Springer Series in Surface Sciences 61

Werner WeschElke Wendler Editors

Ion Beam Modification of SolidsIon-Solid Interaction and Radiation Damage

Springer Series in Surface Sciences

Volume 61

Series editors

Roberto Car, Princeton University, Princeton, NJ, USAGerhard Ertl, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, GermanyHans-Joachim Freund, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin,GermanyHans Lüth, Peter Grünberg Institute, Forschungszentrum Jülich GmbH, Jülich,GermanyMario Agostino Rocca, Università degli Studi di Genova, Genova, Italy

This series covers the whole spectrum of surface sciences, including structure anddynamics of clean and adsorbate-covered surfaces, thin films, basic surface effects,analytical methods and also the physics and chemistry of interfaces. Written byleading researchers in the field, the books are intended primarily for researchers inacademia and industry and for graduate students.

More information about this series at http://www.springer.com/series/409

Werner Wesch • Elke WendlerEditors

Ion Beam Modificationof SolidsIon-Solid Interaction and Radiation Damage

123

EditorsWerner WeschInstitute of Solid State PhysicsFriedrich-Schiller-University JenaJenaGermany

Elke WendlerInstitute of Solid State PhysicsFriedrich-Schiller-University JenaJenaGermany

ISSN 0931-5195 ISSN 2198-4743 (electronic)Springer Series in Surface SciencesISBN 978-3-319-33559-9 ISBN 978-3-319-33561-2 (eBook)DOI 10.1007/978-3-319-33561-2

Library of Congress Control Number: 2016939918

© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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To Mark C. Ridgway

Preface

During the past 50 years ion beam technologies have been proven to be powerfultools in the continuously growing field of materials science. Ion beams are used fortailoring the physical and chemical properties of thin films, surfaces and interfaces.Nanostructures can be formed or modified and nanocomposite materials can besynthesised with new properties which do not exist in natural materials. In the pasttwo decades very high-energy ion beams from accelerators usually used for nuclearand particle research became available with the parameters making them suitablefor materials science. This stimulated intensive research and the use of so-calledswift heavy ions in ion-beam based materials science received much attention. Thiswas an important step to fulfil the demand for modifying more thick or buriedlayers. Moreover, the research in materials science with swift heavy ions inducednew applications as for instance the controlled shaping of embedded nanoparticles,which could not be imagined beforehand. Besides its use in various device tech-nologies, ion beams play an important role in a number of other fields. Examplesfor that are materials research and treatment of radioactive waste in nuclear fissionand fusion technologies, optimization of prosthetic components and the use of ionbeams in cancer treatment. Additionally, ion-beam based analytical techniques arevery important not only in materials science but also in environmental studies andin the field of preservation of cultural heritage.

Ion irradiation of solids has two main effects: the introduction of foreign atomsand the energy deposition into the material. The specific application of ion beamsrequires a thorough knowledge of the interactions of the energetic ions with thecorresponding material. These interactions determine the depth at which the ionscome to rest and cause structural modifications in the material (radiation damage).The radiation damage measured after ion irradiation depends on the primary energydeposition of the ions and the external irradiation conditions and is characteristic fora given material. In some cases radiation damage results in useful changes ofmaterials properties but typically the damage has to be reduced or removed bysubsequent annealing processes in order to achieve the desired results. The choiceof suitable methods for damage annealing often strongly depends on the kind and

vii

concentration of damage produced during ion irradiation in the respective material.Consequently, the investigation of ion-beam induced damage formation is anindispensable part in the field of ion beam physics.

The fundamentals of ion–solid interaction, ion-beam induced damage formationin a broad variety of materials and theoretical description of damage formation havebeen subject of intensive studies of a large number of research groups around theworld. This resulted in an enormous and still growing number of scientific papers.Various excellent monographs about ion beam physics appeared in recent years.Apart from the numerous scientific papers and monographs on the one hand andpure textbooks on the other, a comprehensive description of the theory of ionstopping in matter, a summary of models and the concepts that have been developedover time for characterisation of damage evolution as well as an overview of thestate-of-the-art knowledge on damage formation in various classes of materials isstill missing. With the present book we aim at filling this gap.

The book is organised in four parts. Part I provides the physical basics of ion–solid interaction. This includes a complete treatment of the theory of ion stopping inmaterials, i.e. the treatment of nuclear and electronic energy loss processes. Twofurther chapters give an overview about existing models for the description ofdamage formation due to electronic and nuclear interaction, respectively. If possiblethe general concepts are compared to each other and illustrated with real examples.The last chapter of this part is devoted to the physical basics of ion-beam inducedsynthesis of nanostructures. Part II deals with damage formation, amorphization and(re)crystallisation of semiconductors and ceramics, i.e. of covalent-ionic materials,due to nuclear energy deposition. The effect of high electronic energy deposition insolids is the topic of Part III. Structural modifications and phase transformations incrystalline insulators, metals and semiconductors are summarised. Additionally onechapter of this part reports on effects of electronic energy deposition in amorphoussemiconductors. The final part, Part IV, presents selected applications of ion beams.Here the focus is on shaping and modification of nanoparticles and nanostructuresand on the use of ion-beam induced effects for modification of optical materials.

It should be mentioned that not all existing literature could be taken into consid-eration in detail, but the contents of the various chapters are initially based on scientificresults of the authors and their groups. Additional references to works of other authorsare integrated. Besides well-established experimental results also possible limitationsin their interpretation and open problems are addressed. In this respect the book shouldbe suitable as material for special courses for graduate, postgraduate and Ph.D. stu-dents. Additionally it can be used as a source of information for researchers who areinterested in this field.

Finally we feel the need to thank all co-authors who participated in the projectwith their valuable and highly interesting contributions. With extreme sadness wehad to take note of the early death of our colleague, Mark C. Ridgeway, whosignificantly contributed not only to this book but to the field of ion beam physics ingeneral. We shall always honour his memory.

Jena, Germany Werner WeschFebruary 2016 Elke Wendler

viii Preface

Contents

Part I Physical Basics

1 Ion-Solid Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Konrad Gärtner1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Elastic Ion-Atom Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Ion-Atom Interaction Potential . . . . . . . . . . . . . . . . . . 51.2.2 Scattering Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 81.2.3 Scattering Dynamics. . . . . . . . . . . . . . . . . . . . . . . . . 101.2.4 Scattering Angles and Differential Cross Section . . . . . 111.2.5 Transferred Energy. . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Inelastic Ion-Atom Interaction . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 High Energy Approach of the Transferred Energy . . . . 161.3.2 Low Energy Approach of the Transferred Energy. . . . . 19

1.4 Ion-Amorphous Solid Interaction . . . . . . . . . . . . . . . . . . . . . . 221.4.1 Scattering Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 221.4.2 Elastic Energy Loss and Energy Loss Straggling . . . . . 271.4.3 High Energy Inelastic Energy Loss and Energy

Straggling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.4.4 Low Energy Inelastic Energy Loss and Energy

Straggling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.4.5 Comparison of Stopping Power Theories. . . . . . . . . . . 411.4.6 Energy Loss Distribution. . . . . . . . . . . . . . . . . . . . . . 451.4.7 Angular Distribution. . . . . . . . . . . . . . . . . . . . . . . . . 491.4.8 Spatial Distributions of the Ions and the Damage . . . . . 51

1.5 Ion-Crystalline Solid Interaction. . . . . . . . . . . . . . . . . . . . . . . 551.5.1 Axial Channeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.5.2 Ion Range and Damage Distribution . . . . . . . . . . . . . . 58

1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

ix

2 Models for the Description of Track Formation. . . . . . . . . . . . . . . 63Christian Dufour and Marcel Toulemonde2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.2 Electronic Energy Deposition . . . . . . . . . . . . . . . . . . . . . . . . 65

2.2.1 Electronic Energy Loss . . . . . . . . . . . . . . . . . . . . . . . 652.2.2 Radial Energy Distribution . . . . . . . . . . . . . . . . . . . . 66

2.3 Description of Track Formation . . . . . . . . . . . . . . . . . . . . . . . 692.3.1 Coulomb Explosion . . . . . . . . . . . . . . . . . . . . . . . . . 702.3.2 Bond Weakening (BW) Model . . . . . . . . . . . . . . . . . 712.3.3 Exciton Self-trapping (STX) Model . . . . . . . . . . . . . . 722.3.4 Concept of Reduced Electronic Energy Loss . . . . . . . . 732.3.5 A Transient Thermal Process . . . . . . . . . . . . . . . . . . . 75

2.4 Microscopic Models: Molecular Dynamics Approaches. . . . . . . 942.4.1 Sputtering by Electronic Excitation. . . . . . . . . . . . . . . 952.4.2 Track Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.4.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 99

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3 Modelling Effects of Radiation Damage. . . . . . . . . . . . . . . . . . . . . 105William J. Weber and Elke Wendler3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.2 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.2.1 Origin of Radiation Damage . . . . . . . . . . . . . . . . . . . 1063.2.2 Sources of Radiation Damage . . . . . . . . . . . . . . . . . . 1083.2.3 Measurement of Irradiation Damage . . . . . . . . . . . . . . 1093.2.4 Quantitative Analysis of Radiation Effects . . . . . . . . . . 110

3.3 Modelling of Defect/Damage Accumulation . . . . . . . . . . . . . . 1113.3.1 Defect Reaction Rate Theory . . . . . . . . . . . . . . . . . . . 1123.3.2 Effect of Temperature and Diffusion . . . . . . . . . . . . . . 1163.3.3 Practical Application . . . . . . . . . . . . . . . . . . . . . . . . 117

3.4 Modelling of Amorphisation and Order-Disorder PhaseTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.4.1 Irradiation-Induced Amorphisation Models . . . . . . . . . 1203.4.2 Amorphisation Kinetics. . . . . . . . . . . . . . . . . . . . . . . 125

3.5 Modelling of Complex Processes . . . . . . . . . . . . . . . . . . . . . . 1313.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4 Synthesis of Nanostructures Using Ion-Beams: An Overview . . . . . 137Giancarlo Rizza and Mark C. Ridgway4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384.2 From Thermodynamic to Driven System. . . . . . . . . . . . . . . . . 139

4.2.1 Thermodynamic System . . . . . . . . . . . . . . . . . . . . . . 1394.2.2 Kinetic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.2.3 Driven System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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4.3 On the Strategies to Synthesize Nanostructures UsingIon-Beams: The Case of Metal-Glass Nanocomposites . . . . . . . 1514.3.1 Ion Implantation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.3.2 Limits and Drawbacks of the Implantation Technique

and Some Alternative Approaches . . . . . . . . . . . . . . . 1664.3.3 Ion Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1694.3.4 A Model System to Investigate the Behavior

of Nanoparticles Under Irradiation . . . . . . . . . . . . . . . 1754.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1784.5 Appendix: Elementary Processes in Irradiated Solids . . . . . . . . 179

4.5.1 Principles of Ion-Matter Interaction . . . . . . . . . . . . . . 1794.5.2 Defects and Diffusion . . . . . . . . . . . . . . . . . . . . . . . . 180

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Part II Damage Formation and Amorphization by Nuclear EnergyDeposition

5 Primary Processes of Damage Formation in Semiconductors . . . . . 189Elke Wendler and Werner Wesch5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.2 Rutherford Backscattering Spectrometry for

Damage Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . 1915.2.2 Analysis of RBS Aligned Spectra. . . . . . . . . . . . . . . . 193

5.3 Continuous Damage Evolution up to Amorphisation. . . . . . . . . 1965.3.1 Typical RBS Channelling Spectra of Ion Implanted

Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.3.2 Introduction of Critical Temperatures and Effect

of Ion Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1985.3.3 Depth Distribution of Damage and Effect of Ion

Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2055.3.4 Fluence Dependence and Effect of Ion Mass . . . . . . . . 2085.3.5 Correlation of Damage Cross-Section with Primary

Energy Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . 2145.4 Discontinuous Damage Evolution up to Amorphisation. . . . . . . 217

5.4.1 Damage Formation in AlAs. . . . . . . . . . . . . . . . . . . . 2175.4.2 Damage Formation in GaN . . . . . . . . . . . . . . . . . . . . 224

5.5 Non-amorphisable Materials . . . . . . . . . . . . . . . . . . . . . . . . . 2295.5.1 ZnO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2305.5.2 CdTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

5.6 Summarising Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

Contents xi

6 Damage Formation, Amorphization and Crystallization inSemiconductors at Elevated Temperatures . . . . . . . . . . . . . . . . . . 243James S. Williams6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2436.2 Si Disordering, Amorphization and Crystallisation

Processes at Elevated Temperature . . . . . . . . . . . . . . . . . . . . . 2456.2.1 Dynamic Annealing and Defect Formation

During Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.2.2 Amorphisation Processes at Elevated Temperature . . . . 2486.2.3 Modelling of Ion-Induced Amorphization at

Elevated Temperature . . . . . . . . . . . . . . . . . . . . . . . . 2506.2.4 Ion Beam Induced Epitaxial

Crystallisation (IBIEC) . . . . . . . . . . . . . . . . . . . . . . . 2546.2.5 IBIEC Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

6.3 Irradiation of Ge at Elevated Temperatures . . . . . . . . . . . . . . . 2596.3.1 Disorder Formation, Amorphization and Ion-Induced

Crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2596.3.2 Formation of Porous Amorphous Layers . . . . . . . . . . . 261

6.4 Irradiation of GaAs at Elevated Temperatures . . . . . . . . . . . . . 2646.4.1 Disordering and Amorphization . . . . . . . . . . . . . . . . . 2646.4.2 IBIIA and IBIEC Processes in GaAs . . . . . . . . . . . . . 266

6.5 Other Compound Semiconductors . . . . . . . . . . . . . . . . . . . . . 2686.5.1 SiC Near and Above Tc . . . . . . . . . . . . . . . . . . . . . . 2686.5.2 IBIIA and IBIEC Behaviour in InP . . . . . . . . . . . . . . 2716.5.3 Ternary Semiconductors and Multilayers . . . . . . . . . . . 2726.5.4 Unusual Swelling and Erosion Behaviour in Ion

Implanted GaN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2756.5.5 ZnO Microstructure Following Irradiation . . . . . . . . . . 279

6.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 280References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

7 Defect Accumulation, Amorphization and NanostructureModification of Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287Yanwen Zhang and William J. Weber7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2877.2 Energy Deposition Processes in Crystalline Ceramics . . . . . . . . 289

7.2.1 Damage Production from Atomic CollisionProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

7.2.2 Effects of Electronic Energy Loss. . . . . . . . . . . . . . . . 2917.2.3 Coupled Effects of Ionization and Atomic

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2927.3 Damage Accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

7.3.1 Defect Accumulation . . . . . . . . . . . . . . . . . . . . . . . . 2937.3.2 Irradiation-Induced Phase Transformations . . . . . . . . . 2947.3.3 Modelling Amorphization . . . . . . . . . . . . . . . . . . . . . 298

xii Contents

7.3.4 Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . . . 3007.3.5 Effect of Electronic Energy Loss on

Amorphization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.4 Ionization-Induced Annealing and Ionization-Enhanced

Amorphization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3047.4.1 Ionization-Induced Annealing: Single Beam

Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3047.4.2 Ion-Induced Annealing: Dual Beam Irradiations . . . . . . 3067.4.3 Ionization-Enhanced Amorphization . . . . . . . . . . . . . . 307

7.5 Structural Modifications in Nanostructured CeramicMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3087.5.1 Intrinsically Nanolayered Structures . . . . . . . . . . . . . . 3097.5.2 Nanocrystalline Materials . . . . . . . . . . . . . . . . . . . . . 311

7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Part III Damage Formation and Amorphisation byHigh Electronic Energy Deposition

8 Swift Heavy Ion Irradiation of Crystalline Insulatorsand Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Lionel Thomé8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3218.2 General Considerations About Electronic Energy Loss in

Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3228.3 Experimental Observation of Electronic Energy

Loss Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3248.3.1 Direct Observation of Ion Tracks . . . . . . . . . . . . . . . . 3248.3.2 Indirect Observation of Ion Tracks . . . . . . . . . . . . . . . 327

8.4 Formation of Ion Tracks at Low Fluences . . . . . . . . . . . . . . . . 3298.4.1 Track Formation in Metals and Metallic

Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3298.4.2 Track Formation in Insulators . . . . . . . . . . . . . . . . . . 331

8.5 Structural Transformations at High Fluences . . . . . . . . . . . . . . 3448.5.1 Experimental Observations of Structural

Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3448.5.2 Analysis of the Build-Up of Radiation-Induced

Structural Transformations with PhenomenologicalModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Contents xiii

9 Swift Heavy Ion Irradiation of Crystalline Semiconductors . . . . . . 365Werner Wesch and Claudia S. Schnohr9.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3659.2 Energy Deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3669.3 Track and Damage Formation . . . . . . . . . . . . . . . . . . . . . . . . 369

9.3.1 Materials with High Radiation Resistance . . . . . . . . . . 3699.3.2 Materials with Low Radiation Resistance . . . . . . . . . . 3769.3.3 Threshold Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9.4 Modelling of Track Formation . . . . . . . . . . . . . . . . . . . . . . . . 3869.4.1 Qualitative Estimations . . . . . . . . . . . . . . . . . . . . . . . 3869.4.2 Application of the Inelastic Thermal Spike (i-TS)

Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3889.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 395References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

10 Swift Heavy Ion Irradiation of Amorphous Semiconductors . . . . . . 403Werner Wesch, Tobias Steinbach and Mark C. Ridgway10.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40310.2 Plastic Deformation in Amorphous Materials . . . . . . . . . . . . . . 40410.3 Structural Modification in Amorphous Silicon (a-Si). . . . . . . . . 411

10.3.1 Ion Track Formation in a-Si . . . . . . . . . . . . . . . . . . . 41110.3.2 Plastic Deformation of a-Si . . . . . . . . . . . . . . . . . . . . 413

10.4 Structural Modification in Amorphous Germanium (a-Ge) . . . . . 42010.4.1 Ion Track Formation in a-Ge . . . . . . . . . . . . . . . . . . . 42010.4.2 Porous Layer Formation (Multiple-Ion Irradiation) . . . . 42410.4.3 Plastic Deformation of a-Ge . . . . . . . . . . . . . . . . . . . 43110.4.4 Structural Modification in Other Amorphous

Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43510.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Part IV Selected Applications of Ion Irradiation

11 Ion-Shaping of Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443Giancarlo Rizza and Mark C. Ridgway11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44311.2 Ion-Matter Interaction for a Metal-Dielectric

Nanocomposite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44611.3 Influence of the Embedding Matrix . . . . . . . . . . . . . . . . . . . . 447

11.3.1 Influence of the Ion-Hammering. . . . . . . . . . . . . . . . . 44811.3.2 Influence of the Ion Track. . . . . . . . . . . . . . . . . . . . . 449

11.4 Influence of the Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . 45111.4.1 Deformation Pathways . . . . . . . . . . . . . . . . . . . . . . . 45311.4.2 Kinetics of the Elongation Process . . . . . . . . . . . . . . . 455

xiv Contents

11.4.3 Efficiency of the Ion Shaping Process: The Role of theDeposited Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 458

11.4.4 Elongation as a Function of the Ion Flux . . . . . . . . . . 46011.4.5 Elongation as a Function of the NP Concentration

and Stability of the Ion-Shaped NPs . . . . . . . . . . . . . . 46111.4.6 Role of Thermodynamics in the Shape

Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46311.5 Toward a Phenomenological Description. . . . . . . . . . . . . . . . . 46411.6 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

12 Low Energy Ion Beam Modification of Nanostructures . . . . . . . . . 475Christian Borschel and Carsten Ronning12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47512.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47712.3 Enhanced Dynamic Annealing in Nanostructures . . . . . . . . . . . 48212.4 Semiconductor Nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

12.4.1 Ion Beam Doping . . . . . . . . . . . . . . . . . . . . . . . . . . 48512.4.2 Damage Profiles and Bending of Nanowires . . . . . . . . 489

12.5 Sputtering of Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . 49112.5.1 Static Sputtering Calculations . . . . . . . . . . . . . . . . . . 49212.5.2 Dynamic Sputtering Calculations . . . . . . . . . . . . . . . . 494

12.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498

13 Modification of Structure and Properties of Optical Crystals . . . . . 501Feng Chen and Frank Schrempel13.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50113.2 Ion Beam Induced Damage and its Effects in

Optical Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50213.2.1 Damage Formation Due to Nuclear Energy

Deposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50213.2.2 Damage Formation Due to High Electronic Energy

Deposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50613.3 Methods for the Production of Optical Elements Based on

Crystal Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50913.3.1 Refractive Index Modulation and Waveguides . . . . . . . 51013.3.2 Crystal Ion Slicing (CIS). . . . . . . . . . . . . . . . . . . . . . 51213.3.3 Ion Beam Enhanced Etching (IBEE). . . . . . . . . . . . . . 514

13.4 Application of Ion Beam Induced Effects for theProduction of Optical Elements . . . . . . . . . . . . . . . . . . . . . . . 51713.4.1 Electro-optical Modulators. . . . . . . . . . . . . . . . . . . . . 51713.4.2 Guided-Wave Frequency Doublers . . . . . . . . . . . . . . . 518

Contents xv

13.4.3 Waveguide Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 52013.4.4 3D Optical Micro- and Nanostructures . . . . . . . . . . . . 523

13.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

xvi Contents

Editors and Contributors

About the Editors

Werner Wesch received his Ph.D. (1976) and postdoctoral degree (habilitation1986) in Physics at the Friedrich-Schiller-Universität Jena, Germany. From 1992until his retirement in 2012 he was head of the ion beam physics group at theInstitut für Festkörperphysik in Jena and since 1994 university lecturer. Hisresearch interests include interaction of energetic particles with solids, ion beammodification of semiconductors and insulators due to nuclear and electronic inter-actions as well as structure and defect analysis by ion beams and complementarymethods. He is co-author of more than 230 peer-reviewed journal papers. He wasmember and secretary of the International Committee of the conference series“Radiation Effects in Insulators” and member of the committees “Research with IonBeams and Nuclear Probes” of the Bundesministerium für Bildung und Forschung(BMBF) and of the “Materials Research Program Advisory Committee” (Mat-PAC)of the Gesellschaft für Schwerionenforschung (GSI) Darmstadt.

Elke Wendler received her Ph.D. (1984) and postdoctoral degree (habilitation1999) in Physics at the Friedrich-Schiller-Universität in Jena, Germany, dealingwith radiation damage and optical properties of ion implanted semiconductors.A longer stay at the Ion Beam Centre of the University of Surrey in Guildford, UK,broadened her experience in ion beam analysis. In cooperation with researchersfrom various countries she has published more than 160 papers in peer-reviewedjournals in the field of ion beam analysis and of ion beam modification ofcovalent-ionic materials. In 2014 she became a member of the InternationalCommittee of the conference series “Radiation Effects in Insulators” and “Ion BeamModification of Materials”. Currently the ion beam physics group at the Institut fürFestkörperphysik in Jena works under her leadership.

xvii

Contributors

Christian Borschel Institut für Festkörperphysik, Friedrich-Schiller-UniversitätJena, Jena, Germany

Feng Chen School of Physics, Shandong University, Jinan, China

Christian Dufour Centre Interdisciplinaire de Recherche sur les Ions, lesMatériaux et la Photonique (CIMAP), CEA-CNRS-ENSICAEN-Université deCaen, Caen, France

Konrad Gärtner Institut für Festkörperphysik, Friedrich-Schiller-UniversitätJena, Jena, Germany

Mark C. Ridgway Research School of Physics and Engineering, AustralianNational University, Canberra, ACT, Australia

Giancarlo Rizza Laboratoire des Solides Irradiés, Ecole Polytechnique,CEA/DRF/IRAMIS, CNRS, Université Paris-Saclay, Palaiseau Cedex, France

Carsten Ronning Institut für Festkörperphysik, Friedrich-Schiller-UniversitätJena, Jena, Germany

Claudia S. Schnohr Institut für Festkörperphysik, Friedrich-Schiller-UniversitätJena, Jena, Germany

Frank Schrempel Institut für Angewandte Physik, Friedrich-Schiller-UniversitätJena, Jena, Germany

Tobias Steinbach Institut für Festkörperphysik, Friedrich-Schiller-UniversitätJena, Jena, Germany

Lionel Thomé Centre de Sciences Nucléaires et de Sciences de la Matière(CSNSM), Université Paris-Sud, CNRS-IN2P3, Orsay, France

Marcel Toulemonde Centre Interdisciplinaire de Recherche sur les Ions, lesMatériaux et la Photonique (CIMAP), CEA-CNRS-ENSICAEN-Université deCaen, Caen, France

William J. Weber Department of Materials Science and Engineering, Universityof Tennessee, Knoxville, TN, USA

Elke Wendler Institut für Festkörperphysik, Friedrich-Schiller-Universität Jena,Jena, Germany

Werner Wesch Institut für Festkörperphysik, Friedrich-Schiller-Universität Jena,Jena, Germany

xviii Editors and Contributors

James S. Williams Research School of Physics and Engineering, AustralianNational University, Canberra, Australia

Yanwen Zhang Materials Science and Technology Division, Oak Ridge NationalLaboratory, Oak Ridge, TN, USA

Editors and Contributors xix

Symbols

a Lattice parameter, average displacement distance, screening lengthao Bohr radiusA AreaA0 Deformation yieldbn,e Collision diameter for nuclear (n) and electronic (e) scatteringC ConcentrationCe,a Heat capacity per unit volume of electronic (e) or atomic

(a) systemD Local dose, diffusion coefficiente Elementary chargeE Ion energyEa Thermal activation energyEb Binding energy of an atom in the solidEd Displacement energyEg Gap energyf Damage fractionF Forceg Electron–phonon coupling strengthG Shear modulus, Gibbs free energyħ Planck constantH Enthalpyje Electron fluxJ* Nucleation rateKe,a Thermal conductivity of the electronic (e) or atomic (a) systemm1, m2, m, me Mass (ion, target atom, total, electron)M Mobility

xxi

N Atomic densityNI Ion fluenceNdispl

* Number of displacements per ion and unit depthn Refractive indexne Concentration of free electronsnda Relative number of displaced lattice atomsndpa Number of displacements per lattice atom (dpa)p Momentumq Charge stater DistanceR Radiuss Impact parameter, path lengths(t) Solute concentrationS EntropySn,e Stopping power (energy loss per ion and unit depth) nuclear (n),

electronic (e)Ŝ, Ŝn, Ŝe Stopping cross section (total, nuclear, electronic)Set Threshold value of electronic energy depositionT Temperature, transferred energyTm Melting temperatureTe,a Temperature of electronic (e) or atomic (a) subsystemv Ion velocity, recrystallisation velocityV Ion–atom interaction potential, volumeW, Wn, We Energy straggling cross section (total, nuclear, electronic)Y Yield of backscattered ionsZ Atomic numbera Linear thermal expansion coefficientc Electron–phonon coupling efficiencyC Displacement ratee Dielectric constant, reduced energy of the ion (dimensionless)e* Effective eigenstrainη Shear viscosityk de Broglie wavelength, electron–phonon mean free pathl Reduced mass, chemical potentiallH Hall mobilitym Poisson number, Poisson ratemi Deformation velocity (i = x,y,z)ms Sound velocitymx Shear velocityX Solid angle, atomic volumeX2 Energy stragglingq Resistivity

xxii Symbols

qe Electron densityr Cross section, in-plane stressse Mean free time between two collisions of an electronU Screening function, final scattering angle, ion fluxW2 Angular straggling

Symbols xxiii

Part IPhysical Basics

Chapter 1Ion-Solid Interaction

Konrad Gärtner

Abstract This chapter gives an introduction into the theoretical description of thephysical processes which take place during the ion motion through a solid target.The interaction of the ion with the whole solid is treated as a sequence of binarycollisions with free target atoms at rest. Furthermore, the elastic and inelasticcontributions to the binary collisions are considered to be statistically independentand therefore they are treated separately. The elastic interaction provides changes ofthe energy and the direction of the ion. It is described by classical mechanics anddetermined by the ion-atom interaction potential. The inelastic interaction providesmainly a change of the energy of the ion which has to be described by quantummechanics, however, also some classical approaches are presented. For amorphoussolids, mainly considered here, the binary collisions can be assumed to be statis-tically independent and the physical quantities (e.g. energy and angular distribu-tions) are obtained by statistics. In the case of crystalline solids, only some specialeffects are described.

1.1 Introduction

If an ion beam is directed onto a solid target, the ions move through the targetchanging continuously their direction and energy and maybe also their states ofexcitation and ionization. Finally, the ions come to rest within the target (implanted)or they leave the target (transmitted or reflected). During their motion through thetarget, the ions generate primary recoils (energetic target atoms) which generatesecondary recoils and so on (collision cascade). Due to the ion and the recoils aswell, target atoms can become displaced (point defects, leading also to extendeddefects) and excited (electronic defects) and they can be removed from the surface(sputtering). In the case of a collective energy transfer (e.g. with swift heavy ion

K. Gärtner (&)Institut für Festkörperphysik, Friedrich-Schiller-Universität Jena,Max-Wien-Platz 1, 07743 Jena, Germanye-mail: konrad.gaertner@uni-jena.de

© Springer International Publishing Switzerland 2016W. Wesch and E. Wendler (eds.), Ion Beam Modification of Solids,Springer Series in Surface Sciences 61,DOI 10.1007/978-3-319-33561-2_1

3

irradiation), the structure of the solid can be changed (e.g. track formation) andmore extended defect structures (e.g. voids) can be generated. Furthermore, theimplanted ions and in some cases also the recoils (e.g. mixing in layered targets)cause changes of the local chemical composition.

In order to be able to understand all the possible changes of the target caused bythe ion irradiation, a detailed consideration of the ion-solid interaction is required.First, the following characteristic quantities, important for the simplification of thetheoretical description of the complicated interaction processes, are considered:Eb binding energy of the atoms in the targetdat average distance between the target atomssv duration of vibration of the target atomsk De Broglie wave length of the iontI interaction time of the ion with a target atom

The binding energy Eb and the average distance of atoms dat are in the order ofeV and Å, respectively, and the duration of vibration sv is in the range of about10−13 to 10−12 s. For ions with energies E in the order of 10 keV and above, asconsidered here, the relations

E � Eb

k ¼ h= m1vð Þ\10�2 A � dat

tI . dat=v\10�15 s � sv

are valid, where m1 and v are the mass and the velocity of the ion and h is thePlanck constant. From this follows that the interaction of the ion with the wholesolid can approximately be treated as a sequence of classical (k �dat) binarycollisions with free target atoms (E � Eb) which do not move during theinteraction (tI � sv). Furthermore, as shown later, the elastic and inelastic inter-action between the ion and an atom can be treated separately. They are described inSects. 1.2 and 1.3, respectively. The theoretical description of the interaction of theion with the whole solid, based on the elastic and inelastic binary collisions, ispresented in Sect. 1.4 for amorphous solids which is also valid for polycrystallinesolids (if the grain size is smaller than the ion beam diameter) and for randomincidence of the ion in a crystalline solid. Special effects in the case of alignedincidence of the ion in a crystalline solid are briefly mentioned in Sect. 1.5.

1.2 Elastic Ion-Atom Interaction

With the elastic ion-atom interaction, the configurations of the electrons of the ionand the target atom remain unchanged. This means that the elastic interaction can bedescribed by a classical two-body problem determined by the ion-atom interactionpotential which is a Coulomb potential screened by the electrons of the ion and theatom. The scattering kinematics and dynamics provide the scattering angles of the

4 K. Gärtner

ion and the atom, the differential cross section and the energy transferred from theion to the atom.

1.2.1 Ion-Atom Interaction Potential

During their interaction, the ion and the atom form a quasi-molecule the electronicstructure of which depends on the distance r between the ion and the atom. Theion-atom interaction potential V(r) is defined as the difference of the quantummechanical energies of the quasi-molecule for distance r and the free ion and atom

V rð Þ ¼ Equasi�molecule rð Þ � Eion � Eatom: ð1:1Þ

Let us first consider a single atom with atomic number Z. The correspondingquantum mechanical energy Eatom = E[{ui}] is a functional of the wave functionsui of all electrons i of the atom. Within the statistical model of the atom developedby Thomas [1], Fermi [2] and Dirac [3] and described in detail by Gombás [4, 5],the functional E[{ui}] can approximately be replaced by the functional E[qe],where the electron density qe (number of electrons per unit volume) is the sum overall |ui|

2. There are mainly three contributions to this energy

Eatom � E qe½ � ¼ Eelstat qe½ � þEkin qe½ � þEexch qe½ �; ð1:2Þ

the electrostatic energy, the kinetic energy and the exchange energy.The electrostatic energy as a functional of the electron density is exactly given

by

Eelstat qe½ � ¼ �Z22Z

d3r0qe r0ð Þr0

þ 22

2

Zd3r0d3r00qe r0ð Þqe r00ð Þ

jr0 � r00j ð1:3Þ

with the abbreviation 22 = e2/(4pe0) = 14.39965 eV Å, where e is the elementarycharge and qe is normalized by

Rd3r0qe r0ð Þ ¼ Z.

The kinetic energy can only approximately be expressed as a functional of theelectron density. For this purpose the volume of the atom is subdivided in smallcells d3r′ which are small enough that the potential within d3r′ can be assumed to beconstant and large enough that the electrons within d3r′ can be treated statistically.The number of states dNph in the phase space d3r′ d3p is given by

dNph ¼ 2 d3r0d3p=h3 ¼ 8p d3r0p2dp=h3 for p� pF ð1:4Þ

(the factor 2 takes into account 2 spin states) with the Fermi momentum pFdetermined by the demand that the number of states in d3r′ must be equal to thenumber of electrons in d3r′

1 Ion-Solid Interaction 5

ZpF0

dNph ¼ 8p

3h3d3r0 p3F ¼ qe r0ð Þ d3r0 ð1:5Þ

which provides

pF r0ð Þ ¼ 38p

� �1=3

h q1=3e r0ð Þ: ð1:6Þ

The kinetic energy of all electrons in d3r′ is given by

dEkin ¼ZpF0

dNphp2

2me¼ 4p

5meh3d3r0p5F ¼

310

38p

� �2=3 h2

med3r0 q5=3e r0ð Þ ð1:7Þ

(me is the electron mass) and the total kinetic energy reads

Ekin qe½ � ¼ jkin

Zd3r0 q5=3e r0ð Þ with jkin ¼ 3

103p2� �2=3 22 a0; ð1:8Þ

where a0 = ħ2/(me22) = 0.529177 Å is the Bohr radius and ħ = h/(2p). A similartreatment (see Dirac [3] or Jensen [6, 7]) provides the exchange energy

Eexch qe½ � ¼ jexch

Zd3r0 q4=3e r0ð Þ with jexch ¼ � 3

43p

� �1=3

22 : ð1:9Þ

Now, let us return to the ion-atom interaction potential V(r) defined by (1.1).Within the statistical model of the atom described above, the quantum mechanicalenergies of the ion, the atom and the quasi-molecule are given by (1.2), (1.3), (1.8)and (1.9), where qe is replaced by the electron densities qe,1, qe,2 and qe,12 of theion, the atom and the quasi-molecule, respectively. Now, (1.1) reads

V rð Þ ¼ E½qe;12 r0; rð Þ� � E½qe;1 r0ð Þ� � E½qe;2 r00ð Þ�; ð1:10Þ

withRd3r0qe;1 r0ð Þ ¼ Z1 � nion;

Rd3r00qe;2 r00ð Þ ¼ Z2 and

Rd3r0qe;12 r0; rð Þ ¼

Z1 � nion þZ2, where Z1 and nion are the atomic number and the degree of ion-ization of the ion and Z2 is the atomic number of the target atom. The position vectorsr′ and r″ have their origin in the nuclei of the ion and the atom, respectively. Theelectron density of the quasi-moleculeqe,12(r′; r) for a given distance r between the ionand the atom can be approximated by the superposition of the electronic densities ofthe free ion and free atom at distance r. This is justified because the correct electrondensity belongs to the minimum of E[qe;12(r′;r)] and therefore a deviation from thecorrect electron density influences the energy only slightly. It means, the accuracy ofthe energy is one order of magnitude better than that of the electron density.

6 K. Gärtner

According to the three contributions to the energies E[qe] (1.3), (1.8) and (1.9),the ion-atom interaction potential (1.10) is now given by

V rð Þ ¼ Velstat rð ÞþVkin rð ÞþVexch rð Þ � Z1Z222

rU rð Þ ð1:11Þ

with

Velstat rð Þ ¼ Z1Z222

rþ22

Zd3r0qe;1 r0ð Þd3r00qe;2 r00ð Þ

rþ r0 � r00j j

� Z222Z

d3r0qe;1 r0ð Þrþ r0j j � Z122

Zd3r00qe;2 r00ð Þ

r� r00j j

ð1:12Þ

Vkin rð Þ ¼ jkin

Zd3r0 qe;1 r0ð Þ þ qe;2 rþ r0j jð Þ� �5=3�q5=3e;1 r0ð Þ � q5=3e;2 r0ð Þ

n oð1:13Þ

Vexch rð Þ ¼ jexch

Zd3r0 qe;1 r0ð Þ þ qe;2 rþ r0j jð Þ� �4=3�q4=3e;1 q0ð Þ � q4=3e;2 r0ð Þ

n o:

ð1:14Þ

The four contributions to Velstat are directly provided by electrodynamics and Vkin

(1.13) and Vexch (1.14) are obtained from (1.8), (1.9) and (1.10), where the elec-tronic densities of the free atoms and ions are assumed to be spherical symmetric.Equation (1.11) defines also the screening function U which is unity for r = 0 andtends to zero for r ! 1: The ion-atom interaction potential V(r) [or the screeningfunction U(r)] according to (1.11)–(1.14) must be calculated separately for eachZ1 − Z2 combination by numerical integration. Its accuracy depends mainly on theaccuracy of the electronic densities used and it is less influenced by the uncertaintyof the statistical approximation. In [8], the interaction potential according to (1.11)–(1.14) is calculated using the electronic densities obtained from the electronic wavefunctions for atoms and ions given by Clementi and Roetti [9]. In the following, apotential calculated in this way is called ‘individual potential‘ for short.

For simplicity, in some cases generalized expressions for the screening function areapplied. Using the statistical model of the atom without exchange energy [Eelstat andEkin according to (1.3) and (1.8)] for the calculation of the screening function and forthe determination of the electron density as well, Thomas [1] and Fermi [2] obtainedthe screening function UTF(r/aTF(Z)) for a single atom, where UTF(x) is a universalfunction given numerically and the screening length aTF(Z) = 0.88534 a0 Z−1/3

depends monotonously on the atomic number Z of the atom. Firsov [10, 11] andLindhard [12] extended this screening function approximately to the interaction oftwo neutral atoms only by changing the screening length from aTF(Z) to

1 Ion-Solid Interaction 7

aF Z1;Z2ð Þ ¼ 0:88534 a0

Z1=21 þZ1=2

2

� 2=3 and aL Z1;Z2ð Þ ¼ 0:88534 a0

Z2=31 þZ2=3

2

� 1=2 ; ð1:15Þ

respectively. For convenient practical use, there exist a number of different ana-lytical expressions of generalized screening functions U(r/a) with different universefunctions U(x) and screening lengths a (see e.g. [13, 14]). Here, only two of themare mentioned. The widespread used universal ZBL screening function obtained byZiegler, Biersack and Littmark [13] reads

U rð Þ ¼X41

ai exp �bir

aZBL

� �with aZBL ¼ 0:88534 a0

Z0:231 þZ0:23

2

; ð1:16Þ

with ai = (0.1818, 0.5099, 0.2802, 0.02817) and bi = (3.200, 0.9423, 0.4028,0.2016). It has been obtained by averaging a large number of screening functionscalculated for different Z1 − Z2 combinations similar to the procedure given in(1.11)–(1.14). The OCB screening function suggested by O’Connor and Biersack[15] has the same structure as that given in (1.16). However, there are only threecontributions with ai = (0.35, 0.55, 0.1) and bi = (0.3, 1.2, 6.0) and aZBL is replacedby

aOCB ¼ 0:54þ 0:045 Z1=21 þZ1=2

2

� h iaF Z1;Z2ð Þ; ð1:17Þ

with aF given in (1.15).

1.2.2 Scattering Kinematics

After the elastic collision of the ion of velocity v with a target atom at rest, the newvelocity v01 of the ion is smaller and it deviates from the original direction by anangle u1 and the target atom moves with a velocity v02 at an angle u2 with respect tothe velocity v (see Fig. 1.1).

The conservations of the momentum and the kinetic energy provide alreadysome information about the velocities and angles after the collision which isindependent of the ion-atom interaction potential. Because the correspondingequations become more simple, let us change from the laboratory system to thecenter-of-mass system which moves with the velocity vcm = (m1/m)v, where m1

and m2 are the masses of the ion and the target atom, respectively, and m =m1 + m2 is the total mass. The velocities of the ion and the target atom in thecenter-of-mass system before the collision are

8 K. Gärtner

v1;cm ¼ v� vcm ¼ m2

mv and v2;cm ¼ �vcm ¼ �m1

mv; ð1:18Þ

respectively. The conservation of the momentum in the center-of-mass system

m1 v1;cm þm2 v2;cm ¼ 0 ¼ m1 v01;cm þm2 v02;cm ð1:19Þ

means that the two velocities v01;cm and v02;cm after the collision remain anti-parallel.Under this condition the conservation of the kinetic energy in the center-of-masssystem can only be fulfilled by

v01;cm ¼ v1;cm and v02;cm ¼ v2;cm ¼ vcm ð1:20Þ

(for v01;cm [ v1;cm the momentum conservation requires also v02;cm [ v2;cm whichviolates the energy conservation, similar for v01;cm\v1;cm). This means that the twovelocities in the center-of-mass system can only rotate by the same angle 0 withoutchanging their absolute value (see Fig. 1.1). The velocities v01 and v02 after thecollision in the laboratory system are obtained by adding the velocity of thecenter-of-mass system vcm to v01;cm and v02;cm; respectively: Now, the scatteringangles u1 and u2 in the laboratory system and the energy transferred to the targetatom Tn (nuclear contribution) as functions of the scattering angle 0 in thecenter-of-mass system can easily be obtained from Fig. 1.1 (for tanu1 and forv02 = 2vcm sin 0=2ð Þ see the upper and lower rectangular triangle, respectively)providing

tanu1 ¼v01;cm sin 0

vcm þ v01;cm cos 0¼ m2 sin 0

m1 þm2 cos 0ð1:21Þ

u2 ¼p� 02

ð1:22Þ

Fig. 1.1 Scattering geometry

1 Ion-Solid Interaction 9