidoji et al. were able to simulate the magnetron discharge and its dependence on external parameters, such as the gas pressure, the electrical power [92], and the magnetic field strength [114]. Also the influence of balancing the magnetic field [111] and of depositing an insulating layer [109] were investigated. Vyas and Kushner [31] used a modification of their twodimensional hybrid plasma equipment model [115] to simulate a hollow cathode magnetron discharge. The effect of power and pressure was investigated, as well as the influence of magnitude and orientation of the applied magnetic fields. The limits of validity of hybrid modeling for magnetron discharges are discussed in [112]. It is reported that the application of the hybrid model becomes problematic for strong magnetic fields and/or low gas pressures. Indeed, in such cases the results appear unrealistic: a large amount of ionization occurs at rather large distance from the cathode. The reason is that at high magnetic field strengths the electrons hardly diffuse across the magnetic field lines anymore when using the classical theory. Consequently, an electric field is formed between the sheath region and the anode. Because of this electric field, ionizations occur in that region. Also the simulation results reported in [114] express this behavior: The ionization in the region from 10 to 40 mm above the target is stronger for a magnetic field strength of 360 G than for 180 G but the peak ionization is lower. This is opposite to the behavior reported in [116]: there the simulated peak ionization rate increases with increasing magnetic field. The latter result appears more realistic, as the magnetic field is applied to the discharge to intensify the magnetron discharge, not to spread it out. The problem of the hybrid model to deal with strong magnetic fields is probably due to the fact that it does not account for the anomalous electron transport. Indeed, as mentioned, the slow electron transport is typically treated using a fluid model. This requires the electron diffusion coefficients

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for the transport perpendicular to the magnetic field lines. This coefficient is obtained using the classical theory, i.e. D⊥ =

DB=0 1+

ω2 ν2

,

with D⊥ the diffusion coefficient in the direction perpendicular to the magnetic field lines and DB=0 the one without magnetic field. However, according to experimental evidence (e.g., [117]), the diffusion across the magnetic field lines is larger than predicted by D⊥ . This problem of uncertainties in the electron transport coefficients has been partially overcome by using the data provided by a parametric study performed by numerical solution of the Boltzmann equation for given sets of reduced electric and magnetic fields [113]. Similarly, Vyas and Kushner [31] derived the transport coefficients, for use in the fluid equations, from a kinetic simulation, to enable the fluid algorithm to represent more accurately the low pressure operation. The second limitation of a hybrid model is the fact that the ions are not at all characterized by a Maxwellian distribution, which questions their description as a fluid. The third problem is related to the criterion of selecting which electron can be considered slow. In the nonmagnetized situation [106] the discharge is in a positive space charge mode. Hence, once the electron has traversed the sheath and has an energy below some threshold, it can be transferred to the slow group. In magnetrons, depending on the reduced magnetic field, there can be a negative space charge mode. This means that even if the electron energy is below the threshold at a given moment, the electron can still become fast. This is definitely the situation at high reduced magnetic fields. This situation has led to a modification of the classical hybrid scheme, where all the electrons are treated kinetically, while only the ions are described as a continuum [114]. In this way, the first and the third drawbacks are eliminated, but the computational cost, however, approaches that of a particle model. In conclusion, the main advantage of the hybrid model is that it is faster than the PIC-MCC models discussed below (a solution can be obtained in 2–3 days). The payoff for this advantage is that assumptions need to be made about the slow electron motion in crossed electric and magnetic fields. Because of the latter, this technique seems to fail to describe the anomalous electron transport that occurs for strong magnetic fields and/or low gas pressures. 3.2.6 PIC-MCC Simulations When the calculation of the electric field produced by the external power source and the spatial distribution of the plasma charged particles is added to a MC simulation, the whole simulation becomes self-consistent. It is commonly referred to as PIC-MCC simulations [118]. PIC-MCC simulations are the most powerful numerical tool for the investigation of magnetron discharges. They

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can cope with the low operating pressures and the strongly inhomogeneous field and density distributions. In addition, they are capable of providing a full picture about the processes occurring in the discharge chamber. The drawback of the PIC-MCC simulations is their extensive computational load. This is especially true when two- and three-dimensional simulations are needed, especially for simulating magnetron discharges. The latter is related to the long characteristic times (typically longer than 10−5 s) for achieving a steady state. Simulations of cylindrical postmagnetrons have been performed in one dimension [36,119]. Planar magnetrons can, however, not be simulated with a one-dimensional code. A full three-dimensional PIC-MCC model of a planar magnetron has been reported [97,120]. It has shown the lack of angular dependence in axisymmetrical planar magnetrons. This result has opened the way for simulating this type of magnetrons by a two-dimensional model in cylindrical coordinates. The first very thorough work of that kind has been [116] followed by [121]. The PIC-MCC method has also been successfully applied to RF sputter magnetrons [122]. Moreover, quite some work reports planar magnetron simulations using the OOPIC code [4,6–8]. This OOPIC code was developed by Verboncoeur et al. [123] and can be downloaded freely [124]. Another software package that is used for magnetron discharges is PEGASUS, which combines PIC-MCC simulations with Gaussian fitting, to investigate the erosion [125]. The software itself is described in more detail in [126, 127] and is based on NEPTUNE. The latter is used in [1] to simulate copper deposition by magnetron sputtering. Furthermore, the influence of the dielectric target in an RF sputtered magnetron has been investigated in [128]. Recently, a hybrid PIC-MCC – ion relaxation model has been reported [129]. Here, the electrons are resolved in a standard PIC-MCC algorithm, whereas the ionic distribution is calculated by a continuum relaxation model. In [4] and [6] it is reported that no steady state is found in the PIC-MCC simulations, but the number of particles was continuously increasing. In both cases, however, the time during which the plasma is simulated was very short (5 and 3 μs, respectively). Following the discharge for a longer time period apparently leads to a steady state, e.g., [8, 97, 116]. However, it can be argued that a reliable steady state, i.e., a steady state where the discharge evolves to the correct region of the current–voltage characteristic, can only be reached by including an external electric circuit [37]. The required time for reaching convergence is then typically about 20 μs [37], whereas the results of [8] suggest that convergence was reached after roughly 10 μs. We have developed our own PIC-MCC model for a planar magnetron discharge, which includes the external circuit [37] and accounts for the gas heating and the sputtered atom transport and collisions [38]. This model will be discussed in detail further in this chapter. It is clear that PIC-MCC simulations produce a wealth of data. However, care has to be taken when interpreting the results. Here we mention some shortcomings of the results reported by the group of Nanbu et al. [97,116,121]. First, the time step Δt used for the electron orbits is rather large: in [116]

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it is mentioned that Δt ≈ 0.2 ns. The examined magnetrons have Bmax in the range of 325–650 G. With these Bmax correspond Larmor frequencies of 6–11 GHz, which means that Δt varies from 0.18 to 0.36 times the inverse of the Larmor frequency. In [130] it is argued that such large time steps result in an inaccurate orbit calculation and in a serious artificial electron energy loss. These inaccuracies compromise both the position and the amount of ionization simulated, which seriously questions the reliability of the presented results. This might be the origin of the second shortcoming of these simulations: the simulated cathode sheath thicknesses are very large, in [116] sheath thicknesses vary from 4.5 to 5.8 mm. In reality, a sheath thickness of at most 3 mm is observed (e.g., [91, 130, 131]). Third, the current densities are extremely low for magnetron discharges. From the sketch of the magnetron configuration in [116] we can infer a racetrack surface of roughly 24 cm2 (racetrack length ≈24 cm and width ≈1 cm). The total currents listed in the article are between 35 and 50 mA, which means current densities around 1.5–2.1 mA cm−2 . These values are at least a factor 10 lower than the current densities typically encountered in magnetron sputtering. This discrepancy can again be attributed to the necessity of including an external electric circuit [37]. The fourth remark concerns the simulated current–voltage characteristics: With increasing discharge voltage the discharge current and the electron density are found to decrease, the cathode sheath thickness is found to increase [121]. These simulated dependences are opposite to experimental observations (e.g., [130, 132]). On the other hand, a very interesting result of the PIC-MCC simulations is the magnetic field dependence: in [116] the magnetic field strength Bmax is varied from 325 to 650 G at constant discharge voltage (Vd = 500 V) and gas pressure (0.67 Pa). The simulations show an increase in the plasma density and in the ionization rate. Moreover, the discharge intensifies and the cathode sheath thickness is found to decrease. This is in agreement with common sense: with increasing magnetic field strength, the magnetron intensifies and is better confined. This result seems to indicate that the anomalous electron transport is, at least qualitatively, correctly simulated (see also the discussion about the hybrid model above). In spite of the mentioned shortcomings, the PIC-MCC technique is in principle a very viable method. Hence, we may conclude that PIC-MCC simulations are the most powerful tool to tackle the problem with the full description of the planar magnetrons at all operational conditions. The price for that, however, is the very long computation time.

3.3 Challenges Related to Magnetron Modeling No matter how sophisticated a simulation model is, there are some limitations to what can be expected from the calculation results. The origin of this might be due to inadequate experimental input data or to the specific geometry or

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operating conditions of the process. As such, these situations can be called problem cases or, more positive, challenges related to magnetron modeling. Here, we give a list of the most important ones of these challenges. 3.3.1 Secondary Electron Emission Yield (γ) The secondary electron emission yield γ denotes the number of secondary electrons emitted per incoming ion. The magnetron characteristics depend very strongly on the exact value of the secondary electron emission yield. Unfortunately, these yields are extremely difficult to measure. Consequently, for most materials this yield is not accurately known. This is probably the most fundamental factor limiting the output of magnetron simulations: whatever model used for the magnetron simulation, it can only give accurate results when the applied secondary electron emission yield is correct. In Shidoji et al. it is mentioned that reducing the secondary electron emission yield by 50% can decrease the currents with a factor four to five [9]. In [116] an increase of the secondary electron emission yield from 0.12 to 0.15 leads to a substantial increase in the plasma density (from 1.1 to 1.7 × 1010 cm−3 ) and in the discharge current (from 34.9 to 49.5 mA). Given this situation, the values for the secondary electron emission yield used in magnetron simulations strongly vary. Usually, γ is taken to be independent of the discharge voltage (e.g., in [97,120]). On the other hand, Shidoji et al. use a secondary electron emission yield that depends on the discharge voltage [9]. Sometimes, quite unrealistic values for the secondary electron emission yield are used, e.g., in [133] where for certain conditions a secondary electron emission yield of 5 is assumed for a metal surface. Also in [125] γ is used as a tuning parameter: the authors mention that it is set to 0.03 as a larger value sometimes leads to the unbounded increase of superparticles, i.e., the simulation does not reach a steady state but the plasma density continues to increase. The challenge here is to make the magnetron simulations so reliable that, once the model is calibrated using a material with a relatively well-known secondary electron emission yield, comparison of the experimental and simulation results can be used to determine the secondary electron emission yield of the investigated target material. This is, however, a very ambitious goal, due to the many other uncertainties in the models. Therefore, efforts should also be made for accurate measurements of the secondary electron emission yield. In this respect, Depla et al. recently determined ion induced secondary electron emission yields for several oxides [134]. 3.3.2 Recapture The orbit of a secondary electron emitted from the target follows the magnetic field lines. When the electron does not interact with the discharge gas, this will lead to an electron–target interaction. This event is characterized by

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the reflection coefficient R, which gives the probability that the electron is reflected. If the electron is not reflected, it is recaptured by the target (see also Sect. 3.4 below). To simulate this effect accurately, the initial electron energy may not be neglected, the orbit calculation should be performed very accurately and the reflection coefficient R needs to be known. The required accuracy has important consequences as it can only be reached by reducing the time step used for retracing the electron orbits, which seriously increases the computational effort. In [130] it is shown that the time step needs to be at least a factor of two smaller than currently encountered in magnetron simulations. Second, experimental values of R, needed as input for the simulation, are very scarce. This is highly inconvenient because the process of recapture is quite sensitive to this reflection coefficient [37, 135]. 3.3.3 Electron Mobility The problem of the anomalous electron transport was already mentioned above in Sect. 3.2.5. Indeed, the electron transport in the direction perpendicular to the magnetic field lines is larger than expected from classical diffusion theory. The enhanced mobility is probably due to oscillations in the electric field. This effect has its consequences for magnetron modeling. Obviously, all methods that rely on the classical theory are affected. This is especially true for the models that rely on the fluid theory, as this requires diffusion coefficients (see also the discussion about the hybrid model in Sect. 3.2.5). In principle, the PIC-MCC method should be able to deal with this problem as it uses no assumptions about the particles. However, a culprit here might be the timescale at which the phenomena occur. A broad range of oscillation frequencies have been observed in plasmas, but the typical frequency range is of the order of some 100 kHz, corresponding with a period of 10 μs. Logically, it can be assumed that the discharge only reaches a steady regime after some oscillation periods. This means that the discharge should be followed during a time span of at least 50 μs. Usually, PIC-MCC simulations are not performed for such a long time, implying that they will not accurately describe the effect. A possible way to model the enhanced electron mobility is by adding an extra type of collisions, the so-called Bohm diffusion collisions. The contribution of Bohm diffusion can then be tuned until the simulations correspond with the experimental measurements. This will be briefly explained for the analytical model in Sect. 3.4. For the modeling of Hall thrusters, a similar technique has been used, see e.g., [135, 136]. 3.3.4 Modeling “Industrially Relevant” Magnetron Discharges Of course, the magnetron discharges used for industrial sputter deposition purposes are not fundamentally different from the ones used in laboratories.

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Nevertheless, there are some aspects that make them extremely challenging to simulate. In principle, including these characteristic features in the models is not a fundamental problem, but it will make the simulations much more computational intensive. Given that the computational load is already high (order of several days for both the PIC-MCC and hybrid models), it is clear that an industrially relevant virtual sputter magnetron is not for the immediate future, even taking into account Moore’s law, which states that the computation power doubles, on average, every 18 months. In the following, a few of these challenges will be outlined. Geometry Although it is not a fundamental limit, the geometry of the magnetron considered can make certain models useless. In the glass coating industry, magnetrons with cathodes up to 4 m long are used. For such dimensions, the PIC-MCC technique becomes useless because of the unrealistically high computational effort. Hence, less computational intensive methods are needed. An example is the simulation of the effect of the turnaround region on the uniformity. For this type of simulations the MC method is currently used (see Sect. 3.2.4 above). Although the latter model has the disadvantage that the influence of the anode cannot be modeled self-consistently, it is still the preferred method for such large-scale magnetrons, as the computational load of the PIC-MCC would be forbiddingly high. (High Power) Pulsed Sputtering For industrial applications, pulsed sputtering is frequently used. The typical frequency range is in the order of 10–100 kHz. To simulate such processes, the magnetron needs to be followed for several periods, i.e., several times 100 − 10 μs. At this moment, this cannot be achieved by PIC-MCC modeling. A good candidate for this type of problems would be a hybrid model. Currently, high power pulsed sputtering is gaining interest because of its potential to deposit coatings with unique properties. In such discharges extremely high electrical powers are applied during short intervals. Because of the high plasma density, typical approximations made in magnetron modeling are not valid anymore. First, the magnetic field strength in the magnetic trap region changes because of the large Hall currents induced during the high power on-pulse [137]. In all simulation models known to us, this influence is not accounted for. Second, because of the high ionization degree in these high power discharges, it will not be possible anymore to neglect the interactions between charged particles (Coulomb collisions). Again, most simulation models for planar magnetrons neglect this type of interactions. In [138] a cylindrical postmagnetron is simulated with a PIC-MCC model, taking into account electron–electron interactions, albeit in a simplified way. We have included

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these interactions in our PIC-MCC model [38, 39], based on the method described by Nanbu [139–141]. Third, the gas density reduction or gas rarefaction: in Sect. 3.6, it is discussed how the energetic plasma particles can heat the discharge gas, which leads to a gas density reduction. For standard operating conditions, this effect can be neglected. However, in high power pulsed sputtering this will not be the case. This effect is usually not accounted for in magnetron models. To our knowledge, only the PIC-MCC model described in this chapter (Sect. 3.6) is able to deal with this effect [38, 39]. Accuracy The specifications for sputter-deposited coatings can be very strict. A typical example is the required coating uniformity. Even for large area coating of glass (with cathode lengths of almost 4 m) the required uniformity is usually better than ±0.2%. It is not evident for a numerical technique to reach this level of accuracy. For an MC technique, the only way to reach this is by following a sufficiently large number of particles, i.e., the accuracy comes at the cost of computational load. Apart from this, several other effects influence the accuracy at this scale. Some examples are the gas heating due to sputtered particles and the resulting gas density reduction, a nonuniform gas inlet, the influence of the erosion groove formation on the process, redeposition on the target, heating of the target, the influence of the sputtered particles, etc. Again, all of these examples form no fundamental problem, but they might seriously increase the computational effort. Basically, the mentioned effects are all examples of “extended self-consistency.” Usually, in magnetron modeling the term self-consistency is used with respect to the electric field. However, because of the on-going sputtering and deposition during the process, the environment of the magnetron changes. For a very accurate magnetron simulation, also these changes have to be taken self-consistently into account.

3.4 Two-Dimensional Semi-Analytical Model for a DC Planar Magnetron Discharge In this section, the self-consistent two-dimensional semi-analytical model developed by Buyle et al. for a DC planar magnetron [54–57, 130] will be explained in more detail, because it is very powerful, in spite of its simplicity. Also some characteristic results will be presented.

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3.4.1 Description of the Model Ionization Model First of all, an ionization model was developed, to calculate the ionization rate due to the high energy electrons, i.e., defined as the electrons with energy above the threshold for ionization. For this purpose, the discharge area was split into arch-shaped regions. This can be justified, because the collisionless motion of these high energy electrons is limited to such arch-shaped regions. The effect of electron collisions with the background gas was analytically modeled as a probability for the electrons to hop from one arch to another. Output of this model was the spatial distribution of ions, generated by a single electron emitted from the target. It was found that the ionization distribution is not directly influenced by the gas pressure. This characteristic could be used by future models for magnetron discharges, to reduce the computational load. Another important result was that a substantial amount of ionization occurs in the cathode sheath of the magnetron. This so-called sheath ionization is important because the electrons generated in this way can be further accelerated and give rise to more ionization. This effect was characterized by a multiplication factor, which is defined as the ratio between the total number of ions generated due to the emission of a single electron, and the theoretical number of ions that can be generated without sheath ionization. The multiplication factor was found to be close to one for electrons emitted at the edge of the racetrack, but it can rise to values of 3–5 for electrons emitted close to the racetrack center. Therefore, an average multiplication factor, m, was introduced. Effect of Electron Recapture Furthermore, the process of electron recapture at the cathode was taken into account by introducing the effective gas interaction probability (EGIP), defined as:  s EGIP = f = 1 − exp − , (3.1) l where s is the average distance traveled by the electron before it is recaptured at the cathode and λ is the mean free path length of the electron. For typical magnetron operating conditions (300 V discharge voltage, 600 G maximum magnetic field strength, and 0.5 Pa gas pressure) the EGIP was typically found to be around 1/3, i.e., about 2/3 of the electrons appears to be captured. Self-Consistent Model The model was made self-consistent by considering the ionization distribution from a large number of electrons emitted from the cathode, taking into

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account the multiplication factor and EGIP for each electron. Indeed, from the ionization distribution, obtained in this way, the ion bombardment at the target can easily be deduced. This determines the average number of emitted electrons, based on the ion-induced secondary electron emission yield (γ). In this way, the model is made self-consistent, by iterating through this process, and demanding that the original and the new secondary electron emission profiles should be identical. With this model, the Thornton relation [142] for determining the minimum discharge voltage at which the magnetron can be maintained is extended to a general relation: W , (3.2) Vd = γ < f m > εe ε i where W is the effective ionization energy (∼30 eV for argon), εi is the ion collection efficiency, and εe stands for the theoretical number of ions that the electron efficiently generates before it is lost from the discharge. Both εi and εe can be assumed equal to one for typical magnetron conditions. Note that in the original Thornton relation, sheath ionization is not considered (m = 1) and the EGIP (f ) was assumed equal to 0.5. Because the model developed by Buyle et al. is two dimensional and includes the magnetron configuration (geometry and magnetic field strength), f and m can be calculated more accurately. This extended Thornton relation illustrates that the pressure and magnetic field influence the magnetron discharge through f and m, as all the other quantities in the formula are constant for a given target material and typical magnetron conditions. Calculation of the Discharge Current Furthermore, to deduce the discharge current (Id ) from the obtained magnetron properties, the line current density, jm , was introduced, which is related through the racetrack length, Lrt as: Id = jm Lrt . To determine jm , the Child-Langmuir law had to be extended, leading to the following relation for jm : ! jm = jd A||,t A||,s

B(x)dx,

(3.3)

where jd is the surface current density, the factors A,t and A,s account for the fraction of ions generated within the sheath (typically between 1/4 and 1/2), and for the ions that are generated scattered over the racetrack length, respectively. Finally, the integration is needed to convert the surface current density, jd , into the line current density, jm , and accounts for the nonuniformity across the racetrack (x is the direction across the width of the racetrack). Using this relation, the discharge current could be calculated. The model was able to reproduce the behavior of the magnetron discharge with decreasing pressure and constant current. Also typical current–voltage characteristics

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could be simulated. However, it appeared that the extreme steepness of the current–voltage characteristics was missing, and that the influence of the magnetic field on the gas pressure dependence was not retrieved. This example clearly illustrates the possibilities and limitations of analytical modeling. Effect of Coulomb Collisions and Anomalous (or Bohm) Transport Finally, to resolve the discrepancy between the simulated and experimental results, two processes not included in the model so far were examined, i.e., Coulomb collisions and Bohm (or anomalous) diffusion. The first process could explain the magnetic field dependence of the pressure effect in cylindrical (or post) magnetron discharges, but it can be ruled out for planar magnetrons, because the high energy electrons are too energetic. The latter process was added to the model in an empirical way, by introducing an artificial type of interactions, the Bohm diffusion collisions. The idea comes from Hall thruster simulations, where a similar approach is followed [135, 136]. The Bohm diffusion collisions represent the electron interactions with the electric field oscillations, which are supposed to generate anomalous diffusion. The “amount” of Bohm diffusion is defined as the occurrence of the Bohm diffusion collisions relative to the standard collisions (ionization, excitation, and elastic collisions), and it is proportional to the magnetic field and inversely proportional to the gas pressure, indicating that anomalous electron transport is especially important at strong magnetic fields and low pressures. The influence of increasing the “amount” of Bohm diffusion on the magnetron properties was simulated. These changes could be related to the increased electron mobility in the direction perpendicular to the magnetic field lines. The main advantage of including Bohm diffusion is that it enables to reproduce the extremely steep current–voltage relations, which are characteristic for a magnetron discharge. 3.4.2 Examples of Calculation Results The results of this analytical model were compared with experimental results, using two input values, i.e., the electron reflection coefficient and the relative occurrence of Bohm diffusion collisions, as tuning parameters. Two examples will be discussed in the sections below. In both cases the experimental results were obtained using a commercially available planar circular magnetron (Von Ardenne PPS50, see also Sect. 3.5). Calculated Ionization Distribution vs. Measured Plasma Emission Distribution Figure 3.2 shows the plasma emission recorded as a function of distance perpendicular to the target. The experiment and, hence, the results are very

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Fig. 3.2. The influence of the gas pressure on the optical emission of the magnetron at 50 W, as measured (a) and as simulated (b). Note that the emission has been normalized to the maximum emission at 2.2 Pa (2.0 Pa) for the experiments (simulations). The dashed vertical line (part a) indicates the position of the target. The dotted line (part a) suggests the “true” emission, i.e., without target reflection, for the case of 0.40 Pa

similar to what was measured by Lan Gu and Lieberman [41]. The z  -axis represents the distance along the direction perpendicular to the target (expressed in pixels). The vertical line at z  = 107 indicates the position of a small peak that is due to the light that reaches the camera after reflection on the target. According to [41] the exact position of the target surface is probably at slightly lower z  -values. The z  -axis in pixel-scale could be converted into the z-axis in mm-scale. In the analytical model the light emission of the plasma is not simulated. However, the excitations and ionizations generated by the high energy electrons occur practically at the same positions. Hence, the ionization rate distribution can be used for comparison with the experimentally measured emission intensity and is plotted in Fig. 3.2b. As one can see, the shift of the peak and the decreased intensity with decreasing pressure are reproduced very well. Current–Voltage Characteristics For a second example of the analytical model, the discharge voltage is calculated as a function of the electrical current for different secondary electron emission yields (γ) at constant pressure (0.3 Pa). The variation in γ was experimentally achieved by using different target materials under otherwise identical experimental conditions. Figure 3.3 shows that the increasing steepness of the current–voltage characteristic with increasing γ is reproduced nicely. The relative spacing between the curves is also reproduced very well for γ = 0.06 through 0.14. For γ = 0.18 the experimental curve lies much closer to γ = 0.14 than the simulated one. A possible explanation might be that the choice of yttrium (Y) as material for obtaining the experimental curve corresponding

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0 250 300 350 400 450 500 550 600 discharge voltage Vd (V)

Fig. 3.3. Influence of the secondary electron emission yield γ on the current–voltage characteristics at 0.3 Pa, as measured (a) and as simulated (b). The experimental curves are obtained with different target materials: Au, Al, Mg, and Y. The corresponding γ-values used in the simulation for these materials are 0.06, 0.10, 0.14, and 0.18, respectively, as described in [143]

with γ = 0.18, is based on a calculated value of γ and not on experimental data, see [143]. In view of the relative simplicity of this analytical model, and the fact that certain aspects are completely neglected (such as the effect of bulk electrons), the agreement between the simulations and experimental data is very satisfactory. Therefore, it can be concluded that this analytical model is able to reproduce self-consistently the influence of the main three external parameters (i.e., magnetic field strength, gas pressure, and electrical power) over a wide range. Also the effect of the secondary electron emission yield, γ, could be correctly simulated. This indicates that the model captures the most important processes occurring in the magnetron discharge, and is therefore a valuable “virtual” tool to gain insight in the generic magnetron behavior.

3.5 PIC-MCC Model for a DC Planar Magnetron Discharge This section will give a detailed description of the PIC-MCC model developed by Kolev and Bogaerts for a DC planar magnetron [37–39, 144]. Some characteristic results will also be illustrated. 3.5.1 Particle-In-Cell Model The PIC model is a numerical model describing an ensemble of collisionless charged particles, which can be, but not necessarily, under the influence of an external electromagnetic force. The constituents of the ensemble are not real physical subjects. They are artificial objects, representing the main physical

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properties of the real particles, such as charge, mass, and momentum. These computational particles are referred to as superparticles (SPs). Each SP represents a large number of real particles. This number is called weight and is in the order 106 –109 . Thus, the first approximation of PIC is the replacement of the real particles in a physical system with artificial superparticles. The second approximation is the discretization of time. This means that the simulated system jumps from one temporary state to another, in contrast to the real system, where the time evolution is continuous. The third approximation is the space discretization. Indeed, a mathematical spatial grid is imposed. The electrostatic field that results from the position of all simulated particles at a given moment of time is calculated on the grid only. The external (electric and magnetic) field, if any, is calculated on the same grid. The particles advance is then made by interpolating the forces resulting from these fields from the grid to the particle locations. The trajectory of the SPs is simulated during a large number of time steps. A schematic diagram of a PIC computational cycle, over a time period Δt equal to the discretization pace in time, for the case of an external magnetic field, B and electrostatic field E, is presented in Fig. 3.4. Before the first execution of the cycle, the initial state of the system needs to be specified. This includes the initial distributions of the particles in the coordinate and the velocity spaces. In fact, this is the main guess in a PIC simulation, so it must be a physical one. In following sections, the different steps of the computational cycle will be explained in detail. Integration of equations of motion, moving particles Fi ⎯→ vi’ ⎯→ xi

Weighting (E, B)k ⎯→ Fi

Particle loss / gain at the boundaries (emission, absorption)

Conver gence

Δt

No Integration of Poisson’s equation on the grid (ρ)k ⎯→ (E)k

Yes

End

Weighting (x, v)i ⎯→ (ρ)k

Fig. 3.4. A scheme of a PIC cycle for the case of electrostatic field, E, and an external magnetic field, B. Here v represent the velocity of the SPs, x their coordinate, F the force acting upon them, and ρ is the charge density. The index k refers to the grid and the index i to the SPs. The time step of the cycle is Δt

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Integration of the Equation of Motion In a DC magnetron, the force acting on a charged particle has two components: electric and magnetic. The gravitational force is ignored because its magnitude is very low in comparison to the electric and magnetic forces, due to the small masses of the electrons and ions. Further, the magnetic force can be considered external, because the magnetic field that results from the motion of the charges inside the magnetron is only a small perturbation to the magnetic field created by the magnets. The electric force, on the other hand, is a combination of the applied (external) electric field and the electric field induced by the charged particles. Thus the force is given by: F = F electric + F magnetic = qE + q(v × B),

(3.4)

leading to the equations of motion m

dv = q(E + v × B) dt dx =v dt

where x denotes the coordinate. This is a system of ordinary differential equations. It is discretized by the method of finite differences [145] using central differences for maximum stability and accuracy at low computational cost. A standard way to achieve this is to use the leapfrog algorithm, which is explained in Fig. 3.5.

velocity v-Δt /2

vt-Δt/ 2

vt+Δt / 2

time

coordinate xt Ft - Δt / 2

0

t - Δt /2

t

xt+Δt Ft+Δt

time

t + Δt /2 t + Δt

Fig. 3.5. A sketch of the leapfrog algorithm. The position of a particle is advanced from a moment t to a moment t + Δt, even though the velocity is not known in neither of the two moments, but in between. This represents the time centering. In time t = 0, the initial conditions are specified at the same time. That is why the velocity is initially returned half a time step back in time

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The time centering is clearly seen. The application of the leapfrog algorithm leads to the following system of finite difference equations

v t+Δt/2 − v t−Δt/2 v t+Δt/2 − v t−Δt/2 q = Et + ×B Δt m Δt x t+Δt − x t = v t+Δt/2 . (3.5) Δt The magnetic field is not indexed, because it is constant with time. The system (3.5) is in vector form. When written in components, it produces in general six scalar equations. Those of them that originate from the first equation are heavily coupled because of the rotation term. This makes the direct solution very complicated and computationally ineffective, because a solution is needed for every particle per each time step. The problem can be surmounted by noticing that the first term in the right-hand side of (3.5) is acceleration along the electric field and changes the magnitude of the velocity, while the second term is a rotation of the velocity vector, which does not alter the velocity magnitude. Then the force can be split into pure acceleration and pure rotation. To stick with the time centering, the acceleration can be performed in two stages, each with the half of the time step, while the rotation is performed at once, in between the two half-accelerations. In terms of velocities this can be expressed as half acceleration

f ull rotation

half acceleration

Δt/2

Δt

Δt/2

v t−Δt/2 −−−−−−−−−−−→ v  −−−−−−−−−−−→ v  −−−−−−−−−−−→ v t+Δt/2 , where v  and v  are some dummy velocities. Half-accelerations are trivial for handling since there is no coupling there. The rotation can be handled geometrically, as first suggested by Boris [146] and explained in detail in [144]. Charge Assignment Once the particles’ positions are calculated, the charge density, ρ, at the grid must be obtained. This procedure is called charge assignment. It is done, by ascribing fractions of each point charge to the neighboring grid points. To account for the charge conservation, the sum of all fractions must be equal to the point charge. The resulting charge, assigned to a given grid point is the sum of all fractions assigned to that grid point. The charge density then is the grid charge divided by some specified volume around it. The function, which determines what fraction of the point charge is assigned to a given grid point, is called weighting function or shape factor, S. The latter term arises from the way the grid sees the particles. There are different ways to construct the weighting function. The choice is always a compromise between more accurate physics incorporated and higher computational costs. The simplest scheme for charge assignment is the so-called nearest grid point (NGP) scheme. The NGP scheme assigns the whole point charge to the

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NGP. It is computationally very attractive, but is seldom used, because of its coarseness that can introduce fatal numerical instabilities. The NGP scheme is of zeroth order. The first-order scheme is the so-called cloud-in-cell (CIS) scheme, first introduced by Birdsall and Fuss [147] for one-dimensional plasma simulations. The CIS scheme uses linear interpolation to assign the charge to the NGPs. It is much more accurate and also smoother than the NGP scheme. When still more smoothness or accuracy is desired, higher order schemes can be used. Then a point charge is assigned not only to the closest grid points, but also to one or several distant grid points. The interpolation function is then of second or higher order. The fact that it is not linear anymore leads to often unacceptable computational intensification. That is why the CIS scheme is the most commonly used scheme. In two dimensions, the linear weighting may be performed in two ways. The first one is the complete analogue of the CIS scheme. The second one is area weighting, also known as bilinear weighting. When axisymmetric systems are modeled, the usual choice of coordinate system is cylindrical (r–z). Although formally two-dimensional, it in fact describes a volume. To represent this correctly in the charge assignment, the weighting is to be volumetric (r2 , z), rather than area weighting. The way it is done is shown and explained in Fig. 3.6. Integration of Poisson’s Equation on the Grid The charge distribution, characterized by the charge density, ρ, creates an electrical potential, V , given by Poisson’s equation, which in cylindrical (r, z) coordinates reads ∂ ∂V (r, z) ρ(r, z) 1 ∂ ∂V (r, z) r + =− r ∂r dr ∂z ∂z ε0 ρ(r, z) = q(ni (r, z) − ne (r, z))

(3.6)

where ni and ne represent the ion and electron number density, respectively. The discretized form of (3.6) is obtained by applying the Gauss’ law, for the dashed volume, centered on a grid node (ri , zj ) (see Fig. 3.7): qi,j = 2πri+1/2 ΔzEr i+1/2,j − 2πri−1/2 ΔzEr i−1/2,j ε0   2 2 (Ezi,j+1/2 − Ezj, i−1/2 ), + π ri+1/2 − ri−1/2 where the geometric factors are defined as follows Δri+1/2 ≡ ri+1 − ri " 2# 2 2 Δr i ≡ ri+1/2 − ri−1/2 " 2# 2 Δr 0 ≡ r1/2 .

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z Δr A

B qk

Δz C

zj+1 zk

D

zj

ri

θ

r

rk ri+1

Fig. 3.6. Illustration of the charge assignment in cylindrical (r, z) coordinates, according to the volumetric weighting CIS scheme. The charge qk , located at point (rk , zk ), is distributed among the four grid nodes A, B, C, and D of the grid cell it is located in. The fraction of qk that is assigned to A, for example, is equal to the product of qk and the ratio of the volumes: (1) obtained by full rotation of the shaded area of the grid cell ABCD around the z-axis and (2) obtained by full rotation of the whole grid cell ABCD around the z-axis. Δr and Δz are the grid sizes in r- and z-direction, respectively

z

Δr Ez Er

zj

q, V Δz 0

ri

r

Fig. 3.7. A sketch of the computational grid with grid separations Δr and Δz. The dashed line centered on the grid point (ri , zj ) represents the surface on which the Gauss’ theorem is applied. The discrete values of the charge, q, and the potential, V , are known in the grid nodes, whereas the components of the electric field, Er and Ez , are defined on the Gauss’ surface. The surfaces allocated at the origin, r = 0, (left-dashed box) are half the size of those located in the volume

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Employing single cell finite differencing Er i+1/2,j =

Vi+1,j − Vi,j Δrj+1/2

Ez i,j+1/2 =

Vi,j+1 − Vi,j Δz

and using a standard five-point stencil [146], the discretized form of (3.6) becomes 2ri−1/2 2ri+1/2 (Vi+1,j − Vi,j ) − (Vi,j − Vi−1,j ) (Δr2 )i Δri+1/2 (Δr2 )i Δri−1/2 1 1 + (Vi,j+1 − 2Vi,j + Vi,j−1 ) = ρi,j i > 0 Δz 2 ε0 2r1/2 1 1 (V1,j − V0,j ) + (V0,j+1 − 2V0,j + V0,j−1 ) = ρ0,j 2 2 (Δr )0 Δr1/2 Δz ε0

i=0 (3.7)

The charge density ρi,j is obtained from ρi,j = qi,j /voli,j , where voli,j ≡ π(Δr2 )i Δz.

(3.8)

In the second of (3.7), an implicit boundary condition is implied representing the symmetry around the axis r = 0 $ ∂V $$ = 0. ∂r $r=0 The other boundary conditions are discussed in the next section. Equation (3.6) is an elliptic partial differential equation with separable coefficients. This favors the use of some of the so-called rapid elliptic solvers [148], which are much more efficient than the more frequently used mesh-relaxation methods (MRM’s) [149]. Since there is no periodicity in either of the directions, fast Fourier transform techniques [150] are inapplicable. Instead, the cyclic reduction method (CRM) can be employed. The CRM is more complicated from programmer’s viewpoint, but it is the most efficient numerical method for the case of interest. The description of the general CRM and a possible algorithm for its implementation is given in [151]. If an equidistant grid in the z-direction is used, the CRM algorithm is even faster. An additional advantage of the CRM is that as a direct method, it is presumably more accurate than any of the MRM’s, which are iterative. External Circuit In modeling of magnetron discharges, an external circuit needs to be incorporated, to ensure that the model describes the magnetron discharge in the correct current–voltage regime [37,144]. Figure 3.8 gives a schematic diagram of a simple external circuit, consisting of a constant voltage source, Vext , and

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Cathode

Rball

Vext

Fig. 3.8. An external circuit consisting of a constant voltage source, Vext , and a ballast resistor, Rball , in series with the cathode. The dot-dashed line represents the symmetry axis, r = 0

a ballast resistor, Rball in series with the cathode, together with a sketch of the planar magnetron considered in the present study. The presence of an external circuit leads to the necessity of simultaneous advance in time of the circuit and the discharge. This problem is known as modeling of bounded plasmas with external circuits [152]. Its numerical application in 1D has been given in detail in [152–154]. A comprehensive procedure for the case of 2D Cartesian coordinates can be found in [155]. For 2D cylindrical (r, z) coordinates, certain modifications in the procedure of [155] are necessary. This is explained in detail in [144]. In the following, a brief description is presented. The coupling between the circuit and the discharge is maintained through satisfying the charge conservation at the cathode A

dσ = Iext (t) + Qdisch dt

(3.9)

where σ is the total surface charge density at the cathode, Iext is the external circuit current, A is the cathode surface, and Qdisch is the charge deposited from the discharge on the cathode during a period dt, due to the bombarding fluxes of the charged plasma species. The total surface charge can be determined independently of (3.9) applying the Gauss’s theorem on a number of boxes, closely surrounding the cathode. This is done in the following manner. The cathode surface is divided into nr1 boxes, as shown in Fig. 3.9.

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z Ez ½, 0 i-1

0

Ez ½, nr1

Ez ½, i i

i+1

nr1 E r r 0, nr1+½

Fig. 3.9. The cathode surface (z = 0, r ∈ [0; rnr1 ]) is enclosed by nr1 boxes. For each of them, the Gauss’ law is applied. For all boxes, except the one centered on r = nr1 , the flux of the electric field through the boxes’ surfaces is nonzero only in +z-direction. For the box centered on r = rnr1 , there is also a nonzero flux in r-direction

For the ith box the Gauss’ law is % ε0 E · ds = Qi , Si

where Si is the surface of the ith box, ds is the surface element, and Qi is the total charge inside the box. The total charge, Qi , is a sum of the surface charge, caused by the charging of the cathode by the discharge and the external circuit, and the volume charge that comes from the charged plasma particles. Then the Gauss’ law takes the form % % ! ε0 E · ds = Qi = σi ds + ρi dV , Si

Si

V oli

where σi is the surface charge density at point i, ρi is the charge density in the same point, and V oli is the volume of the box. This is equivalent to εEz i,1/2 =

1 ρi Vol i + σi 2

(3.10)

By use of finite differences, Ez i,1/2 can be expressed as (Φ0 − Vi,1 )/Δz, where Φ0 is the cathode potential. Rearranging (3.10) for σi and summation over all boxes i, yields an expression of the total surface charge density on the cathode (σT ) as a function of the cathode potential (Φ0 ), the potential values at the first grid point in the discharge, for all the boxes, i.e., radial positions (Vi,1 ), and the corresponding volume charge densities in these boxes [144]. Furthermore, (3.9) can be discretized with backward finite differences " # A σTt − σTt−1 = [Iext (t) + Idish ] Δt (3.11)

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Then according to Kirchhoff’s voltage loop law, the cathode potential, Φ0 = Vext − Rext Iext . Expressing from there Iext and introducing it to (3.11) produces

1 Δt σTt = (Vext − Φ0 ) + Qtdish + σTt−1 (3.12) A Rext Equation (3.12) can be combined with the equation for σT (obtained by rearranging (3.10); see above and [144]) in a single equation for Φ0 as a function of the potentials, Vi,1 and charge densities, ρi,0 both taken in time t, and the total surface charge density at time (t − 1). The problem here is that such an equation is at the same time a boundary condition for the Poisson’s equation (3.7) (see above). This leads to the necessity that (3.7) must be solved iteratively until the boundary condition is fulfilled, which is extremely inefficient in computational sense. Alternatively, the potential at any grid point, Vi,j , can be expressed as a superposition of two other potentials P L Vi,j = Vi,j + Φ0 Vi,j .

(3.13)

P , is a result solely of the presence of charges inside the The potential, Vi,j P magnetron. Thus, it is a solution of (3.7) with boundary conditions vi.0 = P P Vi,nz = Vnr,j = 0, where nr and nz represent the grid nodes located at the L walls of the magnetron. The dimensionless potential, Vi,j , accounts for the influence of the cathode potential and is a solution of the Laplace equation L L ΔVi,j = 0 with boundary conditions Vi≤nr = 1 and V L = 0 at the walls. 1 ,0 Replacing Vi,j with (3.13) finally yields an expression for Φ0 as a function of the potentials and charge densities in the plasma, in front of the cathode, at time t, the external (applied) potential, and the total surface charge density at time (t − 1).

−a2 Φ0 =

nr 1 −1 i=0

P P Vi,1 − a3 Vnr − a4 1 ,1

nr 1 −1

ρi,0 − a5 ρnr1 ,0 +

i=0

a1

Qtdisch A

t−1 + a6 Vext + σT

, (3.14)

where a1 −a6 are coefficients, dependent on geometrical factors and on L [144]. Vi,j In this way, Φ0 at time t can be calculated only from known quantities at the same time. Then the overall procedure for simultaneous advance of the circuit and the discharge in time is as follows. In the beginning of the simuL lation, the Laplace equation is solved to determine Vi,j . Next, the coefficients a1 through a6 are calculated. These two steps are performed only once. Then P at each time step, (3.7) is solved with zero boundary conditions to obtain Vi,j . L P With Vi,j and Vi,j known, (3.14) is solved for Φ0 . Afterward, (3.13) is used to produce the discrete potential distribution in the discharge Vi,j . Finally,

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t (3.12) is solved to get σT , which is going to be used in (3.14) during the next time step. Finally, once the potential, Vi,j , is known, the electric field E is calculated from E = −∇V.

Force Interpolation and Smoothing With E i,j calculated above, the force F i,j needs to be interpolated to the particle locations. This operation is known as force interpolation and is inversely identical to the charge assignment (see the section above). What is important is that the same shape factor must be used in both charge assignment and force interpolation. Not doing so usually results in a nonphysical self-force, i.e., a particle experiences a “force” caused by the particle itself [148, 153]. Using the same shape factor secures the total momentum conservation of the simulated system. The procedure of weighting (charge assignment and force interpolation) introduces naturally fluctuations of the grid quantities. The amplitude and the density of the fluctuations are reversely proportional to the number of SPs per grid cell. In magnetrons, the charge distribution is very inhomogeneous (see Sect. 3.5.4 below). This results in a situation where in some grid cells the number of SPs is very small, which can cause strong fluctuations. These fluctuations can evolve in instabilities that can terminate the simulation or bring it to a nonphysical solution. To prevent that from happening, digital filtering or smoothing is needed. The smoothing can be described as the substitution of the grid quantities at every grid node with some-averaged (smoothed) values received by averaging the grid quantities at the adjacent grid nodes. Different methods exist for that. In the current simulation, after the charge assignment is finished the charge density is smoothed before being used for calculation of the potential. There are many digital filters that in principle can be used [156, 157]. However, the filter must be isotropic to represent correctly the physical reality. Additionally, it must be computationally efficient because the filtering is performed at each time step. The filter, adopted in our work, uses the unfiltered values of the closest neighboring grid points (9-point filtering) [153]. It can be represented by the following transformation ρfiltered i,j ←

" raw # raw raw raw raw raw raw raw 4ρraw i,j +2 ρi−1,j +ρi+1,j +ρi,j−1 +ρi,j+1 +ρi−1,j−1 +ρi−1,j+1 +ρi+1,j+1 +ρi+1,j−1 . 16

Stability and Accuracy of the PIC Model The PIC model is a discrete approximation of a continuous physical picture. As such, it always introduces a numerical noise. The magnitude of this noise reaches zero only when the number of SPs per grid cell approaches the number

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of real particles in the same grid cell. Therefore, all kinds of particle simulations are by definition noisy in contrast to fluid models. The noise, however, should be kept under a certain maximum level to have meaningful results. More importantly, the error introduced by the noise must not be allowed to grow with the simulation. Several factors directly influence the noise. These include the time step, the number of SPs per grid cell, the shape factor, and the grid size. The specific mechanism through which this influence is manifested depends on the type of the numerical integrator of the equation of motion. In our model, the leapfrog integrator is implemented. A quantitative analysis for the error introduced by this method is given in [144]. It is found that the standard leapfrog algorithm has a quadratic error term for ω0 Δt 1, where ω0 is the characteristic frequency in the simulated system. In DC magnetrons, the two highest frequencies are the electron plasma frequency  ωpe = ne qe2 /ε0 me and the electron gyro frequency, ωce , given by ωc ≡ |q|B/m. Hence, ω0 = max{ωpe , ωce}. It should be emphasized that the error is cumulative with the number of time steps. This means that the longer the simulation is run, the bigger the overall error is. Consequently, the following paradox exists. Setting the tolerable error too small limits the number of time steps. Allowing a large number of time steps increases the error as the cube of the time step [153]. In the literature, a common compromise has been set ω0 Δt ≈ 0.2.

(3.15)

In magnetrons, the standard leapfrog algorithm is expanded to deal with the magnetic field and the rotation caused by that field. Thus, it is important to keep the calculated rotational angle reasonably close to the real one. An analysis of this problem can be found in [148]. It shows that for magnetic field strengths of interest the condition (3.15) secures that the rotation term of the equation of motion (see section “Integration of the Equation of Motion”) is calculated with an error not exceeding that of the general leapfrog algorithm. The exact condition is ω0 Δt < 0.35. (3.16) Generally, in two and three dimensions, the discussion for the stability and accuracy is much more complex. However, when one of the dimensions is dominant and the stability is achieved for it, it can be accepted that the whole simulation is stable. Typically, the dominant dimension is in the direction of the strongest gradient of the electric field. For our magnetron simulations, it is the z-direction. In line with that, the Courant criterion [158] is vk Δtk /Δz ≤ 1,

(3.17)

where vk is the characteristic velocity of the kth type of SPs and Δtk is its time step. The meaning is that, if (3.17) is violated, too many SPs are jumping

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over field variations, which leads to numerical heating. Thus for a given Δt, defined by (3.16), the formula (3.17) sets a bottom limit for the grid size, Δz. At the same time, the grid must be fine enough to resolve the sheath. This leads to the upper limit for Δz Δz ≤ λD ,

(3.18)

where λD is the Debye length. Relations (3.15), (3.17) and (3.18) define the stability and accuracy domain in terms of time step and grid size for the standard PIC simulation. Adding to that the MCC method (see below) will bring some additional conditions. More details about that are given at the end of Sect. 3.5.2. Modifications of the stability criteria with respect to speeding up the procedure are presented in Sect. 3.5.3. 3.5.2 Monte Carlo Collision Method The PIC method has been originally designed to model collisionless plasmas. In magnetrons, as in other types of glow discharges, collisions sustain the discharge. Therefore, a numerical model must be able to incorporate them. This can be achieved by coupling the PIC model with a Monte Carlo collision (MCC) model. This means that at a certain moment of the time step the SPs should be checked for collisions. The coupling is not just a numerical trick made to account somehow for the collisions. It has sound physical grounds, which follow from the Boltzmann equation [144]. The collision probability for all individual SPs needs to be calculated during every time step. Let us consider here the electrons as example of the SPs. It can be proven [159] that the probability, Pi , for the ith electron to collide binary with any of the particles of type g (gas atom, for example) during time Δt agrees with the probability received from the elementary free-path theory [160]. That is Pi = vΔt, where ν is the collision rate given by v = ng g¯σT with ng being the target density, g¯ the average relative speed between the ith electron and the target particles, and σT is the total cross section. In case of big disparity between the speed of the colliding particle, vi , and the mean speed of the target particles, a common approximation is g¯ ≈ vi . The cross section and therefore the collision rate are generally energy dependent. Since the velocities of the particles change with time, the collision rate is also time dependent. The probability, Q, that an electron, starting at time t = 0, does not collide in the interval (0, t + Δt) is Q(t + Δt) = Q(t)[1 − v(tΔt)],

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where the term in the square brackets is the probability for no collision in the interval (t, t + Δt). When Δt → 0, this equation becomes dQ = −v(t)Q(t) dt with a solution

! t

Q(t) = exp − v(t)dt .

(3.19)

0

Consequently, the probability that a collision takes place within a time t = tc , is given by: Prob{tc } = 1 − Q(tc ). This probability is calculated by generating a uniformly distributed random number (U ) in the interval [0,1] [161] 1 − Q(tc ) = U.

(3.20)

Hence, once U is generated, the time tc , which defines when a collision takes place, can be calculated from (3.19) and (3.20). This constitutes the general Monte Carlo procedure for determining whether a collision takes place. However, it is impractical, unless the collision rate is constant. Instead, the null-collision method, first introduced by Scullerud [162], can be used. Its main feature is the introduction of a constant maximum collision frequency, vmax , that is always higher than v(t) for any t. Hence, when (3.19) and (3.20) are solved with νmax instead of v, the solution is −1 tc = −vmax ln U.

(3.21)

To find a relation between tc and tc , the same U should be used. Then according to (3.20), tc < tc . The collision probabilities for v and vmax , respectively, at time, t = tc , are  !   tc

Pc = 1 − exp −

v(t)dt , 0

(Pc )max = 1 − exp(−vmax tc ).

(3.22)

Consequently, the probability that a collision is regarded real, i.e., it really happens in the model, is equal to Pc /(Pc )max . Note that the condition, Pc < (Pc )max , is always fulfilled. The efficiency of the method arises from the fact that there is no need to calculate the actual collision time tc in (3.20). If tc is small enough, which is always true if the time step of PIC is used, the collision rate can be

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approximated with a quadratic polynomial. Then the integral in the first of (3.22) becomes [139] !

tc

v(t)dt = 0

tc [v(0) + 4v(tc /2) + v(tc )] . 6

(3.23)

Equation (3.23) can be used to determine the probability for collision from (3.22). So far, the total probability of collision, PT has been discussed. When there are m types of collision, all possible in the time interval of interest, the probability for the kth type of collision (0 < k ≤ m), characterized by a collision rate vk is given by  !   tc

Pk = 1 − exp −

vk (t)dt 0

and PT =

m 

Pk .

(3.24)

k=1

Then the sampling of the kth collisional event is made with a probability of Pk /PT . The drawbacks of the null-collision method are connected to the implicit condition that the particles in the ensemble have the same collision rates. Or, what is equivalent, that their mean collision path is the same. Such situation is easily realized in a uniform electric field, for instance. That is why the method is especially efficient in swarm calculations. Its application to PIC simulations has been introduced by Vahedi and Surendra [163]. However, no theoretical analysis has been carried out to determine what strengths of the electric field gradients are allowed for the stability domain of the null-collision method. In magnetrons, strong gradients in both electric and magnetic fields exist, and as it has already been pointed out, the whole magnetron plasma is very inhomogeneous. The amplitude of the electric field gradients is proportional to the strength and geometry of the magnetic field. All this makes the use of the null-collision method in magnetron modeling highly questionable, especially at strong magnetic fields. An additional problem may appear from the selection of the particles (especially the electrons) which are considered for collision in each time step. The procedure of Vahedi and Surendra [163] requires the determination of some maximum number of particles eligible for collision. It is intuitively clear that when the particles are with small dispersion in their velocity distribution they can be selected randomly. Unfortunately, the latter is not the case in magnetrons, where a relatively small number of electrons are responsible for the main processes that sustain the discharge. Consequently, if the colliding electrons are selected purely on random basis it can lead to underestimation of the collisions of the “important” group of electrons.

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The above discussion motivates the introduction of a collision sampling procedure, both capable of dealing with strongly inhomogeneous velocity and spatial distributions and which is computationally effective. Such a procedure has been developed by Nanbu [164]. It makes it possible by use of a single random number to determine not only whether a collision occurs, but also what type of collision takes place. In this way, the computational efficiency is significantly improved. The checking is done on per particle basis, which ensures the equal treatment of all particles. For the case of electron-neutral collisions, where the speed of the neutrals is neglected and they are considered homogeneously distributed in space with a density, ng , the procedure is as follows. The total probability for collision is given by (3.24), where Pk = ng vσk Δt.

(3.25)

Here, v is the electron speed. Equation (3.24) can be rewritten as m

 1 − Pk , Pk + 1 = PT + (1 − PT ) = m k=1

which is visualized in Fig. 3.10. The unit length is divided into m equal intervals, where m stands for the number of different collision types. Each interval has two parts. In the kth interval, e.g., the left part has a length equal to 1/m − Pk and the right side is Pk . The sum of the right parts of all m intervals equals the total collision probability, PT and the sum of the left parts gives the probability of no collision, 1 − PT . The procedure is to generate a uniformly distributed random number U ∈ [0, 1]. The integral part of mU +1 specifies the kth interval corresponding to the kth collisional event, i.e., k = int[mU + 1]. Then only Pk needs to be calculated, which is a significant speed-up in case of big m. Finally, the same U is checked to see in which part of the kth interval it falls. If U>

k − Pk m

holds, the kth collisional event occurs. If not, no collision takes place. P1 0

1 m

P2

Pk

2 m

m−k−1 m

m−k m

Pm

m− 1 m

1

Fig. 3.10. Visualization of the Nanbu’s method [162] for sampling a collisional event. The right-hand side of the kth interval (drawn with thicker lines) represents the probability of the kth type of collision. The sum of the left-hand sides of all intervals (represented by thinner lines) is equal to the probability of no collision. The sum of the right-hand sides equals the total collision probability, PT

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In all MCC algorithms, the collisional time step, Δtc , must be small enough in order not to miss a real collision. This condition stems from the fact that for one time step, the colliding particle is allowed to undergo at maximum one collision. For the Nanbu’s method, this condition is Pk < 1/m for all m, for all electron energies. The electron energies in DC magnetrons are restricted approximately to the energy equivalent of the applied voltage. Then some Pmax can be estimated, that corresponds to the biggest cross section, σmax , of all the included processes for all the possible energies. Then the maximum allowed Δtc can be obtained from (3.25) with σk replaced by σmax and Pk = 1/m. This is practically not a limitation at all, when the method is used in PIC simulations, because the general time step limitation of PIC is more restrictive (see above (3.16)). The coupling of the collision check with the PIC cycle can be done in different moments inside the time step. Our choice is to check for a collision in the middle of the time step. This choice has been dictated by the time centering of the leapfrog integrator (see section “Integration of the Equation of Motion”, and especially Fig. 3.5). At this moment, the positions of the particles are exactly known. At the same time, only half of the acceleration is applied. The rotation may or may not be calculated. It does not matter, because the rotation does not change the speed and hence the energy of the colliding particle. Therefore, the energy is also known at the middle of the time step, though the velocity is unknown. The inclusion of the MCC module to the PIC method is shown schematically in Fig. 3.11. In case of collisions, the precollision velocity of the colliding particle is replaced by its postcollision velocity, which is determined based on the fundamental laws of conservation of energy and momentum during collision, plus geometrical considerations related to the particle’s orientation before the collision, as is explained in detail in [144], for the different types of collisions (electron-neutral, ion-neutral, neutral-neutral, and Coulomb collisions). 3.5.3 Methods for Speeding Up the Calculations As already mentioned before, PIC-MCC simulations of DC magnetrons are very time-consuming. This is caused by two main factors. The first one is the relatively high plasma density leading to a large number of SPs that need to be simulated. The second one is the time needed for convergence to be achieved. The simulation should be run for at least 10−5 s, which is the relaxation time in a DC glow discharge [165]. The high plasma density limits the allowed time step, which results in a large number of time steps (computational cycles) that should be performed to reach the 10−5 s limit. All this motivates the incorporation of computational techniques and physical approximations that can alleviate the computational load. Here, an overview of the methods, used to achieve this goal, is given.

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Integration of equations of motion, moving particles: Fi ⎯→ vi

MCC Yes

Weighting (E, B)k ⎯→ Fi

Integration of Poisson’s equation on the grid (ρ)k ⎯→ (E)k

Post-collision velocities vi ⎯→ vi’

Δt

Weighting (x, v)i ⎯→ (ρ)k

No

No

Convergence Yes End

Fig. 3.11. Scheme of a PIC-MCC cycle. The collision check is performed at time t + (1/2)Δt. The notation vi refers to the postcollision velocity of the ith particle. All other symbols are as in Fig. 3.4

Subcycling Subcycling [158] is the partial advance of the simulated system. This means that different parts of the system are advanced with different time steps. The fast evolving components are normally advanced a fixed number of times per one evolution of the slow components. If the system time step, and therefore the main cycle, is those of the slower components, then the fast ones are moved several times within the main cycle. Or in other words, they are subcycled. The fast particles that require the smallest time step (see section “Stability and Accuracy of the PIC Model”) are the electrons. Ions are roughly 2,000 times heavier than electrons and hardly move during an electron time step. This allows, according to the stability and accuracy criteria of the PIC method, that they can be advanced safely once per 20–50 electron time steps. The electric field, however, is recalculated after each electron time step. In this sense, the term subcycling is to some extent a misnomer in the present study. Variable Time Step The initial density, loaded in the beginning of the simulation, is much lower than the density in a steady state. The time step restriction [(3.16)] must be obeyed at any time. Since the plasma density changes, so can the time step. The classical approach is to fix the time step to obey (3.16) for the maximum expected density. This is a simple and safe decision from a programmer’s

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Time step, [s]

6x10-11

4x10-11

2x10-11

4x10-12 2x10-6

4x10-6

6x10-6

8x10-6

10-5

Time, [s]

Fig. 3.12. Adapting of the time step throughout a typical simulation

point of view, but is computationally ineffective, because unnecessary small time steps are used in the initial stage of the simulation when the discharge is being built up and the plasma density is low. Alternatively, we choose to use a variable time step. This is done in the following way. At the beginning, the time step is set to satisfy (3.16) for the initial electron density. Afterward, at every 5,000 time steps the maximum electron density in the discharge is found and if (3.16) does not hold anymore, the time step is increased by 25%. This is illustrated in Fig. 3.12, where the time evolution of the time step in a typical simulation is shown. As it is seen, during a significant part of the simulation, the time step is larger than the steady state time step. This allows decreasing the necessary number of computational cycles, and hence the overall computation time, by 30–40%. Optimization of the Weighting The weighting has been discussed in previous sections “Charge Assignment” and “Force Interpolation and Smoothing”. Each particle should be assigned to the four NGPs at every time step. If the coordinates of the particles are stored in physical units for distance, a floating-point division should be performed to determine to which cell the particle belongs. This is computationally very costly. Instead, the coordinates of the particles are stored in terms of grid cells. For example, if the ith particle has a coordinate, r = 3.356 expressed in cell units, rather than in meters, the determination of the cell in which the particle is located is done by simply taking the integer part of the coordinate.

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The remaining part specifies the location of the particle inside the cell. When the real coordinate of the particle is needed, a multiplication is performed, which is computationally more effective than a division. Sorting In the classical form of the PIC method, the simulated particles of a given type are indistinguishable. Each particle is just an element of an array. The position of the particle in this array, i.e., its number, is arbitrary and does not provide any specific information for the particle. This means that there is no correlation between the position of the particle in the array, and the particle’s location in the coordinate space. This organization is somehow natural to understand and straightforward to program. The addition of the MCC technique fits into this scheme if the collisions are only between the given type of particles and the background gas, which is homogeneously distributed in the velocity and coordinate space. Examples for such collisions are the electronargon atom collisions. In any other type of collisions (e.g., Coulomb collisions), a collision partner must be found, based on its spatial location. One way to do that is to perform a search among all the particles and to find all collision candidates that are in the vicinity of the incident particle. The usual method here is to approximate this vicinity to the cell in which the incident particle is located. Such procedure is, however, computationally very inefficient. Alternatively, if the array of the particles is not randomly formed, but ordered in such a way that the number in the array is in direct relation to the cell where the particle is located, finding a collision partner will be much faster. The ordering of the particles in such a way is called sorting. It not only facilitates the calculation of collisions, but also has a strong, positive side effect. It speeds up the entire PIC-MCC simulation. This acceleration is in relation with the architecture of the modern computers, as is explained in [144]. It has been reported [166] that for a 2d3v PIC simulation the calculation time decreases with 40–70% when sorting is implemented. Algorithms for ordering of data can be found in text books of numerical methods [167, 168]. However, common sorting techniques are very slow at sorting large arrays of particles, which is the case in a PIC-MCC simulation of a DC magnetron. For example, a quicksort, which is an “in place sort algorithm” [168], needs to pass through the particles approximately log2 N times, where N is the number of particles. This leads to the impressive 8kN log2 N floating point memory loads and stores for the case of a 2d3v PIC-MCC simulation with k types of particles. This inefficiency is a result of the fact that most of the well-known sorting algorithms originate from the past, when the amount of memory has been the main restriction. Contemporary workstations offer sufficient amount of memory and thus there is no need for in place algorithms (algorithms that order an array without using auxiliary arrays). Therefore, we used the “out of place sorting algorithm” [166], as is explained in detail in [144].

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Different Weights for Different Types of Particles Depending on the aim of the particular magnetron simulation, different plasma species can be included that can have rather different number densities. For example, while the electrons and argon ions have always similar densities, the density of the sputtered atoms and fast argon atoms is typically one to three orders of magnitude higher. In this case, it is not feasible to use the same weight for the neutrals and for the charged particles. The condition that the minimum number of followed particles must be such that there are at least several SPs per Debye sphere, combined with the demand for maximal speed of the calculations, determines the weight of the SPs representing the real type of particles with the lowest density. The weight of the SPs representing the real particles with higher density is chosen also higher. How much higher is determined from the practical consideration that the number of SPs from each type must be similar. Using different weights allows obtaining the necessary statistical representation, without paying the price of following an unnecessary large number of SPs. The implementation of different weights is trivial in a PIC code. When a MCC method is added, however, there are two cases. The first one is when plasma species collide with the uniformly distributed background gas, or when they collide with other plasma species with the same weight. In this case, the procedure is the same as when all SPs are equally weighted, because the weight is anyway included in the charge assignment and force interpolation. The second case includes collisions between SPs with different weight. This requires an additional treatment, as explained in detail in [144]. The guiding principle here is that the SPs represent real particles, and all physical relations between the physical particles must be correctly represented by the SPs. 3.5.4 Examples of Calculation Results Operating Conditions and Simulation Data In this section, the PIC-MCC model described in Sect. 3.5, will be applied to a laboratory magnetron, Von Ardenne PPS 50 (see below), operated in argon with a copper cathode, to illustrate the type of results calculated with this model. This includes, among others, the distribution of the electric field and potential, the densities of the plasma species and their energy distribution functions, and the erosion profile as a result of cathode sputtering. The calculation results will be quantitatively compared with existing experimental and numerical data. The scheme of the planar magnetron used for the calculations is shown in Fig. 3.13. It is a Von Ardenne PPS 50 magnetron (commercially available), used with plasma shield (the sidewall on the scheme in Fig. 3.13). The axisymmetric magnetic field is created by two concentric magnets located under the powered electrode, i.e., the cathode. The magnetron is balanced, which means

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30 Substrate 20

z (mm)

10

0

Cathode

-10

S

N

N

S Yoke

-20 0

10

20 r (mm)

30

Fig. 3.13. Scheme of the magnetron Von Ardenne PPS 50 with the spatial distribution of the (measured) magnetic field. The scheme is axisymmetrical relative to the axis, r = 0

that the majority of the magnetic flux lines originates at and returns to the cathode surface without crossing the anode. All walls, except the cathode, are grounded and act as an anode. The smallest separation between the electrodes is equal to 2 mm and the distance between the cathode and the opposite anode plate, where the substrate is mounted, is 24 mm. The cathode is a copper disk with a thickness of 3 mm and a diameter of 58 mm. The discharge is maintained by a DC power supply, which can be run in a constant current or in a constant voltage mode. The magnetic field used in the simulation has been experimentally measured when the discharge has been not operational. In the simulation, the external circuit shown in Fig. 3.8 above, is connected to the cathode. In this numerical study, the magnetron plasma consists of argon atoms, singly charged argon ions and electrons only. All other plasma species are excluded from the analysis. In addition, the argon atoms are considered homogeneously distributed at room temperature (300 K). The choice of these plasma species is dictated from the fact that they are the dominant ingredients and play the main role for the formation and maintenance of the discharge.

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The influence of the sputtered atoms and the presence of the nonequilibrium argon atoms will be discussed in Sect. 3.6 below. The collisions that are taken into consideration are electron elastic scattering from the argon atoms, electron- and ion-induced excitation and single ionization of the argon atoms, and elastic scattering of argon ions from their parent atoms. The last process includes isotropic scattering and charge transfer. The excited argon atoms are not followed in this simulation. The process of their formation, however, is important for a correct representation of the energy balance. In the case of electrons, it also contributes to the conductivity of the discharge, since the electrons cross the magnetic field lines due to collisions. Because the main excitation levels of the argon atom are very closely located in terms of energy [169], all electron-impact excitation processes are grouped into a single collision with an energy loss for the electrons of 11.55 eV. This is done to limit the computational time. The cross sections of the abovementioned collisions have been adopted from the literature [169–173], and can be found back, plotted as a function of energy, in [144]. At the walls, besides sputtering and secondary electron emission, also recapture of electrons at the cathode is included (see also the previous section). The maximum number of the SPs in this simulation is two millions, i.e., one million electrons and one million argon ions. In the beginning of the simulation, they are loaded with uniform density in the coordinate space and with a Maxwellian distribution in the velocity space. The initial density equals 3 × 1014 m−3 . The external circuit is set with a resistor of 1,200 kΩ and with a constant voltage source of −800 V. This corresponds to typical experimental values of this type of magnetron. The maximum magnetic field is 0.045 T. The computational grid has 241 nodes in z-direction and 129 nodes in r-direction. The initial time step is set to 3×10−10 s. The number of electron subcycles per ion cycle is 25. The simulation is run until convergence is obtained in terms of cathode potential and particles’ densities. This is illustrated in Fig. 3.14, where the time evolution of the cathode potential, Φ0 , is shown. Note that, although the external applied voltage is equal to −800 V, the voltage drop across the magnetron discharge is only about −330 V. The gas pressure for this example of simulation results is kept fixed at 0.67 Pa. Calculated Potential and Electric Field Distribution The calculated potential distribution is shown in Fig. 3.15. It has a clearly expressed radial dependence, which follows the pattern of the magnetic field. The gradient in the potential is steepest near the cathode (i.e., in the sheath, see below), and at about r = 18.2 mm, where the radial component, Br , of the magnetic field has a maximum. Above the center of the cathode, where the magnetic field lines are perpendicular to the cathode surface (see Fig. 3.13), the potential shape is identical to that of a nonmagnetized discharge. As a whole, the plasma potential is negative almost in the entire discharge, i.e., the discharge operates in a negative space charge mode. This

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Cathode potential, Φ0 (V)

0

-100

-200

-300

-400 0x100

10-5

2x10-5 Time (s)

3x10-5

4x10-5

Fig. 3.14. Relaxation of the calculated cathode potential, Φ0 , with the physical time of the simulated system. The convergence occurs at time, t ∼ 12 μs

Fig. 3.15. Calculated potential distribution

is in accordance with analytical models [46, 62], a 1d PIC-MCC simulation of a cylindrical postmagnetron [36], and experimental measurements at similar operating conditions [174]. The sheath architecture can also be seen in Fig. 3.15. The sheath itself should be redefined in the case of magnetrons. In nonmagnetized glow discharges, the sheath boundary is frequently defined at the line, which separates the negative from the positive values of the potential. This definition works well in discharges operated in positive space charge mode. In mag-

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netrons, however, the sheath border should be defined as a line where the potential has a well-pronounced inflexion. After the sheath, the next part of the potential can be defined as a presheath. This region is still characterized by a negative space charge and a relatively small electric field (in comparison to the sheath). This presheath is totally absent in discharges with positive space charge. The reason for appearance of the negative space charge and the presheath is in the restricted mobility of the electrons, due to the magnetic confinement. The potential distribution from Fig. 3.15 creates an electrical field, which axial and radial components are shown in Figs. 3.16 and 3.17, respectively.

Fig. 3.16. Axial component, Ez , of the electric field. The white line corresponds to Ez = 10 kV m−1 and may be considered as the end of the sheath

Fig. 3.17. Radial component, Er , of the electric field

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Fig. 3.18. Radial component of the electric field, Er , in the region r ≤ 25 mm

The magnitude of the axial field, Ez , is approximately ten times stronger than the magnitude of the radial field, except in the proximity of the gap between the cathode and the sidewall. In the sheath region above the racetrack, Ez approaches 500 kV m−1 . The sheath border with its strong radial dependence is represented by the white line in Fig. 3.16. Along it, the magnitude of the field is approximately 10 kV m−1 . The sheath thickness is only 1.6 mm above the racetrack, whereas at r = 0 it is about 13 mm. The sheath is thinnest and Ez strongest exactly in the middle between the magnetic poles (r = 18.2 mm). The radial field, Er , changes in the sheath. The field is strongest near r = rmax , where the gap between the cathode and the grounded side wall is located. The structure of Er is such that the region where Er > 0 repels the electrons outward, whereas the region Er < 0 accelerates the electrons outward. This has an effect of electrostatic trap, which enhances the magnetic confinement. The “valley-mountain” structure of Er is more clearly seen in Fig. 3.18, where the region of the discharge away from the sidewall is shown. Calculated Electron and Ar+ Ion Densities Figure 3.19 shows the calculated electron density profile. Most electrons are strongly confined between the magnetic poles. The calculated density profile reproduces satisfactorily the experimental data obtained by Langmuir probe measurements. For example, the calculated peak value at p = 5 mTorr is 1.6 × 1017 m−3 , whereas in [174] it is 9 × 1016 m−3 at p = 2 mTorr. The plasma decay with the distance from the cathode, as well as the radial variation of the electron density, agree very well with the data reported in [172] for p = 5 mTorr. The plasma distribution has a distinctive maximum at z = 1.2 mm. Except in the sheath, the Ar+ ion density profile is identical to the electron density profile. This is so, because although the ions are not magnetized, and therefore not magnetically trapped, they are electrostatically bound to the electrons. The Ar+ ion density profile has a maximum at the same spot as

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Fig. 3.19. Calculated electron density

Fig. 3.20. Calculated profile of the electron-impact ionization rate

the maximum of the electron density. In the sheath, the Ar+ ion density is nonzero, in contrast to the electron density. This gives rise to a positive space charge, and hence strong gradients in the potential distribution and axial electric field distribution. Calculated Collision Rates For the sustainment of the discharge, ionization of the argon atoms is crucial. In the present model, ionization is carried out by electrons and Ar+ ions. The calculated electron-impact ionization rate is illustrated in Fig. 3.20. The Ar+ impact ionization rate is not shown, because it is of minor importance at the operating conditions investigated. It is present only in the sheath where the ions can gain enough energy from the electric field to ionize the atoms [144]. Its peak (5×1021 m−3 s−1 ) is about 1.5 order of magnitude lower than the peak of the electron-impact ionization (1.2 × 1023 m−3 s−1 ). Integrated over the whole computational domain, electron-impact ionization is about 800 times more

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Electron-impact ionization (m-3s-1)

1.2x1023

8.0x1022

4.0x1022

0.0x100 0

5

10 15 z (mm)

20

25

Fig. 3.21. Calculated electron-impact ionization rate, as a function of distance from the cathode, above the racetrack at r = 18.2 mm

important than Ar+ -impact ionization. Therefore, with reasonable accuracy, the Ar+ -impact ionization can be omitted in models and estimates for the current operating conditions. This is like expected, in view of the low voltage across the magnetron. In DC nonmagnetized glow discharges, where the applied voltage is in the order of 1 kV, this process cannot be neglected for an accurate description of the discharge behavior [175]. The axial profile of the electron-impact ionization rate above the racetrack at r = 18.2 mm is shown in Fig. 3.21. It has a maximum at z = 1 mm, i.e., inside the sheath. Its closeness to the cathode can be explained with the magnetic confinement. The Larmor radius for electrons with a speed of 107 m s−1 in a magnetic field of 450 G is 1.26 mm. This means that the electrons, ejected from the cathode, stay long enough in a hemisphere, which center is on the cathode and with a diameter equal to the Larmor radius. The electrons have an energy, which is optimal for ionization, before reaching the surface of the hemisphere. The shape of the ionization profile shows that most of the ionization is carried out by the primary electrons, originating at the cathode, rather than by secondary electrons, created in ionization collisions. There is nevertheless a significant amount of ionization taking place in the presheath and even in the bulk plasma. The borders of the two regions are reflected in the two inflexion points of the ionization profile at z ≈ 4 mm and z ≈ 14 mm. The rates of electron elastic collisions and electron-impact excitation are characterized by a similar profile as the electron-impact ionization rate, but a second maximum is observed at about z = 2–3 mm. This can be explained, because the mean electron energy in this region is around 10 eV, which gives

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Fig. 3.22. Calculated mean energy of the electrons. Note that the z-axis is now reversed with respect to the previous figures, for the sake of clarity of the presented results. The cathode is still located at z = 0. The white line in the bulk corresponds to an energy of 10 eV

rise to a lot of elastic collisions and excitation, but which is a bit too low for efficient ionization. Calculated Mean Energy and Energy Probability Function The collision rates, together with the distribution of the electric and magnetic field determine the spatial distribution of the mean electron and ion energy and the corresponding energy probability functions. The calculated mean energy of the electrons is presented in Fig. 3.22. Note that the z-axis is reversed now, so that the energy spatial distribution is more clearly visible. Above the racetrack, where the electrons are strongly magnetized, their mean energy is about 40 eV. With radial displacement from the racetrack, the mean energy increases to 180 eV above the edges of the cathode. This increase is proportional to the weakening of the radial component of the magnetic field away from the centers of the magnets. Those electrons that enter the bulk are mostly with low energies, i.e., below the inelastic threshold. This fact, as well as the sharp transition of the mean energy profile to a plateau, immediately after the sheath, illustrates the effectiveness of the magnetron: the primary electrons are almost entirely utilized in ionization and excitation events before being lost. The region along the symmetry axis is strongly depleted from electrons, which is reflected by the “valley” in the mean electron energy profile. The distribution of the Ar+ ion mean energy (Fig. 3.23) follows the potential distribution. As it can be expected, the mean energy is high only in the sheath, with a maximum on the cathode surface. In the bulk, it is low, i.e., between 0.2 and 0.6 eV. The mean ion energy in the region between the

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Fig. 3.23. Calculated mean Ar+ ion energy. The cathode is located at z = 0

poles of the magnet is about 225 eV, which is approximately 70% of the cathode voltage. This ratio is slightly higher than the ratio of 60%, calculated in [116]. The difference can be explained from the broader sheath in [116], which allows the ions to spend more time in the sheath and consequently increases the probability for symmetric charge transfer collisions. These collisions are one of the two factors preventing the ions from obtaining energy values equal to the full cathode potential. The second factor is related to the location of the maximum of the ionization (see Fig. 3.20 above); it is between the sheath and the presheath. Thus, most of the produced ions accelerate in a potential difference that is less than the full interelectrode potential. All this shows that the common assumption of “freely falling ions” is not very accurate. Finally, in contrast to the electron mean energy, the ion mean energy does not have a radial minimum above the racetrack. This is a direct result from the fact that the ions are not magnetized. The mean energy does not provide information about the population in the energy spectrum. Such information is given by the energy probability function (EPF), f (ε)(= F (ε)ε−1/2 ), where F (ε) is the energy distribution function and ε is the energy. The normalization of the EPF is: f (ε)ε1/2 dε = 1. The knowledge of the EPF also allows us to determine what the distribution of the velocities is and therefore to define a temperature, which allows simulated data to be compared with data received from probe measurements. As it has been already discussed, probe measurements are normally devoted to study the bulk plasma, because the sheath region compromises the accuracy of the probe readings [176]. For this reason, to compare our calculation results with experimental data, the quasilocal EPF has been sampled in the bulk in the present example. The sampling spot is a volume given by 14 mm < z < 16 mm and 16 mm < r < 20 mm. The calculated electron energy probability function (EEPF) is shown in Fig. 3.24. It represents practically a Maxwellian distribution with a temperature of 6.7 eV. Such a value is in good agreement with the spatial survey of magnetron plasma by means of a Langmuir probe [177]. In

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1012

EEPF (eV-3/2m-3)

1011 1010 109 108 107 106 0

20

40 Energy (eV)

60

80

Fig. 3.24. Calculated electron energy probability function (EEPF) in the bulk (14 mm < z < 16 mm, 16 mm < r < 20 mm)

the latter experiments, for p = 5 mTorr, the electron temperature at z = 3 cm is found to be 5 eV and decreases with increase of z(Te = 2.5 eV at z = 5 cm). Therefore, our calculated Te of 6.7 eV at z = 1.5 cm appears to be consistent with the experiment. The Maxwellian distribution at such low pressure is a result of the magnetic confinement of the electrons, which secures enough collisions with the argon atoms before the electrons diffuse into the bulk. There are reports in the literature for existence of two electron populations with different temperatures, which produces a Bi-Maxwellian distribution [178, 179]. More recent work [177], however, fails to confirm such phenomenon. The calculated results in the present example also do not indicate the existence of two electron populations with different temperatures. The Ar+ ions behave in a very different way than the electrons. Being practically not magnetized, the ions cannot stay for a long enough time in the discharge. This means they cannot suffer enough collisions for bringing them in equilibrium with the background gas. This is reflected in their EPF, shown in Fig. 3.25. As it is clear from this figure, the energy distribution is far from Maxwellian or other equilibrium distributions. In this case, the use of the temperature as a characteristic of the velocity distribution makes no sense. This result indicates that the use of fluid description for the plasma in DC magnetron discharges operated at standard conditions will not be correct. The same refers to hybrid models [110], where a Maxwellian distribution for the ions is normally assumed and they are treated as a continuum. The IEPF in Fig. 3.25 also questions the idea of measuring the ion temperature

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Ar+ IEPF (eV-3/2m-3)

1013

1012

1011

1010

0

4

8 12 Energy (eV)

16

20

Fig. 3.25. Calculated Ar+ ion energy probability function (IEPF) in the bulk (14 mm < z < 16 mm, 16 mm < r < 20 mm)

in DC magnetrons by probe experiments. In a more global aspect, the ion temperature does not bring any valuable information for the state of the plasma in DC magnetrons. It can only be used as mean ion energy, without providing any further insight for the velocity distribution. Calculated Ion Flux Bombarding the Cathode The Ar+ ions that reach the sheath region or are created inside the sheath, accelerate toward the cathode, which they can bombard. This bombarding flux is responsible for the generation of primary electrons that sustain the discharge and for the sputtering. The rate of sputtering is proportional to the energy density of the bombarding flux. The spatial distribution of this flux at the cathode surface determines the sputtering region, i.e., the racetrack. Therefore, the knowledge of the flux is important, to predict the utilization of the target at given operating conditions. The calculated Ar+ ion flux at the cathode is therefore presented in Fig. 3.26. The flux is more or less limited to the region between the magnetic poles. This is in accordance with all experimental data, for instance for the erosion rate due to sputtering (see also Sect. 3.6). Its amplitude is proportional to the Ar+ ion density. The localization of the flux, which follows the localization of the ion density in the radial direction, is a confirmation of the fact that the ions are practically not magnetized. Therefore, their movement in the sheath can be approximated as being directed toward the target, only disturbed by the collisions in the sheath.

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Ar+ flux at the cathode (m-2s-1)

0.0x100

-4.0x1020

-8.0x1020

-1.2x1021

-1.6x1021 0

10

20

30

r (mm)

Fig. 3.26. Calculated Ar+ ion flux at the cathode

As mentioned before, this Ar+ ion flux gives rise to sputtering at the cathode (target). The behavior of sputtered atoms and corresponding ions, as well as of the Ar metastable atoms (which are important for the ionization of the sputtered atoms), will be described in Sect. 3.6.

3.6 Extension of the PIC-MCC Model: To Include Sputtering and Gas Heating In Sect. 3.5, a detailed description has been given about the PIC-MCC model for an argon DC magnetron with copper cathode. The behavior of the sputtered copper atoms and corresponding ions was, however, not yet taken into account, because the Ar+ ions and electrons are the dominant plasma species, determining the magnetron discharge behavior. However, to improve the application of sputter deposition, it is of course important to have a better insight in the behavior of the sputtered atoms. Therefore, the present section deals with an extension of the previously described PIC-MCC model to include the sputtered atoms and ions. Moreover, it should be realized that the region in front of the cathode in a sputter magnetron is a highly dynamic region. Except the charged particles and equilibrium gas atoms, there exist additional energetic species such as reflected, neutralized gas atoms, atoms born in charge exchange collisions, and nonthermal sputtered atoms. All of them can participate in momentum transfer collisions with the background, cold atoms. These collisions are expected to be highly effective, due to the similar masses of the colliding partners. In this way, significant energy and momentum can be deposited from the energetic species to the background gas, giving rise to gas heating, and thereby

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creating density inhomogeneity. These processes will also be treated in the present section. 3.6.1 Description of the Model Species Included in the Model and Their Collision Processes The present model is based on the general PIC-MCC algorithm, presented in the previous section, but some additional species are included, i.e., the fast argon atoms (Arf ), sputtered copper atoms (Cu), singly ionized copper ions (Cu+ ) and argon metastable atoms (Ar∗m ). The latter species are included, because they play an important role in ionization of the sputtered atoms. In addition, several new collision processes are added, such as electron-impact excitation to the Ar∗m levels, and excitation and ionization from these Ar∗m levels, quenching of the Ar∗m levels by collisions with electrons or Ar atoms, electron–ion recombination, ionization of the sputtered Cu atoms by electronimpact, Penning ionization and asymmetric charge transfer. Details can be found in [144]. Besides these collision processes in the plasma, also some additional surface interactions are taken into account, such as sputtering and secondary electron emission caused by fast atoms and copper ions (Cu+ ). Furthermore, deexcitation of the metastable argon atoms and recombination of the argon ions takes place at the walls. The energetic argon atoms are reflected and possibly thermalized at the walls. The coefficient of thermal accommodation, which is a measure for the energy exchange between energetic plasma species and the walls, has been chosen equal to 0.5. Numerical Procedure As it has been explained in the previous section, the PIC-MCC simulation needs to be run until a convergence is obtained. This numerical convergence must be correlated to the attainment of steady state by the real physical system that is the subject of the simulation. The time necessary for the physical system to reach a steady state is determined by the slowest processes in the system. When the heating of the gas is taken into consideration, the characteristic time, τH for thermal equilibrium (i.e., bringing the heat conduction to a steady state) is the longest: τH ∼ 10−2 s. This estimate is based upon the relation [180] L cp ρL2 . = τH = VH k Here, L (0.1 m [101]) is the characteristic diffusion length, VH (= k/(ρcp L)) is the characteristic speed of heat transfer, cp is the specific heat per unit mass at constant pressure, ρ (= 7.97 × 10−5 kg m−3 [181]) is the argon mass density, k (= 0.018 W m−1 K−1 [181]) is the thermal conductivity of argon.

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At the same time, the characteristic time for bringing electrons and ions to steady state does not exceed 10−5 s. All other important processes, such as the relaxation time for thermalization of the energetic neutrals and pressure equalization, are situated in between these two limits. Because of this big difference in the characteristic time steps of the electrons, ions, fast neutrals, and thermal conduction, some modifications in the general PIC-MCC algorithm are necessary to cope with this disparity. Not doing so would result in a huge amount of computational time. This is so, because of the general restriction for the time step (3.16) (see section “Stability and Accuracy of the PIC Model”). The procedure used here [182] is to advance the different sorts of particles with different time steps. The hierarchy being Δte Δti Δtn (with e = electron, i = ion, and n = neutral). This difference in the time steps is accounted for by the weight, W of the produced energetic, charge-exchange neutrals and sputtered atoms, i.e., Wn = Ws = Wi

Δtn , Δti

(3.26)

where s refers to the sputtered atoms. Electrons are subcycled inside the ion time step (see section “Subcycling”) and have We = Wi . In this way, it is assured that the production and loss rates of the real plasma particles are correctly represented, i.e., the global mass conservation is obeyed. The procedure separates the particles into two groups, i.e., fast and slow. The upper size of the SPs from each group can be controlled independently from the corresponding value of the other group. When the upper limit is reached, only the members of the corresponding group are reduced twice, and their weight is doubled. The weight and number of the SPs from the other group are not changed. However, a change in the time step is needed, to maintain (3.26) to be valid. Because Δti is coherent with the stability criterion, the time step that always changes is Δtn (= Δts ). The overall cycle consists of one ion time step. At the end of this time step, the power, P , transferred to and from the feeding gas is accumulated   mg Wn  v  l 2 − vl 2  vl2  v  l 2 − + , P = Vcell Δtn 2 2 2 l

l

l

where vl is the post- and vl the precollision velocity of the lth gas atom, mg is the gas mass, Vcell is the volume of the computational grid cell. The first sum is the contribution from all collisions between the feeding gas atoms from one side and the ions, fast atoms, metastable atoms, and sputtered atoms from the other side. Only collisions, in which the postcollision energy of the gas atoms is less than some threshold, are counted. This threshold is chosen to be [182] (3.27) Eth = 9 × 3/2kb Tg , where kb is the Boltzmann constant.

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The other collisions result in creation of fast gas atoms, which are incorporated by the second sum. The third sum is the contribution of the thermalized fast gas atoms. This calculated power is used as a source term in the heat conduction equation ∂Tg P 1∂ ∂ 2 Tg r =− , (3.28) + ∂z 2 r∂ ∂r k which is solved once per 10−8 s (actually, after each number of time steps divisible exactly by 100 and which sum is greater than or equal to 10−8 s) to calculate the gas temperature, Tg . In the above equation, k is the thermal conductivity of the gas. It should be mentioned that Tg is a dynamic quantity dependent on the coordinates. Consequently the threshold energy given by (3.27) is also dynamic (changes with time) and is a function of the coordinates. The sputtered atoms are followed as particles until being thermalized. Once that happens, they cannot anymore contribute to the gas heating directly; collisions between fast argon atoms and sputtered atoms are disregarded in the model, because of the statistical insignificance of the process. The overall copper density, however, is an important quantity for film deposition purposes. Therefore, a compromise between accuracy of the algorithm and its computational efficiency is to treat the thermalized copper atoms as a fluid. Thus, the overall copper density is a sum of the density of the fast copper atoms and the slow copper atoms. The density of the slow (thermalized) copper atoms, nsl Cu , can be obtained by solving the diffusion equation, which in (r, z) cylindrical coordinates reads DCu Δnsl Cu (r, z) = rloss (r, z) − rprod (r, z),

(3.29)

where DCu is the diffusion coefficient of copper atoms in argon, rloss is the rate of loss of the slow copper atoms, and rprod is their production rate. The −1 diffusion coefficient is taken to be, DCu = 1.44 × 10−2 cm2 s [183]. This value is based on the rigid-sphere collision model and refers to a pressure of 1 Torr and temperature of 300 K in argon. The loss rate of the copper atoms, rloss (r, z), is equal to the rate of ionization of the copper atoms, caused by electron-impact, Penning ionization, and asymmetric charge transfer. The production rate of the copper atoms, rprod (r, z), is equal to the rate of thermalization of the sputtered atoms. Equation (3.29), as well as (3.28), is mathematically the same as the Poisson equation for the potential. Therefore, the same numerical technique, i.e., cyclic reduction [154], is used for their & sl solution. ' The boundary conditions of (3.29) are Δnsl Cu |wall = 0 and ∇r nCu (0, z) = 0. The latter represents simply the cylindrical symmetry of the system. The procedure described above can be represented by the flowchart in Fig. 3.27.

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Initial density and velocity distribution

PIC / MCC for e−, Ar+, Cu+ Production of neutrals Arf, Arm* , Cu with statistical weight Wn = Wi Δtn / Δti Advancing the external circuit

DMCS for Arf, Arm* , Cu, Production of e−, Ar+, Cu+ with statistical weight Wi = Wn Δti / Δtn

Heat Transfer Temperature calculation New gas density

Diffusion transport of Cusl New total Cu density

tn+1 = tn + Δt

No

Convergence

Output

Yes

Fig. 3.27. Flowchart of the simulation procedure for calculation of the gas heating and the behavior of the sputtered (copper) species. The abbreviation DMCS stands for “Direct Monte Carlo simulation”

3.6.2 Examples of Calculation Results The simulation procedure shown in Fig. 3.27 is applied to the magnetron discharge presented in Fig. 3.13. Simulations performed for pressures of 0.13, 0.53, 1.3, 3.3, 6.7, and 13 Pa, and for a maximum radial magnetic field above the cathode of 1,300 G, will be illustrated here. To verify the expected pressure dependence of the gas heating, the parameters of the external circuit (see section “External Circuit”) have been readjusted in the course of the runs to maintain a constant power of 70 W.

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z (mm)

20 15 10 5 0

-30

-20

-10

0 10 r (mm)

20

30

Fig. 3.28. Contour plot of the calculated gas temperature distribution at p = 6.7 Pa

Gas temperature (K)

1000

800

600

400

200 0

2

4

6 8 10 Pressure (Pa)

12

14

Fig. 3.29. Calculated maximum gas temperature as a function of the gas pressure at constant electric power of 70 W

Temperature Distribution Figure 3.28 illustrates the calculated gas temperature distribution for the case of 6.7 Pa. The temperature reaches its maximum in the center of the discharge (both in axial and radial direction. The effect of gas pressure on the maximum gas temperature is plotted in Fig. 3.29. At 0.13 and 0.53 Pa, there is practically no heating of the gas. The rise of the gas temperature is about 1 K, which is within the limit of the expected calculation error. With the increase of the pressure, the gas begins to heat up. For p = 1.3 Pa, the temperature increases to 316 K, at 6.7 Pa, the maximum gas temperature is about 600 K (cf. also Fig. 3.28), and it reaching almost 1,000 K at 13 Pa.

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Hence, it can be concluded that for the given operating conditions (argon gas, copper cathode, input power of 70 W), the gas heating is not significant for pressures of up to 1.3 Pa. These are typical conditions for laboratory magnetrons. Commercial sputtering magnetrons may be up to ten times bigger and with power supplies of several kW. These operating conditions, however, are outside the scope of this study. Indeed, PIC-MCC simulations would become too time-consuming for such large magnetron discharges (see also Sect. 3.3 above). It is interesting to compare the obtained results with other data from the literature. The results for p = 6.7 Pa can be compared to [182], where the gas heating in a sputtering glow discharge is calculated for p = 5.6 Pa and 40 mm electrode separation. The temperature profile obtained there for a copper cathode and three times higher cathode voltage agrees very well with the gas temperature profile illustrated here. The lower voltage in the present case can be easily explained by the magnetic trap leading to much higher discharge efficiency, and to some extent with the smaller volume of the gas. Another comparison can be made with [184], which is the most often cited work in the literature about the gas heating in sputter magnetrons. The author reports a density reduction of about 37% for a pressure of 4 Pa and a current of 300 mA. This corresponds reasonably well with our calculation results, in spite of the larger reactor geometry. Indeed, it can be deduced from Fig. 3.29 that the gas temperature at 4 Pa would be around 440 K, which is about 47% higher than 300 K. More substantial is the difference with the calculations reported in the recent work of Epke and Dew [185]. The maximum temperature is on average a factor of two higher than the values presented here. In addition, the axial temperature profile is with a different shape. The values of the gas temperature in [185] are also higher than those in [184]. The shape of the profile in [185] resembles those obtained for analytical glow discharges in [186]. The common issue between these two models is that they are entirely or partially fluid. In addition, [185] is not a self-consistent simulation, but based on several assumptions, each of which brings uncertainties. The close correspondence of the absolute calculation values presented here, with the kinetic simulation [182] and the early experimental work [184] indicate that a kinetic self-consistent simulation represents more accurately the considered effect of gas heating in sputter magnetrons at typical operating conditions. Density Profiles of the Other Plasma Species The 2D density profile of the electrons (and Ar+ ions) was already presented in section “Calculated Electron and Ar+ Ion Densities” (Fig. 3.19). However, with the extended model, also the density profiles of the sputtered copper atoms, and the corresponding Cu+ ions can be calculated, as well as the density profile of the Ar metastable atoms. Figure 3.30 presents the sputtered Cu atom density distribution at a pressure of 0.67 Pa. It shows a maximum at

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Fig. 3.30. Calculated sputtered Cu atom density

Fig. 3.31. Calculated Cu+ ion density

r = 18 mm, where also the electron (and Ar+ ion) density, the electron-impact ionization rate and hence the ion fluxes toward the cathode reached their maximum. Indeed, the ion flux gives rise to sputtering, hence most sputtering takes place at r = 18 mm, which explains the maximum in the sputtered Cu atom density. However, the peaks in the Cu atom density profile are definitely not so pronounced as in the electron density profile. This is because the Cu atom density profile is much more spread out by diffusion. It was found that at 0.67 Pa, a nonnegligible fraction of Cu atoms with energies of a few eV can reach the opposite wall of this magnetron discharge, where normally a substrate is mounted for deposition purposes. At higher pressures, however, the majority of the fast Cu atoms thermalize before reaching the anode [39]. The density profile of the corresponding Cu+ ions is illustrated in Fig. 3.31. It is characterized by a very similar profile as the electrons and Ar+ ions, but its density is clearly lower, so the main positive charge in the plasma is attributed to the Ar+ ions. Erosion Profile Due to Sputtering Finally, the last calculation result shown here is the calculated erosion profile at the cathode, as a result of sputtering (see Fig. 3.32). Only one half of the axisymmetric target is shown. The erosion rate reaches its maximum at

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Normalised erosion profile

0.4

0

- 0.4

- 0.8 Calculated Measured -1.2 0

10

20

30

r [mm]

Fig. 3.32. Comparison of calculated (solid line) and measured (dashed line) normalized erosion profile

r = 18 mm, as expected. To verify the model, the calculated erosion profile is compared to a measured one at approximately the same operating conditions after 4 h of sputtering. The agreement is very satisfactory. This demonstrates that the model captures the major physical processes of magnetron sputtering, including two-dimensional effects. Therefore, it can in principle be used to support experimental research, for instance for predicting how the target sputtering can be made more uniform and hence the target material can be consumed in a more efficient way.

3.7 Conclusions and Outlook for Future Work In this chapter, we have given an overview of different modeling approaches developed for magnetron discharges, emphasizing their benefits and weaknesses. Furthermore, a two-dimensional self-consistent semi-analytical model and a two-dimensional PIC-MCC model are described in detail, and some examples of calculation results are presented. It was concluded that analytical approaches can give a very rapid prediction of certain discharge characteristics, but they are of course only valid under well-specified conditions. For a full description of the magnetron discharge, the PIC-MCC model is most appropriate, but it requires a very long computation time, which is acceptable for academic research, but unsuitable for industrial applications of sputter magnetrons. Nevertheless, the PIC-MCC simulations can give very useful insights,

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to assist experimental work, and to provide a benchmark for the development of analytical models. There is, however, still room for improvements and further developments of both models. The analytical model can be further improved by taking into account some processes which are neglected up to now, such as the self-consistent calculation of the electron temperature and the presheath, the electron contribution to the discharge current, and the effect of gas density reduction. Furthermore, by making minor, nonfundamental changes to the model, it could also be used to investigate aspects of the magnetron discharge, like the effect of the erosion groove formation on the target, or switching to another (nonreactive) discharge gas. The PIC-MCC model, developed up to now, will also form a basis for further numerical studies. Indeed, the model is now only applicable to DC magnetrons in argon, but for the deposition of metal oxides and nitrides, reactive gas mixtures (argon–oxygen or argon–nitrogen) need to be used. Therefore, we plan to extend the PIC-MCC model, as presented here, to reactive gas mixtures, to predict the discharge behavior for reactive sputterdeposition purposes. Moreover, DC magnetrons often encounter problems when reactive gas mixtures are used, due to the deposition of nonconductive oxides or nitrides onto the cathode surface, and the subsequent charging issues of the cathode, leading to discharge failure. Therefore, we would also like to extend the current PIC-MCC model, developed for a DC magnetron, to pulsed magnetron discharges, so that the model will become more widely applicable to a range of operating conditions.

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4 Modelling of Reactive Sputtering Processes S. Berg, T. Nyberg, and T. Kubart

4.1 Introduction Sputtering is a widely used technique to fabricate thin film coatings. Equipment for large area coatings is easy to set up and film uniformity may be satisfactory controlled over large processing areas. Sputtering is normally carried out in an argon ambient. Not only elemental pure single metal targets may be used, but also metal alloys can be sputtered which offers the possibility to deposit also complex coatings. Common for all inert argon sputtering processes is that the simplicity of the process only requires rather rudimentary process control. These properties make sputtering an attractive and competitive coating process. By adding a reactive gas to the sputtering process, it is possible to form a compound between sputtered metal atoms and reactive gas molecules. In this way, it is possible to form oxides, nitrides, borides, carbides, etc. To distinguish inert argon sputtering from the process where reactive gases are added, the latter process is commonly referred to as a reactive sputtering process. At first glance, one may believe that reactive sputtering processes may be as easy to carry out as the simple and straightforward inert sputtering processes. Unfortunately this is not the case. The addition of the reactive gas significantly changes the behaviour of the sputtering process. Both deposition rate as well as the composition of the film will be heavily influenced by the flow of the reactive gas [1–3]. Typical processing curves are shown in Figs. 4.1 and 4.2. Figure 4.1 illustrates a schematic of a typical processing curve for film deposition rate vs. flow of the reactive gas. Note that the curve takes different pathways for increasing the reactive gas flow than for decreasing this flow. This is indicated by the arrows in the figure. As can be seen, this curve shows a hysteresis behaviour. Increasing the reactive gas flow slightly beyond the processing point A in the figure will result in an avalanche-like transition down to point B. Decreasing the reactive gas flow from high values down to just below point C will result in an avalanche-like transition from C up to D

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Fig. 4.1. Typical experimental curve for a reactive sputtering process (rate indicated by means of optical emission spectroscopy)

Fig. 4.2. The partial pressure of the reactive gas corresponding to the curve in Fig. 4.1

in the figure. It is these avalanche-like transitions that primarily make process control rather complicated for reactive sputtering processes. Figure 4.2 illustrates a typical dependency of reactive gas partial pressure vs. flow of the reactive gas. The processing positions A–B–C–D from Fig. 4.1 are indicated also in this figure. Note that also the partial pressure of the reactive gas will change dramatically at positions A and C and that this curve also exhibits a hysteresis at the same values of the flow of the reactive gas as the curve in Fig. 4.1. Figures 4.1 and 4.2 may illustrate the processing curves for reactive sputter deposition of AlN from an elemental Al target in a mixture of Ar + N2 . It is known that the sputtering yield of Al is significantly higher than the sputtering yield of AlN. This is one of the reasons for the complex behaviour of the processing curves. Originally (with zero flow of nitrogen) the sputter erosion rate from the target is determined by the sputtering yield of the pure Al metal. A slight increase of the flow of nitrogen will not significantly change the composition of the target surface. Therefore the deposition rate does not

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change markedly. Sputtered atoms, however, will be deposited at a far larger substrate surface than the eroded target surface area. This results in a higher fraction of compound at the deposited substrate areas. From Fig. 4.2, it can be seen that, despite that the flow of reactive gas increases (small values of the reactive gas flow), almost no partial pressure is generated in the processing chamber. This illustrates that almost all nitrogen is used up to form AlN at the substrate. From zero and up to position A, the target is said to operate in metallic mode. AlN is formed also at the target but, due to the sputter erosion, less compound will be formed at the target than at the substrate. This explains the slow decrease of the deposition rate up to position A. At this position, conditions are generated that force the partial pressure to increase dramatically to position B. This new nitrogen partial pressure is high enough to completely form AlN also at the whole target surface. From this position, the target is said to be in the poisoned mode. It has been “poisoned” by the nitrogen and a layer of AlN has been formed on the target surface. As a consequence, the deposition rate will be determined by the low sputtering yield value for AlN instead of the much larger sputtering yield value for pure Al. This explains the drastic drop in deposition rate at position A. The nitrogen that is not used up to form AlN will contribute to building up of the partial pressure of nitrogen. Since less Al is sputter eroded after passing the transition A–B, excess nitrogen will show up as an additional nitrogen partial pressure. Consequently, the partial pressure of nitrogen will be higher during decrease of the nitrogen flow. This will be valid until the nitrogen partial pressure will not be large enough to completely form AlN on the target. From this position (C), the target converts to being substantially metallic again (position D) and sputter erosion will be determined by the Al sputter yield.

4.2 Basic Model for the Reactive Sputtering Process As shown in the previous paragraph, the processing curves for the reactive sputtering process do exhibit hysteresis effects. In addition, avalanche-like transitions occur at the edges of the hysteresis region. It is not easy to judge how different processing parameters affect the overall processing behaviour. This calls for a model capable of predicting the influence of different parameters on the process. It should be understood, however, that a model taking into consideration all aspects of this process would be very complex and therefore not so useful for the common user. However, a model has been suggested that has successfully reduced the number of necessary parameters but still enables the users to predict the general behaviour of the outcome of defined reactive sputtering processes. This model is commonly referred to in the literature as the “Berg’s model” [2–9]. Despite the simplicity of this model, it is frequently used as the base for more detailed treatments of reactive sputtering processes. The starting point for developing a model is to try to define the processing conditions as correct as possible. Simplifications must be made to keep the

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complexity down without loosing contact with reality. We start with showing a schematic of a simplified reactive sputtering processing chamber. This is shown in Fig. 4.3. We suggest that, for a mathematical treatment, Figs. 4.4 and 4.5 below may replace Fig. 4.3 to describe the reactive sputtering process. Figure 4.4 illustrates the flux of sputtered particles from the target as well as the flux of reactive gas molecules arriving to the substrate. A fraction (Θt ) of the target area (At ) has reacted with the reactive gas and formed a compound layer at the target surface. This is illustrated so that a fraction Θt of the target surface area consists of the compound. A similar situation will exist at the collecting (= substrate) area (Ac ) but the reacted fraction (Θc ) of this area will not be the same as at the target. During sputtering of the target, an outcoming flux of sputtered metal atoms Fm will be generated. For simplicity we assume that this flux

Fig. 4.3. A simple reactive sputtering system

Fig. 4.4. Particle fluxes to the substrate area Ac

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Fig. 4.5. Flows of reactive gas in the system

originates from the metal fraction (1 − Θt ) of the target area. We also make the simplification that sputtering from the compound area Θt at the target will result in sputter erosion of the compound molecules [10]. During sputtering of the target, a flux of sputtered metal-compound molecules Fc will be generated. In the mathematical description of Figs. 4.4 and 4.5, the flux Fc will originate from the Θt fraction of the target area. The argon ion current density causing sputtering from the target surface is denoted J. The argon ion current is assumed to be evenly distributed over the target surface. Note, however, that sputtering from the Θt fraction of the target will generate a flux of compound molecules, while sputtering from the (1 − Θt ) fraction will result in sputtering of metal atoms. In addition to the sputtered particles, also reactive gas molecules will arrive at the target and the collecting areas. Figure 4.5 illustrates the three possible pathways Qc , Qt , and Qp for the total flow Qtot of reactive gas to the processing chamber. The consumption of reactive gas molecules at the substrate is denoted Qc in the figures, Qt is the consumption at the target, and Qp is the throughput to the vacuum pump. We will use the notations from Figs. 4.4 and 4.5 in the following treatment. Solving for the behaviour of the reactive gas forms the basis of this treatment. The key parameter is the partial pressure P of the reactive gas. There exists a simple gas kinetic equation for the relation between flux of molecules F at all surfaces and the partial pressure [11]. This relation is given by the expression P F = √ , (4.1) 2kT πm where k is a Boltzmann’s constant, T the absolute temperature, and m the mass of the gas molecule.

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4.3 Steady State Equations In the following treatment, it is assumed that the process has reached steady state condition. Thus there are no time-dependent variations in the process. This will make it fairly easy to find balance equations for the target and collecting areas At and Ac . Furthermore, we assume that the partial pressure of argon is much higher than the partial pressure of the reactive gas. Based on this, we assume that the sputtering contribution from ionized reactive gas can be neglected. Sputtering is assumed to be carried out by the argon ions. To avoid introducing confusing constants, we also assume that one reactive gas atom forms a compound molecule with one target metal atom and one gas atom (AlN, TiN, etc.). The reader may adjust for this in the case of forming other compounds (like TiO2 , Al2 O3 , etc.). The target area will be evenly sputter bombarded by the argon ion current density J(A m−2 ). At steady state, a balance must exist between sputter removal of compound material from the poisoned part Θt of the target and formation of compound due to poisoning of the (1−Θt ) metal part by reactive gas. This condition is described by (4.2) J Yc Θt = α2F (1 − Θt ), q

(4.2)

where q is the electron charge. The left-hand side determines the sputter removal of compound molecules, where Yc denotes the sputtering yield of compound molecules from the target by energetic ions hitting the poisoned fraction Θt of the target. The right-hand side defines the formation of compound molecules at the (1 − Θt ) metal part of the target. Here 2F defines the flux of reactive gas atoms (assuming two atoms per gas molecule) at the partial pressure P and α denotes the sticking coefficient, the probability that an arriving gas molecule will react with the target metal and form a metal-compound molecule at the target surface (poisoning). From (4.2) it is easy to solve Θt as a function of F . The steady state condition for the substrate area Ac can be derived by corresponding arguments. At steady state, the sum of the sputtered material (metal atoms and compound molecules) and the reactions with deposited film metal atoms balances so that the substrate surface concentration Θc does not change. The total rate of outsputtered compound molecules Fc from the target will be J Fc = Yc Θt At . (4.3) q A fraction Θc of these molecules will arrive at the compound part Θc Ac of the substrate area. However, depositing compound on compound does not alter the chemical condition of this part of Ac . Depositing compound on the metal fraction (1 − Θc )Ac of the substrate area, however, will convert a metal

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surface to a compound surface. Thus, the rate of converting the metal part of the substrate surface to compound by deposition of outsputtered compound molecules can be described by the expression Fc (1 − Θc ). In addition to this, compound molecules may also be formed by reactions with the reactive gas and the non-reacted metal atoms at the (1 − Θc )Ac fraction of the substrate surface. This will also convert some of the metal fraction to compound. The total conversion rate for formation of these compound molecules is 2Qc , where Qc = αF (1 − Θc )Ac .

(4.4)

The total rate of forming compound molecules at the substrate surface will be the sum of the terms Fc (1 − Θc ) and 2Qc . The factor 2 corresponds to the fact that one reactive gas molecule forms two molecules of compound. Sputtered unreacted metal atoms will also arrive to the substrate surface. The total rate of erosion Fm from the target for these metal atoms will be Fm =

J Ym (1 − Θt )At , q

(4.5)

where Ym denotes the sputtering yield of metal atoms. The fraction Θc of the substrate surface consists of compound and the remaining fraction (1 − Θc) consists of pure unreacted metal atoms. Sputtered metal atoms arriving at the metal-compound fraction of the substrate will convert some metal-compound area to metal area in a corresponding way as described for the arriving metal molecules above. At steady state, the rate of compound formation due to sputtering of compound and gettering of reactive gas onto metal parts of the substrate equals the elimination of compound due to deposition of metal onto compound parts of the substrate. This defines the balance equation of the substrate surface Fc (1 − Θc ) + 2Qc = Fm Θc .

(4.6)

From (4.2) and (4.6), it is relatively simple to solve for Θc as a function of F . In addition to solve for Θt and Θc , we also want to obtain an expression for the deposition rate D. This can be obtained by first calculating the sputter erosion rate from the target. We have assumed that every compound molecule contains one metal atom and one atom from the reactive gas. Thus, the total number of outsputtered metal atoms from the target Rm must be Rm =

J [Ym (1 − Θt ) + Yc Θt ]At . q

(4.7)

A fraction Θc of these metal atoms has reacted with the reactive gas and formed compound molecules at the substrate. The remaining (1 − Θc ) fraction is still unreacted metal atoms. Assuming that Rm will be evenly distributed over the area Ac and taking into consideration the specific densities of the

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Normalized rate, Rm, D (a.u.)

1.2 1.0 0.8 D 0.6 0.4 Rm 0.2 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

Reactive gas pressure (Pa) Fig. 4.6. Sputter erosion rate Rm and substrate deposition rate D vs. the pressure of reactive gas

elemental metal and the formed compound, the deposition rate D may be written as (4.8) D = Rm [c1 (1 − Θc ) + c2 Θc ], where c1 and c2 are the material-specific related constants to obtain the desired unit for D. Typical results for calculated values for D and Rm as a function of the partial pressure P are shown in Fig. 4.6. Note that D initially may increase somewhat for a small flow of the reactive gas before slowly decreasing due to the decrease in Rm . This maximum is caused by outsputtered metal atoms taking up reactive gas atoms thus forming compound molecules at the substrate surface. At this stage, it must be pointed out that D and Rm do not show any hysteresis behaviour when plotted as a function of the reactive gas partial pressure. This was not the case for the deposition rate vs. reactive gas flow as previously shown in Fig. 4.1, where the curve has a pronounced hysteresis. To be able to plot D and Rm as a function of the reactive gas flow Qtot , we must find a relation between Qtot and P . From Fig. 4.5, we may conclude Qtot = Qt + Qc + Qp ,

(4.9)

where Qc , Qt , and Qp as indicated by the arrows represent the reactive gas consumptions at the substrate and target surfaces and throughput to the external vacuum pump Qt = αF (1 − Θt )At ,

(4.10)

Qc = αF (1 − Θc )Ac , Qp = SP,

(4.11) (4.12)

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where S is the pumping speed of the vacuum pump. For simplicity, we choose the same sticking coefficient α at the target and substrate areas. These additional mass flow equations unambiguously define the relation between Qtot and P . From what has been derived so far, it is possible to plot Θc , Θt , P, D, and Rm vs. the Qtot . Typical results from such calculations are shown in Figs. 4.7–4.9.

1.0 B

C

Fractions Θt, Θc

0.8

0.6

0.4

A

Θc

Θt

0.2 D 0.0

0

1

2

3

4

5

Reactive gas flow (sccm) Fig. 4.7. Calculated compound fractions Θt and Θc at the target and substrate, respectively, as a function of the flow of reactive gas

Normalized rate, Rm, D (a.u.)

1.2 1.0 D

0.8

A 0.6 0.4

D C

B

0.2 0.0

Rm 0

1

2

3

4

5

Reactive gas flow (sccm) Fig. 4.8. Calculated sputtering Rm and deposition D rate vs. the flow of reactive gas

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Pressure, P (a.u.)

0.4

0.3

B

0.2

0.1

0.0

C A

D 0

1

2

3

4

5

Reactive gas flow (sccm) Fig. 4.9. Calculated pressure vs. the flow of reactive gas

Processing points corresponding to the positions A, B, C, and D in Figs. 4.1 and 4.2 are inserted in all the calculated curves. A continuous increase or decrease of the reactive gas flow Qtot will directly cause an avalanche from A to B (increasing Qtot ) or from C to D (decreasing Qtot ). By only varying Qtot , it will not be possible to operate the process on the A–C part of the curve. The width of the S shape of the curves represents the hysteresis width. This region is marked by the shadowed region in the figures. It should be pointed out that a practical optimum operation point frequently appears to be as close to the left of point A as possible. Due to small processing fluctuations, however, it is difficult to obtain a stable operation at this position. A small increase in reactive gas flow will cause a dramatic change (decrease) in deposition rate. To return to the high deposition rate close to the left of A, one has to pass through the whole hysteresis loop. This is not a satisfactory solution for a process control system. From Fig. 4.6, it was shown that selecting the reactive gas partial pressure P as the control parameter results in single-valued erosion and deposition rates for all values of P . Due to this, a feedback control system setting a desired value for the partial pressure is the preferred way of carrying out process control for reactive sputtering processes. With the partial pressure control, it is possible to follow the curves in Figs. 4.7–4.9 without avalanching at A and C. Thus, it is possible to operate at any working point. The width and shape of the hysteresis depend on different processing parameters. In the following, it will be illustrated how some material properties and processing conditions affect the hysteresis.

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4.4 Influence of Material Properties and Processing Conditions 4.4.1 Reactivity The value of the sticking coefficient α indicates the reactivity between the gas and the target metal. In Fig. 4.10, the calculated sputter erosion rate Rm vs. Qtot for three reactive sputtering processes differing only in their values of α is shown. The results illustrate clearly that the hysteresis will be less pronounced for low reactivity processes [12]. For some gas/target material combinations, the hysteresis effect may not even occur. 4.4.2 Sputtering Yield The sputtering yield value for the metal (Ym ) is typically significantly higher than the sputtering yield value for the corresponding compound (Yc ). The hysteresis is very sensitive to the ratio Yc /Ym . Figure 4.11 shows the calculated sputter erosion rate Rm vs. Qtot for three reactive sputtering processes differing only in their values of the ratio Yc /Ym . The results illustrate that the hysteresis will be less pronounced for materials where Yc does not deviate too much from Ym [13]. Due to this effect, some target materials may exhibit small or almost no detectable hysteresis. It must be understood that reactivity α and the ratio Yc /Ym are fundamental material properties. It is therefore not possible to freely choose these values in a specific processing condition.

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α=1

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Reactive gas flow (sccm) Fig. 4.10. Influence of the sticking coefficient α on the reactive process. The sputtering rate is calculated for three different values of α

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Reactive gas flow (sccm) Fig. 4.11. Sputtering rates calculated for three different ratios of the compound (Yc ) and metal (Ym ) sputtering yield

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Reactive gas flow (sccm) Fig. 4.12. Sputtering rate vs. the flow of reactive gas calculated for three values of pumping speed

4.4.3 Pumping Speed In contradiction to α and the ratio Yc /Ym , the pumping speed S of the vacuum system may be controlled externally and set to a desired value. In Fig. 4.12, the calculated sputter erosion rate Rm vs. Qtot for three reactive sputtering processes differing only in their values of the pumping speed S is shown. The results illustrate that the higher the value of S the hysteresis will be less pronounced. For S high enough, the hysteresis disappears [14]. This is often used in smaller lab systems where sufficient pumping speed may be

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Sputter erosion rate, Rm (a.u.)

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0.8 3 cm2

0.6

12 cm2

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At=150 cm2 0

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Reactive gas flow (sccm) Fig. 4.13. Influence of the target area calculated for three different areas of the target

achieved. In large-sized industrial systems, however, the required pumping speed value for hysteresis-free operation is often unattainable high. 4.4.4 Target Size In a specific system it may be possible to select the size of the target. Figure 4.13 shows the calculated sputter erosion rate Rm vs. Qtot for three reactive sputtering processes differing only in their values of the target size At . The results illustrate that the hysteresis will be less pronounced for small target areas. When At is small enough, the hysteresis is predicted to disappear [15]. This behaviour is generally valid and is not caused by the fact that a smaller target will be exposed to a higher argon sputtering flux for identical total target current. Altering the target current for the small target does not change the shape of the processing curve. 4.4.5 Mixed Targets It has been found that some target mixtures may affect the reactive sputtering process in such a way that no hysteresis will appear [16]. It has been suggested to fabricate a mixture of a metal and its corresponding metal compound. The metal compound is the one expected to be formed by the metal and the reactive gas in the process. Having different concentrations of compound in such a target may strongly affect the processing curves. Figure 4.14 shows the calculated sputter erosion rate Rm vs. Qtot for three reactive sputtering processes differing only in the fraction of compound in the target.

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Sputter erosion rate, Rm (a.u.)

0.7 0.6

Θb=0%

0.5 0.4 20%

0.3 50%

0.2 0.1 0.0 0

1

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4

5

6

7

Flow of reactive gas (sccm) Fig. 4.14. Sputtering rate vs. the flow of reactive gas calculated for three mixed targets differing in the fraction of compound in the target

Fig. 4.15. Target conditions for a mixed target

The results show that the hysteresis is reduced for increasing fraction of compound in the target. Above a certain compound concentration, the hysteresis disappears. This type of target design offers the possibility to obtain a stable and non-critical process without a need for feedback control of reactive gas partial pressure. The deposition rate depends on the actual composition of the target, but it is always higher than from the corresponding pure metal target operated in compound mode. The results for the mixed target were obtained by introducing a slight modification of the target conditions as originally described in (4.2). The target erosion must correspond to the mixed target composition defined by fraction Θb of the compound in the target bulk. Preferential sputtering of the high sputtering yield material Ym takes place, so the concentration of compound at the target surface would be always higher than Θb . The conditions at the target are schematically shown in Fig. 4.15.

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The term Θt denotes the fraction of compound at the target surface. This fraction is different from the bulk concentration even when sputtering in pure argon. At the remaining part of the target, there is a similar competition between formation of compound and sputter removal of compound as described in the previous sections. Sputter erosion of surface material (compound or metal) will expose a subsurface having Θb compound concentration. Unlike the original basic balance equations, the conditions here may expose, with a probability Θb , a compound after sputter erosion. The compound fraction at the surface may increase due to sputtering of metal with the rate equal to Jq Ym (1 − Θt )Θb and in a corresponding way, the metal fraction at the surface can be increased by sputtering of compound at a rate equal to Jq Yc Θt (1 − Θb ). This gives the following balance equation for the target surface 2αF (1 − Θt ) +

J J Ym (1 − Θt )Θb = Yc Θt (1 − Θb ). q q

(4.13)

The balance equations at the substrate may be treated in the same way as for the simple one metal target one gas system shown in the beginning of this chapter. 4.4.6 Two Reactive Gases When two reactive gases are supplied to the processing chamber, a modification of the original equations has to be done. This is the situation when oxygen and nitrogen may be added to form an oxy-nitride of some target material. It is likely that one of the gases may be more reactive than the other. Normally oxides are formed more easily than nitrides. Due to this difference in reactivity, an additional reaction must be considered. With a certain probability, the most reactive gas may break up the metal compound formed by the less reactive gas and replace this by a metal compound based on the more reactive gas atoms. In Fig. 4.16, a schematic illustration for a mathematical model for the target is shown. Here F1 and F2 represent the fluxes of the two different reactive gases “1” and “2” at their respective partial pressures. Θt1 and Θt2 represent the frac-

Fig. 4.16. Target treatment for the model of reactive sputtering with two reactive gasses “1” and “2”

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tions of the target covered by compound “1” and compound “2”, respectively. We assume that there exists no chemical mixture of the two compounds. Two equations for the target surface are then required, one for each reactive gas. Assuming that gas “1” can replace gas “2” in compound “2” and form compound “1”, the first balance equation will be 2α1 F1 (1 − Θt1 − Θt2 ) + 2α12 F1 Θt2 =

J Yc1 Θt1 , e

(4.14)

where α12 is the replacement coefficient expressing the probability that the incoming reactive gas “1” replaces the existing gas atom in its metal compound thus converting it into the other compound. The first term in the left-hand side in the equation represents the gettering of gas “1” onto the free metal surface fraction (1 − Θt1 − Θt2 ). The second term represents the replacement of gas “2” atoms by gas “1” atoms in the metal compound at the target surface. The right-hand side of the equation represents the sputter erosion of metal-compound “1” from the Θt1 fraction of the target surface. The balance equation for gas “2” will be almost identical with the exception that the fraction area Θt2 of compound “2” will be reduced due to the replacement effect. This will result in the following balance equation: 2α2 F2 (1 − Θt1 − Θt2 ) − 2α12 F1 Θt2 =

J Yc2 Θt2 . e

(4.15)

The balance equations at the substrate may be treated in analogous way to what was outlined for the target including the replacement effect. Results from the calculations may be divided into two groups: 1. Zero replacement probability between the gases The 3D graph in Fig. 4.17 indicates that the two-gas processing system may result in a trapping effect, for a constant flow of gas “2” equal to 4 sccm (dashed curve in graph) and increasing gas “1” to enter the poisoned region. For these processing conditions, it will not be possible to return to the metallic mode (high deposition rate) only by decreasing the flow of gas “1”. Even for zero flow of gas “1”, the system will remain in the poisoned mode caused by gas 2 [17–20]. In this way the process may be trapped in the poisoned mode. 2. A finite replacement probability between the two gases Assuming that gas “1” may react both with the target metal as well as with the compound formed from gas “2” requires non-zero value of replacement coefficient α12 in (4.14) and (4.15). In this treatment, we assume that this reaction will generate the same compound as from reactions between metal and gas 1. Results from calculations based on these assumptions are shown in Fig. 4.18 for different constant supplies of gas “2” and in Fig 4.19 for 5 sccm of gas “2”. Note that the hysteresis may disappear for a gas flow of gas “2” having values above a critical value. Moreover, when approaching high flow levels for gas

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Sputter erosion rate, Rm (a.u.)

Fig. 4.17. Sputter erosion rates vs. flow of reactive gas “1” calculated for different flows of gas “2”. No replacement between gases is assumed

0.6 q2 = 0 sccm 5

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Flow rate of gas "1" (sccm) Fig. 4.18. Sputter erosion rate vs. flow of gas “1” for different flows of gas “2”

“1”, remarkable small amounts of compound “2” are present in the deposited film (indicated by low Θc2 and high Θc1 values). This effect points out a further way of obtaining hysteresis-free reactive sputtering operation [21]. This technique, however, will only be attractive when a small fraction of the extra compound may be accepted in the deposited films. 4.4.7 Reactive Co-Sputtering In certain applications, it may be favourable to use two separate targets as sputtering sources. This is convenient when studying properties of films with different compositions of two materials. Carrying out reactive sputtering from two metal targets, however, increases significantly the complexity of

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Fractions, Θc1, Θc2

1.0 0.8

Θc1 Θc2

0.6 0.4 0.2 0.0 0

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Flow rate of gas "1" (sccm) Fig. 4.19. Fractions of compounds “1” and “2” at the substrate for 5 sccm of reactive gas “2”

Fig. 4.20. Fluxes of sputtered material in the model with two separate targets

the process [22]. The main reason for this is that both target surfaces will be exposed to identical partial pressure of the reactive gas. This implies that two metals having quite different reactivity to the reactive gas will not form the stoichiometric compound at the same partial pressure value. A mathematical treatment of a basic reactive co-sputtering process may be carried out in the following way. Assume that the system may be represented by the schematic shown in Fig. 4.20. Here Θc1 and Θc2 represent the fraction of metal compound formed at the metal “1” (= y) and metal “2” (= 1 − y) fractions, respectively, of the substrate surface and Θt1 and Θt2 represent the corresponding compound formations at the two separate targets. The balance equations for the two targets can be treated as two separate balances having identical partial pressure P of the reactive gas. This treatment will be identical to what has been described previously. “y” is defined as the fraction of material from atoms originating from target “1” and consequently (1 − y) is the fraction of material from atoms

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originating from target “2”. The total number T1 of sputtered metal atoms from target “1” is given by T1 =

J1 [Ym1 (1 − Θt1 ) + Yc1 Θt1 ]At1 . q

(4.16)

Assuming one metal atom and one gas atom in the compound molecule, corresponding number T2 sputtered from target “2” will be T2 =

J2 [Ym2 (1 − Θt2 ) + Yc2 Θt2 ]At2 . q

(4.17)

The fraction y of material “1” in the deposited film will be defined as y=

T1 . T1 + T2

(4.18)

With this definition of y, it is possible to treat the surface fraction yAc as the collecting area for all metal “1”-based material and (1 − y)Ac as the collecting area for all metal “2”-based material. In this way, the treatment of reactive co-sputtering has been reduced to solving two single target processes joined by the partial pressure of the reactive gas. Note that y will be strongly dependent on the partial pressure P . From the above, it is possible to generate calculated curves for y, Θc1 , Θt1 , Θc2 , Θt2 , and P as a function of Qtot . To illustrate that reactive co-sputtering may cause quite complicated relations, we choose to show a curve for y vs. Qtot (Fig. 4.21). As can be understood from this figure, it may be quite complicated to fabricate a mixed film of two metals with arbitrary composition if the desired composition must be generated by carrying out the process inside the hysteresis region.

Fig. 4.21. Calculated fraction y of the material “1” in co-sputtered film vs. the flow of reactive gas

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4.4.8 Comment on Pulsed DC Reactive Sputtering Pulsed DC reactive sputtering has become a popular biasing technique to avoid arcing at insulating spots at target surfaces. Operation frequencies use to be in the 20–300 kHz region. Applying a short negative DC pulse to a sputtering target gives rise to a burst of sputtered material. At first sight, one may believe that these bursts of sputtered material will arrive to the substrate at a rate corresponding to the applied pulse frequency. At these frequencies, however, the transit time for a sputtered atom to reach the substrate will be in the millisecond range (see Chap. 6). Scattering by argon atoms slows down the initial speed and causes additional travelling distance. Due to this effect, the bursts will be spread in time upon arrival to the substrate. The overall result will be that consecutive bursts will overlap and cause an almost continuous flow of sputtered material to arrive to the substrate [23]. Due to the behaviour described above, a pulsed DC reactive sputtering process may be mathematically treated as a continuous DC sputtering process. This is of course only valid for the steady state conditions as have been presented in this chapter. More details of the pulsed DC processes can be found in Chap. 8. 4.4.9 Secondary Electron Emission In the treatment above, we have assumed a constant argon ion current for all levels of reactive gas flow. This is of course quite a simplification. A detailed treatment has to consider that the electron emission efficiency differs between a clean metal surface and a partially oxidized/nitrided target surface. The effect of variation of secondary electron emission will be treated in more detail in Chap. 5 and elsewhere [24, 25]. 4.4.10 Ion Implantation In a self-sustained sputtering plasma, argon ions are generated by electron impact ionization. Upon arrival at the target, a 500 eV argon ion may be implanted some 30–50 ˚ A. The effect of this implantation, however, is assumed not to influence the target material at the penetration depth. In reactive sputtering, however, also the reactive gas may be ionized. These ions may also be accelerated by the electric field and implanted into the target. These implanted reactive gas ions will easily react with the target metal atoms and form compound molecules. Due to this effect, there will always be some compound formation in the whole surface region reaching into the ion penetration depth. This effect will increase the response time for a reactive sputtering process when shifting from one processing point to another. In our treatment so far, we have not included this effect. To correctly carry out a treatment of the dynamic response of a reactive sputtering process, this effect must be included. In the following chapter(s), it will be shown how this effect may influence the reactive sputtering process.

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4.5 Concluding Remarks It must be understood that the assumptions made for the calculations in this chapter are quite crude simplifications. Experience has shown, however, that the general shapes of the predicted curves remarkably well mirror what has been found from a large variety of experimental conditions. This encourages us to assume that we correctly have treated the major parameters that determine the processing behaviour. In this chapter, we have chosen simplicity to observe and point out the major influence of different processing parameters. We believe that this may serve as a good first approximation until the need arises for a more detailed analysis.

References 1. I. Safi, Surf. Coat. Technol. 127, 203 (2000) 2. S. Berg, T. Nyberg, H.-O. Blom, C. Nender, Handbook of Thin Film Process Technology (Institute of Physics Publishing, Bristol, UK, 1998) 3. S. Berg, T. Nyberg, Thin Solid Films 476, 215 (2005) 4. S. Berg, H.-O. Blom, T. Larsson, C. Nender, J. Vac. Sci Technol. A 5, 202 (1987) 5. S. Berg, T. Larsson, C. Nender, H.-O. Blom, J. Appl. Phys. 63, 887 (1988) 6. H. Bartzsch, P. Frach, Surf. Coat. Technol. 142–144, 192 (2001) 7. V.A. Koss, and J.L. Vossen, J. Vac. Sci. Technol. A: Vac. Surf. Films 8, 3791 (1990) 8. H. Ofner, R. Zarwasch, E. Rille, H.K. Pulker, J. Vac. Sci. Technol. A: Vac. Surf. Films 9, 2795 (1991) 9. H. Sekiguchi, T. Murakami, A. Kanzawa, T. Imai, T. Honda, J. Vac. Sci. Technol. A: Vac. Surf. Films 14, 2231 (1996) 10. A.J. Stirling, W.D. Westwood, Thin Solid Films 7, 1 (1971) 11. J.F. O’Hanlon, A User’s Guide to Vacuum Technology (Wiley, New York, 1980) 12. J. Schulte, G. Sobe, Thin Solid Films 324, 19 (1998) 13. S. Zhu, F. Wang, W. Wu, L. Xin, C. Hu, S. Yang, S. Geng, M. Li, Y. Xiong, K. Chen, International Journal of Materials & Product Technology (Inderscience Enterprises, Guilin, China, 2001), p. 101 14. S. Kadlec, J. Musil, J. Vyskocil, Vacuum 37, 729 (1987) 15. T. Nyberg, S. Berg, U. Helmersson, K. Hartig, Appl. Phys. Lett. 86, 164106 (2005) 16. Y. Hoshi, T. Takahashi, IEICE Trans. Electron. E87-C, 227 (2004) 17. W.D. Sproul, D.J. Christie, D.C. Carter, S. Berg, T. Nyberg, in Proceedings of the 46th Annual SVC Technical Conference, Society of Vacuum Coaters, San Francisco, CA, 2003, p. 98 18. N. Martin, C. Rousselot, J. Vac. Sci. Technol. A: Vac. Surf. Films 17, 2869 (1999) 19. P. Carlsson, C. Nender, H. Barankova, S. Berg, J. Vac. Sci. Technol. A: Vac. Surf. Films 11, 1534 (1993) 20. H. Barankova, S. Berg, P. Carlsson, C. Nender, Thin Solid Films 260, 181 (1995) 21. D. Severin, O. Kappertz, T. Kubart, T. Nyberg, S. Berg, A. Pflug, M. Siemers, M. Wuttig, Appl. Phys. Lett. 88, 161504 (2006)

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22. N. Martin, C. Rousselot, Surf. Coat. Technol. 114, 235 (1999) 23. L.B. Jonsson, T. Nyberg, I. Katardjiev, S. Berg, Thin Solid Films 365, 43 (2000) 24. A. Pflug, B. Szyszka, V. Sittinger, J. Niemann, in Proceedings of the 46th Annual SVC Technical Conference, Society of Vacuum Coaters, San Francisco, CA, 2003, p. 241 25. Y. Matsuda, K. Otomo, H. Fujiyama, Thin Solid Films 390, 59 (2001)

5 Depositing Aluminium Oxide: A Case Study of Reactive Magnetron Sputtering D. Depla, S. Mahieu, and R. De Gryse

5.1 Introduction The deposition of aluminium oxide from an aluminium target in DC mode is used in this chapter to illustrate the different aspects of the reactive magnetron sputter process. The choice for this combination of target material and reactive gas is given by the fact that the well-known hysteresis behaviour, described in the previous chapter, is clearly demonstrated for this combination. Two key aspects of the process are responsible for this behaviour. The strong difference in the molecular sputter yield and density for Al and Al2 O3 results in a dramatic decrease of the deposition rate when the target is poisoned [1]. Similarly the ion induced secondary electron emission coefficient of aluminium changes strongly when reacting with oxygen to form aluminium oxide. Hence, the absolute discharge voltage drops in the order of 100 V when the target becomes poisoned [2]. The previous chapter describes the “Berg” model for reactive magnetron sputtering. Several experimental trends can be explained using this model. The model includes most important processes accounting for its success. In short the model describes the gettering of the reactive gas by the target material which influences the reactive gas partial pressure. At low reactive gas flow, the reactive gas is almost completely gettered and hence the target condition remains metallic. When on increasing the reactive gas flow the maximum amount of reactive gas which can be gettered is reached, the reactive gas partial pressure increases and the target becomes completely poisoned. Depending on the experimental conditions this change from metallic to poisoned mode can occur abruptly. Although the model described in the previous chapter explains quite well the hysteresis behaviour some experiments described in this chapter are difficult to understand from this model. This finds its origin to the opinion of the authors in the description of the poisoning mechanism of the target. Indeed, in the “Berg” model, the reaction between the target material and the reactive gas is described by chemisorption. This is in first order a correct approach for the reaction on the substrate but is not a complete description for the target process. Indeed, during magnetron

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sputtering the target is bombarded by ions from the plasma, including reactive gas ions. The ion energy is defined by the discharge voltage and is in the order of 400 eV. Hence, the ions become implanted in the target at a depth in the order of a few nanometers. Two major conclusions can be drawn. First, when reacting with the target material, the reactive ion implantation results in a much thicker layer than modelled in the “Berg” model. Secondly, the reaction mechanism becomes more complicated and hence this could influence the description of the dynamics of reactive magnetron sputtering. These two major conclusions will be addressed in the final part of this chapter where an model is presented describing the reactive ion implantation in an analytical way.

5.2 Some Experiments 5.2.1 A First Series of Experiments The simplest experiment to study the reactive sputtering process is the socalled hysteresis experiment: the discharge is ignited in pure argon, at a given pressure by introducing a certain argon flow to vacuum chamber. The ratio between the argon flow and the argon pressure defines the argon pumping speed. Keeping the discharge current constant, the oxygen flow is stepwise increased. Between each step the process parameters are logged until the process is in steady state. When the target is completely poisoned the measuring procedure is reversed, i.e. the oxygen flow is stepwise decreased until the discharge is again in pure argon. One easy to measure process parameter is the discharge voltage. Although the discharge voltage has a negative value, one generally plots and discusses its absolute value. The tradition will be followed in the remainder of the text. Figure 5.1 (top) shows the result of a hysteresis experiment for the system aluminium/oxygen. We refer to the figure caption for the experimental details. With a sufficient sensitive gas independent pressure gauge the total pressure change can also be measured. Here, a capacitance pressure gauge was used and the result of this measurement is shown also shown in Fig. 5.1 (bottom). Figure 5.1 shows the well-known hysteresis effect which can be explained using the basic model. Hence, when the target is poisoned the discharge voltage drastically decreases. It is also noticed that, when still in metallic mode, the discharge voltage slightly increases as indicated by the dotted line in the figure. Several effects can cause a discharge voltage change. This can be understood from the following equation for the discharge. More details on the derivation of (5.1) can be found in [3]: Vdischarge =

W0 , γISEE εe εi Em

(5.1)

discharge voltage (V)

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480 440 400 360 320 280 0.0

0.5

0.0

0.5

1.0

1.5

1.0

1.5

2.0

total pressure (Pa)

0.65 0.60 0.55 0.50 0.45 0.40

flow (sccm)

2.0 increasing flow decreasing flow

Fig. 5.1. Discharge voltage (top) and total pressure (bottom) as a function of the oxygen flow during reactive sputtering of an aluminium target mounted in a planar 2 in. magnetron at constant current (0.4 A) and constant argon pressure (0.4 Pa). The pump speed was 16 L s−1 . The oxygen flow was stepwise increased after 2 min

with W0 the effective ionisation energy (in eV), εi the ion collection efficiency, εe the fraction of the theoretical amount of ions that the electron effectively generates before it is lost from the discharge, γISEE the ion induced electron emission (ISEE) coefficient (see chapter Baragiola), E the effective gas ion probability and m the multiplication factor. For a magnetron discharge the electrons are trapped in close vicinity of the target. Escape from this trap is only possible through several interactions with the sputter gas. Consequently, εe can be considered close to unity. Because of the trapping of the electrons, most ionisation takes place close to the target surface. Hence, the vast majority of the ions reaches the cathode, which implies εi practically equal to unity. The effective ionisation energy W0 varies only slightly for different magnetron discharge conditions and has a typical value of 30 eV for argon. However, W0 depends on the gas composition of the discharge. Adding a different gas, say a reactive gas, will change its value and hence the discharge voltage. E depends mainly on the gas pressure, and it accounts for the fact that some of the electrons do not interact with the gas but are recaptured by the target. The ionisation caused by electrons that are generated in the cathode sheath is characterised by the multiplication factor. The multiplication factor depends strongly on sheath thickness which is defined by the magnetic field, pressure

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and the discharge power. Finally, the ISEE coefficient γISEE gives the number of electrons emitted per incoming ion on the target. The ISEE coefficient depends strongly on the target condition. Under the experimental conditions, the magnetic field strength is not changed and the work of Buyle [3] shows that the change in the power is too small to strongly affect the sheath thickness. So, in the given context, essentially three parameters influence the discharge voltage, i.e. the gas composition, gas pressure and the target condition. These could explain the measured discharge voltage. However, one must take care since the gas composition and pressure change simultaneously. Hence, other experiments are needed to explain the discharge voltage behaviour. The experiment described in Fig. 5.1 has been repeated under the same conditions but the pumping speed has been changed by opening the throttle valve between the vacuum chamber and the turbomolecular pump. Changing the pumping speed strongly influences the hysteresis behaviour as can be understood from the “Berg” model described in the previous chapter. This experiment shows some interesting features. Using the discharge voltage measured in pure argon as a reference, the discharge voltage difference between a metallic target and a fully poisoned target remains remarkable constant although the pressure change depends strongly on the pumping speed. At low pumping speed (16 L s−1 ) the gas pressure increases on poisoning with 50% and the oxygen gas fraction is approximately 30% while at high pumping speed (130 L s−1 ) the gas pressure increase is only 14% and the oxygen gas fraction is approximately 12%. Hence, one can conclude that the discharge voltage under these experimental conditions is hardly affected by the reactive gas pressure and the plasma composition. This also means that the slight increase of the discharge voltage on addition of oxygen (see Fig. 5.1) is difficult to explain based on gas related effects. The gas pressure and plasma composition before the critical point hardly changes. Although difficult to see in Fig. 5.2, there is clear trend: the discharge voltage increase before the critical point becomes larger with the pumping speed (see also further, Fig. 5.19). To explain the effect of the discharge voltage increase a second set of experiments was performed which is discussed in the next paragraph. 5.2.2 A Second Series of Experiments: Oxygen Exposure and Plasma Oxidation Aluminium [2] As the discharge voltage seems only be affected by the target condition, the discharge voltage was measured under identical conditions except for the target condition. This type of experiments can be obtained by first exposing the aluminium target for a given period to a certain oxygen flow without sputtering the target. Generally an exposure to a gas ambient is expressed in a

pressure difference (ΔPa)

voltage difference (ΔV)

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S = 16 L /s S = 32 L /s S = 65 L /s S = 130 L /s

0 −40 −80 −120 −160 0.25 0.20 0.15 0.10 0.05 0.00 0

1

2 3 flow (sccm)

4

5

Fig. 5.2. Discharge voltage change (top) and pressure change (bottom) as a function of the oxygen flow during reactive sputtering of an aluminium target mounted in a planar 2 in. magnetron at constant current (0.4 A) and constant argon pressure (0.4 Pa) for different argon pumping speeds. The oxygen flow was stepwise increased after 2 min

number of Langmuirs (L), for which 1 L equals 1 × 10−6 Torr s. After the oxygen exposure the oxygen flow is switched off and the discharge ignited in pure argon at constant power (50 W). The discharge voltage is registered during the target sputter cleaning. Using this measuring procedure the discharge voltage is measured under identical conditions and hence its change can be directly related to the target condition. Especially the first measured and stable discharge voltage V30 ms , i.e. 30 ms after the discharge ignition is strongly related to the target condition as the target condition is hardly affected by the sputtering processes. Figure 5.3 (left) shows the discharge voltage behaviour during sputter cleaning after a given exposure. Figure 5.3 (right) shows the discharge voltage difference between V30 ms and VAr , i.e. between the first stable discharge voltage and the discharge voltage measured during sputtering aluminium in pure Ar. This experiment has been repeated under different experimental conditions but the general trend is that V30 ms first increases on oxygen exposure up to an exposure of 300 L and then starts to decrease. This behaviour can be understood as follows. At high oxygen exposure an oxide layer is formed on the target prior to sputtering. The formation of the oxide layer, in the case aluminium, results in an increase of the ISEE coefficient. Hence the discharge voltage decreases. However, for small exposures, no oxide is formed. Indeed, the general mechanism for the oxidation of aluminium at room temperature, for which

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360 340 320

constant time (15 s) −3 constante pressure (3.7 10 Pa) −2 constante pressure (1.7 10 Pa)

25 15 5 −5 −15 −25

oxidation

exposure time 5s 10s 20s 40s 80s 160s 320s 640s 1280s 2560s 4260s

(V30ms − VAr) (V)

target voltage (V)

380

chemisorption

158

−35 −45

300 0.1

0.2 0.3 sputter time (s)

0.4

0.5

102

103 104 105 oxygen exposure (L)

Fig. 5.3. Discharge voltage behaviour during sputter cleaning of an aluminium target which was exposed to a given time (see legend) at a constant pressure of 3.7 × 10−3 Pa (left) and the discharge voltage difference between the discharge voltage measured 30 ms after discharge ignition and the discharge voltage measured during sputtering aluminium in pure argon (right). During the sputtering the argon pressure was constant (6 × 10−1 Pa) and the power was kept constant at 50 W. The experiment has been performed by exposing the target for a constant time at different pressures and for different period at two constant pressures

good agreement in the existing literature [4–10] is found, is the following. Before oxide formation a chemisorbed layer is formed. The onset of oxidation begins only after the surface is completely covered by a chemisorbed oxygen monolayer. When this happens, the aluminium surface becomes comparatively non-reactive as judged by weight gain and resistivity changes. Depending on the method of production, cleaning and (or) pre-treatment of the aluminium, the onset of oxide formation as given by literature [4–10] varies from 50–70 L exposure, over 130–180 L exposure, up to about 200–300 L. Hence, it seems from this experiment that the formation of the chemisorbed layer results in a decrease of the ISEE coefficient of the target. The mechanism for this is not completely clear. Important in the context of this chapter is that (1) the sticking coefficient of oxygen on aluminium target is in the order of 1/100 which is a very low value and that the oxidation of the aluminium target by oxygen exposure is a slow process. Indeed as Fig. 5.3 (right) shows, the discharge voltage difference changes logarithmic with the oxygen exposure. To reach the discharge voltage difference between a metallic target and a poisoned target (see Fig. 5.2) one should expose the target to very high pressures (or for a very long time) before this value can be reached. Of course, one can reason that the sticking coefficient for oxygen during magnetron sputtering is higher than the value derived here. Indeed, in the plasma highly active oxygen, i.e. oxygen radicals, is also present. These radicals have a much higher sticking coefficient on metals in the order 1. Pekker [11, 12] estimates before the critical point the dissociation degree in the order of a few percent while the dissociation can increase in the order of 10–30%, depending on the experimental conditions. In the case of aluminium this would result in average sticking coefficient in the

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discharge voltage (V)

order of 0.1. However, with the given experimental parameters for Fig. 5.2, no hysteresis can be modelled with the basic model. Hence, chemisorption can not be solely responsible for the target oxidation during reactive sputtering. This is an important conclusion which led to the hypothesis that the target poisoning mechanism is also affected by implantation of reactive gas (here oxygen) ions which are formed in the plasma by ionisation of the neutral reactive gas molecules. At this moment this hypothesis has been confirmed in several ways. The most direct proof has been given by G¨ uttler et al. [13]. These latter authors have studied in situ the nitrogen concentration in a titanium target which was sputtered in a mixture of nitrogen and argon. It was clearly shown that the nitrogen concentration stems with several monolayers of TiN at the target surface. They have shown that the thickness of the formed TiN layer on a poisoned target agrees with the ion range of the N2+ ions which are accelerated over the cathode sheath. This latter can also be proven indirectly by using a similar measuring scheme as described above [14]. Before measuring the discharge voltage during sputter cleaning in pure argon, the target is first sputtered in pure oxygen until a constant discharge voltage is measured. As Fig. 5.4 shows, the discharge voltage measured 30 ms after discharge ignition in pure argon has a low value, comparable with the discharge voltage measured in poisoned mode (see Fig. 5.1). After approximately 0.35 s the discharge voltage starts to increase to reach finally the value typically for aluminium sputtered in pure argon. Assuming that during the plasma oxidation the reactive ions become implanted with an energy related to the discharge voltage measured during plasma oxidation, one can calculate the ion range and straggle using SRIM. Following Todorov and Fossum [15], the oxide thickness formed during plasma oxidation can be estimated from the sum of the ion range and straggle and should therefore be in the order of 1.5 nm. Based on the experimental current density (0.03 A cm−2 )

360 340 320 300 280 0.001

0.01

0.1

1

10

time (s) Fig. 5.4. Discharge voltage behaviour during sputter cleaning of an aluminium target which was first plasma oxidized in pure oxygen (0.4 Pa, 0.4 A). The sputter cleaning procedure was performed in pure Ar (0.4 Pa, 0.4 A)

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and the oxide thickness, and the onset of the discharge voltage change, one can calculate a total sputter yield. Repeating the experiments under different experimental conditions results in a value of the sputter yield of 0.231 (error: 0.028), which agrees quite well with published values (0.25 [16]). For sake of completeness let us compare this latter value with the calculated sputter yield of the chemisorbed oxygen on the target [17]. This value can be retrieved from the experiments described in Fig. 5.3 on the assumption that the ISEE coefficient during sputter cleaning of a target covered by a chemisorbed layer changes in the following way γT = γm (1 − θsc ) + γmc θsc ,

(5.2)

with γm the ISEE coefficient of the clean aluminium target, γmc the ISEE coefficient of a covered aluminium target and θsc the fraction of the target covered with oxygen. During the experiment the discharge voltage changes and therefore the ion energy of the impinging ions will change, influencing the ISEE coefficient. However, the discharge voltage change is small, so this influence can be easily neglected. Also, due to the small change in discharge voltage one can also assume that, except for the ISEE coefficient, the parameters in (5.1) remain constant during the sputter cleaning. In this way, by normalizing the discharge voltage as V − VAr x= , (5.3) V30ms − VAr one can easily proof that x equals θsc because the ISEE coefficients hardly change. Following Dawson [18] an exponential change is expected for the removal kinetic with the rate of removal given by dθsc = −Ic+ θsc Q, dt

(5.4)

with Ic+ the ion current density and Q the removal cross section. This results in an exponential decrease of x from which the removal cross section can be calculated. In Fig. 5.5 x is given as a function of the ion fluence (calculated based on the assumption that the ion current density is equal to the current density). An experimental value of 4.95 × 10−17 (error 1.16 × 10−17 ) cm2 is found. This is of the same order of magnitude as the cross section published by others [19]. As the oxygen surface density for a fully covered aluminium surface Nox equals 1.15×1015 atoms cm−2 , the sputter yield Y = (Q Nox ), has a value between 0.03 and 0.06. Compared to the oxygen sputter yield from a plasmaoxidized target this value is much lower. This is somewhat surprising as the binding energy between oxygen and aluminium is in both cases similar and the chemisorbed layer is a surface layer. Indeed, SRIM simulations [20] show a higher yield for oxygen sputtered from a chemisorbed layer (simulation performed with SRIM 2006.02, 0.3 nm O layer on 300 nm Al layer with 0.35 keV Ar bombardment) than from the oxide. A possible reason for this difference

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1.0 0.8

x

0.6 0.4 0.2 0.0 0.0

0.4

ion fluence

0.8

1.2x1017

(ion/cm2.s)

Fig. 5.5. Relative discharge voltage x as a function of the ion fluence (calculated from the current density) during the sputter cleaning of a chemisorbed oxygen layer on an aluminium target. The full represent the exponential fit from which the removal cross section is calculated

can be found in the knock-on of the chemisorbed oxygen into the aluminium target. SRIM shows that this is quite important effect which redistributes the oxygen over a depth of 4 nm (simulation performed with SRIM 2006.02, 0.3 nm O layer on 300 nm Al layer with 0.35 keV Ar bombardment). The same simulation gives a ratio between the sputter atoms to recoil implanted atoms of .1551/.2605 or 1.65. Although this is a strong effect which could result in a slower decrease of the discharge voltage, its influence of the calculation of the sputter yield is not understood yet. Indeed, one would expect that the knock-on implanted atoms would form aluminium oxide resulting in a faster decrease of the discharge voltage. Based on the experiments described in this paragraph, several aspects of the reactive sputtering process can be understood. First, one notices that several mechanisms play a role in the poisoning of the target, i.e. chemisorption, knock-on implantation and direct reactive ion implantation. Second, the poisoning of the target results in a drastic decrease of the erosion rate of the target and as such in drastic decrease of the deposition rate. Indeed, at the same ion energy the calculated sputter yield for aluminium is in the order 0.6 (0.5475 for 380 eV, using SRIM) while the measured total sputter of aluminium oxide is in the order of 0.23. Taking into account the density change, the ratio between the deposition rate in metallic mode and poisoned mode is in the order 10/1, which is confirmed by most authors [1]. Finally, as third conclusion, it noticed that the ISEE coefficient, and as such the discharge voltage, can depend on the interaction mechanism of the oxygen molecules. Chemisorbed oxygen seems to lower the ISEE coefficient while the formation of oxide (subsurface oxygen) increases the ISEE coefficient. However, a word of caution is necessary. One may not generalize this latter conclusion. Indeed, not all oxides have a high ISEE coefficient. Replacing the target by

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another material can lead to different conclusions. To stress this point some experiments, similar as described in Fig. 5.4, are in the next paragraph. Plasma Oxidation of Other Materials [21]

inverse of the discharge voltage (x10−3 1/V)

The performed experiments are similar to the experiment described in Fig. 5.4, i.e. a target is sputtered in pure argon until a constant discharge voltage is noticed. Then argon is replaced by oxygen and again the target is sputtered until a constant discharge voltage is registered. Finally, the oxygen is again replaced by argon and the target is sputter cleaned and the discharge voltage is registered. This experiment was performed for Ag, Al, Au, Ce, Cr, Cu, Li, Mg, Nb, Pt, Re, Ta, Ti, Y and Zr. The discharge voltage measured in pure argon is related to ISEE coefficient of the metal, as given by (5.1). Within a small region of ISEE coefficient, a linear relationship between the ISEE coefficient and the inverse of the measured discharge voltage can be expected. This is exactly what the experiment shows (see Fig. 5.6). Based on this relationship between the ISEE coefficient and the discharge voltage, one can calculate the ISEE coefficient from a discharge voltage measured under identical experimental conditions. Hence, the discharge voltage measured 30 ms after starting up the sputter cleaning procedure of the plasmaoxidized target can give information about its ISEE coefficient. Performing the above described experiment, it is found that the group of studied metals splits up in two groups. For the first group (Al, Ce, Li, Mg and Y) the

Ag Al Au Ce Cr Cu Mg Nb Pt Re Ta Ti Y Zr

4.0

3.5

3.0

2.5

2.0

1.5 0.00

0.05

0.10

0.15

0.20

ISEE coefficient Fig. 5.6. The measured (inverse) discharge voltage in pure argon as a function of the ISEE coefficient of the metallic target material (Argon pressure: 0.3 Pa, discharge current 0.3 A)

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ISEE coefficient increases on oxidation of the target. This corresponds with the generally accepted idea that the ISEE coefficient for oxides is larger than for the corresponding metal. However, for the second group (Ag, Au, Cr, Cu, Nb, Pt, Re, Ta, Ti, Zr) the ISEE coefficient after plasma oxidation is smaller than the ISEE coefficient of the corresponding metal. For some materials (Ag, Au, Cu and Pt) it is questionable if the oxidation procedure actually modifies the target surface composition drastically but for the other materials of this group the low ISEE coefficient is surprising. The origin, but not the complete explanation, for this low ISEE coefficient, is the reduction of the oxide under ion bombardment. Indeed, after plasma oxidation, the metals of the first group form an oxide which sputters congruently, meaning that no reduction occurs on ion bombardment. Hence, their behaviour agrees with the fact that oxides have a higher ISEE coefficient than the corresponding metal. However, for the second group, the oxide reduces under ion bombardment. The effect of the sputter reduction was found after studying TiO2−x target (with x ranging from 2 to 0.25, i.e. from pure Ti towards TiO1.75 ). These targets were manufactured by sintering mixtures of Ti and TiO2 [22]. These compound targets are all reduced titania targets, having an ISEE coefficient lower than Ti. It also shows that the ISEE coefficient is related with the bulk oxygen to metal ratio of the oxide. Figure 5.7 shows this relationship for the two discussed groups.

0.6

Cr2O3 Nb2O5 ReO3 Ta2O 5 TiO2 reduced TiO2-x Al2O3 CeO2 Li2O MgO Y2O3 ZrO2

ISEEox coefficient

0.5 0.4 0.3 0.2

0.12 0.08 0.04 0.00 0.0

0.5

1.0 1.5 2.0 bulk oxygen-to-metal ratio

2.5

3.0

Fig. 5.7. ISEE coefficient of oxidized targets and component targets as a function of the bulk oxygen-to-metal ratio. Filled symbols are used for the oxides with a high ISEE coefficient (no sputter reduction), Open symbols are used for oxides with low ISEE coefficient and grey stars are used for the component targets. The ISEE coefficients were determined by the empirical relationship between the ISEE coefficient and the discharge voltage as shown in Fig. 5.6

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ln [γO-/ N(O)](a.u.)

17

Al Ce Cr Cu Li Mg Nb Pt Re Ta Ti Y

15 13 11 9 7 5 −4

−3

−2 ln [γe]

−1

0

Fig. 5.8. Relationship with the measured ISEE coefficient of the oxide (see Fig. 5.7) and the intensity of the high energy O− peak measured by energy resolved mass spectrometry during reactive sputtering of the targets in an oxygen/argon mixture

The bottom line of this paragraph is that the discharge voltage change during reactive sputtering is material dependent. For aluminium chemisorption results in a discharge voltage increase, while oxidation results in discharge voltage decrease. As shown above, depending on the material, oxidation can also result in a discharge voltage increase. Based on this work, it is possible to predict the direction/magnitude of the change. Knowing this effect is important as the change target in condition during reactive magnetron sputtering also affect the deposited thin film properties. Indeed, the ISEE coefficient of oxides is also related to the emission of negative oxygen ions, which bombard the growing thin film with an energy in the order of the discharge voltage, i.e. several hundred electronvolts [23]. Figure 5.8 shows this relationship and as shown by Ngaruiya et al. [24] the thin film properties are strongly related to this negative oxygen bombardment. 5.2.3 Stability Experiments The experiments have learned that target poisoning during reactive sputtering is influenced by chemisorption, knock-on implantation and direct reactive ion implantation. In these latter two processes, the reactive atoms are distributed in the target over 2 nm in the case of magnetron sputtering. Plasma immersion implantation experiments and ion beam experiments show the importance of the reaction kinetics and diffusion of these implanted ions in the formed oxide layer. To show that the implanted reactive atoms do not react immediately with the target material, one can introduce a waiting period between the oxidation of the target and its subsequent sputter cleaning to check the “stability” of the measured discharge voltage 30 ms after igniting the discharge. Hence, a typical stability experiment was performed as follows. After sputtering the aluminium target in pure Ar until a constant discharge voltage was registered, oxygen was introduced into the chamber. After stabilisation of the discharge

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V30ms-VAr (V)

−30 −32 −34 −36 −38 −40 0

500

1000 1500 2000 waiting period (s)

2500

3000

Fig. 5.9. Change of the discharge voltage difference between the discharge voltage measured at discharge ignition and in pure argon as a function of the waiting period between switching off the discharge (and oxygen flow interruption) and the igniting the discharge in pure Ar (measured at constant power 50 W, argon pressure 0.6 Pa (60 sccm) and 1.7 sccm oxygen). Under the given condition the target was not completely poisoned

voltage, the oxygen flow was interrupted. Simultaneously the magnetron discharge was switched off. After a given waiting period the magnetron was switched on again in pure Ar. During the waiting period the gas atmosphere was pure Ar. Figure 5.9 shows the change of the discharge voltage on discharge ignition (V30 ms ) as a function of the waiting period. Under the used conditions (see figure caption) the target is not fully poisoned. A small but significant decrease of the discharge voltage is noticed which becomes constant after approximately 40 s. This effect is not induced by the residual gasses in the vacuum chamber as an exposure of the same period hardly affects the discharge voltage on discharge ignition [25]. A more provoked effect has been measured by introducing a waiting period between the sputter cleaning of the target and interruption of the reactive gas flow during reactive sputtering of Si in a mixture of nitrogen and argon [26]. Figure 5.10 shows discharge voltage behaviour during sputter cleaning a polycrystalline silicon target which was first sputtered in a 50/50 mixture of argon and nitrogen. After reaching a constant discharge voltage the magnetron was switched off simultaneously with the nitrogen flow. After a given waiting period the discharge voltage was measured in pure Ar. Again one notices that the discharge voltage behaviour changes as a function of the introduced waiting period. In this experiment, the desorption of nitrogen gas from the target could be registered using a mass spectrometer. Both the aluminium and silicon experiments give evidence that the reaction between the reactive gas and the target material can be quite slow, and clearly show that implanted atoms can further react with the target

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target voltage (V)

500 waiting period 0s 1s 12 s 120 s 1200 s 2430 s

480 460 440 420 0

200

400 time (s)

600

800

Fig. 5.10. Discharge voltage behaviour during sputter cleaning a polycrystalline Si target in pure argon (50 W, 0.3 Pa) which was first sputtered in a 50/50 mixture of nitrogen and argon (in reactive mode)

material and/or diffuse out of the target after the magnetron discharge has been switched off. 5.2.4 First Conclusions Based on the Experiments The above-discussed experiments are all based on the discharge voltage behaviour as a function of the experimental conditions. These studies are only valid if one only compares the discharge voltage (behaviour) after carefully controlling the experimental conditions. Indeed, discharge voltages are only interesting when comparing their values or behaviour when measured under identical conditions. It must be stressed that this mistake is often noticed in literature where discharge voltage measured in pure argon is compared with the discharge voltage measured in a mixture of both argon and the reactive gas. This has no use because, as (5.1) shows, the discharge voltage depends not only on the target condition but also on the plasma composition. Nevertheless, the chemisorption experiments (see section “Aluminium”) show the sensitivity of the discharge voltage on discharge voltage ignition, and this effect is due to the sensitivity of the ISEE coefficient of the target. These experiments also show that a more complete model for reactive magnetron sputtering must include reactive ion implantation and knock-on implantation. Moreover, the stability experiments show that the reaction kinetics describing the reaction between the implanted reactive atoms and the target atoms is also an important point of the reactive sputter model. Based on this conclusion, the next part of this chapter describes a first try to extend the model described in the previous model to account for the described experiments.

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5.3 An Extended Model for Reactive Magnetron Sputtering Several models have been developed to describe the reactive sputtering process. Most of these are based on the basic model discussed in the previous chapter. Indeed, these models add specific topics to this model, such as plasma chemistry, the discharge voltage dependence of the target material sputter yield, the chemisorption reaction and many others. However, none of these models can explain the experiments shown in Sect. 5.2. The reason is extremely simple: the basic model includes only chemisorption as a reaction between the target material and the reactive gas. This is in first order a correct approach for the reaction on the substrate but is not a complete description for the target process. However, the approach presented by Berg and Nyberg can still be used but the target processes should be reconsidered. To implement reactive ion implantation in an analytical way, it is advisable to first derive a model that includes only implantation and compares its results with experiments where chemisorption does not affect the target condition. During reactive magnetron sputtering this is difficult to achieve because not only reactive ions but also neutral species are present which can chemisorb on the target surface. However, during reactive ion beam experiments such results can be obtained. Indeed, the compound formation during this type of experiments is studied by change the angle of incidence between the ion beam and the surface. Typically in such experiments the compound is formed at normal incidence (0◦ ) and at small impact angles but above a critical angle no complete oxidation of the target occurs. In this chapter a model is presented which can describe these experiments. It is based on the work of Schulz and Witmaack [27], who described ion implantation in the presence of sputtering. Building further on this model, the influence of chemisorption is included. This results in a more complete model to describe the target processes during reactive magnetron sputtering. The processes which take place on the substrate, i.e. deposition and reaction, are described in the same way as proposed in the previous chapter. Also the approach suggested in the previous chapter, i.e. the calculation of the flow balance is considered. This enables to describe the process parameters as a function of the reaction gas flow. 5.3.1 The Description of Reactive Ion Implantation During Sputtering Simultaneous Reactive Ion Implantation and Sputtering Without Chemical Reaction A classical problem is to calculate the concentration of implanted ions during sputtering. An interesting zero-order approximation has been proposed by Schulz and Wittmaack [27]. Although the model has been described in detail

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by these authors, the derivation is repeated here to make this chapter as complete as possible. The probability that a particle will come at rest in the target at a distance from the surface between x and x + dx is p(x)dx, where p(x) is the normalized distribution, ! D p(x)dx = 1 − B, (5.5) 0

with D the implantation depth and B the back scattering coefficient that accounts for the reflected ions. The depth in the target is represented by x. If I is the ion current density (ion cm−2 s−1 ) and f is the mole fraction of the reactive gas in the plasma or beam, then the number of reactive ions that bombard the target per area and time unit is 2f I. Hence, only molecular ions (R2+ ) are taken into account, and it is assumed in the case of reactive magnetron sputtering that the ionisation probability of the reactive gas and the argon gas in the plasma is equal to each other. It is also assumed that the ion energy is much larger than the binding energy of the molecular ions. Therefore, each molecular ion will dissociate into two reactive atoms, each having approximately half the energy of the original ion. The implantation of the argon atoms is neglected. A similar approach is followed by other models (e.g. TRIDYN [28]) based on the argument that the argon atoms will quickly diffuse from the target. The reactive atoms become implanted following the implantation profile (see (5.5)). So, in a given slab of the target at position x in the target, 2f Ip(x) reactive atoms are implanted or ∂nr (x, t) = 2fIp(x, t)∂t,

(5.6)

with nr the concentration of the non-reacted implanted reactive atoms in the target. In the following x ˆ represents a coordinate with its origin fixed in space. Due to the ion bombardment the surface will recede with a speed defined by the erosion rate vs (t) which is defined as vs (t) =

IYs (t) dˆ xs = , dt n0

(5.7)

with x ˆs (t) the instantaneous position of the surface and n0 the atomic density of the target (atoms cm−3 ). Strictly speaking the atomic density of the target is also a function of time. However, in this first approach the change of the atomic density of the target is neglected. This assumption is made based on the fact that the atomic target density ratio between the compound and metals is on average in the order of 2. As (5.7) shows, the erosion rate depends on the ratio between the sputter yield and the atomic density of the target. Hence by keeping the atomic density of the target constant, the influence of the sputter yield of the model will be too large. However, knowing the large errors on the sputter yields of compounds at small energy (see paragraph A.2.1.1 and Sputtering by Ono in Chap. 1), this approach seems reasonable.

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The coordinate x with its origin in the instantaneous surface is then related to the stationary coordinate x ˆ by x = xˆ − xˆs (t).

(5.8)

Due to erosion, the implanted ions are shifted towards the surface. Simultaneously, additional ions are implanted. Assuming that the implantation profile is not affected by the implantation of reactive ions in the target, i.e. p(ˆ x, t) = p(x) and the range distributions are shifted by the amount x ˆs (t), i.e. p(ˆ x, t) = p(ˆ x−x ˆs (t)). (5.9) The resulting time-dependent concentration of the implanted atoms becomes then ! t nr (ˆ x, t) = 2fI p (ˆ x − xˆs (t ))∂t . (5.10) 0

However, one is not interested in this solution because one needs to know the time-dependent concentration of the implanted atoms nr (x, t). This latter distribution can be derived from (5.10) by realizing that final position of the surface x ˆs (t) is given by ! t I Y (t )dt . (5.11) x ˆs (t) = XS = n0 0 This makes it possible to write the coordinate x as x = xˆ − XS

(5.12)

and the distribution nr (x, t) as !

t

nr (x, t) = 2fI

p (x + XS − x ˆs (t )) ∂t ,

(5.13)

0

with x ˆs (t ) the surface position at time t . It is important to realize that during reactive magnetron sputtering and ion beam experiments, the sputter yield Ys changes as a function of the time, which makes a step-by-step deduction of the accumulated implantation distribution necessary. When the implanted atoms do not react with the target material, the sputter yield remains constant during the experiment. In that specific case, the final surface position XS (see (5.11)) becomes x ˆs (t) = XS =

IY Δt n0

and using (5.7), (5.13) can be rewritten as nr (x, t) =

2fI vs

!

S

p (x + y) dy. 0

(5.14)

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The concentration at the surface (x = 0) equals then with (5.5) nr (0, t) =

2fI 2f (1 − B) = n0 (1 − B). vs Ys

(5.15)

This latter result has been derived by several authors [27, 29, 30]. Simultaneous Reactive Ion Implantation and Sputtering with Chemical Reaction Evidently, in most cases the implanted reactive atoms can react with the target material to form compound molecules. This will influence the target composition and also the sputter yield of the target. Hence, to calculate the time-dependent concentration, the time dependence of the erosion rate must be known. As the latter depends on the surface concentration, (5.13) can only be solved numerically. As shown below, it will be necessary to subdivide the target into two regions, i.e. s, the surface region from which sputtering occurs with a thickness in the order of a monolayer, and b, the subsurface region in which the reactive ions become implanted and react with the target material at a given rate. The thickness of this layer is defined by the implantation depth D. As proposed by Herbots et al. [31] the reaction rate R will be given by R = −kznr(x, t)nm (x, t),

(5.16)

with nm (x, t) the concentration of the non-reacted target material and k the reaction rate constant. The stoichiometry of the formed compound molecules defines z, i.e. MRz . Hence, the change in the concentration of the non-reacted implanted atoms with time can be written by combining (5.6) and (5.16) or ∂nr (x, t) = 2fI p(x) − kznr (x, t)nm (x, t). ∂t

(5.17)

A similar equation can be derived for the change in the concentration of the non-reacted target material with time, i.e. ∂nm (x, t) = −knr (x, t)nm (x, t). ∂t

(5.18)

Based on the definition of the erosion speed, one can rewrite the last two equations as vs (t)

∂nr (x, t) = 2fI p(x) − kznr (x, t)nm (x, t), ∂x

∂nm (x, t) vs (t) = −knr (x, t)nm (x, t). ∂x

(5.19)

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The reaction of the target material will result in a change in the target sputter yield and also in a change of the erosion rate. The sputter yield of the target can be defined as θrs Yr + θms Ym = Ys ,

(5.20)

with θrs and θms the fraction of reacted and non-reacted target atoms at the surface of the target; evidently θrs + θms = 1. Yr and Ym represent, respectively, the sputter yield of the formed compound molecules and the target atoms. Hence, it is assumed that the non-reacted implanted atoms are not sputtered. We assume that these atoms diffuse from the surface into vacuum, just as the implanted argon atoms. Diffusion of non-reacted implanted atoms from the target has been discussed in the previous section. As the sputter yield of the compound and the target material are not equal, preferential sputtering should be taken into account. To describe this effect the fraction of compound molecules at the interface between the surface zone s and the subsurface zone b, i.e. θrb is introduced. This fraction can be calculated from the concentration of non-reacted target atoms nm (s) at the position x = s, with s the thickness of the surface zone: θrb = 1 −

nm (s) . n0

(5.21)

As it is assumed that the non-reacted implanted atoms diffuse from the surface zone, there is no chemical reaction in the surface zone. Based on this assumption the following continuity equation can be derived [32] vs (t)θrb n0 = Iθrs Yr .

(5.22)

This relates the fraction of compound molecules at the interface to their fraction in the surface zone. Indeed, the left-hand side (LHS) of (5.22) describes the transport of compound material from the subsurface zone b to the surface zone s by the movement of the target surface under ion bombardment. The right-hand side (RHS) of the same equation describes the sputtering of the compound from the surface zone. As will be discussed in Sect. 5.4.2, these equations can be solved numerically, which enables to describe the target condition under different experimental conditions. In the case of the ion beam experiments (see further) the target condition can be described as a function of the impact angle between the beam and the surface. In the case reactive magnetron sputtering the target condition can be described as a function of the reactive gas mole fraction or pressure. 5.3.2 The Influence of Chemisorption and Knock-On Effects During reactive magnetron sputtering the target is also exposed to neutral gas species. Hence, chemisorption of these species cannot be neglected. Moreover,

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as discussed by Rosen et al. [28] the chemisorbed atoms can be knocked into the target by ions bombarding the target, resulting in compound formation not only at the surface but also at subsurface region of the target. Indeed, as shown in section “Aluminium” (see Fig. 5.5), knock-on implantation seems to be an important effect. To account for this latter effect and for chemisorption, the model must be further extended. Therefore, the fraction θcs is introduced which is defined as the fraction of compound formed by chemisorption at the target surface. Hence, the surface layer s is now characterized by three fractions, i.e. θcs , θrs and θms for which the sum is by definition equal to 1. θcs , θrs and θms represent, respectively, the surface fraction of compound formed by chemisorption, the fraction of compound formed by reaction between implanted reactive atoms and the target atoms and finally the non-reacted target atoms. Hence it is assumed that no chemisorption occurs on the formed compound. Consequently, the target sputter yield Ys must be redefined as compared with (5.20): θrs Yr + θms Ym + θcs Yc = Ys ,

(5.23)

with Yc the sputter yield of the compound formed by chemisorption. The other sputter yields have been defined before. In this simplified model it is assumed that the compound formed by chemisorption and reactive ion implantation is identical. Hence, the sputter yield Yc = Yr . To modify the model for chemisorption and knock-on implantation, the set of (5.19) should be altered. The knock-on of chemisorbed atoms into the target will modify the amount of reactive atoms which become implanted into the target. Hence, (5.19) should be rewritten as vs (t)

∂nr (x, t) = 2fI p(x) − kznr (x, t)nm (x, t) + Iθcs βpc (x), ∂x ∂nm (x, t) vs (t) = −knr (x, t)nm (x, t), ∂x

(5.24)

with pc (x) the implantation profile of the knock-on atoms. In a first approach it is assumed that this implantation profile does not differ from the reactive ion implantation profile p(x). β is the knock-on yield which is the number of chemisorbed atoms that are knock-on implanted per incoming ion. The fraction of compound formed by chemisorption at the target is defined by three processes: (1) Formation by chemisorption of neutral molecules on the non-reacted target material (2) The removal by sputtering (3) Knock-on of the chemisorbed reactive atoms into the target This results in the following equation which balances the formation and removal by sputtering and knock-on no,s

2F I dθcs = αθms − Yc θcs I − βθcs , dt z z

(5.25)

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with no,s the atomic surface density. F represents the flux of neutral molecules on the target surface given by F =√

fP tot , 2πmkT

(5.26)

with Ptot the total pressure, m the molecular mass, k the Boltzmann constant and T the absolute temperature. The sticking coefficient of the reactive gas on the target surface is defined by α. At steady state (5.25) reduces to I 2F αθms = Yc θcs I + βθcs . z z

(5.25 )

A similar balance equation can be written down for the steady-state fraction of the target material in the surface layer I 2F βθcs + IYs θmb = Ym θms I + αθms . z z

(5.27)

Indeed, on the LHS of (5.27) the first term accounts for the removal of a chemisorbed atom from a compound molecule, while the second term accounts for the transport between the subsurface region and the surface region. On the RHS of (5.27) the sputtering of the target material and the loss of non-reacted target material due to the reaction with the neutral reactive gas molecules by chemisorption is described. For the compound formed by reactive ion implantation only the transport from the subsurface region to the surface region must be taken into account. Hence, (5.22) remains valid. Again this equation can be solved numerically and enables to describe the target condition as a function of the reactive gas mole fraction in the case of magnetron sputtering. However, generally the experiments are performed as function of the reactive gas flow (see Figs. 5.1 and 5.2). Hence, it will be necessary to derive the relationship between the reactive gas partial pressure and the reactive gas flow. 5.3.3 Calculation of the Reactive Gas Partial Pressure as a Function of the Reactive Gas Flow During magnetron sputtering the partial pressure of the reactive gas is controlled by gettering of the reactive gas by the target material. A good approach to this problem has been proposed by Berg and Nyberg in the previous chapter. However, as the description of the target condition has been altered, not all the equations described in the previous chapter can be used. To help the reader, some of the equations are repeated in this chapter. The reactive gas flow qo into the vacuum chamber must be equal to the sum of the reactive gas

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flow into the pump, the flow towards the target (target consumption) and the flow towards the substrate (substrate composition), i.e. qo = qp + qt + qc .

(5.28)

The reactive gas flow towards the pump is defined by the pumping speed S and the reactive gas partial pressure in the vacuum chamber which is at constant total pressure equal to fPtot . Hence, the flow to the pump equals qp = SfPtot .

(5.29)

Similarly to the “Berg” model it assumed that the deposited target material reacts with the reactive gas only by chemisorption. In this way the flow towards the substrate is given by qc = αc F (1 − θc )Ac ,

(5.30)

with F the flux of the reactive gas molecules towards the substrate, αc the sticking coefficient of the reactive gas molecules on the substrate, Ac the substrate area and θc the fraction of compound on the substrate. This latter fraction can be calculated as described in the previous chapter by writing down the following balance equation At At 2αc F (1 − θc ) + I(Yc θsc + Yr θsr )(1 − θc ) = IYm θsm θc , z Ac Ac

(5.31)

where it is assumed that the compound formed by chemisorption and by reactive ion implantation has the same stoichiometry. The reader can find more details in the previous chapter to understand this balance equation. The calculation of the flow towards the target is more complicated. Indeed, besides the chemisorption, reactive ions become implanted in the target by direct implantation and knock-on implantation where they can react with the target atoms. Hence, the total amount of reactive atoms in the target at a given time per unit area can be calculated by integration of the distribution of the implanted reactive atoms and the compound molecules in the subsurface region of the target, i.e. ! D [nr (x, t) + z (no − nm (x, t))] dx. (5.32) N (t) = S

In this way, the flow towards the target can be calculated as z Iβθsc 1 dN + F αθsm − + Ys θrb At . qt = 2 dt 2 2

(5.33)

The first term accounts for the change in the total concentration of reactive atoms in the subsurface region of the target. The second term describes the amount consumed by chemisorption. This part is similar to the equation

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proposed in the previous chapter. The third term of (5.33) describes the atoms which become implanted by knock-on into the target. One must subtract their contribution from the amount which is chemisorbed as the model already accounts for these atoms in the integral given in (5.32). The last term of this equation accounts for the compound molecules present in the surface region. As the non-reacted atoms in the surface region diffuse out of the target, they are not consumed by the target. In steady state, (5.33) simplifies to z Iβθsc + Ys θrb At , qt = F αθsm − (5.33 ) 2 2 as the distribution of the implanted atoms and the formed compound molecules in the subsurface region does not change anymore. Based on (5.28) and the following the same, one can calculate the flow needed to reach a certain mole fraction in the vacuum chamber. In this way, one can plot the target condition and the partial pressure of the reactive gas as a function of the flow, similar to experimental results (see Figs. 5.1 and 5.2).

5.4 Confrontation Between Experiment and Model In Sect. 5.3 a complete description of a reactive sputtering model is given which takes into account several target reactions, i.e. chemisorption, knockon implantation and direct reactive ion implantation. In this section several aspects of the model will be confronted with the experimental results described in Sect. 5.2 together with some specific experiments. In this way, the reader can judge the value of the proposed model. The description of the model has been build up stepwise, taking more processes into account in each step. This approach has been chosen to make the evaluation of the model more transparent by confronting certain facets of the model with the experiment. 5.4.1 Simultaneous Reactive Ion Implantation and Sputtering Without Chemical Reaction Magnetron sputtering of silver (Ag) in a mixture of nitrogen/argon is an example where reactive ions become implanted in the target but do not react with the target material [33]. Moreover, nitrogen does not react on the substrate to form silver nitride. Therefore, this example is interesting to look in a little more detail. Figure 5.11 (top) shows the discharge voltage behaviour during sputter cleaning a silver target which was first exposed to given dose of nitrogen ions. These latter ions were implanted in the target using an ion gun at 5 keV. The discharge voltage is clearly a function of the implantation dose. Figure 5.11 (bottom) shows the discharge voltage behaviour during sputter cleaning a silver target. In this case the target was exposed to the same dose of nitrogen ions but prior to the sputter cleaning procedure a waiting period was introduced. One notices that the effect on the discharge voltage fades away

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46

345

23 12 x1016 ions / cm2 5.8 2.9 1.4

discharge voltage (V)

340 335 330

20 s 25 s 30 s 40 s 50 s 100 s

334 333 332 331 0

20

40 60 time (s)

80

100

Fig. 5.11. Discharge voltage behaviour during sputtering cleaning of a silver target in pure argon (0.4 Pa, 50 W). Using an ion gun the target was bombarded with nitrogen ions prior to the sputter cleaning procedure. In the top figure the total dose was increased. In the bottom figure the same dose (1.4 × 1016 ion cm−2 ) was used but before sputter cleaning a given waiting period was introduced

with an elongation of the waiting period. This example shows that implantation of reactive ions in a target during reactive magnetron sputtering can influence the discharge voltage, even without chemical reaction between the target material and the target atoms. 5.4.2 Simultaneous Reactive Ion Implantation and Sputtering with Chemical Reaction The Target Condition During Oblique Reactive Ion Implantation During this type of experiments, the target is only bombarded by reactive ions. So, the mole fraction f can be set equal to 1. By changing the angle between the reactive ion beam and the surface normal the sputter yield of both the target material and the formed compound will change. This will influence the oxidation level of the target surface. Figure 5.12 shows the results of such an experiment performed by Alay and Vandervorst [34]. The figure shows the oxidation of a Si target by 5 keV O+ 2 implantation as a function of the ion beam angle. In this experiment the target oxidation level has been experimentally measured by following the Si2p line in X-ray photoelectron spectroscopy (XPS) from which the oxidation level can be calculated. Based

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oxidation level

1.0 0.8 0.6 0.4 experimental data (taken from Alay et al.)

0.2

θrb θrs

0

20

40 60 angle of incidence(°)

80

Fig. 5.12. Oxidation level dependence on the angle of incidence of 5 keV O+ 2 ion beam on silicon. Simulation parameters were taken from De Witte (sputter yield ratio of silicon to silicon oxide) or simulated using SRIM (ion range and ion range straggle, sputter yield Si, backscattering coefficient). The reaction rate constant k was used as a fitting parameter between the experimental data and the calculated oxidation levels of the subsurface and surface region (k = 1×10−22 cm3 atom−1 s−1 ) (correlation coefficient 0.9584). The obtained results are independent of the used ion beam current density

on the attenuation length of the photoelectrons the sample depth of XPS is in the order of 5 nm. At normal incidence (0◦ ) the target is completely oxidized (oxidation level 1). At beam angles larger than a critical angle of approximately 22◦ the oxidation level decreases with increasing beam angle. Figure 5.12 shows also the calculation values for θrs and θrb . The values were derived by numerically solving the set of (5.19) and following the same. The sputter yield ratio of Si and SiO2 as a function of the angle of incidence at this energy was taken from the work of De Witte [35]. Changing the angle of the ion beam also modifies the implantation profile. A Gaussian implantation profile was defined. Hence ⎡  2 ⎤ 1 x − Rp ⎦. p(x) = √ exp⎣− √ (5.34) 2πΔRp 2ΔRp The projected ion range Rp and the ion straggle ΔRp were calculated using SRIM. No value of the ion current density (I) was reported in the work of Alay and Vandervorst [34]. However, Wittmaack [36] reported similar results as the former authors using an ion current density in the order of 1 × 1015 ion (cm−2 s−1 ). The same author also reported that increasing the ion current density to 1×1016 ion cm−2 s−1 has no influence on the oxidation level of the target. The independence of the obtained results for the ion current density was checked. The following solution scheme was used. Initial values of θrb and the beam angle were chosen. The chosen beam angle defines the sputter yields Ym and Yr . The erosion rate corresponding with the initial value of θrb was calculated

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based on (5.21) and (5.23). Solving numerically the set of (5.19) using the trivial boundary condition nm (0) = no and nr (0) = 0, a value of nm (s) can be calculated, which defines a certain value of θrb . In steady state the initial value of θrb should correspond to this latter value. Hence, the beam angle is change iteratively until this condition is reached. In the way, a plot of θrb and θrs can be made and compared with the experiment. The reaction rate constant k was used as a fitting parameter to obtain a good correlation between the experiment and the model. The obtained result indicates that the reactive ion implantation is capable of describing this type of experiments. The Target Condition During Reactive Magnetron Sputtering as a Function of the Reactive Gas Mole Fraction Without Chemisorption/Knock-On Implantation During reactive magnetron sputtering the mole fraction of the reactive gas can be modified by changing the reactive gas flow. Of course, the gettering of the reactive gas by the sputtered target material will influence the actual reactive gas mole fraction in the vacuum chamber. So f in the model represents the actual reactive gas mole fraction in the vacuum chamber. With the model described in Sect. 5.3.1, one calculates the mole fraction for a given target condition. Of course, in Sect. 5.3.1 the effect of chemisorption of the reactive gas on the target is not taken into account. Nevertheless, it is instructive to see the influence of the reactive ion implantation on the target condition without taking into account chemisorption. Using the same value for the reaction rate constant k as obtained in the previous section, one can calculate the silicon target condition in a reactive magnetron sputtering experiment as a function of the oxygen gas mole fraction in a mixture of argon and oxygen using the following solution scheme. Again an initial value of θrb is chosen and the erosion rate calculated. In contrast to the ion beam experiments, during reactive magnetron sputtering low energy ions (typically 400 eV) bombard the target always at normal incidence. The sputter yield of Si was calculated with SRIM and set equal to 0.4. The sputter yield of the oxide was set equal to 0.04. The implantation profile of O+ 2 was calculated using SRIM (Rp = 1.7 nm and ΔRp = 1 nm). By numerically solving the set of equations (5.18) using the trivial initial boundary conditions nm (0) = no and nr (0) = 0, a value of nm (s) can be calculated, which defines a certain value of θrb . In steady state the initial value of θrb should correspond to this latter value. Hence, the mole fraction f is changed iteratively until this condition is reached. In the way, a plot of θrb and θrs versus the mole fraction is obtained. Figure 5.13 shows the results. As expected the target becomes oxidized with increasing oxygen gas mole fraction. For reason which will be clear further in text, the influence of the nitrogen mole fraction on the target condition during reactive magnetron sputtering

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degree of oxidation

1.0 0.8 0.6 0.4 θrb θrs

0.2 0.0 0.00

0.01

0.02 0.03 0.04 0.05 mole fraction oxygen

0.06

0.07

Fig. 5.13. Calculated dependence of degree of oxidation of a Si target as function of the mole fraction in the plasma during reactive magnetron sputtering. The used current density was set to a value typical for reactive magnetron sputtering, i.e. 1 × 1017 ions cm−2 s−1 . The other used parameters are discussed in the text

of Si3 N4 will now be discussed. The sputter yield ratio between Si3 N4 (z = 4/3) and Si was experimentally found to be 1/7. The implantation profile of nitrogen ions was again assumed to be Gaussian with the ion range 1.8 nm and the ion range straggle 1.1 nm (SRIM simulation for 400 eV N2 + on Si). Based on an almost linear relationship between the nitrogen flow and the nitrogen partial pressure, it was argued that chemisorption was negligible during this experiment. Further, it was also argued that the chemical reactivity of nitrogen is much lower as compared with oxygen. This effect can be simulated by lowering the value of the reaction rate constant k. Figure 5.14 (top) shows the simulated degree of nitridation of the Si target using the discussed parameters and for different values of k. Figure 5.14 (bottom) shows the discharge voltage behaviour during reactive sputtering of Si as a function on the nitrogen mole fraction. As the experiment shows, the discharge voltage changes abruptly as a function of the nitrogen mole fraction. The same feature is noticed at low values of k in the model. In this specific case, the model can only compare the qualitative behaviour of the target condition as a function of the nitrogen mole fraction and not the quantitative behaviour. Indeed, at these large mole fractions the difference between the sputter yield of the reactive ions and the argon atoms should be taken into account. However, the abrupt change of the target condition as a function of the mole fraction is clearly represented. This strange effect needs some extra explanation to be understood. The interested reader can read Appendix for more details. With Chemisorption/Knock-On Implantation The influence of chemisorption without knock-on implantation (β = 0) has been studied using the same settings as for Fig. 5.14, setting k = 1×10−23 cm3 atom−1 s−1 and by modifying the influence of the sticking coefficient on the

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1.0 0.8 k=100 k=10

0.6

k=3.2

θb

k=1

0.4

θs

0.2

discharge voltage difference (V)

0.0 0.0

0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

40 30 20 10 0 −10 −20 −30 0.4 0.6 mole fraction nitrogen

Fig. 5.14. Top: The influence of the reaction rate constant k on the degree of target nitridation as a function of the nitrogen mole fraction. The simulation parameters are described in the text. Bottom: Discharge voltage difference measured between the discharge voltage measured after nitrogen addition and the discharge voltage measured in pure argon for a polycrystalline silicon target. The target was sputtered at constant power (50 W) [26]

abrupt behaviour. Figure 5.16 shows the result. As the sticking coefficient increases, the S-shape behaviour of the target condition θrb as a function of the reactive gas behaviour disappears, i.e. no abrupt change in the target condition will occur when the sticking coefficient is quite high. This can be understood as follows. As the sticking coefficient increases, the target surface condition becomes more and more defined by the chemisorption, which decreases the erosion rate. Hence, the sputter time increases, and the reactive atoms get more time to react which is similar to a situation with a higher reaction rate constant. Hence a similar situation develops as shown in Fig. 5.14 (left). Figure 5.15 shows also the target surface condition behaviour using the “Berg” model (for the same pressure and a sticking coefficient equal to one). As one notices, the behaviour is quite similar to the described model. This explains the success of the basic model presented by Berg and Nyberg. Indeed,

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1.0 α=0 α=0.001 α=0.01 α=0.05 α=0.1 α=0.5 α=1.0 basic model

fraction metal θms

0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.10 0.15 0.20 mole fraction reactive gas

0.25

0.30

Fig. 5.15. Target surface condition (metal fraction) as a function of the reactive gas mole fraction for different sticking coefficients. Except for the sticking coefficient, the same conditions were used as in Fig. 5.14. The total pressure was set at 0.2 Pa

degree of oxidation

1.0 0.8

sticking coefficient α=0.1 β=0 β=0.001 β=0.01 β=0.05 β=0.1 β=0.5 β=1

0.6 0.4 0.2 0.0 0.0

0.2

0.4 0.6 mole fraction reactive gas

0.8

1.0

Fig. 5.16. Subsurface condition (θrb ) as a function of the reactive gas mole fraction for different knock-on yields β, keeping the sticking coefficient constant at 0.1. All other parameters where identical as in the previous figure

when the reactive gas has a sufficient high sticking coefficient, the target surface condition described by both models is similar. Hence, the well-known hysteresis behaviour will be similar. This latter effect will be discussed in more details in the following section. The difference between both models is that at a mole fraction of 0.15, the target is completely poisoned over a depth of approximately 3 nm in the presented model, while the latter is of course not the case in the basic model. Using the same conditions, the effect of knock-on can be studied, by using a sticking coefficient equal to α = 0.1 and by modifying the knock-on yield β. As shown in Fig. 5.16, increasing the knock-on yield has the opposite effect than chemisorption. At a constant oxidation level, the corresponding reactive mole fraction shifts towards higher values. The same behaviour is noticed for lower sticking coefficients. At low sticking coefficients, one notices that

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an abrupt change in the target becomes possible again with an increasing knock-on yield. The Target Condition During Reactive Magnetron Sputtering as a Function of the Reactive Gas Flow In the previous section, some features of the model were discussed, and it was shown that some experiments can be understood from the model. The aim of this chapter is to discuss the details of the reactive sputtering of aluminium in a mixture of oxygen and argon. Figures 5.1, 5.2 and 5.9 summarize the most important results. Although, some of the details have been already discussed, the model can help the reader to understand this reactive sputtering process in more detail. However, in this first discussion not all details of the reactive sputter process are implemented. These aspects, such as the influence of the deposition and erosion profile, plasma-related topics and the target design, are discussed in Sect. 5.5. The reason for this subdivision is that the model needs further development to describe these aspects also. To model a set of hysteresis experiments as depicted in Fig. 5.2, several parameters, such as the pumping speed, target size, total current and current density, discharge voltage, can be taken from the experimental conditions. Based on this latter parameter, the implantation parameters, and the sputter yield of the target material can be calculated based on SRIM. Of course, some important parameters are difficult to retrieve from the experiments. Although the sticking coefficient of the reactive gas on the target has been determined, this value can only be used as a lower limit because the presence of oxygen radicals can increase the average sticking coefficient. Although the knock yield can also be estimated from simulations (e.g. SRIM, TRIDYN, . . .) its value is difficult to estimate because these models are very sensitive to the input parameters such as surface binding energy. A similar, but slightly more complex problem is the sticking coefficient on the deposited target material. Indeed, as shown by Passeggi et al. [41], the sticking coefficient of the reactive gas during deposition is generally larger than for chemisorption of the reactive gas on an already formed metal layer. The reason for this difference lays in the fact that the chemisorption process during deposition is better described as co-deposition of the reactive gas molecules and the target materials. Another “free” parameter is the substrate area. In (5.31) it is assumed that the target material is deposited uniformly over the substrate (and chamber walls) with a given area. Of course, this is generally not true. A deposition profile should be taken into account. This problem is tackled in the next section but it complicates the model description. However, with (5.31) one actually implements an average flux of the deposited target material on the substrate. The value of this flux is changed by modifying the substrate area in the model. In this way, one accounts for the influence of the flux. This approach will be followed in this section. In the next section, a more detailed discussion will be held on this topic.

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To compare the model results with the experimental results, the sum of the squared differences between the calculated critical points and the experimental critical points, i.e. χ2 , was calculated for different combinations of the sticking coefficient on the target and substrate, the knock-on yield, the reaction rate constant and the substrate area. As a critical point is defined by the oxygen flow and the oxygen pressure at the critical point, two values of χ2 can be calculated, one for the flow and one for the pressure. Figure 5.17 shows the influence of the used parameters on the ratio between χ2 calculated for the flow and the experimental error. Although the parameters were varied over a large interval, only 2.5% of the calculations resulted in a lower χ2 than the experimental error. This sets some restrictions to the value

1000

100

10

1 1

24 579 10 13 15 16 18 sticking coefficient on the target α

1000

100

10

1 1

2

0 0.02 0.03 0.04 0.05 0.06 0.08 0.09 0.1 0.12 0.13 0.2 0.22 0.3 0.33 0.5 0.66 1.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 0.1 0.11 0.15 0.2 0.22 0.3 0.33 0.45 0.5 0.6 0.66 0.99 1

3 5 6 8 912 1314 knock on yield β

1000

100

10

1 1 2 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

58 912 13 16 18 reaction rate constant k

1000 1 2 3 4 5 6 7 8 9 10 11 12 13 14

100

10

1

1 24 56 8 11 12 14 sticking coefficient on the substrate Ac 1000 1 2 3 4 5 6 7 8 9 10 11 12

100

10

1 1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.05 0.053 0.075 0.1 0.15 0.225 0.25 0.273 0.3 0.493 0.5 0.55 0.75 1

256 300 512 600 603.97 900 951.94 1000 1024 2048 4096 16384

2 468 9 101112 substrate area Ac

5 10 14.7 15 17.3 18.5 19.9 20 21.5 24.5 30 40 45 50 60 80 90 500

Fig. 5.17. The ratio of the calculated χ2 and the experimental error. The χ2 is based on the difference between the modelled and experimental critical flows for the experiment depicted in Fig. 5.2. The ratio is plotted as a function of the modified parameters in the model. The numbers on the X-axis refer to the value of the parameter given in the legend. Between two numbers the value of the modified parameter is kept constant while the other parameters are modified. The substrate area is given in cm2 , while the reaction rate constants are given in 10−24 cm3 atom−1 s−1

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Table 5.1. Average value and standard deviation (between brackets) for the modified parameters in the model for which the ratio between the χ2 and the experimental error is smaller than one for both χ2 calculated based on the critical flows and the oxygen pressure at the critical point Sticking coefficient target

Sticking coefficient substrate

Knock-on yield

Reaction Substrate area rate constant (cm2 ) 3 −1 −1 (cm atom s )

0.17 (0.034)

0.35 (0.13)

0.22 (0.093)

2.0 × 10−23 (4.9 × 10−26 )

380 (150)

of the modified parameters. The sticking coefficient on the target should be lower than 0.33. The sticking coefficient on the substrate is lower than 0.75. The limit of the knock-on yield is around 0.33. No good fits are obtained with a substrate area larger than 1, 024 cm2 . The reaction rate constant layers between 14.5 × 10−24 and 24.5 × 10−2 cm3 atom−1 s−1 . Combining this result with the χ2 for the critical pressures, an even more sever reduction of the number of combinations (six out of 3,087 combinations) which results in a good fit is achieved. Table 5.1 shows the average value of the used parameters together with the standard deviation. Although no experimental quantification of these values have been made, the values seem reasonable. The sticking coefficient of oxygen on the aluminium target is much higher than the one measured but as reasoned before the dissociation and excitation of oxygen molecules can explain this higher sticking coefficient. Indeed, it was expected from the work of Pekker [11, 12] that the sticking coefficient should be in the order of 0.1. The higher sticking coefficient for oxygen on the substrate has been discussed before. The ratio between the knock-on yield and the sputtering yield was calculated with SRIM (see above) to be 1.65. As the total sputter yield for the chemisorbed oxygen was set equal to the sputter yield of aluminium oxide, i.e. 0.23, the oxygen sputter yield is 0.14 and hence a value of 1.62 is found for the ratio. A substrate area of 380 cm2 agrees with a circular substrate with a diameter of 11 cm, which seems an acceptable value for deposition from a 2 in. magnetron. The reaction rate constant is may be surprising low but estimated from the small standard deviation its value is a key parameter in the model and can only be changed within a limited interval. Some other experimental trends can be understood from the model. Before the abrupt decrease of the discharge voltage a small increase of the discharge voltage (see Fig. 5.1) is noticed, which was attributed to chemisorption of oxygen on the target. The maximum value of the target coverage by chemisorption θsc is shown in Fig. 5.18 for different pumping speeds. The target coverage by chemisorption increases a function of the pumping speed. Also shown in Fig. 5.18 is the voltage difference between the maximum registered voltage

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0.12 15 0.10 10 0.08 5 16

32

65 pumping speed (L /s)

discharge voltage differerence (V)

25 0.14

185

130

θrb,20s-θrb,steady state

Fig. 5.18. On the left-hand side, the maximum value for the target coverage θsc for different pumping speed calculated by the model using the parameters discussed in the text and given in Table 5.1. On the right-hand side the difference in the maximum discharge voltage registered and the discharge voltage measured in pure Ar

0.15 0.10 0.05 0.00 0.00

0.05

0.10 0.15 0.20 mole fraction oxygen

0.25

Fig. 5.19. Change of the subsurface compound fraction due to waiting without sputtering. θrb,steady state refers to the steady-state compound fraction during reactive sputtering while θrb,20 s refers to the compound fraction calculated 20 s after switching off the discharge, i.e. without sputtering

and the discharge voltage measured in pure Ar for different pumping speed. A similar effect is noticed. It was shown in Sect. 5.2 that the discharge voltage of an aluminium target first sputtered in a mixture of argon and oxygen, can depend on the waiting period before the sputter cleaning in pure Ar. The effect of a waiting period is simulated with the model using the same parameters as used for Fig. 5.19. Figure 5.19 shows the difference between the subsurface θrb compound fraction reached after steady state and the subsurface compound fraction θrb after 20 s waiting without sputtering for different oxygen mole fractions in the plasma. Clearly the non-reacted implanted reactive atoms continue to form compound

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even without sputtering. This could explain the changes in discharge voltage as shown in Fig. 5.9.

5.5 Towards a More Complete Model for Reactive Magnetron Sputtering Of course the model for reactive magnetron sputtering is far too simple to describe all facets of this complex deposition technique. When carefully analysing the assumptions made in the model, it is clear that still quite some progress can be made resulting in more complete model. Some of the needed modifications are quite straightforward and can be easily implemented in the model. 5.5.1 Plasma-Related Topics In the discussion related to Table 5.1, the much higher sticking coefficient of the oxygen molecules on the target as compared to the measured and literature values was attributed to the dissociation of the oxygen molecules into oxygen radicals with a much higher sticking coefficient than the molecular oxygen. Hence, it is important to implement the plasma chemistry. A first attempt to include the plasma chemistry in reactive sputtering modelling was performed by Pekker et al. [11, 12]. To reach this goal a space average model was purposed. The needed reaction rate constants for the plasma reactions have been derived by developing a simple model for the DC magnetron discharge. Once these reaction rate constants are known, the concentration of all species in the plasma should be known, and in this way the equations derived in Sect. 5.3 can be improved by including more species. In this way, the difference in the ionisation probability of argon and the reactive gas can be included which is a better approach than the assumption made in the proposed model. Indeed, in the model it was assumed that the fraction of reactive ions bombarding the cathode is equal to the fraction of reactive gas in the vacuum chamber. Taking into account other species such as reactive gas radicals will influence the importance of for example chemisorption as these species have generally a higher sticking coefficient. The influence of the deposition parameters on the plasma chemistry can be included in the model. Improving the DC magnetron discharge model will be necessary for a better description of the plasma chemistry. Indeed, it is well known that the ISEE coefficient strongly influence the plasma parameters and in this way also the plasma chemistry. As also the target condition changes as a function of the reactive gas flow, a complete model of reactive magnetron sputtering will only be possible when the magnetron discharge is modelled in all its details. We refer to Chap. 3 for more details.

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5.5.2 Deposition Profiles and Erosion Profiles Influence of the Deposition Profile To calculate the substrate condition during reaction magnetron sputtering, i.e. θc , the deposition of the compound molecules and the target atoms which are sputtered from the target should be taken into account. This is given in (5.31). Indeed, the second term of the LHS of this equation describes the sputtering of the compound from the target and the deposition on the substrate. The RHS of (5.31) describes in the same way the sputtering of the metal atoms and their deposition on the substrate. As can be revealed from this equation the sputtered material is deposited uniformly on a substrate with a given dimension. Of course, it is well known that this latter is not valid. In a more realistic approach, a deposition profile can be measured. As the material flux towards a surface element of the substrate is not equal, the composition will not be uniform. Hence, (5.30) and (5.31) should be rewritten. In most cases the deposition profile is radial symmetric and therefore these equations will be modified as an example for this specific situation. The substrate can be subdivided in a series of concentric rings with Δr as width, ! r  2 Ac = 2πr dr  2πn(Δr) . (5.35) 0

n

One can attribute to each ring a θc , and the reactive gas flow towards this ring is, using (5.30) qc,n = αc F (1 − θc,n )Ac,n = 2πn(Δr)2 αc F (1 − θc,n ),  qc,n . qc =

(5.36)

n

The substrate condition can be calculated using (5.31) as 2αc At wn At wn F (1 − θc,n) + I(Yc θsc + Yr θsr )(1 − θc,n ) = IYm θsm θc,n , (5.37) z Ac,n Ac,n with wn a weighting factor which depends on the deposition profile (see further). It is assumed that the weighting factor is not dependent of the deposited material, i.e. oxide or metal. Using a quartz balance the deposition profile of aluminium sputtered from a 2 in. magnetron (see caption for more experimental details) was measured and is shown in Fig. 5.20. Based on this measured deposition profile it is possible to calculate a deposition rate on a planar substrate. The data can be fitted with a sigmoid curve ψ(r). This function must be normalized as ! ∞ ψ(r)2πr dr = 1. (5.38) 0

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15 30 45 60

−15

1

−30

0.8 0.6 0.4

75

−45 −60 −75

0.2 90

0

−90

Fig. 5.20. Normalized deposition rate for the aluminium from a 2-in. magnetron. Experimental conditions: 0.4 Pa Ar, current 0.4 A. The deposition rate was measured with a quartz balance positioned on a circle with a radius of 10 cm, i.e. the balance/target distance was 10 cm and constant

The weighting factors are then calculated as wn = ψ(r)2π(Δr)2 .

(5.39)

Performing this procedure, weighting factors have been calculated for a planar substrate located 6 cm from the target and with a diameter of 32 cm. These conditions have been chosen as the average material flux on the substrate is equal to the homogeneous material flux on a substrate with an area of 380 cm2 (see Table 5.1). Interesting, this results in the same hysteresis behaviour of the pressure as a function of the oxygen flow is simulated in Sect. 5.4. From this analysis one learns that it is the average material flux on the substrate which also influences the critical point. It is also clear that to reach a sufficient degree of oxidation at a high deposition speed one must be able to work between two critical points, or where the hysteresis curve has a negative slope. This can be achieved by using some kind of feedback control of the oxygen flow as shown for the first time by Sproul. An excellent review by this author on this topic is given in [37]. Comparing the top squares in Fig. 5.21, it is clear that to fully oxidize (95%) the deposited material the working point when taking into account the deposition profile shifts even further to higher oxygen pressures and hence to lower deposition rates. Hence, it is important to simulate the deposition profiles which can be done by using the knowledge described in chapter 6 of this book. Influence of the Erosion Profile Similar to the material flux, the ion flux to the target, i.e. the ion current density, is not homogeneous during magnetron sputtering. This is clearly noticed during planar magnetron sputtering by the formation of the erosion track. From the erosion profile (see Fig. 5.22) one can estimate the ion current density distribution on the target. However, this is a crude approximation as the deepening of the race track results in a strong modification of the

5 Depositing Aluminium Oxide θc

Profile 0.9 0.7 0.5 0.3 0.1 0.9 0.7 0.5 0.3 0.1 0.9 0.7 0.5 0.3 0.1 0.9 0.7 0.5 0.3 0.1

oxygen pressure (Pa)

0.2

0.4

0.6

189

Homogeneous

0.8

1.0

0.08 0.06 0.04 0.02 0.00 0.6 0.8 1.0 1.2 1.4 1.6 1.8 flow (sccm)

Fig. 5.21. Simulated hysteresis behaviour using the parameters described in Table 5.1 of the oxygen pressure as a function of the reactive gas flow for a pumping speed of 16 L s−1 . On the RHS the substrate condition is given for four different points on the curve when a homogeneous material flux is used and a substrate area of 380 cm2 . The top square corresponds with on oxide fraction of 95%. On the LHS the substrate condition is given for the same points of the hysteresis curve but now the deposition profile as shown in this figure is taken into account. The full line is the oxidate state for the central line along the substrate. The substrate was at 6 cm from the target and a diameter of 32 cm

0.0

depth (mm)

−0.2 −0.4 −0.6 −0.8 −1.0 −25

−15 −5 5 15 position on the target (mm)

25

Fig. 5.22. Erosion profile measured after 60 h sputtering of an aluminium target at constant current (0.4 A) and constant argon pressure (0.4 Pa) (markers). The simulated target condition (value of θsr ) on the first critical point (see Fig. 5.23) is also shown

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actual magnetic field. Hence, at this moment the presented results are only qualitatively correct. In the model presented in Sect. 5.3, this influence was not implemented as only the measured area of the profile was used to calculate the average ion current density. In this section a method is suggested to include this feature of the deposition process. To account for the non-homogeneous current density, the target (5 cm diameter) is subdivided in ten concentric rings with a width of 2.5 mm and based on the measured erosion profile a current density is attributed to each ring. Then the model is applied for each of these rings to calculate the target condition as a function of the reactive gas pressure or fraction. To calculate the substrate condition and the total flow, the material flux towards the substrate must be calculated. Neglecting at this point the deposition profile (see previous section), (5.30) and (5.31) can again be used. The values of θsc , θsm and θsr needed to solve (5.30) are calculated by averaging over the different target parts using the ion current density as weighting factor. As the model actually calculates the reactive gas mole fraction from the target condition, it is necessary to interpolate to have the target condition of each target part at the same mole fraction. Figure 5.23 shows a calculated pressure/flow hysteresis together for some specific points on this curve the target condition (value of θsr ). As expected the target is first poisoned in the regions where the current density is low. θsr

0.16

0.0

0.2

0.4

0.6

0.8

1.0

oxygen pressure (Pa)

0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0.9

1.1

1.3 1.5 flow (sccm)

1.7

Fig. 5.23. Simulated hysteresis behaviour using the parameters described in Table 5.1 of the oxygen pressure as a function of the reactive gas flow for a pumping speed of 16 L s−1 . To calculate the current density on the target the erosion profile shown in the figure is used. The target condition is represent by the surface fraction of compound formed by reaction between the implanted (direct and knock-on) reactive atoms, i.e. θsr . The larger markers refer to a point on the hysteresis curve for which the target condition is shown. Due to the low resolution of the target subdivision the curve shows some non-realistic jumps

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At the first critical points (at approximately 1.7 sccm) only the region within the highest current density remains more or less metallic. 5.5.3 Rotating Magnetrons To use magnetron sputtering on a large, industrial scale it is necessary to cope with some of the disadvantages inherent to the technique. Particularly, in the case of standard planar magnetrons, the presence of the race track often limits the target consumption to 25 up to 30% and increases the cost of ownership. A sputter source which is able to largely eliminate this problem is the rotating cylindrical magnetron. In this concept, the target is a cylindrical tube, which rotates around a stationary magnet configuration pointing in the direction of the substrate. Although a stationary race track is present during the sputtering process, no groove is formed in the rotating target and a very high target consumption may be achieved [38]. The simulation of this magnetron type during reactive magnetron sputtering is quite challenging because the time dependency is very important. This finds its origin in the fact that a given surface element of the target remains only for a limited time in the race track. Outside the race track the reactive gas can be chemisorbed on the target. Hence, it is not surprising that the rotation speed influences the hysteresis behaviour. Figure 5.24 shows the shift of the hysteresis towards lower oxygen flow as a function of the rotation speed. The simulation of this effect will not be discussed in this chapter as the implementation becomes quite complicated. However, in this context it is

discharge voltage (V)

360

340 0 rpm 2.4 rpm 8.9 rpm 64.1 rpm

320

300

280

0.0

0.5

1.0 1.5 oxygen flow (sccm)

2.0

2.5

Fig. 5.24. Discharge voltage behaviour during reactive sputtering of aluminium with the rotating cylindrical magnetron for different rotation speeds. The arrows indicate the shift of the hysteresis when increasing the rotation speed. Experimental conditions: constant Ar pressure, 0.4 Pa and constant discharge current, 0.3 A [39]

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important to mention that maybe the most important difference between the “Berg” model and the model described in this chapter is the time dependency. To proof this point, a hysteresis is simulated with both models. The simulation conditions of both models have been chosen to reach approximately the same hysteresis curve, explaining the differences between the used sticking coefficients. So, from this point of view, i.e. steady-state conditions, the results are comparable. However, more important is the difference in the kinetics which is introduced by implementing reactive ion implantation. To illustrate this difference, the following experiment is simulated. During partial pressure control, the flow is regulated to maintain a given partial pressure of the reactive gas. When the experiment is started both target and substrate are in metallic mode. So, in the beginning the reactive gas flow must be high because most of the reactive gas is consumed by the target material. The reaction of the gas with the target material will cover both target and substrate with compound and progressively less reactive gas is needed. Hence, as a function of time the reactive gas flow will decrease. This situation is simulated using the “Berg” model and with the presented model. Figure 5.25 shows the results. In agreement with the expectations, implementing ion implantation and knock-on results in more reactive gas consumption during the target oxidation. For the “Berg” model: sticking coefficient reactive gas on the target: αt = 0.7, sticking coefficient reactive gas on the substrate αc = 0.7. For the implantation model: Rp = 2 nm, ΔRp = 1 nm, β = 1, reaction rate −1 constant k = 1 × 10−22 cm3 atom s−1 : αt = 0.03 and αc = 0.35. There is also only a small difference in steady state as the contribution of the target to the total flow is small. However, an important difference is noticed in the path to reach this steady state. From practical point of view

flow (sccm)

100

Berg model presented model

10

1

0.1 0.01

0.1 time (s)

1

10

Fig. 5.25. Reactive gas flow as a function of time during partial pressure control (0.06 Pa) simulated using the “Berg” model and the implantation model. Simulation conditions: current 0.167 A, target surface 10 cm2 , substrate + chamber walls surface 1,000 cm2 , pumping speed 10 L s−1

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this discussion may look not relevant. However, as the thickness of formed compound layer on the target is not just the assumed monolayer in the “Berg” model, one should realize that the process kinetics can not be described accurately, which could influence the understanding of reactive sputtering process using rotating cylindrical magnetron (and/or partial pressure control) where the “Berg” model fails.

5.6 Conclusion In this chapter a model for reactive magnetron sputtering is presented. The main difference between the presented model and the “Berg” model is the implementation of the reactive ion implantation into the target during this process. Taking this effect into account makes it possible to explain in more detail some experimental results during the reactive magnetron sputter deposition of aluminium oxide from an aluminium target. However, this is not the “end” of the modelling work. More work has to be done to make this model more complete. Especially the implementation of the plasma chemistry is important. Also, further experiments are needed to reduce the number of “free” parameters. Acknowledgements. The authors wish to thank the following persons for their contribution to this paper: J. Haemers, A. Segers, K. Eufinger S. Mahieu, S. Heirwegh. X.Y. Li, A. Colpaert and G. Buyle. The authors also wish to express the gratitude to the IWT-Vlaanderen and N.V. Bekaert for the financial support.

Appendix To facilitate the discussion the model described in Sect. 5.3.1 will be simplified. Instead of assuming a Gaussian implantation profile, one can assume that the ions are implanted in a slab with thickness w located at x = D in the target. Outside this slab no ions are implanted. Hence, p(x) = (1 − B)/w, because ! D p(x)dx = 1 − B. (5.40) D−w

If the reaction rate is slow enough, the implantation and the reaction can be decoupled in time, i.e. one can first describe the implantation/sputtering process and follow the reaction of the implanted ions as the slab moves towards the surface of the target. So, the concentration of the ions in the slab before the reaction starts can be calculated as follows. With (5.40) the concentration of the ions in the slab will change as ∂nr =

2fI (1 − B) ∂t. w

(5.41)

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However, as no ions are implanted outside the slab, the implantation period in the slab is defined by the erosion rate of the target Δti = w/vs . Hence, ! ! nr 2fI (1 − B) Δti 2fI (1 − B) w 2fI 2f n0 ∂nr = ∂t = = (1 − B) = (1 − B). w w vs vs Ys 0 0 (5.42) This is not a surprising result as it has already derived before (see (5.14)). Due to the simultaneous sputtering, the surface moves towards this slab. Stated otherwise, with the initial concentration of the reactive atoms described by (5.42), the slab starts to move towards the surface while the reactive atoms react with the target atoms with an initial concentration given by no . Under this assumption the reaction kinetics is given by ∂nr (x, t) = −kznr(x, t)nm (x, t), ∂t ∂nm (x, t) = −knr (x, t)nm (x, t). ∂t

(5.43)

In contrast to (5.23) this latter equation can be solved analytically which   2f results in − zθ rb 1 = kn0 Δtr , (5.44) ln 2fYs 2f − z Ys Ys (1 − θrb ) with Δtr the reaction time. In steady state this reaction time is equal to the time for the slab to reach the surface or equal to D/vs . Hence, one can write   2f Ys D Ys − zθrb = kn20 = C ste , (5.45) ln 2f 2f I Ys − z Ys (1 − θrb )

0.2

−0.6 −0.2

0.6

1.6

1

1.3

0.6 0.5

0

0.4 0.3

-0.3

-0.1 0.1

0.4

0.8

−0.4

0.0 0.00

0.6

0.2

1

-0.2

1.2

b

−1 −0.8

-1.4 -1.2 -1 -1.1 -0.9 -0.7 -0.8 -0.6 -0.4 -0.5

1.8

0.4

0.8 1.4

θ

0.4

1.0

−1.4 −1.2

*

θ

b

0.6

−1.6 0.2

0.8

1.6

1.0

-1.7 -1.3 -1.5

which makes it possible to solve (iteratively) this equation and to plot the target condition as a function of the mole fraction f . The behaviour of the target condition as a function k is similar to the one shown in Fig. 5.26

B

0.2 A

0

0.02

0.04 0.06 mole fraction reactive gas

0.08

0.10 −23

k=1x10

sputter time > reaction time sputter time < reaction time θb implanted concentration too low form amount of compound implanted concentration = amount of formed compound amount of non reacted reactive atoms

−3

(cm /at.s)

0.0 0.00

0.02

0.04 0.06 mole fraction reactive gas

0.08

0.10

k=3x10−23 (cm−3/at.s) sputter time > reaction time sputter time < reaction time θb implanted concentration too low form amount of compound implanted concentration = amount of formed compound amount of non reacted reactive atoms

Fig. 5.26. Ratio between reaction time and sputter time for two different values of k. The used values for the sputter yield were Ym = 0.5 and Yr = 0.05. The current density was set equal to 2 × 1017 ions cm−2 s−1 and the implantation depth D 3 nm. The stoichiometry was set z = 2

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(right). The simplicity of (5.45) makes it possible to understand the abrupt target behaviour as function of the mole fraction for low k values. Indeed, one can calculate, for a given target surface condition, the reaction time needed to reach a certain value of θb . This reaction time can be compared to the time needed to reach the slab located at a depth D in the target. The value of the sputter time Δts can be calculated based on the target surface condition, given by Ys . The ratio Δts /Δtr is shown as a function of θb and the mole −1 −1 fraction for a larger value of k (3 × 10−23 cm3 atom s ) and for a low −1 value of k (1 × 10−23 cm3 atom s−1 ). On both plots the solution of (C.7), i.e. the steady-state value of θb as a function of the mole fraction is also plotted which agrees with the ratio Δts /Δtr = 1. This curve separates the plot into two regions. On the left-hand side (low fractions) the reaction time is longer than the sputter time. On the right-hand side the opposite is true. Also plotted on the same figure is the amount of non-reacted reactive atoms in the target. As one can notice in both situations a certain amount of these atoms is present. This can be understood as follows. When the surface reaches the implantation slab, the reaction must stop, even if not all reactive atoms have reacted with the target material to form compound material. Of course, for the low k value example this amount of non-reacted reactive atoms is much higher than for the higher k value example. So, when increasing the mole fraction, one deviates from the steady state to a situation that the reaction time is shorter than the sputter time (see Fig. 5.26 arrow A). Hence, the reactive atoms have more time to react resulting in more compound formation (see Fig. 5.26 arrow B), which further increases the sputter time. Hence more compound formation becomes possible, on the condition that enough reactive atoms are present. Here the difference between the low and high k value example becomes clear. In the higher k value example the amount of non-reacted reactive atoms is quite small but in the low k value example this amount is much high. Hence, at the critical point (see Fig. 5.26 asterisk) the following situation develops. The deviation from the steady state results in more compound formation as the reaction time is shorter than the sputter time. However, due to large amount of non-reactive atoms present in the target, this results in a strong increase of the compound formation, which lowers the erosion rate, i.e. increases the sputter time with even more compound formation as a result. However, to form more compound more time is needed. Hence, this avalanche ends when the reaction time becomes again equal to the sputter time. This analysis shows another important point. Some simulation codes (e.g. TRIDYN) are used to describe the oxidation of the target by implantation of reactive atoms. These simulations show that the target condition behaviour does not change abruptly. This can be understood from the presented model as these simulation codes do not take the reaction rate into account, i.e. the reaction between the implanted species and the target is immediate or stated differently a very high value for k. From the given analysis

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it is understandable that no abrupt changes can be calculated with these models. Using an infinitive reaction rate in (5.45) shows that 2f /Ys = zθrb . Or stated differently, using (5.22), each implanted atom reacts with the target material to form compound material: Ys I 2f I = Iθrs Yr . θrb n0 = n0 z

(5.46)

Hence, there are no non-reacted reactive atoms present and an avalanche situation can never develop. As a final remark it can be shown that (5.45) is almost similar to a previously published equation. Indeed, in a previous model, the concentration of the implanted reactive ions was calculated in the same way as above. However, as no time dependence was taken into account, the amount of implanted of reactive atoms was generally much larger than the amount needed to form the compound, i.e. z. Hence for 2f /Ys z (5.45) becomes −

Ys2 ln(1 − θrb ) = f. 2C ste

(5.47)

In the previous model the influence of preferential sputtering was not taken into account, and the following equation was derived [40] −

Yb ln(1 − θrb ) = f, 2C ste

(5.48)

with Yb = (Yc − Ym )θrb + Ym . Both (5.47) and (5.48) give a similar S-shaped curve.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

K. Koski, J. H¨ ols¨ a, P. Juliet, Surf. Coat. Technol. 116–119, 716 (1999) D. Depla, R. De Gryse, Plasma Sources Sci. Technol. 10, 547 (2001) G. Buyle, PhD Thesis, Ghent University, 2005 T. Do, N.S. McIntyre, Surf. Sci. 440, 438 (1999) V. Zhukov, I. Popova, V. Fomenko, J.T. Yates Jr., Surf. Sci. 441, 240 (1999) V. Zhukov, I. Popova, J.T. Yates Jr., J. Vac. Sci. Technol. A 17, 1727 (1999) V. Zhukov, I. Popova, J.T. Yates Jr., Surf. Sci. 441, 251 (1999) W.H. Krueger, S.R. Pollack, Surf. Sci. 30, 263 (1972) B.C. Mitrovic, D.J. O’Connor, Surf. Sci. 405, 261 (1998) A. Arranz, C. Palacio, Surf. Sci. 355, 203 (1996) L. Pekker, Thin Solid Films 312, 341 (1998) E. Ershov, L. Pekker, Thin Solid Films 289, 140 (1996) D. Guttler, B. Abendroth, R. Grotzschel, W. Moller, D. Depla, Appl. Phys. Lett. 85, 6134 (2004) 14. D. Depla, J. Haemers, R. De Gryse, Thin Solid Films 515, 468 (2006)

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15. S.S. Todorov, E.R. Fossum, Appl. Phys. Lett. 52, 48 (1988) 16. R. Behrish (ed.), Sputtering by Particle Bombardment II, (Springer, Berlin Heidelberg New York, 1981) 17. D. Depla, R. De Gryse, J. Vac. Sci. Technol, A 20, 521 (2002) 18. P.H. Dawson, Surf. Sci. 57, 229 (1976) 19. E. Taglauer, W. Heiland, U. Beitat, Surf. Sci. 89, 710 (1979) 20. The Stopping and Range of Ions in Matter (SRIM) can be downloaded from www.srim.org 21. D. Depla, S. Heirwegh, S. Mahieu, J. Haemers, R. De Gryse, J. Appl. Phys. 101, 013301/1 (2007) 22. D. Depla, H. Tomaszewski, G. Buyle, R. De Gryse, Surf. Coat. Technol. 201, 848 (2006) 23. S. Mahieu, D. Depla, Appl. Phys. Lett. 90, 121117/1 (2007) 24. J.M. Ngaruiya, O. Kappertz, S.H. Mohamed, M. Wuttig, Appl. Phys. Lett. 85, 748 (2004) 25. D. Depla, R. De Gryse, Plasma Sources Sci. Technol. 10, 547 (2001) 26. D. Depla, A. Colpaert, K. Eufinger, A. Segers, J. Haemers, R. De Gryse, Vacuum 66, 9 (2002) 27. F. Schulz, K. Wittmaack, Radiat. Eff. 29, 31 (1976) 28. D. Rosen, I. Katardjlev, S. Berg, W. Moller, Nucl. Instrum. Meth. B 228, 193 (2005) 29. J.C.C. Tsai, J.M. Morabito, Surf. Sci. 44, 247 (1974) 30. Y. Kudriavtsev, R. Asomoza, Appl. Surf. Sci. 167, 12 (2000) 31. N. Herbots, C. Hellman, O. Vancauwenberghe, in Low Energy ion-surface interactions, ed. by J.W. Rabalais (Wiley, New York, 1994), ISBN 0471938912 32. W. Patterson, G. Shirn, J. Vac. Sci. Technol. 4, 343 (1967) 33. D. Depla, R. De Gryse, Vacuum 69, 529 (2003) 34. J.L. Alay, W. Vandervorst, Phys. Rev. B 50, 15015 (1994) 35. H. De Witte, Ph.D. University of Antwerp, Belgium 36. K. Wittmaack, Surf. Sci. 419, 249 (1999) 37. W.D. Sproul, D.J. Christie, D.C. Carter, Thin Solid Films 491, 1 (2005) 38. J. Musschoot, D. Depla, G. Buyle, J. Haemers, R. De Gryse, J Phys. D. Appl. Phys. 39, 3989 (2006) 39. D. Depla, J. Haemers, G. Buyle, R. De Gryse, J. Vacuum Sci. Technol. A 24, 934 (2006) 40. D. Depla, R. De Gryse, Surf. Coat. Technol. 183, 184 (2004) 41. M.C.G. Passeggi, L.I. Vergara, S.M. Mendoza, J. Ferron, Surface Science 50`e– 510, 825 (2002)

6 Transport of Sputtered Particles Through the Gas Phase S. Mahieu, K. Van Aeken, and D. Depla

6.1 Introduction Knowledge of the transport of sputtered particles through the gas phase is not only a fundamental research topic, but also of main interest to know the properties of the arriving metal flux at the substrate or chamber walls as a function of the deposition geometry and deposition conditions. Characterisation or modelling of this metal flux is, e.g. of interest when a high thickness uniformity is needed [1], when complex substrates are to be deposited [2], or when the thin film composition has to be controlled [3]. However, not only the amount of arriving metallic particles has to be controlled, but also their direction and impact energy are influencing the final properties of the deposited thin films. Although some attempts have been done, characterisation of the incoming direction [4] or the energy [5] of metal particles experimentally is difficult to carry out. A better attempt is to simulate the transport of the sputtered particles, and to verify the simulation code by the experimental much more accessible deposition profile [6]. An overview of the simulation of the transport of the sputtered particles through the gas phase is given in this article. Section 6.2 describes the place wherefrom the sputtered particles leave the target. Section 6.3 discusses the different ways to describe the initial energy and direction of the sputtered particles, when leaving the target. In Sect. 6.4, the mean free path, and thus the distance to a collision with the background gas, is discussed. Also the collision and the interaction potential needed to describe the collision are treated in Sect. 6.4. Section 6.5 briefly discusses the possible ways to include the effect of gas rarefaction. Finally, in Sect. 6.6 a concrete simulation model is described and some typical results are shown.

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6.2 Radial Distribution Where Sputtered Particles Leave the Target Magnetron sputtering is based on the generation of a magnetically enhanced glow discharge. As discussed in detail by A. Bogaerts et al. in Chap. 3 [7], positive discharge gas ions are accelerated towards the target, which is kept at negative voltage. This ion bombardment results in the emission of secondary electrons from the cathode, which are confined in the vicinity of the target due to the electric and magnetic fields. This confinement of the electrons by a non-uniform electric and magnetic field causes a locally enlarged ionisation degree of the sputter gas. Since the ionized gas particles are nearly not influenced by the magnetic field, they are accelerated perpendicular to the target surface by the electrical field. Hence, the locally enlarged ionisation degree results in a locally enlarged ion bombardment of the target and thus a non-uniform erosion of the target resulting in a racetrack [8]. However, one should take into account that this racetrack is not a direct reproduction of the radial distribution wherefrom sputtered particles leave the target. The racetrack is formed by the local erosion of the target, counteracted by the redeposition of some sputtered particles onto the target. Hence, the resulting racetrack is a combination of sputtering and redeposition. The radial distribution wherefrom the sputtered particles leave the target, used for simulating the metal flux towards the substrate can be deduced following two approaches. After sputtering a target, one can measure experimentally the profile of the resulting racetrack, e.g. by profilometry. The inverted profile could be taken as the radial distribution wherefrom sputtered particles leave the target [9–16]. However, following this approach during a simulation, one should take care that redeposited particles are already taken into account. Hence, simulated particles leaving the target, but which finally seems to be redeposited onto the target, should be omitted during the simulation. In another approach, one can simulate the ionisation distribution of the sputter gas and a perpendicular projecting of it onto the target gives the place where ions strike the target [17–25]. This distribution where ions strike the target can also be measured experimentally by measuring the local current into the target region [26, 27]. Because sputtered atoms leave the target in the near vicinity of the ion impact, the distribution where ions bombarded the target corresponds with the distribution where sputtered particles leave the target. In case that the projection of the ionisation distribution is used as input for the radial distribution of leaving particles, redeposited particles onto the target should be taken into account during the simulation.

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6.3 Energy and Angular Distribution of Sputtered Particles Leaving the Target Together with a description of the total sputter yield, Sigmund [28] and Thompson [29] also provided a description of the energy and the angular distribution of the sputtered atoms, respectively. Because nearly all other analytical models of the sputter process are based on the Sigmund–Thompson theory, this theory will be discussed in Sect. 6.3.1. The results of other analytical models, based on the Sigmund–Thompson theory will be summarized in Sect. 6.3.2. Examining experimentally measured energy and/or angular distributions of sputtered atoms indicate that the analytical models not perfectly match with the experiment (see Sect. 6.3.3). A better fit with the experimental energy and angular distribution could be obtained by simulating the sputtering process, or by studying this process by solving the master equations for the linear collision cascade generated by ion impact (see Sect. 6.3.4). 6.3.1 Sigmund–Thompson Theory for the Linear Cascade Regime According to Sigmund, it is convenient to distinguish the collision cascade into three regimes: the single-knock-on regime, the linear cascade regime and the spike regime [28]. In the single-knock-on regime, the bombarding particle transfers energy to target atoms which, possibly after having undergone a small number of one-to-one collisions, are ejected from the surface. In the linear and the spike regime, the recoiled target atoms are energetic enough to generate secondary and higher generation recoils. In this way, a cascade of recoils is generated, and again some atoms at the surface may by ejected from the solid. In the linear regime, the density of recoils is sufficiently low to ensure that most collisions involve one moving and one stationary particle. This regime is called linear since the sputter yield is found to be proportional to the first power of the projectile energy. In the spike regime, the density of recoils is so high that collisions between two moving atoms are frequently occurring. Quantitatively, the single-knock-on regime falls into the lower and medium incident energy region, except for light ions where, because of inefficient energy transfer, it extends up to the lower keV region. The linear cascade regime is characteristic for bombarding particles with an energy from several hundreds eV to some keV and MeV, except for the heavier ions, which tend to cause the spike regime. In case of magnetron sputtering, the lower energy part of the linear cascade regime is of main interest since the bombarding particles have an energy in the range of several hundreds eV. The sputter yield Y is defined as the number of sputtered atoms per bombarding particle. To calculate the sputter yield in function of the energy of the bombarding particle, the number of atoms which suffer recoil and the

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fraction of these recoiled atoms that are able to escape from the surface, should be known. Let n(E, E0 ) be the mean number of atoms set in motion with an initial energy larger than some value E0 in a cascade initiated by a primary particle of initial energy E(> E0 ). Allowing only for elastic collisions and ignoring all binding forces between target atoms, one finds the expression [30]: n(E, E0 ) = Γm

E for large E, E0

(6.1)

with Γm a parameter depending only on the index m, which is a parameter depending on the energy E and which characterizes the scattering cross section. However, a particle bombarding a solid will not only dissipate its energy by elastic nuclear collisions, but also by exciting the electrons. This energy delivered to the electrons will not provide a chance to the atoms to recoil, since this energy will be immediately shared with all other electrons. Therefore, the total number of atoms set in motion after the impact of a primary particle with energy E will not be n(E, E0 ) but n(E, E0 )

FD (E, Ω, r ) , E

(6.2)

where FD (E, Ω, r ) is the density of deposited energy for a particle of energy E impinging under direction Ω at the solid surface at position r , slowed down due to the continuous drain of energy to the electrons. Therefore, the mean number of atoms moving at any time with an energy (E0 , dE0 ) is given by n(E, E0 )

FD (E, Ω, r ) FD (E, Ω, r ) dE0 dt0 = n(E, E0 ) , E E v0 |dE0 /dx|

(6.3)

where dt0 is the mean time needed by a recoil atom to slow down from E0 + dE0 to E0 , v0 is the velocity of a target atom with energy E0 . Hence, the impact of an energetic particle generates a stationary distribution of moving target atoms. Assuming that the direction in which these target atoms move is isotropic, and thus assuming a complete collision cascade, it can be calculated how much of these target atoms are able to reach the surface under a specific direction Ω 0 . This, together with (6.1) gives an expression for the mean number of atoms moving with an energy (E0 , dE0 ) in the direction (Ω 0 , d2 Ω 0 ) in the volume (r , d3 r ): Γm

FD (E, Ω, r ) dE0 d2 Ω 0 Ω0 . E0 v0 |dE0 /dx| 4π

(6.4)

Multiplication with v0 finally yields the current density of atoms reaching the target surface. If these particles can overcome the potential barrier existing at

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the surface, they are considered as sputtered. The simplest model for surface binding is based on a planar surface barrier Us . Finally, an expression for the sputter yield of a particle impinging with an energy E and angle θ at the surface in function of the energy E0 of the sputtered target atoms (r = 0) and their angular direction θ0 is given: d3 Y = Γm

FD (E, θ, 0) dE0 d2 Ω 0 cos θ0 . E0 |dE0 /dx| 4π

(6.5)

Nice about the analytical theory is that the energy and angular distribution of the sputtered atoms can be deduced from (6.5). Assuming a planar surface barrier Us that the recoiled target atoms have to overcome before being sputtered and taking into account the conservation of momentum, the following equations must be fulfilled: E1 cos2 θ1 = E0 cos2 θ0 − Us , 2

2

E1 sin θ1 = E0 sin θ0 ,

(6.6a) (6.6b)

in which the subscript 0 indicates the situation before the target atom reaches the surface and the subscript 1 indicates the situation after the atom leaves the target, as also shown in Fig. 6.1. The energy is indicated by E, v indicates the velocity and θ is the direction with respect to the surface normal. It results from (6.6a) that E1 cos θ1 d2 Ω1 = E0 cos θ0 d2 Ω 0 and hence, (6.5) yields: FD (E, θ, 0) E1 dE1 d2 Ω 1 (6.7) cos θ d3 Y = Γm 1 (E1 + Us )2 |dE0 /dx|E0 =E1 +Us 4π The factor dE0 /dx expresses the loss in energy when a target atom moves through a medium. Hence dE0 /dx is given by NS n (E) in which N is the density of the medium and Sn (E) is the nuclear stopping cross section. According to Sigmund [30] the nuclear stopping cross section is proportional to E 1−2m . The parameter m depends on the energy E and is 0 for E = Us , 0.2–0.3 for E = 1 keV and reaches unity when E is very high. Hence (6.7) gives an

Fig. 6.1. Effect when an atom passes the planar surface barrier Us (left) or is reflected at the target surface (right). The indices 0 and 1 indicate the situation just before and just after the ejection respectively

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expression for the energy and angular distribution of the sputtered atoms: N (θ1 , E1 ) =

d3 Y E1 ≈ cos θ1 . dE1 d2 Ω 1 (E1 + Us )3−2m

(6.8)

The energy distribution has its maximum at Nmax = Us /(2(1 − m)). The angular distribution of the sputtered particles is a cosine function. Thompson [29] derived this expression in case for very low energetic bombardment (m = 0), leading to the well-known energy distribution function: N (θ1 , E1 ) ≈

E1 cos θ1 (E1 + Us )3

and Nmax =

Us . 2

(6.9)

It should be remarked that (6.6) only takes into account that an atom is refracted upon passing through the planar surface barrier. However, it was not taken into account that some atoms (those with E0,x < Us ) are really hindered to be sputtered from the target due to the existence of the surface barrier. Secondly, it is noticed in (6.8) that this analytical model suggests that the energy distribution and the angular distribution are independent of each other. One of the main assumptions of the Sigmund–Thompson theory is that of a complete collision cascade, and thus the assumption of a complete isotropic recoil distribution. However, in case of magnetron sputtering, the energy of the bombarding ions E is often below 400 eV. Hence, the collision cascade generated by the ion impact is often not complete. To make sure that the sputter model is also applicable for magnetron sputtering, some authors have extended or adapted the Sigmund–Thompson theory for the case of low energetic (and/or light) ions, bombarding (heavy) targets. 6.3.2 Other Analytical Models Falcone developed a theory for collisional sputtering, equivalent to the Sigmund–Thompson theory, but that can be extended to other collisional regimes [31]. More explicitly, the Falcone theory should also be applicable in the single knock-on regime. Falcone assumed that a target atom, after having undergone a recoil, arrives at the target surface (and thus can be sputtered) without having undergone elastic collisions. The recoiled target atom only loses energy by inelastic collisions. This is in contradiction to the Sigmund–Thompson theory which assumed that recoiled target atoms only undergo elastic collisions and lose energy by a continuous effect of electronic stopping. In case of bombarding heavy targets by low energy ions, Falcone’s theory resulted in an expression for the energy and angular distribution of sputtered atoms:

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N (θ1 , E1 ) ≈

E1 log (E1 + Us )5/2



γE E1 + Us

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cos θ1 .

(6.10)

Here γE is the maximum transferable in a collision energy between an incident ion of energy E and mass M1 and a target atom of mass M2 : γ=

4M1 M2 . (M1 + M2 )2

(6.11)

However, Kenmotsu argued that inelastic energy loss cannot be dominant for low energy (few hundred eV) ion sputtering [32]. He mentioned that for light ion sputtering, the dominant sputter mechanism is that, after entering the target, an ion is backscattered first by a target atom and then knocks off a target atom near the surface on its way out. Based on this mechanism, and neglecting the inelastic energy loss by electronic stopping, he derived a new expression for the energy and angular distribution of sputtered atoms:

2 E1 γ(1 − γ)E N (θ1 , E1 ) ≈ log cos θ1 . (6.12) E1 + Us (E1 + Us )8/5 Finally, Ono adapted the theory of Kenmotsu by taking into account the inelastic energy loss by electron stopping, which was neglected by Kenmotsu [33]. However, Ono also assumed that primary knock-on atoms produced by ions backscattered at large angles do not lose energy while penetrating the material up to the surface (similar to Falcone). He obtained the following expression for the energy and angular distribution of sputtered atoms due to light ion bombardment: ⎛ ⎞ 1/2 E1 B + E  ⎠ cos θ1 , N (θ1 , E1 ) ≈ log ⎝  1/2 (E1 + Us )5/2 B + (E1 + Us )1/2 / (γ(1 − γ)) (6.13) with B = 2C(γ(1 − γ))1/2 /K, in which 1/2 π 2Z1 Z2 e2 2 M1 C = 0.276a 2 M2 a and

1.216 × 10−2 Z1 Z2 1/2 2  3/2 eV nm , 1/2 2/3 2/3 Z1 + Z2 M1 7/6

K=

where Z1 and Z2 are the atomic numbers of the incident ion and the target atom, respectively, and a is the Thomas–Fermi screening length. All above analytical models assume that the momentum density of the sputtered particles is isotropic. However, Sanders and Roosendaal have shown that due to the conservation of momentum, the momentum density in a collision cascade is not isotropic [34, 35]. Taking into account the conservation

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of momentum, Roosendaal and Sanders developed an analytical model to describe the energy and angular distribution of sputtered atoms [36]. This model provides an analytical solution for three different sputter scenarios. In scenario I, the incident ion causes a primary recoil. This primary recoil is energetic enough to induce further recoils and thus a complete collision cascade. In scenario II, the incident, light ion is backscattered by a target atom. In its way out of the target, the backscattered ion causes a primary recoil and thus also a collision cascade. Finally in scenario III, the energy of the incident ion is so low that it dissipates all its energy in the vicinity of the target surface.  1/2  E1 E1 cos2 θ1 + Us 1 − 3C cos θ1 , (6.14) N (θ1 , E1 ) ≈ E (E1 + Us )3/2 √ with C = 1/ γ (scenario I), 

M1 M1 + M2 M2 M1 − M2  if M1 < M2 (scenario II), or C = M1 /M2 (scenario III). This model shows that the energy distribution of the sputtered atoms depends on their emission angle θ1 . C≈

6.3.3 Comparison to Experimental Results The above-described analytical models give an expression for the energy and angular distribution of the sputtered particles. It can be noticed for all models, except for the Roosendaal model, that the energy and angular distribution are independent of each other. Moreover, these energy-independent angular distributions are always a cosine distribution and do not depend on the energy of the ion E. For the Sigmund–Thompson theory, even the energy distribution of the sputtered atoms does not depend on the energy of the ion E. However, experimentally measured energy and angular distributions reported in literature are in contradiction to these observations. It was observed by Goelich [37] and Chernysh [38] that the angular distribution of the sputtered atoms, due to the impact of low-energy ions, deviates from the cosine distribution. This deviation from the cosine distribution was also reported by Stepanova [39] and Zhou [40]. It was also observed [37, 41] and reported [39, 42, 43] that the angular distribution of the sputtered particles depends on the energy of those particles. This means that, in contradiction to most of the analytical models, the energy and angular distribution are not independent of each other. As a general trend, it was experimentally observed that the most probable energy of sputtered particles emitted closely the normal on the target is lower than the most probable energy of particles leaving the target at high angles.

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To overcome the discrepancies between the analytical models and the experiments, several simulation models have been developed. Such codes simulate the collision cascade generated by ion impact by a Monte Carlo binary collision approximation, e.g. TRIDYN [44], SRIM/TRIM [45], ACAT [46], MARLOWE [47] or by molecular dynamics, e.g. the work by Zhou [40] and Harrison [48]. In literature, the results of these simulation codes have been extensively compared to the experimentally measured data such as sputter yields as well as energy and angular distributions. The simulation codes seemed to be able to reproduce satisfactory the dependency of the energy distribution on the energy of the primary ion E [49, 50]. Also a deviation from the cosine distribution could be simulated [37, 50] as well as the dependency of the angular distribution on the energy of the sputtered particles [37, 50, 51]. It can be concluded that these simulation codes can be used to create an input for the simulation of transport of sputtered particles. However, the use of these codes has the drawback that it increases the computing time, and that for each situation (target material, ion material and ion energy) the energy and angular distribution of the nascent sputter flux has to be simulated. 6.3.4 Numerical Method Stepanova and Dew also studied the energy distribution of sputtered atoms by low energy Ar+ ions [39]. They studied this process by numerically solving the master equations describing the linear collision cascade generated by ion impact. They included only elastic binary collisions. However, beside a planar surface barrier, they also took into account a focusing effect. They calculated the angle-resolved energy distribution of the atoms sputtered by normal impact of low energy Ar+ ions on 26 different target materials. Fitting those results, they could obtain a semi-empirical expression for the angular and energy distributions: q E1 M1 (E1 cosq1 θ1 + Us ) 2 exp −A N (θ1 , E1 ) ≈ (E1 + Us )3−2m M2 E E1 + Us χ(θ1 ), (6.15) 1− Emax + Us with Emax the maximal emission energy and χ(θ1 ) the angular distribution. Although Stepanova mentioned for some conditions that this angular distribution is almost a cosine distribution, they did not discuss it in more detail. A reasonable overall agreement between their numerical calculations and the suggested semi-empirical expression was obtained for: A = 13,

q1 = 2 − M2 /4M1 ,

and q2 = 0.55.

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An expression for the maximal emission energy was given by Moussel et al. [49]: 6 M1 + 2M2 Emax = γE (6.16a) 2M1 + 2M2 while the maximal energy is also sometimes taken from Emax + Us = γE (6.16b) [11, 14]. It should be remarked that the parameter m is the same parameter as in the Sigmund–Thompson theory and thus should have a value between 0 and 0.2 for ion energies E of some hundreds of eV. 6.3.5 Conclusions It was experimentally observed that low energy ion bombardment (E ≈ 200–400 eV) results in the ejection of sputtered target atoms. The angular distribution of these sputtered atoms is ion energy E dependent, and deviates from the simple cosine distribution. Also the energy distribution of the sputtered distribution seems to deviate from the simple Sigmund–Thompson formula and is also observed to depend on the emission angle θ1 of the sputtered atoms. Although there are many variations and adaptations to the Sigmund– Thompson model by various researchers (see Sect. 6.3.2), it continued to be hard to fit all experimental observations with an analytical model. These experimental observations can be reasonably reproduced by the simulation codes based on Monte Carlo binary collision method as well as by the codes based on molecular dynamics. In an attempt to avoid the use of theses simulation codes, Stepanova developed a semi-empirical model. In Fig. 6.2, the energy distributions of atoms sputtered by 200 eV normal impact of Ar+ ions on an Al target are shown. Those energy distributions were calculated from above-mentioned analytical models. However, the results of Kenmotsu, Ono and Roosendaal (scenario II) are not shown, since these models are only valid for light ion bombardment, i.e. M1 < M2 . Also the experimental values, as measured by Goelich are shown for comparison [37]. It can be observed that only the Roosendaal model and the semi-empirical model of Stepanova indicate an angular dependent energy distribution. The Stepanova model can reproduce the experimental values, at least if the parameter m is taken as a fitting parameter and if Emax was calculated from (6.16b). Here m was taken as 0.5, what is high compared to the predictions of Sigmund [30].

6.4 Describing the Collision with the Gas Particle Knowing the place where the sputtered particles leave the target (Sect. 2) and their initial energy and direction (Sect. 3), one can start describing their motion through the gas phase.

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a

b

Fig. 6.2. Energy distributions of Al atoms sputtered by Ar+ 200 eV ion impact, and emitted in a direction of 0◦ (a) or 60◦ (b) which respect to the target normal

Generally, some assumptions are made when describing the collisional transport through the gas phase. Often, only interactions between sputtered particles and the sputter gas are considered. This assumption is justified because the concentration of sputtered particles (ns ) is very small compared to the concentration of the background gas (ng ) at typical magnetron sputter conditions (ns /ng is 10−2 –10−3 ). Secondly, the presence of charged particles (ions, electrons) and the occurrence of non-elastic interactions is neglected because ion and electron concentrations are small in comparison with the neutral atom quantity (typical ionization degree is 10−3 –10−4 ) [52]. Thirdly, for similar reasons only sputtered atoms and no multi-atomic particles are considered. In some cases, it is assumed that the distribution of the background gas is constant in time and space. This latter assumption ignores the pressure variations which may occur during sputtering at higher pressures and sputtering power due to gas heating or gas rarefaction. This effect of gas rarefaction will be treated in more detail in section 6.5.

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6.4.1 The Mean Free Path The collisions between the sputtered particle and the background gas are assumed to be a random process described by a Poisson distribution. Therefore, the distance to the first collision or the length of the free path between two succeeding collisions, λ∗ , is generally determined stochastically from the Poisson distribution according to a standard Monte Carlo procedure: λ∗ = λ ln (1/1 − x), where x is a random number from the interval (0,1) and λ is the mean free path of a sputtered atom [53–57]. Although there is only one physical correct mean free path, there are several mathematical ways to approximate this mean free path. The mean free path λ is related to the collision cross section σ by λ=

kB T 1 = , ng σ pσ

(6.17)

with ng the density, T the temperature, p the pressure of the background gas and kB the Boltzmann constant. In case of the hard sphere approximation σ is calculated from σ = π(rs + rg )2 ,

(6.18)

with rs and rg the atomic radius of the sputtered particle and the background gas, respectively. Taking into account the motion of the background gas, and assuming a Maxwellian gas, the collision cross section as calculated from the kinetic gas theory is given by [59] 1/2 ms , σ = π(rs + rg ) 1 + mg 2

(6.19)

with ms and mg the molecular mass of the sputtered atom and the background gas, respectively. Sometimes, this additional mass factor was simplified resulting in [15, 21]: σ = π(rs + rg )2 (2)1/2 . (6.20) These collision cross sections are independent of the energy or the relative velocity of the colliding atoms, although it has been shown that σ is energy dependent [54, 60]. Robinson showed that this energy dependency can be approximated by an empirical power law: σ(E) = σ(E0 )

E E0

−0.29

σ(E) = σ(E0 ) if E < E0 , where E0 = 1 eV.

if E > E0 ,

(6.20a) (6.20b)

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In stead of multiplying σ with the empirical power law, the energy dependency of the collision cross section can also be approximated by using energy dependent atomic radii, as suggested in the variable hard sphere (VHS) model [12, 61, 62]. In this VHS model, the collision cross section is given by σ = πd2VHS ,

(6.21)

where dVHS , the velocity dependent diameter is given by 1 (ds,VHS + dg,VHS ), 2  ω −1/2 1 2kTref i , = dref i,VHS 2 mr vr Γ (5/2 − ωi )   15 mi kTref /π i = s, = 2(5 − 2ωi )(7 − 2ωi )μi (Tref )

dVHS = di,VHS ref

di,VHS

(6.22) (6.23) (6.24)

with subscript s standing for the sputtered atoms and subscript g for the background gas. μi (Tref ) is the viscosity at the reference temperature Tref , ωi is a temperature exponent describing the velocity dependence of the cross sections and mr is the reduced mass: ms mg /(ms + mg ). In another, but more computation time consuming approach, the collision cross section can be taken as σ = πb2max ,

(6.25)

in which bmax is the value of the impact parameter for which still a small deviation of the sputtered atom trajectory is calculated. The method to calculate this maximal impact parameter bmax is described in Sect. 6.4.2. 6.4.2 The Scattering Angle After a collision between the sputtered atom and the background gas, the sputtered particle is scattered and thus changes its direction of motion. The geometry of this scattering interaction in the centre of mass frame is shown in Fig. 6.3. In case of the hard sphere approximation, the scattering angle θcom is given by (6.26) θcom = 2Arccos(z), with z a random number in the interval (0,1). The polar scattering angle in the laboratory frame θlab is calculated by (6.27), while the azimuthal scattering angle φlab is random between 0 and 2π. sin θcom . (6.27) θlab = Arctg cos θcom + ms /mg

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(a)

(b)

Fig. 6.3. The geometry of the scattering process in the centre of mass frame (a) and the special case for the hard sphere approximation (b)

After collision with the background gas, the sputtered atom has lost energy, and the remaining energy in the laboratory frame Elab is calculated by: 2(1 − cos θcom )mg ms . (6.28) Elab = Elabold 1 − (ms + mg )2 However, this hard sphere approximation is too crude, since the sputtered atom and the background gas interact with each other during the collision process by repulsive and attractive forces. These forces can be described by a spherically symmetric atom–atom interaction potential V (r). Then the scattering angle is given by: ! ∞ dr 2θcom = π − 2b , (6.29) 2 {1 − [V (r)/E 2 2 1/2 r com ] − (b /r )} r0 in which r0 is implicitly determined by V (r0 ) 2 2 , b = r0 1 − Ecom

(6.30)

with b the impact parameter between 0 and bmax , r the inter-atomic distance, V (r) the interaction potential and Ecom the energy of the sputtered particle in the centre of mass frame. Ecom and Elab are related by: Ecom = Elab

mg /ms . (1 + mg /ms )

(6.31)

The maximal impact parameter bmax is then determined as the maximal value of b for which still a deflection is calculated: θcom > θmin where the θmin is an arbitrary value usually taking between 0.2◦ and 2◦ [6, 16, 50]. Sielanko [63] derived an analytic expression for the polar scattering angle θcom using the Lindhard’s differential cross section [64, 65]. sin2

θcom 2





−n P 2 ε2/n = 1+ , (K 2 B)1/n

(6.32)

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with B = (8n − 1)/n2 , and P = b/a with a the screening length and b the impact parameter. ε is given by: Ecom . (6.33) ε = 6.9456 × 106 a Zs Zg By fitting this analytical scattering angle to the numerical evaluation of the scattering integral (6.29) for the screened Coulomb potential with Moliere screening function and Firsov screening length (see Sect. 6.4.3), Sielanko obtained the following expressions for n and K: n = 1 + 4 exp(−1.9ε0.1 ), K = A − DP − C,

(6.34) (6.35)

with

n2.8 √ , A = 0.224[1 + 0.52 ln(1 + ε1.2 )]n1.206 sin 13.8 + 0.12 ε



0.00034 2 B = 0.0283 exp , (6.36) + 0.75 (n − 2.9) 0.001 + ε0.0227n3.8 0.001 C= . 0.0156 + P (ε0.11 +3.962P 0.82 ) To know the new direction of the sputtered atom, the scattering direction (θlab and φlab ) should be added to the original direction (θ and φ) as schematically shown in Fig. 6.4 what finally results in the new direction of the sputtered particle (θnew and φnew ). This addition of two directions is done by the following matrix multiplication [55, 66]: ⎤ ⎡ ⎤ ⎡ cos(θ) cos(φ) − sin(φ) sin(θ) cos(φ) sin(θnew ) cos(φnew ) ⎣ sin(θnew ) sin(φnew ) ⎦ = ⎣ cos(θ) sin(φ) cos(φ) sin(θ) sin(φ) ⎦ cos(θnew ) − sin(θ) 0 cos(θ) ⎤ ⎡ sin(θlab ) cos(φlab ) × ⎣ sin(θlab ) sin(φlab ) ⎦ . cos(θlab ) In this way, the new direction in which the sputtered particle moves is known. 6.4.3 The Interaction Potential The interaction between two atoms is described by an interaction potential V (r). A large number of interaction potentials can be found on literature, and a good overview is given by Eckstein [67]. To describe the interaction between a thermalized background gas atom and a sputtered atom with an energy ranging from thermal up to a few tens of eV, not only the short range repulsive forces, but also the long range attractive forces should be taken into account. However, most of the available interaction potentials only count for one of both forces.

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Fig. 6.4. Definition of the original (θ and φ) and the additional direction (θlab and φlab ) which has to be added to obtain the new direction of the sputtered atom

Attractive Potentials An often applied attractive potential was proposed by Morse: V (r) = D[exp(−2α(r − r0 )) − 2 exp(−α(r − r0 ))].

(6.37)

A list of all available parameters D, α and r0 for homonuclear interactions is given in [67]. Also the Lennard–Jones potential, in most cases the 6–12 potential, describes the attractive forces and is given as: V (r) = λ6 r−6 − λ12 r−12 .

(6.38)

Repulsive Potentials Most of the potentials describing the repulsive forces are of the group of screened Coulomb potentials. These potentials can be written as: V (r) =

Z1 Z2 e2 Φ(r/a), r

(6.39)

with Φ(r/a) the screening function and a the screening length. This screening function is often approximated by  r Φ(r/a) = ci exp −di , a i=1 n 

with

n  i=1

ci = 1.

(6.40)

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215

Table 6.1. Parameters for the different screening functions [67]

n c1 c2 c3 c4 d1 d2 d3 d4

Bohr

Moli`ere

Kr-C

Ziegler– Biersack– Littmatk

Lenz–Jensen

1 1

3 0.35 0.55 0.1

3 0.1909 0.4737 0.3354

3 0.0102 0.2433 0.7466

1

0.3 1.2 6

0.2785 0.6372 1.9192

4 0.0282 0.2802 0.5099 0.1818 0.2016 0.4029 0.9423 3.1998

0.206 0.3876 1.038

Table 6.1 shows the used constants for different screening functions. The Bohr, Moli´ere, Kr-C and Lenz–Jensen screening function can be used with several screening lengths: – The Firsov screening length [68]: 2 1/3  −2/3 9π 1/2 1/2 aB Z 1 + Z 2 aF = 128 – The Lindhard–Scharff screening length [69]: 2 1/3  −1/2 9π 2/3 2/3 aB Z 1 + Z 2 . aLS = 128

(6.41)

(6.42)

In case of the homonuclear case (Z1 = Z2 ), Robinson suggested to use the “universal” value [67]: A. (6.43) aF = 0.0750 ˚ The Ziegler–Biersack–Littmark screening function should be used with their own screening length [70]: 2 1/3 " #−1 9π aB Z10.23 + Z20.23 , (6.44) aZBL = 128 with aB is the Bohr radius. There is no strong justification why to use the Firsov screening length or the Lindhard–Scharff screening length. So, the screening length can be seen as a fitting parameter. Lenz and Jensen proposed another screening function, to be used with the Lindhard–Scharff screening length [67]: ΦLJ (r/a) = (1 + y + 0.344y 2 + 0.485y 3 + 0.002647y 4)e−y  with y = 9.67r/aLS .

(6.45)

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Also Nakagawa and Yamamura suggested a screening function [71]: ΦNY (r) = exp(−Ar + Br3/2 − Cr2 ).

(6.46)

The parameters A, B and C where calculated for all individual elements [71] and include the screening length. Applying the geometric mean rule for heteronuclear collisions lead to the following constants:



 2/3 1 1 3/2 3/2 A12 = A11 + A22 (B11 + B22 ) , , B12 = 2 2

  4/3 1 3/4 3/4 C11 + C22 . (6.47) C12 = 2 Beside these individual potentials, Nakagawa and Yamamura also provided an average screening function [71]: ' & (6.48) Φaverage (r/as ) = exp −α(r/as ) + β(r/as )1.5 − γ(r/as )2 , with

 as =

and

9π2 128

1/3 aB (6.49)

2/3

(Z10.307 + Z20.307 )

Z10.169 + Z10.169 , β = 0.763 Z10.307 + Z20.307



4/3 Z10.0418 + Z20.0418 γ = 0.191 α = 1.51, . Z10.307 + Z20.307 (6.50) Another, widely used [72, 73] repulsive potential, but not a member of the screened Coulomb family, is the Born–Mayer potential which is given by:

r

V (r) = ABM e−BBM .

(6.51)

In which ABM is an energy parameter and BBM is a screening length. Abrahamson [74] calculated the parameters ABM and BBM for 104 homonuclear interactions, by fitting the Born–Mayer potential with the Thomas– Fermi–Dirac interaction potential. Based on the combining rule, the parameters for heteronuclear interactions can be calculated:  (6.52) V12 (r) = V11 (r)V 22 (r) and thus:

 A11 A22

B11 + B22 . (6.53) 2 Nice about the Born–Mayer potential is that the collision cross section can be approximated by A12 =

σ = π(rsg )2 ,

and B12 =

with rsg =

−2 Ecom ln  , Bs Bg As Ag

(6.54)

with Bs , As and Bg , Ag the homonuclear Born–Mayer constants for the sputtered atom and the background gas, respectively [5, 55].

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Combined Potentials Some potentials combine as well as the attractive as the repulsive potentials. These combined potentials are usually a combination of a standard repulsive potential and a standard attractive potential, fitted by an intermediate potential. The procedure to obtain this intermediate potential is described by Eckstein [67]. Myers et al. [51] developed a combined potential by fitting the Born–Mayer potential with the attractive part of the Lennard–Jones potential: V (r) = A exp(−Br) + (C/r)D − (E/r)6 ,

(6.55)

with (C/r)D a term used to produce a smooth transition between the repulsive part and the attractive part. Yamamura [50] provided a combined potential by adding the Morse potential to the screened Coulomb potential, without using an intermediate potential. In stead of using an attractive potential, combined with a repulsive potential, Kuwata et al. [75] calculated a quantum-chemical interaction potential for Argon–Copper and Argon–Aluminium using the Kohn–Sham density functional theory, utilizing the PW91 potentials. These quantum-chemical interaction potential describe as well as the attractive as the repulsive part nicely.

6.5 Gas Rarefaction In the introduction of the previous section, it was stated that the distribution of the background gas is often assumed to be constant in time and space. However, the interaction between energetic particles and the background gas results in a dynamic behaviour of the volume in front of the magnetron. This dynamic behaviour was first measured by Hoffman [76], who called this a “sputter wind” and attributed it to the pressure variations into a vacuum chamber resulting from collisions between the sputtered particles and the background gas. Later on, Rossnagel [77] and Dr¨ usedau [78] suggested that the pressure inside the vacuum chamber remains unaltered, but claimed that the density of the background gas reduces and called this gas rarefaction or gas heating. They experimentally measured the reduction of the gas density as a function of system pressure, discharge power, target material, by means of a local pressure probe. As shown in(6.17), the mean free path of a sputtered particle depends on the density of the background gas. Also the interatomic interaction and the scattering angle depend on the energy of the background gas ((6.29), remark that Ecom is used). Hence, this gas heating or gas rarefaction in front of the target should be taken into account, when describing the collisional transport of sputtered particles through the gas phase.

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Rossnagel [77], Dr¨ usedau [78] and Palmero [79, 80] proposed an analytical model to describe the gas rarefaction. Although (1) they assumed that only sputtered atoms transfer energy to the background gas, (2) they made other simplifying assumptions and (3) the uncertainty in several of the input parameters, their calculated gas densities behave similar to the measured gas densities. Several authors developed a Monte Carlo (or particle-in-cell Monte Carlo) model to simulate self-consistently the DC magnetron discharge, paying attention to the gas heating or gas rarefaction [81–86]. By taking into account that the background gas can not only be heated by collision with sputtered particles, but also by collisions with ions, or reflected neutrals, and even thermal radiation from hot bodies or optical emission from the plasma, they revealed the influence of several deposition parameters on the effect of gas rarefaction. It was, e.g. generally concluded that the gas temperature (and thus also the mean free path) increases with increasing discharge power, system pressure or target sputter yield. The same tendencies were observed by Ekpe and Dew, who simulated the gas heating effects by numerical modelling [87]. Like they also discuss in another chapter of this book, they mentioned that the gas heating effect not only influences the transport of the sputter particles, but also significantly influences the global energy flux towards the substrate and thus influences the thin film growth in general [88].

6.6 Typical Results of a Binary Collision Monte Carlo Code Based on the above-mentioned theories and ideas, a binary collision Monte Carlo simulation program has been developed by our research group. The radial distribution wherefrom sputtered particles leave the target was deduced from an experimentally measured racetrack in case that a circular, planar 2 in. magnetron is used as sputter source. In case that a rotating, cylindrical magnetron is used as sputter source, the radial distribution wherefrom sputtered particles leave the target was obtained by a perpendicular projection of the simulated ionisation distribution in front of the target [89]. The initial energy and angular direction of the nascent sputter flux were simulated by SRIM. The mean free path was calculated from (6.25), and the scattering angle in the centre of mass frame from (6.29). The code allows the user to choose between several screened Coulomb potentials (Moliere, Kr-C, Ziegler–Biersack– Littmark = ZBL) with Firsov screening length and, in case of the Ar−Al and Ar−Cu combination, the combined potential as calculated by Kuwata (see Sect. 6.4.3). No gas rarefaction was taken into account, i.e. the background gas distribution was assumed to be constant in space and time. The motion of the

6 Transport of Sputtered Particles Through the Gas Phase

a

b

c

d

219

Fig. 6.5. Comparison of the simulated and experimentally measured deposition profiles, of a circular, planar c2 inch Cu target, sputtered by Ar ions. The simulated data were obtained by using the ZBL (a), the Kr-C (b), the Moli`ere (c) and the Kuwata (d) potential

background gas was taken into account, assuming that the background gas is a Maxwellian gas of 400 K. When the energy of the sputtered particles decreased below an arbitrary value, it was assumed to be thermalised, and the further collision process was taken as a random motion. Figure 6.5 shows the simulated and experimentally measured deposition profiles for a circular, planar 2 in. Cu target, sputtered in Ar, at 0.55 Pa. An ion energy of 400 eV was taken to carry out the simulations. The influence of the chosen interaction potential is investigated. For each interaction potential (Moli`ere, Kr-C, ZBL or Kuwata), one calibration factor has been introduced to be able comparing the experimental and simulated results. This because the experimental data represent a deposition rate (a thickness divided by the deposition time), while the simulations are time independent and only give a thickness profile. More precisely, the simulated data have been multiplied by a factor, in such a way that the simulated and experimental deposition rate at a target-substrate distance of 16 cm and at radial distance R of 0 m are nearly the same. This calibration factor is only necessary to overcome the time independence of the simulation, and does not have to be changed in function of the pressure, T-S distance or vertical height.

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The full lines represent the simulated data, while the experimental data are shown by the dotted lines and open symbols. It is observed that all of the used interaction potentials can nicely reproduce the experimental deposition profiles as a function of the target-substrate distance. Moreover, no real difference between the several simulated profiles can be observed. Nevertheless, the choice of a specific interaction potential influences the description of the collision process, and thus the properties of the simulated arriving metal flux. These properties of the metallic particles arriving at the central place of the substrate (R ≤ 0.01 m) are shown in the Figs. 6.6–6.9. The total amount of arriving particles (in arbitrary units) and their average energy is given as a function of the target-substrate distance for the several interaction potentials. Also the number of collisions that the non-thermalised particles suffered is shown. It is observed that the average energy of the arriving particles is higher when using the Kuwata potential, although these particles suffer more collisions. This indicates that the Kuwata potential, which not only takes into account the repulsive forces but also the attractive forces, is softer than the screened Coulomb potentials. As already mentioned, also the material flux during rotating, cylindrical cathode magnetron sputtering can be simulated by the developed code. A scheme of the setup of the rotating, cylindrical magnetron and where the deposition profiles were measured is shown in Fig. 6.9. The material flux was measured at a radial distance of 114 mm from the central axis of the

Fig. 6.6. Total number of particles arriving at the central place of the substrate (R ≤ 0.01 m) as a function of the target-substrate distance and the used interaction potential

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Fig. 6.7. The average energy of the particles arriving at the central place of the substrate (R ≤ 0.01 m) as a function of the target-substrate distance and the used interaction potential

Fig. 6.8. The number of collisions that the non-thermalised particles arriving at the central place of the substrate (R ≤ 0.01 m) suffered as a function of the targetsubstrate distance and the used interaction potential

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Fig. 6.9. Front and side view of the rotating cylindrical magnetron and the place where the material flux was registered by a quartz crystal oscillator

Fig. 6.10. Experimental and simulated deposition profiles during rotating, cylindrical magnetron sputtering of a Cu cathode in Ar (400 eV) at 0.3 Pa

cylindrical target. These fluxes were measured at a lateral distance of 96, 132 and 190 mm away from the left side of the target. Figure 6.10 shows the simulated and the experimental deposition profiles, for a Cu cathode, sputtered in Ar at 0.3 Pa. The simulations were carried out with the Kuwata potential. Again, the simulated data have been multiplied by a factor, in such a way that the simulated and experimental data at the

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largest deposition rate are more or less the same. A good correlation between both curves is observed.

6.7 Specific Example: In-Plane Alignment of Biaxially Aligned Thin Films Biaxially aligned thin films are polycrystalline thin films with not only a preferential crystallographic out-of-plane orientation, but also with alignment along a certain reference direction parallel to the substrate plane. This type of films has been obtained by unbalanced reactive magnetron sputter deposition on both amorphous glass and randomly textured polycrystalline substrates tilted with respect to the incoming material flux [90–93]. The quality of the in-plane alignment can be evaluated from X-ray diffraction pole figures, and is usually expressed by the full width at half maximum (FWHM) of the diffraction pole. The influence of several deposition parameters (target-substrate distance, target-substrate angle, deposition pressure and substrate bias) on the degree of in-plane alignment has been discussed [93]. It was shown that the influence of these parameters can be traced to the influence of two main properties, i.e. the mobility of the adatoms at the growing surface and the angular spread of the incoming material flux. This angular spread of the incoming material flux, as a function of the deposition parameters could be simulated by the above-described binary collision Monte Carlo model. The Moli´ere screening function with Firsov screening length was used. Figure 6.11 shows the influence of an increasing target-substrate angle α on the simulated angular spread of the incoming flux and the resulting inplane alignment for YSZ thin films, as measured from the XRD pole figures. It is observed that the tendency of both curves are similar, indicating that the improvement of the in-plane alignment with increasing target-substrate angle α is caused by a decreasing angular spread of the incoming material flux.

Fig. 6.11. Influence of the target-substrate angle α on the degree of in-plane alignment for 1.3 μm thick YSZ films deposited by unbalanced magnetron sputtering on positively biased polycrystalline stainless steel (Vs = +15 V) at a pressure of 0.425 Pa, a T-S distance of 13 cm, floating temperature, and an Ar/O2 ratio of 60/2.5

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Fig. 6.12. Influence of the pressure on the degree of in-plane alignment for 1.3 μm thick YSZ films deposited by unbalanced magnetron sputtering on positively biased (+15 V) polycrystalline stainless steel at a target-substrate angle of 55◦ , a T-S distance of 13 cm, floating temperature and an Ar/O2 ratio of 60/2.5

The influence of the deposition pressure on the simulated and experimentally measured angular spread is shown in Fig. 6.12. The angular spread of the incoming material flux decreases with decreasing pressure because the sputtered particles undergo fewer collisions with Ar and are thus less scattered. The difference between the simulated angular spread and the experimentally measured in-plane alignment at low pressure is caused by a too strong variation in total energy flux towards the substrate [93]. This influence of the deposition parameters on the energy flux towards the substrate is discussed in Chap. 7 of this book.

6.8 Conclusions The several steps and approaches to simulate the collisional transport of sputtered particles through the gas phase by a binary collision Monte Carlo method have been treated. The nascent sputter flux, and thus the initial energy and angular distribution of sputtered atoms leaving the target have been discussed in detail. It could be concluded that the analytical models can not fully describe the experimentally observed energy and angular distribution of sputtered particles, in case of low energy (E < 400 eV) ion bombardment. The use of a numerical model and even better a simulation code such as ACAT, SRIM, TRYDIN. . . is believed to approach better the reality. The collision between the sputtered particles and the background gas is described in detail. Also the use of several kinds of interaction potentials was summarized. It could be suggested that the use of a combined potential, taking into account the attractive as well as the repulsive forces, is recommended. The mean free path can be deduced from the used interaction potential, by calculating the maximal impact parameter.

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In the ideal case, a global model, simulating the magnetron discharge, together with the transport process would allow the user to take into account the effect of gas heating on both processes. Although the gas heating effect was not yet included in the described model, general results about the transport of sputtered particles are shown. Finally, a clear correlation between the properties of the incoming material flux, in specific the angular spread on the incoming flux, and the in-plane alignment of biaxially aligned thin films has been shown.

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65. J. Lindhard, M. Scharff, Phys. Rev. 124(1), 128 (1961) 66. A. Bogaerts, Ph.D. Thesis, University of Antwerp, 1996 67. W. Eckstein in Computer Simulation of Ion-Solid Interactions, Springer Series in Materials Science 10, ISBN: 3-540-19057-0 68. O.B. Firsov, Sov. Phys. -JETP 6, 534 (1958) 69. J. Lindhard, M. Scharff, Phys. Rev. 124, 128 (1961) 70. J.P. Biersack, J.F. Ziegler, Nucl. Instrum. Meth. 194(1–3), 93 (1982) 71. S.T. Nakagawa, Y. Yamamura, Radiat. Eff. 105, 239 (1988) 72. V.A. Vol’pyas, E.K. Gol’man, Tech. Phys. 45(3), 298 (2000) 73. V. Abhilash, R. Balu, S. Balaji, S.S. Nathan, S. Mohan, Comput. Mater. Sci. 30, 523 (2004) 74. A.A. Abrahamson, Phys. Rev. 178(1), 76 (1969) 75. K.T. Kuwata, R.I. Erickson, J.R. Doyle, Nucl. Instrum Meth. Phys. Res. B 201, 566 (2003) 76. D.W. Hoffman, J. Vac. Sci. Technol. A 3(3), 561 (1985) 77. S.M. Rossnagel, J. Vac. Sci. Technol. A 6(1), 19 (1988) 78. T.P. Dr¨ usedau, J. Vac. Sci. Technol. A 20(2), 459 (2002) 79. A. Palmero, H. Rudolph, F.H.P.M. Habraken, Thin Solid Films 515, 631 (2006) 80. A. Palmero, H. Rudolph, F.H.P.M. Habraken, Appl. Phys. Lett. 89(21), 211501 (2006) 81. T. Kobayashi, Appl. Surf. Sci. 169–170, 405 (2001) 82. A. Kersch, W. Morokoff, Chr. Werner, J. Appl. Phys. 75(4), 2278 (1994) 83. V.V. Serikov, K. Nanbu, J. Appl. Phys. 82(12), 5948 (1997) 84. V.V. Serikov, S. Kawamoto, K. Nanbu, IEEE Trans. Plasma Sci. 27(5), 1389 (1999) 85. G.M. Turner, J. Vac. Sci. Technol. A 13(4), 2161 (1995) 86. I. Kolev, A. Bogaerts, IEEE Trans. Plasma Sci. 34(3), 886 (2006) 87. S.D. Ekpe, S.K. Dew, J. Phys. D. Appl. 39, 1413 (2006) 88. Chapter in this book: S. Ekpe and S. Dew 89. J. Musschoot, D. Depla, G. Buyle, J. Haemers, R. De Gryse, J. Phys. D Appl. Phys. 39(18), 3989 (2006) 90. S. Mahieu, P. Ghekiere, D. Depla, R. De Gryse, O.I. Lebedev, G. Van Tendeloo, J. Cryst. Growth 290, 272 (2006) 91. P. Ghekiere, S. Mahieu, R. De Grys, D. Depla, Thin Solid Films 515, 485 (2006) 92. S. Mahieu, G. Buyle, P. Ghekiere, D. Depla, R. De Gryse, Thin Solid Films 515, 416 (2006) 93. S. Mahieu, P. Ghekiere, D. Depla, R. De Gryse, Thin Solid Films 515(4), 1229 (2006)

7 Energy Deposition at the Substrate in a Magnetron Sputtering System S.D. Ekpe and S.K. Dew

7.1 Introduction In any deposition process, the effective energy deposited onto a surface of a substrate material by the depositing and reactive particles is essential to understanding the mechanism of film growth on the surface [1–4]. For low-pressure plasmas, such as with the magnetron sputtering process, these particles include electrons, ions, neutrals, etc. [2,5–9], which interact with the surfaces and each other through collisions and or chemical reactions. It should be evident therefore that the energy flux onto a substrate depends on the process conditions, such as magnetron power, pressure, geometry, etc. [2, 5, 8]. The deposited energy causes the effective temperature of the growing film to rise [10]. As the particles arrive at the substrate, they transfer momentum, and increase the mobility of the particles at the surface of the growing film [3, 4]. They also cause peening, increasing density and comprehensive stress [11–17]. Apart from surface mobility, the reaction rates/pathways are greatly influenced by this incident energy, and hence the micro- and nanostructure and the properties of the growing film depend strongly on the energy management [18–24]. The relationship between substrate temperature and film properties, such as columnar structure, grain size, etc., has been well established [25–31]. The density and temperature of the plasma species at the substrate region have been shown to also affect the properties of the deposited film [32–34]. Gas pressure, as a process parameter, affects the kinetic energy of the deposited species as well as the characteristics of the plasma [7, 8, 35, 36], hence it may be said to have an important indirect effect on the micro- and nanostructural evolution of the film. Since these effects determine the effective energy deposited [36], then, energy flux onto the substrate is one of the important parameters in many high performance thin films. Several factors contribute to integral energy flux to the substrate. Some of these include the energy of the sputtered atoms – kinetic and potential (heats

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of condensation and reaction), plasma radiation, energy transfer from gas collision and heating effect, reflected gas neutrals, and the kinetic and potential (recombination) energies of ions and electrons at the substrate [2, 5–8, 37]. Recombination energy of the charged particles (electrons and ions) is greatly determined by the substrate potential, and the surface electrical conductivity [38, 39]. At potentials different from the floating potential, where the number densities of ions and electrons are significantly different, the recombination energy becomes insignificant compared with the other energies. Wendt et al. [37] showed that the kinetic energy of ions accelerated in the potential drop in front of the substrate is not the main contribution to the thermal power at the floating potential. The reflected neutrals have been considered to depend on the mass of gas ion relative to that of the target material [2, 5]. The velocity of the incident ion relative to the target material, vrel may be expressed as: Mi − Mt vi , vrel = (7.1) Mi + Mt where vi is the ion velocity just before striking the target, Mi and Mt are, respectively, the masses of the gas ion and the target material. If the mass of the incident ion is less than that of the target material, it is likely that the ion will be reflected backward in a single collision, and the energy of the reflected ion may be a significant fraction of its initial energy [40]. If the mass of incident ion is greater than that of the target material, the ion can be reflected backward only as a result of more than one collision, and of course with a reduced energy. Thus for argon on aluminum, the reflected neutrals would contribute a negligible effect to the integral energy at the substrate. However, for Ar/N2 atmosphere, the reflected neutralized nitrogen may contribute a significant factor to the energy flux. Energy exchange within the reactor may be said to be determined by two sets of parameters: external and internal. The external parameters are the controllable experimental factors, such as power, voltage, current, DC or RF excitation, the geometry, and the composition/pressure of the background gas. The internal parameters are factors, such as the degree of ionization, potential drop in the plasma sheath, masses of the component particles, and the free paths of the various different particles. These internal parameters influence the flux of particles as well as the energy flux toward the substrate. These two sets of parameters interact in a complicated manner that determines the total energy flux to the substrate. Figure 7.1 shows the schematic representation of the interaction of the parameters resulting in the different sources of energy flux in the sputter reactor. The determination of the energy flux can be used as a tool to infer the relationship between the parameters. The following sections discuss the measurement of energy in a typical deposition system, process factors affecting the deposited energy, the interpretation of the measured energy in terms of energy per deposited atom, and a model for the estimation of the energy per deposited atom.

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Etc Ec Ea

Substrate Etr Ee

Eiz Ep

Etr Er Etc

Plasma Ei Target

Fig. 7.1. The sources of energy flux in a sputter deposition system. E is the energy flux and the subscripts i, a, r, c, p, e, iz, tr, and tc represent incident ion, sputtered particle, reflected neutral, condensation, plasma radiation, electron, positively ionized gas atom, thermal radiation, and thermal conduction through the substrate holder and probably to the gas

7.2 Energy Measurement When energy is deposited onto a surface, the temperature of the surface rises. Thus, by measuring the change in temperature, the deposited energy can be inferred. The measurement of the increase of the in situ substrate temperature has been a well-established method for the determination of the integral energy flux to the substrate [2, 5, 41–43]. Another method for the determination of the thermal flux at the substrate is the steady-state determination of the temperature gradient along the sample holder [6] although this does not account for the front-side heat losses due to, for example, radiation or gas transport. Many techniques for determining temperature have been developed over the years. These are significantly complicated by the vacuum system, reducing thermal contact, and uniformity. Some require physical contact between the measuring device and the specimen, and others employ noncontact methods such as radiometry and luminescence. For the contact methods, thermocouples have been widely used [2, 10]. This has a major advantage of making low temperature measurements (900 K) and requires corrections for the surface emissivity [48] and other factors. A solution to the above issues would be the use of the temperaturedependent resistance of homogeneously heated thin films or other sensors embedded in the substrate surface [6, 7, 49, 50]. The main advantage of this technique is that the sensing element is buried inside the absorber, and hence does not require attachment of the thermal sensor to the specimen. Selfheating effects [51, 52] by the sensing current can be minimized by applying a low-bias voltage [6] across the terminals of the thermistor, and by the use of a high resistance material. Silicon thermistors appear to be nearly ideally suited as thermometers for high-resolution microcalorimeters [51]. A polysilicon thin film thermistor created using a modified complementary metal-oxide semiconductor (CMOS) process is reported in [7, 36, 49]. It consists of a four point serpentine polysilicon structure suspended on a membrane for reduced thermal capacity. A thin silicon dioxide (SiO2 ) layer covers to act as an absorber and to prevent electrical shorting when depositing metals films. The factors of interest are the temperature sensitivity of the sensor, thermal capacity of the absorber, and the thermal conductance linking the substrate to the ambient. The temperature-dependent resistance of the sensor is given by the power series [7]: & ' R(T ) = R0 1 + α(T − T0 ) + β(T − T0 )2 + γ(T − T0 )3 + · · · . (7.2) Figure 7.2 shows the temperature dependent resistance of two iterations of the sensor designs. As reported [7, 36, 49], it was found that for low temperature measurements (∼250◦ C), second and higher orders can be neglected, and a linear dependence assumed. Figure 7.3 is a photo of one of the cells of the second iteration of the sensor design (details can be found in [36]).

7.3 Factors Affecting Energy Flux The integral energy deposited onto the substrate depends on several factors, which can be controlled through the process conditions, such as magnetron power (DC/RF), gas (inert/reactive) pressure, substrate-target distance and substrate bias. The different individual contributions to the total deposited energy respond/react differently to changes in the process conditions. For instance, at constant power mode, increase in pressure of the process gas may result in a decrease in voltage. The effect of this would be an increase in the plasma densities, including negative ions (where electronegative materials are used), increase in the reflection probability of gas neutrals, decrease in the

Relative change in resistance of sensor

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0.25 polyfit (sensor 1)

0.2

expt (sensor 1) polyfit (sensor 2)

0.15

expt (sensor 2)

0.1 0.05 0

0

25

50 75 100 Temperature (oC)

125

150

Fig. 7.2. Dependence of the resistance of sensor on temperature. The solid line and dashes are the polynomial fit to the experimental data for sensors 1 and 2, respectively

Fig. 7.3. Photo of one of the cells of the second iteration sensor showing the arrangement of the sensing elements and the etched silicon underneath the SiO2 covered polysilicon

kinetic energy of the particles (sputtered, reflected neutrals, charge carriers), and of course increase scattering effects. Increased scattering leads to greater gas heating – higher temperature gradient [53–57], and hence increase heat transfer through conduction. The effect on the transport of the particles is, however, complicated by the inhomogeneous rarefaction of the process gas. Figure 7.4 shows a simulated inhomogeneous rarefaction of the process gas for the sputtering of Al at a pressure of 1.33 Pa and magnetron power of 300 W from a 7.6 cm circular planar target [57]. The theoretical and experimental impact on the energy flux of several such factors will be explored below.

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Fig. 7.4. Rarefaction of the process gas for the sputtering of Al in (a) Ne, (b) Ar, and (c) Kr at a pressure of 1.33 Pa and magnetron power of 300 W. The data were taken along a vertical plane through the center of the chamber with the substrate located at a distance of 10 cm in front of the target [57]

7 Energy Deposition at the Substrate in a Magnetron Sputtering System 0.025 Total energy flux (W/cm2)

0.05 Total energy flux (W/cm2)

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(a) 0.04 0.03 0.02

0.67 Pa 1.33 Pa

0.01

(b) 0.67 Pa 1.33 Pa

0.02 0.015 0.01 0.005

0

0 0

100 200 300 Magnetron power (W)

400

0

100 200 300 Magnetron power (W)

400

Fig. 7.5. On-axis steady-state energy flux at the substrate region as a function of magnetron power at a substrate-target distance of (a) 10.8 and (b) 21 cm for two different pressures during the deposition of Al

7.3.1 Magnetron Power and Pressure Not surprisingly, the total energy deposited at the substrate generally increases with magnetron power [5–7] for both DC and RF operation modes. It has been shown that increasing pressure results in reduction of the total energy at the substrate due to energy losses from increase interactions of the particles in transport to the substrate region [7, 8]. Figure 7.5a shows the variation in the total energy flux at a distance of 10.8 cm from a 76 mm target during the deposition of Al, with power and pressure. For the range of process conditions considered in that study, energy flux increases linearly in direct proportion to power. The power transfer efficiency to an area at the substrate was determined as 1.6 × 10−4 and 1.4 × 10−4 cm−2 for pressures of 0.67 and 1.33 Pa, respectively [36]. This value varies with pressure probably because of changes in the ionization efficiency in the plasma and in the energy absorption rate of the gas for the energetic particles from the target. In a related study using Cu [49], it was observed that the variation in energy flux with power indicates a nonlinear dependence. The deviation was attributed to the effect of increased reflected neutrals, sputtering yield, gas heating effect, and the thermalization of energetic particles. Cu shows a reduced thermalization compared with Al at the same process conditions [57–59], hence is associated with higher power transfer efficiency than Al is. In a study with molybdenum [60], total energy deposited onto the substrate was measured experimentally, and shown to reduce with pressure. It was further shown that while some factors vary appreciably, some vary minimally or remain constant with pressure. 7.3.2 Substrate-Target Distance Figure 7.5b shows the variation in the total energy flux with power for pressures of 0.67 and 1.33 Pa at a substrate-target distance of 21 cm (long throw) for the sputtering of Al. The energy in this case is lower than that at a

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distance of 10.8 cm. Similarly, the power transfer efficiency reduces greatly to 7.0 × 10−5 and 5.2 × 10−5 cm−2 for the respective pressures. Experimental results presented in [6] show a similar trend for the deposition of Al at magnetron powers of 38 and 65 W. Initial increase in distance results in a rapid decrease in energy, however, at greater distances, further increase in distance results in a minimal decrease in energy. The greater separation between pressures at longer throw distances indicates the greater importance of the kinetic energy and scattering under these conditions. Since at these distances, most of the particles are thermalized. Such particles would contribute negligible further kinetic energy to the total deposited energy. 7.3.3 Electrical Effects Biasing of the substrate can be used in studying the effect of charge carriers (electrons and ions) on the total energy deposited onto the substrate [61, 62]. The contribution of these charge carriers depend on their densities and temperatures and the electric field in the substrate region. The electric field in the sheath at the substrate region may be inferred from the potential difference, Vb = Vpr − Vpl , where Vpr and Vpl are, respectively, the probe and plasma potentials. The probe potential at which the probe current is zero defines the floating potential. The plasma characteristics – plasma and floating potentials, charge carrier densities, and temperatures – can be determined using a Langmuir probe [6,63–66], and are essential in the estimation of the deposited energy due to the charge carriers. Where electronegative species are present, negative ions may exist. Given the masses of these negative ions, while the electrons are confined by the magnetic field in front of the target, the negative ions are accelerated away from the target. Given the distance of flight toward the substrate and the accelerated energy of the negative ions, these ions can reach the substrate with significant energy. The density of these ions depends strongly on the partial pressure of the electronegative specie(s), and the total pressure of the system. The energy flux at the probe for probe potentials greater than the plasma potential is given approximately by [7, 37, 67]: p = po + je Vb ,

(7.3)

where po is the energy flux at a probe potential equal to the plasma potential and je is the saturation current density at the probe for a given bias. Figure 7.6 shows the estimated energy flux at the substrate at a distance of 10.8 cm from the target for Al sputtered from a 7.6 cm target at 200 W and 0.67 Pa. Also presented are the directly measured energy fluxes at lower potentials. Higher biases tended to damage the sensor (first iteration). This figure clearly indicates a very strong sensitivity of energy flux to probe voltage, and hence the important role electrons can play in heating the growing film and the substrate during deposition. On the other hand, ions are clearly not as

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Total energy flux (mW/cm2)

250

200 estimated measured

150

100

50

0

-15

-10

-5 0 5 Probe potential (V)

10

15

Fig. 7.6. Total energy flux at the substrate as a function of probe potential for a magnetron power of 200 W, substrate-target distance of 10.8 cm and gas pressure of 0.67 Pa

important to the energy flux. However, for probe potentials much lower than the floating potential, the sensitivity of the energy to ions becomes apparent [67]. Substrate biasing therefore offers an important tool in the control of energy flux to the growing film.

7.4 Total Energy per Deposited Atom To fully understand the effect of energy on the mechanism of film growth, it is useful to interpret the energy at the substrate on a per deposited atom basis. As already noted, several factors contribute to the total energy flux. Those factors of course depend on the process conditions. The total energy per deposited atom Etot may be evaluated by normalizing the total energy flux at the substrate to the atomic deposition rate φat . Thus, Etot = qtr /φat ,

(7.4)

where φat = (dr NA ρ)/M , and dr is the average deposition rate determined from the film thickness, NA is Avogadro’s constant, ρ is the density, which is usually less than the bulk density and M is the molar mass. The dependence of the total energy per deposited atom on magnetron power for two different gas pressures for Al deposition is shown in Fig. 7.7. Values of the determined total energy per deposited atom depend strongly on the magnetron power and gas pressure regimes. The results show that, for

S.D. Ekpe and S.K. Dew Energy per deposited atom (eV)

238

60 55 50

0.67 Pa

45

1.33 Pa

40 35 30 25 20

0

100 200 300 Magnetron power (W)

400

Fig. 7.7. Total energy per deposited aluminum atom as a function of magnetron power at gas pressures of 0.67 and 1.33 Pa using a substrate-target distance of 10.8 cm

both 0.67 and 1.33 Pa, the energy per atom decreases with increasing magnetron power. Of course, both the energy flux and deposition rate increase with power. However, the deposition rate increases more quickly than the energy flux, hence the downward trend observed in Fig. 7.7. In part, this is due to the gas rarefaction effect as higher powers result in significant gas heating that reduces particle scattering and hence increases the flux of sputtered particles to the substrate. However, the rarefaction of the filling gas adjacent to the cathode [53, 56] increases the impedance of the plasma, affects the current– voltage characteristics of the discharge [68], and reduces the contribution of charge carriers to the total energy. Dickson et al. [69] noted that a possible effect of gas heating is a decrease in electron density in ionized physical vapor deposition (IPVD). Thus, increased gas heating with magnetron power limits the increase in the electron contribution to the total energy. Also, gas density reduction affects the ionization of the gas through reduced electron collisions, as well as a reduction in the contribution to the energy flux due to plasma radiation. Figure 7.7 also shows that by increasing the gas pressure from 0.67 to 1.33 Pa, the energy per deposited atom increases. However, as the magnetron power is increased, the difference in the energy per deposited atom between the two pressures decreases such that at higher magnetron powers it trends toward being constant irrespective of both power and pressure [7, 8]. Thornton [2] suggested that, for a given metal, the energy per atom deposited is independent of the deposition rate and gas pressure. However, in a related study [5], it was shown that the deposited energy per atom decreases with increasing magnetron power, but with decreasing process gas pressure. Results for the sputter deposition of AlN in a RF discharge [70] also shows that the energy per deposited particle increases with decreasing discharge power, but also with increasing sputtering pressure. In comparison with that

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of a DC discharge, it was noted that energy per deposited particle in a RF discharge was significantly higher [2, 70]. The observed trend shows that the plasma effects are more important, at least at lower power levels. The above shows the dependence of energy flux toward the substrate on the process conditions. Individual contributing factors respond differently and in a complicated manner with changes in the process conditions. As the deposited energy affects the mechanism of film growth, it is essential to model the effects of the process conditions on the energy flux. This would provide the tool for understanding of the film growth mechanism, as well as controlling the film growth for desired properties.

7.5 Energy Model Thornton [2] proposed that the total energy per incorporated atom may be estimated from the following model: Etot = U + Ek + Eo

RE + EP . Y

(7.5)

The first and second terms are the contributions of the heat of condensation (and, by extension, reaction) and the average kinetic energy of the deposited atoms, respectively. The third term represents the contribution of the reflected gas neutrals, Er (determined per sputtered atom in terms of the sputtering yield, Y ) as a function of the incident ion energy, Eo and energy reflection coefficient, RE (which depends on the energy of the incident ion [5, 40]). This contribution depends greatly on the masses of the sputtered atom and that of the process gas. The fourth term represents the contribution due to radiation within the plasma. As already shown, the contribution of the charge carriers must not be ignored. Thus, to the above model, we would add contributions due to electrons and ions bombardment Ech and thermal radiation Et from hot bodies within the reactor (both in terms of per incorporated atom). Etot = U + Ek + Eo

RE + EP + Ech + Et . Y

(7.6)

The nature of Ech depends strongly on the substrate bias. At floating and higher potentials, it would be reasonable to assume that it is mainly from electrons [7, 37, 60], and probably negative ions [71]. Since the last terms in (7.6) can be determined from the bulk plasma properties and the power input, respectively, it may be necessary to rewrite (7.5) in the form of the total transferred energy qtr to the substrate: qtr = φat (U + Ek + Er + EP ) + qch + qt ,

(7.7)

where qch and qt are, respectively, the total contributions of the charge carriers and the thermal radiation.

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7.5.1 Sputtered Particles The contribution of the sputtered particles to the total deposited energy is mainly from their potential energy (condensation/reaction) and kinetic energy. The condensation energy, U per atom of aluminum, copper, and tungsten are, respectively, 3.33, 3.5, and 8.9 eV [2, 72]. For the reactive sputter deposition process, energy of reaction must be considered. The average kinetic energy of the ejected atoms may be determined using the modified Thompson model (with a correction factor as detailed in [73]). The modification takes care of the high-energy tail observed with high ion energies which is usually found to drop off sharply with the relatively low ion energies associated with magnetron sputtering. The final expression for the energy distribution which combines the Thompson distribution [74], modified anisotropy correction, and cut off factor (which sets the distribution to vanish at E = Emax ) is as follows [73]: q

Mg (E cosq11 ϑ + Ub ) 2 f (E)dE ∝ E(E + Ub )−3+2m exp −A Mt Ei E + Ub dE, (7.8) × 1− γEi where ϑ is the emission angle, Ub is the surface-binding energy, A = 13, q2 = 0.55, and q1 ≈ 2 − Mt /4Mg . Emax is the maximum emission energy, which is related to the sputtering threshold, Eth , by: Emax = (E0 Ub )/Eth . The maximum recoil energy γEi is sometimes taken to be equal to (Emax + Ub ). The parameter m, which depends on the species, was found to be generally between 0.20 and 0.23 [73]. The average initial energy Eki determined using the semi-empirical relation of (7.8), for sputtered Al and Cu atoms as a function of the ion energy are shown in Fig. 7.8. In transport toward the substrate, this energy is degraded due to collisions with the gas to Ekf , and may be expressed in the form: μd μηP d = Eki exp − , (7.9) Ekf = Eki exp − λ kT where η is the collision cross section for momentum exchange between the sputtered particles and gas molecules, λ is the mean free path and μ is related to mass ratio of the gas to the particle, and is given by [75]:   (1 − M )2 (1 + M ) μ=1− ln for M < 1, 2M (1 − M )   (1 + M ) (M − 1)2 ln for M > 1, μ=1− 2M (M − 1) with M = Mg /Mt .

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Fig. 7.8. Average kinetic energy of Al and Cu atoms sputtered normally to the surface as a function of the argon ion energy obtained from the semi-empirical model of (7.7)

Clearly, the contribution per sputtered particle to the total energy flux increases with ion energy (power), but decreases with pressure due to scattering, and of course is affected by the changes in the local temperature [53, 56]. 7.5.2 Reflected Neutrals The contribution of the reflected neutrals to the total deposited energy depends on the mass ratio of the target material to that of the incident ion. For a given gas, both the energy and particle reflection coefficients have been shown to increase with the mass of the target material [5]. In [5], it was reported that the reflection coefficients decrease monotonically with increase in the incident energy, since the probability for backscattering decreases with the penetration depth. However, above 400 eV, the differences in values for different energies are small, and the coefficients may be assumed constant with energy. The functional dependence of the energy reflection coefficient on the target voltage, VT , within the range of study [5] is then given by: ε VT , (7.10) RE = 1 − VE where VE is a fitting parameter, which was found to be around 20,000 V for the heavy elements, but lower for the lighter ones. The power ε was found to be around 0.006 for germanium and to increase to about 0.037 for tungsten. The initial energy of the reflected neutral is equal to the product of the ion energy

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and the reflection coefficient. Because of collisions with the gas, the deposited energy of the reflected neutrals decays exponentially with pressure–distance product as: ηr P d RE E0 exp − , (7.11) Er = Y kT where ηr is the cross section for momentum transfer between the reflected neutral and the gas. Since this energy contribution is scaled with the ejected atoms at the target, and the interest is in terms of the deposited atoms at the substrate, it would be necessary to factor (7.11) by the deposition efficiency, which is defined as the ratio of the deposited particle flux at the substrate to the emitted flux at the target. 7.5.3 Plasma Radiation The contribution of the plasma radiation is due to the radiation emitted as a result of ionization and excitation events. These events are said to be caused by electrons with energies above the inelastic collision cross section threshold [2]. The average energy spent by an electron in making an ion is about 26.4 eV for argon [76]. The difference between this energy and the ionization energy of the gas (15.8 eV for argon) is radiated from the plasma. For an event involving an electron and argon atom, this energy is roughly estimated to be emitted in either the forward or reverse direction at about 5.33 eV [2]. In terms of the energy per sputtered atom, this energy is normalized to the sputtering yield as EPS = 5.33 eV/Y, (7.12) which assumes every ion is collected at the cathode. However, the interest is in terms of the energy per deposited atom; therefore, EPS is further normalized to the deposition efficiency [7] as: EP = EPS /χde ,

(7.13)

where χde is the deposition efficiency. It depends on scattering events, and is determined by pressure and the distance of the substrate from the target. Since the spatial profile of depositing flux depends on the process conditions and system geometry [58], then the deposition efficiency would also be determined by the process conditions and system geometry. Gas heating effects would result in improvement in the deposition efficiency through reduced scattering of the sputtered particles [53, 56, 57]. In the low power regime of operation, this efficiency reduces greatly with pressure, and hence a large increase in the contribution of the plasma radiation to the total deposited energy per atom. This contribution to the total energy per deposited atom decreases with power, but at high power densities, it tends toward being constant. This may be attributed to the gas heating effects, especially at high power densities [53, 54, 57, 77].

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7.5.4 Charge Carriers Substrate bias is a significant factor in determining the contribution of ions and electrons to the total energy deposited onto the growing film [6, 78–82]. For instance, when the substrate is negatively biased, positive ions (essentially from the process gas) are accelerated to the substrate, on the other, when it is biased positively, electrons and negative ions are accelerated to the substrate. In either case, the charged particles transfer their energy or a part to the growing film [72, 80, 81]. If the ion energy is high, it may also result in resputtering of the deposited film, leading to a loss in energy [83]. At a substrate bias potential equal to or greater than the floating potential, the primary positive ions contribute an insignificant energy to the growing film, whereas electrons contribute significantly to the total energy [5, 7, 37]. Positive ions have been shown to be significant only at potentials lower than the floating potentials [67, 78, 81]. The energy contribution of the charge carriers depends on their temperatures, kT , and densities, n, as well as the potential, Vbias , i.e., the substrate potential Vsub relative to the plasma potential, VP . Apart from their kinetic energies, recombination energy of the carriers at the substrate also contributes to the total deposited energy. This energy, however, is significant mostly at floating potentials (in which case the net current of the electrons and ions is zero). This energy is determined by the flux of the ions ji , ionization potential Eion of the process gas (which is about 15.8 eV for Ar), and work function Φ of the substrate. It may be expressed as [67] qrec = ji (Eion − Φ). (7.14) For a Maxwellian energy distribution, the energy flux of electrons, and ions are, respectively, given by [6, 43, 67, 84]:

0.5 e(VP − Vsub ) kTe Eee = 2kTe ne , exp − 2πme kTe 0.5 kTe Eei = 0.61neVbias , mi

(7.15) (7.16)

where mi and me are, respectively, the mass of the ions and electrons. The sum of these contributions gives the contribution from the charged carriers Ee . The electron density in the substrate region is proportional to the discharge power [78], and for a given power depends on the pressure. The spatial distribution of electron density is highly influenced by the pattern of the magnetic field in a magnetron system, as shown in Fig. 7.9 [66]. The electron density is significantly higher above the target etch tracks than along the central axis at those regions close to the target. However, at large target-substrate distances the distribution tends to flatten out. That of the ions is similar, though of a lower density at some distances from the edge of the sheath. For a balanced magnetron system, the shape profile of the spatial distribution of electrons

244

S.D. Ekpe and S.K. Dew 7.00E+10

Density, ne (cm-3)

6.00E+10

40 mtorr 3cm

5 mtorr 3 cm

40 mtorr 5 cm

5 mtorr 5 cm

40 mtorr 10 cm

5 mtorr 10 cm

5.00E+10 4.00E+10 3.00E+10 2.00E+10 1.00E+10 0.00E+00 0

2

4

6

8

10

12

14

16

Lateral displacement (cm) Fig. 7.9. Langmuir probe lateral scan of the profile of electron density for different pressures and different distances from the target [66]

seems to be similar to that of the deposition rate. The densities increase with pressure and power. Electron temperature depends on the gas-target combinations and discharge voltage. For a given experimental system, the dependence of electron temperature on the discharge voltage Vd is given by [68]: kTe = υVd2 ,

(7.17)

where υ is constant for a given gas-target combination, magnetic field and system geometry. The spatial profile of the electron temperature was also shown to follow that of the electron density, being higher in regions of high electron density than at regions of low density. However, it was also observed that the spatial electron temperatures decrease with pressure [66], probably as a result of more frequent collision and excitation of the process gas. The dependence of these plasma properties on the deposition properties can generally be estimated from Langmuir probe measurements. 7.5.5 Thermal Radiation Usually, the target in a magnetron sputtering system is water cooled. However, a weak thermal link has been observed between the target shield and the water-cooling system. Thus, during operation, the shield heats up and results in a significant heat transfer to the substrate by thermal radiation [8]. There is also evidence of significant temperature gradient between the target surface and the cooling surface, when the target material is nonconducting or has low thermal conductivity [46]. These and other hot surfaces could certainly

7 Energy Deposition at the Substrate in a Magnetron Sputtering System

245

contribute significant energy to the substrate in a low temperature deposition. The main source of heating in this case is from the system input power. The steady-state temperature of these surfaces is therefore expected to increase with magnetron power. The net radiation Qtr between the two grey bodies separated by nonabsorbing medium is given as [85] Qtr = F12 σ(ε1 A1 T14 − ε2 A2 T24 ),

(7.18)

where ε and σ are, respectively, the surface emissivity and Stefan–Boltzmann constant, F12 is the view factor of body 2 relative to that of 1, and T is the surface temperature. In the sputter deposition system, the target and the shield around the target may be treated as a small grey body in a grey enclosure [86], if the difference in temperature between the substrate and the chamber walls is small. The grey enclosure behaves as a black body through the opportunities for ultimate absorption offered by repeated reflection from its walls [86]. All the emission from the target q12 will ultimately be absorbed by the enclosure. However, only a fraction of the emission from the walls will be absorbed by the target, depending on the view factor, F12 , of the wall with respect to the target and the emissivity of the target. Bayley et al. [86] showed that the net radiation flux qtr from the heating source to the walls of the enclosure is given by: qtr = ε1 σ(T14 − T24 ),

(7.19)

where ε1 is the emissivity of the hot body whose surface temperature is T1 . For a typical case study [7, 36], T2 was assumed at room temperature, while the shield temperature, T1 was estimated by considering that almost all the incident energy is converted into heat and the main sources of heat lost from the shield are by thermal conduction through the gas as well as through the shield material to the cooling surface and radiation. As already stated, the energy source to the shield is mainly from the system input power. The emissivity used was found to be different from the bulk emissivity for steel. A value of 0.79 was used, which is higher than the bulk value. This value is close to the value for oxidized steel [87]. There has been evidence that the surface emissivity of material does change with temperature and surface roughness [48,88–90]. For example, Love [89], cites an increase in emissivity for chromium steel from about 0.2 at 600◦ C to over 0.7 at 900◦C. Simonsen [90] gave values for the emissivity of steel at 100◦C ranging from 0.066 to 0.97, depending on whether the surface is polished or rough. Figure 7.10 shows the comparison of the modeled shield temperature with the measured values. In the magnetron sputtering system, the ground shield around the target and the surface of the target are always subjected to bombardment by energetic particles, and hence their surfaces are rough. At a magnetron power of 200 W, the contribution of thermal radiation to the deposited energy during the sputter deposition of Al was estimated to be about 11% of the total

246

S.D. Ekpe and S.K. Dew

Fig. 7.10. Comparison of the model with the measured temperature of the shield around a 76 mm planar target located in the magnetron sputter chamber of radius 24 cm during the sputtering of Al at a magnetron power of 200 W and gas pressure of 0.67 Pa

energy [7]. This is significant. For the deposition of reactive materials such as AlN, where the target surface might be coated with the reactive material (which in some cases may not result in a conducting surface) during the operation, the contribution is expected to be higher. 7.5.6 Model Results The following presents the results of the model, which shows good agreement with experimentally determined values for different process conditions. The spatial profile of energy per deposited particle is also presented. Total Energy Flux at the Substrate The steady-state total energy deposited as a function of the magnetron discharge powers for gas pressures of 0.67 and 1.33 Pa at substrate-target distances of 10.8 and 21 cm along the central axis of the target are shown in Fig. 7.11. A similar trend was obtained for Cu [36]. As expected, the total deposited energy increases with power, but reduces with pressure and distance. The trends seem to suggest a nonlinear dependence on power, probably due to gas heating effects. Figure 7.12 shows the variation in the on-axis total energy with distance for Al at 0.67 Pa sputtered from a 7.6 cm target at three different powers [36]. The deposited energy follows the pattern of particle deposition [58], especially at distances not too far off from the target. This suggests that a major contribution to the total energy at such distances may be from sputtered particles.

7 Energy Deposition at the Substrate in a Magnetron Sputtering System 0.04

0.1 (a) 0.08

0.67 Pa (model)

Total energy flux (W/cm2)

Total energy flux (W/cm2)

247

1.33 Pa (model) 0.67 Pa (expt) 1.33 Pa (expt)

0.06 0.04 0.02 0

0.67 Pa (model)

(b)

1.33 Pa (model)

0.03

0.67 Pa (expt) 1.33 Pa (expt)

0.02

0.01

0 0

100 200 300 400 Magnetron power (W)

500

0

100 200 300 400 Magnetron power (W)

500

Fig. 7.11. Steady-state energy as a function of magnetron power at substrate-target distances of (a) 10.8 and (b) 21 cm during the sputter deposition of Al at 0.67 and 1.33 Pa

Total energy flux (W/cm2)

1 0.8 0.6

100 W

0.4

300 W

200 W

0.2 0

0

5

10 15 z (cm)

20

Fig. 7.12. Variation in the total energy flux at the substrate with substrate-target distance, z as a function of magnetron power at a gas pressure of 0.67 Pa during the sputter deposition of Al

The on-axis deposited energy initially increases with distance to a maximum at some point close to the target, then, thereafter falls off. The trend of the fall is steep initially, then leveling off afterward. This behavior is attributed to the erosion profile of the magnetron target, as it is consistent with the profiles of sputtered and plasma particles [58, 66]. The trend of the modeled result at some distances off the target is consistent with experimental results of [43]. At long distances, most of the sputtered atoms are thermalized after involving several collisions with the background gas. Thus the energy contribution of the sputtered atoms to the total deposited energy is greatly reduced to just about their condensation energy. In addition to the reduction in the energy of the sputtered atom, the flux of the sputtered atoms is also reduced due to scattering and spreading.

S.D. Ekpe and S.K. Dew Energy per deposited atom (eV)

248

90 80 0.67 Pa (model)

70

1.33 Pa (model)

60

0.67 Pa (expt)

50

1.33 Pa (expt)

40 30 20

0

100 200 300 400 Magnetron power (W)

500

Fig. 7.13. Total energy per deposited Al atom as a function of power at distance of 10.8 cm for different pressures

Total Energy per Deposited Atom Variation in the total energy per deposited atom with power and pressure for Al is shown in Fig. 7.13 [36]. Although the model may overestimate the energies at very low powers or very high pressures by underestimating the deposition flux [36,58], the trend suggests that the total energy per deposited atom decreases strongly with power, but increases with pressure. However, at high power densities, it remains constant with power and pressure. The power threshold of this saturation depends on the spatial location of the substrate relative to the target. It should be expected that the power threshold would increase with spatial distance. The size of the target is also a determining factor in the spatial profile of the total energy per deposited atom. Figure 7.14 compares the effect of circular target size on the simulated spatial profile of the total energy per deposited atom, for the same power density (9.5 W cm−2 ) and pressure (0.67 Pa) for Al deposition [36]. Increase in the size of the inner radius of the etch track as well as the overall target size results in a shift in the regions of uniform film away from the target. As can be noticed in (b), this shift results in a lower energy per deposited atom compared with those of the smaller target (a) over the same area. However, the simulated energy per deposited atom at regions of uniform film is about the same for both cases under the same power density and other operating conditions. For Al and Cu, this energy is about 28 and 25 eV atom−1 , respectively, given the DC planar magnetron considered in that study [36] and limits of uncertainty. The atomic weight of the sputtering gas has been shown to have a strong influence on the energy per deposited atom [91]. This energy decreases with atomic weight of the process gas. For instance, for the deposition of molybdenum using cylindrical-post magnetron sputtering source with the sensor located at a radius of 14.2 cm from the cathode axis, values of the energy per deposited atom has been shown to decrease from about 85 to below

7 Energy Deposition at the Substrate in a Magnetron Sputtering System

249

Energy per deposited atom (eV) (a) 120 100 80 60 40 20 0

10 5 5

0 10 z (cm)

15

20

25 −10

−5

R (cm)

Energy per deposited atom (eV) (b) 120 100 80 60 10

40 20 0

5 5

0 10 z (cm)

15

20

25 −10

−5

R (cm)

Fig. 7.14. The effect of the size of target and etch track on the energy per deposited atom, for targets with erosion tracks of radii between (a) 0.9 and 3.32 cm and (b) 3.15 and 7.0 cm

30 eV atom−1 when the process gas was changed from neon to xenon [91]. Apart from the effect of reduced sputtering yield when using gases of lower atomic weight, the high energy per deposited atom measured with the use of neon is attributed to the effect of reflected neutrals. The energy per deposited atom in the case of reactive deposition has been shown to be generally higher than that of elemental deposition [70, 91]. This is attributed to, among others, the atom reflection at the cathode, a reduced sputtering yield (reduced deposition rate) [92–94] due to compound formation on the surface of target by adsorbed working gas species [95, 96], a reduced ion component in the discharge current because of higher secondary electron emission coefficient at the target [91], and the effect of negative ions accelerated from the sheath in front of the target toward the substrate. However, the effects of induced secondary electron emission (ISSE) coefficient and negative

250

S.D. Ekpe and S.K. Dew 35

30

(a)

25

Contributions to the energy per deposited atom (eV)

Contributions to the energy per deposited atom (eV)

35 sputtered atom reflected neutral plasma radiation

20

electrons

15

thermal radiation

10 5 0

(b)

30

sputtered atom reflected neutral

25

plasma radiation

20

electrons

15

thermal radiation

10 5 0

0

100 200 300 400 Magnetron power (W)

500

0

100 200 300 400 Magnetron power (W)

500

Fig. 7.15. Contributions by mechanism to the total energy per deposited Cu atom as a function of magnetron power at a substrate-target distance of 10.8 cm and gas pressure of (a) 0.67 and (b) 1.33 Pa

ions may not generally hold in all cases. For instance, it has been shown that the ISSE coefficient decreases with oxygen content in cases where there may be reduction behavior of the oxides under ion bombardment [96]. Thornton and Lamb [91] reported values of 230 and 100 eV atom−1 , respectively, for Al2 O3 and Cr2 O3 , which are several times higher than values reported for the elemental metals at similar process conditions. To control the deposition process for a desired film property, it is necessary to understand the effects of process conditions on the individual energy contributions. Figure 7.15 shows the estimated contributions to the on-axis total energy per deposited atom as a function of power and pressure, using Cu as a case study, at a substrate-target distance of about 10.8 cm. The trend of the results suggests that the plasma effects (electrons and plasma irradiation) are more important at lower power levels. The contribution of the sputtered atom increases with power but decreases with pressure as would be expected. In the low power regime, the contribution of electrons decreases rapidly with power, but at high powers, it is nearly constant with power. That of the plasma irradiation has a similar trend to that of the electron, though the initial decrease is not as rapid as that of the electron. Both contributions increase with pressure. At about 300 W and 0.67 Pa, both electrons and plasma irradiation contribute about 30% each to the total energy per deposited Al atom [36]. This percentage contribution changes with pressure and power. In obtaining the above, it was assumed that all incident energy at the substrate is absorbed. However, in practical context, some of the radiant energy may be reflected away from the surface, though some through multiple reflections may be deposited back onto the substrate [97]. It should also be noted that the reflection coefficient decreases with energy [87], and since we are dealing with low energy here, the associated error in the estimation, of course reduces. The contribution of the reflected neutrals decreases marginally with power, but increases marginally with pressure. This is consistent with the results reported in [60]. Though the energy and particle reflection coefficients

7 Energy Deposition at the Substrate in a Magnetron Sputtering System

251

increase with pressure through reduced discharge voltage for a constant power mode of operation, the effect is limited by the decrease in the incident ion energy.

7.6 Conclusions The effective energy deposited onto the surface of a growing film plays an important role in determining the property and structural formation of the film. The different sources of the deposited energy in a magnetron sputter deposition system are reviewed. Embedded sensors, such as microfabricated polysilicon thin film thermistor, offer a useful tool to measuring the energy flux at the substrate. This eliminates the problems associated with discrete sensors, such as thermocouples. Experimentally determined values of energy flux are also presented for typical case studies. A theoretical model for estimating the energy flux at the substrate is also reviewed and results compared with reported experimental data. The model takes into consideration the effects of process conditions on the various contributions to the total deposited energy. The total deposited energy has been shown to vary with the process conditions. At large distances from the target (long throw), sputtered particles contribute negligibly to the total deposited energy. Estimated total energy per deposited atom is found to decrease with power, but increase with pressure at low power regimes. This energy trends toward being independent of power and pressure for given spatial location relative to the target. It is generally shown that, the energy per deposited atom is highly dependent on the spatial location. The energy per deposited atom is greater in RF systems than in DC systems. For the reactive deposition process, the energy per deposited atom is generally higher than that of elemental deposition. This is attributed to the reduced deposition rate as a result of formation of compound on the target surface, among others.

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8 Process Diagnostics J.W. Bradley and T. Welzel

8.1 Introduction Reactive sputtering of materials can be carried out using a number of different types of plasma excitation methods including DC, RF, Pulsed DC and AC, with the power applied to a sputter cathode. The usual configuration adopted in industry is the magnetron, operated typically in the so-called unbalanced mode. Although DC and RF excitation are often employed in non-reactive plasmas, with each technique having its own particular advantages (e.g. DC is high rate, RF can sputter many types of targets), it has become common when introducing reactive gases like oxygen or nitrogen to modulate the discharge power by pulsing the driving voltage waveform. Modulating the applied power in the form of pulses in the frequency range 5–350 kHz was introduced in reactive magnetron sputtering as a solution to the problem of arcing of insulating material at the target. This sputtering system has been used for almost 20 years as a method for the rapid deposition of various dielectric thin films for a variety of technological applications. In both industrial and academic research, two pulsing configurations are typically utilised: unipolar and bipolar modes [1]. The latter may be asymmetrical or symmetrical. The asymmetrical bipolar mode differs from the unipolar mode in that a positive voltage (typically +30 to +50 V) at the cathode is introduced instead of grounding the target within ‘off’ phase. The technique is widely used, for instance, for a preparation of high quality insulating dielectric films [2–4]. A typical representation of the target voltage and current traces for this pulse mode is shown in Fig. 8.1. It is now well recognised that asymmetrical bipolar pulsing is more efficient than the symmetrical configuration in its ability to discharge insulating spots on the target by plasma electron bombardment when the polarity of the magnetron target is reversed [3]. This chapter reviews most of the recent investigations of reactive sputter deposition processes using plasma diagnostic techniques. Special emphasis is placed on the increasingly popular method of mid-frequency pulsed magnetron sputtering. Despite being closely related, the relatively new technique

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Fig. 8.1. Typical pulsed magnetron discharge waveforms: (a) voltage and (b) current. After the large positive transient at t = 5 μs (duration 200 ns) the remainder of the ‘off’ phase is known as the reverse time (from [77])

of high-power pulsed ionised magnetron sputtering (HPPMS or HIPIMS, see, e.g. [5, 6]) with extremely short voltage pulses compared to pulse repetition time, is not covered in much detail here. The chapter is subdivided into different plasma diagnostic methods such as electric probes, mass spectrometry, optical emission spectroscopy (OES) and laser-induced fluorescence (LIF). Each method is briefly described and its application to particular aspects of reactive sputtering is presented.

8.2 Electrical Probes Since their introduction by Langmuir and Mott-Smith [7, 8] electrical probes have developed to be one of the most frequently used techniques to characterise plasma-processing discharges, including sputtering discharges. This is due to the fact that, experimentally, probe techniques are comparatively easy

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to apply and that current-voltage data can be readily obtained. However, correct interpretation of probe characteristics is not so simple. 8.2.1 Probe Techniques A typical electrical single probe, as introduced by Langmuir and Mott-Smith, is a small conducting electrode of well-defined geometry – mostly in the form of a cylindrical wire – which is variably biased with respect to a fixed potential, e.g. the grounded wall of the processing chamber. If its potential is positive relative to the plasma potential Vp , then the probe draws an electron current, from which the electron density can be determined. If its bias is slightly negative with respect to Vp electrons begin to be repelled, and from the attenuation of electron current with increasing negative bias in this retardation region, the electron energy distribution function (EEDF) or under the assumption of an approximately Maxwellian distribution the electron temperature Te can be calculated [9]. Biasing the probe potential more negatively, still leads to the point where the diminishing current is compensated by the positive ion current at the floating potential Vf , which is consequently determined from the zero current value of the probe characteristics. Finally, if the probe is biased even more negatively, the ion saturation current is reached from which in principle, the ion density can be obtained. The general problem with electrical probes is that they are physically in contact with the plasma. This may lead to significant disturbances of the plasma environment and therefore it is an usual requirement that probes have very small dimensions. Adjacent to a biased probe surface, a space-charge sheath will form, through which the charged particles have to pass to be collected. For this reason, in most cases, there is no simple relationship between the measured current and the density of the charged species in the plasma bulk, particularly true when cold ions move through the positive space-charge region. This necessitates complex theories for different probe geometries when considering the ion collection part of the characteristics (see, e.g. [10–12] and references therein). Another form of electrical probes applied to reactive discharges is the double probe. In this case, the probe consists of two – normally identical – electrodes and the voltage is applied between both probe tips instead against a fixed reference [13]. The complete system is insulated which has the advantage that no reference is needed and that the probe “floats” with any fluctuations in the potential of the surrounding plasma. This makes double probes advantageous for measurements in pulsed or strongly fluctuating discharges, allowing average values to be gleaned. However, as the probe has no reference, no values of the plasma or floating potentials (Vp , Vf ) can be obtained, and due to the isolated set-up, the current is always limited by the ion saturation current. It has therefore to be assumed that the density of the electrons equals that of the positive ions, which must be calculated from the ion saturation current. An experimental approach to do this without complex calculations has been

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J.W. Bradley and T. Welzel

given by Sonin [14]. Another drawback of the method is that only a small part of the EEDF can be probed and from this it has to be extrapolated to a Maxwellian distribution. Under this assumption, an electron temperature can be determined [15, 16]. A single probe can also be built as a so-called emissive probe. This method is based on the fact that, once the probe is heated enough, it starts not only to emit photons but also electrons. To achieve this heating, the probe wire is mostly formed into a loop, which carries a separate conduction current. If the probe is above the plasma potential, the thermal electrons cannot overcome the potential barrier to the plasma and the emission is prevented. From a probe below the plasma potential, the electrons can leave the surface and upon moving away from the probe form a virtual additional ion current [17, 18]. If this current is of the order of the electron saturation current the probe characteristic is strongly modified: At the plasma potential a sharp kink occurs and the apparent ion current forces the floating potential to move close to the plasma potential [18]. The kink in the characteristics gives a very precise measurement of the plasma potential with a theoretical error of the order of kTW /e (k – Boltzmann constant, e – elementary charge, TW – wire temperature) compared to the single probe evaluation. If the wire temperature is chosen appropriately, the apparent floating potential may be used as a direct measure of the plasma potential. This gives a convenient method to access time-varying plasma potentials as the floating potential is quite easily measured with a high-impedance oscilloscope. All probe methods have the advantage that they measure the plasma at one particular point, i.e. they give three-dimensionally resolved results. The measurement position can be chosen simply by an appropriate positioning of the probe. On the other hand, special care has to be taken with respect to surface modification of the probe surfaces in reactive sputtering. As a probe material, mostly wires of tungsten, platinum or gold are used that withstand the reactive gas component. Special constructions of ceramics or glass prevent contact with conductive surfaces when conducting layers are deposited. To avoid insulating layer formation on the probe, the tips are heated or frequently cleaned by sputtering, i.e. the application of a high negative potential. An advantage of the double probe technique for reactive sputtering is that the characteristic is symmetric, any falsification of the probe due to contamination can be immediately observed by deviations from the symmetry. This is significantly more difficult with asymmetric single probe characteristics. 8.2.2 Use of Electrical Probes in Pulsed Reactive Sputtering Recently, pulsed magnetron discharges have been widely used for reactive sputtering of thin films. Apart from discharges which are aimed at the achievement of a highly ionised flux of sputtered material in a quite short time (high-power pulse magnetron sputtering, see, e.g. [5, 6]) such discharges are typically operated with frequencies between about 1 and 500 kHz and a duty

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cycle (i.e. the duration of the sputter pulse to the total period) of several 10’s of %. Such discharges have been characterised by time-averaged single Langmuir probes which average the plasma parameters over many periods [19–21]. Glocker [21] reports that the average electron density is elevated by a factor of 4 and the electron temperature by a factor of 1.3 compared to a DC discharge accompanied by an increase in the energy influx to the substrate. Lee et al. [22] found that with increasing pulse frequency from 75 to 250 kHz the ion density is increased as well as the electron temperature. The results are hard to compare because Glocker ran the magnetron discharge in constant power mode but Lee et al. with constant current. Due to the probably varying plasma potential as reference these measurements are uncertain and it is necessary to measure the quantities with a single probe time resolved. The first such measurements to be reported were by Mahoney et al. [23] who found an increase in the electron temperature compared to DC and with increasing frequency but no significant increase in the density when they afterwards averaged their time-resolved values. A similar result was published by Seo et al. [24] who observed a significant increase in Te  with decreasing duty cycle, but only a weak dependence of ne  at constant power. Bradley et al. [25,26] have investigated an asymmetric bipolar-pulsed magnetron with a time-resolved single probe on the discharge axis and found an increase in Te  and ne  compared to the DC case by about 30% and 20%, respectively (see Fig. 8.2). Moreover, they observed very high-electron temperature values just at the beginning of the pulse ‘on’ phase after the target voltage was switched to negative values. Though these values were probably not due to a Maxwellian distribution, they indicate the presence of a ‘burst’ of high-energy electrons during the first μs of the ‘on’ phase. Even in the following quiescent part of the ‘on’ phase Te has values above that of the DC case as shown in Fig. 8.2. For a similar magnetron sputtering system with the same power supply Belkind et al. [27] and with a different pulse power supply Bradley et al. [28,29] and B¨ acker et al. [30] observed a similar ‘burst’ with the effective electron temperature reaching values sometimes as high as 38 eV (see Figs. 8.3 and 8.4). The phenomenon is explained by the rapidly advancing sheath in front of the negatively biased target, which accelerates electrons into the bulk plasma causing this ‘burst’. Through inelastic collisions this results in a developing electron density on the scale of some μs (cf. Figs. 8.2 and 8.4) with the electrons still keeping energies slightly higher than in a DC discharge. Initially, high electron temperatures in the ‘on’ phase were also observed in discharges with longer timescales and longer ‘on’ phases [24,31–36]. In some cases, where the EEDF was thoroughly studied with a single probe [24, 31, 33, 34], it has been shown that Te,eff actually consists of a high-temperature electron group and a cold-temperature electron group with the hot group containing this significant temperature peak [31] (see Figs. 8.5 and 8.6).

260

J.W. Bradley and T. Welzel 25

15

(a) 50 kHz 12 Ne/1015m−3

Teff / eV

20 15 10 5

9 6 3

0 0

2

4

6

(a) 50 kHz

0

8 10 12 14 16 18 20

0

2

4

6

Time / μs 8

12 10 Ne/1015m−3

6 Teff / eV

8 10 12 14 16 18 20 Time / μs

4 2

(b) 100 kHz 0 0

1

2

3

4

5

6

Time / μs

7

8

9 10

8 6 4 2

(b) 100 kHz

0 0

1

2

3

4 5 6 Time / μs

7

8

9 10

Fig. 8.2. Effective electron temperature (left) and electron density (right) for two different pulse frequencies at the same duty cycle of 80%. The average value is given by the horizontal lines in the figures and compared to the values obtained for a DC discharge at the same power of 200 W (the lower horizontal lines). The figures start with the ‘off’ phase at 0 μs (from [50])

Fig. 8.3. Effective electron temperature during one pulse (f = 100 kHz) for different axial positions of the probe along the axis of symmetry. The duty cycle is 50% with the ‘off’ phase starting at 0 μs and the ‘on’ phase starting at 5 μs (from [28])

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Fig. 8.4. Plasma density and effective electron temperature in a pulsed-DC discharge at 20 kHz with a duty cycle of 80% in metal and reactive mode. The target material used was Al, the pulse ‘on’ phase starts at 10 μs (after [27])

In all these cases, after the temperature peak, a rise of in the electron density is observed. The timescale for this strongly depends on the pulse parameters but is always longer than for the temperature rise (cf. Figs. 8.2 and 8.4). Time-resolved double probe measurements in a discharge comparable to that of Bradley et al. [28] but in reactive mode and with a different target material (Mg) reproducibly revealed two peaks in the charge carrier density at the beginning of the ‘on’ phase [37–42] for different conditions (see Fig. 8.7). They do not show, however, significant temporal changes in Te . This is not surprising as double probes are rather insensitive to the EEDF as discussed above. The argumentation for the formation of the two peaks in the density is basically the same as discussed for the temperature peaks. The fast advancing cathode sheath rapidly accelerates electrons, which are ‘left over’ from the previous pulse. Subsequent ionisation collisions lead to the first density peak. Afterwards, the heavy ions move to the cathode, release secondary electrons which are accelerated and upon collisions start to form a stationary discharge with a high density in form of the second maximum.

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(a)

1011 1 ms 3 ms 5 ms 7 ms 9 ms 11 ms 15 ms 25 ms

EEPF (eV-3/2 cm-3)

After pulse-on 1010 25 ms 109

108

107 1 ms 106

0

5

10

15

20

25

Kinetic energy (eV)

EEPF (eV-3/2 cm-3)

(b)

1011 26 ms 28 ms 29 ms 31 ms 32 ms 34 ms 38 ms 50 ms

After pulse-off 1010

26 ms

109

108

107

106

50 ms

0

5

10

15

20

25

Kinetic energy (eV) Fig. 8.5. Electron energy distribution functions for different times within a pulse in (a) the ‘on’ phase and (b) the ‘off’ phase for a 20 kHz discharge with 50% duty cycle. Times refer to the beginning of the ‘on’ phase at 0 μs (from [34])

The subsequent drop in the density before the stationary state is a consequence of the power supply behaviour which is different to that of Bradley et al. [26, 28]: the target voltage waveform exhibits a strong initial overshoot (see Fig. 8.8a) whereas the one used by Bradley et al. has an almost rectangular form (see Fig. 8.8b). The relaxation of the target voltage also leads to much reduced ionisation in the discharge volume. A two-dimensional survey of the discharge cross section has confirmed this picture [43]. The first maximum was found primarily in the middle of the discharge (see Fig. 8.9) because in the initial stage of the ‘on’ phase electrons can

8 Process Diagnostics

263

12

z = 2 cm z = 4 cm z = 6 cm z = 8 cm z = 10 cm z = 12 cm

10

Te (eV)

8

6

4

2

0 0

100

200

300

400

500

Time (μs)

Fig. 8.6. Electron temperatures at different heights above the racetrack in a 2 kHz discharge with 50% duty cycle. The figure starts with the ‘off’ phase at 0 μs and shows two groups of differently hot electrons during the whole ‘on’ phase with especially the hot group peaking at the start of the ‘on’ phase (from [31])

Charge Carrier Density [1010 cm-3]

10

on

off

8 6

C

A

D

E

B

4 2 0 10 8 6 4 2 0 0

1

2

3 Time [µs]

4

5

Fig. 8.7. Temporal evolution of the charge carrier density in a reactive 200 kHz discharge for different Ar/O2 mixtures and a duty cycle of 60%. The timescale starts with the ‘off’ phase. The traces show two density peaks (A, C) at the beginning of the ‘on’ phase (2 μs) with their intensity depending on the gas mixture F(O2 )/(F(Ar) + F(O2 )) of (top) 44% and (bottom) 9% (from [38])

Ud / V

264

J.W. Bradley and T. Welzel 200 100 0

−100

V ~ 110 V V ~ 21 V

time / µs

0 1 2 3 4 5 6 7 8 9 10 11

−200

V ~ −175 V

−300 −400 −500

V ~ −540 V

−600

Fig. 8.8. Target voltage waveforms for a pulsed magnetron discharge driven by a standard Pinnacle Plus (left) and a modified Pinnacle Plus (right) (both Advanced Energy Inc.) unit. The rectangular one was obtained for a Ti target in argon, the overshooting one with a Mg target in Ar/O2 but the waveforms are rather due to the power supply than due to the operating conditions. The marks given by Bradley et al. (right) are ‘A’ for the ‘on’ phase, ‘B’ for the stable ‘off’ phase and ‘C’ for the overshoot in the ‘off’ phase (after [26])

10

Axial Position z [mm]

n [10 40

>12 10 8 6 4 2 60 50 40 35 30 20 10 0, ω = 20 is a constant, and Xi and Vij are defined as Xi = χi + kc

iN 

Zj Pij (rij )

(14.20)

j=i1

and



iN (j)

Vij (rij ) = Ji δij + kc

Qik (rij ).

(14.21)

k=i1 (j)

Here the notation i1 , i2 , . . . , iN represents a list of i’s neighbors. For long range ionic interactions, the dimension of the computational cell may be small compared with the cutoff distance of the potential. In such a case, the ith atom’s neighbors may include numerous image atoms introduced by periodic boundary conditions. The notation of i1 , i2 , . . . , iN concisely addresses this problem by including all image atoms in the list. The pair wise functions Pij (r) and Qij (r) depend on the charge parameters of atoms i and j, ξi and ξj . For ξi = ξj = ξ, 3 2 1 3 2 3 ξ + ξ r + ξ r exp(−2ξr), (14.22) Pij (r) = 8 4 6 11 3 1 1 1 1 + ξr + ξ 2 r2 + ξ 3 r3 exp(−2ξr), (14.23) Qij (r) = − r r 8 4 6 and for ξi = ξj , ξi ξj4 exp(−2ξi r) 1 ξj ξi4 exp(−2ξj r) exp(−2ξi r) + Pij (r) = − ξi + + r (ξi + ξj )2 (ξi − ξj )2 (ξj + ξi )2 (ξj − ξi )2 + Qij (r) =

(3ξi2 ξj4 − ξj6 ) exp(−2ξi r) (3ξj2 ξi4 − ξi6 ) exp(−2ξj r) + , r(ξi + ξj )3 (ξi − ξj )3 r(ξj + ξi )3 (ξj − ξi )3

(14.24)

ξi ξj4 exp (−2ξi · r) 1 ξj ξi4 exp(−2ξj r) − − r (ξi + ξj )2 (ξi − ξj )2 (ξj + ξi )2 (ξj − ξi )2 −

(3ξi2 ξj4 − ξj6 ) exp(−2ξi r) (3ξj2 ξi4 − ξi6 ) exp(−2ξj r) − . r(ξi + ξj )3 (ξi − ξj )3 r(ξj + ξi )3 (ξj − ξi )3

(14.25)

In (14.20) and (14.21), δij = 1 when i = j and δij = 0 when i = j, kc = −2 14.4 eV ˚ A e is the Coulomb constant (e represents the electron charge), Zi is an effective charge of atom i, χi is its electronegativity [76], Ji (Ji > 0) is referred to as an “atomic hardness” [78] or a self-Coulomb repulsion [75], and the notation i1 (j), i2 (j), . . . , iN (j) means all the j atoms (i.e., j and its images) that are within the cutoff distance from atom i. Vij (rij ) defined in (14.21), (14.23), and (14.25) involves a summation of 1/rij . The direct summation of the slowly decaying function 1/r imposes a

14 Atomic Assembly of Magnetoresistive Multilayers

513

serious divergence problem. Note that the total contribution of these divergent terms to the system energy is N qi qj 1 . 2 i=1 j=i rij

N

i

1

It can be replaced by fast convergent Ewald summations [72,79,80] under the system neutral condition. The Ewald approach was therefore used to modify Vij to eliminate the divergence problem. Equation (14.19) is a function of both atom positions and atom charges. During simulations, charges are dynamically solved from the energy minimization conditions at each time step. Once charges are known, (14.19) becomes a function of atom positions only. Because charges are solved from energy minimization conditions, their derivatives have no effects on forces or stresses. Equation (14.19) can then be used to calculate forces and stresses as if the charges were constant. Equation (14.19) is a quadratic type of potential with respect to qi . It has been demonstrated that the minimum ionic energy and  N the equilibrium charges defined by (14.19) for a neutral system i=1 qi = 0 can be effectively solved using an integrated conjugate gradient technique and a Newton–Raphson method [72]. A relatively long cutoff distance of 12 ˚ A was used for the potential. This cutoff distance addresses the long range Coulomb interactions relatively well. It does not affect the properties of the existing EAM potential [58, 62, 63] because (14.11) and (14.12) enable the potential to be virtually cutoff at 7 ˚ A or below. All the charge parameters determined for the O−Al−Ni−Co−Fe system are listed in Table 14.2. Since the modified CTIP model has no effect on the potential that describes the metal system, the existing EAM potential database [58, 62, 63] can in principle be used directly for metal alloys. To better fit the experimental cohesive energies, lattice constants, elastic constants, and crystal structures of the four binary metal oxides: corundum Al2 O3 and Fe2 O3 , and B1 CoO and NiO, however, we have adjusted the EAM parameters for metals. The revised EAM parameters for the metals of interest here are shown in Table 14.3. In Table 14.3, F3 is split to F3 − and F3 + , the former is used when ρ ≤ ρe , and the latter is used when ρ > ρe . Compared with Table 14.1, relatively large Table 14.2. CTIP parameters for selected elements Element O Al Ni Co Fe

qmin (e) −2 0 0 0 0

qmax (e) 0 3 2 2 3

χ (eV e−1 ) 2.00000 −1.47914 −1.70804 −1.67765 −1.90587

J (eV e−2 )

−1 ξ (˚ A )

Z (e)

14.99523 9.07222 9.10954 8.65773 8.99819

2.144 0.968 1.087 1.055 1.024

0.00000 1.07514 1.44450 1.54498 1.28612

514

H. Wadley et al. Table 14.3. Revised EAM parameters for metals Element r e (˚ A) fe ρe ρs α β A (eV) B (eV) κ λ Fn0 (eV) Fn1 (eV) Fn2 (eV) Fn3 (eV) F0 (eV) F1 (eV) F2 (eV) F3 − (eV) F3 + (eV) η Fe (eV) μ ν

Al

Ni

Co

Fe

2.86392 1.20378 17.51747 19.90041 6.61317 3.52702 0.31487 0.36555 0.37985 0.75969 −2.80760 −0.30144 1.25856 −1.24760 −2.83 0 0.62225 −2.48824 −2.48824 0.78591 −2.82453 0.85 1.15

2.48875 2.21149 30.37003 30.37137 8.38345 4.47117 0.42905 0.63353 0.44360 0.82066 −2.69351 −0.07644 0.24144 −2.37563 −2.70 0 0.26539 −0.15286 4.58568 1.01318 −2.70839 0.85 1.15

2.50598 2.31544 31.89166 31.89166 8.67963 4.62913 0.42138 0.64011 0.5 1.0 −2.54180 −0.21942 0.73338 −1.58901 −2.56 0 0.70585 −0.68714 3.09213 1.07702 −2.56584 0.85 1.15

2.48199 2.31453 24.59573 24.59573 9.81827 5.23641 0.39281 0.64624 0.17031 0.34061 −2.53499 −0.05960 0.19306 −2.28232 −2.54 0 0.20027 −0.14877 6.69465 1.18290 −2.55187 0.85 1.15

adjustments are made to parameters F3 and η for Ni, Co, and Fe. These adjustments do not affect the predictions of equilibrium properties of the elements as such parameters are only used in an electron density range significantly higher than the equilibrium electron density, (14.16). The adjustments, however, are essential to improve the fitting results. All the pair energies involving oxygen are expressed using (14.11). Instead of using the alloy EAM model, (14.13), (14.11) is also used to fit the pair cross potential between aluminum and nickel to the experimental cohesive energy of the B2 phase of NiAl. All of the fitted EAM pair energy parameters are summarized in Table 14.4. The electron density of oxygen is described using ,  fe exp −Γ rre − 1 f (r) = (14.26)  20 . 1 + rre − Ψ The corresponding parameters are listed in Table 14.5. A cubic spline is used to fit the oxygen embedding energy so that it can smoothly describe the energy of four binary metal oxides: corundum Al2 O3

14 Atomic Assembly of Magnetoresistive Multilayers

515

Table 14.4. Parameters for EAM pair potentials between oxygen and oxygen, oxygen and all other metals, as well as between aluminum and nickel atoms Pair

r e (˚ A)

α

β

A (eV)

B (eV)

κ

λ

O−O O−Al O−Ni O−Co O−Fe Al−Ni

3.64857 2.98520 2.95732 2.59586 3.07992 2.71579

5.44072 8.49741 7.96528 8.25224 7.52309 8.00443

3.59746 4.52114 4.42411 4.37548 4.13330 4.75970

0.34900 0.09738 0.13521 0.25714 0.17108 0.44254

0.57438 0.38121 0.25332 0.34029 0.39869 0.68349

0.08007 0.18967 0.47077 0.37419 0.22335 0.63279

0.39310 0.95234 0.65524 0.50843 0.34380 0.81777

Table 14.5. EAM parameters for oxygen electron density function fe 1.39478

Γ

Ψ

2.11725

0.37457

Table 14.6. EAM parameters for the splined oxygen embedding energy function i 0 1 2 3 4

Fo,i (eV) −1.56489 −1.58967 −1.54116 −1.51798 −1.19082

F1,i (eV) −1.39123 1.30636 2.02821 2.30979 4.12936

F2,i (eV)

F3,i (eV)

ρe,i

ρmin,i

ρmax,i

1.77199 9.81033 6.56240 7.69582 10.32338

1.59833 0.00000 0.00000 0.00000 0.00000

54.62910 64.26953 66.21202 66.92391 74.23105

0 54.62910 65.24078 66.56797 70.57748

54.62910 65.24078 66.56797 70.57748 ∞

and Fe2 O3 , and B1 CoO and NiO. The cubic spline functions for the oxygen embedding energy have the form: F (ρ) =

3 

Fi,j

i=0

ρ −1 ρe,j

i for ρmin,j ≤ ρ < ρmax,j ,

j = 0, 1, 2, 3, 4

(14.27) and the corresponding parameters are listed in Table 14.6. The EAM + CTIP potential is well fitted to cohesive energies, lattice constants, and single crystal elastic constants of both metals and their oxides. 14.2.2 MD Model The atomic scale structure of a thin film is the end result of many atomic assembly mechanisms. For instance, depending on the initial kinetic energy of the “adatoms” and the latent heat release during their condensation, a variety of impact induced assembly mechanisms occur, including thermal spike induced (athermal) atomic hopping [81], adatom biased skipping [81], resputtering [82], adatom reflection [82], and adatom-surface atom exchange [60,83]. These impact events are then followed by thermally activated diffusion. Here

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complexities arise because each atom on the surface can exist in many different local atomic configurations and hence a large number of different diffusion paths and diffusion rates are encountered [56, 84]. The morphology, texture, grain size, and many of the defects within a film are all dependent upon a combination of the impact events and the subsequent diffusion routes. In some applications, energetic inert gas ion impacts with the surface are utilized to modify the structure of the already deposited film. Successful modeling and simulation methods of these deposition processes must accurately represent the transport of vapor atoms, their impact with surfaces, and their thermal diffusion under various environmental conditions. In a MD simulation, the positions and velocities of all the atoms are solved from Newton’s equations of motion. If high fidelity interatomic potentials are used to calculate the interatomic forces, MD methods can accurately represent atomic events during vapor condensation, especially those associated with energetic impacts. Additionally, it allows the exploration of the relaxed nanostructure, and the calculation of the stress and strain. MD methods are therefore of great utility for exploring the growth of nanostructures. In principle, molecular dynamics allows the atomic environment dependent atom motion, the temperature, the defect content, and the stress states of the medium to be tracked. In a MD simulation of growth, a computational substrate is created by assigning positions of atoms according to equilibrium crystal, lattice constants, and crystallographic orientation of the substrate. Initial substrate temperature is introduced by assigning velocities to atoms based upon the Boltzmann distribution. To minimize the effect of small systems that must be employed with practical computational resources, periodic boundary conditions are used in the two horizontal (e.g., x- and z-) coordinate directions. Free boundary conditions are used in the vertical (y-) coordinate direction. Growth of the top y surface is simulated by injecting adatoms into the top surface at an injection frequency that corresponds to the simulated growth rate. Newton’s equations of motion are then used to solve for the positions of both the substrate atoms and the adatoms as a function of time. To prevent a shift of the computational system due to the momentum imported by impact of the adatoms with the top surface, the positions of the bottom monolayer of atoms are fixed during simulations. During growth, both the remote kinetic energy of adatoms and the latent heat release during adatom condensation onto the surface cause a continuous increase in system temperature. To mimic the isothermal growth conditions that are usually used in reality, an intermediate region that is above the fixed region but several monolayers below the surface is maintained at a desired substrate temperature by adding Nose–Hoover dragging forces [85] to the atoms. This leaves a free surface zone that allows realistic simulations of events induced during adatom impacts. It also naturally introduces a heat conduction zone through which the energy created at the surface during impacts is transported to the temperature controlled region where it is then removed. This MD model of growth allows the study of effects of adatom incident energy, adatom incident angle, and adatom species on the atomic scale

14 Atomic Assembly of Magnetoresistive Multilayers

517

structures of the multilayers. The approach has one significant deficiency. An accelerated rate of deposition must be used since the real-time covered in the simulation is short (1–10 ns).

14.3 Growth of Metal Multilayers Using a 10 ˚ A thick Ni80 Fe20 substrate crystal containing 120 (22¯4) planes in the x-direction, and 16 (2¯ 20) planes in the z-direction (see Fig. 14.2b), a threedimensional MD model has been used to simulate the [108] direction growth A)/Cu(15 ˚ A)/Co90 Fe10 (40 ˚ A) multilayer unit at an adatom of a Co90 Fe10 (58 ˚ energy of 3.0 eV, a normal adatom incident angle, a substrate temperature of 300 K, and a deposition rate of 1 nm ns−1 . The multilayer stack employs similar materials to those used in experiment [86]. Because the intent is to explore the atomic scale structure of the interfaces, the layer thickness was somewhat arbitrarily chosen and the multilayer structure is not the one that would necessarily give rise to the best GMR properties. An adatom energy of 3.0 eV was used because a direct simulation Monte Carlo analysis indicated that this is the typical energy encountered during sputter deposition in chambers at pressures between 1 and 10 mTorr [51]. The use of normal adatom incidence and substrate temperature of 300 K also mimics the typical low temperature GMR sputter deposition processes where adatoms impact the growth surface with a cosine distribution peaking at normal incidence. The detailed atomic structure of the simulated multilayers is shown in Fig. 14.2b, where Ni, Fe, Co, and Cu atoms are marked green, yellow, blue,

Fig. 14.2. A comparison of (a) three-dimensional atom probe image and (b) a molecular dynamics simulation of the growth of permalloy/copper/permalloy multilayers on Ni80 Fe20 (111) substrates. (c) Example of a simulation where the atom impact energy is varied as the layers are deposited

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and red, respectively. It is interesting to note that the Cu-on-CoFe interface appears relatively sharp, whereas the CoFe-on-Cu interface is quite diffuse. A significant number of Cu atoms are seen within the top CoFe layer, several atomic planes from the nominal CoFe-on-Cu interface. This indicates that Cu incorporation in the subsequently deposited CoFe layer is much greater than in the previously deposited CoFe layer. Atomic configurations like the one shown in Fig. 14.2b have also been directly measured using a three-dimensional atom probe (3DAP) approach [87]. Such a 3DAP technique has been used to obtain the atomic configuration A)/Co90 Fe10 (40 ˚ A)/Cu(30 ˚ A)/Co90 Fe10 (40 ˚ A) multilayof the Ni82 Fe18 (50 ˚ ers [58]. The result of the 3DAP experiment for a film grown under similar conditions to that simulated is shown in Fig. 14.2a. A remarkable similarity is found between the simulated and experimental images. For instance, the roughness (amplitude and wavelength) for both the Cu-on-CoFe interface and the CoFe-on-Cu interface seen in Fig. 14.2b is very close to that of the experiment in Fig. 14.2a. The experiment and simulation both indicate that the Cu-on-CoFe interface is relatively sharp while the CoFe-on-Cu interface is diffuse, and that Cu atoms are mixed in the subsequently deposited CoFe layer. Additional MD model simulations have been performed on a simple Ni/Cu system to establish the functional dependence of the interfacial roughness and interlayer mixing on the energy of the depositing atoms. Ni substrate crystals consisting of 120 (22¯ 4) planes in the x-direction, 3 (111) planes in the y-direction, and 16 (2¯ 20) planes in the z-direction were used. Alternative 20 ˚ A Cu and 20 ˚ A Ni was deposited on the substrate at different adatom incident energies Ei between zero and 5.0 eV, but fixed substrate temperature of 300 K, normal adatom incident angle, and deposition rate of 10 nm ns−1 . The simulations reveal a strong dependence of atomic scale structure upon the incident adatom energy. Figure 14.3 shows a typical set of results, where the Cu and Ni atoms are marked by light and dark spheres, respectively. It can be seen that at low incident energies (∼0.1 eV or less) typical of either thermal evaporation (e.g., MBE) or high pressure sputtering (e.g., diode sputtering), the interfaces exhibit both significant roughness and Cu mixing in the subsequently deposited Ni layer. Interestingly, little mixing of Ni is seen in either of the deposited Cu layers. Increasing the incident energy from 0.1 to 5.0 eV significantly reduces the roughness of both the Cu-on-Ni and the Nion-Cu interfaces. Examination of Fig. 14.3 reveals that the Ni-on-Cu interface appeared to be flatter than the Cu-on-Ni interface at incident energies below 1.0 eV, but when the incident energy reached 3.0 eV, the roughness difference appears to be reversed. Increasing the incident energy from 0.1 to 1.0 eV also reduced the interlayer mixing of Cu in the Ni layer, but as the incident energy was increased to 3.0 eV and above, the mixing began again to increase. Again Cu atoms were dispersed in the subsequently deposited Ni layer much more significantly than Ni atoms were dispersed in the subsequently deposited Cu layer.

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Fig. 14.3. Molecular dynamics simulations of the deposition of Cu/Ni/Cu layers on a (111) nickel substrate using different vapor atom incident energies

Since both the amplitude and wavelength of interfacial roughness are expected to influence the GMR ratio, an interfacial roughness parameter, r1 , was defined that depends both on the height and the width of a surface asperity [60]: n  hil +hir r1 =

i=1

n 

2

,

(14.28)

wi

i=1

where hil and hir are, respectively, the height measured from the left and right sides of the ith asperity, and wi is the width of this asperity (see Fig. 14.4). Summation is conducted over the n asperities in the x-direction. The value of r1 as a function of incident energy is plotted in Fig. 14.5a for both the Cu-on-Ni and the Ni-on-Cu interfaces. Figure 14.5a shows that the interfacial roughness initially decreased with increasing incident energy, reached a minimum between 2.0 and 3.0 eV for the Ni-on-Cu interface and at about 4.0 eV for the Cu-on-Ni interface, and then began to rise as the incident energy was increased beyond 4.0 eV. When surface roughness is experimentally measured by atomic force microscope, the average deviation from the mean height of local asperities, r2 , is usually obtained [88]. This definition ignores the (lateral) wavelength of

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Fig. 14.4. Definition of parameters used to define the surface roughness of a thin film

asperities. However, for sinoidal asperities with an asperity wave length w, r2 , and r1 are simply related by wr1 . (14.29) r2 = π The r2 surface roughness parameter is also shown (on the right coordinate) in Fig. 14.5a. Kools [88] has experimentally measured the r2 surface roughness of Ni80 Fe20 /Cu/Ni80 Fe20 exchange-biased spin valve, and his results are shown in Fig. 14.5b. The samples used in Kools’ experiments were prepared by magnetron sputtering with a background Ar pressure between 1 and 10 mTorr and a (constant) target substrate distance of 109 mm. Using MD predictions of the energy spectrum of sputtered Cu and a binary collision model for Cu transport in an Ar atmosphere [51], the average incident atom energy was estimated to be about 0.1 eV at 10 mTorr and about 4.0 eV at 1 mTorr. These energies and the equivalent pressures are marked in Fig. 14.5. It can be seen that the decrease of roughness with incident energy in the low (thermal) energy range, the existence of a minimum roughness in the intermediate energy range, and an increase of roughness with incident energy in the high energy range were all also observed in the experiments. Since Cu mixing in the Ni layer is significant, a parameter was defined to quantify it. The approach used here is based upon an identification of the atomic neighbors of each atom. If a Cu atom is isolated on a Cu/Ni interface, it must have at least three Cu nearest neighbors in order for it to be attached to a cradle from the Cu side of the interface. In an fcc structure, each atom has 12 nearest neighbors, and so a Cu atom located on a perfect (flat {111}) Cu−Ni interface will have nine (or less) Ni nearest neighbors. The only way for a Cu atom to have more than nine Ni neighbors is for it to be dispersed in a Ni layer. A more strict criterion for identifying a Cu atom mixed in Ni is that it has more than ten Ni nearest neighbors. A mixing probability of Cu, p, can therefore be defined as the number of the Cu atoms with at least ten Ni nearest neighbors divided by the total number of the Cu atoms deposited. The calculated mixing of Cu as a function of incident energy is shown in Fig. 14.6a. It can be seen that Cu mixing initially decreased with increasing

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Fig. 14.5. Comparisons between the simulated and experimentally measured surface roughness of NiFe/Cu/NiFe multilayers

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Fig. 14.6. Comparisons between simulated and experimentally deduced copper distributions in a NiFe overlayer

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incident energy, reached a minimum at an incident energy between 1.0 and 2.0 eV, and then increased significantly with increasing incident energy. Serious Cu mixing (p = ∼1.7%) occurred when the incident energy approached 5.0 eV. A direct experimental measurement of the Cu mixing probability, p, has not been made. However, in Kools’ experiments with exchange-biased spin valve based upon Ni80 Fe20 /Cu multilayers [88], the extent of Cu mixing into a Ni rich permalloy layer was related to a thickness, td , within which the permalloy lost its magnetism due to Cu contamination. Values of 2td were measured as a function of Ar background pressure [88] and are reproduced in Fig. 14.6b, where the Ar pressure range has been converted into an estimated incident energy range as described above. The predicted trend of Cu mixing with incident energy is clearly seen in the 2td data of Kools’ experiment. In particular, the existence of a minimum mixing at a pressure of about 8 mTorr (near 1 eV) is in good agreement with the simulations. The experimental data in Fig. 14.6b also indicate that the degree of mixing after high energy deposition is significantly greater than after a low energy deposition as seen in the MD simulations. It is clear from Fig. 14.3 that Cu mixing was caused by the migration of previously deposited Cu atoms into subsequently deposited Ni layers. Earlier Monte Carlo calculations and a variety of experiments [89–92] have shown a strong tendency for Cu to segregate on the surface of Cu−Ni alloy. This is understandable because the surface energy of Cu is lower than that of Ni [93] and the larger atomic size of Cu releases the surface “tension” stress. Both effects favor the segregation of Cu onto a growth Ni surface. These segregated Cu atoms can be gradually trapped in the Ni layer, resulting in regions of mixing near the Ni-on-Cu interface. This process is significant for low deposition rate, high substrate temperature deposition where atomic hopping is dominant. However, during the low temperature, high rate deposition simulated, thermally activated atomic hopping was kinetically constrained, and a purely thermal diffusion mechanism was unable to significantly disperse Cu into a Ni layer. Also a purely thermal diffusion mechanism is unable to rationalize the observed dependence of mixing upon the impact energy of atoms (Fig. 14.6). As a result, the surface segregation cannot fully account for the Cu mixing observed. Stop-action MD simulations indicated that the mixing of Cu in subsequently deposited Ni layers resulted from the ejection of Cu atoms from a Cu layer during energetic Ni atom impacts [60]. The probability of this exchange mechanism increases with the adatom energy, and decreases with the perfection of the surface. It has been found that the probability for a Ni adatom to exchange with a Cu surface atom is always much higher than that for a Cu adatom to exchange with a Ni surface atom. MD simulation indicated that when the surface element has a low cohesive energy and a large lattice interstice, it is more likely that the impacting atom can penetrate the surface and

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cause the exchange. The impact of a Ni atom on a Cu surface is then much more likely to eject a Cu atom than vice versa. The exchange mixing mechanism can be used to now rationalize the dependence of mixing upon incident energy observed in Fig. 14.6. At low incident energies, the growth surfaces are rough and defective, and exchanges are therefore able to occur relatively easily. As a result, Cu mixing was moderately high at low incident energies. As the incident energy increases, the surface roughness decreases, and the Cu intermixing was reduced. This trend continued until the incident energy reached 2.5 eV. Above this energy, further surface flattening was small. However, the probability of exchange continued to increase with incident energy, and so the Cu mixing began to increase with energy. The atomistic scale calculations above have indicated that high energy depositions promote flat interfaces, but this is achieved at the expense of mixing by an exchange mechanism. Figures 14.5 and 14.6 indicate that the incident energy for the best combination of interfacial roughness and intermixing lies between 1.5 and 2.5 eV. This appears to be one reason why energetic sputter deposition processes have produced better GMR films than those grown by MBE where the atoms have low energies during impact with surface. Youssef et al. studied the effects of sputtering pressure on the GMR properties of RF diode sputtered [Co(11 ˚ A)/Cu(9 ˚ A)]25 multilayers [30]. Their results indicated that the GMR ratio is relatively low at both low and high pressures, and the highest GMR ratio occurred at an intermediate pressure. In addition, their results also indicated that the lowest saturation field was obtained at an intermediate pressure. These results are in good agreement with simulations. Simulations also indicated that during RF diode sputtering, a fraction of high energy inert gas ions may reach the substrate surface and also promote interlayer mixing [49, 51, 94]. They also indicate that heavier Xe ions usually cause more significant damage (exchange) than Ar ions at a given ion energy. This suggests that during RF diode or magnetron sputtering of GMR multilayers, the use of Ar instead of Xe is preferential. Gangopadhyay et al. [86] explored the effects of sputtering inert gases on the GMR properties of the [CoFe(15 ˚ A)/Cu(dCu )]8 multilayers deposited using DC magnetron sputtering and found that the GMR ratio obtained using Ar sputtering gas was significantly higher than that obtained using the Xe sputtering gas. This result is again corrugates the insights gained from simulations. These studies emphasized the need to reduce high energy inert gas ion flux at the substrate. It can be achieved by optimizing RF power, chamber pressure [51], or the substrate bias voltage. In an IBD process, the adatom energy increases with ion beam gun voltage that accelerates the inert gas ions used for sputtering. Saito et al. [95] measured the GMR ratio of [Ni80 Fe20 (20˚ A)/Cu(50 ˚ A)/Co(10 ˚ A)/Cu(50 ˚ A)]10 multilayers at different Ar acceleration voltages. Their results indicate that the GMR ratio is indeed highest when an intermediate Ar acceleration voltage of 700 eV was used. Direct molecular dynamics simulation of high energy

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inert gas ion impact on the target has shown that both the energy and flux of reflected neutrals are lower for Xe ions than for the Ar ions [49]. Since the high energy reflected neutrals can significantly damage the multilayer, the simulations predict that the use of Xe instead of Ar should produce better GMR multilayers during IBD deposition. Schmeusser et al. [53] measured the GMR properties of the IBD deposited [Co(10 ˚ A)/Cu(dCu )]16 multilayers using Ar and Xe as the working gases. Their results indicate that the saturation field of the multilayers deposited using Xe gases is in general significantly lower than that deposited using Ar gas which again, appears in good agreement with the simulations. The detailed atomistic picture that has emerged from the analyses above enables the exploration of other deposition strategies that might further better both roughness and intermixing. Since flat surfaces are promoted by high energy deposition, high impact energies should be used to terminate the growth of a metal layer (e.g., Ni or Cu). Because mixing decreases as the incident atom energy is reduced, the use of a very low energy to deposit the first few monolayers of a new metal (Cu or Ni) avoids mixing by the exchange mechanism. Once complete coverage by 4–5 monolayers has been achieved, the subsequent deposition of the energetic atoms needed to flatten the layer is much less likely to cause interlayer mixing. It should then be possible to use intralayer energy modulation during growth to promote flatness without inducing interlayer mixing between different metal layers. To test this notion, growth of the multilayers shown in Fig. 14.2b was simulated using the modulated energy scheme. The calculated atomic structure and modulated energy scheme are shown in Fig. 14.2c. It can be seen that the structure is indeed significantly improved. The experiments by Hylton et al. [96] may have come closest to achieving energy modulation deposition for published work to date. In their experiments, different chamber pressures were used to deposit the NiFe and the Ag layers of the NiFe(40 ˚ A)/Ag(40 ˚ A) multilayers using magnetron sputtering. They found that at a constant high pressure of about 25 mTorr, the multilayers had the lowest GMR ratio. Because a low adatom energy was obtained at a high pressure due to background gas scattering, the multilayers thus produced were likely to have a high interfacial roughness. The GMR ratio was significantly increased when the multilayers were deposited at a constant lower pressure of 3 mTorr. This occurred because the low pressure resulted in a relatively high adatom energy, which flattened the interfaces. However, the highest GMR ratio was obtained when a low adatom energy (high pressure) was used to deposit the NiFe layer on Ag surface and a high adatom energy (low pressure) was used to deposit the Ag layer on NiFe surface. As will be discussed below, the NiFe(40 ˚ A)/Ag(40 ˚ A) system is special in that Ag has a very strong tendency for surface segregation and the deposition of NiFe on Ag is associated with a very high probability for the exchange. The use of a low energy to deposit NiFe on Ag becomes critical in reducing the mixing of

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Ag into the subsequently deposited NiFe layer. This experiment can certainly be well explained by the simulations. While GMR effects were first discovered for simple A/B/A systems [7, 8], significant improvement has been achieved by introducing additional elements in the multilayers [31]. Hyperthermal impacts may flatten the interfaces, but its use is limited by the interlayer mixing. If there is a third element C that can always segregates on to the surface of the A/B/C system and the A and B elements deposited on C can always be reconstructed to flat configurations, then an addition of C to the system may significantly reduce the interfacial roughness of the multilayers. An element C that has such an effect is called here a surfactant. To explore this idea, the MD approach was used to simulate multilayer dispositions with the addition of elements that might act as surfactants. The NiCo/Cu multilayer system was chosen for study because this simple system has practical applications as spin valves. The same crystal geometry shown in Figs. 14.2 and 14.3 was used to simulate the growth of the Ni75 Co25 /Cu/Ni75 Co25 multilayers in the [108] direction. A substrate temperature of 300 K and hyperthermal adatom energy of 1.0 eV typical of sputtering deposition processes were used. Simulations were first carried out at a normal incident atom angle of 0◦ . The resultant atomic structure is shown in Fig. 14.7a. It can be seen that the use of moderate hyperthermal energy deposition at normal incidence results in interfaces in the Ni75 Co25 /Cu/Ni75 Co25 multilayers that are relatively smooth. However, the adatom incident angles in real sputtering deposition processes are hardy all normal. They typically satisfy a cosine angular distribution where a significant fraction of adatoms makes impact at oblique incident angles. To induce significant interfacial roughness so that the surfactant effects can be clearly revealed, we simulated growth of the same multilayers using an oblique adatom incident angle of 45◦ . The

Fig. 14.7. (a) Molecular dynamics simulation of a Cu/NiCo/Ni multilayer on a NiCo substrate. The atoms impacted at normal incidence to the surface with an impact energy of 1.0 eV. (b) Atomic structures of simulated multilayers grown using oblique deposition. The strong flattening effect associated with the addition of a small number of silver atoms can be clearly seen in (c)

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resulting atomic configuration is shown in Fig. 14.7b. It can be seen that both the Cu-on-Ni75 Co25 and Ni75 Co25 -on-Cu interfaces are rougher than the ones in Fig. 14.7a. The roughness is even more significant on the Cu surface. This is caused by shadowing effects during oblique angle deposition. We have explored the effects of adding Ag and Au into the system. The growth of the Ni75 Co25 /Cu80 Ag15 Au5 /Ni75 Co25 multilayers under the oblique incident angle condition is shown in Fig. 14.7c. It indicates that the roughness of the interfaces and the free surface is dramatically reduced. Interestingly, it can be seen that the Ag has segregated to the surface. No Au surface segregation is seen in Fig. 14.7c. More detailed simulations with Ag and Au added separately have verified that the surface flattening effect of Au is minor compared to that resulting from Ag. Magnetron sputtering has been used to deposit Co/Cu multilayers with and without Ag by Yang et al. [97] and with and without Au by Egelhoff et al. [98]. In the experiments by Yang et al., in situ X-ray photoelectron spectroscopy (XPS) indicated that Ag atoms float out to the surface during successive growth of Co and Cu layers. Their TEM analysis further indicated that Ag leaves behind smooth interfaces without pinholes. These effects resulted in an order of magnitude increase in GMR ratio compared to Ag-free samples [97]. In the experiments by Egelhoff et al., no Au surface segregation was observed and Au was shown to slightly reduce the GMR ratio [98]. We have also experimentally explored the effects of surfactants. A Randex Model 2400-6J diode sputtering system was used to deposit Cu and Cu80 Ag15 Au5 films on silicon wafers at an Ar pressure of 20 mTorr and a plasma power of 175 W. The AFM surface morphology as a function of film thickness for two typical samples with and without the addition of Ag and Au are shown, respectively, in the left and the right columns of Fig. 14.8. The films with Ag and Au added are seen to be smoother. The RMS surface roughness for the two samples was measured and the results are shown in Fig. 14.9. It can be seen that the surface roughness increased as the film thickness increased. However, over the entire film thickness range explored, the surface roughness was always significantly lower for the films grown with the Ag-Au surfactant. Detailed AFM analysis of a surface profile along a cross section going through the black dots in Fig. 14.8d indicated that these black dots are pinholes with an average depth possibly well above 40 ˚ A. Scanning Auger microscopy has been used to analyze the surface compositions of the films. A focused ion beam technique was used to etch the sample to different thickness so that a surface composition vs. film thickness profile could be obtained. Figure 14.10 shows the Cu, Ag, and Au composition profiles along the thickness of a Cu80 Ag15 Au5 film. Figure 14.10 clearly shows a very strong Ag surface segregation effect. The significant interfacial roughness in the Ni75 Co25 /Cu/Ni75 Co25 multilayers shown just above is anticipated to cause a significant deterioration of the GMR properties. Pinhole formation in Cu layers seen in the simulation would also allow the two adjacent ferromagnetic layers to be connected which in turn

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Fig. 14.8. Surface roughness maps for Cu thin films grown on silicon substrates with and without Ag−Au surfactant additions

would prevent the magnetic moment of either magnetic layer from being easily rotated. To investigate these suppositions, both Ni75 Co25 /Cu/Ni75 Co25 and the Ni75 Co25 /Cu80 Ag15 Au5 /Ni75 Co25 multilayers were deposited using RF diode sputtering and their GMR properties were measured. The GMR ratio and saturation field as a function of the number of GMR sandwich repeats are shown in Fig. 14.11a, b for the two types of multilayers. Figure 14.11 verifies that the addition of Ag and Au in the Cu layer significantly improves the GMR properties, due to the improved interfacial structures and the elimination of pinholes in the Cu layers. Simple two-dimensional EAM calculations of the energetics of the Cu−Ag−Au system sufficiently helped better understanding of the mechanisms responsible for strong surfactant effects [59]. These calculations indicate that the exchange between a Ag atom on a Cu surface and an underlying Cu atom resulted in an increase of energy. The exchange of an Au atom on a Cu surface with an underlying Cu atom, however, decreased energy. As a result,

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Fig. 14.9. Surface roughness of copper thin films grown with and without the presence of a Ag−Au surfactant

Fig. 14.10. Compositional profiles for a copper film grown using a Ag−Au surfactant. Strong surface segregation exits at the top face surface (right) of the film

Ag atoms tend to stay on the surface while Au atoms are easily buried in the bulk. While it appeared that lower surface energy materials can segregate to the surface, further analysis of the atomic structures has indicated that other factors could affect surface segregation. For instance, embedding a bigger atom into a lattice composed of smaller atoms was found to be energetically unsta-

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Fig. 14.11. Effect of surfactant additions to the GMR transport properties of permalloy/Cu/permalloy multilayers

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ble. If the bigger atom was embedded at a near surface site, it could be easily exchanged with an atom on the surface. Ag has a size significantly bigger than that of Cu, Ni, and Co, Ag hence segregates to the surface. For atoms of the same size, the critical factor for surface segregation was found to be the cohesive energy of the atoms. During energetic impact, the incoming atom more easily penetrates a lattice of lower cohesive energy atoms. This then results in an exchange between the incoming atom and an underlying atom, leaving the lower cohesive energy atom at the surface. Thermodynamically the surface segregation of lower cohesive energy atoms on the surface (breaking some bonds) also incurs less of an energy penalty than that of more strongly bonded atoms. Ag and Au are of the same size, but Ag has a much lower cohesive energy [93], it should surface segregates. Likewise, Ni and Cu have about the same size, but since Cu has a much lower cohesive energy [93], so Cu segregates to the surface of Ni [89–92]. Compared with Cu, Au, Ni, and Co, Ag has a lower cohesive energy and relatively big size, so Ag has a strong tendency to surface segregate. These results suggest that during the deposition of multilayers, Ag atoms continuously segregate to a surface. Other metal atoms deposited on this Ag-rich surface can then quickly exchange with Ag atoms. These calculations also indicated that Ag atoms have lower energy barriers for surface diffusion than Cu, Ni, or Co atoms. As a result, Ag atoms can rapidly migrate to lower energy ledge, kink and pit sites on a surface. The presence of these Ag atoms was found to reduce the Schwoebel barriers for other atoms on top of the terrace and to greatly increase the rate at which they could hop down to the ledge sites. Consequently, Ag promoted the step flow growth that resulted in much smoother surfaces. Furthermore, the energetic impact induced flattening more easily occurs on a surface composed of more mobile atoms. The hyperthermal energy processing conditions that can lead to smooth growth are readily accessible if an Ag-rich surface is formed during deposition. The efficient smoothing of local surface asperities also greatly reduces the probability of surface shadowing during oblique deposition which synergistically enhances the effects of the surfactant. Experiments show that multilayers using Ag as the spacer do not exhibit good GMR properties [99]. This is at first surprising since Ag is highly conductive. The results discussed above suggest that this anomaly might be a consequence of Ag segregation to the surface as other materials are deposited on it. This would make the thickness of an Ag spacer extremely difficult to control during multilayer deposition. Extensive mixing of Ag into the subsequently deposited magnetic layer is also likely to occur. In the worst case, this results in a thin layer of the magnetic layer near the interface completely losing its ferromagnetism, forming the so-called “dead layer.” While Ag is a good surfactant candidate, it does not appear to be a good choice for the spacer layer material. Good surfactants must be able to both segregate to the surface and to reconstruct surface asperities into flat configurations. Ideally, their presence

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at the surface and in the interior of the layers should not affect transport properties. Atomistic simulations have shown that mobile materials present on the surface are likely to flatten the growth surface. These materials usually have a low surface energy, a low bonding energy, and a low melting temperature. Elements that have relatively large atomic sizes also favor segregation to a surface to reduce the energy. Roughly speaking, elements with low melting temperature and relatively large size may be the most promising surfactants candidates. This includes metals such as Ag, Pb, Sb, Ga, etc.

14.4 Ion-Assisted Growth of Metal Multilayers When multilayers are grown with many layers, it becomes increasingly difficult to keep an atomically flat growth surface during deposition. Ion-assisted deposition is often used to resolve this problem. During RF diode sputtering, a substrate bias voltage can be applied to induce low energy inert gas ion impacts with the growth surface while in IBD, a secondary ion beam gun can be used to inject inert gas ions toward the substrate surface. Atomistic simulations have been applied to explore ion assistance schemes for growing Ni/Cu [100–102], Co/Cu [103–105], and Ta/Cu [106] multilayers. MD analysis of inert gas ion impact effects during the growth of Ni/Cu/Ni multilayers [100–102] indicates that the energy of the assisting ions must be very low ( 0.02) after a high ion fluence of 0.5 ions ˚ A . It should be pointed out that values of the flattening and mixing critical energies depend on the designated fluence, but this dependence is relatively weak when the −2 fluence is 0.5 ions ˚ A or above. The effects of ion incident angle have been explored in detail. To uniquely define the incident angle, the impact ions were assumed to move in the (1¯10) plane and the angle θ was measured from the [¯1¯1¯1] direction. To examine the role of incident ion angle, critical energies for flattening and mixing were calculated as a function of the ion incident angle for the different ion species and the results are shown in Fig. 14.15 for both flattening and intermixing. It is evident from Fig. 14.15 that it is relatively more difficult to find an ion energy that can cause flattening without mixing when ions strike

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Fig. 14.13. The effects of ion impact energy upon the surface roughness and intermixing of a Co-on-Cu interface

the surface at a normal incident angle. Flattening without mixing can be more easily achieved when a higher ion incident angle is used. The results described above assume that the ion assistance occurs concurrently with deposition of Co on Cu. During this simultaneous ion assistance, it is difficult to avoid mixing. Better ion assistance strategies are possible. One is to use modulated ion assistance where the first half of a new material layer is grown without ion assistance while the remainder of that layer is grown with ion assistance. Because the interface is already buried below the surface, the ion impacts are less likely to cause mixing. Another strategy is sequential ion assistance, where the entire layer is grown without ion assistance. Upon the completion of a layer, ion irradiation is then introduced to flatten the surface

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Fig. 14.14. The effect of ion fluence upon the surface roughness and intermixing at a Co-on-Cu interface

before another layer is deposited. This strategy is even less likely to result in mixing. Direct MD simulations of the Ar ion-assisted growth of the Cu/Co/Cu multilayers were carried out to explore the three ion assistance strategies. The model system analyzed consisted of a Cu substrate made up of 72 (22¯4) planes in the x-direction, eight (111) planes in the y-(growth) direction, and 42 (2¯ 20) planes in the z-direction. A Co layer and then a Cu layer were sequentially deposited onto the surface at a substrate temperature of 300 K, a deposition rate of 1 nm ns−1 , an adatom energy of 0.5 eV, and an adatom incident angle of 40◦ . An oblique rather than normal adatom incident angle was used to promote the formation of a rough surface so that the effects of ion assistance could be revealed. Ion/adatom flux ratios of two and four were used for simulations of simultaneous and modulated energy ion assistance schemes, −2 respectively. An ion fluence of 0.5 ions ˚ A was used to assess sequential ion assistance scheme. The atomic configurations obtained from the three

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Fig. 14.15. The effect of the ion incident angle upon the surface flattening and intermixing critical energies

deposition strategies are shown in Fig. 14.16, where Cu, Co, and Ar are marked by white, gray, and black spheres, respectively. Figure 14.16a shows that without ion assistance (ion energy of zero), the Cu-on-Co interface is very rough. When 8 eV, normally incident Ar ion assistance was used (Fig. 14.16b), the interfacial roughness was significantly reduced. However, when the Ar ion energy was increased to 20 eV (Fig. 14.16c), significant interlayer mixing occurred at the Co-on-Cu interface and some mixing even occurred at the Cu-on-Co interface. Our simulations over a wide variety of ion incident energies and incident angles indicate that the lowest combination of roughness and mixing occurs at an ion energy of 10 eV and an ion incident angle of 50◦ .

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Fig. 14.16. Atomic structures of Cu−Co−Cu multilayers grown in (a) without ion assistance. (b) and (c) show the effect of increasing the ion energy during deposition. The structure shown in (d) was made by using ion assistance during only the last half of each layer. The structure shown in (e) used an ion treatment to smooth each layer after its deposition was complete

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Fig. 14.17. The effects of ion energy upon the structure of copper films deposited on Ta using simultaneous ion assistance

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Fig. 14.18. The effects of ion energy upon the structure of copper films grown on Ta using modulated and sequential ion assistance schemes

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The multilayer structure shown in Fig. 14.16d was obtained using modulated ion assistance, where the first five atomic layers of Co were deposited without ion assistance, and the remainder of the Co layer was deposited using an ion assistance at 20 eV. Flat interfaces without mixing were obtained. The multilayer structure shown in Fig. 14.16e was obtained using sequential ion assistance, where complete metal layers were first deposited without ion assistance, and then ion impacts with a relatively high energy of 30 eV were used −2 to flatten the finished metal surfaces at an ion fluence of 0.5 ions ˚ A . The results show that the sequential ion assistance scheme resulted in a multilayer structure with planar interfaces and little interfacial mixing even when a moderately high ion energy of 30 eV was used. The above analysis applies to Co/Cu system. MD simulations were also used to study the growth of low energy Ar ion-assisted deposition of Cu on Ta [106]. Here both metals and ions were assumed to impact the growth surface at an incident angle of 45◦ . The effects of ion energy and ion to metal ratio on film packing density, surface roughness, and Cu/Ta intermixing are shown in Fig. 14.17 for simultaneous ion assistance scheme. Figure 14.18 shows film density, surface roughness, and interlayer mixing as a function of argon ion energy for modulated energy and sequential ion assistance schemes. During the modulated energy ion assistance simulations, the first half of the Cu layer was deposited without ion assistance and the remainder was deposited using an ion/metal flux ratio of 6. During the sequential ion assistance simulations, −2 an ion fluence of 1.0 ions ˚ A was applied after completion of deposition of the Cu layer. These results indicate that increasing the ion energy and ion to metal ratio can increase the film density and reduce the growth surface roughness, but it causes an increase in interlayer mixing. The modulated energy and sequential ion assistance schemes delay the occurrence of the intermixing to higher ion energy levels. As a result, higher ion energies can be used to improve film density and surface smoothness without causing significant mixing. Several groups have tried to explore low energy ion-assisted deposition approaches and have shown significant ion assistance effects upon thin film structures [107]. We utilized the above simulations to recently help develop a biased target ion beam deposition (BTIBD) system [108,109] and have used it to explore the low energy ion-assisted growth of model Ta/Cu bilayer films at room temperature. The BTIBD system utilized a similar low energy ion source consisting of an end-Hall ion source and a hollow cathode electron source for both sputtering and ion assistance. These ion sources reliably produce a high density, very low ion energy (up to 100 eV) inert gas ion beam [110]. When a negative bias voltage is applied to the sputtering target, the inert gas ions are attracted and make a near-normal incidence with the target surface. An advantages of this technology is that it enables an easily controlled low sputtering energy and it creates a low flux of reflected neutrals.

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Fig. 14.19. Effect of argon ion energy upon the roughness and resistivity of copper films deposited on a Ta surface

Model 5 nm Ta/30 nm Cu films were deposited on 6-in. silicon wafers using the three ion assisting methods (simultaneous deposition with ion assistance, modulated ion assistance, and sequential ion assistance). An oblique incident angle of θ = 45◦ was used for the assisting argon ions. All samples were grown at a base pressure of 3×10−8 Torr, an Ar working pressure of about 4.5×10−4 Torr, and a target bias voltage of 300 V. Different assisting ion energies and ion fluxes were investigated. The surface roughness and in-plane electrical resistivity of films grown using simultaneous ion assistance at an ion to metal ratio of about four are shown as a function of the assisting ion energy in Fig. 14.19a, b. It can be seen from Fig. 14.19a that the film surface roughness initially decreased as the ion energy was increased from 0 to 7 eV, and then remained constant with further increases in ion energy. This is in good agreement with simulations (Fig. 14.17b). Figure 14.19b shows that the film electrical resistivity at first decreased as the ion energy was increased from 0 to 10 eV but then started to increase as the ion energy was further increased. Many studies [111, 112]

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Fig. 14.20. Effect of ion flux upon the surface roughness and resistivity of Cu films deposited on Ta surfaces

have shown that the resistivity of thin films increases due to defects such as vacancies and voids [113], or impurity atoms [114] like those introduced by intermixing at film interfaces. Surface and interfacial roughness can also contribute to increases in resistivity [114–118]. It is therefore anticipated that the resistivity changes observed in the experiments are related to changes in surface (and interfacial) roughness, the film’s packing density (vacancy and void content), and alloy scattering due to interlayer mixing. The initial drop in resistivity is consistent with a decrease in roughness while the subsequent resistivity rise may be a manifestation of mixing at the Cu on Ta interface. The measured film roughness and resistivity as a function of the assisting ion flux at an assisting ion energy of 10 eV, are shown in Fig. 14.20. It can be seen from Fig. 14.20a that an increase of ion to atom ratio from 0 to 8 resulted in an ∼50% reduction in surface roughness. The roughness then reached a near plateau as the flux ratio was increased above 10. This is again in good agreement with the simulation studies. Figure 14.20b shows that the film resistivity initially decreased as the flux ratio was increased to around 6. A further increase in the ion flux then caused an increase in film resistivity.

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Fig. 14.21. Copper concentration profiles at a Cu on Ta interface for films grown with various ion assistance strategies

The initial decrease in resistivity is believed to result from impact flattening which is accompanied with reduction in defects (Fig. 14.12). The subsequent increase can again be explained by the increase in intermixing. To further verify that intermixing indeed occurs, Auger electron spectrometry was used to measure the film composition as a function of depth for samples grown using the simultaneous ion assistance scheme. The results are shown in Fig. 14.21 for three ion assistance conditions: (a) no ion assistance, (b) an ion energy of Eion = 10 eV and an ion to metal flux ratio of 5, and (c) an ion energy of Eion = 20 eV and an ion to metal flux ratio of 14. The data reveal that the sharpest interface with a minimal combination of interfacial roughness and interlayer mixing was obtained using an intermediate ion energy of 10 eV and an ion to atom ratio of 5. Direct TEM observations of the structures grown above (Fig. 14.22) also indicate that the sample grown with 10 eV ion assistance had the sharpest interface. The prediction that the low energy ion-assisted deposition can improve the interfacial structures has also been experimental verified in other experiments. A group at Veeco investigated the effects of ion assistance on the transport properties of GMR stacks [119]. They found that ion energies in the range 10– 60 eV significantly improved the GMR ratio of their films. Kools also found that slight increases in kinetic energy (>5 eV) could lead to biased diffusion and the suppression of defect formation [120]. Another example is the growth of X-ray mirror multilayers that involve many repeats (400) of Cr/Sc bilayers with the thickness of the layers similar to that of GMR multilayers (around 10 ˚ A) [121]. Flat sharp interfaces are critical for a high reflectivity X-ray mirror. Low energy ion-assisted deposition was found to significantly improve the quality of such multilayers [121]. The ion energy for the best performance device was reported to be around 9 eV [121].

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Fig. 14.22. Transmission electron micrographs of Cu films grown on Ta surfaces using various low energy, ion assistance strategies. Film (b) had the highest quality interface

14.5 Dielectric Layer Deposition The computational substrates used for oxidation are similar to those used for vapor deposition. However, instead of injecting adatoms toward the growth surface, oxidation is simulated by introducing an atomic oxygen vapor above the metal surface. The oxygen vapor is maintained at a constant pressure and a constant temperature. To maintain a constant vapor pressure, the number of oxygen atoms in the vapor zone is monitored. Once oxygen atoms are found to have condensed on the metal surface, they are refilled in the vapor zone. The growth of an AlOx tunnel barrier layer on a fcc Ni65 Co20 Fe15 substrate can be simulated using the MD methods with a charge transfer potential approach described earlier. An initial fcc Ni65 Co20 Fe15 single crystal substrate was made from 120 (22¯ 4) planes in the x-direction, 3 (111) planes in the ydirection, and 16 (2¯ 20) planes in the z-direction. It was created using the equilibrium (bulk) lattice parameter (a = 3.604 ˚ A) for fcc Ni65 Co20 Fe15 . The usual MD simulation approach was then used first to deposit approximately six atomic layers of Ni65 Co20 Fe15 on the initial (111) substrate surface. This

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Fig. 14.23. Atomic structures showing the effect of aluminum metal layer thickness upon morphology of reactively grown AlOx layers (a, b). (c) Shows the formation of a hole in the aluminum oxide layer and its self-healing as oxidation progresses

was followed by the deposition of aluminum layers of various thickness. The metal vapor atoms were injected perpendicular to the growth surface, the substrate temperature was maintained at 300 K, and the deposition rate was 10 nm ns−1 as films were grown at various adatom energies between 0.1 and 5.0 eV. The geometry of the system can be seen in Fig. 14.23a, b. Examples of aluminum films grown on Ni65 Co20 Fe15 crystals can be seen on the left of Fig. 14.23a, b. They were obtained by depositing the Ni65 Co20 Fe15 layer at a 4.0 eV adatom energy and the aluminum layer at a 0.2 eV adatom energy. Increasing the adatom energy during deposition of the permalloy layer resulted in much smoother Ni65 Co20 Fe15 surfaces consistent with earlier findings with metal/metal multilayers [58, 60]. Relatively smooth aluminum surfaces were always obtained regardless of its adatom energy. Further analysis indicates that this occurs because aluminum has a low Schwoebel barrier [122] facilitating the step flow mode of growth. In addition, aluminum has a relatively low cohesive energy (3.58 eV) compared to Ni (4.45 eV), Co (4.41 eV), and Fe (4.29 eV). As a result, the binding between aluminum and the underlying Ni, Co, Fe atoms (with higher cohesive energies) is stronger than the binding between the aluminum atoms themselves. This also promotes wetting of aluminum on the Ni65 Co20 Fe15 surface. However, epitaxial

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aluminum growth was complicated by the large lattice mismatch (∼15%) between aluminum and Ni65 Co20 Fe15 . Misfit dislocations are formed in the bilayer system by mechanisms similar to those recently analyzed in detail for the (111) gold/permalloy system [123, 124]. Oxidation was initiated by introducing an atomic oxygen atmosphere above the Al/Ni65 Co20 Fe15 multilayer. To ensure that a sufficiently thick oxide layer formed within the real time simulated (∼1 ns), both a high oxygen −3 vapor temperature (8,000 K) and vapor density (0.0003 oxygen atoms ˚ A ) were used. This corresponds to a vapor pressure of ∼12 atm. of pure (atomic) oxygen. The heat of the Al + O reaction [125] released about 8.4 eV per atomic oxygen atom that reacted with the aluminum crystal. The thermostatically temperature controlled molecular dynamics algorithm [85] conducted this thermal energy away from the surface and maintained a surface temperature close to 330 K during oxidation even though the heat of reaction and atomic oxygen collision rate with the solid surface were both very high. The atomic configurations of two typical oxidized aluminum layers whose thickness prior to oxidation was about 12 and 2.5 ˚ A are shown on the right of Fig. 14.23a, b. The formation of amorphous aluminum oxide layers was observed in all the simulations as expected from experimental observations [126]. The composition of the AlOx oxide layer increased from x = 0 before oxidation to x = 1.55 ± 0.05 by the time oxidation ceased. The surface morphology of the AlOx layers was found to be highly sensitive to the aluminum layer thickness prior to oxidation. It can be seen from Fig. 14.23a that the oxidation of the thicker aluminum layer resulted in the formation of a compositionally uniform, continuous and flat AlOx film. However, when the aluminum layer thickness was reduced to 2.5 ˚ A (Fig. 14.23b), the AlOx film became discontinuous, and in some places much thicker than anticipated even though the aluminum layer prior to oxidation had been continuous in thickness and relatively smooth. Oxidation of the thin aluminum film clearly results in the formation of holes in the AlOx layer, exposing areas of the underlying Ni65 Co20 Fe15 crystal. The AlOx oxide layer roughness could be quantified by taking the maximum difference in the layer thickness and dividing it by the average layer thickness. This relative roughness was calculated for many simulations and is shown in Fig. 14.24 as a function of the average aluminum layer thickness (prior to oxidation). When the aluminum layer thickness exceeded 6 ˚ A, the AlOx oxide layer roughness was low. It can be seen that as the aluminum layer thickness was decreases below 6 ˚ A, the AlOx oxide layer roughness increases rapidly, which corresponds to the formation of pin holes in the AlOx layer. Further simulations using different random number seeds to initiate the runs indicated that at an aluminum layer thickness of between 6 and 10 ˚ A, holes were formed with a probability that increased as the aluminum layer thickness was reduced. For all the simulations carried out at aluminum layer thicknesses above 10 ˚ A,

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Fig. 14.24. Effect of aluminum layer thickness upon roughness of a reactively formed AlOx layer

Fig. 14.25. The effect of the preoxidized aluminum layer thickness upon the magnetoresistance change of a magnetic tunnel junction

smooth oxide layers were obtained and no holes were observed. As a result, we conclude that for the crystal surfaces analyzed here, the critical aluminum layer thickness for smooth oxidation is around 10 ˚ A. This finding appears to be consistent with the experimental observation of the loss of TMR when the preoxidation aluminum layer thickness was decreased below ∼8 ˚ A, Fig. 14.25 [42], and with other experimental studies [43]. To explore the mechanism of hole formation, an initial Ni65 Co20 Fe15 substrate crystal containing 54 (22¯4) planes in the x-direction, 3 (111) planes in the y-(growth) direction, and 32 (2¯20) planes in the z-direction was assembled. About six additional atomic layers of the Ni65 Co20 Fe15 alloy were deposited on the initial substrate using an incident atom energy of 4 eV, a growth temperature of 300 K, and a deposition rate of 10 nm ns−1 . Between two and three

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˚) of aluminum were then atomic layers (corresponding to a thickness of ∼6 A deposited on the Ni65 Co20 Fe15 surface using an incident atom energy of 0.2 eV, a growth temperature of 300 K, and a growth rate of about 1.5 nm ns−1 . Oxidation of the aluminum surface was simulated by exposing the surface at a substrate temperature of 300 K to a high pressure (12 atm.) atomic oxygen vapor held at a high vapor temperature of 8,000 K. Time-resolved atom position images obtained during MD simulation of oxidation are shown in Fig. 14.23c to enable the detailed mechanisms of the oxide layer formation to be examined. It can be seen that after 40 ps of oxidation, oxygen vapor atoms had reacted with the aluminum surface to form aluminum oxide. The initial oxidation was not uniform and an aluminum depleted region developed near the center of the simulated region. This arises because the cohesive energy (eV atom−1 ) in oxides (such as Al2 O3 ) is much higher (more negative) than that of either pure aluminum or pure oxygen [72, 77], and the first nucleated oxide regions then grow by drawing in nearby aluminum atoms. This results in the depletion of aluminum in the nearby surface, leading to exposure of the underlying Ni65 Co20 Fe15 . To access the stability of the aluminum depleted zone, the oxygen vapor was removed and the system was annealed at 600 K for 100 ps as shown in Fig. 14.23c. It can be seen that at least within the simulated timescale, the hole formed was quite stable and changed little during annealing. Oxidation was then resumed. As can be seen, after another 20 ps oxidation at 300 K, the aluminum depleted zone began to shrink. After a total of 80 ps of oxidation, the aluminum depleted zone had completely disappeared and the nickel alloy substrate was more or less covered by the aluminum oxide. To quantify the observations, the surface oxide layer of the system shown in Fig. 14.23c was divided into a 14 × 14 grids. If a grid element contains neither an oxygen nor an aluminum atom, the substrate area beneath the element was considered to be uncovered by either (the unreacted) aluminum or AlOx . The fraction of the grid elements that contained either aluminum or oxygen atoms can then be determined and is referred to as the coverage parameter. An oxygen fraction for a grid element can also be defined as the ratio of the number of oxygen atoms to the total number of oxygen and aluminum atoms in the grid. Let the oxygen fraction for grid element i be XO,i (i = 1, 2, . . . , ng , where ng is the total number of grids containing either aluminum or oxygen atoms). The overall state of oxidation can be represented by the average oxygen fraction ng 1  ¯ XO,i . XO = ng i=1

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The uniformity of the oxide can then be represented by the standard deviation of the oxygen fraction . / ng ¯ O 2 / 1  XO,i − X . σ=0 ¯O ng X i=1

¯ O , and the standard deviation of the oxygen The average oxygen fraction, X fraction, σ, are shown in Fig. 14.26a as a function of oxidation time. The corresponding surface coverage parameter is shown in Fig. 14.26b. Figure 14.26a indicates that during oxidation, the oxygen fraction in the oxide layer gradually increased, eventually approaching a value of 0.7 (corresponding to AlO2.33 ). Formation of this highly oxidized state is consistent with highly reactive oxygen vapor conditions used for the simulation. Figure 14.26a also shows that the deviation in oxygen fraction was greatest early in the oxidation process and decreased with elapsed oxidation time. It indicates that the oxide formed during the earliest stages of oxidation was the least uniform. This occurs at a stage when holes have their highest nucleation probability. Figure 14.26b indicates that the initially fully aluminum covered Ni65 Co20 Fe15 surface gradually became locally uncovered once the oxidation began. This was associated with the formation of holes in the oxide layer such as that in Fig. 14.23c. However, the surface coverage parameter began to recover after an oxidation time of about 50 ps. The surface was once again fully covered (now by AlOx ) after about 90 ps of oxidation, consistent with the healing of the holes. Because the structure changed little during annealing, the rapid elimination of holes such as those shown in Fig. 14.23 appears to be a consequence of continuous oxidation rather than atom diffusion. Unlike annealing, continued oxidation caused increases in both thickness and oxygen composition of the oxide layer. Once oxidation starts, local regions with rich oxygen concentrations randomly form on the aluminum surface due to the random arrival of oxygen atoms on the surface. As has been discussed in the above, these oxygen-rich regions can attract nearby aluminum atoms to form expanding oxide nuclei because aluminum in the oxide has a much lower energy than aluminum alone. When the aluminum layer is very thin, this process can easily cause aluminum depleted zones around the oxygen-rich regions. As a result, holes are most likely to form at the earliest stage of the oxidation. The observation that holes begin to shrink in the fully oxidized layer can be easily rationalized by surface energy arguments. Suppose a round hole with a radius r forms in an AlOx layer with a thickness h, Fig. 14.27. The formation of such a hole results in (1) the creation of additional oxide surface area (on the interior of the hole), (2) the elimination of an oxide surface area (at the top of the hole), and (3) the creation of a Ni alloy surface area and the

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Fig. 14.26. Variation of average oxygen fraction (a) and AlOx coverage (b) with time during the reactive oxidation of aluminum to form an AlOx tunnel barrier on permalloy

elimination of some Ni alloy/oxide interface area (at the bottom of the hole). The resulting energy change, ΔE, can be written as ΔE = 2πrhγAlOx + πr2 (γNi65 Co20 Fe15 − γAlOx /Ni65 Co20 Fe15 − γAlOx ), (14.30) where γAlOx and γNi65 Co20 Fe15 are the surface energies of AlOx and Ni65 Co20 Fe15 , respectively, and γAlOx /Ni65 Co20 Fe15 is the interface energy

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Fig. 14.27. Model circular hole in an AlOx film on a permalloy substrate

between AlOx and Ni65 Co20 Fe15 . Equation (14.30) indicates that holes with small radius, r, and large thickness, h, have a positive energy increase, ΔE. They are therefore unstable and can be healed. This is driven by a competition between the area increases of the aluminum oxide–nickel alloy interface and the area reductions of the aluminum oxide and nickel alloy surfaces. Under the accelerated oxidation conditions required for the short timescale simulations, the transient hole phenomenon was found to occur within a very short time (80 ps) (Fig. 14.23). This means that during experiments where the timescale is significantly longer, the phenomenon is not likely to be constrained by kinetics. As a result, holes are expected to always form during the early stage of oxidation of very thin (say, 6 ˚ A thick) aluminum layer under conditions commonly used in experiments. The healing of the holes is expected to occur when the aluminum layer is relatively fully oxidized. This suggests that the transient holes never form during oxidation of thick aluminum layers because the surface aluminum is always fully oxidized before any aluminum region is completely depleted to initiate a hole. The simulations also suggest that holes cannot be healed during prolonged oxidation of very thin aluminum layers because when the aluminum is quickly fully oxidized, the driving force for the hole healing is saturated and therefore the hole ceases to further shrink. A direct experimental observation of the atomic scale structure of the AlOx -on-Ni65 Co20 Fe15 system has not been found in literature. However, Petford-Long et al have carried out extensive high resolution transmission electron microscopy (HRTEM) and 3DAP experiments to examine atomic scale structure of the AlOx oxide formed from oxidation of 6 ˚ A thick aluminum layer on Co90 Ni10 [127]. Their experiments indicated that with under-oxidation conditions, the AlOx layer exhibits discontinuous islands with significant areas of the Co90 Ni10 surface uncovered. This phenomenon closely corresponds to the formation of big holes in the oxide layer in the under-oxidized samples seen above. Annealing of these under-oxidized samples was found to cause the AlOx islands to spread to form a network along the grain boundaries. Note that

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the experimental annealing involved further oxidation. These experimental observations appear similar to the results of the simulations. The experiments further indicated that more complete oxidation of the aluminum surface produces a more continuous AlOx layer. However, these as-grown films still contained holes on the scale of roughly 10 nm. The oxygen composition in the AlOx layer, on the other hand, was found to be still far below the one defined by the fully oxidized AlO1.5 (Al2 O3 ). These holes were eliminated and a continuous AlO1.5 layer finally formed when the samples were annealed. This suggests that oxygen may have diffused from further away to complete the oxidation and that the layer then laterally spread to fill the holes. Dewetting of a very thin metal layer on another metal surface due to the oxidation is clearly a very potent and perhaps quite widespread phenomenon. Low energy electron microscopy (LEEM) and diffraction (LEED) experiments have been used to study the oxidation of 1–2 flat monolayer’s of chromium deposited on a (100) tungsten surface [128]. The results indicate that the chromium oxide layers become unstable and form three-dimensional islands as the Cr2 O3 was formed at elevated temperatures, T ≥ 790 K.

14.6 Ion-Assisted Reactive Growth of Dielectric Layers The various simulations and experiments discussed above indicate that inert ion impacts with the rough surface of a thin film are highly effective at redistributing the material contained in asperities leading to formation of a flatter surface. This occurs because high energy inert gas impacts break the bonds between the atoms forming the asperities, and when this is performed during deposition, it reduces the probability of island nucleation and the significance of flux shadowing. Because highly constrained kinetic conditions are used in the deposition of metallic multilayers, the broken surface asperities are also slow to reform through thermally activated diffusion, even when these structures are thermodynamically more favorable. There has been significant speculation that ion assistance during the reactive formation of dielectric tunneling barriers might be advantageous. However to our knowledge, little has been reported on the benefits of inert gas ion assistance during the reactive growth of tunnel barrier layers used in TMR structures. It is possible that the idea has not been pursued because of concern that the ion assistance used to flatten the surface of a very thin aluminum oxide layer will also cause damage to the underlying interface. This stems in part from a realization that the cohesive energy of Al2 O3 is about 6.5 eV [72, 77], which is much higher than those of the underlying metals (Al: 3.58 eV, Ni: 4.45 eV, Co: 4.41 eV, and Fe: 4.29 eV). This implies that ion energies sufficient to break the Al2 O3 bonds are likely to cause significant mixing at the underlying Al2 O3 –metal interface since the Al2 O3 layer is very thin. We have also seen that, there is

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a strong tendency for the holes to form when the aluminum oxide thickness is between 10 and 20 ˚ A. In this regime, it is likely that even if the assisting ion impacts flatten the oxide layer without causing damage to the underlying interface, a rough oxide layer surface is likely to quickly reform as the process is driven by energetic considerations and not significantly impeded by kinetic’s at ambient or perhaps even cryogenic temperatures. The simulation methods developed in this chapter allow the beginning of a quantitative assessment of these speculations. The MD approach used to simulate the oxidation of the single-layer aluminum sample shown in Fig. 14.23b can be used for this assessment except that an additional xenon ion flux is injected toward the growth surface translating an ion energy of 8.0 eV. The xenon energy and the xenon to oxygen flux ratio can be easily varied but only results for a flux ratio of 6.67 are discussed here. The atomic configurations obtained before and after the ion-assisted oxidation are shown in Fig. 14.28a, b, respectively. If these are compared with the nonassisted results in Fig. 14.23b it can be seen that a rough AlOx layer has still formed even with an ion assisting energy of 8.0 eV. This is consistent with the very strong dewetting phenomena present in very thin oxide layers. Careful comparisons of Figs. 14.28 and 14.23 indicate that the ion impacts have had some effect and appear to have slightly reduced the height of the AlOx regions and reduced the underlying metal surface area that has been exposed. It should be pointed out that the dewetting effect is most severe for the very thinnest oxide layers. As a result, the observation here does not

Fig. 14.28. Ion-assisted reactive growth of an AlOx layer on a permalloy substrate. Ion assistance has failed to modify the dewetting phenomenon driving the transient hole formation during the oxidation of thin aluminum layers

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exclude the possibility that ion-assisted deposition could be used to more effectively flatten thicker aluminum oxide or other reactively formed films that have a lesser tendency for dewetting. The flattened islands on these thicker films are also less likely to reform under appropriately constrained kinetic growth conditions. An additional advantage for thicker oxide films is that higher ion energies could be used without risk of damaging the underlying oxide–metal interface.

14.7 Conclusions Atomistic simulation approaches capable of accurately predicting atomic scale structures of vapor deposited metal and metal oxide multilayers have been developed. Simulations using these approaches revealed fundamental insights determining the quality of GMR metal multilayers and TMR metal/metal oxide multilayers. Increasing adatom energy is found to both flattens the interface and causes interlayer mixing. As a result, a low roughness/mixing combination is obtained at an intermediate adatom energy. The use of modulated energy deposition, where low energy is used to deposit the first a few monolayers of a new layer to minimize mixing and a high energy is used to complete that layer to flatten the finished surface, is found to further minimize interfacial roughness and interlayer mixing. Surfactants can also reduce interfacial roughness. Low energy ion-assisted deposition is additional means to reduce interfacial roughness. To minimize inert gas impact induced mixing, modulated ion assistance and sequential ion assistance can be used. Due to a strong dewetting phenomenon, pin holes always form during growth of tunneling barrier layers provided that the barrier layer thickness is below ∼6–10 ˚ A. These pin holes cannot be avoided using ion assistance. Smooth uniform barrier layers can be grown when the barrier layer thickness is increased beyond 10 ˚ A.

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Index

β-alumina, 375 Absorption coefficient, atoms, 312 Ag-based delafossite materials, 424 All-TCO p–n junction diode, 444 delafossite based I–V characteristics, 442 transparency of, 444 spinel based I–V curve, 441 lattice mismatch, 442 ZnO epitaxial layer, 442 Ambipolar diffusion, HiPIMS, 330 Amorphous semiconductors, 442 Analytical and experimental models, 81 discharge voltage, 82 ionization distribution vs. plasma emission distribution, 81 Apatite, 388 Apatite structure, 371 Ar-N2 gas mixture, emission spectrum in, 308 Argon neutral emission, 329 Asymmetrical bipolar discharge, 255, 268 electron temperature, 260 insulating dielectric films, 255 potential drop, 270 target potential and plasma potential, 268 Atomistic simulations, 505 advantages of the approach, 555 analyses results, 525

energy modulation deposition experiments, 525 interatomic potentials, 505 charge transfer ionic potential (CTIP), 510 CTIP and EAM parameters, 513, 514 embedded atom method (EAM) potential, 505, 508 inert gas impacts and ionic interactions, 507 ionic energy, calculations, 510 oxygen electron density, 514, 515 molecular dynamics (MD) model, benefits of, 515 surfactant effects, 526 Ag atoms surface diffusion and influence, 531 compositional profiles, 527, 529 experimental approach with Au-Ag, 527–529 mechanisms responsible, 528 molecular dynamics approach, 526 Aurivilius phase structure, 371 Barium cerate, 390–392 Bipolar pulsed dual magnetron discharges, 350, 358 charge carrier density and electron temperature, 350 CME and plasma properties of, 350 ITO film properties and CME, 358 optical emission intensity of, 352 Brownmillerite structure, 370

562

Index

Carbon nitride, RF sputtering and deposition of, 273 Cathode poisoning, 319 CCE, see Cross corner effect Ceramic Na+ conductors, 375 Cerates, 390 Ceria based electrolytes, dopant and ionic conductivity of, 384 Charge carrier density, 264, 349 and electron temperature, 349 spatial distribution of, 262, 264 unipolar and bipolar pulsed single magnetron discharges, 349, 350 Charge carrier density and deposition rate, ratio of, 360 Chemical Modulation, p-type conductivity, 450 Chromogenic building skins, energy efficiency, 486 Coating densification and stack integrity, 380 Computation illustrations, 103 calculation process, 105 collision rates, 109 electric field, 107, 108 electron and Ar+ ion densities, 108 ion flux bombarding the cathode, 114, 115 mean energy and energy probability function, 111 operating conditions and simulation data, 103 potential distribution, 105, 106 sheath architecture, 107 Computational load allieviation, 99 sorting, 102 subcycling, 100 superparticle (SP) weight determination, 103 variable time step, 101 weighting optimization, 101 Computer simulation codes, 2 molecular dynamic (MD) types, 4 Monte Carlo (MC) types, 2 Construction of GMR structures, sputtering methods, 504 Copper aluminum oxide (p-CuAlO2 thin films), 459 delafossite structure, 469

direct bandgap of, 465 Fermi level and Fermi energy, 471 field emission property, research into, 475 film orientation, 459 future research topics, 475 Hall coefficient of, 468 I–V characteristics of, 466 percentage of excess oxygen in, 461 residual stress, 460 Seebeck coefficient (SRT) of, 468 strain and particle size determination of, 461 strain broadening, 460 synthesis by reactive DC sputtering, 457 polycrystalline nature, 474 reflectance and transmittance measurement, 457 target preparation and film deposition, 457 temperature variation of conductivity, 467 thermoelectric properties, enhancement, 469 transmittance, reflectance and absorption coefficient, 462, 463 Cross corner effect, 337, 339 asymmetry of, 339 E × B drift, 340 in literature, 337 ionisation points, 342 magnets, size and shape of, 341 prevention, 344 Cross magnetron effect, twin or dual magnetron, 346 Cu-based delafossite materials, 424 CuAlO2 thin film, deposition parameters of DC sputtering, 428 D10 -based delafossite materials, 424 Cu-based and Ag-based, 424 space group, 424 Delafossite materials, 416, 417 applications, 417 defect reaction for, 451 ion exchange method, 432 temperature variation of conductivity of doped and undoped, 453

Index ternary and quaternary metal oxides, 416 thermoelectric properties, research, 475 Delafossite structure, 416 Dissociated oxygen, 360 atomic oxygen and total fluxes, ratio of, 361 deposition process and jO− /jtot values, 361 Doppler broadening, 313 Double probes, 257 Electrical probes, plasma-processing discharges, 256 Electrically floating surface and plasma potential, 270 Electrochemical stability, 393 Electrochromic display, 400 Electrochromic foils, 490 applications, present and future, 490, 492 properties, 490, 491 Electrochromic materials, 399 deposition of, 400 optical properties, 399 Electrochromics, 487 devices, 487 materials in EC devices, 488 present state and future promise, 493 technological progress, 488 film porosity, need for, 489 non-conventional technologies, 488 Electron drift velocity and magnetic field, 340 Electron emission, 43 electron transport, 46 electron yields, 46 angular distributions, 48 clean and contaminated surface areas, 48, 49 ion velocity dependence, 47, 50 oxidation effects on, 57 excitation mechanisms, 44 kinetic electron emission (KEE), 45 minimum energy, U , 44 potential electron emission (PEE), 45 separation of PEE and KEE, 46

563

factors influencing, 46 research directions, 58 Electron energy distribution functions (EEDF), 259, 261, 262 Electron temperature and pulse frequencies, 259, 260 Electronic collisions, 323 Emissive probe, plasma potential, 258 Encapsulation layer, 392 Energy-resolved mass spectrometry, 272 Excitation spectrum, LIE, 322 Excited level 3, creation terms and loss terms for, 323, 324 Film forming particles, total flux of, 361 Film thickness distribution, targets, 354, 355 Film transparency and deposition process, 337, 354 Fluorescence spectrum, LIE, 322 Fluorite structure, 369 Gadolinia doped ceria (GDC), 385 coating deposition by DC sputtering, 385 deposition on YSZ membranes, 385 gadolinium doping, 385 ionic conductivity, 385 Gas temperature measurement, 313 Giant magnetoresistive effect, 497 micro electronic devices, 497 tecnological developments, 497 Giant magnetoresistive multilayers, 498 construction of GMR structures, 504 magnetron sputtering methods, 504 vapor deposition methods, 504 discovery and use, 498 magnetic tunnel junctions devices, 500 perpendicular or parallel current flow, 499 spinvalve devices, 501 magnetotransport properties sensitivity, 503 performance optimization, 501 TMR effects, 503 tunneling magnetoresistance (TMR) effect, 501 Glow discharges

564

Index

1D hybrid model simulation, 50, 51 discharge conditions, 52 effective electron yield, 54 electron recapture, 54 incidence angle dependence, 52 sputter deposition, 53 chemisorption of the cathode, 56 discharge voltage change, 56 electron emission yield discrepancies, 57 ion-induced electro emission, role of, 50 GMR materials, sputtering pressure, 524 IBD process, 524 ion beam deposition process, 524 simulations vs experiment, 524 GMR properties, multilayer types, 527 Half-intensity broadening of Ti lines, 317 High-Power Impulse Magnetron Sputtering, 327 dissociation rate for, 332 electric impulsions production, 327 emission spectra, 328 HiPIMS, see High-Power Impulse Magnetron Sputtering Hollow cathode lamp, 311 pulsing, 311 spectrum, 317 Homo-junctions and lattice matching, 445 Indium tin oxide (ITO), 354 CCE and CME effects on, 355 incorporation of oxygen in, 360 Inorganic electrochromic thin films, 399 Instrumental broadening, 460 Ion energy distribution functions, 274 effective stopping potential, 274, 277, 278 of argon, oxygen and titanium species, 274 pulse frequency effects on, 276 time-resolved measurements of, 278, 279 Ionisation collisions, 261

Ionisation points, spatial distribution of the, 343 ITO films, 355 FWHM values and lattice spacing of, 359, 360 optical absorption and electrical resistivity of, 355–358 K2 NiF4 structure, 374 KH2 PO4 (KDP) structure, 374 La-Si coatings, 389 LAMOX (Lanthanum-Molybdenum Oxide) structure, 372 LAMOX coatings, 388 Lanthanum molybdate coatings, 387 Lanthanum silicates, 388 Large bandgap spinels, 415 Laser beam energy density, 325 Laser excitation, 325 Laser-induced fluorescence (LIF), 293, 320 experimental set-up for, 326 physical principle of, 320, 322 uses of techniques, 293, 295 Li+ conductor elecrolytes, 377 LIF, see Laser-induced fluorescence LiNbO3 , 402 LISICON fast ionic conductors, 393 Lithium all solid state sputter deposited microbattery, 392 Lithium conductor electrolytes, 393 Low-pressure magnetron plasma, 307 Magnetic field distribution, target, 339 Magnetic field tuning, see Cross corner effect Magnetically shielded anode, effects of, 351 Magnetron and plasma density distribution, 342 Magnetron discharge modeling analytical approach evaluation, 123 future research scope, 124 industrial challenges, 76 accuracy, 78 geometry issues, 77 pulsed sputtering, 77 laboratory challenges, 75

Index electron recapture, 76 electron transport issues, 76 secondary electron emission yield γ, 75 planar semi-analytical model, 78 Coulomb collisisons and Bohm diffusion, 81 discharge current calculation, 80 ionization model, 79 self-consistent model, 79 Magnetron discharge modeling approaches analytical models, 65, 66 disadvantages, 67 model types and their uses, 66 Boltzmann equation, use of, 68 fluid models, 67 hybrid models, 71 features and uses, 71 validity limitations, 71 Monte Carlo (MC) simulations, 69 advantages, 70 types and their contribution, 70 PIC-MCC simulations, 65, 72 disadvantages, 73 features and advantages, 72 requirements in the model, 65 Magnetron discharge simulations, 62 discharge model, 63 electrons and discharge gas particles, 65 ion bombardment, 63 magnetic field, 62 sputtered particle transport, 64 Magnetron plasmas, 301, 311, 346 emission spectroscopy of, 302 laser-induced fluorescence of, 326 optical emission spectroscopy, 301 resonant absorption spectroscopy of, 311 time dependent optical emission of, 346 Magnetron source, 347 Magnetron sputtering, 61, 200, 229 biaxially aligned thin films, 223 deposition pressure influence, 224 in-plane alignment, 223 binary collision Monte Carlo code, 218

565

deposition profiles comparison, 219 material flux simulation, 220, 222 sputter sources, 218 cost reduction needs, 61 energy and angular distribution of particles, 201 alternate analytical models, 204 analytical models and experiments, 206 collision cascade regimes, 201 comparitive merits of models, 208, 209 numerical method, 207 Sanders and Roosendaal model, 205 Sigmund-Thompson theory, 201 simulation model development, 207 sputter yield calculation, 201 energy deposited on substrate, 229, 231 electrical effects, 236 energy exchange parameters, 230 integral energy flux, factors, 232, 234 magnetron power, 235 measurement, 231 particles’ influence, 229 per atom basis, 237, 238 probe and plasma potentials, 236, 237 target distance, 235 temperature determination techniques, 231, 233 energy model representation, 239 electron density, 243, 244 electron temperature, 244 ions and electrons contribution, 243 plasma radiation contribution, 242 reflected neutrals contribution, 241 shield temperature, 245, 246 sputtered particle contribution, 240 thermal radiation effects, 244 energy model results, 246 per atom basis, factors, 248 plasma, power and pressure effects, 250 reactive vs elemental deposition, 249 total energy flux, 246, 247

566

Index

film structure, role of energy deposited, 251 gas particle collision, 209 attractive potential, 214 Born–Mayer potential, 216 collision cross section, σ, 210 combined potential, 217 hard sphere approximation, 211 mean free path, 210 repulsive potential, 214 scattering interaction geometry, 211, 212 screening functions, 215 sputtered atom direction, 213, 214 Thomas-Fermi-Dirac interaction potential, 216 gas rarefaction, 217 radial distribution of particles, 200 simulation process inputs, 61 simulation vs experiment results, 224 virtual sputter magnetron, 62 Metal multilayer growth, 532 atomic layer structure, HRTEM experiments, 552 dielectric layer deposition, 545 aluminum film growth, 546 aluminum layer thickness effect, 547, 548 mechanism of hole formation, 548, 550 quantification of oxidation, 549, 551 ion assisted deposition, 532 Auger electron spectrometry results, 544 flat growth of surface, 532, 533 in-plane electrical resistivity, 542, 543 ion incident angle effect, 534, 537 low energy approaches, 539–541, 544 model system analysis, comparison, 536, 538 rough surface reconstruction, 533 roughness and intermixing parameters, 534–536 ion assisted dielectric layer growth, 553 formation of a flatter surfaces, 553, 554

MD approach to simulation, 554 molecular dynamics simulation, 516 atomic configurations, 517, 518 copper (Cu) mixing, 520, 522 exchange mixing mechanism, 524 incident adatom energy, 518, 519 interfacial roughness, amplitude and wavelength, 519 surface roughness, 519–521 micro accumulator, 392 electrolyte (LIPON), 393 positive electrode, 394 Microbatteries, 392 Mixed conductors, 377 Monochromator, 303 Monte Carlo collision (MCC) model, 95 features and benefits, 95 null collision method, 96 collision sampling procedure, 98 merits and drawbacks, 96 Monte Carlo simulation, magnetron system, 340 NASICON (Sodium (Na) Super Ionic Conductor), 375 NASICON sputter deposition, 398 Neodymium cobaltite thin films, 399 Ni-based spinel films (NiCo2 O4 ), deposition of, 421 Ni/R, see Charge carrier density and deposition rate, ratio of O− and O− 2 negative ions, energy distribution of, 280–281 OES, see Optical emission spectroscopy Optical depth and absorption coefficient, relationship between, 312 Optical diagnostics, 301 Optical emission line, 382 Optical emission measurement, 347 Optical emission spectroscopy (OES), 286 plasma emission monitoring (PEM), 287 reactive sputter processes, characterisation of, 286 sputter gas argon, emission lines of, 288

Index Optical imaging, 292, 293 Optical thickness kσ0 L, 312 Oxygen partial pressure (pO2 ), powering modes, 363 Oxygen partial pressure and substrate, 361 P-TCO thin films, 424 copper gallium oxide and copper indium oxide, 424 deposition parameters and electrooptical parameters of DC-reactive sputtered, 440 deposition parameters of delafossite, 428 doping concentrations, 426 fabrication and characterization, 475 optical and electrical properties of spinel and delafossite, 435 P-type transparent conducting oxides delafossite structure based, 422, 424 deposition parameters, 428 electro-optical properties of, 420, 425 oxygen intercalation in, 453 pulsed laser deposition (PLD) of, 427 reactive sputtering of, 436 space group, 424 materials, 420 p-type conductivity, causes of, 448, 451 reactive sputtered, 432 spinel structure based, 420 conductivity and transparency, 420 deposition of, 421 structural requirement for designing, 450 substitutional doping of, 453 Particle-in-cell (PIC) model, 83 approximations and computational cycle, 84 charge assignment, 86, 88 external circuit incorporation, 89 force interpolation digital filtering, 93 leap-frog algorithm, 85 Poisson’s equation, cyclic reduction method (CRM), 87 stability and accuracy, 93

567

Particle-size broadening, 459, 460 Perovskite structure, 370 Perovskites, 398 PIC-MCC model extension computation illustrations, 115, 119 electron density profile, 121, 122 erosion profile, 122, 123 gas temperature distribution, 120 description, 116 numerical procedure, 116 species included, 116 Plasma broadening, 313 Plasma density, decay of, 266 Plasma emission monitoring (PEM), 287 Plasma mass spectrometry, 270 Plasma potential measurements, 266 PLD, see Pulsed laser deposition Potential drop, substrate and plasma, 270 Probe techniques, plasma-processing discharges, 257 Proton conductors, 389 Pulsed laser deposition, delafossite p-TCO thin films, 427 Pulsed magnetron discharges, 258, 264 electrical probes, 258 target voltage waveforms, 262, 264 Pyrochlore phase, cathode and YSZ film, 381 Pyrochlore structure, 370 Radiative deexcitation, 323 Reactive magnetron sputtering, 255, 367 laboratory set-up for, 271 trends to control, 367 unipolar and bipolar modes, 255 Reactive magnetron sputtering process, 131, 153, 240 aluminium oxide deposition, 153 discharge voltage, 156, 157 hysteresis behaviour and experiment, 153–155 parameters influencing, 156 Berg’s model, 133, 153 complete model, 186 comparison with Berg’s model, 192 deposition profile influence, 186, 188

568

Index

erosion profile influence, 189 plasma chemistry implementation, 186 rotating cylindrical magnetron, 191 energy of reaction, 240, 241 experiment conclusions, comparison validity, 166 gas addition effects, 131 ion induced electron emission (ISEE), 155 material properties and process conditions, 141 electron emission and ion implantation, 150 pumping speed and sputtering yield, 141, 142, 156 reactive co-sputtering, 147 sticking coefficient α, 141 target size and mixtures, 143 two reactive gases, 145, 147 mathematical treatment, 134, 135 oxygen exposure, discharge voltage, 157, 158 plasma oxidation, 158 difference from oxygen exposure, 160 discharge voltage, 159 experiments with other materials, 162, 164 relative discharge voltage, 160, 161 total sputter yield, 160 poisoning of target, mechanisms and effects, 161 reactive gas flow calculated χ2 and the experimental error, 183 experimental trends, 184, 185 parameters involved, 182 reactive gas partial pressure calculation, 174 reactive gas flow, 173 reactive ion implantation, with chemisorption chemisorption influence, 171 erosion rate and target sputter yield, 170, 171 knock-on effect, 181 oxidation level dependence, 176, 177 target condition, 176

target surface condition, 180, 181 reactive ion implantation, without chemisorption, 175 experiment vs. model, 175, 176 implanted ion concentration, 167 mole fraction calculation, 178, 179 reaction rate constant, k, 179, 180 time-dependent concentration, 169 stability experiment, 164–166 steady state equations, 136 assumptions of conditions, 136 deposition rate D, 138 derivation of equations, 136 reactive gas flow vs. compound fractions, Rm and D, 138, 140 Reactive sputtering technique, 473 applications to p-type thin films, 473 bandgap CuAlO2 thin films, 474 research needs, 474 conductivity increase, 474 cost-effective fabrication, 475 spinel and delafossite oxide films, 474 Reactively DC sputtered YSZ-based sensor fabrication process, 396 RF coil and magnetron cathode, 303 samaria doped ceria (SDC), 385 Secondary electron emission coefficient γ, 265 Silver cobalt oxide thin film, 425 direct bandgap of, 422 optical transmission of, 427 XRD pattern of, 430 Silver nitride, 175 Sine wave driven dual magnetron, deposition rate of, 364 Sine wave powering, 351 Single probes, 266 SOFC, see Solid oxide fuel cells Solid electrolytes, 369 H + Carriers, 374 O2− Carriers, 369 ionic transport properties, 369 Li+ Carriers, 377 Na+ Carriers, 375 Solid oxide fuel cells, 377 Solid state electrochemical gas sensors, 394 amperometric sensors, 395

Index magnetron sputtering considerations, 396 potentiometric gas sensors, 395 Source broadening, 313 Space charge sheath formation, 257, 269 Spinel materials, 414 binary, ternary and quaternary metal oxides, 415 large bandgap spinels, 415 Spinel structure cations, 415 fcc array of anions, 414 Sputtering processes, 1 angular distribution of sputtered atoms, 23 low and high incident-energy, 21, 22 Sigmund-Thompson theory, 201, 204 sputtering mechanism, 24, 25 atomically rough surface, 28, 29 compound materials, 2 computer codes, 1, 6 differential yields, 2 direct and indirect knock-out process, 12, 14 energy spectrum of sputtered atoms, 16 inelastic and elastic energy loss, 20 light-ion bombardment, 17, 18 Thompson formula, 16 incident energy Bohdansky formula, 8 sputtering mechanisms, 5 Thomas-Fermi potential, 8 Yamamura formula, 5, 9 incident-angle energy, density distribution, 9, 14, 15 low energy sputtering, randomization, 28 modeling of surfaces, 11 multi-component material surface ion-fluence dependence, 30, 35 Monte Carlo simulation codes, 33 multiple component target theory, 32 radiation-induced diffusion, 35 spectroscopies used, 30 surface composition, understanding of, 31

569

surface segregation, 30 time constant of kinetic process, 37 preferential ejection angle βp , 26 surface roughness effects, 2 Stimulated emission, 324 Strain broadening, 460 Strontium zirconate, 391 Substitutional doping and p-type conductivity, 453 Substrates, heat load of, 353 Tantalum pentoxide (Ta2 O5 ) films, 401 Target erosion inhomogeneity, 337 and homogeneous target erosion, 342 plasma density distribution, 342 rectangular magnetron’s, 338 Target potential and plasma potential, 268 TCO, see Transparent conducting oxides Ternary metal oxide crystal structure, 414 Thin film growth techniques, 427 Thin films, 420 electrical conductivity and optical transmittance of ZnRh2 O4 , 420, 422 resistivity of Ni−Co−O, 420 Ti and Ti+ energy levels, 303, 304 Titanium oxynitride thin films, 272 Transparent conducting oxides, 417 bandgap designing for, 418, 419 characteristics and applications of, 417, 418 p-type, 418 delafossite structure based, 422 spinel structure based, 420 Transparent electronics, 418 p-TCO materials for, 420 transparent field-effect transistors (TFET), fabrication of, 447 Tungsten oxide films, 400 properties of, 400 sputtering, 400 Unipolar pulsed single magnetron discharges, 349 CCE and plasma properties of, 349 ITO film properties, 356

570

Index

Vacuum chamber, 302 Valence band edge, metal-oxygen bondings, 449 Y doped barium cerate coatings, 390 YSZ, see Yttrium stabilized zirconia YSZ ceramic films, deposition methods of, 379 YSZ coatings, 380 deposition rate and emission line, 382 RF sputtered, 380 Yttrium stabilized zirconia, 379 coatings on large area substrates, 382

deposition rates of, 381 properties, 379 sputtering deposition parameters, effects of, 397 thin-film morphology, 384 Zirconia based electrolytes, 378 dopant concentration, 378 vacancies rate, 378 Zn-based spinel p-TCO films, deposition of, 422 Zr-Y metallic alloys, sputter-deposition rates of, 382

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