Oxford Handbook of Nanoscience and Technology: Volume 2: Materials: Structures, Properties and Characterization Techniques 9780199533053, 0199533059

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THE OXFORD HANDBOOK OF

NANOSCIENCE AND TECHNOLOGY

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The Oxford Handbook of Nanoscience and Technology

Volume II of III Materials: Structures, Properties, and Characterization Techniques

Edited by

A.V. Narlikar Y.Y. Fu

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Great Clarendon Street, Oxford ox2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York c Oxford University Press 2010  The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2010 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data The Oxford handbook of nanoscience and technology : frontiers and advances : in three volumes / edited A.V. Narlikar, Y.Y. Fu. p. cm. Includes bibliographical references and index. ISBN 978–0–19–953305–3 (hardback) 1. Nanotechnology—Handbooks, manuals, etc. 2. Nanoscience—Handbooks, manuals, etc. I. Narlikar, A.V., 1940– II. Fu, Y.Y. III. Title: Handbook of nanoscience and technology. T174.7.094 2010 620’.5–dc22 2009036761 Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by CPI Antony Rowe, Chippenham, Wiltshire ISBN 978–0–19–953305–3 1 3 5 7 9 10 8 6 4 2

Preface Wolfgang Pauli is known to have remarked, “God made solids, but surfaces were the work of the Devil.” This Handbook deals with the Devil’s work. As the size of the material is reduced, surfaces acquire increasing importance, and indeed override the bulk when one of the dimensions of the material shrinks to nanometers. Simultaneously, at the nanoscale, quantum effects come into play and the properties of matter confined to nanodimensions are dramatically changed. Nanoscience and nanotechnology are all about relating and exploiting the above phenomena for materials having one, two or three dimensions reduced to the nanoscale. Their evolution may be traced to three exciting happenings that took place in a short span from the early to mid-1980s with the award of Nobel prizes to each of them. These were the discovery of the quantum Hall effect in a two-dimensional electron gas, the invention of scanning tunnelling microscopy (STM) and the discovery of fullerene as the new form of carbon. The latter two, within a few years, further led to the remarkable invention of the atomic force microscope (AFM) and, in the early 1990s the extraordinary discovery of carbon nanotubes (CNT), which soon provided the launch pad for the present-day nanotechnology. The STM and AFM have emerged as the most powerful tools to examine, control and manipulate matter at the atomic, molecular and macromolecular scales and these functionalities constitute the mainstay of nanotechnology. Interestingly, this exciting possibility of nanolevel tailoring of materials was envisioned way back in 1959 by Richard Feynman in his lecture, “There’s plenty of room at the bottom.” During the last 15 years, the field of nanoscience and technology has expanded internationally and its growth has perhaps been more dramatic than in most other fields. It has been transformed into an intense and highly competitive research arena, encompassing practically all disciplines that include theoretical and experimental physics, inorganic, organic and structural chemistry, biochemistry, biotechnology, medicine, materials science, metallurgy, ceramics, electrical engineering, electronics, computational engineering and information technology. The progress made in all these directions is truly spectacular. In this edited Handbook of Nanoscience and Technology, we have attempted to consolidate some of the major scientific and technological achievements in different aspects of the field. We have naturally had to follow a selective rather than exhaustive approach. We have focused only on those topics that are generally recognized to have had a major impact on the field. Inherent in this selection process is the risk of some topics inadvertently getting overemphasized, while others are unavoidably left out. This is a non-trivial problem especially in the light of the great many developments that have taken

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place in the field. However, a great diversity of important developments is represented in this Handbook and helps us overcome some of these risks. The present Handbook comprises 3 volumes, structured thematically, with 25 chapters each. Volume I presents fundamental issues of basic physics, chemistry, biochemistry, tribology, etc. at the nanoscale. Many of the theoretical papers in this volume are intimately linked with current and future nanodevices, molecular-based materials and junctions (including Josephson nanocontacts) and should prove invaluable for further technology development. Self-organization of nanoparticles, chains, and nanostructures at surfaces are further described in detail. Volume II focuses on the progress made with a host of nanomaterials including DNA and protein-based nanostructures. This volume includes noteworthy advances made with the techniques of improved capability used for their characterization. Volume III highlights engineering and related developments, with a focus on frontal application areas like Si-nanotechnologies, spintronics, quantum dots, CNTs, and proteinbased devices, various biomolecular, clinical and medical applications. The other prominent application areas covered in this chapter are nanocatalysis, nanolithography, nanomaterials for hydrogen storage, nanofield emitters, and nanostructures for photovoltaic devices. This volume concludes the Handbook with a chapter that analyses various risks that are associated in using nanomaterials. We realize that the boundaries separating a few of the topics of the above three volumes are somewhat shadowy and diffuse. Some chapters of Volumes II and III could have also provided a natural fit with Volume I. For instance, some of the novel molecular devices of Volume III could have alternatively been included in the realm of basic studies that form a part of Volume I. The three volumes together comprise 75 chapters written by noted international experts in the field who have published the leading articles on nanoscience and nanotechnology in high-profile research journals. Every chapter aims to bring out frontiers and advances in the topic that it covers. The presentation is technical throughout, and the chapters in the present set of 3 volumes are not directed to the general and popular readership. The set is not intended as a textbook; however, it is likely to be of considerable interest to final-year undergraduates specializing in the field. It should prove indispensable to graduate students, and serious researchers from academic and industrial sectors working in the field of nanoscience and technology from different disciplines like physics, chemistry, biochemistry, biotechnology, medicine, materials science, metallurgy, ceramics, electrical, electronics, computational engineering, and information technology. The chapters of the three volumes should provide readers an analysis of the state-of-the-art technology development and give them an opportunity to engage with the cutting edge of research in the field. We would like to thank all the contributors for their splendid and timely cooperation throughout this project. We are grateful to Dr Sonke Adlung for being most cooperative and considerate and for his important suggestions to help us in our efforts, and acknowledge with thanks the efficient assistance provided by Ms April Warman, Ms Phaedra Seraphimidi and Mr Dewi Jackson. Special thanks are due to Mrs Emma Lonie and

Preface

Ms Melanie Johnstone for commendably coordinating the proof correction work with over 200 contributors. One of us (AVN) thanks the Indian National Science Academy, New Delhi for financial assistance in the form of a Senior Scientist fellowship and the UGC-DAE Consortium for Scientific Research, Indore, for providing infrastructural support. He thanks the Consortium Director, Dr Praveen Chaddah, and the Centre Director (Indore), Dr Ajay Gupta, for their sustained interest and co-operation. He further acknowledges with thanks the technical assistance provided by Mr Arjun Sanap, Mr D. Gupta, Dr N.P. Lalla, Mr Suresh Bharadwaj and Mr U.P. Deshpande on many occasions. He is particularly grateful to his wife Dr Aruna Narlikar for her invaluable help, patience, and support throughout, and especially for her useful suggestions on many occasions during the course of the present project. He acknowledges the commendable technical support of his daughter Dr Amrita Narlikar at Cambridge, and also of Dr Batasha who remains a close and valued friend of the family. YYF extends his thanks to the National Natural Science Foundation of China (Contracts No. 60776053 and No. 60671021), and the National High Technology Research and Development Program of China (Program 863 and Contract No. 2007AA03Z311) for financial support. He remains indebted to his father, who passed away many years ago, for his invaluable guidance, advice and help to build his life and career, and to his mother, wife and son, for their sustained patience and support. November 2008

A.V. Narlikar Y.Y. Fu

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Contents List of Contributors 1 Optical properties of carbon nanotubes and nanographene R. Saito, A. Jorio, J. Jiang, K. Sasaki, G. Dresselhaus, and M.S. Dresselhaus 1.1 Overview 1.2 Definition of carbon nanotubes and nanographene 1.3 Experimental setup for confocal resonance Raman spectroscopy 1.4 Raman signals and sample evaluation 1.5 Excitons in single-wall carbon nanotubes 1.6 Summary and future directions References 2 Defects and disorder in carbon nanotubes Philip G. Collins 2.1 Introduction and outline 2.2 Categorization of defect and disorder 2.3 Experimental identification of defects 2.4 Physical consequences of defects and disorder 2.5 Concluding remarks Acknowledgments References 3 Roles of shape and space in electronic properties of carbon nanomaterials Atsushi Oshiyama and Susumu Okada 3.1 3.2 3.3 3.4

Introduction Nanospace in carbon peapods Boundaries in planar and tubular nanostructures Double-walled nanotubes: Peculiarity in cylindrical structure 3.5 Defects in carbon nanotubes 3.6 Hybrid structures of carbon nanotubes 3.7 Summary Acknowledgments Appendix: Total-energy electronic-structure calculations References

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1 3 6 7 19 23 25 31 31 32 50 68 80 81 81 94 94 95 102 112 119 124 134 135 135 136

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4 Identification and separation of metallic and semiconducting carbon nanotubes ´ Katalin Kamar´as and Aron Pekker 4.1 Introduction 4.2 Basic properties determining metallic or semiconducting behavior of carbon nanotubes 4.3 Characterization techniques sensitive to metallic or semiconducting type 4.4 Specific chirality-selective growth techniques 4.5 Physical postgrowth selection methods 4.6 Enrichment by chirality-sensitive chemical reactions 4.7 Modification of transport properties without change in chirality 4.8 Applications as transparent conductive coatings 4.9 Summary/Concluding remarks Acknowledgments References 5 Size-dependent phase transitions and phase reversal at the nanoscale Amanda S. Barnard 5.1 Introduction 5.2 Phase reversal at the nanoscale 5.3 Concluding remarks References 6 Scanning transmission electron microscopy of nanostructures S.J. Pennycook, M. Varela, M.F. Chisholm, A.Y. Borisevich, A.R. Lupini, K. van Benthem, M.P. Oxley, W. Luo, J.R. McBride, S.J. Rosenthal, S.H. Oh, D.L. Sales, S.I. Molina, K. Sohlberg, and S.T. Pantelides 6.1 Introduction 6.2 Aberration correction in electron microscopy 6.3 Semiconductor nanocrystals 6.4 Semiconductor quantum wires 6.5 Nanocatalysts 6.6 Magnetism in gold and silver nanoclusters 6.7 Charge ordering in manganites 6.8 Summary Acknowledgments References 7 Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications J.D. Taylor, B. Elliott, D. Dickel, G. Keskar, J. Gaillard, M.J. Skove, and A.M. Rao 7.1 Introduction 7.2 Mechanical vs. electrical responses

141 141 142 151 161 162 163 170 173 176 177 177 183 183 186 200 200 205

205 207 215 221 226 237 240 243 244 244 249

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7.3 Analytic modelling 7.4 Applications 7.5 Cantilevered multiwall carbon nanotubes (MWCNT) 7.6 Conclusion Acknowledgments References 8 Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research Francis L. Martin and Hubert M. Pollock 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Introduction Some existing mid-IR microspectroscopy techniques Development of near-field techniques Towards a brilliant benchtop IR source Possible advantages of using normal AFM probes Experimental procedures for PTMS Prospects for high spatial resolution in near-field FTIR spectroscopy 8.8 Spectroscopic detection of small particles 8.9 Data analysis 8.10 The analysis paradigm to discriminate nanomolecular cellular alterations in biomedical research 8.11 Standardization 8.12 A “biochemical-cell fingerprint” or phenotype 8.13 Medium-term goals from a long-term objective 8.14 Conclusion Dedication Acknowledgments References 9 Holographic laser processing for three-dimensional photonic lattices Satoru Shoji, Remo Proietti Zaccaria, and Satoshi Kawata 9.1 9.2 9.3 9.4

Introduction Theoretical Experimental Combination of holographic laser processing and multiphoton direct-writing technique 9.5 Conclusion References 10 Nanoanalysis of materials using near-field Raman spectroscopy Norihiko Hayazawa and Prabhat Verma 10.1 Introduction 10.2 SERS analysis of strained silicon 10.3 TERS analysis of ε-Si and GaN thin layers

261 271 279 281 282 282 285 285 286 291 297 299 301 304 310 311 319 323 323 325 328 329 329 330 337 337 340 352 358 361 362 364 364 365 370

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10.4 Improvement of TERS sensitivity for crystalline materials 10.5 Controlling the polarization in detection for crystalline materials 10.6 Towards higher spatial resolution 10.7 Conclusions References 11

Scanning SQUID microscope study of vortex states and phases in superconducting mesoscopic dots, antidots, and other structures T. Nishio, Y. Hata, S. Okayasu, J. Suzuki, S. Nakayama, A. Nagata, A. Odawara, K. Chinone, and K. Kadowaki 11.1 Introduction 11.2 Development of a high-resolution scanning SQUID microscope 11.3 Direct observation of quantized flux in superconducting rings 11.4 Vortex confinement in microscopic superconducting disks, triangles, and squares 11.5 Direct observation of an extended penetration depth in thin films 11.6 Vortex states in unconventional superconductors 11.7 Final observations Acknowledgments References

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379 385 397 401 402

405

405 410 414 419 423 429 434 435 435

New phenomena in the nanospace of single-wall carbon nanotubes Z.J. Shi and Z.N. Gu

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12.1 Introduction 12.2 Filling MWNT 12.3 Filling SWNT 12.4 New phenomena in the nanospace of SWNTs 12.5 Filling of DWNTs 12.6 Summary Acknowledgments References

438 439 442 443 465 470 470 470

Thermopower of low-dimensional structures: The effect of electron–phonon coupling M. Tsaousidou

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13.1 Introduction 13.2 Thermopower of two-dimensional semiconductor structures 13.3 Phonon-drag effect on the thermopower of semiconductor quantum wires

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13.4 Phonon-drag thermopower of doped single-wall carbon nanotubes 13.5 Conclusions Acknowledgments References 14

ZnO wide-bandgap semiconductor nanostructures: Growth, characterization and applications E. McGlynn, M.O. Henry, and J.-P. Mosnier 14.1 Introduction 14.2 ZnO wide-bandgap semiconductors: Crystalline and electronic structure and optical and materials properties 14.3 Growth of ZnO nanostructures 14.4 ZnO nanostructure characterization 14.5 ZnO nanostructure applications 14.6 Conclusions and acknowledgments References

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Selective self-assembly of semi-metal straight and branched nanorods on inert substrates X.-S. Wang, S.S. Kushvaha, X. Chu, H. Zhang, Z. Yan, and W. Xiao 15.1 15.2 15.3 15.4

Introduction Experimental and drift-correction procedures Previous studies of semi-metal growth on inert substrates Growth and morphology of Sb nanorods and Bi nanobelts on inert substrates 15.5 Summary Acknowledgments References 16

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Nanostructured crystals: An unprecedented class of hybrid semiconductors exhibiting structure-induced quantum confinement effect and systematically tunable properties Jing Li and Xiao-Ying Huang 16.1 Introduction 16.2 II–VI-based inorganic–organic hybrid nanostructures 16.3 Conclusions Acknowledgments References

507 516 517 518 523 523 527 538 549 556 562 564 572

572 575 577 582 592 593 593

598 598 602 626 626 627

Nanoscale Ge1−x Mn x Te ferromagnetic semiconductors J.F. Bi and K.L. Teo

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17.1 17.2 17.3 17.4

632 634 636 639

Introduction Growth procedure and characterization Structure analysis Optical properties

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17.5 Magnetic properties 17.6 Transport properties 17.7 Conclusions Acknowledgments References

641 650 655 656 656

Synthesis, characterization and environmental applications of nanocrystalline zeolites Vicki H. Grassian and Sarah C. Larsen

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18.1 Introduction 18.2 Synthesis and characterization of nanocrystalline zeolites 18.3 Applications in environmental catalysis 18.4 Applications in the adsorption of environmental contaminants 18.5 Hierarchical zeolite structures 18.6 Outlook for environmental applications of nanocrystalline zeolites Acknowledgments References 19

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Unusual properties of nanoscale ferroelectrics Huaxiang Fu 19.1 Introduction 19.2 Critical questions on low-dimensional ferroelectricity 19.3 Theoretical approaches 19.4 2D ferroelectric structures: Surfaces, superlattices and thin films 19.5 1D ferroelectric nanowires 19.6 Ferroelectric nanoparticles 19.7 Conclusions References Magnetic properties of nanoparticles Steen Mørup, Cathrine Frandsen, and Mikkel F. Hansen 20.1 Introduction 20.2 Magnetic domains 20.3 Magnetic anisotropy 20.4 Magnetic dynamics in nanoparticles 20.5 Magnetic structures in nanoparticles 20.6 Summary and conclusions Acknowledgment References Structural, electronic, magnetic, and transport properties of carbon-fullerene-based polymers A.N. Andriotis, R.M. Sheetz, E. Richter, and M. Menon 21.1 Introduction

659 660 666 675 679 683 683 684 688 688 692 693 696 700 703 708 709 713 713 714 716 719 735 740 740 741 745 745

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23

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21.2 Computational methods 21.3 Defect-induced structural and electronic features 21.4 Electronic, magnetic and transport properties 21.5 Magnetic coupling among magnetic moments 21.6 Conclusion Acknowledgments References

748 752 757 764 768 769 769

Magnetic nanowires: Fabrication and characterization Kleber Roberto Pirota, Marcelo Knobel, Manuel Hernandez-Velez, Kornelius Nielsch, and Manuel V´azquez 22.1 Introduction 22.2 Metallic nanowire fabrication, the state-of-the-art 22.3 Structural characterization 22.4 Magnetic reversal process: Single nanowire 22.5 Magnetic anisotropy and interactions: Role of geometrical arrangement 22.6 Transport measurements 22.7 Temperature-driven effects 22.8 Dynamic properties of magnetization 22.9 Future perspectives References

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772 773 784 788 794 799 804 811 815 817

Iron-oxide nanostructures with emphasis on nanowires U.P. Deshpande, T. Shripathi, and A.V. Narlikar

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23.1 Overview of iron-oxide nanostructures 23.2 Iron-oxide nanowires 23.3 Synthesis of α-Fe2 O3 nanowires and nanosheets by thermal oxidation route 23.4 Preferential bending of [110] grown α-Fe2 O3 NWs about the C-axis 23.5 Quantitative estimation of NW alignment using XRD and GIXRD 23.6 Summary Acknowledgments References

825 829

DNA-based self-assembly of nanostructures Joshua D. Carter, Chenxiang Lin, Yan Liu, Hao Yan, and Thomas H. LaBean

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24.1 Introduction 24.2 DNA building blocks and assembly strategies 24.3 DNA-directed assembly of heteromaterials 24.4 Conclusion References

867 868 878 886 887

838 848 852 859 860 860

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Properties and potential of protein–DNA conjugates for analytic applications M. Adler 25.1 Introduction 25.2 DNA–protein conjugates and their applications 25.3 Conclusions Acknowledgments References

Subject Index

891 891 892 921 922 923 929

List of Contributors Adler, M. Chimera Biotec GmbH, Department of Immunoanalytics, EmilFigge-Str 76a, 44227 Dortmund, Germany. [email protected] Andriotis, A.N. Institute of Electronic Structure and Laser, FORTH, P.O. Box 1527, 71110 Heraklio, Crete, Greece. [email protected] Barnard, A. School of Chemistry, University of Melbourne, Parkville, 3010, VIC, Australia (now at CSIRO Materials Science and Engineering, Clayton, 3168, VIC, Australia. [email protected]) Bi, J.F. Information Storage Materials Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 2 Science Drive 3, Singapore 11754. [email protected] Borisevich, A.Y. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA. [email protected] Carter, J.D. Department of Chemistry, Duke University, Box 90345, Durham, NC 27708, USA. [email protected] Chinone, K. SII NanoTechnology Inc., Matsudo, Chiba 270-2222, Japan. [email protected] Chisholm, M.F. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA. [email protected] Chu, X. Department of Physics, and Nanoscience and Nanotechnology Initiative, National University of Singapore, 2 Science Drive 3, Singapore 11754. [email protected] Collins, P.G. Department of Physics and Astronomy, University of California, 4129 Frederick Reines Hall, Mail Code: 4576, Irvine, CA 92697, USA. [email protected] Deshpande, U.P. UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore-452001, India. upd [email protected] Dickel, D. Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0312, USA. [email protected] Dresselhaus, G. Massachusetts Institute of Technology, Cambridge, MA 02139, USA. [email protected] Dresselhaus, M.S. Massachusetts Institute of Technology, Cambridge, MA 02139, USA. [email protected]

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Elliott, B. Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0312, USA. [email protected] Frandsen, C. Department of Physics, Nanostructured Materials Group, Building 307, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark. [email protected] Fu, H. Department of Physics, University of Arkansas, 226 Physics Building, Fayetteville, AR 72701, USA. [email protected] Gaillard, J. Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0312, USA. [email protected] Grassian, V.H. Department of Chemistry, University of Iowa, Iowa City, IA 52242-1419, USA. [email protected] Gu, Z. College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, P.R. China. [email protected] Hansen, M.F. DTU Nanotech, Department of Micro- and Nanotechnology, Building 345 East, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark. [email protected] Hata, Y. Department of Applied Physics, National Defense Academy, 1-1020, Yokosuka, Kanagawa 239-8686, Japan. [email protected] Hayazawa, N. Nanophotonics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan, and CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan. [email protected] Henry, M.O. School of Physical Sciences, and National Centre for Plasma Science & Technology, Dublin City University, Glasnevin, Dublin 9, Ireland. [email protected] Hernandez-Velez, M. Departamento de Fisica Aplicada, Universidad Aut´onoma de Madrid, and Instituto de Ciencia de Materiales de Madrid CSIC Campus de Cantoblanco, 28049, Madrid, Spain. manuel.hernandez@ uam.es Huang, X.-Y. State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences, Fuzhou, Fujian 350002, P.R. China. [email protected] Jiang, J. North Carolina State University, Raleigh, North Carolina 276957518, USA. [email protected] Jorio, A. Department of Physics, Universidade Federal de Minas Gerais Belo Horizonte, Brazil, and Instituto Nacional de Metrologia Normalizac¸a˜ o e Qualidade Industrial Duque de Caxias, Brazil. [email protected] Kadowaki, K. Institute of Materials Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan. [email protected] Kamar´as, K. Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences, P.O. Box 49, Budapest, H-1525 Hungary. [email protected] Kawata, S. Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan, and Nanophotonics Laboratory, RIKEN Hirosawa, Wako, Saitama 351-0198, Japan. [email protected]

List of Contributors xix

Keskar, G. School of Materials Science and Engineering, Clemson University, Clemson, SC 29634-0312, USA. [email protected] Knobel, M. Instituto de F´ısica Gleb Wataghin (IFGW), Universidade Estadual de Campinas (UNICAMP), C.P. 6165, Campinas, 13.083-970 SP, Brazil. knobel@ifi.unicamp.br Kushvaha, S.S. Department of Physics, and Nanoscience and Nanotechnology Initiative, National University of Singapore, 2 Science Drive 3, Singapore 11754. [email protected] LaBean, T.H. Departments of Computer Science & Chemistry, Duke University, Box 90345, Durham, NC 27708, USA. [email protected] Larsen, S. Department of Chemistry, University of Iowa, Iowa City, IA 52242-1419, USA. [email protected] Li, J. Department of Chemistry and Chemical Biology, Rutgers University, 610 Taylor Road, Piscataway, NJ 08854, USA. [email protected] Lin, C. Biodesign Institute, and Department of Chemistry and Biochemistry, Arizona State University, University Drive and Mill Avenue, Tempe, AZ 85287, USA. [email protected] Liu, Y. Biodesign Institute, and Department of Chemistry and Biochemistry, Arizona State University, University Drive and Mill Avenue, Tempe, AZ 85287, USA. yan [email protected] Luo, W. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA, and Department of Physics and Astronomy, Vanderbilt University, 2201 West End Avenue, Nashville, TN 37240, USA. [email protected] Lupini, A.R. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA. [email protected] Martin, F.L. Centre for Biophotonics, Lancaster Environment Centre, University of Lancaster, Lancaster LA1 4YQ, UK. [email protected] McBride, J.R. Department of Chemistry, Vanderbilt University, 2201 West End Avenue, Nashville, TN 37240 USA. [email protected] McGlynn, E. School of Physical Sciences, and National Centre for Plasma Science & Technology, Dublin City University, Glasnevin, Dublin 9, Ireland. [email protected] Menon, M. Center for Computational Sciences, University of Kentucky, Lexington, KY 40506-0045, USA, and Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0045, USA. [email protected] Molina, S.I. Departamento de Ciencia de los Materiales e I.M. y Q.I., Facultad de Ciencias, Universidad de C´adiz, Campus R´ıo San Pedro, 11510 Puerto Real, C´adiz, Spain. [email protected] Mørup, S. Department of Physics, Nanostructured Materials Group, Building 307, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark. [email protected]

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Mosnier, J.-P. School of Physical Sciences, and National Centre for Plasma Science & Technology, Dublin City University, Glasnevin, Dublin 9, Ireland. [email protected] Nagata, A. SII NanoTechnology Inc., Matsudo, Chiba 270-2222, Japan. [email protected] Nakayama, S. SII NanoTechnology Inc., Matsudo, Chiba 270-2222, Japan. [email protected] Narlikar, A.V. UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore-452001, India. [email protected] Nielsch, K. Institute of Applied Physics, University of Hamburg, Jungiusstrasse 11, Hamburg 20355, Germany. [email protected] Nishio, T. Institute for Nanoscale Physics and Chemistry, INPAC, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium. [email protected] Odawara, A. SII NanoTechnology Inc., Matsudo, Chiba 270-2222, Japan. [email protected] Oh, S.H. Division of Electron Microscopic Research, Korea Basic Science Institute, Yeoeun-dong. 52, Yusung-gu, Daejeon 305-806, Korea. [email protected] Okada, S. Center for Computational Sciences, University of Tsukuba, Tennodai, Tsukuba 305-8577, Japan, and CREST, Japan Science and Technology Agency, Sanbancho Bldg., 5, Sanbancho, Chiyoda-ku, Tokyo 102-0075, Japan. [email protected] Okayasu, S. Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan. [email protected] Oshiyama, A. Department of Applied Physics, The University of Tokyo, 7-31 Hongo, Tokyo 113-8656, Japan. [email protected] Oxley, M.P. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA, and Department of Physics and Astronomy, Vanderbilt University, 2201 West End Avenue, Nashville, TN 37240, USA. [email protected] Pantelides, S.T. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA, and Department of Physics and Astronomy, Vanderbilt University, 2201 West End Avenue, Nashville, TN 37240, USA. [email protected] ´ Research Institute for Solid State Physics and Optics, HunPekker, A. garian Academy of Sciences, P.O. Box 49, Budapest, H-1525 Hungary. [email protected] Pennycook, S.J. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA. [email protected]

List of Contributors xxi

Pirota, K.R. Instituto de F´ısica Gleb Wataghin (IFGW), Universidade Estadual de Campinas (UNICAMP), C.P. 6165, Campinas, 13.083-970 SP, Brazil. krpirota@ifi.unicamp.br Pollock, H.M. Department of Physics, University of Lancaster, Lancaster LA1 4YB, UK. [email protected] Rao, A.M. Department of Physics and Astronomy, and Center for Optical Materials Science and Engineering Technologies, Clemson University, Clemson, SC 29634-0312, USA. [email protected] Richter, E. Daimler Chrysler AG FT3/SA, Wilhelm-Runge-Str. 11, 89081 Ulm, Germany. [email protected] Rosenthal, S.J. Department of Chemistry, Vanderbilt University, 2201 West End Avenue, Nashville, TN 37240, USA. [email protected] Saito, R. Department of Physics, Tohoku University, 980-8578, Sendai, Japan. rsaito@flex.phys.tohoku.ac.jp Sales, D.L. Departamento de Ciencia de los Materiales e I.M. y Q.I., Facultad de Ciencias, Universidad de C´adiz, Campus R´ıo San Pedro, 11510 Puerto Real, C´adiz, Spain. [email protected] Sasaki, K. Department of Physics, Tohoku University, 980-8578, Sendai, Japan. sasaken@flex.phys.tohoku.ac.jp Sheetz, R.M. Center for Computational Sciences, University of Kentucky, Lexington, KY 40506-0045, USA. [email protected] Shi, Z. College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, P.R. China. [email protected] Shoji, S. Department of Applied Physics, Osaka University, Suita, Osaka 5650871, Japan. [email protected] Shripathi, T. UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore-452001, India. [email protected] Skove, M.J. Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0312, USA. [email protected] Sohlberg, K. Department of Chemistry, Drexel University, 3141 Chestnut St, Philadelphia, PA 19104, USA. [email protected] Suzuki, J. Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan. [email protected] Taylor, J.D. Department of Physics and Astronomy, Clemson University, Clemson, SC 29634-0312, USA. [email protected] Teo, K.L. Information Storage Materials Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 4 Engineering Drive 3, Singapore 117576. [email protected] Tsaousidou, M. Department of Materials Science, University of Patras, University Campus, 26504, Rio, Greece. [email protected] van Benthem, K. Dept. Chemical Engineering and Materials Science, University of California at Davis, 1 Shields Ave., Davis, CA 95616, USA. [email protected]

xxii

List of Contributors

Varela, M. Materials Science and Technology Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6030, USA. [email protected] V´azquez, M. Instituto de Ciencia de Materiales de Madrid - CSIC, Campus de Cantoblanco, 28049 Madrid, Spain. [email protected] Verma, P. Department of Applied Physics, and Graduate School of Frontier Bioscience, Osaka University, Suita, Osaka 565-0871, Japan. [email protected] Wang, X.-s. Department of Physics, and Nanoscience and Nanotechnology Initiative, National University of Singapore, 2 Science Drive 3, Singapore 11754. [email protected] Xiao, W. Department of Physics, and Nanoscience and Nanotechnology Initiative, National University of Singapore, 2 Science Drive 3, Singapore 11754. [email protected] Yan, H. Biodesign Institute, and Department of Chemistry and Biochemistry, Arizona State University, University Drive and Mill Avenue, Tempe, AZ 85287, USA. [email protected] Yan, Z. Department of Physics, and Nanoscience and Nanotechnology Initiative, National University of Singapore, 2 Science Drive 3, Singapore 11754. [email protected] Zaccaria, R.P. Department of Applied Physics, Osaka University, Suita, Osaka 565-0871, Japan. [email protected] Zhang, H. Department of Physics, and Nanoscience and Nanotechnology Initiative, National University of Singapore, 2 Science Drive 3, Singapore 11754. [email protected]

Optical properties of carbon nanotubes and nanographene R. Saito, A. Jorio, J. Jiang, K. Sasaki, G. Dresselhaus, and M.S. Dresselhaus

1.1 1.1.1

Overview Shape of graphene and nanotubes

Graphene (GR) is a single planar sheet of on hexagonal lattice of graphite.1 A single-wall carbon nanotube (SWNT) is a monolayer graphene sheet rolled up seamlessly into a cylinder (Iijima 1991; Saito et al. 1998a; Jorio et al. 2008). When the constituent graphene sheets are two (many), GR and SWNT become bilayer graphene (graphite) and double (or multi)-wall carbon nanotubes (DWNTs, MWNTs). Carbon nanotubes and graphene are now widely investigated as nanomaterials since the thickness of a graphene sheet or the diameter of a carbon nanotube is on the order of a nanometer. When we put electrodes at both ends of a GR or a SWNT with a gate electrode in between, we can make a semiconductor device. A graphene sheet possesses many unique transport (Novoselov et al. 2005; Zhang et al. 2005), mechanical, and optical properties (Pimenta et al. 2007). Thus, in order to get the high performance that these materials can provide, we should know how to evaluate the sample. Especially for carbon nanotubes, depending on how we roll up the GR sheet, there are more than 200 different geometrical structures that a SWNT can assume, which we call the SWNT chirality (Saito et al. 1998a). An important finding of SWNTs is that the electronic properties of a SWNT can be either metallic or semiconducting depending on the chirality (Hamada et al. 1992; Saito et al. 1992a,b; Tanaka et al. 1992). Thus, the electronic properties of either semiconducting or metallic materials can be obtained simply by changing the chirality.

1.1.2

Raman spectroscopy

In the following, starting from the basic definition of a carbon nanotube and graphene, we will show the basic concepts behind graphene and SWNTs. The

1 1.1 Overview

1

1.2 Definition of carbon nanotubes and nanographene

3

1.3 Experimental setup for confocal resonance Raman spectroscopy

6

1.4 Raman signals and sample evaluation

7

1.5 Excitons in single-wall carbon nanotubes

19

1.6 Summary and future directions

23

References

25

1 A graphene ribbon is a graphene sheet with

a finite width in one direction in the plane.

2

Optical properties of carbon nanotubes and nanographene

structure of a SWNT can be seen by many spectroscopic techniques, such as resonance Raman spectroscopy, photoluminescence, electron diffraction, and scanning probe microscopy (SPM) (Jorio et al. 2008). Among them, resonance Raman spectroscopy is widely used by many groups for nanotube characterization since Raman spectroscopy is a non-destructive, non-contact measurement that is carried out under ambient air pressure and room temperature and can be used to characterize both semiconducting and metallic tubes (Rao et al. 1997; Doorn et al. 2003; Fantini et al. 2004; Telg et al. 2004). Using an optical microscope, the excitation laser light can be focused down to the wavelength of the light, and thus a spatial resolution up to 1 μm can be easily obtained, which we call confocal micro-Raman spectroscopy. When the excitation energy is equal to the transition energy of the materials, the Raman signal is enhanced significantly (say by an order of 1000) and this is known as resonance Raman spectroscopy. Thus, if we use many laser excitation energies, we can plot the Raman intensity as a function of laser excitation energy, which we call a Raman excitation profile, from which we gain information about the transition energy of each unique SWNT (Doorn et al. 2003; Fantini et al. 2004; Telg et al. 2004). Figure 1.1 shows a typical Raman spectrum of SWNT bundles (Fantini et al. 2005). In graphene and SWNTs, a strong Raman-active mode is the so-called G-band, appearing at around 1580–1595 cm−1 (Jorio et al. 2003a; Dresselhaus et al. 2005). The G-band in graphene is an optic phonon mode in which the longitudinal optic (LO) and transverse optic (TO) phonon modes are degenerate. In the case of SWNTs, these two G-band phonon modes are split mainly into two, which we call G+ and G− , whose frequency separation is proportional to the inverse of the square of the diameter (Jorio et al. 2002). Group theory tells us that the Raman-active modes of SWNTs have A, E 1 and E 2 symmetries for the rotational symmetry around the nanotube axis, and only the A modes have a strong intensity (Jorio et al. 2000; Saito et al. 2001). 10 G

Elaser = 2.41 eV

Fig. 1.1 Raman spectrum of SWNT bundles (electric-arc) obtained with excitation laser energy E laser = 2.41 eV. The wellstudied features, namely the radial breathing mode (RBM), the disorder-induced (D) band, and the graphite-like (G) band features are assigned. The intensity for the intermediate frequency mode (IFM) region is multiplied by 30 to clearly show the richness of the low intensity IFM spectral region that is the focus of this work (Fantini et al. 2005). Copyright American Physical Society.

Intensity (arb. units)

8

6 600

4

800

1000

1200

×30 RBM

2

D

IFM 0 0

400

800

1200

Raman shift (cm–1)

1600

2000

1.2 Definition of carbon nanotubes and nanographene

3

Around 1350 cm−1 , we can see the D-band phonon mode whose intensity depends on the amount of defect structure in the SWNTs (Pimenta et al. 2007). The D-band to G-band intensity ratio (ID /IG ) is proportional to 1/L a (the 4 (where E laser crystalline size) (Tuinstra and Koenig 1970a,b) and to 1/E laser denotes the laser excitation energy) (Cancado et al. 2006; Sato et al. 2006). One important issue for the D-band is its dispersion, whereby the frequency of the D-band increases by 53 cm−1 /eV with increasing E laser (Matthews et al. 1999). SWNTs have a radial breathing mode (RBM) in which the C atoms of the SWNT are vibrating coherently in the radial direction. The frequency of the RBM ωRBM is inversely proportional to the diameter dt . There are many important Raman modes appearing between 100 and 3200 cm−1 (Jorio et al. 2003a; Dresselhaus et al. 2005). If the reader knows the characteristics of each phonon mode, the information provided by Raman spectroscopy is very rich, and thus the Raman spectroscopy of graphene and SWNTs have become a standard for characterizing such samples.

1.1.3

Exciton physics

When we discuss an optical process for SWNTs, an exciton (a bound pair of a photoexcited electron and a hole) is important (Ando 1997; Pedersen 2003; Spataru et al. 2004; Zhao and Mazumdar 2005; Dresselhaus et al. 2007). In conventional semiconductors, an exciton exists only at low temperature, below 10 K. Since the exciton binding energy for a SWNT is very large (up to 1 eV) because of the low dimensionality of the SWNT, an exciton can exist even at room temperature. Thus, the energy gap for a single particle (transport) and the optical transition energy for an exciton are different from each other because of many-body effects (Dresselhaus et al. 2007). Because of the localization of the wavefunction in real space, the delocalized Bloch wavefunctions labelled by the wavevector k are mixed with each other by the Coulomb interaction, and we solve for the mixing term by the Bethe–Salpeter equation (Jiang et al. 2007a). Since an electron or a hole can exist on one of the two energy dispersion curves around the degenerate Fermi energy, there are four possibilities (16 if we include spin) for the excitonic states.2 Because of symmetry, three of the four excitonic states are not optically allowed and we call them dark excitons and only one of the four excitons is optically allowed (a bright exciton) (Barros et al. 2006a,b). Since one of the dark exciton states has a smaller transition energy than the bright exciton state, the quantum efficiency of SWNTs cannot be so high.3 In Section 1.5 we will discuss how to calculate the excitonic states and their relationship to resonance Raman spectroscopy for SWNTs.

1.2 1.2.1

Definition of carbon nanotubes and nanographene Crystal structure of graphene

The crystal structure of monolayer graphene is a two-dimensional hexagonal sheet (Saito et al. 1998a). The bond length of the two nearest- neighbor carbon

2 In the case of an armchair nanotube, since

there are two energy minima in the 1D Brillouin zone, we have 16 possible pairs of excitons without considering spin.

3 The energy separations between the dark

and the bright excitons states are smaller than the thermal energy at room temperature. Thus, we expect a reasonable population of bright exciton states at room temperature. However, since the dark exciton states increase the number of phonon-emission processes, the quantum efficiency is low.

4

Optical properties of carbon nanotubes and nanographene

y

(a)

(b) b1

x A

K

B

Γ

a1

ky

a2 Fig. 1.2 (a) The unit cell and (b) Brillouin zone of graphene are shown as the dotted rhombus and the shaded hexagon, respectively. ai , and bi , (i = 1, 2) are unit vectors and reciprocal lattice vectors, respectively. (c) Parallel equidistant lines represent the cutting lines for the (4, 2) nanotube. The cutting lines are labelled by the cutting line index μ, which assumes integer values from 1 − N /2 = −13 to N /2 = 14. Saito et al. (1998a) Copyright Imperial College Press.

M K

b2

kx

(c)

ky b1 14 0 –13 K2

Γ M

by

Γ

bx

K1

K K′

kx

b2

˚ We define the unit vectors a1 and a2 as shown in Fig. 1.2, atoms is 1.42 A. √ ˚ The unit cell of graphene is given ˚ 3=2.46 A. whose lengths are each 1.42 A× by a rhombus (or a hexagon) containing two carbon atoms (A and B). The corresponding reciprocal lattice is obtained by rotating the hexagonal lattice in real space by 90 degrees and the reciprocal lattice vectors are given by b1 and b2 in Fig. 1.2(b). Thus, the first Brillouin zone is a hexagon in k space and , K and M denote high-symmetry points in the 2D Brillouin zone, respectively, at the zone center, hexagonal corner, and the center of the edge. A carbon atom has atomic orbitals 1s, 2s, 2p in which the 2s and inplane 2px and 2p y orbitals hybridize with one another to make a covalent bond (σ bond) between the three nearest-neighbor carbon atoms, forming the strongest chemical bond among materials. The remaining 2pz orbitals make a π bond to each other to form the valence electrons. The π energy band consists of the two 2pz orbitals of the A and B atoms in the unit cell. Using a simple tight-binding model, the Hamiltonian matrix, H and the overlap matrix S are given by 2 × 2 matrices:  H=

2 p

t f (k)

t f (k)∗

2 p



 ,S =

1

s f (k)

s f (k)∗

1

 ,

(1.1)

where t and s are, respectively, the nearest-neighbor hopping and overlap integrals between A and B atoms and f (k) is the phase factor appearing in the Bloch wavefunction consisting of A and B atoms √

f (k) = eik x a/

3

+ 2e−ik x a/2

√ 3

 cos

kya 2

 .

(1.2)

1.2 Definition of carbon nanotubes and nanographene

(a)

15.0 10.0 5.0 0.0 –5.0 –10.0

E [eV]

π*

K

(b) Fig. 1.3 (a) The energy-dispersion relations for 2D graphite are shown throughout the whole region of the Brillouin zone. The inset shows the energy dispersion along the highsymmetry directions of the triangle  M K shown in Fig. 2(b) (see text). (b) Around the K (or K ) point, the energy dispersion is expressed by two cones (the Dirac cones).

π

Γ

K M

M K

K′

 is Solving the secular equation det(H − ES) = 0, the energy band E(k)  obtained as a function of w(k), k x and k y :  = E g2D (k)

 2 p ± tw(k) ,  1 ± sw(k)

 is given by: where the function w(k)  √  k a k a  = | f (k)|  2 = 1 + 4 cos 3k x a cos y + 4 cos2 y . w(k) 2 2 2

(1.3)

(1.4)

The π energy bands of graphene consist of an occupied π band and an unoccupied π ∗ band whose energy gap at the K point is zero. The energy bandwidth for the π ∗ band is larger than that for the π band because of the s parameter. The energy dispersion near the K point is linear around the K point, and thus the energy bands near the Fermi energy E F at the K point have two cones whose apexes touch each other (see Fig. 1.3(b)). Near E F , we have two sets of cones, one set around the K point and the other around the K point. Thus, the energy is always degenerate for k and −k states (due to time-reversal symmetry).

1.2.2

5

Electronic structure of SWNTs

When we unroll a SWNT, we get a graphene sheet with a fixed width (a nanographene ribbon). Thus, when we neglect the curvature effect of the cylinder, the electronic structure of the nanographene and a SWNT are the same, except for the electronic states at the edges of the graphene. In the equatorial direction, there is a periodicity that connects the same atom by a vector around the equator of the cylinder, which we call a chiral vector, C h = na1 +√ma2 ≡ (n, m). The diameter of a nanotube is given by dt = |C h |/π = a n 2 + m 2 + nm. In the direction of the nanotube axis, there is a periodicity of T that is perpendicular to C h ; and T = t1 a1 + t2 a2 ≡ (t1 , t2 ), where t1 , and t2 are given by t1 = (2m + n)/dR , and t2 = −(2n + m)/dR with dR = gcd(2m + n, 2n + m) (gcd = the greatest common divisor of two integers). The unit cell of a SWNT is a rectangle specified by C h and T , in which we have N ≡ 2(m 2 + n 2 + nm)/dR hexagon and 2N carbon atoms. Thus, we will get 2N one-dimensional (1D) energy bands (Saito et al. 1998a).

6

Optical properties of carbon nanotubes and nanographene

K

(a) M

K

(b) SI

The reciprocal lattice vectors of a SWNT for C h and T are, respectively, K 1 and K 2 , where K 1 = (−t2 b1 + t1 b2 )/N and K 2 = (mb1 − nb2 )/N . We have N discrete values of kc in the direction of K 1 , (μK 1 , μ = 1, . . . .N ) and a continuous value of k is taken in the direction of K 2 , −π/T < k < π/T (see Fig. 1.2(c)). When we plot the possible k values in the 2D Brillouin zone, there are N line segments that we call cutting lines, separated by |K 1 | with a length, 2π/T . Thus, the electronic energy band consists of N π and π ∗ energy subbands that cut the 2D energy dispersion curve of graphene in the direction of K 1 ,    K2 π π . (1.5) E μ (k) = E g2D k + μK 1 , μ = 1, . . . , N ; − < k < |K 2 | T T If a μth cutting line goes through the K point, the SWNT energy bands becomes metallic, while if no cutting line goes through the K point, the SWNT will have an energy gap (semiconducting). When we calculate the distance of K in units of K 1 , we get −→

 K ·K 1 2n + m = . 2 3 |K 1 | K

(c)

SII

Fig. 1.4 Cutting lines around the K point for (a) Metallic (M), (b) Type-I (SI) semiconducting, (c) Type-II (SII) semiconducting single-wall carbon nanotubes. In the case of semiconducting SWNTs, the K point exists at a 2/3 or 1/3 position between two nearest cutting lines (Samsonidze et al. 2003). Copyright American Scientific Publishers. 4 In the case of an exciton, the energy subband

is a function of the center of momentum and again we have a singular density of states at the energy minimum.

(1.6)

When 2n + m = 3 p + r , ( p is an integer and r = 0, 1, and 2), the corresponding SWNTs are, respectively, metallic, type I semiconductor and type II semiconductor NTs (nanotubes). The difference between type I and type II semiconductor nanotubes is the position of the K point between two nearest cutting lines (see Fig. 1.4). When the 1D energy subband has a minimum √ (or maximum), the density of states becomes singular at that energy (∝ 1/ E − E 0 ), and the optical absorption becomes strong at this energy.4

1.3

Experimental setup for confocal resonance Raman spectroscopy

The basic setup for the Raman scattering experiment is an excitation laser, a monochromator and a detector. An interference filter is needed before the sample to remove plasma lines (or luminescence) from the lasers. Another filter (a notch filter) has to be placed before the monochromator to filter the elastic scattered light. Resonance Raman measurements require many excitation laser lines to get in and out of resonance with different SWNTs. In this case, the filters have to be tunable. The interference filter can be replaced by a set of prisms, although this is not very efficient. Alternatively, they can be replaced by another monochromator. There are already tunable notch filters available, but they are rather expensive. To filter the scattered light of a tunable system, usually researchers use a triple-monochromator micro-Raman setup. The two prior gratings, usually used in the fore-monochromator stage, serve to eliminate the light originating from elastic scattering processes, and the last grating, usually called a spectrograph, is responsible for the dispersion of the light originating from the inelastic scattering.

1.4 Raman signals and sample evaluation 7

For a one laser line experiment, a single-monochromator micro-Raman system can be used. In this case the scattered light goes directly from the sample to the spectrograph stage passing through a notch filter that blocks the elastic scattered light. A more intensive Raman signal is obtained with a single-monochromator (usually 10–100 times stronger). For the excitation, typically both discrete excitation energies from Ar:Kr and He:Ne lasers, and tunable Ti:sapphire and dye lasers are used. Both Ti:sapphire and dye lasers are usually pumped by an Ar ion laser with output power ∼6 W. The Ti:sapphire laser provides tunable output wavelengths in the near-infrared (NIR) range and the dye laser allows us to change the output wavelengths in the visible region and the specific wavelength range depends on the kind of dye solution used. In micro-Raman measurements, an optical microscope is attached to the system so that we can get a spatial resolution of the order of the wavelength of light. Usually the samples are focused using 10×, 50× and 100× objectives for SWNTs in solution, in bundles and isolated on a SiO2 substrate, respectively. In experiments with carbon nanotube samples on SiO2 substrates or suspended in aqueous solution, high laser power density (up to 40 mW/μm2 ) can be used sometimes without heating or damaging the samples. For nanotube samples in bundles or as isolated SWNTs suspended in air, a low power density ( 0. At the G-band phonon energy h¯ ωλ = 0.2 eV, Im(h(E)) becomes very large, thus giving rise to a large broadening of the Raman spectra. As for Re(h(E)), the value of h(E) changes sign from (0) positive (phonon hardening) to negative (phonon softening) at h¯ ωλ = 0.2 eV. The dark area denotes the forbidden transition region up to 2|E F |. Thus, when |E F | becomes large, the phonon-hardening contribution becomes small and (0) thus the phonon frequency becomes soft up to 2|E F | = h¯ ωλ . In Fig. 1.10, we plot the calculated G-band phonon frequency as a function of the Fermi energy for (a) (10,10) armchair, (b) (10,4) chiral, and (c) (15,0)

1.4 Raman signals and sample evaluation

T = 300 K

1600

TO

1560

LO

1520

(b) 1620 T = 300 K

Egap = 0.045 eV

0

0.1

EF (eV)

0.2

T = 300 K

1640

LO 1580

TO 1540

(10,10) –0.3 –0.2 –0.1

(c)

0.3

hw (cm–1)

1640

hw (cm–1)

hw (cm–1)

(a)

(10,4) 1500

–0.3 –0.2 –0.1

0

0.1

0.2

0.3

EF (eV)

1600

TO

1560

LO

1520

15

(15,0)

Egap = 0.045 eV –0.3 –0.2 –0.1

0

0.1

0.2

0.3

EF (eV)

Fig. 1.10 Calculated G-band phonon frequency as a function of the Fermi energy for (a) (10,10) armchair, (b) (10,4) chiral, and (c) (15,0) zigzag SWNTs. The error bars correspond to the self-consistently calculated phonon spectral width. In the case of armchair SWNTs, there is no electron– phonon interaction for TO phonon modes, which shows the bases for the chiral angle dependence of the electron–phonon interaction (Sasaki et al. 2008).

zigzag SWNTs (Sasaki et al. 2008). The error bar in the figure corresponds to the self-consistently calculated phonon spectral width, which comes from Im(h(E)). All LO phonons become soft for |E F | < 0.2eV and have a mini(0) mum value for |E F | = 0.1 eV, which corresponds to 2|E F | = h¯ ωλ . Around |E F | ∼ 0 eV, the phonon has a finite lifetime (phonon self-energy) due to the electron–phonon interaction, and thus at the Dirac point there is an uncertainty relation spectral width for the phonon frequency. Therefore, the corresponding Raman spectra give an intrinsic spectral width due to the Kohn-anomaly effect. For the case of the TO phonon mode, since the TO phonon of the armchair nanotube does not couple to electrons, there is no spectral width and no frequency shift, while the chiral and zigzag nanotubes show some electron– phonon coupling. This clearly shows that the electron–phonon coupling has a chiral-angle dependence. Sasaki et al. showed that the el–ph coupling of TO phonon modes for chiral and zigzag nanotubes comes from the curvature effect through which the one-dimensional cutting lines shift away from the K points of the Brillouin zone (Sasaki et al. 2008). In the case of armchair nanotubes, the cutting line remains at the K point, so that there is no curvature-induced tiny energy gap for armchair SWNTs that are strictly metallic (Saito et al. 1992b). For a detailed discussion of the el–ph interaction see (Sasaki et al. 2008). In Figs. 1.11(a) and (b), we show experimental results for the G-band intensity as a function of applied gate voltage for two different isolated SWNTs. For the given excitation energy of 1.91 eV and the observed radial breathing mode (RBM) frequencies of 196 and 193 cm−1 for the two individual metallic SWNTs of Figs. 1.11(a) and (b), respectively, we can assign (n, m) by using the conventional assignment technique that we previously developed for single-nanotube Raman spectroscopy (Jorio et al. 2001). As a result, we assign the (n, m) values of (12,6) for the SWNT in Fig. 1.11(a) and (15,3), (16,1), or (11,8) for the SWNT in Fig. 1.11(b). When we compare the experimental results of Fig. 1.11 with the calculated results, we should consider that the G-band Raman intensity depends on chiral angle, whereby for zigzag nanotubes the LO (TO)-phonon mode

16

Optical properties of carbon nanotubes and nanographene

(b) 1650

1650

1600

1600

Raman Shift [cm–1]

Raman Shift [cm–1]

(a)

1550 1500 1450

Fig. 1.11 Experimental results of the Raman G-band intensity as a function of applied gate voltage for two metallic SWNTs (a) and (b). A strong (weak) intensity peak is denoted by the bright (dark) color; Farhat et al. (2007); Sasaki et al. (2008).

1400 –2

1550 1500 1450

2 0 Gate Voltage [V]

–1

–0.5 0 0.5 Gate Voltage [V]

is strong (suppressed), while for armchair nanotubes the TO (LO) phonon mode is strong (comparable in intensity) (Saito et al. 2001). This chirality dependence of the G-band intensity is also observed in single-nanotube Raman spectroscopy (Yu and Brus 2001; Jorio et al. 2004a). This chirality dependence of the G-band mode Raman intensity is exactly opposite to the chirality dependence of the phonon-softening in which the TO-phonon mode is suppressed in armchair nanotubes. It should be noted here that the corresponding el–ph interaction for phonon-softening and for Raman processes are independent of each other. The el–ph interaction for phonon-softening works for electrons to be excited from the valence energy band to the conduction band, while the el–ph interaction for Raman processes works for electrons to scatter within the conduction bands. Because of the different symmetry of the wavefunction as a function of k between the valence and the conduction band, the opposite chiral angle dependence appears for the two cases (Sasaki et al. 2008). A chirality-dependent Raman intensity was observed for metallic SWNTs as shown in Fig. 1 of (Wu et al. 2007). Although the E F positions for the observed metallic SWNTs are unclear, the experimental results for the Raman spectral width are consistent with the calculations in the following way. For example, the calculated TO mode in a (15, 15) SWNT gives a sharp lineshape and the calculated LO mode shows a broad feature for a (24, 0) SWNT. It is noted that the Raman intensity is proportional to the el–ph coupling for an optically excited electron in the conduction energy band. Since this el– ph coupling depends on the chiral angle due to the trigonal warping effect (Saito et al. 2001), one of the two optical modes may be invisible in the Raman intensity (Reich and Thomsen 2004; Wu et al. 2007).

1.4.5

Elastic scattering of electrons and the D-band

The photoexcited electron can be scattered elastically (measurements show that the energy is not changed in the scattering process) by defects, such as point defect and the edge of the crystal (boundary scattering) (Cancado et al.

1.4 Raman signals and sample evaluation

17

2004; Sato et al. 2006; Pimenta et al. 2007). An elastic scattering process can thus be substituted for one of the two scattering events in a second-order Raman process (see Fig. 1.5). In this case, only one phonon frequency with q = 0 participates in the scattering event. If one of the two scattering events of the G band process is an elastic scattering event, we instead see the doubleresonance Raman band at around 1350 cm−1 for 2.41 eV laser energy, and we call this phonon, which is associated with a defect, the D-band (disorderinduced feature). The D-band Raman intensity is strong when the number of defects in the sample is large or the sample size of the crystal is small and boundary scattering becomes important. Since the G and G bands are intrinsic Raman signals that we can see even when there are no defects, the intensity ratio of the D-band related to the Gband (ID /IG ) is frequently used to characterize a sample in terms of defect density or the crystalline size (Pimenta et al. 2007).

1.4.6

Elastic scattering of photons

Rayleigh scattering of light, which is known as the reason why the sky is blue, is the elastic scattering of light. When we see the optical process of Rayleigh scattering in a solid, the scattering of a photoexcited electron does not change in energy for the scattering event. The Rayleigh scattering is in one sense an annoying signal for Raman measurements since a strong tail of the spectral line at 0 cm−1 interferes with the observation of the inelastic Raman signals. Thus, we usually cut out the elastically scattered light by using laser-specific filters (notch filters). Similar to resonance Raman spectroscopy, the Rayleigh intensity is enhanced for the resonance condition for optical transitions (see for example, the contribution by T. F. Heinz in Jorio et al. 2008). The resonance Rayleigh scattering technique is now also being used for determining the E ii value for specific (n, m) SWNTs (Wang et al. 2006).

1.4.7

Sample evaluation by Raman spectroscopy and photoluminescence

Here, we list the ways that a sample is characterized using Raman spectroscopy. If we want to know whether or not the sample contains SWNTs, we can check for the existence of an RBM signal. For given E laser , the RBM signal would tell us that a resonance with a particular (n, m) SWNT exists. When the diameter range for SWNTs in the sample does not match the resonance condition, we can not see any signal. By checking the typical diameter distribution in a sample by TEM or by SPM, we can determine which E laser is suitable for observing the RBM spectra for a given sample. A sample synthesized by chemical vapor deposition (CVD) usually has a large diameter distribution from 0.8–2 nm, and we can easily find the RBM signal. If we can wrap a SWNT by some surfactant10 such as SDS (sodium dodecyl sulfate) or DNA, we will see the distribution of SWNTs in the photoluminescence (PL) spectra for the semiconducting SWNTs in the sample. In PL spectroscopy, we can change the excitation energy easily if we use a lamp with a continuous

10 There are many known surfactants that can

be used for suspending SWNTs in water.

18

Optical properties of carbon nanotubes and nanographene

Fig. 1.12 Photoluminescence spectra of isolated single-wall carbon nanotubes as a function of emission wavelength and excitation wavelength. Each peak corresponds to a M ener(n, m) semiconductor SWNT. For E 11 gies, we do not have a PL peak (Bachilo et al. 2002; Jorio et al. 2004b).

range of wavelengths. The photoabsorption occurs at E 22 for a conventional range of photon wavelengths, and the photoexcited carriers are relaxed quickly by emitting phonons to reach E 11 and the PL is then observed at E 11 . When we now make a 2D map of the PL intensity as a function of the energies of photoabsorption and PL, many peaks appear, each peak corresponding to (n, m) for that tube (Bachilo et al. 2002; Jorio et al. 2004b) (Fig. 1.12). Thus, as far as we have a sample for which the SWNTs are suspended in water through a surfactant or wrapping agent, PL measurements are convenient and provide a powerful tool for determining the distribution of (n, m) semiconducting SWNTs in the sample. Raman measurements, on the other hand, have the merit that (1) we can measure a sample as it is prepared (no surfactant treatment is needed), and that (2) we can measure not only semiconductor SWNTs but also metallic SWNTs. In the case of metallic SWNTs, we do not see the PL spectra, since the photoexcited carriers non-radiatively decay into the metallic energy band at a speed (1 ps) that is much faster than the optical transition (1 ns). Thus, in a SWNT bundle, we expect a rapid energy transfer from a semiconducting SWNT to a metallic SWNT, and no SWNTs remain in an excited state long enough for PL emission to be observed (Jiang et al. 2005c). Thus, individual SWNTs that are isolated from other SWNTs by a surfactant or suspended between two electrodes in the air are needed to observe PL. The (n, m) assignment is generally done by measuring the RBM frequency that is used in comparing with the prediction coming from the Kataura plot (see Fig. 1.7). The metallicity (metal or semiconductor) of a SWNT can usually be determined by the G− band spectra. In the case of the G band, the G− feature for a metallic SWNT can go down to as low as 1550 cm−1 (see Section 1.4.4) and the spectra becomes broad. For a conventional laser energy

1.5 Excitons in single-wall carbon nanotubes 19

(from 1.5–2.5 eV), the G− band intensity relative to the G+ band intensity is strong for metallic SWNTs or graphene. The chiral angle of a SWNT can be estimated in two ways: (1) the RBM signal is generally strong for zigzag nanotubes,11 (2) the G− band intensity relative to the G+ band intensity becomes strong for armchair nanotubes. These behaviors come from the diameter and chirality dependence of the electron–phonon coupling matrix element for each phonon mode (Jiang et al. 2005a, 2007b; Jorio et al. 2005a). For an isolated single nanotube laid on a TEM grid, we can directly determine its (n, m) values by the electron diffraction method, which can be used for SWNTs with dt up to 3 nm (Paillet et al. 2006). In addition to the RBM frequency, the RBM intensity also provides information regarding the (n, m) assignment and population estimation. Due to the unique exciton–phonon and exciton–photon matrix elements in SWNTs, the RBM intensity shows regular (2n + m) family patterns, i.e. (1) the intensity has a large value near zigzag tubes, while it has a small value near armchair tubes, (2) the intensity has a larger value for a VHS along K M than that along K , (3) the intensity has a strong diameter dependence due to the exciton– photon matrix elements (Jiang et al. 2005a,b, 2007b; Jorio et al. 2005b; Popov et al. 2005).

1.5 1.5.1

11 The PL intensity is strong for large chi-

ral angles close to armchair nanotubes and for Type I relative to Type II semiconductor SWNTs (Bachilo et al. 2002).

Excitons in single-wall carbon nanotubes Dark and bright excitons

As we discussed in the overview section, the exciton of a SWNT is always formed upon photoexcitation even at room temperature (Chou et al. 2005; Maultzsch et al. 2005; Wang et al. 2005). Although the story for excitons for SWNTs is complicated, we try to offer here a basic explanation that is as simple as possible.12 An exciton is a bound pair of a photoexcited electron (e) and a hole (h). For a photoexcited e–h pair, we expect the wave vector of the electron and the hole to appear at the same position around the K or K point. However, since we expect the scattering of the electron (or the hole) by the phonon, the electron and the hole can have different k vectors. We have two regions for possible k vectors around points K and K and so we have four possibilities of the positions of the electron and the hole for an exciton, such as (K, K), (K, K ), (K , K) and (K , K ) (see Fig. 1.13). The excitonic states for (K, K ) and (K , K) are energetically degenerate, and we call them E symmetry excitons because they transform into one another by time-inversion symmetry. Since the position of the electron and hole are different from each other for the E exciton, the E exciton can not be recombined to emit a photon, even though such an e–h pair feels an attractive Coulomb interaction in real space. Thus, the E exciton is a dark exciton. As for the (K, K) and (K , K ) e–h pairs, the two states have the same energy and are therefore strongly mixed with each other to make symmetry-adopted non-degenerate exciton states, such as (K, K) ± (K , K ), which we denote by the A∓ excitons.13 The symmetry operation is a C2 rotational symmetry operation around the axis perpendicular to the nanotube axis, which turns the SWNT upside down. The exciton wavefunction for A− (A+) is odd (even) under this symmetry operation and the

12 We will consider here only the case of light

polarized parallel to the nanotube axis. For a detailed discussion of the symmetry of the exciton, see Dresselhaus et al. (2007) and Jiang et al. (2007a,b).

13 The notation of A+ is used for a bright

exciton by some other groups. In any case, the bright exciton has an odd symmetry under C2 rotation (Jiang et al. 2007a).

20

Optical properties of carbon nanotubes and nanographene

K

K E=

& K

K

K

A± = K

Fig. 1.13 E and A symmetry excitons in SWNTs. Solid (open) circles denote an electron (a hole).

K – +

K

electron–photon Hamiltonian is odd. Therefore, only the A− exciton is optically allowed ( exciton (odd) | Hamiltonian (odd) | ground state (even) = 0). An important point is that the even symmetry A+ exciton has a lower energy than the A− exciton. Thus, even though the A− exciton is formed by light, the A− exciton easily decays to the A+ exciton to reach the thermal equilibrium distribution. This is the main reason why the quantum efficiency of a SWNT is so low (several per cent at most). The energy gap between the lowest-energy exciton A+ and the bright exciton A− is of the order of 1–10 meV and thus the PL intensity has a temperature dependence around 10–100 K.

1.5.2

GW correction and Bethe–Salpeter equation

The coupled electron–hole excitation energies and wavefunctions can be obtained by solving the Bethe–Salpeter equation (BSE) (Perebeinos et al. 2004; Spataru et al. 2004; Ando 2006b; Dresselhaus et al. 2007; Jiang et al. 2007a),       [E(kc ) − E(kv )] + K kc kv , kc kv n (kc kv ) = n n kc kv , (1.9) where kc and kv denote wave vectors of the conduction and valence energy bands and E(kc) and E(kv ) are the quasi-electron and quasi-hole energies,   respectively. K kc kv , kc kv is the electron–hole interaction kernel. The Coulomb effect appears not only for e–h pairs but also for the photoexcited electron (hole) and many valence electrons. The latter interaction increases the energy of the electron (decreases the energy of a hole), and this interaction thus contributes to the increase in the energy gap compared with the single-particle energy difference between an electron and a hole. This energy difference is called the self-energy. The self-energy is calculated by the GW approximation (Spataru et al. 2004; Ando 2006b; Dresselhaus et al. 2007; Jiang et al. 2007a), where the self-energy is a convolution of the non-interacting Green functions G and the screened Coulomb interaction W (Aryasetiawan and Gunnarsson 1998; Aulbur et al. 2000). The corresponding

1.5 Excitons in single-wall carbon nanotubes 21

(a)

(b)

Kc Kc

W Kc Kc

Kc

K

åGW =

=

–W+

Kv Kv

G

Kc V

Kv

Kv

Kv Kv Fig. 1.14 Feynman diagrams for the (a) GW approximation for the self-energy  and the (b) electron–hole interaction Kernel in the BSEs. Wiggle and dashed lines represent screened and bare Coulomb interactions, respectively.

Feynman diagram is shown in Fig. 1.14(a). Because of the electron–electron interaction, the wave vector k in the lattice is no longer a good quantum number and has a lifetime. Thus, the electron (hole) is treated as a quasi-particle within its lifetime and an energy uncertainty for the quasi-particle energy occurs. The kernel K in eqn (1.9) has direct and exchange terms (Onida et al. 2002; Spataru et al. 2004; Jiang et al. 2007a). Figure 1.14(b) shows the Feynman diagram for the kernel. The exchange term determines the energy splitting between the spin-singlet and triplet states. The exciton excitation energy is thus given by (single-particle energy) + (self-energy) – (exciton binding energy) ≡ (quasi-particle energy) – (exciton binding energy). We call (self-energy) – (exciton binding energy) the many-body effect. In SWNTs both the self-energy and exciton binding energy show (2n + m) family patterns (see Fig. 1.15) (Dresselhaus et al. 2007; Jiang et al. 2007a). The many-body correction to E 11 and E 22 energies is around 0.2 eV, while the correction to E 33 and E 44 2 13 E11

1.5 16 E [eV]

14

13 S

1 16 14 16 14 0.5

13 Ebd

2n + m = 17 S – Ebd

0 0.5

1.5

1 1/dt [1/nm]

2

Fig. 1.15 The excitation energy E 11 , selfenergy , binding energy E bd and energy corrections  − E bd based on the extended tight-binding method for E 11 bright exciton states. Open and filled circles are for SI and SII SWNTs. The dashed line is calculated by E log = 0.55(2 p/3dt )log[3/(2 p/3dt )] with p = 1 (Jiang et al. 2007a). Copyright American Physical Society.

22

Optical properties of carbon nanotubes and nanographene

energies can be 0.6 eV (Dresselhaus et al. 2007; Jiang et al. 2007a; Sato et al. 2007a). Moreover, the many-body correction does not contribute to the family spread in E 11 and E 22 in the Kataura plot, while it contributes significantly to the family spread in E 33 and E 44 (Dresselhaus et al. 2007; Jiang et al. 2007a; Sato et al. 2007a).

1.5.3

Localization of the wavefunction

The delocalized Block functions with different k are mixed with one another. Since the exciton wavefunction has a Gaussian shape along the nanotube axis direction, the mixing coefficient for k shows a Gaussian distribution whose center is a van Hove singular k point (Jiang et al. 2007b). Since the size of the exciton wavefunction is larger than the diameter dt of a SWNT (around 1–10 nm depending on dt ) (Spataru et al. 2004; Dresselhaus et al. 2007), the mixing coefficient has a large value for only one cutting line of the 1D Brillouin zone and thus the wavefunction is almost constant around the circumferential direction of a nanotube. Because of the localization of the exciton wavefunction, optical transition matrix elements for an exciton are enhanced by one order of magnitude relative to the matrix element for a free e–h pair (Jiang et al. 2007b). This enhancement facilitates the observation of single- SWNT spectroscopy by Raman and PL spectroscopy.

1.5.4

14 Recent experiments show that enhanced

PL intensity occurs by the mixing between dark and bright excitons within S < 0 under the magnetic field (Matsunaga et al. 2008; Srivastava et al. 2008).

Spin of the exciton

When we consider an exciton, there is an additional freedom for the exciton states associated with the spin (Dresselhaus et al. 2007). We have a spinsinglet (S = 0) and three spin-triplet (S = 1) exciton states for each of the four symmetry exciton states. Thus, the exciton states have 16 possible states for a given E ii . Since the singlet exciton is generated by an optical transition, the spin-triplet exciton is a dark exciton. However, when we consider a spinflipping mechanism such as a magnetic field, the triplet exciton can be coupled with the singlet exciton and made bright. This can be demonstrated by the spectral change in the PL spectra in the presence of a magnetic field (Ando 2006b; Zaric et al. 2006).14 The triplet states are somewhat (1–100 meV) below the singlet states in energy and thus they also contribute to the low PL quantum efficiency of a SWNT (Spataru et al. 2005).

1.5.5

Exciton phenomena in the Raman and PL spectra

Recent Raman and PL experiments show that exciton properties are essential for understanding these photophysical observations. Here, we list the important phenomena related to excitons. For RBM and G-band Raman intensities, the chiral-angle dependences that are mainly determined by exciton–phonon matrix elements, are not sensitive to the exciton effects. However, exciton effects enhance the magnitude and

1.6 Summary and future directions

diameter dependence of the Raman intensities through the exciton–photon matrix elements (Jorio et al. 2006; Jiang et al. 2007b). The E ii values can be modified by contact with nearby materials, which we call environmental effects (Miyauchi et al. 2007). Whatever energy upshifts or downshifts are observed depend on whether the semiconducting SWNT is type I or type II, and the value of i in E ii , and the SWNT diameter. The magnitude of the shift in energy depends on the dielectric constant of the environmental material. This situation is similar to the behavior of the exciton energy for the hydrogen atom, which depends on the dielectric constant by substituting e2 / for e2 in me4 /2h¯ 2 in the hydrogen 1s orbital. The part of the electric field between the e–h pair of the exciton that lies outside of the SWNT will affect the exciton energy. Since the electric field also penetrates into the nanotube itself, the dielectric response of the π electron is important, too. An important point is that even a metallic SWNT has an exciton binding energy, though it is relatively small in comparison to that of semiconducting SWNTs. The environmental effect is proportional to the many-body effect values that are scaled by the reduced mass of the e–h pair and the dielectric constant of the surrounding materials. Since the effective mass of the electron and hole depend significantly on the type (type I or II) of the semiconducting nanotube, which is understood by the trigonal warping effect, the many-body effect depends on the type of the semiconducting SWNTs. Another example of an experiment that is sensitive to the excitonic nature of the photophysical processes is the two-photon absorption experiment (Maultzsch et al. 2005; Wang et al. 2005). Since the two-photon absorption has a different allowed symmetry for the initial and final states, the measurements can probe the first excited exciton states (even symmetry) directly. This measurement was able to clearly distinguish between the exciton story and bandto-band excitation absorption and emission, to make it clear that excitonic processes dominate the photophysics of nanotubes. Scanning transmission spectroscopy and transport experiments are related to one-particle excitation, where the carriers have a self-energy but no exciton binding energy. Thus, the observed energy gap from such measurements provide a clear distinction between the single (quasi-) particle picture and the exciton-bound electron picture.

1.6

Summary and future directions

In summary, we have discussed here the unique electronic structure of graphene and SWNTs. One-dimensional materials are expected to have a singular joint density of electronic states that gives rise to a sharp spectral feature associated with transition energies E ii appearing in resonance Raman and PL spectra. From the RBM spectra and by using the Kataura plot, we can determine the nanotube diameter and (n, m) values for a resonant SWNT. The double-cone structure of the energy dispersion at the Fermi energy around the K and K point is responsible for many exotic phenomena observed in the Raman spectra, not only for the intravalley scattering but also for the intervalley scattering phonon modes. The existence of a Fermi surface in doped

23

24

Optical properties of carbon nanotubes and nanographene

graphene and metallic SWNTs is also relevant to the Kohn anomaly for the G-band phonon mode. The intervalley scattering effect for the q = 0 phonon modes can be seen in the second-order Raman scattering for which a doubleresonance condition for the Raman spectra provides as strong a signal as that of the first-order Raman spectra. Because of the linear energy dispersion of the electronic structure of SWNTs and graphene, a dispersive ∂ω/∂ E laser behavior can be seen in the second-order overtone and combinational phononmode frequencies. When one of the two scattering processes in the secondorder Raman spectra is an elastic scattering event induced by defects or the boundary of the crystal, defect-induced Raman features, such as the D-band can be seen, which is useful for evaluating the defect density or the crystallite size from measurement of the ID /IG intensity ratio. Excitons are essential for describing optically excited e–h pairs and the existence of not only bright excitons but also dark excitons, and this has already been demonstrated by many high magnetic field and low-temperature experimental results (Zaric et al. 2004, 2006). The localization of the exciton wavefunction contributes significantly to the enhancement of optical transitions in nanotubes (Jiang et al. 2007b). Since Raman spectroscopy clearly reveals many aspects of nanotube physics, it is now possible to go into great detail about the photophysics of nanotubes in the frequency domain. On the other hand, many open issues remain to be addressed about the dynamics of photoexcited carriers (Hiroki et al. 2006) that are expected to be more intensively investigated in future work. When the laser pulse is sufficiently short compared to the time of a single vibration, we can see coherent phonons (RBM, and even G-band vibrations in the case of SWNTs) excited by the light through the electron– phonon interaction (Lim et al. 2006). Transmission spectroscopy studies of such coherent phonons are now providing another spectroscopic technique to observe specific phonons. When the light pulse is sufficiently strong, the optical absorption is saturated and the material becomes transparent. Saturated optical absorption provides an interesting technique for studying non-linear optical phenomena. This approach is also important for possible applications in a mode-locked laser for a fast optical switch (on the order of 10 fs) by using SWNTs as a saturable absorber especially designed for 1.54 μm operation at the telecommunication wavelength of optical fibers (Rozhin et al. 2006). One weak point about confocal micro-Raman spectroscopy is its relatively poor spatial resolution of around 1 μm, which is not sufficient for nanotechnology applications. Using a non-propagating light beam at the tiny aperture of the tip of an optical fiber, or resonantly coupled to a tiny metal tip, a much smaller spatial resolution (∼10 nm) can be achieved in a Raman-spectroscopyrelated experiment. This approach has matured in the past few years and is called near-field optics (near-field Raman and PL, see for example the contribution by A. Hartschuh et al. in Jorio et al. 2008). It is expected that near-field optics will become an important technique for the next generation of photophysics. Carrier injection and exciton formation are important for semiconductor-SWNT photodevices and for this reason exciton dynamics in SWNTs will also become an interesting subject. Biexciton formation in a

References

SWNT is a new topic at an early stage of investigation, while an Auger process has been identified as an important process for dissociating two excitons (the contribution by Y-Z. Ma et al. in Jorio et al. 2008). Nanometer devices made with a graphene ribbon are now becoming popular. Because of the singular behavior of the quantum Hall effect in graphene, many unusual transport properties have been observed (Geim and Novoselov 2007). Excitonic effects should be observed in a narrow graphene ribbon in analogy to the many excitonic effects that have been seen in SWNTs. The exciton physics of a metallic SWNT is an especially interesting subject in which the dynamic dielectric response of the exciton will be a key issue for gaining further understanding of environmental effects in metallic SWNTs. A new nanostructure of a linear single-atom carbon chain structure encapsulated in the core of a SWNT is an ultimate 1D carbon material (Zhao et al. 2003; Nishide et al. 2006, 2007). The study of the photophysics of the nanochain of carbon would be a very interesting fundamental subject for future study. From the standpoint of metrology, measurements of the absolute value of the Raman intensity will be important as a standard for evaluating the signal from a nanotube (or graphene) sample observed under many different conditions (see, for example, the contribution by A. Jorio et al. in Jorio et al. 2008). The fabrication of photodevices based on SWNTs on Si or other semiconductors will become more important in the future. In this case, the energy transfer of photoexcited carriers between two SWNTs whose length and distance is of the order of the wavelength of light will become an important subject not only for nanoscience but also for nanotechnology. Experimental techniques for nanoscience and technology should be improved, making use of the many possibilities introduced by carbon nanotubes and other new materials now available at the nanoscale. For example, in resonance Raman spectroscopy, a quick and continuous change of E laser without changing the focused position in the sample is highly desired. The technique for making fiducial marks on a sample and combining spectroscopy with TEM and SPM will become increasingly important. Thus, there are many opportunities in nanoscience and nanotechnology for both SWNTs and graphene to continue to have a large impact on other fields of science and on other areas of technology. The authors thank their many valuable collaborators and coworkers for many discussions and comments. R.S. acknowledges a Grant-in-Aid (Nos. 16076201, 20241023) from the Ministry of Education, Japan. A.J. acknowledges financial support by FAPEMIG, CAPES and CNPq, Brazil. MIT authors acknowledge support under NSF Grant DMR 04-05538.

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Defects and disorder in carbon nanotubes

2

Philip G. Collins

2.1

Introduction and outline

The presence of defects subtly alters all of a material’s properties. Chemical reactivity, mechanical strength, optical absorption, and electronic transport all vary with defect concentration. Despite these wide-ranging effects, however, it is very rare for defect control to be critical to a particular application. For example, electronic circuits perform perfectly well in the presence of defects. Even though a defect might trap or scatter charge carriers, a typical conductor will electrostatically screen such sites and still maintain a preponderance of unaffected carriers and conduction channels. In reduced dimensions these generalizations begin to fail. In onedimensional (1D) systems, defects play the most significant roles, and they can dominate physical property measurements. For example, the strength of a 1D chain cannot exceed its weakest link. Charge carriers in 1D are confined to a single trajectory and cannot avoid a scattering center nor scatter into nearby momentum states. Such consequences are critically important to the newly developing field of nanoscience, in which low-dimensional materials are synthesized, characterized, and integrated into applications. Carbon nanotubes (CNTs) represent an ideal materials system for studying and probing the possible effects of defects. CNTs, and single-walled nanotubes in particular, are nearly 1D materials exhibiting a wide range of interesting physical behaviors. Enormous progress has been made over the last decade in characterizing and studying this material system, to the point that the strength, chemical activity, and electrical resistivity of individual CNTs have all been intensively investigated. However, despite the progress in CNT research generally, CNTs with point defects represent a rich and mostly untapped system. Experimental investigations have only begun to address the loss of mechanical strength, the change in optical activity, or the increase in electrical resistance that can be attributed to point defects. At present, no clear consensus has emerged on the properties of CNT defects, nor especially any quantitative correlation between these properties and different defect types. This topic is likely to demand more attention as the field matures away from claims of CNTs as perfect and defect free.

2.1 Introduction and outline

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2.2 Categorization of defect and disorder

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2.3 Experimental identification of defects

50

2.4 Physical consequences of defects and disorder

68

2.5 Concluding remarks

80

Acknowledgments

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References

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The purpose of this chapter is therefore to provide a practical introduction to the subfield of CNT defects, and to provide the background necessary to begin clarifying the physical consequences of defects, especially wherever these effects can be distinguished from the underlying CNT properties. For this context, it is appropriate to narrowly define CNTs as highly crystalline materials sharing common physical and chemical properties, and to attribute all departures to varying degrees of modification. The most narrow definition of pristine CNTs, to be used throughout this chapter, is of seamless and undistorted graphitic cylinders with aspect ratios exceeding at least 100. Single-walled nanotubes (SWNTs) consist of a single cylinder, and multiwalled nanotubes (MWNTs) have multiple concentric cylinders separated by nearly the same distance as the van der Waals stacking of sheets in graphite. Throughout the chapter, separate abbreviations will be used to distinguish SWNTs from MWNTs, with carbon nanotubes in general being abbreviated as CNTs. A great deal of theoretical modelling has been completed on the topic of CNT defects, and much of it is reviewed by other chapters in this Handbook. This chapter has a distinctly different purpose. As a practical introduction, this chapter aims to emphasize promising directions and practical techniques, in order to help researchers gauge future directions of the field. The chapter also attempts to focus on the middle ground where experimental and theoretical techniques have tractable overlap, and it is therefore somewhat biased towards SWNTs and single defects or very low defect densities. Despite this focus, many potential CNT applications require high degrees of disorder. For these, extreme oxidation of CNTs is an enabling first step towards interconnecting CNTs with polymers, catalytic particles, biomolecules, or other functional elements to make composite hybrids. Many reviews exist on the chemical methods for further tailoring highly disordered CNTs (Hirsch 2002; Banerjee et al. 2005; Lu and Chen 2005). The organization of the chapter is into three main sections. Section 2.2 is a pedagogical categorization of the types of defects and disorder found in CNTs. Beginning with lattice vacancies and bond rotations, the discussion progresses through extrinsic disorder and concludes with a brief section on highly disordered CNT materials. Section 2.3 next describes the most effective experimental methods for locating defects based on their short-range effects. The section focuses on precision techniques useful for individual point defects, but also touches upon lower-precision methods suitable for bulk characterization. Section 2.4 concludes with a review of the long-range consequences that CNT defects have on physical properties. Certain types of defects disproportionately perturb physical properties. Besides being consequential to CNT applications, these types of defects are also the easiest to observe experimentally, making them overrepresented in the field’s current literature. As defects having more subtle effects are observed and characterized, this natural bias will fade.

2.2

Categorization of defect and disorder

In the period of 1995–96, opinions differed on the crystalline quality of carbon CNTs. Ebbesen declared, “Evidence is accumulating that carbon nanotubes are rarely as perfect as they were once thought to be,” (Ebbesen and Takada

2.2 Categorization of defect and disorder

1995) only a few months apart from the publication of “Crystalline Ropes of Single Walled Nanotubes” (Thess et al. 1996). The second publication, combined with the worldwide sharing of material by R. E. Smalley’s research group, helped to popularize the latter judgment. Over the ensuing decade, the prevailing conception of CNTs and particularly SWNTs continued to be one of perfectly crystalline wires, with every major advance in SWNT synthesis (Thess et al. 1996; Cassell et al. 1999; Nikolaev et al. 1999; Hata et al. 2004) reinforcing this belief through claims of defect-free material. Of course, it is thermodynamically impossible for defect densities to go to zero, even in highly pure crystalline systems. The ground state of pure carbon systems is the sp2 conjugated lattice of the graphene sheet, but many possible topological perturbations are possible. Defects that are not too energetically costly will exist in equilibrium proportions determined by their Boltzman factor exp(−E d /kb T ). One might imagine that the strong sp2 -bonding network of graphitic carbons would energetically preclude defect formation, and in fact atomic vacancies are quite unfavorable. But line and screw defects, interstitials, and bond-rotation defects are all observed in graphites in considerable numbers. An extensive literature has developed around defects in graphite, much of which is applicable to newer materials such as graphene and CNTs. Telling and Heggie have recently completed an extensive review of radiationinduced damage in graphites (Telling and Heggie 2007), a field dating back to the beginnings of the nuclear industry over 50 years ago. Even though this research has focused on high-energy processes, it still provides an excellent starting point for understanding CNTs. Two special differences between graphites and CNTs arise. First, the allowed categories of defects are restricted by dimensionality. SWNTs obviously cannot contain higher-dimensional defects like line and screw dislocations. Isolated SWNTs also cannot support many of the common interstitial defects, which are stabilized by intersheet bonding in the graphites. On the other hand, the point defects found in graphites are more complicated in CNTs because of circumferential, curvature-induced strain. Since carbon bonds exist at various angles to this strain, defect stabilities depend precisely on position within the lattice, as well as CNT diameter and helicity (Carlsson 2006). This section begins by continuing along these lines, analyzing CNT defects likely to exist in the most pristine materials. Next, the effects of postsynthesis processing will be considered, since most mechanical and chemical processing serve to nucleate even more defects. The section concludes by considering two types of disorder that are not lattice defects per se—carbonaceous and noncarbonaceous material in the surrounding environment—but that, nonetheless, can be nearly indistinguishable from defects in the ways that they perturb CNT properties.

2.2.1

Intrinsic defects in highly ordered CNTs

2.2.1.1 Vacancies The most typical type of defects in crystalline lattices are point vacancies, interstitials, and bound complexes of the two. A missing or extra atom is a small perturbation in weakly bonded metal crystals, but the same is not true in

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graphene. A graphene vacancy breaks three short, strong C=C bonds and costs nearly 7.8 eV (Kaxiras and Pandey 1988). Even though CNT synthesis is a high-temperature process (up to 3000 ◦ C), the likelihood of incorporating such sites during crystal growth remains extremely low. Furthermore, the carbonrich synthetic environment excludes the likelihood of vacancy survival, beyond mere thermodynamic considerations. Vacancy defects are not uncommon, however, and are produced postsynthesis. For example, knock-on events by high-energy electron, ion, or neutron radiation can dislodge or fully remove a carbon atom. This type of vacancy initially results in three dangling bonds that will immediately rehybridize or react with surrounding molecules (Fig. 2.1). One rearrangement amenable to theoretical study is the so-called 5-1DB vacancy defect, in which two of the dangling bonds bridge to form a strained, pentagonal 5-membered ring, leaving only a single dangling bond “1DB” (Ajayan et al. 1998), confirmed by Lu and Pan (2004). The resulting structure costs only 5–6 eV, though substantial disagreement continues over whether this rearrangement occurs spontaneously and whether it is even the lowest-energy configuration (Berber and Oshiyama 2006). Further theoretical effort has focused on how the dangling bonds associated with one or more vacancies might serve as sites for interconnecting nanotubes, providing chemical sensitivity, or for incorporating dopants (Liu et al. 2006a,b; Kotakoski et al. 2007). Considerations of dangling bonds are somewhat academic when CNTs are surrounded by a typical experimental environment. Unlike vacancies produced deep in a graphite crystal, CNT surfaces interact with adsorbed gases, moisture, supporting substrates, and nearby amorphous carbons, all of which provide spontaneous reaction pathways to saturate dangling bonds. In all but ultrahigh-vacuum conditions, most intermediates are susceptible to nucleophilic attack by H2 O, making −OH-terminated vacancies one of the most likely, and physically relevant, configurations. The metastable chemistry of a single vacancy also drives a tendency towards vacancy coalescence. In graphite, a divacancy formed by two missing atoms only costs ∼ 1 eV more than the monovacancy, and nearly 6 eV less than two separated monovacancies. Thus, particularly during annealing processes, single defects are observed to merge and grow into larger voids in graphene

Fig. 2.1 Small vacancy defects in the graphene system. Monovacancy (a) before and (b) after reconstruction and H-termination of the remaining dangling bond. Divacancy (c) before and (d) after reconstruction.

2.2 Categorization of defect and disorder

sheets. In SWNTs, the vacancy migration barrier is only 1 eV, suggesting mobility at temperatures as low as 100–200 ◦ C (Krasheninnikov et al. 2006). The divacancy has a few notable properties, including the ability to reconstruct into a pentagon, octagon, and pentagon (5-8-5) structure that is free of dangling bonds. With the additional strain of curvature, divacancies in SWNTs are believed to have smaller formation energies than monovacancies by nearly 1.5 eV (Krasheninnikov et al. 2006). 2.2.1.2 Interstitials Interstitials form a second important category of defects generally. An interstitial defect consists of an extra atom not on any lattice site, bonded within an otherwise perfect lattice. Oxygen interstitials, for example, limit the purity and performance of the world’s best silicon crystals. In diamond and graphene, very short lattice bonds prohibit the inclusion of interstitials, and even atomic hydrogen cannot freely diffuse through a graphene sheet. In the case of graphites, however, the interstitial nomenclature is relaxed to include out-of-plane, covalently bonded carbon atoms. Such defects cost nearly 5.5 eV, and they will not be incorporated during CNT synthesis except by arc or laser ablation, where extremely high-temperature, carbon-rich plasmas are used. Instead, the primary source of interstitials is likely to again be knockon damage, since the production of each vacancy also releases a carbon atom. These atoms usually remain confined within a graphite crystal, and they will be accommodated by producing a covalent link across two neighboring graphene sheets. This type of interstitial, driven by aromaticity and low coordination, will migrate within and between graphene layers and until it binds to a vacancy site to produce a stable, vacancy–interstitial complex called a Frenkel defect. In CNTs, the carbons freed by knock-on damage are not so well trapped, particularly when the material is SWNTs diluted for imaging purposes. Furthermore, carbons on surfaces remain highly mobile, since they cannot be stabilized by bonding among two graphene sheets. A carbon bound to a single sheet has a binding energy of 1.2 eV, but a migration energy less than 0.1 eV (Xu et al. 1993; Nordlund et al. 1996). Thus, an interstitial is highly mobile and, since the barrier to Frenkel recombination is only 0.2 eV, these defects are likely to be short lived in CNTs (Telling and Heggie 2007). If they do not recombine, candidate interstitials probably agglomerate or bind with adsorbates to form small, physisorbed clusters of graphitic or amorphous carbon. Thus, Frenkel defects, like vacancies, are not likely to play large roles in the properties of CNTs experimentally. When discussing CNTs, it is important to distinguish between interstitials, adducts, adsorbates, and intercalants. As defined here, CNT interstitials are atomic carbons covalently bonded between two carbon shells and, rarely, to just one. Covalent attachments by other atoms or molecules are termed adducts and are treated below in Section 2.3. Chemisorbates and physisorbates are more weakly bound than adducts, comprising a loosely defined continuum of binding strengths associated with different degrees of charge transfer to the CNT surface. When these adsorbates sit between two or more layers of graphene, they are termed intercalants. The intentional intercalation of graphite

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crystals has an extensive history (Enoki et al. 2003), since it provides a means to tailor graphite’s physical properties. In CNTs, such intercalants are described as a source of extrinsic disorder in Section 2.2. 2.2.1.3 Bond rotations and non-hexagonal rings While vacancies and interstitials are highly disfavored, bond rotations are not, and these constitute the most prevalent type of defect in high-quality graphites. A single bond rotation can be incorporated into graphene at a cost of approximately 3.5 eV without disturbing the sheet’s topology or sp2 conjugation. The rotation only affects four adjacent hexagons, converting two into pentagons and two into seven-sided heptagons. This particular 5-7-7-5 configuration has been studied extensively and is known in the literature as a Stone–Wales (SW) defect (Dienes 1952; Stone and Wales 1986). Despite their prevalence, SW defects are comparatively inconsequential to many physical properties and difficult to observe experimentally. In CNTs, the SW defect is presumed to be as pre-dominant as it is in graphite, even though CNT synthesis, especially chemical vapor deposition (CVD) synthesis, proceeds at lower temperatures than typical graphitization. The SW defect has a complex energy scale, including a formation barrier of no less than 9 eV and a dissolution barrier of 5.5 eV (Dumitrica et al. 2006). The former might seem to preclude formation of any SW defects at all, but this barrier drops rapidly in the presence of interstitials (Ewels et al. 2002). As the synthesis of most CNTs occurs in a surplus of adsorbed, reactive carbons, it is reasonable to estimate that the SW concentration approaches the thermodynamic limit of its 3.5 eV net cost. In this case, SWNTs synthesized at 3000 K will contain 1 SW defect per μm, on average. As noted by Ewels et al. this mechanism also increases the possibility of SW creation due to ion beam or electron radiation in the presence of amorphous carbon, which often coat CNTs. Furthermore, these SW defects are long-lived after the initial synthesis, being trapped in the lattice by the high dissolution barrier. A slightly simpler defect than the SW configuration is a single “5–7” pair, in which a pentagon adjacent to a heptagon neatly replaces two hexagons (Fig. 2.2). The 5–7 defect introduces only a slight pucker to a graphene sheet and it is the irreducible building block for higher-order defects like the SW. However, a single 5–7 defect breaks rotational symmetry and modifies the orientation of the graphene lattice. This defect exists as a special type of dislocation dipole, in which two misoriented graphene sheets are seamlessly joined by hexagons up to a single point of residual disorder. Alternately, one can imagine the progressive growth of a graphene sheet, in which the incorporation of a 5–7 defect results in a permanent rotation of the primary lattice vectors of all subsequently added carbons. This visualization is important for understanding that isolated 5–7 defects cannot be annealed from graphene layers without a massive, plastic reconstruction of every carbon bond in a half-plane. The incorporation of a 5–7 defect results in a loss of commensurability between the layers in graphite, making its effective cost much higher than a SW defect. One way to lower this cost is by incorporating a second 5–7 defect and restoring the lattice to commensurability with its underlying layer. The SW defect is, in fact, a pair of 5–7 defects having zero separation. Thus, even

2.2 Categorization of defect and disorder

37

Fig. 2.2 (a) A single 5–7 defect and (b) a 5-7-7-5 Stone–Wales configuration. Sighting along the zigzag edges in a side view visualization clarifies the nucleation of a dislocation by the 5-7 defect, and the absence of long-range disorder in the Stone–Wales case.

though a 5–7 defect is not readily removed by annealing, if it becomes mobile it can combine with another to form a SW defect. Alternately, a SW defect under strain can separate into two, counterpropagating 5–7 defects (Walgraef 2007). In single-layer graphene and SWNTs, the mobility barrier for an isolated 5–7 defect is low enough that high temperatures and/or strains can nucleate and then dissociate SW defects into separated 5–7 defects (Xia et al. 2000; Dumitrica et al. 2006). Unlike graphite layers, CNT growth appears to have no preferred registry, and there is no impetus for 5–7 defects to occur in pairs during synthesis. Instead, they are just as likely to be incorporated singly as paired into a SW defect. In the cylindrical CNT geometry, the dislocation introduced by a 5–7 defect is manifested as a change in helicity. A SWNT with indices (n, m) will seamlessly change to (n ± 1, m ∓ 1); the incorporation of d 5–7 defects around a SWNT circumference can change its indices to (n ± d, m ∓ d) (Chico et al. 1996a). Since the SWNT band structure is sensitive to helicity, one or more 5–7 defects results in the equivalent of a SWNT–SWNT heterojunction, a topic discussed in more detail in Section 2.4.2. This effect is a remarkable departure from the properties of graphite. Similar junctions can also be constructed from isolated pentagons or heptagons not joined in a 5–7 pair. In these case, the helicity changes are accompanied by local concave or convex distortions to the SWNT sidewall. The additional curvature is allowed in SWNT and fullerene topologies, but forbidden in planar graphite, making these types of defects a category unique to CNTs. The addition of heptagons around a SWNT circumference will cause the diameter to flare out, and with an equal number of pentagons the distortion can be seamlessly joined to a larger diameter SWNT, and in fact these were the very first properties of CNTs ever studied (Iijima et al. 1992b). In general, equal numbers of pentagons and heptagons allow extended structures where both chirality and diameter may vary, though some chiralities may only be joined with accompanying kinks or bends. An elegant demonstration of this effect has been reported by Yao et al. (Yao et al. 2007). The synthesis of different SWNT diameters is partly determined

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Defects and disorder in carbon nanotubes Table 2.1 Formation energy E d and relative concentrations of common intrinsic defects. Defect type

Monovacancy Divacancy Interstitial or other covalent sp3 adduct Stone–Wales 5-7-7-5 Single 5–7 defect

E d , eV

Equilibrium defect concentrations for synthesis at T = 1200 K (CVD)

at T = 3000 K (plasma)

7.0–7.8 8.7 (HOPG) 4.5–5.5 (SWNT) 5.5

10−33 10−38 10−22 10−24

10−13 10−16 10−9 10−10

3.5 3.4 (SWNT)

10−15 10−15

10−6 10−6

by growth temperature (Bandow et al. 1998), and by changing temperature midgrowth one can move the optimum diameter from one value to another. Yao et al. nucleated SWNTs in one diameter range and then changed the growth temperature, making a different size more preferable for continued growth. The SWNTs responded by changing diameter through the incorporation of 5–7 defects (Yao et al. 2007). The technique promises to improve the intentional creation of SWNT heterojunctions for further investigation. 2.2.1.4 Summary and caveat In summary, the types of defects intrinsic to CNTs provide some interesting surprises. The most conventional types of defects, like vacancies and interstitials, are unlikely to be found in pristine CNTs. The Stone–Wales 5-7-7-5 configuration, unique to graphitic systems, is predicted to be a relatively common type of defect. Equally likely are various topology-changing combinations of 5-7 pairs, including physically separated pentagons and heptagons that are uniquely stable in CNTs. These latter defects can have significant, long-range consequences. Table 2.1 below summarizes estimated defect energies E d of each defect type and provides an estimate for the expected equilibrium concentration of each at two common CNT synthesis temperatures. Small SWNTs contain roughly 106 carbons per micrometer of length, so that even the most common types of defects have mean separations of 1 μm. Note that predicted values of E d vary widely in the literature and, particularly in CNTs, are not readily accessible experimentally. The equilibrium concentrations in Table 2.1 set lower limits on experimental defect concentrations, and are in reasonably good agreement with some highly crystalline SWNTs. Many experiments, however, observe much higher defect concentrations. Carlsson has suggested that heats of formation, which are smaller than the E d values, might provide concentrations in better agreement with experiment (Carlsson 2006). As described in the following parts of Section 2.2, however, many additional sources of defects and disorder also contribute. Finally, having reviewed the possible intrinsic sources of disorder, it must be pointed out that the CNT end-cap is itself an unavoidable defect or cluster of defects. With a possible exception for very long CNTs, the concentration of

2.2 Categorization of defect and disorder

two end defects per CNT is overwhelmingly larger than all other categories of intrinsic defects combined. While the point may seem pedantic, Sections 2.3 and 2.4 will demonstrate how these end effects can dominate chemical and optical attempts to find and to quantify defect densities in CNTs.

2.2.2

Environmental disorder

Physics predicts special consequences for systems confined to 1D, to the extent that they can be extraordinarily different from higher-dimensional materials. In principle, a SWNT is a promising candidate to test these predictions and uncover new physics. In practice, even a pristine and defect-free SWNT exists in an imperfect and disordered 3D world, and this coupling to a 3D environment affects the degree to which unusual 1D physics might be observed. This section considers two primary sources of disorder, the environment surrounding a CNT and the substrate supporting it. Like the lattice defects described above, these two sources can be optimized but never wholly removed experimentally. In this sense, the environment is an intrinsic part of CNT research. Depending on the strength of the interactions, environmental consequences may be as strong as for defects within the carbon lattice. This fact is a particular difficulty for comparing experiment with theory, since theory often treats CNTs in isolation and experiment never does. 2.2.2.1 Weakly bound adsorbates The graphite surface is a relatively inert and clean one, and freshly cleaved graphites were a favorite substrate for early scanning tunnelling microscopy work because of its ease of preparation. The graphite surface is hydrophobic, and it is not susceptible to appreciable charge transfer from most adsorbates. Nevertheless, it would be incorrect to conclude that CNTs in ambient are clean or adsorbate free. Even hydrophobic surfaces can have adsorbed monolayers of H2 O on them. Soluble airborne gases and contaminants, including alkali salts and light hydrocarbons, will also adsorb onto these surfaces. These mobile, low atomic number species are difficult to image by either TEM or STM, and practitioners of these techniques understand that “cleanlooking” images do not always capture the real extent of surface chemical disorder. Three differences between CNTs and graphite suggest ways that CNTs are even more sensitive to adsorbates than graphite surfaces are. First, the CNT curvature results in partial sp3 hybridization, accentuated in small SWNTs, that enhances the π electron density on the cylinder’s outer surface. Common dissolved species like Na+ and H+ will dynamically interact with this surface electron density, even in the absence of chemical bonding or static charge transfer (Kuhn and Silversmith 1971; Bradley et al. 2003). As recently demonstrated electrochemically, these effects are strong enough to turn the insulating surface of a diamond into a conductor (Chakrapani et al. 2007). Second, curvature also frustrates dense packing of CNTs. Whether bundled together, settled onto a surface, or packed into a pellet, CNT materials have enhanced specific surface areas comprised of physically interconnected, interstitial voids. Adsorbates in these voids can be better coordinated than on

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flat graphene surfaces, and the voids readily accommodate a wider range of molecular shapes and sizes than do interlayer graphite interstices (Stan and Cole 1998; Eswaramoorthy et al. 1999; Cole and Hernandez 2007). Thus, even though a graphite crystal is not intercalated by air or moisture at room temperature, the interstitial pockets between SWNTs must be considered filled unless specifically degassed. Finally, every atom in a hollow SWNT is a surface atom, and the addition of adsorbates is a proportionally larger perturbation in SWNTs than in solid materials. The system can no longer be considered a simple carbon lattice if it is, in fact, a carbon lattice interacting with H-bonded water dipoles, chemisorbed O2 − , or physisorbed hydrocarbons. In the context of novel, low-dimensional physics, the effects of these adsorbates can be enormous because they break the remaining rotational and translational symmetries of a SWNT and physically extend the system away from an idealized, 1D line. Placing CNTs into vacuum is insufficient to remove these adsorbates, in the same way that it does not remove chemisorbed molecules from the internal surfaces of a vacuum apparatus. Extended baking in ultrahigh vacuum, combined with annealing or surface milling, are the standard surface-science techniques for preparing atomically clean crystal surfaces. SWNTs are compatible with high-temperature vacuum degassing, and a small number of experiments have observed substantial electronic effects from even mild treatments (Collins et al. 2000a; Bradley et al. 2000; Sumanasekera et al. 2000; Derycke et al. 2002; Kim et al. 2003b; Kruger et al. 2003; Kingrey et al. 2006). This literature indicates the need to consider environmental disorder when interpreting CNT results, and similar effects may be equally important for ongoing graphene research (Ishigami et al. 2007; Moser et al. 2008). Rather than trying to achieve perfect vacuum, a second method of eliminating environmental disorder is to encapsulate CNTs into a homogeneous and uniform chemical environment. A breakthrough in the field of CNT optical spectroscopy occurred when SWNTs were solubilized by hydrophobic surfactants and effectively separated into isolated, uniform micelles (Bachilo et al. 2002; O’Connell et al. 2002). Cocooned into individualized, highly uniform pockets, SWNTs finally began to exhibit spectroscopic fingerprints associated with their different electronic structures, fingerprints that had been quenched or otherwise hidden in previous measurements. Subsequent work, however, has proven that these encapsulating environments must be included in the modelling and interpretation of optical data (Lefebvre et al. 2008). And while encapsulation or vacuum processing can minimize the effects of environmental disorder, they do not remove other common forms of disorder such as amorphous carbon adsorbates or the substrate effects described below. 2.2.2.2 Substrate effects As a component of the surrounding environment, substrates deserve special consideration as a source of strong perturbations. A substrate has a different bulk chemistry, electron affinity, and work function from a SWNT, and furthermore will have its own surface electronic structure and morphology. Substrate–SWNT interactions can have multiple unintended physical and

2.2 Categorization of defect and disorder

chemical consequences, only one of which is to help trap the adsorbates described above. For example, CNT devices are often fabricated by placing CNTs on a dielectric or on a thin oxide grown on a semiconductor. For electrical measurements, this CNT-oxide-semiconductor architecture allows electrostatic coupling between a semiconducting CNT and an underlying electrode to vary the CNT Fermi level and produce transistor-like switching (Biercuk et al. 2008). This same coupling, however, is decidedly 3D. The dielectric properties of the substrate screen a portion of the long-range, electron-electron interactions that are predicted to cause special, non-Fermi liquid behaviors in 1D conductors (Kane et al. 1997). If a 1D conductor strongly coupled to electrodes is a 3D system, then it is well suited to substrate-bound applications like electronics but no longer a prime candidate for probing novel physics. Furthermore, even traditional applications like transistors cannot take full advantage of CNTs without special care. Modelling of “needle contact,” quasi1D electrodes suggests that order-of-magnitude performance improvements remain to be observed if CNT dimensionality can be more effectively managed and integrated into appropriate architectures (Heinze et al. 2005). Besides electrostatic gating, the principal CNT–substrate interaction is generally attributed to be van der Waals adhesion. This adhesion, which is insensitive to lattice mismatch with the SWNTs, is strong enough to quench long-wavelength SWNT phonons like twistons and keep SWNTs securely attached to surfaces. However, the premise that SWNTs are largely inert and only weakly adsorbed by substrates is open to ongoing research. Modelling and experiment both suggest that more complex electronic rehybridization occurs spontaneously, even on relatively stable substrates such as SiO2 (Czerw et al. 2002; Maiti and Ricca 2004; Tsetseris and Pantelides 2006). This type of rehybridization is driven in part by CNT strain, and therefore is a diameterdependent effect. Another strong effect occurs at the interfaces between small SWNTs and metals. Metal coatings on SWNTs can very effectively shunt the SWNT’s 1D electrical conductivity, invalidating the normal experimental technique of 4-point resistance measurements (Bezryadin et al. 1998). These effects may also explain the diameter-dependent, interfacial resistance between metallic SWNTs and metal electrodes, which complicate measurements of ballistic conductance in small-diameter SWNTs (Heinze et al. 2005; Kim et al. 2005). Thermal conductivity indicates similar effects on insulating substrates. Freely suspended, current-carrying CNTs will self-heat to the point of oxidation, but on substrates decidedly different characteristics are observed (Yao et al. 2000; Collins et al. 2001; Pop et al. 2006). In fact, thermal emission microscopy is unable to measure any temperature rise in the latter case, suggesting that heat is very effectively transported out of the carbon lattice by strong substrate interactions. In summary, a number of consequential effects occur when CNTs sit on substrate surfaces, even chemically inert ones. In addition to these general surface interactions, CNTs have enhanced coupling to a substrate when defects are present. Defects in a CNT will bind covalently to many substrates (Krasheninnikov et al. 2002a; Kotakoski and Nordlund 2006), complicating the interpretation of their effects. And even if

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the CNTs are defect free, defects in the substrate will interact with the CNT. Non-crystalline substrates like thermally grown SiO2 have extensive defect populations that are nearly continuous in energy, and even defects buried below the surface will interact dynamically with surface atoms, with surface adsorbates, and with a CNT’s conduction electrons (Freitag et al. 2007). When cooled to cryogenic temperatures, this substrate disorder becomes static and its modulation of the CNT potential can be directly imaged (Tans and Dekker 2000; Woodside and McEuen 2002). At room temperature, these defects dynamically couple to many degrees of freedom and can participate in inelastic, dissipative scattering, even though lattice defects are typically only considered sources of elastic scattering. For example, defects that are energetically close to the Fermi level dynamically capture and release free carriers from the CNT. This charge trapping and the electronic noise associated with it is a primary engineering challenge of oxide semiconductor interfaces, and the problem is aggravated in CNTs by their small carrier numbers (Collins et al. 2000b; Lin et al. 2007). Substrate defects that are energetically deep and/or physically distant are nominally inconsequential to CNTs. Their long-range, perturbing potentials only minimally contribute to sensitive measures such as electron backscattering (McEuen et al. 1999; Ando 2005). However, this generalization changes when large electric fields are present from gate biasing voltages. When gated, the small diameter of a CNT produces excessive electric fields that can initiate chemical redox processes in surface contaminants and cause field emission into and out of deep oxide traps (Fuhrer et al. 2002; Radosavljevic et al. 2002). Under these conditions, these defects can become strongly interacting perturbations resulting in memory effects and hysteresis (Bradley et al. 2003; Kim et al. 2003b; Gruneis et al. 2007), distinct from the chargetrapping noise described above. Furthermore, substrate defects and chemical variability may play a primary role in some cases of CNT chemoresistance, an actively investigated area that is poorly understood at present (see Section 2.4.2). Thus, substrate defects both near to and far from the Fermi level interact with CNTs and give rise to distinguishable dynamical properties. By performing measurements on isolated, freely suspended CNTs, the research community continues working to differentiate between the properties of CNTs and CNT– substrate interactions. However, the appreciation of these interactions is relatively recent, and the literature broadly attributes properties to CNTs that may be specific to suspended SWNTs (Cao et al. 2005; Mann et al. 2007) or suspended MWNTs (Frank et al. 1998; Poncharal et al. 2002). A similar process is repeating itself in the current area of graphene research, though at much greater speed. In 2008, a rapidly growing consensus is developing that bilayer graphene may be uniformly superior to single-layer graphene because of the consequences on the initial layer of substrate disorder (Lin and Avouris 2008). So far, Section 2.2 has treated two types of disorder that are intrinsic to performing measurements on CNTs. Next, this section considers extrinsic variables that vary more widely from sample to sample and from technique to technique.

2.2 Categorization of defect and disorder

2.2.3

Disorder introduced by processing

Once synthesized, the manipulation of CNTs into a useful state can involve extensive mechanical and chemical processing. This processing changes the CNT surface and often creates new defect sites. Any meaningful evaluation of defects and their consequences must be in the context of processing history and, consequently, of the targeted application. For example, an isolated, mmlong SWNT is ideal for building multicomponent circuits because it is likely to have long, defect-free regions; the chemist, on the other hand, might prefer the same SWNT chopped into short segments and solubilized. This section addresses some of the most common processing techniques, including the intentional production of new types of defects. Park provides a more extensive review of purification techniques (Park et al. 2006). First, though, there is one type of processing that tends to improve CNT crystallinity. Vacuum heating anneals CNTs effectively at 1200–1500 ◦ C, even though bulk graphitization of carbons requires temperatures exceeding 2200 ◦ C. The temperature difference is related to the high degree of crystallinity pre-existing in most CNTs. Virtually all oxygen-containing functionalities can be desorbed from a CNT lattice by 1000 ◦ C, and remaining defects and vacancies become mobile by 1200 ◦ C. Above this temperature, 5–7 defects can migrate and annihiliate, and monovacancies left by desorption of carboxylic groups can be healed (Krasheninnikov and Nordlund 2002; Krasheninnikov et al. 2006). An elegant demonstration of the opening and closing of sidewall vacancies was performed using SWNTs decorated with C60 molecules. Under TEM observation, molecules were observed moving in and out of sidewall holes in individual SWNTs (Smith et al. 1998; Monthioux et al. 2001). At still higher temperatures, a structural relaxation can occur in which pairs of SWNTs merge into larger diameter tubes (Terrones et al. 2000; 2002a). Of course, vacuum annealing tends to bring the material into thermodynamic equilibrium, meaning that the density of SW defects will typically be reduced, but not to zero. 2.2.3.1 Purification The least damaging purification technique employs filtration to separate CNTs from carbon and metal contaminants. Fullerenes and carbon onions, metal clusters, and polyaromatic carbons all readily wash through fine microfilters, whereas high aspect ratio CNTs generally do not. Consequently, a relative pure “buckypaper” of CNTs can be formed on such filters, dried into a free-standing film, and further studied (Bandow et al. 1997; Bonard et al. 1997; Eisebitt et al. 1998). The filtering process, while relatively gentle, involves viscous and capillary forces at the nanoscale that generate not only mechanical entanglements but also highly strained kinks and bends. The SW defect is believed to be the primary mechanism for plastic strain release in these conditions (Nardelli et al. 1998), and the further accumulation of such defects ultimately ruptures the carbon lattice and produces reactive multivacancies. In experimental studies, a relatively high resilience and resistance to damage is observed in SWNT bundles and MWNTs, which benefit from multiple parallel shells that enhance rigidity (Falvo et al. 1997).

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Defects and disorder in carbon nanotubes

A harsher purification scheme involves oxidation in air at temperatures of 400–750 ◦ C. At the low end of this temperature range, amorphous carbons rapidly convert to CO2 through co-operative reactions with oxygen and water vapor. Graphitic carbons burn more slowly, since in these only the carbon edges are reactive. Often, bulk CNT material contains a significant fraction of graphitic impurities in the form of spherical onions, mesoscopic graphitic flakes and scrolls, and incomplete carbon shells. These carbons will oxidize slowly along their perimeter edges, just like graphite does. Therefore, it is difficult to remove them selectively from CNTs, since they burn at roughly the same rates that CNT end-caps are opened and the CNTs shortened (Ajayan et al. 1993; Ebbesen et al. 1994). In some cases of purification, it is acceptable to lose a significant fraction of the CNTs from a sample in order to fully remove these graphitic impurities. For example, shortening and thinning MWNTs can uncover inner MWNT cylinders that are highly crystalline. In these cases, partial oxidations in wet air may be run at 750 ◦ C (Hiura et al. 1995), though a majority of the CNT material is lost at such high temperatures. Acid oxidation is a more selective technique that removes amorphous carbons, graphitic mesocarbons, and contaminant metals without causing a substantial loss of CNTs. Widely used, commercial SWNTs such as HiPCO are processed in this manner (Rinzler et al. 1998). Investigation of the interaction between graphitic materials and acids has a long history. The strong acids (e.g., HNO3 , H2 SO4 , HClO4 , and HPO4 ) do not continuously etch graphene’s surface and instead covalently add to its edges and basal planes (Kinoshita 1988). On the CNT sidewall, these covalent adducts constitute a new type of defect quite different from the intrinsic defects described above. The adduct defect occurs when a single carbon atom rehybridizes to an sp3 conjugation, forming a new bond perpendicular to the sp2 plane as illustrated in Fig. 2.3(a). These adducts are not believed to be populated by high-temperature CNT growth, because at high temperatures sp3 carbons are less stable than sp2 ones. In fact, diamond-like edges spontaneously graphitize. Given this stability of the graphite lattice, CNT adducts are often metastable, and many adducts are chemically reversible alterations (Boul et al. 1999; Cui et al. 2003). This reversibility has led to recent investigation producing graphene sheets from oxidized graphite. For example, consider the variety of defect configurations depicted in Fig. 2.3. An initial acid treatment, perhaps with electrochemical assistance, adds conjugate base anions like NO3 − to the sidewall (Fig. 2.3(a)). Chemical reduction of the NO3 adduct may only strip an NO2 − ion, leaving behind surface epoxides (Fig. 2.3(b)) or ethers (Fig. 2.3(c)). Epoxides are the most common functionality observed by FTIR on oxidized graphites, but CNT curvature tends to convert them into ethers (Lee and Marzari 2006). Ethers and epoxides, along with hydroxyls, ketones, and other functional groups that do not break C=C bonds, can be titrated from graphitic surfaces at 250–450 ◦ C (Kinoshita 1988). Therefore, a purification cycle of acid etching followed by heating exists in which no carbons are lost from the lattice and a SWNT surface is returned to a defect-free state (Mickelson et al. 1998; Strano et al. 2003; Ramesh et al. 2004). In practice, however, bulk processing does not always drive these cycles to completion, instead leaving various degrees of residual

2.2 Categorization of defect and disorder

45

Fig. 2.3 (a) An adduct defect, in which the carbon lattice is not broken but a single atom is sp3 hybridized, slightly puckering the CNT wall. Here, an NO3 adduct is depicted. (b) Stripping an adduct often leaves a residual oxygen, which binds two carbons in an epoxide configuration. The epoxide is stable on flat graphites or when its axis is parallel to the SWNT axis. (c) Along a circumference, the additional strain from curvature breaks the C−C bond in an epoxide, converting it to an ether. The ether is less perturbative because each carbon is sp2 hybridized (Lee and Marzari 2006). (d) A carboxylate defect is not an adduct. It requires cleavage of two C−C bonds, both of which must be terminated (−H or −OH groups not shown), and upon annealing a C atom is lost from the system. O atoms are black in all four images.

sidewall disorder that can evolve with time. Especially following extensive oxidation, remaining contaminants or adducts can be driven by aromaticity to cluster and cooperatively produce irreversible disorder or fragmentation. 2.2.3.2 Additional oxidation and functionalization Chemical processing can be divided into two categories: attempts to purify the CNTs and attempts to modify their surfaces. In the latter, the processing is designed to chemically attack the CNT sidewall, perhaps for the purpose of covalently functionalizing it further. The acid oxidation described above can be used as a starting point for introducing additional disorder, albeit disorder with physical consequences that are different from vacancies or SW defects. Practically, acid treatments are often combined with additional oxidants, heating, and/or ultrasound treatments to produce different stable and permanent functional groups. One very important such functional group is the carboxylate, a versatile chemical handle for further derivitization (Banerjee et al. 2005) and a starting point for a wide range of research seeking to covalently link CNTs to electrically, optically, or chemically active species. Careful analysis has suggested that even extensive acid oxidation does not generate new carboxylate defects or vacancies in the pristine CNT sidewall (Ziegler et al. 2005a,b; Coroneus et al. 2008). This is due, in part, to steric hindrance limiting multiple anions from attacking the same carbon ring. Instead, the further addition of permanganate ions (or, alternatively CrO4 − , OsO4 − , or RuO4 − ) helps to

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Defects and disorder in carbon nanotubes

initiate 2 + 2 cycloaddition reactions that can break lattice bonds and produce carboxylates (Hwang 1995; Biro et al. 2002). This situation is different from the mere addition of a new bond perpendicular to the carbon lattice. The carboxylate defect (Fig. 2.3(d)) can result in the removal of carbon atoms from the system and the growth of sidewall holes, making it a much less reversible modification. The extensive use of such treatments, often combined with heating and ultrasound, is standard processing to separate SWNTs, shorten them into more convenient lengths, and etch away metal and amorphous carbon contaminants (Liu et al. 1998). Typically, the resulting materials are highly defective. Besides having a higher concentration of ends, a pre-ponderance of sidewall hydroxyl- and carboxyl-terminated vacancies are created (Mawhinney et al. 2000; Hamon et al. 2001; Monthioux et al. 2001), and as many as 5% of the C atoms in a SWNT are adjacent to these defects after processing. These heavily modified materials are not the focus of this review; instead, other reviews of SWNT chemistry begin from the premise of high SWNT defect densities (Hirsch 2002; Banerjee et al. 2005). 2.2.3.3 Mechanical processing CNTs are somewhat resistant to mechanical damage from low-power ultrasound, and after short processing times no accumulation of damage is typically observable (Furtado et al. 2004). As the processing time increases, however, there is no question that damage accumulates, and ultimately it can be identified by bulk techniques like optical spectroscopy (Benedict et al. 2005; Grossiord et al. 2007), gas adsorption (Dagaonkar et al. 2002), or others (Satishkumar et al. 1996; Monthioux et al. 2001). The exact resistance to lowpower ultrasound has not been carefully mapped for individual CNTs, in part because it depends on temperature, solvent, and other factors. Commercial horn sonicators operate at high powers (e.g., >15 W) and readily damage CNTs. Bends, buckles, and multivacancies rapidly accumulate with processing time (Lago et al. 1995; Lu et al. 1996), though the effects can be partly mitigated by working in different solvents. Similiary, ball milling is effective at mechanically disintegrating CNTs into onions and amorphous carbons (Li et al. 1999). Cleverly used, mechanical processing can do more than just destroy CNTs. At the other extreme from ball milling, an AFM tip has been used to “knick” a pristine SWNT and produce a single site of disorder for futher study (Park et al. 2002). One of the most intriguing examples of mechanical processing involved the repeated dipping of a MWNT into a Hg bath (Frank et al. 1998; Poncharal et al. 2002). Over the course of thousands of dipping cycles, visible quantities of amorphous carbons and incomplete graphitic shells were observed sloughing off MWNT bundles onto the Hg surface. Once cleaned in this manner, the MWNTs exhibited ballistic electrical conducances matching theoretical limits. 2.2.3.4 Other modifications While robust, the graphitic carbon lattice is susceptible to damage by many means. In a TEM, knock-on events by high-energy electrons rapidly create

2.2 Categorization of defect and disorder

vacancies and can completely fragment MWNTs (Chopra et al. 1995a). Ion irradiation, too, produces vacancies and reactive sites for energies as low as 50 eV (Nordlund et al. 1996). In a focused ion beam, even low-dosage Ga+ irradiation used for imaging will tear holes in carbon sidewalls. When put into an energetic plasma, CNTs are rapidly etched by O2 , covalently functionalized by H2 (Buchs et al. 2007), or fragmented by heavy ions. All of these mechanisms are useful for the intentional study of the effects of defects (Osvath et al. 2005; Robinson et al. 2006), and completely misleading when not properly accounted for. The wary experimentalist questions whether each technique in a process is intrusive or perturbative. For example, CNTs are routinely imaged by SEM and exposed to high electron doses in the electron-beam lithography process. Some troubling observations of beam-induced changes have been observed in CNT devices, with the suggestion that few-keV electrons can initiate reactions including sidewall oxidation by water (Suzuki and Kobayashi 2005; Vijayaraghavan et al. 2005). Analysis suggests that the electronic effects are due to chemical processes on the SiO2 support, not the CNT itself (Rius et al. 2007), providing a specific example of the substrate effects described in Section 2.2.2.2. In any case, a number of research groups rely wholly on AFM imaging and optical lithography in the fabrication of CNT devices in order to avoid unintentional consequences of electron beams.

2.2.4

Disorder in CNT materials

A final category of disorder is due to impure or non-crystalline starting material. Unlike the point defects that are thermodynamically intrinsic to CNTs, amorphous carbons and distorted or incomplete graphitic shells are unnecessary but common products of CNT synthesis, especially resulting from efforts to produce CNTs at low temperatures or in bulk. A long-standing problem in CNT synthesis continues to be the quantitative evaluation of purity, either before or after additional purification or processing (Niyogi et al. 2002; Itkis et al. 2005; Park et al. 2006). Beyond the difficulty of determining appropriate measures, however, is the even more difficult problem of specifying a practical definition for CNT disorder. The literature accepts a wide range of carbonaceous cylinders under the term “nanotube,” ranging from centimeterlong, 1-nm diameter single-walled cylinders to 200–300 nm diameter cylinders composed of herringbone-stacked graphitic sheets. This span fosters misunderstandings and misappropriations, as properties such as high strength, ballistic transport, and chemical inertness measured on one material certainly do not pass on to others. Disorder includes a wide range of departures from the perfect cylindrical CNT, with the most common ones described below. Tradeoffs exist between material quality and synthetic cost, and in many cases the presence of disorder can be inconsequential or even beneficial depending on the application. However, better informed decisions require more accurate characterizations of disorder and its specific consequences, and empirical batch-by-batch characterization is currently the only way to qualify CNT materials for particular commercial applications.

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Defects and disorder in carbon nanotubes

2.2.4.1 Mesoscopic carbon and non-carbon constituents The disorder that affects the immediate appearance of CNTs in high-resolution imaging is mainly due to materials adsorbed on their outside walls or incorporated into the cylindrical hollows. Beyond mere appearances, these contaminants substantially change the chemical and physical properties of bulk material. For example, amorphous carbons are more reactive than pristine CNTs, they have more varied functional groups, and they have different surface areas and adsorption characteristics. The processing described in Section 2.2.3 attempts to remove most such contaminants, but sustained efforts over the last decade have not yet produced high-purity CNT material in bulk. This failure, in turn, severely limits attempts to apply bulk techniques to the study of defect concentrations and their properties. Metal contamination of CNTs is a noteworthy problem that continues to plague many types of bulk CNT characterization (Park et al. 2006) and continues to be an area of active research (Ding et al. 2006; Jurkschat et al. 2007; Xiang et al. 2007). Transition metals help catalyze CNT growth and are required in all methods of SWNT growth. Postsynthesis, the metals are difficult to remove: they are often encapsulated by many spherical layers of graphitic carbon, and they can be in the form of pure metals, carbides, or oxides. Some metal sits at the tips of SWNTs, or inside the end-caps of MWNTs, where complete removal requires extensive etching of the caps and concurrent sidewall damage. When present, these residual metals modify most observable bulk properties—electrochemical activity, thermal stability, surface area and density, magnetic susceptibility, etc.—even when they are only incorporated at the CNT ends (Itkis et al. 2005). Mesoscopic graphitic carbons are a second primary contaminant in CNTs. While amorphous carbons and fullerenes are easily removed, graphitic flakes are more difficult to selectively oxidize or dissolve because they have the same chemistry as CNTs. Furthermore, these small flakes include incomplete or damaged cylinders partially wrapping around a CNT, in which case it becomes impossible to distinguish contaminated CNTs from highly defective ones. In general, MWNTs are described as highly crystalline and defect free when highresolution TEM observes smooth, straight, and continuous inner layers, even though one or more disordered outer layers are also resolved. Mild oxidation of the most reactive carbons can eliminate these mesoscopic graphites, but only with a loss of CNT ends, small-diameter SWNTs, and an expansion of point defects into larger sidewall holes (Ajayan et al. 1993; Tsang et al. 1993). An alternate approach is to solubilize and dilute the CNTs, which when fully separated can be fractionated (Arnold et al. 2006). Residual mesoscopic flakes are particularly problematic for optical characterization, because they provide a high density of optically active functional groups that are not necessarily present in the underlying CNTs. 2.2.4.2 MWNT structural defects In some ways, MWNTs represent an intermediate material between graphite and SWNTs. Their multilayered structure resembles graphite crystals, and like graphite a MWNT stably supports the type of defect known as an interstitial– vacancy bound pair. On closer inspection, though, MWNTs are more complex

2.2 Categorization of defect and disorder

49

than either crystalline graphites or SWNTs, and subtle forms of disorder exist. For example, curvature forces adjacent layers in a MWNT to nearly always be incommensurate, and this intrinsic broken symmetry has various consequences (Roche et al. 2001). Analysis also suggests that some synthesis techniques produce multilayer scrolls, or MWNTs composed of both scrolls and cylinders, rather than merely the purely cylindrical structure (Zhou et al. 1994; Bursill et al. 1995; Feng et al. 1996). However, the most important contributions to MWNT disorder are much less subtle. MWNT synthesis occurs under rapid, non-equilibrium growth dynamics. The resulting disorder can include tapering cylinders, variable numbers of carbon layers, and partial interior filling, examples of which are shown in Fig. 2.4. The cylindrical crystalline structure can also be severely compromised by certain additives. So-called “herringbone” and “bamboo” defects mix the physical properties of inplane and c-axis graphite by introducing graphitic layers misaligned with the primary MWNT axis. The herringbone structure, depicted in Fig. 2.4(e), consists of stacked layers or conical sections that are tilted with respect to the main axis. Herringbone ordering is a common morphology for carbon fibers and nanohorns (Yudasaka et al. 2008), and materials having this structure may or may not have a hollow interior. Bamboo defects consist of several transverse, internal walls segmenting the interior

Fig. 2.4 TEM images of (a) a pristine SWNT, (b) a clean-walled MWNT, and (c–f) MWNTs with varying degrees of disorder. Layers of amorphous carbon, fullerenes, and mesoscopic graphitic sheets are common MWNT impurities. Lattice disorder includes (e) “herringbone” and (f) “bamboo” defects described in the text. (d) Reprinted with permission from Iijima et al. (1992a). Copyright 1992 American Physical Society. (e) Reprinted with permission from Park et al. (1999). Copyright 1999 American Chemical Society. (f) Reprinted with permission from Jia et al. (2005), copyright 2005 Wiley-VCH.

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of a MWNT into independent pods or isolated volumes. Bamboo defects are often quasi-periodic, as shown in Fig. 2.4(f), and by nucleating new shells they maintain roughly constant outer diameters. 2.2.4.3 Substitutional dopants The high-strength carbon lattice is not generally susceptible to substitutional doping, and cannot be tailored as widely as most technological semiconductors. Therefore, it is unlikely that substitutional dopants exist inadvertently in CNTs. Nevertheless, both B and N atoms can replace C atoms in the graphite lattice without severely disrupting the bonding network. Substitutional doping of graphites by B or N atoms, as well as BN dimers, has been pursued since the 1960s (Lowell 1966; Marchand and Zanchetta 1966), and concentrations up to 5 atomic% are achievable (Oya et al. 1979; Belz et al. 1998). When similar techniques are applied to CNTs, the range of possible stoichiometries include lightly doped CNTs, various Bx C y Nz line phases (WengSieh et al. 1995; Terrones et al. 1996), and pure BN nanotubes (Chopra et al. 1995b). Small, pristine SWNTs seem highly resistant to inplane substitutional dopants, perhaps because their stability during synthesis is already strained by curvature. MWNTs more readily incorporate dopants, though experiments are often at concentrations >0.1 atomic%, far exceeding the degree of disorder usually considered to be a defect concentration. Both B- and N-doped MWNTs display large concentrations of herringbone and bamboo defects, suggesting that a delicate balance is achieved during growth. The N dopants in particular introduce pyridine rings and pentagons, both of which increase the local curvature of the CNT sidewall and tend to cap off the cylindrical structure. The structural disorder that results makes it difficult to distinguish between true lattice substitutions and dopants incorporated between carbon layers, and analytical TEM suggests a non-uniform distribution of the elements and possible phase separations between BN and C shells (Suenaga et al. 1997; Carroll et al. 1998; Golberg et al. 2002). Due to the high concentrations involved and the extensive disorder that is induced, these materials will not be considered further in the following sections. A review by Terrones et al. describes successful synthesis techniques and characterizations of substitutionally doped CNTs (Terrones et al. 2008).

2.3

Experimental identification of defects

The next section of this review summarizes the development of experimental methods for locating defects in CNTs. The intent is to provide both historical context and a guide for continued investigation. The techniques described in this section are roughly organized in order of precision, from atomic-resolution scanning tunnelling microscopy (STM) to more indirect measures of defect density by optical and electronic techniques. It is immediately apparent that locating defects with high precision is inversely correlated with yield: the highest precision techniques are painstaking and unable to categorize defect densities with good statistics. As a group, the techniques are highly complementary and progress in the identification, categorization, and control of CNT defects is likely to take advantage of the entire suite of methods.

2.3 Experimental identification of defects 51

2.3.1

Scanning tunnelling microscopy

STM is one of the highest resolution tools in the experimentalists’ toolkit. The issue of finding and characterizing defects therefore seems naturally suited to STM, especially for an all-surface material like SWNTs. Where conventional bulk techniques might fail to distinguish a single point defect or its physical effects, one might expect STM to provide decisive insights. In practice, however, this idealization has proven very difficult. Early work imaging CNTs routinely failed to achieve atomic resolution, in part because of the difficulties of clean sample preparation. A SWNT’s diameter is equal to only a few monatomic metal steps, and when deposited from solution, all CNTs assemble with codeposited carbonaceous adsorbates and solubilized contaminants. By using highly purified suspensions, one can achieve dilute dispersions in which isolated CNTs are only bound to the surface by weak van der Waals forces. However, in this “pristine” state, small electrostatic forces readily move CNTs on a surface (Falvo et al. 1999), often precluding the tunnelling conditions necessary for atomic resolution. Researchers developed various solutions for overcoming these experimental challenges. In the earliest work, MWNTs (Ge and Sattler 1993) and SWNTs (Ge and Sattler 1994) were directly condensed from a carbon plasma onto cleaved graphite surfaces for imaging. A more versatile technique used deposition from suspensions of SWNTs. Partly solubilized SWNTs in parallel bundles are stabilized by increased van der Waals attractions that help anchor them in place during imaging, reducing their tendency to roll on the surface and allowing for atomic resolution. This solution has been widely employed to study the correspondence between SWNT chirality and electronic band structure (Hassanien et al. 1998; Odom et al. 1998; Wildoer et al. 1998), and has resulted in the observation of CNT defects (Clauss et al. 1998, 1999). A third solution to sample preparation has leveraged progress in synthetic CVD techniques to grow clean and isolated SWNTs in place on a substrate (Kong et al. 1998). Using CVD, CNTs free of any chemical processing can be imaged on various substrates (Biro et al. 1997) or even freely suspended across gaps (LeRoy et al. 2004b). Advantages of pristine SWNTs include the opportunity to directly image as-grown defects, and the ability of very long SWNTs to more effectively pin themselves to a surface for imaging. Unfortunately, experimental STM results on such pristine SWNTs are limited because CVD growth of SWNTs is mostly confined to insulating substrates. Even with appropriate samples, the STM imaging of SWNTs remains complicated by curvature and electronic delocalization (Kane and Mele 1999; Orlikowski et al. 2000; Lambin et al. 2003). The high sensitivity of the z-axis tunnelling feedback is poorly suited to imaging curved surfaces like cylindrical CNTs. Uniquely identifying the CNT indices (n, m) requires algorithms that can recreate the true, curved surface from measured images (Venema et al. 2000; Ouyang et al. 2002). The fact remains that barely 25% of a SWNT’s atoms are accessible to an STM tip, and any bundling of SWNTs further limits tip access. The sensitivity of STM to electronic states is another complication in SWNTs, since these states are fully coherent around the tube circumference. In flat graphite crystals, defects are observed as complex, extended Moir´e

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Fig. 2.5 A semiconducting SWNT simultaneously imaged at (a) positive and (b) negative biases exhibits chiral striping that is the result of interference patterns of the injected electrons, not merely the underlying atomic positions. Disruptions in these patterns are interpreted as defects (c). Analysis of this example determines the chirality to be constant, ruling out the 5–7 type of defect that might change the SWNT (n, m) indices. The observed change in interference pattern on either side is likely due to different backscattering conditions from an asymmetric defect. Reprinted with permission from Clauss et al. (1999). Copyright 1999 EDP Sciences.

patterns resulting from the interference of incident and reflected electron waves, even when the scattering site is tens of nm from the tip (Kobayashi 1994). Identical effects occur in SWNTs, but with the complication that only a portion of the surface is accessible (Clauss et al. 1999; Kane and Mele 1999; Ouyang et al. 2001, 2002). This means that detecting a defect from long range is possible, but pinning down the location of that defect is challenging. The consequence of these difficulties is that atomic defects are exceedingly difficult to unambiguously identify. STM images taken at different biases routinely fail to resolve pentagonal or heptagonal structures, even when performed at low temperature and with atomic resolution. Instead, the tunnelling current non-locally probes hidden atoms, and provides long-range detection of a defect without identifying its exact position or atomic arrangement. Figures 2.5 and 2.6 clearly demonstrate these problems. Enhanced contrast arising from bias-dependent electron-interference patterns obscures the underlying lattice and is particularly complex in the vicinity of a defect. Far from the defect,

Fig. 2.6 In these examples, a change in chiral indices determines the presence of unpaired 5–7 defects. (a) At least two, separated 5–7 defects are present (i.e. not paired together in an SW configuration). (b) Three 5–7 defects are required to produce this change in chirality. Scale bar is 1 nm in both. Reprinted with permission from Ouyang et al. (2001). Copyright 2001 American Association for the Advancement of Science.

2.3 Experimental identification of defects 53

resolution of the lattice contains some information about the defect constituents: Fig. 2.5(c) must contain no unpaired 5-7 defects, whereas Figs. 2.6(a) and (b) must contain two and three of them, respectively. Analysis of such patterns (Kane and Mele 1999; Orlikowski et al. 2000) concludes that a proper interpretation requires solving the inverse scattering problem, without a priori knowledge of the defect type or exact orientation. More recently, Yang et al. reached similar conclusions after modelling the long-range interference patterns that result from different SW orientations (Yang et al. 2005). In order to determine the exact nature of a defect observed by STM, a spatial map of tunnelling spectroscopy must be compared to theoretical calculations based on possible atomistic models. Beginning from the observed chiral indices of a SWNT, one can model combinations of pentagons and heptagons to try to reproduce the experimental image and spectra. Experimental SWNTs with diameters of 1.0–2.5 nm provide a large number of possible arrangements to test, especially since typical samples like SWNT–SWNT junctions have very low symmetries. In practice, however, qualitative agreement can usually be obtained after testing a relatively small number of configurations. For example, in 2003 Kim et al. inferred the positions of two pentagons and two heptagons in a SWNT (Kim et al. 2003a). The SWNT was observed to change indices from (15,2) to (19,3), ruling out the likelihood of a Stone– Wales defect and indicating the need for at least one pentagon and heptagon. Reasonable, though not necessarily unique, agreement with experiment was obtained by modelling the junction to have an isolated pentagon, with the heptagon adjacent to a second pentagon–heptagon pair. Ideally, one would quantitatively optimize agreement between the model and experiment, but in this junction the unit cell consists of ∼5000 atoms. Limited by this size to tightbinding techniques, a rigorous optimization among all possible geometries would not necessarily achieve a clearer result. A similar result obtained by Ishigami et al. in 2004 attempted to use an iterative algorithm to speed up the selection and refinement of possible models. In this case, shown in Fig. 2.7, modelling suggested two pentagon–heptagon pairs that were diametrically opposite each other, placing one completely out of sight of the STM’s imaging (Ishigami et al. 2004). In each of these examples, complex, localized states were directly observed with particular energies and spatial extents of 1–3 nm or more. Finding defects becomes much easier when they are intentionally incorporated, and some studies have created defective CNTs to aid STM study. Ar+ -ion irradiation produces vacancies and vacancy–interstitial pairs in graphite, so irradiation of CNTs allows these defects to be studied experimentally (Osvath et al. 2005) and theoretically (Krasheninnikov and Nordlund 2002). Alternately, a hydrogen plasma can be used to produce vacancies and covalent H–C adducts (Buchs et al. 2007). Unfortunately, images of these defect types are not noticeably different from the figures shown above, with the exception that they do not produce topological changes in chirality. A less perturbative approach is to investigate defects that are intentionally introduced by the STM itself. Venema et al. first employed voltage pulses to locally modify a SWNT (Venema et al. 1997). By applying a 5-V pulse, metallic SWNTs were fragmented into shorter segments with the properties of

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Fig. 2.7 (a) STM-resolved point defect between two chirally distinct segments in a 1.4-nm SWNT. The same location, as denoted by the black dot, appears different when imaged at different biases because of the complex electronic structure. Experimental conditions are 77 K, 0.2 nA, and +0.5 V (a,b) or +0.3 V (c). Continuous evolution of the image complicates any straightforward atomistic interpretation, but provides an extensive dataset to compare against theoretical models. This site was determined to be a 5–7 defect, physically separated from a second 5–7 defect located on the hidden backside of the SWNT. Reprinted with permission from Ishigami et al. (2004). Copyright 2004 American Physical Society.

a coherent quantum dot (Venema et al. 1999). Park et al. employed similar, oxidative pulses using a conducting-tip AFM to both “nick” and completely break SWNTs (Park et al. 2002). In this case, the goal was to introduce modest electronic barriers without necessarily cutting through the SWNT. The work was completed on insulating substrates using electrically connected SWNT devices. Recently, Berthe et al. combined this principle with STM spectroscopy to investigate the structural consequences of modest voltage pulses on pristine SWNTs (Berthe et al. 2007). In some cases, these pulses merely deposited material onto the SWNT, but in others localized states were produced within the SWNT lattice. These states gave rise to interference patterns like those shown above, and a localized pair of electronic states at −0.45 and +0.26 eV in the vicinity of the disorder. Point defects that are added after synthesis cannot consist of topology-changing 5–7 defects, and instead might consist of SW defects, covalent sidewall adducts, or ad-atom–vacancy pairs. The authors successfully removed their features using smaller magnitude pulses and concluded that the defects are of the SW type, since bond rotations are in principle reversible modifications of the lattice. Further modelling would be required to distinguish between Stone–Wales defects and covalently bound adducts. An alternate use of STM spectroscopy is to look at inelastic tunnelling spectroscopy (IETS). Unlike the elastic spectroscopy, which primarily measures the electronic van Hove singularities, IETS is sensitive to dissipative channels including localized phonon modes. Experimentally challenging, IETS measurements on SWNTs have, nonetheless, measured low-energy modes associated with defects or dissipation through the radial breathing mode (LeRoy et al. 2004a; Vitali et al. 2004). Recent theoretical modelling predicts energetic shifts of +20 cm−1 for the C–C bond in a Stone–Wales defect, and −50 cm−1 shifts around isotopic C13 impurities (Vandescuren et al. 2007). The latter

2.3 Experimental identification of defects 55

type of disorder is particularly difficult to study by any other experimental means. Despite providing extraordinary precision, the STM techniques are not building a library of ready images that can categorize different defect types. If anything, the defects characterized by Ouyang, Kim, and Ishigami were readily modelled because of a change in SWNT chirality; but the majority of SWNT defects, including adducts and SW bond rotations, do not change the underlying SWNT lattice. These may be much more difficult to locate and uniquely identify. Finding the rare, tractable defect by STM, combined with the degree of work required to model it, may never improve into a particularly efficient way of categorizing CNT defects and defect densities. While promising, Ishigami et al. note that their modelling only accounts for “the atomic structure of the most dominant defect,” leaving open the possibility of additional atomic disorder hidden within these complex images.

2.3.2

Electron microscopy

STM may be most closely associated with atomic-resolution imaging, but transmission electron microscopy (TEM) is unquestionably the primary tool for characterizing CNTs. TEM is responsible for the initial identification of CNTs, for characterizing CNT growth, and for understanding the complex morphologies of MWNTs. Until recently, however, TEM imaging has not been associated with the kind of resolution necessary for studying individual point defects in any material, much less CNTs. CNTs present a special challenge to TEM resolution because of carbon’s small atomic number. Most common TEM instruments remain blind to a single graphene layer normal to the electron beam. Image contrast for CNTs instead arises fortuitously from their cylindrical geometry (Iijima 1991). While the beam’s interaction with much of a CNT is negligible, the extreme CNT edges provide a lattice parallel to the electron beam for which diffraction conditions exist. Iijima and other early practitioners took advantage of these conditions to produce the first clear CNT images and to convince the community that these materials were indeed hollow cylinders (Fig. 2.8). In time, the common images of two parallel lines became synonymous with the accepted CNT structure. The presence of extraneous material, however, complicated image

Fig. 2.8 Early images of CNTs from Iijima’s landmark publication. In the schematic, H and V correspond to regions where the beam is normal and parallel to the graphene crystal lattice, respectively. Scale bar in middle panel is 3 nm. Reproduced with permission from Iijima (1991). Copyright 1991, Nature Publishing Group.

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interpretation, and particularly claims of lattice purity or crystallinity. Whereas some claimed that CNTs were perfect and defect free but coated with amorphous carbons, there was also clear evidence of structural deficiencies (Iijima et al. 1992a,b; Ebbesen and Takada 1995). In this context, the demands of CNT science have helped forment the development of TEM expertise. Resolving individual SWNTs, determining chirality from SWNT diffraction patterns, and distinguishing DWNTs from SWNTs, are three problems that have each in turn pushed the limits of TEM techniques over the past decade. The current challenge in CNT imaging remains resolving individual defects and characterizing their nature. As in the past, Iijima and his research group at Japan’s AIST Research Center have led the way in demonstrating stateof-the-art capabilities and new techniques. For example, a Fourier-transform filtering technique has been devised to separate foreground carbon atoms from background ones, allowing the independent imaging of the carbon lattices making up the forward and rear walls of a SWNT (Suenaga et al. 2007). Recent work summarized below has provided fantastic images of defect creation, mobility, and annealing in CNTs with atomic resolution. First, however, it must be noted that TEM investigation is perturbative. The study of CNT defects is inherently hindered by the fact that the electron beam itself induces changes in a CNT (Chopra et al. 1995a; Kiang et al. 1996). Knock-on events routinely produce ad-atom–vacancy defects in which carbon atoms are removed from the CNT lattice to ad-atom sites, and adjoining CNTs intermix and anneal into differently sized structures (Banhart 2002; Yudasaka et al. 2003). The low threshold of these mechanisms (∼ 120 keV) limits the use of higher accelerating voltages, which might otherwise provide higher-resolution imaging. Even when using short doses at lower voltages, the electron beam can heat the CNT lattice, instigate chemical changes, or inadvertently contaminate the material. These types of damage are routinely observed, closely coupling TEM imaging to questions of defect creation. Taking advantage of defect-creation processes provided the first clear TEM images of SWNT point defects (Hashimoto et al. 2004). In this work, defects were produced in situ by electron-beam irradiation and then imaged. In order to observe 5–7 defects, SWNT–SWNT junctions were produced by focusing the beam onto a small portion of a SWNT, resulting in its local modification. Near these sites, the authors observed stable deformations consistent with 5–7 defects, as shown below in Fig. 2.9. In other graphitic sheets, knock-on

Fig. 2.9 Before (a) and after (b) TEM images of a single graphene layer with a beam-induced 5–7 defect. An edge dislocation is unambiguously visible at the middle of the network where one zigzag chain is missing through it. (c) An atomistic model of the pentagon–heptagon pair in the graphitic network. (d) A simulated TEM image, showing good comparison with the TEM image shown in b. Scale bar, 2 nm. Reproduced with permission from Hashimoto et al. (2004). Copyright 2004, Nature Publishing Group.

2.3 Experimental identification of defects 57

Fig. 2.10 TEM images of a SWNT region containing a SW (5-7-7-5) defect after heat treatment at 2273 K. The region enclosed by the black line is enlarged in the center image. Each carbon ring appears to have a bright spot at its center. A composite image on the right places gray dots inside hexagonal regions with six neighbors. The two white dots have seven neighbors, and the two black dots have five neighbors. Reproduced with permission from Suenaga et al. (2007). Copyright 2007, Nature Publishing Group.

damage more directly produced atomic vacancies. Adjacent, unreconstructed monovacancies were observed, at least for short durations, along with adatom–vacancy pairs. Building on these techniques, Suenaga proceeded to clearly resolve both 5–7 and SW defects in SWNTs as shown in Fig. 2.10 (Suenaga et al. 2007). In this work, resistive heating was used to enhance thermodynamic defect formation in SWNTs, rather than merely imaging irradiation damage. In principle, this allows the results to more closely match pristine SWNTs. SWNTs were heated and then quenched to capture the large defect densities that occur during massive structural reconstructions (Yudasaka et al. 2003). Suenaga and coworkers observed nucleation kinks surrrounded by clusters of defects, including mobile SW defects. In this case, the mobility of SW defects was observed at room temperature, though with the possible activation by beam interactions. Shortly afterwards, Jin et al. used similar techniques to directly observe the migration and coalescence of ad-atom–vacancy defects. Migrations of the ad-atoms and the vacancies were independently monitored, and the authors noted the lower mobility of the vacancies (Jin et al. 2008). A surprisingly high mobility was observed for large vacancy clusters formed by the coalescence of 10 or more atomic vacancies, and the annealing of these clusters into uniform holes in a CNT sidewall.

2.3.3

Electrochemical and chemoselective labelling of defects

While STM and TEM may be the best techniques for directly imaging defects with atomic resolution, these techniques are not able to rapidly characterize material. As described in the review by Itkis et al., the highest-resolution techniques might only characterize 1 picogram of material, and no bulk CNTs today are homogeneous to that degree (Itkis et al. 2005). Short of creating excess damage, STM and TEM and not well suited for scanning many CNTs in an atomic-resolution search for rare sites. Yet addressing the

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assertion that CNTs are perfect or defect free requires finding extremely rare sites, and doing so with sufficient statistics that concentrations are unambiguously determined. Section 2.3 next continues by addressing this shortcoming through lower-resolution techniques that are, nevertheless, sensitive to individual point defects. Because dilute defects must first be located in order to be imaged, combinations of these techniques can lead to very effective characterization. One way to address the characterization of low defect densities is to consider similar problems faced by the semiconductor industry. In modern, high-quality Si crystals, the concentrations of defects is exceedingly small, approaching one interstitial defect per 1013 Si atoms (Huff 2002). To measure these concentrations and characterize their properties, the industry relies on a combination of optical spectroscopy and chemical labelling techniques. Spectroscopy provides a different fingerprint from each category of defect, while the labelling provides a quantitative enumeration of their densities (Huff 2002). 2.3.3.1 Electrochemical labelling The principle of electrochemical identification is to take advantage of the enhanced, or different, chemical reactivity of a defect site. A substitutional dopant or interstitial has a different charge density and coordination than the surrounding lattice atoms, providing a means for chemical differentiation. By precisely controlling reactive potentials electrochemically, the experimentalist attempts to work within a narrow parameter window where particular reactions are driven at defect sites without affecting the remainder of the crystal surface. Two categories of selective electrochemical processes are regularly used by the semiconductor technology: etching and deposition. Standard semiconductor technology relies heavily on the etching technique, in which point, screw, and line defects all nucleate the removal of surface material to produce “etch pits” or channels. Alternately, defects can nucleate the deposition of material from solution. In both cases, the physical size of a point defect is amplified a thousand-fold or more, with the final pit or deposit size being solely determined by duration. After converting atomic defects into pits or deposits 50 nm, 500 nm, or 5 μm in size, these sites can be readily counted by scanning electron microscopy (SEM) or even optical microscopy over cm2 areas. Similar deposition techniques applied to high-quality graphites have quantitatively determined vacancy concentrations as low as one per 1010 atoms (Hennig 1964; Evans et al. 1971). Such a high, quantitative yield is ideally suited for testing the assertion that CNTs are “molecules” lacking structural disorder. Beginning in 1996, numerous groups have observed spotty nucleation while studying electrochemical deposition on oxidized, highly defective MWNTs (Ebbesen et al. 1996; Satishkumar et al. 1996), and proven a direct correlation between decoration and oxidation extent. Entirely omitting the initial oxidation can reduce the density of deposits, but not below the level of experimental contaminants. Gross disorder or contamination, non-covalent functionalization, and amorphous carbonaceous deposits all provide efficient nucleation sites that undermine the electrodeposition technique, producing continuous coatings rather than isolated, countable deposits. Furthermore, diffusion-limited transport can limit

2.3 Experimental identification of defects 59

Fig. 2.11 Dilute labelling of SWNTs using Ag electrodeposition. All three images are from the same, large network of SWNTs, but different effective potentials result in different degrees of coverage. Reprinted with permission from Day et al. (2005). Copyright 2005, American Chemical Society.

the efficiency with which neighboring deposits grow, leading to underestimates of the true nucleation site density. These problems make electrochemical labelling generally less useful for CNT materials than for clean semiconductor surfaces. In order to access the types of intrinsic defects described in Section 2.2.1, the starting material must be exceptionally clean. This realization has led to more recent experiments using SWNTs grown in place on substrates, with no additional processing or manipulation. In the absence of gross contaminants, electrodeposition onto conducting networks of SWNTs still leads to three possible types of results. At low potentials, no deposition occurs and the SWNTs remain clean. At high potentials, SWNTs may be uniformly coated, or nearly so. In between, a window exists in which the nucleation is highly selective (Austin et al. 2002; Fan et al. 2005b; Quinn et al. 2005). On interconnected films, it is difficult to precisely control the electrochemical potential everywhere, but gradients can fortuitously result in some SWNTs being in the proper bias window (Day et al. 2005, 2007). This problem of precise chemical control has been solved in at least three different ways. In one unique case, H2 Se gas dilutely nucleated nanocluster growth of Se nanoparticles on what were presumed to be SWNT defects (Fan et al. 2002). While not controlled by an electrolyte potential, the technique appears to progress by the same principle and can be used to rapidly assess large numbers of SWNTs on a surface. In other work, an electrochemical microelectrode was scanned across a SWNT film, allowing deposition to be independently tested at various different sites serially (Day et al. 2007). While not as rapid as the Se deposition, this technique has the potential advantage of being able to electrochemically characterize different defects in addition to labelling them. A third solution is to perform the bulk electrochemical technique on single SWNTs, one at a time. Fan et al. implemented selective electrochemical labelling to perform a quantitative defect enumeration, using homogeneous potentials to determine the appropriate electrochemical conditions on individual SWNTs (Fan et al. 2005b). A nominal defect density of one defect per 4 μm was measured, with higher densities in regions of kinks and bends as

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Fig. 2.12 (a,b) Single-defect electrochemical labelling using Ni deposition on single, isolated SWNTs. The single-tube technique allows defect identification (c) to be directly correlated with two-terminal electronic behaviors (d). In this example, the SWNT device acts like a field effect transistor with the entire gate dependence (d) localized at the same spots where decoration occurs (c). The scanning gate imaging technique is described further in Section 2.3.5. Adapted from Fan et al. (2005b). Copyright 2005, Nature Publishing Group.

might be expected. While the slowest of the three techniques, the single SWNT characterization allowed one-to-one correspondences to be drawn between point defects and their electronic consequences (Fig. 2.12). In addition, the high level of control allowed a single site to be labelled, stripped, and reproducibly labelled again, providing a convincing demonstration of selectivity towards the defect site. Using the electrochemical labelling technique, SWNTs synthesized in different CVD runs were observed to have widely varying defect densities, even when the CVD parameters were nominally identical. However, a strong correlation was observed between the defect density and the median SWNT length. CVD runs that resulted in shorter SWNTs (100 μm) had the smallest densities. Across all of the different synthesis conditions, the product of mean length and mean density uniformly averaged 4 sidewall defects per SWNT (Fan et al. 2005a,b). This result suggests that defect incorporation may play a controlling role in the termination of SWNT growth, and that defect enumeration could be a useful technique for process control and optimization of SWNT synthesis, just as is done for Si crystals. Even on isolated SWNTs, a degree of imprecision remains because different defect types are indistinguishable. Certain bond rotations or adducts might only be slightly different in reactivity, and experimental errors limit the practical electrochemical windows for selecting individual types. Furthermore, it is impractical to rigorously clean surfaces supporting SWNTs. By comparison, etch pits on Si crystals are relatively straightforward to interpret because depth profiling is first used to expose pristine surfaces within the crystal. This step has the benefit of removing extraneous surface contaminants that might mimic the reactivity of defects. In the case of SWNTs, any species that is electronically well connected to the SWNT can be labelled. Thus, in addition to lattice defects and covalent adducts, nucleation may occur because of charged contaminants trapped at the SWNT substrate interface or also shallow charge traps in the dielectric immediately supporting a SWNT. As described in Section 2.2.2,

2.3 Experimental identification of defects 61

these sites are not defects in the conventional sense but they are present in the majority of CNT research. Thus, labelling and counting these sites is appropriate in practice if they interact with and affect chemical and electronic behaviors. 2.3.3.2 Chemoselective labelling While the electrochemical techniques described above are not particularly sensitive to different defect types, other reactions are more selective. A careful series of titrations using NaHCO3 , Na2 CO3 , and finally NaOH can quantitatively determine the concentrations of different oxygen-containing groups in bulk graphites, distinguishing between carboxyls, phenols, and carbonyls (Donnet 1968; Kinoshita 1988). Even SW defects have a higher reactivity than their surrounding hexagonal rings (Liu et al. 2006b; Wang et al. 2006; Horner et al. 2007). Though small defect concentrations in CNTs cannot be effectively titrated in the presence of end-caps and other disorder, selective reactions can be used as an initial step to activate defect sites for subsequent labelling and detection. For example, ozone is believed to be highly reactive at SW defect and vacancy sites, whereas the pristine sidewall forms only an unstable, shortlived complex (Banerjee and Wong 2002; Liu et al. 2006b). Other oxidants, including KMnO4 − , OsO4 , and RuO4 , are relatively efficient at cleaving C−C bonds in the vicinity of a defect or end-cap, but do not readily attack the sidewall at room temperature (Hwang 1995; Coroneus et al. 2008). These oxidations can produce carboxylic groups, a unique form of defect that can be chemically tailored. In particular, carboxylates have a highly selective reaction with the reagents N -ethyl-N  -(3-dimethylaminopropyl) carbodiimide (EDC) and N -hydroxysuccinimide (NHS). Following reaction, the NHS ester is readily displaced from the carboxylate by any molecule with an amineattachment site, providing a direct, versatile, and high-yield route for selective labeling by a wide range of molecules (Grabarek and Gergely 1990; Banerjee et al. 2005). Figure 2.13 demonstrates an example of this attachment process. The EDC/NHS protocol has been used to covalently bind the free amines found on streptavidin or lysozyme to isolated, carboxylic defects on individual,

Fig. 2.13 Chemoselective labelling of SWNTs can locate defects for low-resolution imaging techniques like SEM. In these images, each SWNT device has been prepared with a single carboxylate defect, which was subsequently activated by EDC/NHS and linked to Au-labelled streptavidin. Each bright dot corresponds to a 25-nm Au particles, and some non-specific adsorption occurs on the surrounding substrate (SiO2 ). Scale bar is 500 nm. Adapted and reproduced with permission from Goldsmith et al. (2007). Copyright 2007, American Association for the Advancement of Science.

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electrically connected SWNTs (Goldsmith et al. 2007). By using Au-labelled proteins, the defect sites are then easily located and counted by SEM. Other than these types of selective labelling, most conventional chemical analyses are of limited use for identifying and locating defects. A wide variety of chemical techniques are routinely applied to bulk CNT material, especially for the purpose of characterizing purity by distinguishing between CNTs, amorphous carbon, and metallic content. But signal-to-noise limitations severely limit the reliable measurement of low concentrations of defects in otherwise pristine graphitic lattices, especially in the presence of amorphous carbons or other disorder. Section 2.4.1 further addresses the bulk chemical reactivity of defective CNTs.

2.3.4

Optical spectroscopy

The earliest scientific studies of defects categorized different types based on the color characterizations of wide-bandgap crystals. In a crystal, a defect does not modify the overall band structure but it does introduce a localized state within the bandgap. These localized states provide many semiconducting crystals with additional color, luminescence, and electro-optic qualities. Optical spectroscopy, combined with the electrochemical decoration described above, has therefore become a key tool in the semiconductor industry for identifying defects. This section describes attempts to locate CNT defects using similar, far-field spectroscopies and more recent, near-field techniques. CNTs prove to be a much more complex and difficult system to characterize than Si crystals, but much can be learned, nonetheless. 2.3.4.1 Far-field spectroscopy Traditional optical spectroscopy is central to the characterization of carbon materials, including mesoscopic carbons and CNTs. In addition to the unique spectral fingerprints that identify CNT diameter, chirality, and electronic structure, spectral features also help to qualitatively evaluate purity. For example, many chemical functionalities are identified by unique peaks in FTIR. Single C−O bonds due to hydroxyl terminations occur at 1190 cm−1 . The double C=O bonds of carboxylic groups are found at 1720 cm−1 . Both peaks are well separated from other carbon IR modes and easily identified when these types of defects exist in substantial quantities. Furthermore, changes in peak height can provide a straightforward measurement of changes in a material, e.g. before and after a particular processing step. Unfortunately, however, the quantitative evaluation of FTIR peak heights has not been established as a means of measuring absolute disorder, much less of defect concentrations or locations. Moreover, the absence of these special modes only indicates a “low level” of disorder, since the signal-tonoise in FTIR is insufficient to resolve single defects. As with electrochemical decoration, the determination of defect densities is experimentally limited by the presence of contributing contaminants, which produce a relatively high noise floor. Even using perfectly purified SWNTs would not be sufficient to remove these contributions, since SWNT end-caps also

2.3 Experimental identification of defects 63

Fig. 2.14 The Raman D-band peak around 1300 cm−1 is a sensitive but qualitative measure of disorder. Here, Ar+ irradiation has been used to introduce defects, increasing the D-band peak intensity ID 8-fold. Reprinted with permission from Skakalova et al. (2006). Copyright 2006 Wiley-VCH.

contribute to FTIR signals. Therefore, it is extremely difficult to isolate the IR contributions of sidewall defects and unambiguously measure sidewall crystallinity. Similar effects occur in Raman characterization of CNTs. In Raman spectroscopy, a broad spectral band associated with symmetry breaking occurs around 1350 cm−1 . This “disorder” band, or D-band, is associated with nonhexagonal rings including S–W defects, 5–7 defects, and vacancies. As shown in Fig 2.14 the intensity of this D-band can be enhanced by intentionally introducing disorder through chemical processing (Skakalova et al. 2005; Barros et al. 2007) or Ar+ irradiation (Skakalova et al. 2006). A recent review by Pimenta et al. describes the theory and application of D-band measurements to the evaluation of commercial carbons and disordered CNTs (Pimenta et al. 2007). As with FTIR, however, the technique is more limited when studying pristine CNTs with very few defects. In highly crystalline CNTs, the D-band shrinks and its height can be difficult to estimate above the background spectrum, which still includes substantial contributions from the pentagons at CNT bends and end-caps (Duesberg et al. 2000; Lamura et al. 2007). Especially in shorter CNTs where these end effects can dominate, the diminishing contribution of sidewall defects is difficult to isolate or quantify (Pimenta et al. 2001; Chou et al. 2007). With appropriate sample preparation, photoluminescence, resonant Raman, and Rayleigh scattering spectra may all be collected from single, isolated SWNTs or dilute SWNT dispersions (Bachilo et al. 2002; O’Connell et al. 2002; Dresselhaus et al. 2003; Sfeir et al. 2004). In each case, spectral lines can be uniquely associated with a particular SWNT and distinguish its chiral indices (n,m). All of these techniques are therefore sensitive to index-changing defects, though the Raman D-band remains the most direct optical measurement of disorder. For example, Anderson et al. measured D-band intensities on individual SWNTs produced by both arc-discharge and CVD growth methods, and observed the CVD SWNTs to have disorder intensities about three times lower (Anderson et al. 2005).

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2.3.4.2 Near-field spectroscopy In recent years, remarkable progress has been made in developing microRaman and near-field spectroscopies. By working in the near field on the isolated SWNT samples described above, spectral variations can be distinguished within particular regions of long SWNTs. The development of these less-common techniques is perfectly suited to locating features like defects. In the case of resonant near-field Raman spectroscopy, spatial resolution approaching 15 nm has been obtained using confocal (Doorn et al. 2005) or apertureless architectures (Anderson et al. 2005). At this resolution, both direct and indirect measurements of defects become possible in principle. The D-band intensity as a function of SWNT position serves as a direct map, definitively associating disorder with particular SWNT locations. Indirect indicators include any Raman mode sensitive to changes in a SWNT’s (n,m) indices. For example, the radial breathing mode (RBM) and the G- and G -bands are Raman features that directly measure diameter and symmetry, respectively, and change in response to 5–7 defects. In practice, the RBM and G modes are spatially extended with large oscillator strengths. Mapping changes in them is relatively straightforward, and an indirect but convincing way of determining the presence of one or more 5–7 defects (Hartschuh et al. 2003; Anderson et al. 2005; Doorn et al. 2005). Directly observing increases in the D-band intensity, on the other hand, proves very difficult. Doorn et al., for example, mapped SWNTs longer than 100 μm in length and in the rare-transition regions where chiral indices changed, no extra D-band intensity could be resolved (Doorn et al. 2005). Anderson et al. achieved the necessary signal enhancement by replacing the confocal geometry with a Au-coated metal tip (Anderson et al. 2007). The Au provided plasmonic, local field enhancements that could be scanned or manipulated in the region of a defect. With the combination of 40 nm lateral resolution and enhanced optical intensity, D-band increases were in fact resolved: a twofold increase in D-band intensity is reproduced in Fig. 2.15. The RBM transition region in this SWNT appears to extend over >100 nm, 260

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Fig. 2.15 Direct and indirect Raman spectroscopy of a defect in a single SWNT. The RBM mode abruptly shifts from 250 to 190 cm−1 (left axis), indirectly identifying a defect through its effect on the SWNT diameter. Within the transition region, the D-band intensity approximately doubles (right axis). Note that the D-band intensity is much higher at both SWNT ends, indicating one difficulty of resolving sidewall disorder. This apertureless measurement achieved 40 nm lateral resolution by using plasmonic enhancement from an Au probe. Reprinted with permission from Anderson et al. (2007). Copyright 2007 American Chemical Society.

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2.3 Experimental identification of defects 65

Fig. 2.16 Stepwise quenching of SWNT photoluminescence during the stochastic addition of covalent adducts, in this case 4-chlorobenzene-diazonium tetrafluoroborate. Each panel represents a different SWNT, with two curves for two adjacent, independently measured segments (670 nm in length). Each asterisk indicates one unit of PL decrease. Adapted and reproduced with permission from Cognet et al. (2007). Copyright 2007, American Association for the Advancement of Science.

suggesting perhaps two or more 5–7 defects separated by long distances. This particular arrangement may have aided the experiment, and further work will determine whether adjacent 5–7 defects and SW defects can be resolved in the D-band. Whereas Raman spectra are sensitive to particular phonon modes, photoluminescence (PL) reflects the electronic band structure of a semiconducting SWNT (Bachilo et al. 2002; O’Connell et al. 2002) or MWNT (Uemura et al. 2006). Band structure is closely related to a SWNT’s (n,m) indices (Chou et al. 2004), so that a 5–7 defect that changes these indices can be mapped indirectly through its effect on electronic transitions, just as with the Raman RBM mode described above. In principle, a careful comparison of RBM and PL transitions surrounding a 5–7 defect might suggest different length scales, since the electronic and phonon localization lengths need not be identical. In practice, however, the interpretation of SWNT PL is much more complicated than a phonon map (Lefebvre et al. 2008). The primary source of PL in SWNTs is from radiative excitonic recombination. In the absence of disorder, SWNT excitons are relatively long-lived, diffusing over a 90-nm range and sampling approximately 10 000 lattice sites. The PL signal, then, is essentially a map of exciton diffusion and lifetime. Dilute disorder, whether from adsorbates, substrate charge traps, or lattice defects, amplifies non-radiative recombination channels and quenches this PL signal. Finally, as near-field optics gets closer to resolving individual defects, it becomes possible to monitor defect creation in real time. In general, the Raman and PL techniques described above are sensitive to (n,m)-changing defects and not SW or vacancy defects that can be produced postsynthesis. Nevertheless, Cognet et al. studied the PL of individual SWNTs exposed to sulfuric acid or diazonium salts, and observed stepwise drops occuring during exposure (Cognet et al. 2007). The steps shown in Fig. 2.16 were attributed to the creation of adduct defects, much like conducance experiments to be described in Section 2.4.2. These adducts do not change the gross electronic spectrum of a SWNT, but they do pin excitons and amplify their non-radiative recombination channels. In similar work, defects have been observed to produce localized emission from unbound electron–hole pairs electrically injected from opposite ends of a SWNT (Freitag et al. 2006; Avouris et al. 2008).

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2.3.5

Electrical conductance

Of the different methods for locating defects described in this section, electrical conductance is probably the least well known outside the CNT field. Conductance is not generally sensitive to individual defects, and it is merely a measure of global disorder and doping in 3D and bulk materials. This remains true as dimensionality is reduced to 2D films, though in specialized cases point defects give measurable effects (Pelz and Clarke 1985; Wong 2003; Punnoose and Finkel’stein 2005). Further reduction to the 1D limit, however, opens up new characterization possibilities with exceptional sensitivity. Quantum wire conductances are strongly disorder dependent, and even arbitrarily small disorder can induce localization in these systems (Egger and Grabert 1995; Auslaender et al. 2002). This section describes techniques that take advantage of this amplification to locate defects with good spatial resolution. Additional measurements of the long-range electronic effects of defects are described further in Section 2.4.2. The invention of STM was rapidly followed by numerous extensions known collectively as “scanning probe microscopy” (Kalinin and Gruverman 2006). The most common of these, atomic force microscopy (AFM), is sensitive to sample topography and in special cases can achieve atomic resolution. AFM does not, however, resolve individual defects such as the disorder on MWNTs (Bachtold et al. 2000), small gaps in SWNTs (Park et al. 2002), or point adducts (Goldsmith et al. 2007), perhaps because the technique is not sensitive to electronic structure. Augmenting AFM with a metalized cantilever, on the other hand, allows electronic measurements to be made with nanometer precision. In its simplest implementation, the AFM tip acts as a moveable contact probe to measure CNT resistances. Dai et al. demonstrated an early measurement of this type, mapping the length-dependent resistance of a CNT and finding it to be roughly proportional to length (Dai et al. 1996). This type of measurement requires one fixed contact electrode on the CNT, and it can be hampered by variability and irreproducibility of the tip–CNT contact point. Kelvin force microscopy (KFM), electrostatic force microscopy (EFM), and scanning impedance microscopy (SIM) are all variations of this principle of mapping electronic features (Kalinin and Gruverman 2006). Using a CNT contacted by two electrodes and appropriately biased, these techniques directly map electrostatic surface potentials or their gradients along the CNT length, thereby avoiding instabilities associated with trying to use the tip itself as an electrical contact. In pristine SWNTs, these techniques have confirmed the absence of potential gradients and verified the general lack of diffusive scattering (Bachtold et al. 2000). Furthermore, they have directly imaged the electronic consequences on a SWNT caused by disorder in supporting substrate (Tans and Dekker 2000; Woodside and McEuen 2002). By far the strongest effects, though, occur whan a SWNT contains a defect. In this case, sharp potential drops can be resolved at room temperature at fixed positions along the tube length (Freitag et al. 2002; Goldsmith 2002; Goldsmith and Collins 2005). Figure 2.17 shows example data taken from a KFM measurement of surface potentials.

2.3 Experimental identification of defects 67

Fig. 2.17 KFM mapping of the electostatic potentials along single SWNT devices. (a) In a pristine SWNT, all of the potential drops are observed to be at the contact interfaces and the SWNT itself is nearly equipotential. (b) With strongly scattering defects, the potential drop at the interfaces is insignificant compared to those in the vicinity of the defect(s). Adapted and reproduced with permission from Goldsmith (2002).

Another local conductance measurement is known as scanning gate microscopy (SGM) (Bachtold et al. 2000; Staii and Johnson 2005). SGM proves to be a relatively simple and reproducible technique for locating defects, though its resolution and applicability is somewhat limited. In SGM, the conductive cantilever is biased with respect to a CNT device and then employed as a movable, electrostatic gate electrode. For CNTs that have field-sensitive conductances, the SGM technique provides a spatial map of the sensitivity. For instance, many SWNTs have transistor-like responses to gating, and the SGM technique maps the device location(s) that contribute. SGM has been tremendously helpful for proving that many SWNT transistors switch because of gate-sensitive Schottky barriers, rather than solely due to bulk carrier depletion (Heinze et al. 2002). Many types of defects produce similar gate dependencies because their localized electronic states scatter in narrow, resonant-energy ranges. This makes defects readily observable in SGM (Bachtold et al. 2000; Goldsmith 2002; Staii and Johnson 2005; Goldsmith et al. 2007). In CNTs with metallic bandstructure, defects may be the sole contributors to gate sensitivity, so that an SGM image is a very simple map of defect positions. A previous example of this was presented in Fig. 2.12(d). With semiconducting CNTs, gate-dependent carrier concentrations and contact interfaces also contribute to the SGM image (Fig. 2.18). In principle, these effects limit SGM’s ability to resolve defects, but the defect scattering is often strong enough to be distinguished (Freitag et al. 2002; Zhou et al. 2005). As a consequence, defect sites are often quite obvious, even in semiconducting SWNTs, as long as they are physically distant from the Schottky barriers. The lateral resolution of these techniques rarely exceeds 20 nm, making them somewhat imprecise among the different scanning probe methods. However, they have the additional advantage of being able to perform electronic spectroscopies. The potential drop at a defect site is not simply proportional to the total bias applied to the CNT, but is instead energy dependent. Careful mapping of this dependence, which might reflect scattering processes that are resonant with the defect’s energy levels, may allow different types of defects to be electronically classified. This advance would be an important step for these techniques, because at present the structure of different defects reported by KFM or SGM has been decidedly vague. In fact, most measurements

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Fig. 2.18 SGM (a–e) and SIM (f–j) images taken at varying biases of a SWNT device with electrostatically sensitive sites. Positions of the contact electrode interfaces are drawn in (c). Under some conditions, gating is solely observed at the Schottky barriers at both ends of the device (a,b). Additional inhomogeneity is observed with larger and more perturbative tip biases (c,d). Reproduced with permission from Freitag et al. (2002). Copyright 2002, American Physical Society.

have been unable to distinguish between sidewall defects, adducts, and the more extrinsic effects discussed in Section 2.2.2: a charge trap in the CNT’s supporting oxide may locally enhance gate sensitivity as much as a SW defect. Differentiating between these possible mechanisms is key to the usefulness of the measurements, and energy-dependent mapping is a potential solution. Similar techniques have electronically distinguished different SWNT chiralities (Heo and Bockrath 2005), but the application to defect spectroscopy remains unproven. Ultimately, different defect types may be too similar to be distinguished (Orlikowski et al. 2000). Mobile surface contaminants present substantial difficulties to the techniques described here. In ambient conditions, these contaminants dynamically follow the scanning probe and screen its effectiveness (Kalinin and Bonnell 2004). More accurate electronic information is recovered by performing measurements in UHV conditions, but with a low throughput that undermines the effectiveness of the techniques for locating defects. Furthermore, sample preparations like baking and degassing possibly modify a defect’s interesting aspects. An intermediate solution to UHV is to cool samples below 0 ◦ C for imaging. Rather than removing all surface ions, cooling fixes the ions in place by freezing the surface moisture that contributes to their high mobility. Comparison of images taken at 20 ◦ C and −70 ◦ C indicate virtually no change in electrical characteristics, despite substantial contrast enhancements in EFM, KFM, and SGM.

2.4

Physical consequences of defects and disorder

The final main section of this chapter focuses on the long-range consequences of defects and disorder. Of course, some of the physical consequences

2.4 Physical consequences of defects and disorder 69

of defects are the enabling mechanisms for the experimental methods just described, and every CNT defect of course has its individual, localized effects. This section focuses on additional effects that, while not used primarily as methods for locating defects, are, nonetheless, sensitive to their presence. In particular, a single defect can be consequential far out of proportion to its atomic concentration. This is particularly true for electronic and mechanical properties, for which the extended properties of a 1D system are especially sensitive to weak links. The main emphasis in this section is on these longrange and disproportionate consequences, where for example two identical CNTs become measurably distinguishable because of the presence of a single defect in one. Ultimately, these physical effects are the main motivation for studying defects in CNTs. Defects in highly confined, 1D materials remain poorly understood experimentally, despite their potential for novel physics and their looming importance in nanometer-scale electronic and mechanical devices. Section 2.4 is organized into three parts, separately treating the cases of chemical, electronic, and mechanical properties.

2.4.1

Chemical reactivity of CNT defects

Because of the strain on its curved surface, one might guess that the presence of a CNT defect could lead to dramatic long-range chemical effects, perhaps the exothermic unravelling or melting of the cylindrical structure. However, the carbon cylinder is an energy minimum compared to narrow graphene strips, and such disintegration is only believed to occur for extreme conditions such as large tensile strains, extensive oxidation, or irradiation (Chopra et al. 1995a; Cabria et al. 2003; Li et al. 2006). Defects may in fact lower the melting temperature of CNTs, for example from 4500 K to 2600 K (Zhang et al. 2007), but the temperatures remain extremely high for most practical considerations. Instead, the presence of point defects appears to have minimal effects on the measurable chemical reactivity of CNTs, especially when assayed by bulk analytical techniques. For example, thermal analysis is widely employed to measure CNT content and purity. Thermogravimetric analysis (TGA) involves monitoring sample weight during heating, typically at rates of 1–3 ◦ C/ min. In air, amorphous carbons convert to CO and CO2 at temperatures below 400 ◦ C, whereas CNTs burn at higher temperatures of 400–750 ◦ C (Dillon et al. 1999). Above 1000 ◦ C, any remaining mass can be attributed to metal contaminants such as transition-metal catalysts. The highest-quality SWNTs exhibit relatively sharp weight loss profiles, as demonstrated in Fig. 2.19. Highly defective or contaminated materials, on the other hand, burn over much wider ranges of lower temperatures. Similar thermal analysis conducted in vacuum is readily integrated with mass spectrometry (TGA-MS) to identify the specific desorption products. The mass spectrometry increases the dynamic range of this type of measurement, from approximately one part per thousand sensitivity to one part per million. TGA and TGA-MS both readily quantify purity in disordered carbons. Typical chemical surface groups can even by titrated by their different gas evolutions,

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Fig. 2.19 (a,b) Normalized TGA spectra for SWNTs produced by laser ablation and then purified by a 2-step technique of acid reflux and air oxidation. After purification, the SWNT fraction resists oxidative weight loss up to 550 ◦ C and has an inflection point at 735 ◦ C (a). Acid reflux durations exceeding 16 h are more effective at removing amorphous carbons, but a substantial reduction in SWNTs is also seen (b). (c) By coupling TGA with mass spectrometry, oxygen-containing functional groups may be titrated from acid-oxidized graphitic carbons. (b) Reproduced with permission from Dillon et al. (1999). Copyright 1999 Wiley-VCH. (c) Reproduced with permission from Barton et al. (1973). Copyright 1973 Elsevier Limited.

with CO2 evolution peaking at 250 ◦ C and 600 ◦ C, and CO peaking at 700 ◦ C (Fig. 2.19(c)) (Barton et al. 1972, 1973). In graphite, the latter CO peak results from basal-plane oxidation, initated from edges or defects. A comparison of the SWNT (Fig. 2.19(a)) and graphite (Fig. 2.19(c)) results suggests that similar CO evolution occurs from highly purified SWNTs. In this case, reactive endcaps constitute the pre-dominant contribution to gas evolution, and this end-cap burning dominates the width of the TGA profile and precludes direct observation of the enhanced reactivitity of other dilute, sidewall defects. Without purification, the residual metals common in bulk SWNTs reduce the oxidation threshold in this type of measurement. Because these metals catalyze oxidation processes as well as CNT growth, their presence substantially reduces and broadens the temperature range over which bulk material burns. These factors indicated that neither TGA nor TGA-MS is particularly applicable to measuring defect densities in the presence of any appreciable degree of disorder. Another similar and widely reported characterization technique involves electrochemical cycling of CNT materials. Redox measurements, in contrast to the depositions described in Section 2.3.3, are widely used to compare the electrochemical activities of bulk carbon electrodes (Taylor and Schultz 1996), and a rapidly growing field of literature compares the different redox activities of heavily oxidized CNT electrodes. Without oxidative treatments, however, the baseline electrochemical activities of CNTs are orders of magnitude smaller. The electrochemical activity of the pristine sidewall is small, but not beyond the range of sensitive electronics when coupled with the proper architecture of localized probing (Burt et al. 2005) or isolated CNTs (Heller et al. 2005, 2006). In principle, the chemical activity of a defect should be readily detectable in such measurements, perhaps by using redox couples having high turnover rates. Similar work on graphitic electrodes has led to a better appreciation of the activity of edge sites and defects (Ji et al. 2006; Punbusayakul et al. 2007).

2.4 Physical consequences of defects and disorder 71

Many additional techniques have the ability to characterize CNTs, especially for the purpose of characterizing purity by distinguishing between CNTs, amorphous carbon, and metallic content. Various functionalization schemes, measurements of BET surface area, neutralization, titration, and fractionation are all useful tools that have played roles in the quantitative comparison and optimization of carbon fibers, activated carbons, and carbon electrodes (Kinoshita 1988). But all of these carbons contain substantial proportions of oxidized edges that are not available in pristine CNTs, and the techniques are generally unable to distinguish the very small concentrations of sidewall defects in pristine material. In addition, measurements of CNT bundles are dominated by interstitial voids. While those measurements are useful, they further restrict the ability to resolve sidewall defects. These limitations in turn complicate efforts to buttress empirical knowledge with a microscopic understanding of the individual effects of different types of defects. Overcoming those limitations requires that the CNTs first be disordered and then oxidized, for example by the processing described in Section 2.2.3. In this case, an abundance of surface groups can be readily detected by bulk analytical methods. In fact, a remarkable property of CNTs is their ability to maintain a cylindrical structure in the presence of tremendous defect densities. Mawhinney et al. studied an ozone titration of oxidized SWNTs, and observed that 1 out of every 20 carbons was located at a defect site (Mawhinney et al. 2000).

2.4.2

Electrical transport and CNT defects

The potential usefulness and novelty of CNT electronic properties have motivated an aggressive pursuit of electrical measurements over the past decade. Many interesting effects have been experimentally observed, reproduced, and theoretically explained, and several up to date reviews describe the current understanding of the field (Biercuk et al. 2008). This section excludes most of these phenomena, focusing instead on the transport effects that are specific to SWNT defects. Before the incorporation of defects, pristine SWNTs are high mobility, quasi-ballistic conductors with exceptional properties. The SWNT conductance G is often discussed in terms of the quantum of conductance, G o = 2e2 / h, appropriate for a single, current-carrying, 1D channel or quantum state with spin degeneracy. The pristine, metallic SWNT has two such states, located at the K and K momentum points of graphene’s Brillouin zone. Mixing between these states, as well as electron–phonon scattering and backscattering generally, is strongly suppressed (McEuen et al. 1999; Ando 2005), so that short, clean SWNTs behave like ballistic, 1D conductors. 2.4.2.1 Conductance of topology-changing 5–7 defects Before the first SWNT devices had been successfully fabricated or measured, the possibility of novel SWNT heterojunctions had been considered theoretically (Charlier et al. 1996; Chico et al. 1996a,b; Saito et al. 1996). These “junctions,” described above in Sections 2.2.1 and 2.3.1, consist of SWNTs with 5–7 defects that change the topological indices (n, m)

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(a)

(b)

(c)

(d)

+2V

Vg = –4V 400

7

5

/ (nA)

300 200 100

SiO2 Au

0 –4

–3

–2

–1

0 V (V)

1

2

3

4

Fig. 2.20 Atomistic models of a metal–metal (a) and two metal–semiconductor (b,c) SWNT heterojunctions. Orientation of the 5- and 7-membered rings determines each junction’s geometric length and angle. (d) The kinked junctions are more readily located experimentally (AFM image in inset) and fabricated into devices. A measurement on a semiconductor–metal junction has an asymmetric, rectifying I–V. (a,b) Reproduced with permission from Saito et al. (1996). Copyright 1996 American Physical Society. (c,d) Reproduced with permission from Yao et al. (1999). Copyright 1999 Nature Publishing Group.

midway along the SWNT. Because electronic band structure is sensitive to these indices in SWNTs, this type of defect is remarkably consequential. As confirmed by STM spectroscopy (Ouyang et al. 2001; Kim et al. 2003a; Ishigami et al. 2004), different defect configurations can change semiconducting SWNTs into metallic ones and vice versa, thereby producing 1D semiconductor–semiconductor, metal–semiconductor, and metal–metal heterojunctions, as depicted in Fig. 2.20. While these junctions can be atomically resolved on metallic surfaces, testing their transport properties requires an insulating substrate that precludes STM imaging. Nevertheless, certain defect configurations result in characteristic kinks, and these kinks are easily resolved by AFM. Yao et al. found approximately 1% of 500 SWNT devices to contain sharp kinks, and proceeded to electrically characterize these (Yao et al. 1999). Some kinks exhibited asymmetric rectification appropriate for metal–semiconductor junctions (Fig. 2.20(d)). Other, more conductive devices had non-linear I–V characteristics and temperature dependences suggestive of metal–insulator–metal junctions. In both types, the transport is dominated by a tunnelling barrier formed at the heterojunction interface, i.e. by localized states surrounding the defect site(s). Yao et al. analyzed their results in terms of tunnelling between Luttinger liquids, an aspect of SWNT junctions that is wholly different from conventional 2D heterojunctions. Unfortunately, due to both the rarity of such devices and the lack of precise indexing, there has been limited opportunity to model the particular characteristics of specific atomic arrangements. Recent progress indexing junctions optically, as described in Section 2.3.4.2, combined with the deterministic synthesis of heterojunctions described in Section 2.2.1.3, should lead to renewed activity and progress in this area. Other notable cases of rectification have been observed in different SWNT junctions. A portion of a SWNT decorated by an adsorbed impurity was

2.4 Physical consequences of defects and disorder 73

found to rectify current (Antonov and Johnson 1999), as are many Y-branched MWNTs having three terminals (Papadopoulos et al. 2000). A definite metal– semiconductor junction was fabricated at the crossing point between a semiconducting SWNT and a metallic one (Fuhrer et al. 2000). The rectification in such experiments is usually attributed to a particular location in the device, though this is impossible to prove using fixed electrodes. Using a sliding STM tip to study transport at different points, Collins et al. observed the transition from symmetric to rectifying I–V behavior along a SWNT bundle (Collins et al. 1997). Each of these examples continues to generate interest in SWNT heterojunctions as possible nanoscale electronic components. 2.4.2.2 Conductance of other point defects Most other point defects break the SWNT symmetry and introduce acceptor and/or donor states, but they do not change the entire SWNT topology. Instead, the electronic perturbations are spatially localized. Nevertheless, these point defects still impact every passing charge carrier and can therefore substantially change G. For example, decay lengths of 0.5–3.0 nm are observed for typical defect states by STM spectroscopy (Kim et al. 2003a; Ishigami et al. 2004; Ruppalt and Lyding 2007). These states, which wrap entirely around a SWNT circumference, are extended barriers or trapping potentials for free carriers to traverse. SW defects are mildly disruptive, producing shallow pairs of donor and acceptor states with energies near the band edges (Lee et al. 2005). Adduct and vacancy defects, on the other hand, tend to produce higher potential barriers. In either case, the potentials are short range, and they therefore promote the large-momentum-transfer backscattering to which SWNTs are not normally susceptible (Ando 2005). Modelling of these barriers and their effects on G has been completed for an extensive range of defect types. The theoretical literature deserves a separate review to be thorough, since it covers SW defects, vacancies, substitutional dopants, and adducts of various molecules, all performed in an assortment of symmetries with respect to SWNT axis and chirality. Unfortunately, experimental confirmation of these predictions lags far behind, as very few experiments combine the resolution and defect control necessary to make quantitative comparisons. Furthermore, model configurations are chosen for computational conveniences such as symmetry, rather than for their chemical appropriateness. Even so, numerous models predict the same common feature, independent of the techniques used, which is a defect-induced G suppression on the order of 50%. Two examples are shown in Fig. 2.21. A simple, hand-waving interpretation of this frequent outcome is that a defect disrupts transport in one but not both of a SWNT’s conduction channels. Modelling further suggests a rich assortment of behaviors as two or more defects promote interference effects and interchannel scattering, some of which are shown in Fig. 2.22. Higher defect densities, of course, generally degrade a SWNT until it is insulating. Experimentally, many measurements have clearly resolved G decreases in SWNT films as they are chemically attacked. Usually, accompanying changes of optical properties are used to determine the effectiveness of different reagents. Covalent sidewall reactions like fluorination appear to have the

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Fig. 2.21 (a) G of an (18,18) armchair nanotube with a Stone–Wales defect, determined using tight-binding (dashed line) and k · p (solid lines) models. Different symmetry defects have substantially similar consequences, merely shifted in energy. (b) G of a (5,5) armchair SWNT with different covalent adducts, determined using density-functional calculations. Each adduct decreases G by approximately 50%, though the position of the minimum shifts due to differences in each adduct’s electron affinity. (a) Reproduced with permission from Matsumura and Ando (2001). Copyright 2001 Physical Society of Japan. (b) Reproduced with permission from Park et al. (2005). Copyright 2005 IOP Publishing Limited.

strongest effects on conductivity, in agreement with the predictions for sidewall adducts (Mickelson et al. 1998; Sumanasekera et al. 1999; Pehrsson et al. 2003). A recent review provides a full summary of the electrical properties of chemically modified CNTs (Burghard 2005). The modelling, however, suggests that G is sensitive to single sidewall defects, and that it should be possible to clearly resolve their creation or other dynamics. G measurements can be performed in any environment where a SWNT device is stable—at high temperature, in acidic electrolytes, under radiation, etc.—and this provides a fair degree of experimental versatility for trying to resolve defects’ electronic effects. Changes in G have, in fact, been observed before and after Ar+ irradiation (Woo et al. 2006) and electron-beam irradiation (Bachtold et al. 1998; Kasumov et al. 1998; Suzuki and Kobayashi 2005; Vijayaraghavan et al. 2005), but it is difficult to attribute these induced changes to any particular mechanism. Alternately, researchers have investigated changes in G due to chemical reactions, and these experiments are more

Fig. 2.22 (a) Theoretical G for two vacancies versus their separation D (in units of the SWNT diameter L). Of three orientations shown, only one completely blocks both conductance channels. (b) Theoretical G of a (6,6) armchair nanotube containing substitutional oxygen defects. The first atom decreases G by 30%, but the effect of the second (dashed line) or third oxygen (bold line) depends strongly on the defect’s separation. (a) Reproduced with permission from Nakanishi et al. (2000). Copyright 2000 Elsevier Limited. (b) Reproduced with permission from Rochefort and Avouris (2000). Copyright 2000 American Chemical Society.

2.4 Physical consequences of defects and disorder 75

Fig. 2.23 (a) G(t) during oxidative incorporation of adduct defects in HClO4 , followed by electrochemical reduction of the same SWNT. Sharp steps of 10–30% are attributed to individual chemical events. (b) G(Vg ) for a metallic, pristine SWNT oxidized and then reduced. Substantial recovery of G following reduction is due to the formation of ethers, which have sp2 conjugation and high conductance. Note that measurements are at low bias (100 mV), and that tunnelling across the defect state dominates G at higher bias. Reproduced with permission from Goldsmith et al. (2007). Copyright 2007, American Association for the Advancement of Science.

promising. Cui et al. looked at OsO4 oxidation, and observed hundredfold reductions in G that could be reversed by UV exposure (Cui et al. 2003). In this case, osmylation presumably forms an adduct to the SWNT sidewall, which UV can photocleave. Other researchers have investigated the reactions of diazonium salts with SWNTs (Balasubramanian et al. 2003; Wang et al. 2005). Diazonium reactions are of particular interest because they are more selective to the carbon lattice, and to small-diameter metallic SWNTs in particular (Bahr et al. 2001; Strano et al. 2003). Researchers have observed G degradations on both metallic and semiconducting SWNTs. In 2007, Goldsmith et al. resolved discrete steps in G using electrochemical oxidation in acids like HNO3 and HClO4 (Goldsmith et al. 2007). Whereas previous work had observed gradual degradation, the measurements in Fig. 2.23 capture individual sidewall oxidation and reduction reactions. The important difference is the adoption of electrochemical techniques from graphite and graphite oxides (Kinoshita 1988; Sumanasekera et al. 1999), whereby the reaction rates can be exactly tailored. By biasing the SWNT at its threshold of reactivity, individual, stochastic events occur well separated in time. Oxidizing conditions drop G as covalent adducts introduce sp3 conjugations, and reducing conditions step G back up as ethers are produced (see Fig. 2.3). The presence of the residual ethers dominates the gate dependence G(Vg ), especially when experiments are performed on metallic SWNTs as shown in Fig. 2.23(c). The ability to resolve sidewall reactions allows SWNTs to be studied before and after incorporation of single defects. Furthermore, devices fabricated with single defects can be tailored as described in Section 2.2.3. It is likely that this level of precision will be particularly useful for comparing experiment with theory. For example, Fig. 2.24 shows G(Vg ) measurements taken on a metallic SWNT and a semiconducting SWNT before and after the production of a carboxylate defect (Coroneus et al. 2008; Goldsmith et al. 2008). Subsequent probing of these defects with EDC allowed the direct observation of binding and unbinding events as EDC molecules interacted with the carboxylate. G modulation on the order of 50% was observed from individual chemical events, and analysis of the binding statistics provided a measure of the EDC

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Fig. 2.24 A semiconducting (a) and a metal SWNT (b) both exhibit a reduced, gate-sensitive G following the incorporation of a carboxylate defect. Note that the gate voltage on the x-axis is applied to a liquid electrolyte. (c) G fluctuations of 50% indicate ongoing binding dynamics between the carboxylate defect and a surrounding electrolyte containing EDC. Adapted and reproduced with permission from Goldsmith et al. (2008). Copyright 2008, American Chemical Society.

turnover rates in good agreement with bulk measurements. The large signal suggests SWNT defects could have promising applications in the study of single-molecule dynamics and chemistry. Finally, three-terminal SWNT devices in field effect transistor (FET) geometries are typically analyzed in terms of their gate dependences. The behavior of G(Vg ) determines primary device benchmarks such as on/off ratio and transconductance. But the presence of defects can produce misleading results, since defects introduce additional gate sensitivities shown in Section 2.3.5 and Fig. 2.23. For example, extracting an effective carrier mobility from the device transconductance requires knowledge of the relevant capacitance, a parameter that varies greatly depending on whether the entire SWNT or just a point defect is modulating the current. While SWNTs do have outstanding carrier mobilities at room temperature, defects account for some of the very large values reported in the literature (Zhou et al. 2005; Woo et al. 2006). 2.4.2.3 Chemoresistance The chemoresistive sensitivity of CNTs was not anticipated theoretically and its initial observation (Collins et al. 2000a; Kong et al. 2000) seemed at odds with the principle of inert, graphitic conductors. Subsequent study has identified multiple possible mechanisms for the observed effects, but no unified understanding of their contributions has emerged, and theory and experiment remain separate. Nevertheless, one point has achieved broad theoretical and experimental consensus, and that is the importance of defects and disorder. All of the conductance effects described above are sensitive to a defect’s charge, and this localized charge is chemically active. Reactive defects help promote charge transfer, chemisorption, and covalent bonding between a CNT and its environment, thus enhancing their response as chemical sensors. Oxidations generally improve the sensitivity of many types of CNT electrodes, including electrochemical ones. Acid treatments enhance the sensitivity of MWNTs to different analytes, though also to common gases like O2 and water vapor (Watts et al. 2007). Mendoza et al. observed a 100% increase in sensitivity to NO2 after a long treatment with hot HNO3 (Mendoza et al. 2007). These researchers concluded that oxygen-containing defects,

2.4 Physical consequences of defects and disorder 77

Fig. 2.25 G(t) for two SWNT sensor devices electrochemically decorated with Pd, as they are repeatedly probed by H2 gas in air. G decreases ∼ 40% in the first device (solid line), a semiconducting SWNT liberally decorated as shown in the micrograph. G decreases more than 99% in the second device (dashed line), composed of a semi-conducting SWNT with a single, selectively decorated defect. Reproduced with permission from Khalap (2008).

and particularly carboxylates, play a key role. Theoretical modelling generally agrees with this result, especially in light of the weak interactions and charge transfer associated with defect-free SWNTs, though some calculations also predict enhancements for SW defects (Maiti et al. 2006). The field of chemical sensing is complicated by a competition among multiple possible mechanisms. Exposed interfaces at the CNT–electrode contacts, for example, have a clear and measurable chemoresponse that is distinct from any sidewall response (Shim 2005; Shim et al. 2005; Zhang et al. 2006). Multi-CNT electrodes, such as the thin films that are commonly used, also include highly resistive intertube junctions. These junctions are sensitive to physisorbates and are also repositories for reactive contaminants and amorphous carbons. Robinson et al. completed a careful study of SWNT films by measuring chemoresponses before and after mild oxidations in UV/ozone (Robinson et al. 2006). Despite mapping tenfold response enhancements that accompany carboxylate densities, the relative importance of interfacial effects, CNT–CNT contacts, and CNT sidewalls remained ambiguous because the chemical oxidation changed all three. CNT-based H2 sensors are one of the few demonstrations of chemoresistivity that have progressed to the commercial market, and yet despite this relative maturity, the H2 -sensing mechanisms remain poorly understood. Typically, Pd metal is deposited onto SWNT devices (Kong et al. 2001) in order to sensitize them to H2 gas. Empirically, sputtered Pd gives the desired result, but thermal deposition and electrodeposition do not (Yuan et al. 2007). Using the selective electrodeposition technique described in Section 2.3.3, it can be proven that H2 sensitivity only results when Pd particles decorate defect sites (Fig. 2.25). Pd that otherwise decorates pristine sidewalls or contact interfaces does not result in H2 sensitivity (Khalap 2008). The effectiveness of Pd sputtering indicates that it probably introduces low levels of damage.

2.4.3

Mechanical effects of CNT defects

One of the most notable properties of CNTs is their mechanical strength. Very early calculations suggested high tensile strengths for CNTs (Overney et al.

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1992; Kiang et al. 1995; Ruoff and Lorents 1995), and beginning in 1996 a number of mechanical measurements were successfully made on individual CNTs. A team at NEC laboratories used a TEM technique to measure the Young’s modulus of MWNTs and SWNTs to be approximately 2 TPa and 1.2 TPa, respectively (Treacy et al. 1996; Krishnan et al. 1998). Similar values were obtained independently using AFM techniques (Falvo et al. 1997; Wong et al. 1997). Subsequently, a variety of strength measurements have been performed on individual CNTs, bundles and yarns of CNTs, and CNTs incorporated into composites. In general, the tensile strength of CNT ropes can exceed 1 GPa and withstand strains of 5% or more. These high values, combined with outstanding strength-to-weight ratios, make CNTs technically superior to metal alloys, Kevlar, or other carbon-based polymers (Baughman et al. 2002), and have even supported speculation about cabling for Earthbound space elevators. Obviously, this extraordinary mechanical strength is sensitive to defect densities, but it is important not to overemphasize their role. Many factors limit the ultimate performance of high-performance fibers, and the presence or absence of CNT defects is only relevant in the limit that no other weaknesses exist. Experimental fibers may contain voids, inhomogeneous microstructure, and CNTs weakened by kinks, entanglements, and twists. Furthermore, mechanical testing on macroscopic samples require efficient strain transfer from one CNT to another, and early mechanical experiments routinely observed CNTs pulling out of their matrices rather than tearing (Ajayan et al. 1994). To address these failure modes, researchers have actively pursued chemical oxidation and functionalization to provide covalent linkages, either among the CNTs or between them and a supporting matrix. The relatively inert CNT surfaces are poorly suited to covalent intertube strengthening, though, and the net effect of these treatments has usually been to compromise the intrinsic strength of the CNT lattice. Much of the existing literature therefore places lower bounds on the actual strain or strength of individual CNTs, and in the context of such difficulties the consequences of dilute defects are completely obscured. Defects like divacancies and interstitials are mechanically important in graphites and graphitic carbon fibers because they help to form interlayer bonds among the carbon sheets. Similar cross-links increase the shear strength of MWNTs and can reduce the “sword-in-sheath” failure in which the core of a MWNT pulls out of an outer shell (Cumings and Zettl 2000; Yu et al. 2000c; Deshpande et al. 2006). Cross-linking interstitials also seem to play a role in intertube welding or coalescence (Terrones et al. 2000, 2002b; Krasheninnikov et al. 2002b). However, as described previously, such defects are not thermodynamically likely in pristine material. Instead, they usually result from intentional modifications, such as those that serve to weaken the CNTs but effectively bind them within a matrix. This type of disorder, including surface functionalization, is treated in more detail by Hauke and Graupner in this Handbook (see Chapter 16 of Volume I). Despite these problems, progress has been made preparing high-quality fibers from CNTs. Their fabrication, testing and optimization has progressed rapidly by adopting technologies like spinning and extrusion from the mature

2.4 Physical consequences of defects and disorder 79

industries of carbon and polymer fibers. A particularly advantageous aspect of SWNTs is their tendency to bundle into long, parallel ropes during growth (Thess et al. 1996). This bundling, driven by van der Waals interactions, helps eliminate tight kinks and entanglements, and can even help provide efficient strain transfer when the SWNTs are long enough to have large contact areas. These surface interactions also lead to a modest cicumferential deformation that further enhances intertube adhesion (Ruoff et al. 1993; Tersoff and Ruoff 1994). Recent results suggest that pristine SWNTs, properly wound into large bundles, do not need additional cross-linking chemistries to achieve high strengths or stiffnesses (Li et al. 2004; Zhang et al. 2004; Koziol et al. 2007). Contrary to earlier assumptions, the only chemical processing needed to enhance SWNT fiber strength is a densification step that enhances SWNT–SWNT ordering and optimizes stress transfer. Both acetone and water have been proven effective at driving SWNT densification (Hata et al. 2004). A large number of interesting studies have tested the tensile strengths of ropes made from non-cross-linked SWNTs (Salvetat et al. 1999; Walters et al. 1999; Vigolo et al. 2002; Li et al. 2004; Zhang et al. 2004; Koziol et al. 2007) or MWNTs (Pan et al. 1999; Yu et al. 2000a,b). Ideal fibers must effectively eliminate voids and have very long, highly aligned CNTs. In this limit, the strength of a fiber can be accurately interpreted in terms of the constituent CNTs. Moreover, short fibers composed of high-purity, low defect concentration CNTs can in principle achieve less than one defect per fiber, on average. In this limit, fiber strengths should split into a bimodal distribution, with a high-strength peak for the defect-free fibers and a lower-strength peak for fibers containing one defect. In 2007, Koziol et al. investigated this limit using CNT fibers spun directly from the growth zone of a CVD synthesis process (Koziol et al. 2007). In addition to fibers with a specific strength of 1 GPa/SG (strength per unit of specific gravity), a separate population of fibers was measured with a specific strength averaging 6.5 GPa/SG and peaking at 9 GPa/SG. The bimodal distribution, shown in Fig. 2.26, agrees with a weak-link model of single point defects. As the gauge length is increased, the tensile strengths of the two populations do not shift; rather, the fraction of higher-strength fibers quickly decreases. The conclusion is that defects, incorporated randomly per unit of length, exist as rare weak links in the fibers and account for a >80% loss in strength when present. It is notable that the high-strength fibers also achieved strains of 6–8% and stiffnesses of 150 to 400 GPa/SG (Fig. 2.26(b)). The decrease in strength observed experimentally agrees with a recent review of various theoretical estimates (Pugno 2007). In this review, the practical strength of CNT cables is estimated to be less than 30% of a CNT’s ultimate strength due to vacancies. A single unreconstructed vacancy accounts for as much as a 50% drop in a SWNT (Sammalkorpi et al. 2004). These theoretical estimates do not consider further reductions due to bond rotations or 5–7 defects, though the theoretical literature does further consider the effects of vacancies on strength, strain, fatigue, and crack propagation in SWNTs. Pugno reviews a wide range of experimental strength measurements and, rather than highlighting disagreements with the predictions for ideal SWNT strengths,

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Fig. 2.26 (a) Composite strength of SWNT fibers versus gauge lengths. At the smaller gauge lengths, the distribution clearly evolves into a bimodal distribution. The “high-strength” fibers are presumed to differ from “low-strength” ones by the complete absence of SWNT defects. (b) The highstrength SWNT fibers also have high stiffness and compare favorably to other high-strength fibers. Reprinted with permission from Koziol et al. (2007). Copyright 2007, American Association for the Advancement of Science.

highlights the reasonable agreement with theoretical predictions focused on the consequences of vacancies. Other than vacancies, SW and 5–7 defects also play important roles in CNT mechanical properties. In particular, the spontaneous nucleation of these defects plastically relieves stress in CNTs strained above 5% (Nardelli et al. 1998). This relief mechanism increases the maximum strain achievable in practical fibers, as well as tempering the consequences of buckles and kinks. While the introduction of these defects is plastic and generally non-reversible, their mechanical consequences are not as substantial as their electronic effects. That is, while SW defects change chirality and electronic structure, the tensile properties of SWNTs are not strongly dependent on either chirality or diameter (Lu 1996). Once present, these defects nucleate additional mechanical changes as CNTs are strained further (Yakobson et al. 1996; Falvo et al. 1997; Lourie et al. 1998). The variety of allowed morphological deformations reduces the likelihood of brittle fracture in CNTs, and results in a physical behavior remarkably similar to macroscopic polymers. Yakobson and Avouris have reviewed the non-linear and inelastic deformations that occur at very high strain (Yakobson and Avouris 2001).

2.5

Concluding remarks

This chapter has attempted to highlight ongoing needs and opportunities in the research area of CNT defects and disorder. The growing appreciation of defects and their consequences is merely the beginning of research in this

References

area and a sign of continued progress in the CNT research field. With precise measurement and control of individual defects, researchers may in the future have sensitive probes of novel physics, and also versatile scaffolds for construction at the molecular scale. However, the main emphasis in this chapter has been the extent to which techniques still need development. While some techniques can highlight point defects, the main emphasis in Section 2.3 remains the difficulty of the task. Finding and characterizing defects, especially in the presence of any other intrinsic and extrinsic disorder, continues to be a challenging task. Furthermore, similar challenges are faced by all nanoscale materials. This chapter has focused on defects in CNTs, but inhomogeneity and imperfection is a continuing problem for the commercial adoption of nanomaterials generally. The development of high-resolution metrologies is one aspect to long-term solutions, but this chapter has also highlighted the need for efficient and high-yield characterizations. New and appropriate vocabulary for nanomaterials is also necessary, so that the complexity of these high surface area objects can be accurately described. This chapter has attempted to distinguish between different types of disorder and, in particular, types that have different consequences. Ultimately, these subtleties will be captured by terms in common usage. This development would help improve communication between experts in disparate parts of the research and development communities, and help nanoscience fully mature into nanotechnologies.

Acknowledgments The author is indebted to mentors, colleagues, and coworkers who have contributed to the content of this chapter over many years through fruitful discussions, critical reviews, and the completion of difficult experiments. The author also acknowledges financial support from NSF DMR-0239842 and EF-0404057.

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3.3 Boundaries in planar and tubular nanostructures

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Appendix: Total-energy electronic-structure calculations

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Roles of shape and space in electronic properties of carbon nanomaterials Atsushi Oshiyama and Susumu Okada

3.1

Introduction

The discovery of carbon 60 molecules (Kroto et al. 1985) and carbon nanotubes (Iijima 1991) has triggered a great expansion in the amount and variety of research on the carbon nanomaterials (CNMs). One of the fascinating characteristics of CNM is its wealth of physical properties that are closely related to its atom-scale structures and consequently its nanoscale shapes. For instance, a carbon nanotube (CNT) becomes semiconducting in some cases and metallic in others, depending on tiny differences in its atomic arrangement. It was predicted theoretically (Hamada et al. 1992; Saito et al. 1992) in the early stage and then confirmed later by scanning tunnelling microscopy measurements (Odom et al. 1998; Wild¨oer et al. 1998). This remarkable property is due to the anisotropy of energy bands of graphite, the mother material, and to the nanoscale geometry, in other words, nanoshape that imposes additional boundary conditions on electron wavefunctions. It is generally true that surfaces and interfaces become more and more important when sizes of materials become tiny. Shapes of nanometer-scale materials therefore naturally affect physical properties of the nanomaterials substantially. In addition, one of the structural peculiarities of carbon nanomaterials is their structural hierarchy. Carbon atoms are assembled to form, e.g. C60 and then C60 clusters like superatoms form some particular bulk forms. CNT consisting of carbon atoms also form multiwalled CNTs or CNT bundles. This structural hierarchy naturally generates internal space on the nanoscale. The space inherent to carbon nanomaterials also plays an important role in their physical properties. C60 fullerenes, for instance, are condensed in cubic crystalline structure (Kr¨atschmer et al. 1990), leaving space among fullerenes and exhibiting semiconducting behaviors with an energy gap of a few

3.2 Nanospace in carbon peapods 95

eV (Saito and Oshiyama 1991a; for a review of earlier theoretical works, see Oshiyama et al. 1992). By filling the space with foreign elements such as alkaline metal and alkaline-earth metal elements, C60 fullerides become ionic metals (Saito and Oshiyama 1991b) and eventually superconductors (Hebard et al. 1991; Tanigaki et al. 1991). Further, what has been clarified from theoretical studies in these years is the existence of a certain class of electron states that distribute in the internal space. They are occasionally called intercluster, intracluster, intertube, intratube, or nearly free-electron states. Whatever the name is, a common feature is that they distribute mainly in the internal space of the hierarchical materials and become important in nanoscale structures. Morphology of CNMs actually induces pleasant surprises. In addition to the soccer ball and the straw, nanopeapods (Smith et al. 1998), nanohorns (Iijima et al. 1999), nanoshuttlecocks (Sawamura et al. 2002) made of carbon and other light elements have been discovered or synthesized continually. This is probably due to the chemical feasibility of carbon elements to form a variety of structures. In a view that the nanoscale geometry certainly affects electronic properties, the wealth in morphology is expected to produce a wealth of physical properties of CNMs. The computational science approach has contributed a lot to the progress in the science of carbon nanomaterials, as is evidenced by a variety of literature that presents the so-called first-principle calculations. In this chapter, we review some of those studies, emphasizing roles of shape and space in CNMs to manifest their charming electronic properties. Some description as to the methodology of the calculations is given in the Appendix.

3.2

Nanospace in carbon peapods

One of the characteristics of CNMs may be space within the materials. The space may be used to accommodate foreign species. CNTs indeed accommodate fullerenes that are in turn occasionally filled with different chemical

(a) C60@(10, 10)

(b) C60@(9, 9)

Fig. 3.1 Atomic structures of carbon peapods: Total-energy optimized structures of C60 encapsulated in (a) (10, 10) and (b) (9, 9) carbon nanotubes (reprinted in part from Okada, S., Saito, S., Oshiyama, A. (2001) c 2001 The AmerPhys. Rev. Lett. 86, 3835,  ican Physical Society).

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elements. This hierarchical structure is called carbon peapods (Fig. 3.1). A variety of peapods have been experimentally reported: C60 (Burteaux et al. 1999; Smith et al. 1999), higher fullerenes Cn with n = 70, 78, 80, 84 and also metallofullerenes (Hirahara et al. 2000, 2001; Sloan et al. 2000; Kataura et al. 2001) are encapsulated in CNTs. In this section, we discuss electronic structures of the peapods and elucidate the important role of the internal space: i.e. peculiar intratube electron states generated in the space cause substantially unusual electronic properties in the peapods. Figure 3.1 shows total-energy minimized atomic geometries of C60 encapsulated in the armchair (9, 9), and (10, 10) nanotubes obtained in the densityfunctional theory (DFT) (Okada et al. 2001a). Some details of the calculations are described in the Appendix. In the (10, 10) peapod, C60 @(10, 10), C60 inside and the nanotube outside almost keep their original shapes before encapsulation. The calculated distance between the wall of the (10, 10) nanotube ˚ which is close to the interlayer distance and the nearest atom of C60 is 3.31 A, ˚ or the inter-C60 distance in the fcc C60 . On the other of graphite (3.34 A) hand, the space provided by the (9, 9) CNT is insufficient for C60 : Both the tube and the fullerenes are substantially distorted in C60 @(9, 9) peapod. The fullerenes inside are elongated along the tube direction and the tube silhouette itself becomes undulating. The calculated distances between the ˚ in a C60 @(9, 9) wall of the nanotubes and the nearest atoms of C60 is 2.75 A peapod. These characteristics in the geometry are reflected in the total energies of the peapods. Figure 3.2 shows eneregetics in formation of a variety of peapods. The energy difference ΔE in the reaction, CNT + Fullerene → Fullerene@CNT − E, is plotted as a function of tube radii (Okada et al. 2001a, 2003a; Otani et al. 2003). The reaction is exothermic for thick enough CNT. For C60 , the (10, 10) CNT is the most favorable pod with the encapsulation energy of ΔE = −1.7 eV per C60 . The zigzag (17, 0) CNT has a comparable encapsulation ˚ in this case and the energy of ΔE = −1.4 eV. The interwall distance is 3.11 A slight squeezing causes the lower encapsulation energy. The space in (9, 9) and (15, 0) CNTs is too small for C60 to be encapsulated so that the reactions are endothermic. With increasing tube radius, the energy gain upon encapsulation becomes vanishing. This is because C60 is placed at the center of CNT in Fig. 3.2. It is indeed found that in thicker CNTs C60 approaches to the wall ˚ and therefore gains the energy of about 1 eV with the distance of about 3 A (Okada et al. 2003a). Encapsulation in chiral nanotubes has not been studied theoretically. Yet the energetics for achiral (armchair and zigzag) CNTs shown in Fig. 3.2 clearly indicates strong correlation between the encapsulation energy and the space provided by CNT. The encapsulation of C60 in CNT is due not to chemicalbond formation but to the relatively weak interaction with van der Waals character. It is therefore concluded that the minimum radius of CNT for the ˚ and the most favorable radius is about encapsulation of C60 is 6.1–6.2 A ˚ 6.6 A.

3.2 Nanospace in carbon peapods 97

4 (15,0)

DE [eV/fullerene]

3 2 1 (17,0) 0

(10,10)

(9,9)

(21,0)

(20,0) (19,0)

–1

(18,0)

(16,0)

(11,11)

(13,13)

(12,12)

–2 5.5

6.0

6.5

7.0 7.5 Tube radius [Å]

8.0

9.0

8.5

Fig. 3.2 Reaction energies ΔE (see text) per fullerene as a function of CNT radius in the encapsulation reaction of the fullerene for armchair (n, n) and zigzag (n, 0) CNTs. Circles, squares, triangles and reverse triangles denote ΔE for C60 , C70 (standing), C70 (lying) and C78 . The lines are guides for the eye.

The energetics for encapsulation of higher fullerenes is also shown in Fig. 3.2. For C70 , there are two possibilities for geometries in encapsulation: In one geometry, the long axis of the oval is parallel to the CNT axis (lying geometry) and in the other it is perpendicular (standing geometry). For the stable peapods, the lying geometry is energetically favorable since the contact area is larger in the lying geometry than in the standing geometry. On the other hand, for thicker tubes the standing geometry is favorable since C70 in the geometry can be close to the CNT wall. This geometrical factor peculiar to elongated fullerenes adds an interesting variation in energetics of the encapsulation. C60@(10,10)

Energy (eV)

(a)

(10,10)

C60@(9,9)

(b)

(9,9)

2

2

2

2

1

1

1

1

0

0

0

0

–1

–1

–1

–1

–2

–2

–2

–2

–3

G

X

–3

G

X

–3

G

X

–3

G

X

Fig. 3.3 Energy-band structures of C60 @(10, 10) (a) and C60 @(9, 9) (b) peapods along with the energy bands of corresponding pristine CNTs. Energies are measured from the Fermi-level energy E F (reprinted in part from Okada, S., Saito, S., Oshiyama, A. (2001) Phys. Rev. Lett. 86, 3835, c 2001 The American Physical Society). 

98

Roles of shape and space in electronic properties of carbon nanomaterials

Electronic energy bands of C60 @(10, 10) and C60 @ (9, 9) peapods calculated by Okada et al. (2001a) are shown in Figs. 3.3(a) and (b), respectively. In an isolated C60 , there are a fivefold degenerate highest-occupied state labelled as h u and a threefold degenerate lowest-unoccupied state labelled as t1u . In the fcc C60 , both h u and t1u show a dispersion of ∼0.6 eV (Saito and Oshiyama 1991a) and become valence and conduction bands, respectively. The energy gap remains finite. As for (n, n) nanotubes, there are two energy bands near the Fermi energy E F , and the two bands cross E F at the k  2π/3a of the Brillouin zone (Hamada et al. 1992). Therefore, it may be expected in C60 @(n, n) that the two bands originated from the nanotube cross E F and are located in the gap of C60 , and that the peapod is metallic. However, Fig. 3.3(a) clearly shows a new feature that is missing in the discussion above. Four bands cross the Fermi level in the C60 @(10, 10) peapod: Two of them have large linear dispersion, keeping the character of the π orbitals on the nanotube, whereas the other two have less dispersion, similar to the π state on the C60 chain. It is found that the latter two bands keep the character of a t1u state of C60 . The C60 @(10, 10) peapod is thus a metal with multicarriers, each of which distributes mainly either on the tube or on the C60 chain. On the other hand, the energy bands of C60 @(9, 9) peapod do not exhibit such a feature (Fig. 3.3(b)). Only two bands that have π character of the nanotube cross the Fermi level. The energy bands of a C60 @(9, 9) peapod are understandable in terms of the electron states of the constituents before the encapsulation. The C60 @(9, 9) is thus a metal and, in sharp contrast to C60 @(10, 10), the charge density at the Fermi level E F distributes only along the walls of the nanotube. Dr–

total charge

Dr+

0.2

(a) C60 @(10,10)

2.5E-4 2.5E-4

Dr

total charge

–

Dr+ 2.5E-4

(b) C60 @(9,9)

5.0E-4

0.2

Fig. 3.4 Contour plots of the total valence charge density of C60 @(10, 10) (a) and C60 @(9, 9) (b). The contour plot of more negatively charged (electron rich) area, Δρ − and that of more positively charged area Δρ + , than a sum of two self-consistent electron densities of CNT and C60 are also shown. Each contour represents twice (or half) the density of the adjacent contour lines. The values shown in figures are in unit of electron/(atomic c 2001 The American Physical Society). unit)3 (reprinted in part from Okada, S., Saito, S., Oshiyama, A. (2001) Phys. Rev. Lett. 86, 3835, 

3.2 Nanospace in carbon peapods 99

inside

The difference in energy bands between the C60 @(10, 10) and C60 @(9, 9) peapods is attributed to the space inherent to the hierarchical solids, as we explain below. Figure 3.4(a) shows the electron density of C60 @(10, 10). We observe the low charge density between the C60 and CNT, which indicates that the constituent units are bound weakly. In order to examine the electron density more carefully, we plot the difference between the electron density of C60 @(10, 10) and the sum of the electron densities of the nanotube and of C60 chain. It is clear that electrons are transferred mainly from the π orbitals of CNT and C60 to the space between the CNT and C60 . The electron-rich region shows a peculiar space distribution (Fig. 3.4(a)). Figure 3.5 shows the distribution of a squared wavefunction of a certain electron state in CNT (Okada et al. 2000a). In contrast to the usual electron states in condensed matters, the state has an amplitude not on atomic sites but around internal space in matters. These states are called intratube or intertube states or occasionally nearly free-electron (NFE) states. The state is located in energy at about 3 eV above E F , depending on radius of the tube. These states are common and peculiar to spacious matters. In graphite, Holtzwarth et al. (1982) and Posternak et al. (1983, 1984) have found electron states that distribute between the layers of graphite. These states, called interlayer states, are located at several eV above E F in energy and play important roles in the excitation spectrum in graphite intercalation compounds. In Sr6 C60 and Ba6 C60 , intercluster states that distribute among C60 fullerenes contribute enhancement of density of states and thereby contribute to the occurrence of superconductivity (Saito and Oshiyama 1993). Also, Miyamoto et al. (1995) have shown that intratube states are coupled with alkaline-metal elements inside CNT. We now come back to the peapod. The electron-rich region in the C60 @(10, 10) peapod shown in Fig. 3.4(a) shows a peculiar distribution in the space between C60 and (10, 10) CNT. This space coincides with the space where the NFE states distribute. The energy level of the NFE state is generally located at a couple of eV above E F in CNT. This state is still above E F even in C60 @(10, 10). The hybridization between the NFE state and the π orbitals of C60 is, however, substantially induced upon encapsulation of the fullerenes in CNT. The hybridization occurs not with particular states of an isolated C60 but with most π and σ orbitals. Analysis of the wavefunctions of the Kohn– Sham orbitals in a range of −4 −3 eV in C60 @(10, 10) confirms this situation. As a result of this, most of the electron states originated from π orbitals of

Fig. 3.5 Contour map of the charge distribution of the nearly free-electron (NFE) state of an isolated (6, 0) CNT at  point on a crosssection including the tube axis (left panel) and another cross-section perpendicular to the axis (right panel). Gray dashed lines represent the position of the tube wall. Each contour line represents twice (or half) the density of the adjacent contour lines (reprinted in part from Okada, S., Saito, S., Oshiyama, A. c 2000 The (2000) Phys. Rev. B 62, 7634,  American Physical Society).

100

Roles of shape and space in electronic properties of carbon nanomaterials

Fig. 3.6 Energy band structures of zigzag peapods, (a) C60 @(16, 0), (b) C60 @(17, 0), (c) C60 @(18, 0), (d) C60 @(19, 0), and (e) C78 @(19, 0). Arrows indicate the t1u -like electron states. Energies are measured from the mid-gap energy of each empty nanotube (reprinted in part from Otani, M., Okada, S., Oshiyama, A. (2003) Phys. Rev. B c 2003 The American Physical 68, 125424,  Society).

Energy [eV]

(a) C60@(16,0) (b) C60@(17,0) (c) C60@(18,0) (d) C60@(19,0) (e) C78@(19,0) 1

1

1

1

1

0

0

0

0

0

–1

–1

–1

–1

–1

–2

–2

–2

–2

–2

–3

Γ

X

–3

Γ

X

–3

Γ

X

–3

Γ

X

–3 Γ

X

C60 shift downward upon encapsulation. These downward shifts of C60 states renders the C60 @(10, 10) peapod a metallic system with different characters of multicarriers, as is shown in Fig. 3.3(a). The electron densities of the thinner peapods are quite different from that of C60 @(10, 10). We certainly observe charge redistribution upon encapsulation in a C60 @(9, 9) peapod, as is shown in Fig. 3.4(b): Electrons are transferred from π orbitals of the nanotube to the spacious region inside the tube and around the fullerenes. Yet no single specific state such as the NFE state is responsible for this charge transfer. This is because the space between the constituents in C60 @(9, 9) is not sufficient to generate the NFE state located at a couple of eV near E F . Probably the NFE state is above the vacuum level. In this case, the relative location of electron states originated from CNT and from C60 is essentially unchanged by the encapsulation. The resulting energy bands are thus given in Fig. 3.3(b). When the NFE state appears in a suitable energy range as in C60 @(10, 10), it selectively hybridizes with π orbitals of C60 and thereby produces the interesting variation of energy bands at E F . This is an example of the general theoretical finding that space inherent to spacious matters decisively affects the electronic properties of nanomaterials. The downward shifts of the energy levels originated from C60 due to the nanoscale space in C60 @(10, 10) infer an interesting possibility: Semiconductor CNTs can be metallic upon encapsulation of fullerenes when suitable space is provided, and thus hybridization between π states of C60 and the NFE states takes place. Otani et al. (2003) have explored this possibility. Figure 3.6 shows their calculated energy bands of C60 @(n, 0) peapods. It is found that the t1u -like states shift sensitively to the radius of CNT. The t1u -like states are located above the energy gap of the empty tube in the C60 @(16, 0). The states shift downwards on increasing the radius and take the lowest value in C60 @(19, 0). The energy band in this case is not a simple sum of energy bands of the constituents, the C60 chain and the nanotube. In C60 @(19, 0), the t1u -like state are located at the mid-gap position of the (19, 0) empty nanotube

3.2 Nanospace in carbon peapods 101

so that the energy gap of the peapods decreases by half from the empty ˚ change of the tube nanotubes. The t1u -like states shift by 0.5 eV with the 1-A radius. The C60 peapods consisting of zigzag CNTs are also energetically stable and are expected to be synthesized in certain experimental conditions. Indeed, the transport experiments on the C60 peapods in the field effect transistor (FET) clearly exhibit the bandgap modulation that depends on radius and chirality of CNTs (Shimada et al. 2002). Metallization of semiconductor CNTs upon encapsulation is not found for the C60 peapods in the calculations above. Otani et al. (2003) have further explored a possibility of the metallization of semiconducting nanotubes by encapsulating the large fullerenes possessing the deeper lowest unoccupied (LU) state. The fullerene C78 with C2v symmetry is a possibility. Figure 3.6(e) shows calculated energy bands of the C78 @(19, 0). The most interesting feature is found in the energy bands near the Fermi level. The originally unoccupied states of the C78 (depicted by the arrow in Fig. 3.6(e)) shift downwards and are located near the top of the valence bands of the corresponding semiconducting (19, 0) nanotube. Consequently, there are three states almost degenerate near the valence-band top at the  point. Analysis of the Kohn– Sham orbital of the valence-band top at the  point has clarified that the wavefunction has amplitude not only on the tube but also on the C78 , indicative of the hybridization between the π states of the CNT and the LU states of the C78 . The remaining two states near the valence-band top also distribute both on the tube and the C78 to some extent. At the X point, on the other hand, the flat bands near the valence-band top preserve the character of the C78 states. The calculated energy bands indicate that C78 @(19, 0) is a semi-metal in which electrons with the C78 character are at the X point, whereas holes with the mixed character are at the  point. It should be remarked that the local density approximation (LDA) used in the calculations underestimates the fundamental gap of the semiconducting materials so that the calculated values are of a semi-quantitative nature. Yet these calculations have certainly clarified that the electronic structures of the peapods are indeed tunable by the nanoscale space between peas and pods. One-dimensional conductors are generally expected to show the metal– insulator transition associated with lattice distortion (Peierls 1955). An earlier theoretical calculation (Mintmire et al. 1992) shows that the nanotube itself is robust against the Peierls instability. In C60 @(10, 10) or in C78 @(19, 0) discussed above, some carriers are distributed along the chain of fullerenes. This may cause displacements of fullerenes. The calculated energy bands of C60 @(10, 10) near E F in Fig. 3.3(a) show that the filling is incommensurate. Thus, incommensurate charge density waves are generally expected. Since the space is decisive to control energy bands, however, we expect that the filling is sensitive to the radius of the encapsulating nanotube. Hence, displacements of fullerenes in peapods may exhibit variation depending on the tube radii and, in another cases, on the sizes of fullerenes. Some experimental images by the transmission electron microscope indeed show dimerization of fullerenes in the peapods, but further efforts are necessary to reveal details.

102

Roles of shape and space in electronic properties of carbon nanomaterials

3.3

Boundaries in planar and tubular nanostructures

Shape is another factor that affects the electronic properties of CNMs. In this section, we focus on shapes of boundaries in nanostructures. In planar structures such as graphene, the boundary is an edge of a sheet. In tubular structures, it is an end of a CNT, for instance. We may also consider boundaries of two different nanostructures. These boundaries have been considered just as morphology of materials in the past. In nanomaterials, however, they play important and in some cases decisive roles in the electronic properties, as will be discussed in this section. We may also raise the question of what boundaries are. For some electron states, not the geometrical boundary but some modification of structures can be boundaries. This aspect we also discuss in this section.

3.3.1

Flat-band state and its magnetism in zigzag-edged graphene

We first discuss peculiar electron states in a zigzag-shaped graphene sheet that is occasionally called a zigzag-shaped graphene ribbon. The atomic arrangement of the graphene ribbon is shown in Fig. 3.7. A pioneering study was done by Fijita et al. in 1996. They discovered that peculiar electron states exist in a finite-size graphene ribbon with zigzag-shaped edges. The states are located in energy at the Fermi level E F , where the π states of a graphene sheet stick together at the K point in Brillouine zone. An unusual feature is that the states show no dispersion when the wave vector k varies from the K point to the inner k point along the K line: i.e. there are flat bands at E F in zigzag-shaped graphene ribbons. Analyses of their wavefunctions have revealed that the states distribute mainly along zigzag edges and the amplitudes decay inside. Thus, they are called edge states. It is noteworthy that only zigzag-edged graphene ribbons exhibit such flat bands and the energy bands of other graphene ribbons such as armchair-edged ribbons are obtained simply by projection of the energy bands of infinite-size graphene on the direction of

(a)

(b)

Fig. 3.7 Atomic arrangements of a graphene ribbon with zigzag-shaped edges. (a) A side view and (b) a top view of the graphene sheet along with contour plots of its spin density n ↑ ( r ) − n ↓ ( r ). In (a) the edges are perpendicular to the plane and C atoms on the plane are depicted by shaded circles. Positive and negative values of the spin density are shown by solid and dashed lines, respectively. Each contour represents twice (or half) the c 2001 The American Physical density of the adjacent contour lines (reprinted from Okada, S., Oshiyama, A. (2001) Phys. Rev. Lett. 87, 146803,  Society).

3.3 Boundaries in planar and tubular nanostructures 103

the ribbon. It is also important that the edge states are of π -orbital character so that even if the edges are terminated by hydrogen or by other molecules they are located at E F and show no dispersion (Nakada et al. 1996; Miyamoto et al. 1999). The Fermi-level density of states and electron correlation are two major factors that induce the ferromagnetic ordering in itinerant electron systems. Each role, along with effects of orbital degeneracy and van-Hove singularity inherent to specific lattices, has been examined for more than half a century with several model Hamiltonians (for a review see Dagotto 1994 or for real materials Asada and Terakura 1992). What is unequivocally clarified as to the mechanisms of the ordering is not many, however. The flat-band ferromagnetism is one the such mechanism that have been proved exactly in the Hubbard model under certain conditions (Lieb 1989a,b; Mielke 1991, 1992; Tasaki 1992, 1998): For several specific lattice structures, a delicate balance of transfer integrals results in an energy band that lacks dispersion; introduction of the Hubbard U between the band electrons induces the ferromagnetic ordering. Yet it is uncertain whether the flat-band ferromagnetism is real or fictitious in nature. Edge states found in zigzag-shaped graphene ribbons provide suitable real stages to explore flat-band magnetism. Figure 3.7 shows the calculated spin r ) − n ↓ ( r ) of a graphene ribbon that has straight zigzag edges density n ↑ ( (Okada and Oshiyama 2001b). As explained above, there are flat-band states (edge states) at E F in graphene ribbons with zigzag edges. It is thus expected that a certain magnetic ordering appears on the graphene ribbons. The calculations based on DFT have clearly uncovered the existence of the magnetic ordering that is mainly located along the zigzag edges and decays gradually inside. On each edge, the spins are aligned ferromagnetically. However, it is not the ferromagnetic ordering in total. There are two atomic sites in the hexagonal primitive cell of graphite and all the atomic sites are classified into two sublattices A and B (bipartite lattice). The number of carbon atoms in each lattice of the ribbon is identical (NA = NB ). It is clearly observed in Fig. 3.7 that the spin is polarized in one direction on one sublattice and in the opposite direction on the other. As a result, the total spin S of the graphite ribbon is vanishing. This corresponds to a theorem S = (NA –NB )/2 in the bipartite lattice, which is proved in the Hubbard model under some conditions (Lieb 1989a,b).

3.3.2

Nanometer-scale magnetic ordering in zigzag carbon nanotubes with finite length

The lengths of CNTs are usually finite and the open ends of the nanotubes are indeed observed in transmission electron microscope (Ajayan et al. 1993). In such nanotubes with finite lengths, zigzag-shaped edges are naturally expected. CNTs with zigzag-shaped edges are formed by rolling zigzag-edged graphene ribbon. The edge states and the resulting magnetic ordering in the graphite ribbons therefore infer that CNTs with zigzag-shaped edges may be nanometer-scale ferromagnets. Okada and Oshiyama (2003b) have explored

104

Roles of shape and space in electronic properties of carbon nanomaterials

(a)

(b)

(c)

Fig. 3.8 Contour plots of spin density n ↑ ( r ) − n ↓ ( r ) of the finite-length (a) (7, 0), (b) (10, 0), and (c) (8, 0) CNTs. Each contour represents twice (or half) the density of the adjacent contour lines. Positive and negative values of the spin density are shown by solid and dashed lines, respectively c 2003 The Physical Society of Japan). (reprinted in part from Okada, S., Oshiyama, A. (2003b) J. Phys. Soc. Jpn. 72, 1510, 

the possibility based on DFT calculations, choosing (7, 0), (8, 0), and (10, 0) tubes as representatives of the zigzag CNTs. r ) − n ↓ ( r ) of the finite-length (7, 0), (10, The calculated spin densities n ↑ ( 0), and (8, 0) CNTs are shown in Figs. 3.8(a), (b), and (c), respectively. The spin densities clearly exhibit the occurrence of certain magnetic ordering in the finite-length CNTs. It is also found that the magnetic ordering is sensitive to the tube index. Most of the electron spins are polarized in the same direction for the finite-length (7, 0) and (10, 0) nanotubes, whereas the spin is polarized in one direction at one of two edges and in an opposite direction at the other edge for the finite-length (8, 0) nanotube. The magnetic state is lower in total energy than the non-magnetic state: The calculated total energy difference for a finitelength (7, 0) nanotube is 0.17 eV. Electronic energy levels (Kohn–Sham energy levels) of the finite-length (7, 0), (10, 0), and (8, 0) nanotubes for majority and minority spins are shown in Figs. 3.9(a), (b), and (c), respectively. We have found that fourfold and sixfold nearly degenerate electron states emerge for the (7, 0) and (10, 0) CNTs, respectively, at the Fermi level. Since the states are occupied by four electrons for the (7, 0) nanotube and by six electrons for the (10, 0) nanotube, significant exchange splitting for the states results in the spin polarization on the finite-length nanotubes. The calculated total spins are S = 2 and 3 for (7, 0) and (10, 0) nanotubes, respectively. In sharp contrast to these finite-length CNTs, the finite-length (8, 0) CNT does not exhibit a highspin state. The calculated number of electron spins S is zero. However, it is clearly demonstrated in Fig. 3.8(c) that certain magnetic ordering takes place around their edge carbon atomic sites. The peculiar index dependence of the total spin S is understood by applying the zone-folding analysis to the electronic structure of the corresponding zigzag graphite ribbon. Let us begin with the electronic structure of the zigzagedged graphite ribbon. There are doubly degenerate flat dispersion bands at E F near the zone boundary of the one-dimensional (1D) BZ (Fig. 3.9(d)) When we rely on the tight-binding model for the sufficiently wide ribbons,

3.3 Boundaries in planar and tubular nanostructures 105

MJ

–1

(b) (10,0)

MN

–1

–3

–1

α α

–2

α3 –2.5 α2 α1 –3

MJ

MN

–1.5 α3 α2 α1

Energy (eV)

–2 –2.5

MN

MJ

(d) Ribbon 1

–1.5 Energy (eV)

Energy (eV)

–1.5

(c) (8,0)

–2 –2.5 α

α

0 Energy (eV)

(a) (7,0)

–3.5

–3.5

–4

–4

–4

EF

–2 –3 –4

–3

–3.5

–1

–5 – π/a (7,0) (10,0) (8,0)

0

π/a

Fig. 3.9 Electronic energy levels for majority (MJ) and minority (MN) spins of (a) (7, 0), (b) (10, 0), and (c) (8, 0) CNTs. α states denote the states possessing the edge states character. (d) Electronic energy bands of the zigzag graphene ribbon. A dashed line denotes the Fermi level energy. The k points allowed by the periodic boundary condition along the circumference for the (7, 0), (10, 0), and (8, 0) are shown by the circles below (d). States depicted by solid circles have the character of the flat bands (reprinted in part from Okada, S., Oshiyama, A. (2003b) J. Phys. Soc. Jpn. 72, c 2003 The Physical Society of Japan) 1510, 

the two bands are completely degenerate in the region 2π/3a ≤ |k| ≤ π/a of the 1D BZ, where a is the periodicity along the edge (Fujita et al. 1996). On decreasing the width of the ribbon, the degeneracy is partly lifted and the two bands are degenerate in the smaller region, k0 ≤ |k| ≤ π/a (k0 > 2π/3a). This situation is unchanged even if we perform more elaborate, e.g. DFT, calculations. Actually, in Fig. 3.9(d) k0 is about 0.8π/a. The edge states are entirely localized at but extended along an edge of the graphene ribbon at the zone boundary k = π/a. On the other hand, the edge states lose their edge-localized character and the wavefunction penetrates inside the ribbon with decreasing wave number. Then, at k = k0 , the edge states are smoothly connected to π and π ∗ states the characters of which are identical to the π and π ∗ states of bulk graphite around the K point. When we consider electronic structures of zigzag CNTs with finite length, a periodic boundary condition is imposed along the circumference. This results in a discrete set of allowed wave numbers along the circumference: The wave numbers allowed in the finite-length (n, 0) nanotubes are kN =

2π N a n

(N = 0, ±1, ±2, · · ·).

In the finite-length (7, 0) CNT, seven k points are allowed under the periodic boundary condition along the circumference (Fig. 3.9(d)). Near the Fermi level, two states labelled by k = ±6π/7a that originate from the flat-band states of the graphene ribbon are allowed (Fig. 3.9(d)). The fourfold nearly degenerate α states in the (7.0) CNT (Fig. 3.9(a)) correspond to these flat-band states with k = ±6π/7a. In the finite-length (10, 0) CNT, three k points are allowed on the flat bands around the E F (Fig. 3.9(d)). One of three k points corresponds to the zone boundary of the 1D BZ of the grapene ribbon, k = π/a, where the wavefunctions are completely localized at the edge carbon atoms. This is the α1 state in Fig. 3.9(b). On the other hand, the states, α2 and α3 , corresponding to the remaining k points, k = ±4π/5a, are extended inside the nanotube to some extent. For the finite-length (8, 0) nanotube, since electron

106

Roles of shape and space in electronic properties of carbon nanomaterials

states at the zone boundary of the 1D BZ of the graphene ribbon are allowed by the periodic boundary condition (Fig. 3.9(d)) the electron states around the E F of the (8, 0) CNT are completely localized at the edge carbon atoms. The above zone-folding analyses lead us to the following natural interpretation of the radius-sensitive magnetic ordering obtained above. In the (7, 0) and (10, 0) nanotubes, sizable amplitude overlap between orthogonalized degenerate electron states results in the high-spin state excluding the double occupancy of the π orbital of each carbon atomic site, as in the case of Hund’s rule for the d orbital in the transition-metal magnets. In the (8, 0) nanotube, the degenerate states labelled k = π/a at E F are localized at the edges. The distribution leads to the local ferromagnetic spin ordering at each of the two edges. The coupling between the polarized spins at both edges is related to generation of electron–hole pairs, as is explained below. The electron states labelled k = ±6π/8a of the ribbon are allowed in the (8, 0) CNT (Fig. 3.9(d)). These are located near E F but split to some extent (the finite-length splitting). When the spins at both ends are coupled in a ferromagnetic way, the electron states for majority and minority spins are split in energy (the exchange splitting). This exchange splitting is beneficial for the states originated from k = ±π/a. Yet other states originated from k = ±6π/8a also undergo this exchange splitting. Therefore, when the finite-length splitting is larger than the exchange splitting, this ferromagnetic coupling results in the generation of holes in the bonding π states with minority spin and of electrons in the antibonding π states with majority spin. This reduces the energy gain due to C–C bonding. In the antiferromagnetic coupling, on the other hand, such generation of electron–hole pairs does not take place. This is why the antiferromagnetic state is energetically favorable for the (8, 0) nanotube. It should be mentioned that the coupling between the spins at two edges of the graphene ribbon is also related to the formation of holes in the bonding π state inside the BZ. As stated above, the ferromagnetic coupling between the two edge states is generally favorable to avoid the double occupancy at each atomic site inside. In the ferromagnetic state, there is the exchange splitting between the majority and minority spins. In this case, a hole at the bonding π state with minority spin may be generated inside the BZ so that the energy gain due to the bonding is lost. This is why the antiferromagnetic rather than the ferromagnetic ordering is favorable in the graphene ribbon. Spin ordering in graphene ribbons or zigzag CNTs is closely related to the peculiarity of the hexagonal bond network that consists of two sublattices, A and B. A general trend found in the DFT calculations is that spins in each sublattice are coupled with each other ferro-magnetically, whereas spins belonging to different sublattices are coupled antiferromagnetic-ally. In the normal situation, the numbers of sites in the sublattices, NA and NB , are identical. When imperfections such as vacancies or wavy edges are introduced, the numbers, NA and NB , may be different. In such cases, we expect ferrimagnetism, which has been discussed in detail by Okada and Oshiyama (2001b, 2003b). Two-dimensional hexagonal atomic arrangements induce interesting electronic properties, as shown above. It is noteworthy that the (111) surface in the diamond structure has hexagonal symmetry. Okada et al. (2003d) have

3.3 Boundaries in planar and tubular nanostructures 107

performed DFT calculations for hydrogen-covered Si (111) surfaces. They found that particular arrangements of Si dangling bonds that are formed by removal of hydrogen atoms exhibit ferromagnetic states. It is of interest to pursue the relation between the graphene sheet and Si (111) surface.

3.3.3

Border states and their magnetism in BNC nanosheet

The flat bands in graphene ribbons that cause magnetism in graphenes and zigzag CNTs are due to the peculiar edge states. This state is now regarded as an example of a new class of electron states peculiar to borders in nanosheets with hexagonal symmetry, as will be discussed below. Experimentally, syntheses of hexagonal networks of boron, nitrogen and carbon atoms have been achieved both in planar and tubular forms (Stephan et al. 1994; Watanabe et al. 1996; Suenaga et al. 1997; Kohler-Redlich et al. 1999; Bengu and Marks 2001). Experimental results and DFT-based totalenergy calculations (Blase et al. 1999) are indicative of phase separation of graphene and hexagonal BN (h-BN) strips in BNCx hexagonal-network compounds. The phase separation in general produces an interface, or a border in planar structures. Figure 3.10 shows energy bands of BNC2 and BNC4 sheets in which h-BN and graphene strips are separated and the borders are of zigzag shapes (Okada et al. 2000b). It is found that these sheets are semiconducting with direct energy gaps and that the energy gap monotonically decreases an increasing the width of graphite strips. It is remarkable that the highest-occupied π and the lowest-unoccupied π ∗ bands show almost flat dispersion around the zone boundary J point. Figure 3.11 shows the wavefunctions of these flat-band states in BNC4 , which clarify the characteristics of such states in the BNC sheets. It CB*

(b)

(a)

5

5

CN*

b

b 0

Energy (eV)

0

CB CN

–5

–5 a J

–10

–15

Γ Γ

J

K

a

–10

K

–15

Γ

J

K

Fig. 3.10 Energy bands of (a) BNC2 and (b) BNC4 sheets. A part of each sheet is shown in each inset where white, black, and shaded circles are C, N and B atoms, respectively. Periodicity in each sheet is represented by vectors a and b. Symmetry lines in the 2-dimensional BZ is shown in the inset. The origin of the energy is the top of the occupied bands. See text for labels in (b) used for some bands at the BZ boundary (reprinted in c 2000 The American Physical Society). part from Okada, S., Igami M., Nakada, K., Oshiyama, A. (2000b) Phys. Rev. B 62, 9896, 

108

Roles of shape and space in electronic properties of carbon nanomaterials

Fig. 3.11 (a) Contour plots of squared wavefunctions of the flat-band states of BNC4 at J, which are labelled as CB, CB∗ , CN∗ , and CN in Fig. 3.10(b). Plots are on the vertical cross-section of the sheet that contains the vector b in Fig. 3.10(b). White, black, and shaded circles denote C, N and B atoms, respectively. (b) A schematic energy diagram for the hybridization of the edge states of graphene and of h-BN in BNC4 sheets. Each contour line represents twice or half the density of its adjacent lines (reprinted in part from Okada, S., Igami, M., Nakada, K., Oshiyama, A. (2000b) Phys. Rev. B 62, 9896, c 2000 The American Physical Society). 

(a) CB

CB*

(b) CB* B C

CN* CB

CN*

CN

N

CN

is found that the highest π state at J (labelled CB in Fig. 3.10) is localized at C and B atoms, which constitute the border of h-BN and graphene strips. The state has π bonding character with substantial hybridization of π orbitals of C and B atoms. It is then naturally expected that the antibonding state also exists. We find that a state in the conduction bands has such a character (CB∗ in Fig. 3.10). The wavefunction of CB∗ shown in Fig. 3.11(a) is located at the border and clearly manifests its antibonding character. The wavefunction of the lowest conduction band at J (CN∗ in Fig. 3.10) is also shown in Fig. 3.11(a). It is localized at C and N atoms in this case, which constitute the other border of h-BN and graphite strips. Again, substantial hybridization of π orbitals of C and N atoms is observed. The state has antibonding character. We have explored the valence bands and found a state labelled CN. The wavefunction of CN is located along the border formed by C and N and is of bonding character (Fig. 3.11(a)). Figure 3.11(b) shows an energy diagram of the edge states of the ribbons hybridized in the heterosheet: The edge states of the graphene ribbon and of the h-BN ribbon are hybridized at the borders in BNC4 heterosheet, becoming the 4 border states. Due to the energy-level difference of three π orbitals of C, N and B atoms, the 4 border states are in the order of CN, CB, CN∗ and CB∗ in increasing energy, and the 2 middle states appear near E F . The borders consisting of different atom elements introduce different energy levels and then the hybridization renders the energy gap open at J. The electronic structures of BNCx described above have been obtained after complete optimization of atomic geometries. The calculated bond length of ˚ and 1.50 A, ˚ respectively. The length C−N and B−C at the border is 1.37 A ˚ ˚ and 1.43 A. ˚ The of C−C bonds is 1.40–1.42 A and that of B−N is 1.40 A bond lengths of C−C and B−N are similar to those in graphite and h-BN. The calculated bond energies of B−C and N−C are smaller than that of C−C by 1.52 eV and 0.81 eV, respectively. On the other hand, the bond energy of B−N is smaller than that of graphite by only 0.31 eV. Hence, the phase separation leading to the striped structures of the h-BN and graphite discussed above is energetically favorable. It is now clarified that zigzag borders of hexagonally networked heterosheets induce a new class of electron states that are localized along the borders (border

3.3 Boundaries in planar and tubular nanostructures 109

(b)

(a)

B N C BNC-I

BNC-II

states). It is emphasized that the states are peculiar to the zigzag borders. We indeed find that the border states are absent in BNCx sheets where the borders between graphite and h-BN strips are of armchair shape. The border state is extended along the borders and exhibits almost no dispersion at the same time. Hence, it is of interest to consider magnetism of such heterosheets (Okada and Oshiyama 2001b). An example is shown in Figs. 3.12(a) and (b). In these hetero-sheets, graphite ribbons are separated from each other intervened by honeycomb structures consisting of B−N bonds, thereby producing zigzag borders; the borders are undulating and thus the number of C atoms belonging to each sublattice is different. In the heterosheet shown in Fig. 3.12(a) (labelled BNC-I hereafter), a unit cell contains 13 B, 14 N atoms, and 21 C atoms that are grouped into A-sublattice and B-sublattice C atoms (NA = 11 and NB = 10). Similarly, in the heterosheet shown in Fig. 3.12(b) (labelled BNC-II), a unit cell contains 11 B and 13 N atoms, and 24 C atoms that are grouped into NA = 13 and NB = 11 C atoms. For those representatives of hexagonal heterosheets consisting of B, N, and C atoms, BNC-I and BNC-II, it is found that both ferro (ferri)-magnetic and non-magnetic states exist as solutions of the Kohn–Sham equations in DFT. The ferromagnetic state is lower in total energy than the non-magnetic state by 20 meV and 11 meV per unit cell for BNC-I and BNC-II, respectively. This clearly indicates the occurrence of the ferromagnetic ordering in the BNC heterosheets. The calculated values of the polarized spin per unit cell are one and two for BNC-I and BNC-II, respectively. Figure 3.13 shows the calculated r ) − n ↓ ( r ) (the up-spin is the majority) of BNC-I and BNCspin density n ↑ ( II. It is clear that the majority spin is distributed exclusively on the sublattice A of graphene, whereas the minority spin is on the sublattice B. Boron sites correspond to the A sublattice sites in the present honeycomb network so that the majority spin is also on the B sites near the border. The calculated total spin is identical to S = (NA − NB )/2 for both BNC-I and BNC-II. This indicates that there is ferrimagnetic ordering in flat bands in the bipartite lattice. BNC-I and BNC-II have periodic structures along the borders. This is unnecessary, however, to realize the ferro- (ferri-) magnetic ordering. The zigzag borders between chemically different elements and the imbalance between the numbers of carbon sublattice sites suffice.

Fig. 3.12 Top views of BNC heterosheets, (a) BNC-I and (b) BNC-II.White, shaded and black circles denote C, B, N atoms, respectively (reprinted in part from Okada, S., Oshiyama, A. (2001) Phys. Rev. Lett 87, c 2001 The American Physical 146803,  Society).

110

Roles of shape and space in electronic properties of carbon nanomaterials

(a)

(b)

BNC-I

BNC-II

Fig. 3.13 Contour plot of spin density n ↑ (r ) − n ↓ (r ) on the BNC heterosheet. (a) BNC-I and (b) BNC-II. Positive and negative values of the spin density are shown by solid and dashed lines, respectively. White, shaded and black circles denote positions of C, B, and N atoms, respectively. Each contour represents twice (or half) the density of its neighboring contour lines (reprinted in part from Okada, S., Oshiyama, A. (2001) Phys. c 2001 The American Physical Society). Rev. Lett 87, 146803, 

3.3.4

Nanoshuttlecocks: Decoration of C60 works as scissors of π electrons

As is explained above, the shapes of edges or borders are important in electronic properties in nanomaterials. Borders are not necessarily caused by differences in chemical elements. When we focus on certain electron states, attachment of foreign species can also generate borders for the particular electron states and thus modify electronic properties. In CNMs, π electron states are usually decisive in electronic properties. In this subsection, we demonstrate that attachment of aromatic molecules to C60 works as atom-scale scissors and thereby generates borders for π electrons. A new form of C60 derivatives with conical shapes has been synthesized under controlled modifications of C60 by attaching aromatic molecules (Sawamura et al. 2002; Matsuo et al. 2004). Each derivative consists of C60 with five foreign molecules (phenol groups) attached to five carbon atoms surrounding one of the twelve pentagons in C60 so that its shape is a nanometerscale badminton “shuttlecock”. It has also been found that the shuttlecock molecules are self-assembled into solid or liquid-crystal phases, where the shuttlecocks are stacked in a head-to-tail manner forming one-dimensional columns, which are in turn arrayed in a pseudohexagonal packing. The exotic structure is fascinating, and the properties of their condensed phases are interesting in relation to those of the known C60 solids, e.g., the face-centered cubic (fcc) (Saito and Oshiyama 1991) and the polymerized phases of C60 (Iwasa et al. 1995; N´un˜ ez-Regueiro et al. 1995; Okada and Saito 1999), since both the constituent units and their arrangements differ vastly from the conventional solid fullerenes. Okada et al. (2004) have clarified a new feature in the electron states of the nanoshuttlecock. Figure 3.14(a) shows atomic structures of a one-dimensional

3.3 Boundaries in planar and tubular nanostructures 111

(a)

(b)

(c)

C50

C5

Fig. 3.14 (a) Total-energy optimized atomic configuration of C60 H-(biphenyl)5 . (b) Structure of the C60 shuttlecock seen from the bottom. White spheres indicated by the white arrow heads are the atomic sites where the “feathers” are attached. A dark sphere on the pentagon represents an atomic site where an H atom is attached. (c) A side view of the atomic arrangement depicting the segmentation of the π -electron network on C60 into C5 and C50 (reprinted in part from Okada, S., Arita, R., Matsuo, Y., Nakamura, E., Oshiyama, A., Aoki, H., (2004) Chem. Phys. c 2004 Elsevier B. V.). Lett. 399, 157, 

chain of the shuttlecocks, C60 H-(biphenyl)5 , where each shuttlecock consists of C60 , five biphenyls (C12 H10 ), and an H atom. Since C60 (biphenyl)5 possesses an unpaired electron on the pentagon surrounded by the feathers (Fig. 3.14(b)), an extra H atom is usually attached to the pentagon to avoid the occurrence of a free radical. The DFT-based total-energy calculations have ˚ which is larger clarified that the optimum intermolecular distance is 10.6 A, ˚ than that (10.0 A) in the solid C60 . Figure 3.15(a) shows the electronic band structure of the C60 H(biphenyl)5 chain. The structure significantly differs from those for the fcc and polymerized phases of C60 . The chain is a semiconductor with an indirect gap E g of 1.4 eV that is larger than that of the fcc C60 by 40%. The valence and conduction bands around the energy gap are found to be very narrow (a few tens of meV): They are narrower by an order of magnitude than that (∼0.4 eV) for the conventional solid fullerenes. The results indicate that an important effect of the attachment of the biphenyls is the segmentation of the spherical π electron network on the C60 . Indeed, the highest-occupied (HO) state is localized on the C atoms belonging to the pentagon surrounded by the five biphenyls (Fig. 3.15(b)), whereas the (a)

(b)

(c)

2.0 HO

LU

Energy (eV)

1.5 1.0 0.5 0.0 –0.5 –1.0

Γ

X

Fig. 3.15 Band structure for the chain of C60 H(biphenyl)5 (a). The origin of the energy is the top of the valence band. Right panels are squared wavefunctions at the  point of the highest-occupied (b) and the lowest-unoccupied (c) bands projected on a plane cutting C60 . Each contour represents twice (or half) the amplitude or the density of the adjacent contour lines (reprinted in part from Okada, S., Arita, R., Matsuo, Y., c 2004 Elsevier B. V.). Nakamura, E., Oshiyama, A., Aoki, H., (2004) Chem. Phys. Lett. 399, 157, 

112

Roles of shape and space in electronic properties of carbon nanomaterials

lowest-unoccupied (LU) state is distributed on the remaining part of the C60 (Fig. 3.15(c)). The second lowest-unoccupied and the second highest-occupied electron states (not shown) also exhibit features similar to those of the LU and HO states, respectively. Thus, as far as the π electron states are concerned, the C60 shuttlecock chain is regarded as a new one-dimensional system where the unit consists of the C5 ring and the C50 cage (Fig. 3.14(c)) rather than the C60 or the attached (biphenyl) molecules. Physically, the segmentation occurs because each C atom on the pentagon to which the molecules are attached has a fourfold coordination unlike the threefold co-ordination in the pristine C60 , so that the biphenyls effectively divide the icosahedral π electron network into the C5 and C50 segments. An unexpected behavior is that the π electron amplitudes on the segments, i.e. the C5 ring and C50 cage across adjacent molecules, ˚ are orthogonal to each other, although they are only separated by about 3 A (Fig. 3.14(c)). The narrow bandwidths cause large peaks in the density of states (DOS) around the energy gap. The calculated peak values of DOS for LU and HO states are 15 and 10 states/eV, respectively. If we can achieve band-filling control, these large DOS around the gap are expected to induce magnetic, or possibly superconducting properties. Okada et al. (2004) have explored a possibility of magnetism theoretically. They remove the H atom attached to the C5 to dope carriers. This modification results in the half-filled energy band and the large DOS at the Fermi level. They have indeed found spin polarization on the nanoshuttlecock. The spin density is found to distribute mainly to C atoms on the pentagon surrounded by the biphenyls, whose distribution is similar to that of the LU state of the C60 H(biphenyl)5 . The calculated total energies indicate that the spin-polarized state is the ground state, whereas a spin-unpolarized metallic state is metastable with the total energy higher by 98 meV per molecule than that of the polarized state. Further calculations are indicative of the occurrence of antiferromagnetic ordering in the carbon nanoshuttlecocks.

3.4

Double-walled nanotubes: Peculiarity in cylindrical structure

Double-walled carbon nanotubes (DWCNTs) have been synthesized by using carbon peapods as starting materials (Smith et al. 1998; Sloan et al. 2000). Electron-beam irradiation on the carbon peapods induces coalescence of encapsulated C60 clusters and results in the DWCNTs, the diameters of the ˚ and 13–14 A, ˚ respectively. We might inner and outer shells are about 6 A expect that the electronic properties of DWCNTs are deduced from simple sums of electron states of constituent CNTs. DFT-based calculations have clarified that it is not true in some cases: The difference in curvature of the two constituent CNTs plays an important role. In this section, we discuss curvature-difference-induced metallization in DWCNTs. It is again regarded as an example in which nanoscale shape affects electronic properties in CNMs. We also discuss the capacitance of DWCNT that is certainly important in its application to electronics.

3.4 Double-walled nanotubes: Peculiarity in cylindrical structure 113

3.4.1

Curvature-induced metallization of double-walled semiconductor carbon nanotubes

In graphite, electron states are classified into two groups; σ (sp 2 orbital) and π ( pz orbital) states. In CNT the π states are rehybridized with the σ states due to its lack of mirror symmetry or, in other words, addition of curvature. Thus, the rehybridization causes downward shifts of the π electron states of CNTs and the amount of the shift depends on the curvature. Indeed, theoretical calculations have revealed that thin (n, 0) nanotubes smaller than (6, 0) become metallic due to the significant π − σ rehybridization (Blase et al. 1994). Considering this downward shift due to π − σ rehybridization, Okada and Oshiyama (2003c) predicted a possibility of metallization in DWCNTs in which constituent CNTs are both semiconducting. They first explored the stability and the preferable interwall spacing of the zigzag DWCNTs, (7, 0)@ (n, 0), then the calculated energy gain ΔE, which is defined in the reaction scheme, (7, 0) + (n, 0) → (7, 0)@(n, 0) − ΔE as a function of the interwall spacing. It is found that the (16, 0) is the most favorable outer nanotube for the (7, 0) nanotube and gives the value of ΔE = −0.94 eV/cell with the interwall ˚ A (17, 0) CNT is also favaorable with the comparable value spacing of 3.52 A. of ΔE = −0.75 eV/cell. By interpolating the results for zigzag DWCNTs, the ˚ most preferred interwall spacing for the thin DWNTs is estimated to be 3.56 A, ˚ The larger which is larger than the interlayer spacing of the graphite (3.34 A). interwall spacing has indeed been found in the electron diffraction pattern of the DWCNTs. It seems that the small tube radius and the incommensurability of atomic arrangements between inner and outer nanotubes causes the larger interwall spacing than the graphite. Since the interwall spacing experimentally observed exhibits substantial distribution around their optimum value, the DWNTs other than the most stable (7, 0)@(16, 0) are expected to be synthesized under normal conditions. Figures 3.16(a), (b), and (c) show the energy bands of zigzag (7, 0), (16, 0), and (17, 0) CNTs, respectively. As shown there, all the zigzag nanotubes considered here are semiconductors with moderate direct energy gaps at  point. The energy gap of the (7, 0) is narrower than those of the (16, 0) and (17, 0): The calculated values are 0.49 eV, 0.60 eV, and 0.52 eV, for (7, 0), (16, 0), and (17, 0), respectively. The decrease in the energy gap of (7, 0) is a consequence of the fact that the π−σ rehybridization increases with decreasing tube radius, so that the π -like states shift downwards. It is therefore expected that the DWCNTs consisting of the semiconducting zigzag nanotubes exhibit interesting variation of electronic structures around the energy gap, since the amounts of energy shift of the π and π ∗ states of the zigzag nanotubes strongly depends on their radii. The energy bands of the DWCNTs, (7, 0)@(16, 0) and (7, 0)@(17, 0), are shown in Figs.3.16(d) and (e), respectively (Okada and Oshiyama 2003c). The energy bands of these DWCNTs show surprising features: The energy gap vanishes, although each constituent nanotube is a semiconductor with a finite energy gap. In (7, 0)@(16, 0), the π and π ∗ states merge and thus the density of states at the Fermi level becomes finite. In (7, 0)@(17, 0), the π and π ∗ states even overlap and a substantial number of carriers is generated. The formation

Roles of shape and space in electronic properties of carbon nanomaterials

(a)

(b)

3

(c)

3

(d)

3 2

2

1

1

1

1

1

–1 –2 –3 Γ

0 –1

–3

–1 –2

–2 X

0

Γ

–3 X

Energy (eV)

2 Energy (eV)

2

0

0 –1 –2

Γ

–3 X

(e)

3

2

Energy (eV)

Energy (eV)

3

Energy (eV)

114

0

β α

–1 –2

Γ

X

–3 Γ

X

Fig. 3.16 Energy-band structures of (a) (7, 0), (b) (16, 0), (c) (17, 0) CNTs and (d) (7, 0)@(16, 0), and (e) (7, 0)@(17, 0) DWCNTs. Energies are measured from the top of the π band. The α and β denote the electron states of the highest branch of the π band and the lowest branch of the π ∗ c 2003 The American Physical Society). band, respectively (reprinted from Okada, S., Oshiyama, A. (2003) Phys. Rev. Lett. 91, 216801, 

of the double-walled structures that introduces a new class in the nanotube systems causes the metallization of the semiconductors. Moreover, the results suggest that the multiwalled nanotubes containing the innermost nanotubes ˚ become coated metallic wires. The metallization of with a diameter of 4–6 A the (7, 0)@(16, 0) and (7, 0)@(17, 0) is totally due to the difference in the downward shift of the π and π ∗ electron states between the inner and outer nanotubes: Downward shifts of the π and π ∗ states of the inner tube are larger than those of the outer tube because of the stronger rehybridization of π and σ states; the bottom of the conduction band possessing the π ∗ character of the inner nanotube is thus located near or below the top of the valence band distributed on the outer nanotube. Overlap between the conduction band of the inner nanotube and the valence band of the outer nanotube results in the metallization of the DWCNTs. Yet it should be mentioned that the manybody correction might be important for the unoccupied energy bands of the semiconducting nanotubes and is to be studied in the future. The distribution of the wavefunction unequivocally reveals that the top of the π band (the doubly degenerate α band) is distributed on the outer nanotube, while the bottom of the π ∗ band (the β band) is distributed on the inner nanotube (Fig. 3.17). In the (7, 0)@(17, 0), it is found that 5% of the α band is unoccupied, whereas 10% of the β band is occupied. When the (7, ˚ separation, this corresponds to 0)@(17, 0) DWCNTs are bundled with 3 A the electron and the hole concentrations of about 1.5 × 1020 cm−3 . Thus, the DWNTs consisting of the semiconducting nanotubes with thin radius are semimetals in which two kinds of carriers are isolated in space. The curvature of the inner tube itself is also an interesting factor to possibly cause the metallization in DWCNTs. The DFT-based calculations for (8, 0)@(19, 0), (8, 0)@(20, 0), (10, 0)@(19, 0), and (10, 0)@(20, 0) DWCNTs have been also performed (Okada and Oshiyama 2003c). It has been clarified that the downward shifts of conduction and valence bands of the inner tubes are substantial but that (8, 0) and (10, 0) are too thick to cause the

3.4 Double-walled nanotubes: Peculiarity in cylindrical structure 115

a

b

Fig. 3.17 Contour plots of the squared wavefunctions at the  point of the highest blanch of the π band (α) and the lowest branch of the π ∗ band (β) on the cross-sectional plane of (7, 0)@(17, 0). Each contour represents twice (or half) of the density of the adjacent contour lines. The solid circles denote the atomic positions (reprinted from Okada, S., Oshiyama, A. (2003) Phys. Rev. Lett. 91, c 2003 The American Physical 216801,  Society).

metallization: The calculated bandgaps for (8, 0)@(19, 0), (8, 0)@(20, 0), (10, 0)@(19, 0), and (10, 0)@(20, 0) are 0.26 eV, 0.23 eV, 0.50 eV, and 0.45 eV, respectively. Hence, the metallization of the DWCNTs is a consequence of a subtle balance between the curvature difference of the constituent tubes and the π −σ rehybridization of the inner tube.

3.4.2

Quantum effects on capacitance of double-walled carbon nanotubes

CNT is considered to be a candidate material for several applications in future technology. Using CNT in field effect transistors (FET) is a possibility (Chen et al. 2006). This is because CNT is robust in atomic structures and capable of carrying more current than conventional semiconductors, being a candidate material for channels or gates in future electronics. CNT is also in accord with a trend in current electron devices: The thickness of gate stacks now reaches the order of a nanometer; in order to assure enough currents, multigate structures called fin-FETs (Hisamoto et al. 1991) or coaxial cylindrical gate-channel structures called surrounding gate transistors (SGTs) (Masuoka et al. 2002), are regarded as boosters of future semiconductor technology. Double-walled carbon nanotubes may thus be the ultimate elements of SGTs. In application of electron devices, conductance and capacitance are two fundamental quantities that determine the performance of circuits. Yet capacitance in nanostructures has been less studied compared with conductance. The importance of the capacitance becomes more prominent in current electron devices, where a variety of capacitors are fabricated in complicated and miniaturized built-in structures. Hence, the capacitance of DWCNT is an interesting and important issue that should be clarified. Uchida et al. (2007) have challenged the issue based on the DFT-based calculations and revealed important quantum effects.

116

Roles of shape and space in electronic properties of carbon nanomaterials

–Q

+Q Fig. 3.18 Double-walled carbon-nanotube capacitor consisting of a inner (4, 0) nanotube and an outer (24, 0) nanotube and an outer (24, 0) nanotube. One of the tubes is biased with μ relative to the other, and ±Q charges are stored (reprinted from Uchida, K., Okada, S., Shiraishi, K., Oshiyama, A. c 2007 (2007) Phys. Rev. B 76, 155436,  The American Physical Society).

m (4,0)CNT

(24,0)CNT

Figure 3.18 shows the DWCNT capacitor investigated by Uchida et al. The inner is (4, 0) CNT, the minimum zigzag tube that has ever been observed experimentally (Peng et al. 2000), and the outer one is a (24, 0) zigzag CNT. ˚ and 9.4 A, ˚ respectively. The radii of these two CNTs, Rin and Rout , are 1.7 A ˚ The separation of the walls of the two CNTs is thus 7.7 A, which assures negligible overlap of electron densities distributed in the two CNTs, thereby constituting a nanoscale capacitor. The (4, 0) and (24, 0) CNTs are semiconducting and metallic, respectively, when we apply a simple theory in which electron states are determined by putting a circular boundary condition to the wavefunctions in a single graphite sheet. Yet the sp 3 hybridization caused by the finite curvature makes the isolated (4, 0) CNT metallic and (24, 0) semiconducting with a very narrow bandgap. When investigating the responses of the capacitor to the bias voltage μ, Uchida et al. performed electron-state calculations of the capacitor in which the Fermi levels of the inner CNT E F in and the outer CNT E F out are forced to satisfy the relation of E F out = E F in + μ. The scheme is explained further in the Appendix. When μ > 0, the outer and inner CNTs are charged negatively (−Q < 0) and positively (+Q > 0), respectively. Total charge neutrality of the capacitor is preserved in the scheme. According to the calculated Q − μ relation, we obtain the capacitance, C=

dQ , dμ

of the DWCNT capacitor. Figure 3.19(a) shows the calculated capacitance C, which exhibits two important quantum effects. First, the calculated capacitance strongly depends on the bias voltage μ. This is in contrast to the bias-independent capacitance in classical electromagnetism. Second, the capacitance is larger than the expected value, Ccl , which is derived from classical electromagnetism as Ccl =

2π ε0 ˚ ∼ 3.4 × 10−21 F/A, log(ROUT /RIN )

3.4 Double-walled nanotubes: Peculiarity in cylindrical structure 117

DOS ( m = 2.2 eV)

(b) (iv)

5

(i)

4

4

out

3

(ii)

Ccl

2 1 0

(24,0)CNT

(iii)

Energy (eV)

Capacitance (10–21F/A)°

(a)

m = 2.2 eV –4

0 4 Bias m (eV)

2

(iii)

EF

(ii)

2.2 eV (i)

0 8

(iv)

Fig. 3.19 (a) Calculated capacitance of (4, 0) @(24, 0) DWCNT as a function of bias voltage. (b) Density of states (DOS) of the inner (4, 0) CNT and outer (24, 0) CNT of (4, 0)@(24, 0) under the bias voltage of μ = 2.2 eV (reprinted from Uchida, K., Okada, S., Shiraishi, K., Oshiyama, A. (2007) Phys. c 2007 The American Rev. B 76, 155436,  Physical Society).

in

EF

(4,0)CNT (arb. unit)

where ROUT and RIN are the radii of the outer and the inner tubes, respectively, and ε0 is the dielectric constant in vacuum. We will discuss the origins of these quantum effects below. In order to consider the first quantum effect, the strong bias dependence of the capacitance, we show density of states (DOS) of constituent (4, 0) and (24, 0) CNTs with and without the bias voltage (Figs. 3.20(a) and (b)). We observe characteristic spiky peaks, which are van-Hove singularities reflecting the quasi-one-dimensionality of the CNTs. It is found that these DOS structures of the inner and outer CNTs are almost rigid except for their relative shift under the bias application. Under the bias application (Fig. 3.20(b)), the excess ΔN (= Q/e) electrons are stored in the outer (24, 0) CNT, filling the electron states in the energy range of E˜ Fout ≤ ε ≤ E Fout . The electron state at E˜ Fout in Fig. 3.20(b) corresponds to that at E F out without bias voltage (Fig. 3.20(a)). As for the inner (4, 0) CNT, on the other hand, the electron states in the energy range of E Fin ≤ ε ≤ E˜ Fin are empty under bias application, which corresponds (a)

(b) DOS(μ = 2.7 eV)

(c) μ = 2.7 eV Uout

DOS (μ = 0) CNT(24,0)

out

EF

0

4

~out EF –Q

EF

+Q=eDN

(arb. unit)

(arb. unit)

(4,0)CNT

+1 +2

–8

in

CNT(4,0)

–4 –8 –1 –2

–4

~in

EF

EF

+Q

(Å) 0

μ

(eV)

2

–Q

8

(24,0)CNT

Uin

–8

–4

0 (Å)

C atom 4 8

Fig. 3.20 Density of states of inner (4, 0) and outer (24, 0) CNTs without bias application (a) and under bias application of μ = 2.7 eV (b). (c) Redistribution of electron density in (4, 0)@(24, 0) DWCNT under bias application of μ = 2.7 eV (reprinted in part from Uchida, K., Okada, c 2007 The American Physical Society). S., Shiraishi, K., Oshiyama, A. (2007) Phys. Rev. B 76, 155436, 

118

Roles of shape and space in electronic properties of carbon nanomaterials

to the depletion of N electrons in the inner CNT. The relative shift of the DOS between the inner and outer CNTs induced by the bias application corresponds to E˜ Fout − E˜ Fin . Uchida et al. found that this quantity coincides with the difference V in the potential energy between the outer and the inner CNTs. It is therefore meaningful to decompose the bias voltage μ as μ ≡ E Fout − E Fin = V + U out + U in , where U out ≡ E Fout − E˜ Fout

and U in ≡ E˜ Fin − E Fin

are the Fermi-level shifts originated from the Fermi statistics of electrons: Addition or removal of a finite number of electrons causes the Fermi-level shift in general. In macroscopic capacitors, V is much larger than U in and U out , and thus μ is almost equal to V . However, in nanoscale systems such as the present DWCNT capacitor, we are unable to neglect the contributions from U in and U out to the bias voltage μ. From the two equations above, we obtain the following expression of the inverse capacitance. C −1 ≡

dμ −1 −1 = C0−1 + Din + Dout . dQ

The first term is the inverse of the usual potential-difference capacitance C0 ≡ dQ/dV . The second and the third terms are defined as Din ≡ dQ/dU in and Dout ≡ dQ/dU out , which reflect the Fermi-level shifts in nanostrcutures. We recognize that C is smaller than C0 and approaches C0 when Din and Dout are large enough. Also, C is vanishing when Din and Dout are small enough. ˚ which is The calculated electrostatic capacitance C0 is 5.4 × 10−21 F/A, in good agreement with the maximum value of C in Fig. 3.19. The bias dependence of C0 is small, so that the bias dependence of the total capacitance C comes from the contributions from the DOS terms, Din and Dout . Actually, the singular points of C depicted as (i)–(iv) in Fig. 3.19(a) appear when the Fermi levels E F in and E Fout cross the singular points in the DOS, shown in Fig. 3.19(b), of the inner and outer tubes. In Fig. 3.19(b), (i), (iii), and (iv) are van-Hove singularities peculiar to quasi-one-dimensional systems, whereas (ii) is a dip, i.e. a very small bandgap, between the conduction-band bottom and the valence-band top of the outer (24, 0) CNT. The plateau regions in C also correspond to those in Din and Dout , and the magnitude of C is correlated with those of Din and Dout . We next discuss the second quantum effect: the enhancement of the calculated capacitance C compared with the classical capacitance Ccl . The capacitance C is composed of the potential-origin capacitance C0 , and the DOS contribution of the inner and outer CNTs, Din and Dout . It is obvious that Din and Dout render the capacitance small. The potential contribution C0 is thus the origin of the enhancement of the capacitance and the calculated value of C0 indeed corresponds to the maximum value of the calculated C. The enhancement of C0 compared with the classical value Ccl is a consequence of the spill of stored electrons. Figure 3.19(c) shows the electron redistribution in the DWCNT capacitor under the bias application. It is clear that the density of

3.5 Defects in carbon nanotubes

stored electrons in the inner (outer) CNT spills outward (inward). This reduces the effective electrode–electrode separation and thus enhances the electrostatic capacitance.

3.5

Defects in carbon nanotubes

Imperfections such as point and line defects in materials are inevitable. The imperfection may be a minority in terms of its numbers. Yet it is well known that the minority decisively affects the physical properties of materials. An example is a point defect in semiconductors that induces deep levels in the energy gap and thereby modifies electronic properties substantially. As for CNMs, the properties of defects have been less studied, presumably because new properties without imperfections are still at the exploration stage. It should be emphasized, however, that understanding the effect of imperfections on the physical properties of materials forms the mainstay of science and technology of materials. In this section, we first present recent calculations about the divacancy in CNTs. The importance of edge or border shapes in hexagonally bonded networks have been discussed in previous sections. Line defects with particular shapes in CNTs therefore have peculiar properties that we also discuss in this section.

3.5.1

Divacancy in carbon nanotubes

Vacancies in CNTs have long been viewed as the least probable defect types because of their prohibitively large formation energies in graphitic systems. However, vacancies are unavoidable in non-equilibrium conditions, e.g. during the nanotube growth or under irradiation. Although monovacancies had been the first choice in theoretical works (Lu and Pan 2004; Ma et al. 2004; Miyamoto et al. 2004; Berber and Oshiyama 2006b), transmission electron microscopy experiments (Hashimoto et al. 2004) indicated multivacancies, rather than monovacancies, appear under moderate irradiation conditions. When we consider the robustness of the hexagonally bonded carbon networks, the three neighboring C atoms induced by a monovacancy are incapable of being rebonded and a dangling bond is left around the monovacancy. On the other hand, four neighboring C atoms around the divacancy easily form rebonds, which may decrease its formation energy. Extending the discussion, even-number vacancies are energetically favorable compared with odd-number vacancies. Quantitative calculations for atomic and electronic structures of the divacancy in CNTs have been performed recently (Berber and Oshiyama 2008), which we discuss in this subsection. The structural optimization of divacancies has been performed based on the DFT. It is found that divacancies in CNTs self-heal by spontaneously reconstructing into stable structures, which are depicted in Fig. 3.21. Four under-coordinated C atoms around an ideal divacancy are rebonded to form a new network, where two pentagons and a single octagon are present. There are 2 different divacancy orientations with respect to the tube axis for each chirality as shown in Fig. 3.21. We

119

120

Roles of shape and space in electronic properties of carbon nanomaterials

Fig. 3.21 Atomic structures of reconstructed divacancies in achiral nanotubes. New bonds formed during the reconstruction are highlighted by the darker shading. Atoms at the far side are omitted for clarity. The orientation of a divacancy is defined as the relative direction of the new bonds with respect to the tube axis. (a) Parallel and (b) tilted configurations of the divacancy in armchair nanotubes. (c) Tilted and (d) perpendicular c 2008 The American orientations of the divacancy in zigzag nanotubes (reprinted from Berber, S., Oshiyama, A. (2008) Phys. Rev. B 77, 165405,  Physical Society).

here define the orientation of a divacancy as the direction of the new bonds relative to the tube axis, indicated by darker colors in Fig. 3.21. In an armchair CNT, a divacancy has either a parallel or a tilted orientation as depicted in Figs. 3.21(a) and (b), respectively. In a zigzag CNT, a perpendicular orientation, shown in Fig. 3.21(d), is possible in addition to the tilted divacancy, shown in Fig. 3.21(c). Self-healing of a divacancy is a concerted bond-formation mechanism, where 2 new bonds are formed simultaneously. The lengths of these 2 bonds are equal to each other in fully relaxed atomic structures. Hence, the length d of the new bonds is a unique indicator of the structural stability of a reconstructed divacancy. The bond lengths d are shown in Fig. 3.22(a) for different orientations of divacancies, where dashed lines correspond to armchair tubes, and solid lines to zigzag nanotubes. In Fig. 3.22, squares are used for data points of parallel, crosses for data points of perpendicular, and circles for tilted divacancies. The structural stability suffers from the diameter increase mostly in the parallel divacancy as d increases sharply with increasing tube ˚ This orientation is expected to be the least diameter, approaching to ∼1.6 A.

Bond length (Å)

1.65 1.60

zigzag

1.55 armchair

1.50

zigzag

1.45 1.40

(b)

parallel tilted perpend.

4

5

6 7 8 Diameter (Å)

9

Formation Energy (eV)

(a)

7.0 6.0 armchair

zigzag

5.0 4.0 armchair 3.0

zigzag parallel tilted perpend.

2.0 1.0 0.0 4

5

6 7 8 Diameter (Å)

9

Fig. 3.22 The energetics and structural properties of divacancies in CNTs. (a) The length d of the new bonds in reconstructed defects. (b) The divacancy formation energy E DV for different orientations. The labels in the plots correspond to the direction of the divacancy with respect to the c 2008 tube axis. Solid lines and dashed lines are guides for the eye (reprinted from Berber, S., Oshiyama, A. (2008) Phys. Rev. B 77, 165405,  The American Physical Society).

3.5 Defects in carbon nanotubes

stable due to the extended character of the new bonds. The other divacancy orientations show a smaller dependence of d on the tube diameter, as seen in Fig. 3.21(a). Zigzag nanotubes exhibit a wavy pattern for d as a function of the tube diameter, which shows local minima for small-gap zigzag nanotubes. Note that the bond length d for the favorable orientation in defective ˚ only slightly larger than the bond length in zigzag nanotubes is ∼1.45 A, graphene. The formation energies E DV of reconstructed divacancies are shown in Fig. 3.22(b) for different divacancy orientations. The divacancy formation energy E DV is ∼3.5–3.8 eV in zigzag nanotubes and ∼3.9–4.3 eV in armchair nanotubes for moderate nanotube diameters. The energetics of divacancies exhibits a strong dependence on their orientation: E DV is at least 1.5 eV lower for the preferred orientation. The tilted orientation is more favorable in armchair nanotubes and the perpendicular orientation in zigzag nanotubes. Thus, the smaller angles between the new bonds and the tube axis results in the higher formation energies E DV . Divacancies show smaller formation energies in small-diameter nanotubes because the energy gain by the reconstruction is higher in narrower nanotubes that have more flexible atomic structures. Therefore, narrow nanotubes are more likely to acquire defects during the growth. The formation energy E DV increases monotonically on increasing the diameter in armchair nanotubes, while a wavy pattern occurs in zigzag nanotubes, as shown in Fig. 3.22(b). The periodicity of the wavy pattern is the same as that of the bandgap modulations (Hamada et al. 1992). Similar bond-length modulations were reported before for both mono- and divacancies in zigzag nanotubes (Lu and Pan 2004; Ma et al. 2004; Berber and Oshiyama 2006b) although the connection between formation energies and bandgaps is still unclear. The most important factor in understanding the reconstruction of divacancies in CNTs is the formation of 2 new bonds. In an ideal vacancy geometry, ˚ This the 2 atoms responsible for each new bond are separated by ∼2.46 A. ˚ value decreases to 1.45–1.60 A in the reconstructed geometries, thus causing a substantial tensile stress on surrounding atoms in the direction parallel to the new bond. The dependence of the formation energy E DV on the orientation of the new bond is explained in terms of the relaxation patterns of this tensile stress. In the perpendicular divacancy in zigzag CNTs, depicted in Fig. 3.21(d), the tensile stress is relaxed by the local shrinkage in the diameter of CNT. Our calculations show that atoms located on a belt surrounding the perpendicu˚ In tilted divacancies lar divacancy exhibit inward relaxation by 0.3–0.35 A. (Figs. 3.21(b) and (c)), the tensile stress is distributed in a spiral way along CNTs. This causes helical dimples where the helix periodicity is determined by the orientation of the new bonds. It is found that the atoms located on ˚ This long-range helical the helical dimples are relaxed inwards by ∼0.1 A. dimple costs more energy than the local diameter shrinkage. As for the parallel divacancy in armchair CNTs, depicted in Fig. 3.21(a), the tensile stress is on one side along the tube axis. The only way to relax this tensile stress is the bending of the tube toward the divacancy side. When the diameter of the tube increases, the tube bending requires the stretching and the bending of local bonds at the backside, where atoms are arranged in a hexagonal network. This

121

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Roles of shape and space in electronic properties of carbon nanomaterials

relaxation pattern costs more energy with increasing diameter. The calculated formation energy of the parallel divacancy indeed increases with increasing diameter, as shown by the squares in Fig. 3.22(b), and approaches a certain value (> 7 eV), which corresponds to the formation energy of the divacancy in graphite. We also expect that the energetics of the parallel and the tilted divacancies shows a more pronounced dependence on the number of defects per length. The DFT-based calculations clearly show peculiar relaxation patterns around the divacancy in CNTs. Formation of two pentagons and a single octagon heals the divacancy and thereby leads to a substantial decrease in formation energies that are much smaller than the typical formation energy, 5 eV, of the monovacancy. Images of scanning tunnelling microscopy have already been calculated (Berber and Oshiyama 2008) and experimental identification of the divacancy is awaited.

3.5.2

Magnetic properties of topological line defects in carbon nanotubes

Defects could be regarded as internal edges or borders where shapes of the bond network are modulated. Hence, there emerges a possibility that a particular line defect produces flat bands and thus induces magnetism in CNTs without termination of the bonding network by edges. An interesting example has been proposed by Okada et al. (2006, 2007). They consider the armchair (n,n) CNT and zigzag (7, 0) CNT with the topological line defect, denoted by (n, n)D and (7, 0)D , respectively, in which octagons and fused pentagons are alternately aligned in parallel form for (n, n)D (Fig. 3.23(a)) and in spiral form for (7, 0)D (Fig. 3.23 (b)), respectively, to the tube axis. In the (n, n)D nanotubes, the topological defect can be obtained by insertion of a C2 cluster in a hexagon per double periodicity of the pristine armchair CNT. In the (7, 0)D CNT, the topological line defect has a spiral arrangement in which the C2 clusters are implanted at an angle of 30 degree to the tube axis direction. From another viewpoint, these CNTs can be (a)

(b) Fig. 3.23 Optimized geometries of CNTs with the topological line defect. (a) (8, 8)D CNT and (b) (7, 0)D CNT. Gray spheres denote the C2 clusters inserted in the perfect CNTs.

3.5 Defects in carbon nanotubes

123

Fig. 3.24 Isosurfaces of the spin-density distribution (Δn = n ↑ − n ↓ ) for the CNTs with the topological defect: (a) (8, 8)D and (b) (7, 0)D CNTs.

regarded as a zigzag graphite ribbon rolled into a cylinder by adding an array of C2 clusters to glue the zigzag edge sites of the ribbon. Thus, the peculiar localized states similar to the edge state are expected to arise. Figure 3.24(a) shows the spin density (Δn = n ↑ − n ↓ ) of fully optimized geometry of the (8, 8)D CNT. It is found that the (8, 8) CNT with the topological defect exhibits magnetic ordering: Along the circumference of the CNT, the polarized electron spins are strongly localized to the atoms connected to the C2 clusters (Fig. 3.24 (a)), and the distribution of the spins rapidly decreases with increasing distance from the defect. Along the tube axis, the polarized spins are coupled ferromagnetically, showing long-range ordering. Thus, the armchair nanotubes with the topological line defect are ferromagnetic nanowires consisting solely of threefold-coordinated C atoms without carrier doping or network termination imposed by vacancies or edges. As for the (7, 0)D nanotube, the spin density also exhibits similar characteristics. The spins are localized to the atoms adjacent to the inserted C2 and ferromagnetically aligned along the direction of the topological line defect resulting in a spiral spin distribution (Fig. 3.24(b)). This is a theoretical finding that intrinsic ferromagnetism may be possible by introducing line defects with particular shapes. Corresponding magnetic moments are listed in Table 3.1. It is clear that finite magnetic moments do exist on the CNTs with the topological line defects and that the number of polarized electron spins is insensitive to the tube diameter. Due to the spiral distribution of the polarized electron spin on the (7, 0)D CNT, the number of polarized electron spins is slightly larger than those on the (n, n)D CNTs. The energy gains are also shown in Table 3.1. The ˚ of the CNTs with topological line defects. Energy Table 3.1 Magnetic moments μ [μB /A] gains ΔE [meV] upon ferromagnetic spin polarization from non-magnetic states in (n, n)D CNTs are also shown.

μ ΔE

(7, 0)D

(5, 5)D

(6, 6)D

(7, 7)D

(8, 8)D

(9, 9)D

0.138 —

0.035 2.9

0.053 6.3

0.041 4.2

0.037 3.9

0.037 3.8

(10, 10)D 0.045 2.3

124

Roles of shape and space in electronic properties of carbon nanomaterials

ferromagnetic states of the (n, n)D CNTs are slightly lower in total energy than the non-magnetic ones. They clearly indicate that the ferromagnetic spin ordering takes place on the (n, n)D CNTs below the Curie temperature of about a few tens of K. Stability of the topological defects has also been examined by Okada and colleagues. They have calculated total energies of the CNTs with the topological line defects as a function of the tube radius. It is found that the total energy of the CNT with the defect monotonically decreases with increasing tube radius and is slightly higher than that of the perfect CNT with the same radius by 40 meV/atom. Hence, the nanotubes with the topological defect are energetically quite stable so that they keep their tubular structures under certain conditions.

3.6

Hybrid structures of carbon nanotubes

CNT, having fascinating electronic and mechanical properties, is a promising material in a variety of applications. Electron devices such as field effect transistors (FETs) have indeed been fabricated, albeit not on a mass-production scale. On the other hand, the current electronics industry benefits from huge expertise in Si technology, which is certainly to be inherited in the future. The inclusion of CNT in nanoelectronics devices therefore requires integration with Si substrates, metal gates and contacts. The electronic properties of such hybrid structures are expected to be sensitive to local chemical environments. The atomistic understanding of the adsorption of CNT on Si and metal surfaces and then the resulting modification of properties is therefore imperative. In this section, we present DFT-based calculations for metal and Si surfaces that contribute to such atomistic understanding. One of the other possibilities may come from the internal space of CNTs. The space works as a container of foreign species, as is shown in Section 3.2 for peapods. In medical engineering, the space may be used as a tool of drug delivery. Or more generally, the space may be used as a template to forge new phases of foreign materials. In this section, we also present the calculation for ice in CNTs in which CNT works as a template.

3.6.1

Carbon nanotubes on metal surfaces

It has been demonstrated that the individual semiconducting CNTs can work as field effect transistors in which CNTs form contacts with various metal surfaces such as Pt, Au, Ca, Al and so forth (Martel et al. 1998; Tans et al. 1998; Nosho et al. 2005). It was reported that the FETs exhibit different properties depending on the contact metal species; e.g. they exhibit n-type and p-type properties for Ca and Pd electrodes, respectively. While the fabrication technology regarding nanotubes is advancing steadily, little is known of the fundamentals of the formation and properties of the hybrid structures, compared with the current stage of the semiconductor technology. In particular, stability and properties of the interface between nanotubes and metals are of principal importance in the next-generation technology.

3.6 Hybrid structures of carbon nanotubes

125

In the previous sections, we have demonstrated that, in complexes consisting of CNTs and foreign atoms or molecules, the mixing between wavefunctions of CNTs and those of the foreign atoms/molecules is decisive in geometric and electronic properties. On the other hand, there are peculiar electron states that are localized near surfaces of matter (surface states). On the one hand, these electron states are related to surface atomic reconstructions, and on the other hand, they affect atomic and electronic structures of foreign species adsorbed on the surfaces. It is thus expected that each of the CNT states and the metalsurface state plays its own role in the atomic and electronic structures of a hybrid object. Okada and Oshiyama (2005) have performed the DFT calculations that unravel the interplay of the tube-origin and surface-origin electron states in determining stability and properties of nanotubes attached to metal surfaces. They take a semiconducting (10, 0) CNT adsorbed on the metal surfaces that is considered to be a structural model of the contact between CNTs and metal electrodes. For the metal surfaces, they consider aluminum (100) and calcium (100) surfaces. The binding energies of the semiconducting (10, 0) CNT on Ca(100) and Al(100) surfaces are first calculated. It is found that the optimum spacing ˚ between the wall of a CNT and the topmost surface atomic position is 3.0 A ˚ for Ca(100) surfaces. Calculated absorption energies for for Al(100) and 4.5 A Al and Ca are 3.21 eV and 0.47 eV, respectively, showing a drastic difference caused by chemical-element difference of metals. Large absorption energy for the Al surface is due to the substantial hybridization between electron states of the CNT and the Al surface, whereas the CNT is bound to the Ca surface due to rather weak electrostatic and dispersive attractive interactions. Calculated densities of states (DOSs) of the (10, 0) CNT on Ca(100) and Al(100) surfaces are shown in Fig. 3.25. The characters of the π and π ∗ states of the CNT are more or less preserved, as is evidenced in the vanHove singularity shown by arrows in Fig. 3.25. In the CNT adsorbed on the Ca surface, it is found that the Fermi level is located by 0.2 eV above the bottom

(a)

(b)

30

40 30 20 10 0 –2.0

35 DOS (states/eV)

DOS (states/eV)

50

25 20 15 10 5

–1.5

–0.5 –1.0 Energy (eV)

0.0

0 –1.5

–1.0

–0.5 Energy (eV)

0.0

0.5

Fig. 3.25 Density of states for the (10, 0) CNT adsorbed on (a) Ca(100) and (b) Al(100) surfaces. Dotted vertical line denotes the Fermi level. Solid and pale shaded arrows indicate the electron states originated from π and π ∗ states of the CNT, respectively (reprinted from Okada, S., Oshiyama, A. c 2005 The American Physical Society). (2005) Phys. Rev. Lett. 95, 206804, 

126

Roles of shape and space in electronic properties of carbon nanomaterials

(a)

(b)

(c)

(d)

Fig. 3.26 Contour plots of the accumulated and diminished charges for the (10, 0) CNT on (a) the Ca(100) surfaces and (b) the Al(100) surfaces. Solid and dashed contour lines denote the negatively charged and positively charged areas. Contour plots of valence total charge densities for (c) the (10, 0) CNT on the Ca(100) surface and (d) the (10, 0) CNT on the Al(100) surface. Gray circles denote the C and metal (Ca or Al) atoms. Each contour represents twice (or half) the density of the adjacent contour lines (reprinted from Okada, S., Oshiyama, A. (2005) Phys. Rev. Lett. c 2005 The American Physical Society). 95, 206804, 

of the conduction band of the (10, 0) CNT and crosses several unoccupied π electron bands. Thus, electron transfer from the Ca surface to the nanotube takes place. These results explain the experimental result that the nanotubes on the Ca contact electrode can work as an n-type FET. Figure 3.26(a) shows the difference between the charge density of the CNT on the surface and the sum of the charge densities of the isolated CNT and of the CNT-free Ca surface (Δn = n CNT+Ca − n NT − n Ca ). It is clear that electrons are transferred mainly from the s state of the Ca to the π state of the CNT: Moreover, the diminished charge of the Ca slab is mainly distributed around the interface region between the CNT and the surface. Accumulated charges are distributed near the bottom of the nanotube facing the surface. The calculated number of electrons injected into the CNT is 5.2 × 106 cm−1 . This charge transfer also affects the energy gap between the bottom of the conduction band and the top of the valence band of the pristine CNT (so-called E 11 gap). The E 11 gap is wider by 20 meV than that of the isolated (10, 0) CNT. Al surfaces provide other features that are in contrast to the Ca surfaces. It is found that the Fermi level is located just below the bottom of the conduction band of the (10, 0) CNT that is adsorbed on the Al surface (Fig. 3.26(b)). This difference from the Ca surface is partly due to the different work functions: The work function of Al is larger than that of Ca by 1 eV. Owing to this larger work function, the charge transfer from the metal surface to the CNT does not take place for the Al surface. Instead, adsorption of the nanotube results in the charge redistribution in both Al surfaces and the CNT. In Fig. 3.26(b), the difference between the charge density of the CNT on the Al surface and the sum of the charge densities of the isolated CNT and of the tube-free Al surface is shown. It is clear that electrons are transferred mainly from the π orbital of the CNT and the p orbital of the topmost Al atoms to the space between the tube and the Al surface. The distribution of the electron-rich region reflects the fact that the formation of weak covalent bonds between the CNT and

3.6 Hybrid structures of carbon nanotubes

the surface Al atoms takes place. Valence electrons are evidently distributed between the CNT and the topmost Al atoms (Fig. 3.26(d)). This charge distribution exhibits different characteristics from that of the CNT on Ca surfaces (Fig. 3.26(c)). For CNT on Ca, p orbitals of the metal are unavailable for the bond formation. The hybridization in the CNT on Al modulates the electronic structure of the nanotube: The E 11 gap value decreases by 40 meV compared to that of the isolated (10, 0) CNT in contrast to the CNT on Ca system. Experimental measurements of E 11 are thus useful for identification of CNT–metal contacts. The present calculations clearly show that the charge redistribution near the tube surface interface and also the line-up between the nanotube and the metalsurface electron states substantially depend on metal species, thereby providing rich variation of properties even with the same semicoinductor CNT. This is in interesting contrast with the case of conventional semiconductor metal interfaces in which the band line-ups of the two materials and thus the Schottky barrier heights are insensitive to metal species presumably due to intrinsic interface states (Tersof 1985): when we take a semiconductor and examine a variety of metal contacts, the observed derivative of the Schottky barrier height with respect to the metal workfunction is 0.1 or less for elemental and III-V semiconductors. This also causes difficulty in predicting and forming ohmic contacts. In nanotube metal interfaces, the band line-up is rather predictable from the electron affinities and work functions of the two materials, as is shown here. This difference between CNTs and conventional semiconductors is ascribed to the morphology of tube structures that modify the mixing with intrinsic states in metals. In addition to the charge redistribution at the interface, we find substantial charge oscillation inside the nanotube and the Al slab. In particular, in the Al slab, we can clearly see the oscillation with the periodicity that corresponds to the spacing between adjacent atomic layers (Fig. 3.26(b)). The electrons in both the CNT and the Al slab undulate to screen the charge redistribution caused by the hybridization of bond reformation near the interface.

3.6.2

Carbon nanotubes on SI surfaces

Si surfaces have premier status in semiconductor technology. Fabrications of sophisticated structures of electronic devices are mostly done on the Si (001) surface. When Si(001) is slightly misoriented along the 110 direction by a few degrees, double-atomic-layer steps named DB are present and are almost equally separated from each other (Alerhand et al. 1990; Poon et al. 1990; Oshiyama 1995). This offers a possibility to produce well-aligned metallic CNT arrays on a technologically important substrate. Berber and Oshiyama (2006a) have performed DFT calculations to reveal atomic and electronic structures of CNT on Si (001) stepped surfaces. They take a (5, 5) metallic CNT as a representative. The possible adsorption sites of CNTs on Si(001) stepped surfaces explored by the DFT calculations are depicted in Fig. 3.27. Adsorption sites, A–D are terrace adsorption sites, and could be grouped into two, based on the adsorption direction, namely

127

128

Roles of shape and space in electronic properties of carbon nanomaterials

Fig. 3.27 Adsorption sites of carbon nanotubes on Si(001) stepped surfaces. The Si dimers formed as the reconstruction of the surface are highlighted by dark shading. The left panel contains rebonded DB edge, and the right panel unrebonded DB edge. The positions of the nanotube are depicted by transparent cylinders, and the tube axes are indicated by dashed lines. Adsorption sites are labelled by the capital letters referred c 2006 The American Physical Society). throughout the text (reprinted from Berber, S., Oshiyama, A. (2006) Phys. Rev. Lett. 96, 105505, 

perpendicular (A and C) or parallel (B and D) to the dimer rows. The nanotube axis at site A lies in between dimers, while at site C it lies on top of the dimers. In the case of the adsorption parallel to the dimer rows, the CNT could adsorb on the trench of the surface, site B, or on top of the dimer rows, site D. The adsorption at step edges also possesses 2 metastable adsorption sites with unequal stability that could be distinguished by the existence or lack of covalent bonding with the lower terrace atoms. All adsorption sites depicted in Fig. 3.27 are at least metastable configurations. We define the adsorption energy of CNT on a Si surface as the energy difference between the total energy of the combined system and the summation of the total energies of isolated CNT and the clean surface. Note that the surface resumes an asymmetric dimer geometry in our reference relaxed structure. Positive adsorption energies indicate that adsorption is favorable starting from the reference structures. Adsorption at site A has the highest calculated adsorption energy of 2.77 eV. The least favorable adsorption site, C, has almost zero adsorption energy, and the adsorption energies for the other less favorable adsorption sites D, F, and H are 0.89 eV, 0.6 eV, and 1.0 eV, respectively. As for the adsorption at step edges, the site at the rebonded step, E, is the most energetically favorable with an adsorption energy of 1.88 eV. The site at the unrebonded step, G, also has a comparable adsorption energy of 1.76 eV. The optimum atomic structure of a (5, 5) CNT adsorbed at site A is depicted in Fig. 3.28(a). The Si dimer atoms capped by the CNT displays a symmetric configuration, i.e. the heights of the dimer atoms are the same within the numerical accuracy. The CNT adsorption prevents energy gain due to the charge transfer upon the buckling of dimers. On the contrary, other Si dimers on the surface keep their asymmetric geometry. Close inspection of the total charge density reveals that 4 C−Si bonds with covalent characters are formed, which are highlighted by the dark sticks in Fig. 3.28(a). The bond lengths of ˚ are comparable with but larger than the Si−C these C−Si bonds dA ≈ 1.97 A ˚ bond length of 1.89 A in silicon carbide. The carbon atoms that make bonds

3.6 Hybrid structures of carbon nanotubes

(a)

(b)

1.0

129

(c) 1.5 1.0

dNT dA

Energy (eV)

0.5

NT2

EF NT1 –0.5

–1.0 J'

m3

0.5

m2

EF

m1

–0.5

S2

–1.0

S1

–1.5 Γ

J

–2.0

Γ

J

Fig. 3.28 The atomic and electronic structure of (5, 5) CNT adsorbed on the terraces site of Si(001). The (a) relaxed structure and (b) band structure of the CNT adsorbed on Si(001) at site A, which compares with (c) the band structure of 4 H atoms bonding to the side wall of the CNT. The Fermi c 2006 levels E F are shown by the horizontal dashed lines (reprinted in part from Berber, S., Oshiyama, A. (2006) Phys. Rev. Lett. 96, 105505,  The American Physical Society).

with the surface atoms, highlighted by dark balls in Fig. 3.27(a), are displaced outwards. The C−C bonds dNT around those outward moving C atoms are ˚ extended to ≈ 1.5 A. In the band structure of the CNT adsorbed at site A, shown in Fig. 3.27(b), the tube axis is in the  to J direction. The most striking feature is the disappearance of the metallic character along the tube axis. The Fermi level E F does not cross any energy bands along  to J, indicating a semiconducting behavior in the tube direction. The second main feature is the substantial rehybridization of CNT and surface states. The bands labelled by m1 , m2 , and m3 are considerably mixed CNT-surface states, while S1 and S2 could be assigned to the Si surface. Due to substantial rehybridization, state NT1 , which has a large amplitude on the CNT, fails to show a flat-band line perpendicular to the tube axis, J to . The adsorption induces structural deformations on the CNT, which may cause drastic changes in the electronic structure. In order to understand the origin of the gap opening, we calculate the electronic structure of this deformed CNT that is isolated by removing Si and H atoms from the optimized combined structure. Even the deformed tube shows a metallic character. Next, the effect of sp 3 -type bonding is investigated by attaching 4 H atoms to the deformed nanotube at 4 carbon atomic sites that are highlighted by dark shading in Fig. 3.28(a). Only this explicit introduction of sp 3 bonding reproduces the gap opening, as shown in Fig. 3.28(c). The bands around the E F of isolated (5, 5) CNT are π bands, and the introduction of sp 3 -bonded atoms into π electron network results in the gap opening and the flattening of the π bands. Let us now discuss the adsorption of (5, 5) CNT near DB step edges at the site E. The CNT at the step edge forms only 3 covalent bonds with the Si surface atoms, as depicted in Fig. 3.29(a), because of geometric restrictions. The optimum adsorption geometry of the CNT at the rebonded step edge has 2 covalent bonds with the lower Si terrace atoms, and 1 with the edge atoms,

130

Roles of shape and space in electronic properties of carbon nanomaterials

1.0

(a)

(b)

dD Fig. 3.29 The atomic structure of (5, 5) CNT adsorbed (a) at rebounded DB step edge, site E and (b) the corresponding energy bands near Fermi level E F (reprinted in part from Berber, S., Oshiyama, A. (2006) Phys. Rev. c 2006 The American Lett. 96, 105505,  Physical Society).

du

dL

Energy (eV)

0.5 S

EF

NT –0.5

–1.0 Γ

J

although it is possible to identify weak bonds between the nanotube atoms and the other step-edge atoms. As a result of these weak bonds, not shown in Fig. 3.29(a), all dangling bonds at the step edge are more or less terminated, and the dimers at the rebonded step edge are in a symmetric configuration ˚ is in contrast to the clean rebonded DB step. The C−Si bond du ≈ 1.99 A ˚ and the bond slightly longer than the bonds with terrace atoms d L ≈ 1.96 A, ˚ The C−Si bond lengths length of rebonding dimer at the edge dD is ≈ 2.5 A. of the CNT adsorbed at the unrebonded edge G are also in the same range. In spite of the fact that the number of covalent bonds formed with the surface is smaller, the additional weak extended bonds with the edge atoms make the adsorption energy as large as, or even larger than that for the terrace trench, site B. The band structures of the CNTs adsorbed at step edges calculated along the tube direction,  to J, are shown in Fig. 3.29(b). The E F crosses both a CNT and a surface band, labelled by NT and S, respectively. The electrical conduction along the CNT should preserve the metallic character of an isolated CNT. The CNT used in this study has relatively small diameter for computational reasons. The adsorption energies of moderate-diameter CNTs should be proportional to the number of C−Si bonds allowed by geometrical constraints. The smaller curvature in larger-diameter CNTs should reduce the energy gain per C−Si bond, while more bonds are possibly made. The adsorption at the step edge could become more favorable than the adsorption on a terrace in moderate diameter ranges once the bonding with the upper terrace becomes effective. In that case, ordered alignments of CNTs at DB step edges may be self-organized.

3.6.3

Ice in carbon nanotubes

Water is familiar and indispensable to human life and a lot of work has been carried out to understand its properties. In particular, the phase diagram of water has been explored and, interestingly, more than ten polymorphs have been identified in its solid form (ice) depending on temperature and pressure

3.6 Hybrid structures of carbon nanotubes

(for a review see Savage 1986). In these ice polymorphs, there is a common structural characteristic: i.e. each water molecule is linked to four neighboring water molecules through hydrogen bonds (H bonds) with each hydrogen and oxygen participating in a single H bond and two H bonds, respectively. The quest for a new polymorph is still of interest (Lobban et al. 1998). Carbon nanotubes (CNTs) offer a new stage where novel structures may arise: The nanometer-scale internal space provided by CNTs as well as their speculated hydrophobic characters may cause new ice polymorphs in CNTs. Recent X-ray diffraction measurements (Maniwa et al. 2002, 2005) have indeed found that water molecules are in CNTs with diameters ranging from ˚ to d = 14.4 A, ˚ and the liquid phase is transformed to several d = 11.7 A distinct solid phases at around room temperature. It is argued that stable structures have the shapes of polygonal ice nanotubes (ice NTs) from the observed periodicity of the ordered phases. Polygonal-shaped ice CNTs were originally proposed by molecular-dynamics simulations (Koga et al. 2000, 2001, 2002) in which water molecules are described by classical model potentials and CNT– water interactions by simplified Lennard-Jones potentials. Another moleculardynamics simulation using a similar empirical model (Noon et al. 2002) has been also performed and a different structure consisting of water columns is ˚ recent neutronproposed. For thicker CNT with the diameter of d ∼ 14 A, diffraction measurements combined with MD simulations (Kolesnikov et al. 2004, 2006) infer the existence of another structure in which a chain of water molecules is wrapped by a cylindrical ice tube in CNT. In spite of those efforts, however, little is known as to the detailed structures of ice in CNTs, and further it is of importance to clarify the salient features of ice confined in nanospace on the basis of quantum theory, such as DFT. Recently, Kurita et al. (2007) have performed DFT calculations for ice in CNTs to reveal the energetics and the electron states. They have examined adsorption of a water molecule inside CNTs and found that the adsorption energy is much less than a typical value of the hydrogen-bonding energy ∼0.2 eV. Then, they explored possible tubular ice structures in which the number of hydrogen bonds becomes maximum, and reached a conclusion that structures consisting of polygon units are the most promising candidates for stable ice tubes in CNT. When water molecules constitute a polygon, as in Fig. 3.30(a), one hydrogen atom of each water molecule forms a hydrogen bond with an oxygen atom of an adjacent water molecule. When polygons are arranged along the tube axis, the remaining hydrogen atom of each water molecule participates in an additional H bond between polygons (Fig. 3.30). As a result, each water molecule is fourfold co-ordinated with neighboring water molecules through H bonds. Kurita et al. have found three possibilities for such tubular structures consisting of polygons, as shown in Fig. 3.30 for the case of pentagon. In the stacked-polygon structure (Fig. 3.30(b)), a polygon ring is formed through H bonds and then the polygon rings are stacked along the tube direction, thereby forming interpolygon H bonds. In the helix–polygon structure (Fig. 3.30(c)), one of the H bonds in a polygon is broken and reformed with each water molecule of two adjacent polygons, and then a helix ice nanotube (NT) is formed. In the double-helix–polygon structure (Fig. 3.30(d)), two helices are combined to form a double-helix structure of ice, as in DNA.

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Roles of shape and space in electronic properties of carbon nanomaterials

Fig. 3.30 Water polygons and three possible structures of ice NTs consisting of the polygons. A water pentagon, a hexagon, a heptagon and an octagon (a). Tubular structures for the case of pentagons are demonstrated in (b), (c) and (d). Black and gray balls depict oxygen and hydrogen, respectively. The stacked polygon (b), the helix polygon (c) and the double-helix polygon (d). The top figures are views from the tube axis, whereas the lower figures are from the perpendicular directions in (b), (c) and (d) (reprinted in part from Kurita, T., Okada, S., Oshiyama, A. (2007) Phys. c 2007 The American Physical Society). Rev. B 75, 205424, 

As stated above, in all these structures, each water molecule is fourfold coordinated with neighboring water molecules. As for CNT, which accommodates pentagonal ice nanotubes (NTs), the ˚ which is appropriate (14, 2) chiral CNT is adopted. Its diameter is 11.8 A, to accommodate pentagonal ice NTs. The period of (14, 2) along the tube axis has almost perfect commensurability with that of ice NTs, when the lengths of ˚ By analyzing diameters of the three pentagonal ice H bonds are dH = 2.7 A. NTs with this dH , distances between the oxygen atom of a water molecule and ˚ respectively, for stacked-pentagon NT, the CNT wall are 3.61, 3.66 and 3.80 A, helix–pentagon NT and double-helix–pentagon NT. Table 3.2 shows the cohesive energies of naked pentagon ice NTs that are not encapsulated in CNT in order to reveal the stability of ice NTs compared with a normal polymorph of ice. The cohesive energy E c here is defined as the total energy of the constituting isolated water molecules minus the total energy of the ice NT. It is found that E c values of the ice NTs are 60–76% of the corresponding value of the most stable bulk polymorph Ih under normal conditions. Ice NTs are therefore stable even without encapsulation and are new members of the polymorphs of ice. Encapsulation energy is also calculated for (14, 2) CNT. Each ice tube is placed in a CNT and geometry optimization has been performed. Encapsulation of an ice NT in CNT indeed produces an energy gain. The encapsulation energy E e , that is defined as the total energy of the ice NT encapsulated in

3.6 Hybrid structures of carbon nanotubes

133

Table 3.2 Cohesive energies E c of naked ice NTs and their encapsulation energies E e in (14, 2) CNT. For comparison, the cohesive energy of bulk ice Ih, which is the most stable under ambient pressure, is shown. The energies are per water molecule and in units of meV. Encapsulation energies obtained in the generalized gradient approximation (GGA) and the local density approximation (LDA) are shown (see Appendix). E e (meV) Structure Stacked-pentagon NT Helix–pentagon NT Double-helix–pentagon NT Ih ice

E c (meV)

GGA

LDA

543 509 431 713

9.22 10.2 8.33 –

57.5 54.7 60.7 –

CNT subtracted from a sum of total energies of the empty CNT and the naked ice NT, are shown in Table 3.2. In the GGA, the calculated E e is about 10 meV per water molecule, being insensitive to the structures of ice NTs. Figure 3.31 shows the redistribution of electron density upon encapsulation. It is found that H2 O units render the carbon wall polarized and thereby dipole– dipole interactions are generated. This is the origin of the energy gain upon encapsulation. The density-functional scheme may be insufficient to describe interactions between water and carbon hexagonal networks, since the interaction includes the van der Waals character. It is indeed known that the local density approximation (LDA) provides the layer–layer distance of graphite in agreement with the experiment, whereas the layers are not bound in the generalized gradient approximation (GGA). We thus calculate the encapsulation energies using LDA. The obtained values are shown in Table 3.1. They are in the range of 55–61 meV. These are substantially larger than the corresponding values obtained in GGA. We expect that real values are somewhere between the GGA and the LDA values, since LDA generally overestimates binding energies. Even if we take the LDA encapsulation energies, they are still much smaller than the cohesive energies. We have found that the encapsulation (a)

(b)

Fig. 3.31 Contour plot of the redistribution of the electron density of stacked-pentagon ice NT in CNT, (a) a view along the tube and (b) a side view. The solid and dashed lines represent positive and negative values. Each contour represents twice or half the value of the adjacent line. Dark gray, black and light gray circles represent carbon, oxygen and hydrogen atoms, respectively. The black circular line in (a) and the thick line in (b) indicate the wall of CNT (reprinted in part from Kurita, T., Okada, S., Oshiyama, A. c 2007 The (2007) Phys. Rev. B 75, 205424,  American Physical Society).

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Roles of shape and space in electronic properties of carbon nanomaterials

energies are insensitive to structural difference among ice NTs and that they are one order of magnitude smaller than the cohesive energies. It is thus likely that the hydrogen-bonding configuration plays a principal role in determining the structures of ice NTs in CNT. The DFT-based calculations have revealed that the tubular ice NTs are stable with a cohesive energy comparable with that of the most stable ice Ih. Structural analyses of different tubular ice structures are performed in detail, and now it is clarified that the total-energy difference among the ice structures is a consequence of the difference in angles of the hydrogen bonds. The energy bands of tubular ice have also been calculated and a similarity among ice polymorphs is found (Kurita et al. 2007). The tubular ice NT is a new form of ice that is realized by the nanoscale template, the carbon nanotube.

3.7

Summary

We have reviewed total-energy electronic-structure calculations based on the density-functional theory that clarify and predict fascinating properties of carbon nanomaterials. An emphasis is put on the roles of internal space and boundary shapes in the electronic properties of carbon nanomaterials. In carbon peapods, nearly free-electron states that are inherent to the internal space hybridize with carbon orbitals and then make the peapod a new multicarrier system. It has been shown that the space as well as peas (fullerenes) is decisive in electronic properties. The magnetic properties of zigzag-edged graphene ribbons and carbon nanotubes are discussed and the essential role of the edge state has been clarified. The edge state belongs to a new class of electron states that is inherent to zigzag borders in hexagonally bonded networks. Such an example is also found in topological line defects in carbon nanotubes where ferromagnetic spin ordering is found. We have also demonstrated that curvature in double-walled carbon naotubes also plays an important role and occasionally induces metallization of semiconductor naotubes. This is a consequence of π −σ rehybridization in the tubular structure. The abundance of divacancies in carbon nanotubes and some interesting relaxation patterns of neighboring atoms have also been discussed. A carbon nanoshuttlecock has been introduced. The calculation has shown that decoration of foreign molecules works as scissors for particular electron states and causes magnetic properties in the case of the shuttlecock. The possible application of carbon nanotubes, atomic and electronic structures of carbon nanotubes on metal and Si surfaces have been discussed. It is found that interface properties crucially depend on chemical elements that form hybrid structures with carbon nanotubes. We have also shown that tubular structures cause peculiar quantum effects in a nanocapacitor consisting of double-walled carbon nanotubes. These findings infer the importance of firstprinciples calculations in order to predict properties of the hybrid structures. Finally, carbon nanotubes may be a nanoscale template to forge new phases of foreign materials. We have demonstrated that this is indeed the case for formation of tubular ice in carbon nanotubes.

Appendix: Total-energy electronic-structure calculations

Acknowledgments We thank S. Saito, M. Otani, K. Uchida, S. Berber and T. Kurita for discussions and collaboration.

Appendix: Total-energy electronic-structure calculations In this Appendix, we briefly explain the density-functional theory and related schemes that are used to obtain results presented in this chapter. Materials consist of nuclei and electrons. Hence, the Hamiltonian is well defined. Nuclei move and interact with each other, which is described by a r ) from the nuclei, Hamiltonian Hn . Electrons move under the potential vn ( which is represented by Hen . Electrons themselves are in motion interacting with each other, which is described by He . The total Hamiltonian is the sum of these ingredients so that H = Hn + Hen + He . In the adiabatic approximation that we usually take, electron motion always follows nucleus motion and thus nuclear co-ordinates are treated as classical variables. On the other hand, electrons are Fermions and quantal objects. Consequently, we have to consider exchange and correlation effects among electrons. A basic theorem in the density-functional theory is that the total energy of an interacting electron system, which is defined as E = Ψ |He + Hen |Ψ using the exact many-electron wavefunction Ψ , is given as a functional of electron density n( r ) (Hohenberg and Kohn 1964; Levy 1979):  E[n] = vn ( r )n( r ) d r + F[n]. (A1) Further, introducing an effective one-electron system in which the electron density is identical to that in the real interacting system, or equivalently r ) as expressing the electron density as a sum of single-electron orbitals φi (

|φi ( r )|2 , (A2) n( r) = i

and minimizing the total energy with respect to φi ( r ), we obtain a variational equation called the Kohn–Sham equation as    [n] n(r ) δ E ∇2 XC dr + + vn ( (r ) φi (r ) = εi φi (r ), r) + − 2 δn | r − r | (A3) in atomic units. Here, the third term of the left-hand side is a normal electrostatic potential from other electrons and the E XC [n] in the fourth term is a purely quantum-mechanical factor called the exchange-correlation energy. The exchange-correlation potential is given as a functional derivative of E XC [n]. For the exchange-correlation energy, we need approximations. A typical approximation is the local density approximation (LDA) (Kohn and Sham 1965; Perdew and Zunger 1981) or the generalized gradient approximation (GGA) (Perdew et al. 1996). Eqns (A2) and (A3) are solved self-consistently

135

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Roles of shape and space in electronic properties of carbon nanomaterials

and the resulting {φi ( r )} is used to construct the electron density, which is in turn substituted into (A1) to obtain the total energy. The energy functional F[n] in eqn (A1) is written in the form, 

∇2 n( r )n(r ) d r dr + E XC [n]. |φi + φi | − (A4) F[n] =  | 2 | r − r i The force acting on each nucleus is obtained by the derivative of the total energy with respect to the corresponding nuclear coordinates. Using the calculated forces, we can reach total-energy minimized atomic structures. Or we may calculate dynamical aspects using, for instance, the molecular-dynamics technique. r ) and then the differential In practice, we introduce a basis set to express φi ( equation (A3) is converted to the corresponding matrix equation that is usually solved by iterative techniques. The basis set should be practically complete and the plane-wave basis set is convenient in a sense that we may improve the completeness in a systematic way. A real-space technique is an alternative way to solve eqn (A3). The frozen-core approximation drastically reduces computational time. In the approximation, the electron density of core states in the material is assumed to be identical to the density of the isolated atom or ion. To implement the idea, we use the pseudopotential to simulate the effects from the nucleus and core electrons. Norm-conserving pseudopotentials (Hamann et al. 1979; Troullier and Martins 1991) and ultrasoft pseudopotentials (Vanderbilt 1990) are frequently employed in practical calculations. In calculations of the capacitance, minimizing the total energy (A1) does not provide the required quantities. We have to add the additional energy that represents the external bias voltage μ, i.e. −μ N [n],

(A5)

where ΔN [n] is the functional of the electron density and represents the charge stored in each electrode (Uchida et al. 2006). By minimizing eqn (A1) plus eqn (A5) with respect to n( r ), we obtain the capacitance, C=

d(eN ) . dμ

(A6)

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Roles of shape and space in electronic properties of carbon nanomaterials Tasaki, H. Phys. Rev. Lett. 69, 1608 (1992). Tasaki, H. Prog. Theor. Phys. 99, 489 (1998). Tersoff, J. Phys. Rev. B 32, 6968, and references therein (1985). Troullier, N., Martins, J.L. Phys. Rev. B 43, 1993 (1991). Uchida, K., Kageshima, H., Inokawa, H. Phys. Rev. B 74, 035408 (2006). Uchida, K., Okada, S., Shiraishi, K., Oshiyama, A. Phys. Rev. B 76, 155436 (2007). Vanderbilt, D. Phys. Rev. B 41, 7892 (1990). Watanabe, M.O., Itoh, S., Sasaki, T., Mizushima, K. Phys. Rev. Lett. 77, 187 (1996). Wild¨oer, J.W.G., Venema, A.G., Rinzler, A.G., Smalley, R. E., Dekker, C. Nature 391, 59 (1998).

Identification and separation of metallic and semiconducting carbon nanotubes ´ Katalin Kamar´as and Aron Pekker

4.1

Introduction

Carbon is special among the elements. Its unique electronic structure, which makes hybridization possible, leads to the known variety in organic chemistry and biology. The 2s2 2p2 electrons can assume different configurations and form various multiple bonds. This variety appears not only in compounds, but also in the allotropes of elemental carbon. Carbon atoms in the sp3 -hybrid state appear in diamond, and sp2 -hybridized carbon is contained in graphite, graphene, fullerenes and carbon nanotubes. In graphene, the three sp2 electrons per carbon form planar structures of σ bonds with 120◦ bond angles. The remaining 2p electrons on each atom combine into a delocalized π -electron system (with electron densities concentrated in two planes above and below the atomic plane), which is responsible for many interesting physical and chemical properties. In fullerenes and nanotubes, the electron system is similar, but perturbed by the curvature. These perturbations cause the additional manifold of possibilities leading to special properties and applications. Electrical properties vary over a very wide range in carbon allotropes: from insulators (as diamond and fullerenes) to semi-metals and metals. Carbon nanotubes exhibit the whole range among themselves, depending on geometry: they can be narrow- or wide-gap semiconductors, semi-metals or metals. This wide range of properties is of huge advantage in applications; however, it can cause substantial difficulties when it comes to working with “real-life” macroscopic samples, which most of the time contain an uncontrollable amount of each type. Therefore, one of the most heavily pursued directions in nanotube research has been the identification and separation of nanotubes according to their electric properties. This chapter summarizes the methods used for both. We begin with an overview of the electronic structure of nanotubes, explaining how the metallic and semiconducting properties arise. Then, we present

4 4.1 Introduction

141

4.2 Basic properties determining metallic or semiconducting behavior of carbon nanotubes

142

4.3 Characterization techniques sensitive to metallic or semiconducting type

151

4.4 Specific chirality-selective growth techniques

161

4.5 Physical postgrowth selection methods

162

4.6 Enrichment by chirality-sensitive chemical reactions

163

4.7 Modification of transport properties without change in chirality

170

4.8 Applications as transparent conductive coatings

173

4.9 Summary/Concluding remarks

176

Acknowledgments

177

References

177

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Identification and separation of metallic and semiconducting carbon nanotubes

the most widely used characterization techniques to determine metallic or semiconducting behavior. We focus on macroscopic samples, but throughout the whole chapter, “metallic” or “semiconducting” character will be used to describe individual nanotubes within the sample. Therefore, we will not cover direct resistivity measurements on macroscopic samples whose interpretation is still problematic because the measured resistivity is influenced by tube–tube contacts besides the individual properties; we will also exclude the vast literature on individual nanotube devices. We will focus instead on optical methods: optical transmission, Raman and photoluminescence, the most successful characterization techniques employed for nanotube networks. We continue with the methods of selective growth of specific nanotubes, and subsequently with separation methods based on different physical properties of tubes with different chirality. The next section describes the chemical reactivity differences of various types of nanotubes and how these can be used for separation; finally, we present a method of changing the transport properties of tubes of any chirality by doping or dedoping. Numerous excellent review articles have appeared in many of the topics mentioned above; we will refer the reader to those where appropriate and will go into more detail where, to our knowledge, such reports do not yet exist.

4.2

Basic properties determining metallic or semiconducting behavior of carbon nanotubes

The introduction to the properties of carbon nanotubes in the language of solid-state physics is contained in (Dresselhaus et al. 2001) and is summarized by Saito et al. in the present Handbook (see Chapter 1 of this volume). An alternative explanation from a chemistry point of view has been presented by Joselevich (2004). Since the topic is such that at least a fundamental understanding of both physics and chemistry is required, we will try to relate to the latter as well throughout this section.

4.2.1 1 A triangular lattice with two atoms in the

unit cell.

The structure of CNTs

Carbon nanotubes can be derived from the single graphite layer called graphene. Carbon atoms in graphene are arranged in a honeycomb structure1 (Fig. 4.1). We can consider the nanotubes as a rolled-up piece of this graphene sheet. To do this theoretical derivation we introduce the chiral vector (c). This vector connects two unit cells and can be described by the base vectors (a1 , a2 ), c = na1 + ma2 ,

(4.1)

where n and m are integers. Cutting the graphene sheet perpendicular to the chiral vector through its endpoints, we get a stripe of graphene. Its sides match each other if we form a tube. The circumference of this tube is equal to the length of the chiral vector. In addition to the chiral vector we introduce the

4.2 Basic properties determining metallic or semiconducting behavior of carbon nanotubes

143

Fig. 4.1 Honeycomb structure of the graphene sheet. The chiral vector is constructed by the lattice vectors a1 , a2 and the (n, m) indices. c = na1 + ma2 . Achiral nanotubes correspond to the (n, n) armchair and (n, 0) zigzag directions. The nanotube unit cell (gray rectangle) is defined by the translational vector t and the chiral vector c.

chiral angle (θc ), which is the angle between the chiral vector and the base vector a1 . All of the nanotube structural properties: chiral angle, diameter, translational period can be expressed by the (n, m) integers. The diameter, which determines important optical parameters, is: d=

|c| a0  2 n + nm + m 2 , = π π

(4.2)

where a0 is the length of the base vector of the graphene lattice. We are free to choose the (n, m) pairs, but due to the sixfold symmetry of the graphene we get structurally different nanotubes only in the (0◦ , 30◦ ) chiral angle range.2 Although there exists an infinite number of nanotubes in this region, we can sort them into three groups. Most of them are chiral tubes, but there are two types of achiral tubes: the ones with indices (n, 0) and θc = 0◦ are called zigzag tubes, and those with indices (n, n) and θc = 30◦ are called armchair tubes.

4.2.2

The Brillouin zone of nanotubes

The simplest model to construct the nanotube’s Brillouin zone is the zonefolding method. This method takes into account only the confinement effect, and neglects the contribution of curvature. The aforementioned rolling-up process is equal to the introduction of a periodic boundary condition along the circumference. The hexagonal Brillouin zone of graphene is shown in Fig. 4.2(a). Deriving the nanotube from the graphene sheet has two main consequences. First, the size of the first Brillouin zone in the direction related to the tube axis is determined by the translational period t (Fig. 4.1): k =

2π . t

(4.3)

Due to the nearly infinite size of the nanotube in this direction the wave vector is continuous. The second consequence is that the wave vector is quantized in the direction related to the circumference (k⊥ ) as a result of the periodic

2 Due to the sixfold symmetry we get unique nanotubes in the (0◦ , 60◦ ) range, but the nanotubes in the (0◦ , 30◦ ) and (30◦ , 60◦ ) ranges

differ only in the handedness of their helicity. Here, we restrict ourselves to the (0◦ , 30◦ ) range.

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Identification and separation of metallic and semiconducting carbon nanotubes

Fig. 4.2 (a) Hexagonal Brillouin zone of graphene with the special points. (b) The consequence of zone folding. The Brillouin zone is quantized in the direction related to the tube circumference (k⊥ ) and continuous in the direction related to the tube axis (k ).

boundary condition: k · c = 2πl,

k⊥ =

2π 2 l = l, |c| d

(4.4)

where l has integer values: l = −q/2 + 1, . . . , −1, 0, 1, . . . , q/2, where q is the number of graphene unit cells in the nanotube unit cell,   2 n 2 + nm + m 2 . (4.5) q= NR Thus, the zone-folding method reduces the two-dimensional Brillouin zone of graphene to q equidistantly separated parallel lines. These lines are perpendicular to the direction related to the nanotube axis, and their lengths are determined by the translational period (Fig. 4.2(b)).

4.2.3

Tight-binding model of graphene

While the zone-folding method modifies only the Brillouin zone, we can use the graphene electronic structure to produce the nanotube band structure. The simplest model is the first-order tight-binding calculation. The pz electrons form the π-bonds of graphene. We concentrate on these electrons because from the point of view of optical spectroscopy the σ electrons deeply below the Fermi energy are irrelevant. The π band structure of graphene is shown in Fig. 4.3(a). We can see that at the K-point (1/3k1 ,−1/3k2 ) the valence

Fig. 4.3 (a) Tight-binding band structure of graphene. The conductance and valence band cross at the K points. (b) Contour plot of the graphene electronic band structure with the allowed lines in case of the (12,3) chiral tube.

4.2 Basic properties determining metallic or semiconducting behavior of carbon nanotubes

145

Fig. 4.4 Real-space LCAO representation of selected graphene orbitals built of atomic pz functions. Dark spots represent pz orbitals with positive coefficients, light spots those with negative coefficients. Adapted from Joselevich (2004). Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

and conductance bands are connected. This gives graphene its semi-metallic behavior. In organic chemistry, molecular orbitals are more often represented in real space, combined from atomic orbitals (linear combination of atomic orbitals, or LCAO representations). Figure 4.4 shows the graphene orbitals corresponding to the special points in the graphene Brillouin zone (Fig. 4.3(a)). These pictures represent the top view of the pz orbitals according to their coefficients in the combination: dark points are positive, and light ones are negative. If the coefficient changes sign between two atoms, that means there is no charge density and thus no chemical bond present. The low-energy (filled) states in the valence band form a continous charge density over the whole or a large part of the graphene plane; these are called bonding orbitals. The high-energy (empty) states in the conduction band contain mostly localized charge density and are called antibonding. The two degenerate orbitals at the K points are called nonbonding, because they lie exactly at the Fermi level and represent no energy gain with respect to carbon atoms without a π -bond (Joselevich 2004). These orbitals will determine the conducting properties when a nanotube is formed from the graphene sheet.

4.2.4

Metallic and semiconducting nanotubes

As we have seen in Section 4.2.2, the allowed k-points in the Brillouin zone are confined to parallel lines in the zone-folding approximation. The idea of

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Identification and separation of metallic and semiconducting carbon nanotubes

the model is that the band structure of the nanotube is given by the graphene electronic energies along the allowed lines (Fig. 4.3(b)). The length and orientation of these lines are determined by the (n, m) pair of integers. Although this approximation is rather rough, it provides us with many useful details about the electrical properties of nanotubes. Likewise, the picture of localized orbitals in real space provides us with more insight to connect the nanotube properties with those of conjugated aromatic systems. The most interesting property of nanotubes is that they can be metallic or semiconducting merely due to the way the carbon atoms are arranged on their surface. In the zone-folding picture this essentially different behavior depends on whether any of the allowed k-lines cross the K point. The allowed lines fulfil condition (4.4); their properties are governed by the (n, m) indices through the chiral vector c. The coordinate of the K point is ( 13 k1 , − 13 k2 ). The condition for a nanotube to be metallic is: K · c = 2πl =

1 2π (k1 − k2 ) (na1 + ma2 ) = (n − m) , 3 3 3l = (n − m) .

(4.6) (4.7)

This means that one-third of the nanotubes are metallic in an ensemble that contains all possible chiralities. (This is not the case with all growth methods, as we will see later.) In the tubes for which the (n, m) pair does not fulfil condition (4.7), the K-point is not on the allowed lines, therefore we have no valence- and conduction-band crossing in the nanotube band structure. These tubes have finite energy gaps and behave as semiconductors (Fig. 4.5). Rolling up the graphene sheet can also be represented using the real-space LCAO orbitals introduced above (Fig. 4.6). If 3l = (n − m), the atoms with

Fig. 4.5 Schematic band structure and density of states of different nanotubes. (a) Metallic nanotube: the crossing bands at the Fermi level result in finite density of states and metallic behavior. The other noncrossing bands cause Van Hove singularities. (b) Semiconducting nanotube: there are no allowed states at the Fermi energy, the tube behaves as a semiconductor. Van Hove singularities appear at the band minima and maxima as a result of the 1D electronic system.

4.2 Basic properties determining metallic or semiconducting behavior of carbon nanotubes

147

Fig. 4.6 Schematic of rolling up a nonbonding orbital of graphene corresponding to the Fermi level. (a) Orbital symmetry is preserved, resulting in a metallic nanotube; (b) Orbital symmetry is not preserved, resulting in opening of a gap and forming a semiconducting nanotube. Adapted from Joselevich (2004). Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

similar LCAO coefficients will overlap (Fig. 4.6(a)), and the structure shown in Fig. 4.4 will be preserved. This means allowed states at the Fermi level and consequently metallic behavior. If, on the other hand, the overlap occurs between atoms with different coefficients (Fig. 4.6(b)), the electronic structure is not preserved because the orbital symmetry is not conserved, and a gap will open with the lower orbital filled. Such a tube will exhibit semiconducting properties. This representation is not only helpful to illustrate the electronic structure of nanotubes to those trained in chemistry, but it also helps to understand why metallic and semiconducting nanotubes exhibit different chemical affinity to certain types of reagents. Following Joselevich (2004) we can extend the concept of aromaticity from the strictly taken H¨uckel rules to a situation where an aromatic system is one where the non-bonding orbitals are not preserved but the electrons are fully occupying the lower-lying bonding orbitals, thereby stabilizing the conjugated π -electron system. This condition is fulfilled in the case of semiconducting tubes. In contrast, the system where the nonbonding orbitals are occupied (metallic tubes) is antiaromatic. Organic chemistry has several rules for typical reactions involving aromatic systems and we will see in Section 4.6 that indeed semiconducting nanotubes are more readily participating in those type of reactions. On the other hand, when the reactions involve charge transfer, the decisive factor is the accessibility of electronic states, which is usually more favorable in the case of metallic nanotubes. Representation of the electronic structure through the density of states is very common because it is equally understandable for scientists coming from either a physics or chemistry background. The density of states connects the macroscopic and microscopic properties of the system: the band structure and the possible energy levels (Fig. 4.5). Depicted in this way, the analogy to chemical energy-level diagrams is obvious. The vertical axis is the energy of the electronic states and the horizontal axis on the DOS figures on the right is the number of allowed states having that energy. At the flat maxima and

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Identification and separation of metallic and semiconducting carbon nanotubes

minima in the band structure, many k values correspond to the same energy and the number of allowed states shows an abrupt increase; these spikes are called Van Hove singularities and are analogous to discrete molecular energy levels. This structure is the consequence of quantum confinement of the electrons within the graphene sheet in the radial direction. At the same time, there is a continous background in almost the whole energy range, corresponding to the electronic states on the surface of even one single nanotube, which behaves as a solid. This continuous background is finite at the Fermi level for metallic tubes, where the bands cross, but the number of states is less due to the steeper dispersion; for semiconducting tubes, there are no allowed states at the Fermi level and the highest occupied and lowest unoccupied states are those of the first Van Hove singularities. The distance between these two levels depends inversely on the diameter (Mintmire and White 1998).

4.2.5

3 We consider the system non-magnetic.

Optical transitions in nanotubes

In macroscopic systems the interaction of light with the material can be characterized by the polarization (P) induced by the external electric field. The response of the system can be characterized by the dielectric function that connects the external field (E) and the dielectric displacement (D),3 D = 0 E + P = E = (1 + i2 )E.

(4.8)

With optical techniques we can measure different properties like reflectivity or transmission. These functions are related to the dielectric function but always depend on the measurement geometry. To rule out this geometry we can use simple approximations or rigorous calculations. The final goal is to find a simple relation between the measured data and the imaginary part of the dielectric function. The imaginary part is in close connection with the microscopic properties of the system. It has maxima at the energies of allowed optical transitions, which are defined by selection rules. It is connected to the joint density of states, the density of pairs of states with the same k value and given energy separation. Nanotubes placed in an electromagnetic field absorb photons with energy corresponding to the peaks in the joint density of states. The allowed states involved in the transition are defined by selection rules. The symmetry of nanotubes can be described by one-dimensional line groups (Damnjanovi´c et al. 1999) and the selection rules are therefore dependent on the polarization of the exciting light. For light polarized along the tube axis, the transitions between states are allowed that lie symmetrically with respect to the Fermi energy. These transitions are commonly labelled v1 → c1, v2 → c2 etc. (for “1st valence band level to 1st conduction band level”), or S11 , S22 . . . for semiconductors and M11 , M22 . . . for metals. The energy sequence of the first three transitions seen in optical spectra of most nanotubes is S11 < S22 < M11 . Perpendicular to the tube axis, selection rules are different, but because of preferential absorption (antenna effect), their oscillator strength is negligible compared to the parallel intensity.

4.2 Basic properties determining metallic or semiconducting behavior of carbon nanotubes

149

Fig. 4.7 Transition energies versus nanotube diameter. Full circles represent semiconducting tubes. Open circles represents metallic tubes. (From the homepage of S. Maruyama: http://www.photon.t-u-tokyo. ac.jp/∼maruyama/kataura/katauran.pdf, with permission from the author.)

If we plot the transition energies versus the diameter of the tubes we get the so-called Kataura plot (Fig. 4.7) (Kataura et al. 1999). Due to development in single-nanotube investigation methods and theoretical models taking excitonic effects into account, Kataura plots have undergone tremendous improvement in the last few years (see the chapter by Saito et al. in this Handbook). The simple plot shown here, however, is still used to gain an intuitive picture about the excitation profile of our sample if we know the diameter distribution.

4.2.6

Corrections to the zone-folding picture

4.2.6.1 Curvature effects In the zone-folding picture the nanotube is simply a stripe of graphene concerning electronic states.4 Obviously, this model cannot predict all physical properties of nanotubes. The main difference between the nanotubes and the zone-folding model system is the curvature. The simplest method to introduce the curvature into the nearest-neighbor tight-binding calculation is to allow changes in the nearest-neighbor distances. 4.2.6.1.1 Secondary gap The condition for metallic behavior (an allowed line crosses the K point of the graphene band structure) is sensitive to changes in the electronic structure. Therefore, we expect fundamental differences in the case of metallic tubes due to the curvature. The different bond lengths invoke the shift of the Fermi point with respect to the graphene K-point. Since the allowed k-lines of the nanotubes are unchanged, we expect that it might not cross the new Fermi point. If we use the linear approximation of the dispersion relation in the vicinity of the Fermi point, we get a simple relation between the energy of

4 Apart from the edge states.

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Identification and separation of metallic and semiconducting carbon nanotubes

the gap and the (n, m) indices (Kleiner and Eggert 2001): γ0 a02 γ0 π 2 2 2 (n − m)(2n + 5nm + 2m ) = cos . (4.9) 8c5 4d4 We can see that only the armchair nanotubes (n = m) remain metallic, in this case the shift of the Fermi point occurs along the allowed k-line. In zigzag and chiral “metallic” tubes a secondary gap opens at the Fermi level. The energy of this gap is predicted of the order of 10 meV. The presence of this gap was verified by STM measurements, and it is also observed that armchair nanotubes preserve their metallic behavior (Ouyang et al. 2001). Eg =

4.2.6.1.2 Rehybridization Another consequence of the curvature is the rehybridization of the π and σ states. It can change dramatically the electronic structure of nanotubes, especially the small-diameter ones. Without curvature the π and σ states are orthogonal and thus cannot be mixed (Haddon 1988). In the curved system the overlap is non-zero and the mixed states repel each other: the σ ∗ state shifts upward and the π ∗ downward in energy. This shift increases with decreasing diameter. In the case of small diameter ( 37.2 nm, b-TiO2 between 9 nm < D < 37.2 nm, and a-TiO2 for particles with D < 9 nm (Jiang and Li 2008). This sequence is in agreement with experimental results (Zhang and Banfield 1998; Li and Ishigaki 2004). Once again, however, the particles where assumed “spherical” and surface chemistry was ignored. Equally active are theorists Barnard, Zapol and Curtiss who have used a more general description of the a-TiO2 → r-TiO2 phase transition that included the nanocrystal shape and surface chemistry (as well as size), based on the Gibbs free energy of formation of nanocrystals as a function of size and shape (Barnard and Zapol 2004a; Barnard 2006b), as defined above in (5.6). Using this model a phase transition was identified to be in the range ∼7 nm

5.2 Phase reversal at the nanoscale 197

to ∼23 nm, varying in this case due to additional dependencies identified including the nanomorphology and surface chemistry (Barnard and Zapol 2004b,c). To illustrate how the size-dependent a-TiO2 → r-TiO2 phase transition is affected by changes in particle shape, a range of a-TiO2 and r-TiO2 nanocrystals were examined (with “clean” surfaces) with different shapes, including cubes, prisms, bipyramids, bifrustums and spheres (Barnard and Zapol 2004b). The value of G x (where x = a for anatase, r for rutile) was calculated using eqn (5.6) and plotted as a function of the number of TiO2 formula units, as shown to the right of Fig. 5.7. The points of intersection were determined for each combination. Shown in Fig. 5.7 is a subseries of a-TiO2 (top) and r-TiO2 (bottom) nanocrystals, with varying aspect ratio. In this particular subseries the equilibrium (lowest energy) shapes are denoted by a-TiO2 -B and r-TiO2 -D, and although it is difficult to discern from the graph, the transition size decreases as the r-TiO2 aspect increases, provided the a-TiO2 aspect decreases. The intersection point for the two equilibrium shapes corresponds to a-TiO2 nanocrystals with average diameters of ∼10.4 nm, but this graph contains 25 possible cross-overs in the range 8 nm < D < 18 nm. In general, the results of Barnard et al. definitely showed that the description of faceted nanocrystals is crucial to obtaining a value of size at the phase transition that correlates well with experiment, as the free energy of the “spherical” nanocrystals is considerably higher than that of the faceted nanocrystals and the “sphere-tosphere” phase transition corresponded to only ∼2.6 nm in diameter (Barnard and Zapol 2004b). Of course, the final issue that has been neglected in all of the theoretical work described above, is the relationship between size-dependent phase transitions and surface chemistry. Barnard and Zapol (2004c)

Fig. 5.7 The free energy (with respect to bulk r-TiO2 ) of a-TiO2 and r-TiO2 nanocrystals of various shapes as shown, calculated using (5.6), with cross-overs indicating the phase transition sizes and energies.

198

Size-dependent phase transitions and phase reversal at the nanoscale

1 Note that this does not correspond to solu-

tion acidity, as there are a number of reactions that occur in the vicinity of surfaces, and surface charge transfer was excluded.

were the first to examine this issue, beginning with a comparison of the phase and phase stability of a-TiO2 and r-TiO2 with either clean, partially hydrogenated or fully hydrogenated surfaces. By again plotting the value of G a and G r as a function of the number of TiO2 units, and determining the crossing of the free-energy curves, the phase stability of the faceted titania nanocrystals was investigated as a function of surface hydrogenation. Both the equilibrium shape and the critical size for the phase transition were affected by the presence of hydrogen on the surface. Although the degree of hydrogenation has almost no effect on the minimum energy shape of the a-TiO2 nanocrystal, the effect of surface hydrogenation on the shape of r-TiO2 at the nanoscale was clearly evident. The results predicted that while clean r-TiO2 surfaces produce longer prismatic nanocrystals, the nanocrystals become more stunted as the degree of hydrogen coverage is increased, finally resulting in a squat crystal when all under-co-ordinated surface atoms are terminated. The corresponding phase-transition sizes were found to occur at (a-TiO2 nanocrystals with average) diameters of ∼8.9 nm and ∼23.1 nm with partial or fully hydrogenation, respectively. This is not to say that a-TiO2 nanocrystals may not exist above these sizes, but above these sizes they will be metastable with respect to transformation to r-TiO2 . The role of hydrogen on a-TiO2 and r-TiO2 surfaces is still somewhat unclear, although it is possible that so-called “experimentally clean surfaces” may to some extent be covered with hydrogen (Diebold 2003). Thermogravimetric analysis in combination with infra-red and X-ray photoemission spectroscopies has shown the a-TiO2 nanocrystals at different sizes to be composed of an interior a-TiO2 lattice with surfaces that are hydrogen bonded to a wide set of energetically non-equivalent groups (Li et al. 2005). Although it has been found that r-TiO2 surfaces do not interact strongly with hydrogen (Henrich and Kurtz 1981), and that molecular hydrogen does not absorb at ambient temperatures (Pan et al. 1992), low-energy ion-scattering experiments indicate that atomic hydrogen does bind to the r-TiO2 (110) surface at room temperature (Pan et al. 1992). Moreover, it is highly likely that titania nanocrystals in highly acidic solutions effectively have H-terminated surfaces. This brings us to the issue of surface pH and solution pH, which is an important factor in the synthesis of titania nanocrystals (Chemseddine and Moritz 1999; Penn and Banfield 1999; Gao and Elder 2000; Zaban et al. 2000; Sugimoto et al. 2003a). Following on from the previous work (above) the link between surface acidity and the shape and phase of titania was investigated by Barnard and Curtiss (2005a) using the ab-initio surface energies and surface stresses calculated by Barnard, Zapol and Curtiss (2005). These surfaces were passivated with O, OH, H2 O, H2 O+H and H, to represent strongly alkaline, alkaline, neutral, acidic and strongly acidic surface acidities.1 The results revealed an interesting relationship between surface acidity and nanocrystal shape (Barnard and Curtiss 2005a). When hydrogen is dominant on the surface (or there is a greater fraction of hydrogen present in the adsorbates) there is little change in the shape of the nanocrystals with respect to the (neutral) water-terminated nanocrystals; however, when oxygen is dominant on the surface, the nanocrystals of both polymorphs become elongated and display a more nanorod-like

5.2 Phase reversal at the nanoscale 199

morphology. This is consistent with experimental findings (Chemseddine and Moritz 1999; Penn and Banfield 1999; Gao and Elder 2000; Zaban et al. 2000; Sugimoto et al. 2003a). In each case, the value of G a and G r (from eqn (5.7)) was calculated, by using the appropriate γxi and σxi from Barnard et al. (2005) for each characteristic surface acidity, in combination with corresponding equilibrium shape. From these plots the surface-pH-dependent phase transition sizes were identified at approximately 6.9 nm, 13.2 nm, 15.1 nm, 18.4 nm, and 22.7 nm, for the strongly alkaline, alkaline, neutral, acidic and strongly acidic surface chemistries, respectively. The a-TiO2 nanocrystals are stabilized by surface adsorbates containing a large fraction of hydrogen, whereas r-TiO2 nanocrystals are stabilized by surface adsorbates containing a large fraction of oxygen. These results clearly demonstrated a dependence of the phase transition size on composition of the adsorbates, and were fitted to the form S = S0 exp{5s/4}, where S is the phase-transition size in nanometers, s is the fraction of hydrogen in the adsorbates, and S0 is the phase transition size when s = 0. A consequence of the dependence of the a-TiO2 → r-TiO2 phase transition on the surface chemistry, is that the possibility is introduced for phase transitions to be induced by a change in the absorbed groups on the surfaces (Barnard and Curtiss 2005a). However, although the opportunity exists for phase transitions induced by surface chemistry, this is a very complicated area, since the results mentioned above (Barnard and Curtiss 2005a) also indicated that these types of phase transitions are themselves, size dependent. An experimental observation of such a phase transition induced by changes in surface acidity has been observed in titania (Saponjic et al. 2005), though the phase transformation was accompanied by a shape transition (between and nanotubes and nanosheets), and the size dependence could not be characterized. However, in these experiments there is no driving force other than the changes in chemical environment. An experimental investigation of the affect of solution pH (as opposed to surface pH) has been recently undertaken by Finnegan and colleagues (Finnegan et al. 2007), by carrying out a series of hydrothermal treatments over in the range 1 < pH < 12, at temperatures of 105, 200 and 250 ◦ C. The phase-reversal experiments were carried out subsequent to hydrothermal treatments. Samples hydrothermally treated at a low pH value were then retreated hydrothermally at a high pH value. The purpose of the subsequent treatment was to see if the conversion can be reversed, in the event that there was a conversion of titania to a stable phase during the initial treatment. The final results of this systematic study predicted that the phase stability of titania is also dependent on the pH of the surrounding solution, when an additional driving force is supplied (such as in this case, a temperature greater than 200 ◦ C). In strongly acidic solutions, these experiments showed that formation of small r-TiO2 is favored. In contrast, in strongly basic solutions, a-TiO2 nanocrystals are formed. These results indicate that a-TiO2 nanocrystals are likely to be stabilized at a high pH, and rutile at a low pH, in agreement with reports of several prior studies (Sugimoto et al. 2003b; Li et al. 2004) but

200

Size-dependent phase transitions and phase reversal at the nanoscale

are inconsistent with the simple model results described above (Barnard and Curtiss 2005a). Possible reasons for this discrepancy include the failure of the model results to include: (i) particle-particle interfaces due to agglomeration (which are likely to be present in the experiments), (ii) the failure of the surface energies (Barnard et al. 2005) used in the model to include the influence of surface charge, or (iii) the fact that the model study examined the relationship between stability and surface acidity/chemistry, whereas the experimental study examined the relationship between stability and solution acidity/chemistry. In addition, the model calculations assumed that all the nanocrystals had the opportunity to adopt the equilibrium shape (at low temperature), but this may not have been the case experimentally and it is possible that the shapes of the real samples were governed by kinetics (at ≥200 ◦ C). Ideally all of these factors should be tested individually, and then recombined appropriately to form a complete description, but it is a good example to highlight the importance of reactions at surfaces and how “characteristic surface chemistry” may not be characteristic of the surrounding solution.

5.3

Concluding remarks

Although there is a large amount of information here, we have only just begun to scratch the surface when it comes to understanding the structural stability of nanomaterials, and phase reversal at the nanoscale. The introduction to the field presented here outlines some of the critical issues, and highlights some of the influential physical parameters including temperature, pressure, shape, solution chemistry, surface chemistry and surface charge. To date, there has been no systematic experimental or theoretical/computational study that incorporates all of these parameters in one system, and although it is clear that surfaces play an important part it is not entirely clear which physical parameters will ultimately tip the balance and promote phase reversal in any particular nanoscale material. It is also important to remember that while the phase stability of nano- (and macro-) scale materials is determined by thermodynamics, both the formation of these nanocrystals and the structural transformations themselves are kinetic processes. The thermodynamic studies outlined here offer a valuable guide as to the critical size at which transitions may be expected (due to a reversal in relative stability), but do not guarantee that a transition will occur. This will, of course, depend on the energetic barriers, which may also be a function of the same thermodynamic parameters (listed above) and the particle size.

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Scanning transmission electron microscopy of nanostructures S.J. Pennycook, M. Varela, M.F. Chisholm, A.Y. Borisevich, A.R. Lupini, K. van Benthem, M.P. Oxley, W. Luo, J.R. McBride, S.J. Rosenthal, S.H. Oh, D.L. Sales, S.I. Molina, K. Sohlberg, and S.T. Pantelides

6.1

Introduction

In the last few years the field of transmission electron microscopy (TEM) has seen a rate of instrumental advance unparalleled since the invention of the microscope in the 1930s. This advance has been enabled by the successful correction of the dominant aberrations present in electron lenses (Haider ˚ et al. 1998; Batson et al. 2002). Resolution below one Angstrom is now routine. The benefits are much greater, however, than just the ability to resolve smaller atomic distances. It is now possible to image individual heavy atoms on surfaces (Nellist and Pennycook 1996; Sohlberg et al. 2004) or inside a bulk material (Voyles et al. 2002; Lupini and Pennycook 2003), and even to perform a spectroscopic identification of a single atom (Varela et al. 2004). These instrumental advances allow nanostructured materials to be probed with unprecedented sensitivity to determine their atomic and electronic structure. An ideal complement to this electron microscopy data is provided by densityfunctional theory, which allows structures suggested from the microscope to be refined, and the origin of their properties to be investigated. The combination can provide insights into the atomistic details of nanomaterial growth, their structural stability and their functionality, whether they be nanostructures for catalysis or for their size-tunable optical properties. It is a unique capability of the electron microscope to investigate nanostructures one by one, to find the differences between functional and non-functional nanostructures. Such insights are difficult or impossible to determine from average structural or electronic measurements. It is interesting that this revolution in electron microscopy is parallel to that occurring in nanoscience in general, and particularly, that both were

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6.2 Aberration correction in electron microscopy

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6.3 Semiconductor nanocrystals

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6.4 Semiconductor quantum wires

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6.5 Nanocatalysts

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6.6 Magnetism in gold and silver nanoclusters

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6.7 Charge ordering in manganites

240

6.8 Summary

243

Acknowledgments

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References

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0.1

1

Resolution (Å)

10

100

1000

104 Fig. 6.1 The evolution of resolution; squares represent light microscopy, circles TEM, triangles STEM, solid symbols before aberration correction, open symbols after correction, after Rose (1994).

105 1800

1840

1880

1920 Year

1960

2000

2040

anticipated by Richard Feynman in his prophetic 1959 lecture “There’s Plenty of Room at the Bottom” (Feynman 1959). In that lecture, besides his predictions for things small, he challenged us to “improve the resolution of the electron microscope by 100 times”, in order to “just look at the thing”. It was understood that the resolution of the electron microscope was fundamentally limited by the intrinsic spherical aberration in the primary, image-forming magnetic objective lens, and also that spherical aberration is unavoidable with rotationally symmetric magnetic fields. Feynman’s response was simply “why must the field be symmetric?” Today, aberration correction has been successfully achieved using electron-optical components that break rotational symmetry, just as Feynman suggested. The impact of these correctors on electron microscopy is evident from Fig. 6.1, which shows the improvement in resolution from the era of optical microscopy to the present. This review begins with a brief introduction to aberration correction in electron microscopy and follows with several examples of insights into nanomaterials and the atomic origins of their functionality. The references are not intended to be complete, and the original papers should be consulted for more details. For other recent reviews of applications in materials science see Varela et al. (2005), Pennycook et al. (2007, 2008) and Lupini et al. (2007).

6.2 Aberration correction in electron microscopy

6.2

Aberration correction in electron microscopy

In this section we present a brief review of aberration correction in the TEM and scanning TEM (STEM) and a comparison of the two modes of microscopy. We then focus on the STEM, and present some recent examples of the benefits of the smaller, brighter probe enabled by aberration correction, demonstrating single-atom spectroscopy, simultaneous phase contrast and Z-contrast imaging, and the ability to depth section a specimen through a focal series.

6.2.1

Correction of lens aberrations

The main objective lens in an electron microscope is a round lens, which has an intrinsically high spherical aberration, of the order of 50 wavelengths compared to the roughly one wavelength typical of a light optical lens. The high aberration is due to physics, not poor-quality components. This has been well understood for almost the entire history of the electron microscope, starting with the classic proof by Scherzer that spherical aberration is unavoidable in a round lens with static fields and no charges on the axis (Scherzer 1939). Scherzer also proposed the first design for an aberration corrector by relaxing one of these conditions, specifically by breaking the rotational symmetry of the system (Scherzer 1947). This is the principle used today to successfully correct spherical aberration. The major reason that this has taken over five decades to achieve is essentially instrumental, the need for highly stable electronics and the need to tune all 40 or more optical elements individually. In the era of fast computers and efficient charge-coupled-device detectors it is possible to measure and correct aberrations iteratively, a process referred to as autotuning, essentially a multidimensional form of autofocusing. There are presently two designs of aberration corrector available for electron microscopes, a quadrupole/octapole design produced by Nion and a hexapole design by CEOS. The former has been exclusively used in STEM, whereas the latter has been used in both STEM and TEM. Correcting for the loworder aberrations allows the probe-forming aperture to be increased. In an aberration-free system resolution is given by 0.61λ/θ , where λ is the electron wavelength and θ is the semi-angle of the probe-forming aperture, and so it is clear that increasing θ will directly result in improved resolution. In practice, the aperture is increased until the next higher aberrations limit the resolution. In an uncorrected system, defocus can be used to partially balance the effect of spherical aberration and there is an optimum aperture size that gives an optimum probe shape not too different from that of an aberration-free system, an Airy disc. In this case the resolution is given by 1/4

d = 0.43λ3/4 CS ,

(6.1)

where CS is the coefficient of (third-order) spherical aberration. Similarly, in a (third-order) corrected system, defocus and CS can both be used to partially compensate the next higher aberration, 5th-order spherical aberration C5 , and

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(a)

(c)

(b)

Fig. 6.2 Illustration of certain lens aberrations. (a) A perfect lens focuses a point source to a single image point. (b) Spherical aberration causes rays at higher angles to be overfocused. (c) Chromatic aberration causes rays at different energies (indicated by light and dark gray lines) to be focused differently. Reproduced from Varela et al. (2005).

the resolution becomes (Krivanek et al. 2003) 1/6

d = 0.43λ5/6 C5 .

(6.2)

These correctors are able to shape the electron wave front to a degree of perfection better than a quarter wavelength (∼0.5 pm) over 70 micrometers, a level of performance that compares favorably with that of the Hubble Space Telescope. However, the correctors only compensate for the geometric aberrations of electron lenses, not for any chromatic aberration, which is the focusing of electrons of different energies at different points (see Fig. 6.2). Chromatic aberration correction has been successfully demonstrated for scanning electron microscopy (Zach and Haider 1995) that operates at significantly lower accelerating voltages, and efforts are currently underway to extend this to the higher accelerating voltages normally needed for TEM (Rose 1994).

6.2.2

Comparison between TEM and STEM

In conventional TEM a thin sample is illuminated by a near-parallel electron beam, as shown in Fig. 6.3(a). Scattered waves transmitted through the specimen are focused by the objective lens to form an image, and all image points are recorded simultaneously. In STEM, the incident beam is focused to a fine probe, which is scanned across the sample. Images are obtained using detectors to record various signals as a function of probe position, as shown in Fig. 6.3(b). Therefore, in STEM, image points are obtained sequentially. TEM, with its parallel recording, is ideal for capturing specific signals rapidly from large fields of view, whereas STEM is optimal for extracting maximum information from a single point, by locating the beam over a feature of interest (an atomic column or impurity atom, for example) and detecting multiple signals simultaneously. Typical signals would include bright-field and annular darkfield (ADF) or high-angle ADF (HAADF), commonly referred to as Z-contrast or Z-STEM images, also spectroscopic images formed from electrons that have lost specific energies. Today, both STEM and TEM modes are available on a

6.2 Aberration correction in electron microscopy

209

Fig. 6.3 Ray diagrams showing the important optical elements for (left) TEM and (right) STEM. The TEM image is obtained in parallel, the STEM image pixel-by-pixel by scanning the probe. The diagrams are shown with the electron source at the top for TEM and at the bottom for STEM to show the reciprocal nature of the optical paths. STEM also provides simultaneous bright field and ADF imaging, or, simultaneous ADF and spectroscopic imaging. Actual microscopes have several additional lenses and the beamlimiting aperture positions may differ. Reproduced from Pennycook (2006).

single microscope column and it is straightforward to switch from one mode to the other. In STEM, the primary focusing and aberration correction takes place before the specimen, whereas in TEM it occurs after the specimen, as shown in Fig. 6.3. These two modes of operation may therefore appear entirely different, but in fact they are closely related by a reciprocity principle (Cowley 1969; Zeitler and Thomson 1970). The primary imaging mode in TEM is brightfield imaging, but a bright-field image can also be obtained in STEM using a collector aperture that is optically equivalent to the condenser aperture in TEM, see Fig. 6.3. The major difference between the two geometries is the ray direction, and since contrast in a bright-field image arises primarily through elastic scattering, which is independent of the direction of ray propagation, the two microscopes give identical bright-field image contrast. However, STEM is much less efficient than TEM for bright-field imaging because of its sequential mode of acquisition.

6.2.3

˚ Sub-Angstrom resolution through STEM

It is interesting to observe that before the era of aberration correction it was always the TEM that showed the highest resolution (see, for example, Smith 1997, 2008, also Fig. 6.1). However, with aberration correction, not only is the STEM probe size reduced substantially, but the maximum intensity is correspondingly increased, as shown in Fig. 6.4. This results in significant gains in contrast and signal-to-noise ratio as well as resolution. The HAADF image has intrinsically double the resolution of a phasecontrast image, but before aberration correction was severely limited by noise. It remains noise limited, but much less so after aberration correction. Furthermore, at high resolution the HAADF mode of imaging is much less sensitive

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Fig. 6.4 Comparison of typical reduction in probe sizes from an uncorrected case (a,c) to a corrected case (b,d) for 100-kV and 300-kV microscopes. Simulations include no source size, energy spread or non-round aberrations. Corrected and uncorrected probes are normalized to the same total intensity passing through the probe-forming aperture. Along with the reduction in full width halfmaximum comes a concomitant increase in peak probe intensity. Reproduced from Pennycook et al. (2008).

to chromatic instabilities such as fluctuations in high voltage or objective lens current (Nellist and Pennycook 1998). It was the STEM, therefore, that ˚ first demonstrated a direct image with sub-Angstrom resolution, a Z-contrast image, as shown in Fig. 6.5. Aberration correction has allowed some of the intrinsic benefits of STEM to be realized in practice, which in turn has stimulated significant development of commercial instruments, with most manufacturers now offering instruments with this level of performance or even better. As an example, we present some

Fig. 6.5 Z-contrast image of Si taken along the 112 zone axis, resolving columns of ˚ apart. Image recorded atoms just 0.78 A with the ORNL 300-kV VG Microscopes HB603U STEM equipped with Nion aberration corrector. Image has been filtered to remove noise and scan distortion, adapted from Nellist et al. (2004).

6.2 Aberration correction in electron microscopy

211

Fig. 6.6 Z-contrast images of GaAs in various projections in the same specimen by tilting. (a) [110] with intensity profile, (b) distinguishing the less bright Ga columns from the brighter As columns, (c) [231], (d) [121] and (e) [111]. Images are raw data from the ORNL Titan 80–300, courtesy A. R. Lupini.

results obtained on the Oak Ridge National Laboratory (ORNL) FEI Titan 80–300 aberration-corrected STEM equipped with CEOS DCOR and FEI high-brightness gun, installed as part of the Transmission Electron Aberrationcorrected Microscope (TEAM) Project of the US Department of Energy (see http://ncem.lbl.gov/TEAM-project/). A major advantage of this instrument is that it has a standard SuperTwin objective lens that allows high specimen tilts to be achieved so that several zone axes can be imaged in the same specimen ˚ with sub-Angstrom resolution, as shown in Fig. 6.6. An example of the highest resolution demonstrated at the time of writing is shown in Fig. 6.7, resolution of Ga dumbbells in wurtzite GaN in the 211 orientation. Similar performance has also been reported on other microscopes (Sawada et al. 2007).

6.2.4

Simultaneous phase-contrast and Z-contrast imaging in STEM

A key advantage of the STEM is its ability to use multiple detectors simultaneously. Figure 6.8 shows a focal series of images using STEM brightfield and HAADF detectors simultaneously, giving pixel-to-pixel correlation between the two forms of image. The bright-field images, as in the TEM, are interference patterns, and image contrast depends on the relative phases of

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Scanning transmission electron microscopy of nanostructures

Fig. 6.7 A Z-contrast image (left) of wurtzite GaN in the 211 projection, with the structure shown in the inset, Ga atoms in white, N atoms as small gray spheres, with a line trace from the dashed area (below) showing ˚ resolution of the Ga dumbbells just 0.63 A apart (white arrows), and the N atoms also just visible (black arrows). The Fourier transform (right) shows information transfer to ˚ Images are raw data from the ORNL 0.63 A. Titan 80–300, courtesy A. R. Lupini.

Fig. 6.8 Comparison of incoherent and coherent imaging of SrTiO3 in the 110 projection using a 300-kV VG Microscopes’ HB603U STEM with Nion aberration corrector. (a) A through focal series with 4-nm defocus steps in which the phasecontrast images show complex variations in contrast, while the incoherent ADF image contrast slowly blurs. All images are raw data and show some instabilities. (b) Under optimum conditions, the phase-contrast BF image shows the O columns with high contrast, while the ADF image shows the Sr columns brightest, the Ti columns less bright and O barely visible. Microscope parameters are optimized for the smallest probe with, nominally, a probe-forming aperture of ∼22 mrad, Cs ∼ −0.037 mm and C5 ∼ 100 mm. Courtesy of M. F. Chisholm, A. R. Lupini and A. Borisevich, adapted from Pennycook (2006).

6.2 Aberration correction in electron microscopy

the scattered or diffracted beams. They are coherent phase-contrast images, and even with aberration correction their form depends sensitively on specimen thickness and microscope focus. The detailed form of phase-contrast images can be compared to image simulations and the improved sensitivity after aberration correction allows quantitative information to be extracted even on the composition of light columns such as oxygen (Jia et al. 2003; Jia 2004, 2005, 2006). While the STEM counterpart will be noisier, there is an advantage in having the cation positions clearly delineated in the simultaneous Z-contrast image. The normal STEM imaging mode uses an ADF detector that collects a large fraction of the scattered electrons (Crewe et al. 1970). If the inner detector angle is sufficiently high, the scattered intensity varies approximately as Z 2 , where Z is the atomic number. In this case the HAADF detector integrates a large number of diffracted or scattered beams and the Z-contrast image therefore represents the total scattered intensity falling on the detector. This is an example of an incoherent imaging mode, such as obtained with an optical camera. The Z-contrast image is also capable of atomic resolution in crystalline materials, but the image has the advantage of being intuitive in nature (Pennycook and Boatner 1988; Pennycook and Jesson 1990, 1991, 1992). Contrast does not vary dramatically with specimen thickness or microscope focus, as seen in Fig. 6.8. The Z-contrast image shows the expected Z dependence, with Sr atom columns appearing brightest, Ti less bright, and with O columns barely observable above the background noise.

6.2.5

Atomic resolution spectroscopy

By replacing the bright-field detector with an electron spectrometer, simultaneous Z-contrast imaging and electron energy-loss spectroscopy (EELS) becomes possible. The first demonstration of atomic-resolution chemical analysis was in 1993, with spectra taken plane by plane across an atomically abrupt interface (Browning et al. 1993), using the HAADF signal to accurately maintain the probe over the desired plane. In a similar manner, it later became routine to perform column-by-column spectroscopy (Duscher et al. 1998). Now, aberration correction has enabled dramatic gains in sensitivity due to the availability of smaller probes containing the same current. More current can channel down an atom column of interest with less wasted illuminating neighboring columns. This has allowed the first spectroscopic identification of a single atom inside a bulk material, as shown in Fig. 6.9 (Varela et al. 2004). Recently, the first true two-dimensional spectroscopic maps have been obtained that show atomic resolution (Bosman et al. 2007). An example of spectroscopic imaging in SrTiO3 in the 110 projection is shown in Fig. 6.10 using the aberration-corrected VG Microscopes HB603U with an extended column and a custom-designed objective lens winding to allow all of the bright-field disk to be admitted to the spectrometer without the use of additional coupling modules. For a recent review see Pennycook et al. (2009).

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Fig. 6.9 Spectroscopic identification of an individual atom in its bulk environment by EELS. (a) Z-contrast image of CaTiO3 showing traces of the CaO and TiO2 {100} planes as solid and dashed lines respectively. A single La dopant atom in column 3 causes this column to be slightly brighter than other Ca columns, and EELS from it shows a clear La M4,5 signal (b) Moving the probe to adjacent columns gives reduced or undetectable signals. Adapted from Varela et al. (2004).

Fig. 6.10 Two-dimensional spectrum imaging of SrTiO3 in the 110 projection using the VG Microscopes HB603U with Nion aberration corrector and Gatan Enfina operating at 300 kV. The left-hand panel shows the Z-contrast image with the Sr/O, Ti and O columns shown as black circles, white circles and white squares, respectively. The center and right-hand panels show the integrated signal from the simultaneously acquired Ti L and O K edges (40-eV integration windows were used with a large 260-eV background fitting window to reduce possible artifacts). In the O image, the pure O columns (squares) are seen bright but the Sr/O columns (circles) are not visible due to scattering of the beam by the Sr. A beam current of about 50 pA and an exposure time of 0.025 s per pixel were chosen to minimize damage. The convergence angle was about 25 mrad. Courtesy A. R. Lupini, reproduced from Pennycook et al. (2009).

6.2.6

Three-dimensional imaging and spectroscopy through optical sectioning

One of the unanticipated advantages of aberration correction has been the greatly reduced depth of field. Just as in a camera, as the aperture is opened up the depth of field decreases as the square of the aperture angle. Although the aperture angles are still quite small by light optical standards, the typical depth of field has reduced dramatically as a result of aberration correction, becoming significantly less than the thickness of a typical transmission specimen. Therefore, electron microscopy no longer provides the simple twodimensional projection that it did in the past, but now can provide a view of a slice through the specimen at the depth where the beam is focused.

6.3 Semiconductor nanocrystals

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Fig. 6.11 A series of frames from a through focal series of images, recorded with 0.5-nm focal steps, showing the appearance and disappearance of an individual Hf atom (circled) in a thin layer of SiO2 between the HfO2 dielectric (left) and the Si gate region (right). Reproduced from van Benthem et al. (2006).

A through focal series becomes a through-depth series of images, providing three-dimensional information on the specimen with depth resolution at the scale of a few nanometers and atomic resolution laterally (Borisevich et al. 2006a,b). A spectacular example of this capability is shown in Fig. 6.11 in which individual Hf atoms in a sub-nm wide region of SiO2 in a highK semiconductor device structure have been located to a precision of about 0.1 × 0.1 × 1 nm (van Benthem et al. 2005). Recently, a true confocal mode of operation of the STEM has been demonstrated (Nellist et al. 2006).

6.3

Semiconductor nanocrystals

Colloidal semiconducting nanocrystals are an emerging nanotechnology that electron microscopy has played a critical role in developing. Due to quantum confinement, the nanocrystals exhibit size-tunable optical properties as well as an enhanced molar absorptivity compared to their bulk counterparts (Brus 1984; Murray et al. 1993; Alivisatos 1996). Current efforts are focused on implementing nanocrystals in a variety of fields including photovoltaics, lightemitting diodes and fluorescent probes for biological studies (Colvin et al. 1994; Schlamp et al. 1997; Huynh et al. 2002; Rosenthal et al. 2002; Alivisatos et al. 2005; Mueller et al. 2005; Robel et al. 2006; Schaller et al. 2006).

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With diameters ranging from 1–10 nm, every atom in the nanocrystal becomes important. Subtle differences in size, shape, and even stoichiometry can affect their photoluminescence properties. In a regime where every atom counts, aberration-corrected STEM is an ideal method of characterization. In particular, the sub-Angstrom probe afforded by the VG HB603U 300-kV STEM at ORNL in conjunction with HAADF imaging provides atomic-level characterization with chemical sensitivity. The following section provides a review of our work involving Cd-based nanocrystal systems. The STEM images obtained revealed subtle structural differences that result from changes in surfactant composition, the true nature of epitaxial shells and helped identify the primary growth surface. The specific systems to be discussed are conventional CdSe nanocrystals, CdSe/ZnS and CdSe/CdS/ZnS graded core/shell nanocrystals and CdSx Se1−x alloy nanocrystals (McBride et al. 2004, 2006; Rosenthal et al. 2007).

6.3.1

Sample preparation

For information on nanocrystal synthesis see the review by Rosenthal et al. (2007). STEM samples were prepared by drop-casting dilute solutions of clean nanocrystals in toluene onto an ultrathin coated lacey carbon support TEM grid (Ted Pella Inc.) and allowing them to dry. The commercial water-soluble quantum-dot samples were prepared by dipping the TEM grid into the solution and allowing it to air dry. Samples were checked for quality using conventional TEM prior to imaging using STEM. In order to reduce contamination from the residual organics associated with colloidal nanocrystals, samples were exposed to a light-bulb treatment in the airlock of the VG HB603U prior to sample insertion into the microscope column. An exposure of about 10–15 min generally eliminated contamination and allowed for repetitive imaging over the same area. The tradeoff, however, is that the nanocrystal surface generally becomes oxidized during this process. Also, it takes several hours for the stage to cool to allow for imaging without serious sample drift. An alternative approach of using a plasma cleaner for 12 s has shown little success.

6.3.2

TOPO vs. TOPO/HDA CdSe nanocrystals

It was reported that an improvement in nanocrystal fluorescence quantum yield could be achieved with the addition of a long-chain primary amine, such as hexadecylamine (HDA) to the traditional surfactant, trioctylphosphine oxide (TOPO) (Talapin et al. 2001). Aberration-corrected STEM was used to determine if the additional surfactant had an effect on the nanocrystal structure (McBride et al. 2004). In addition to being the first aberration-corrected STEM images of CdSe nanocrystals, the images showed a clear difference in shape, as seen in Fig. 6.12. The ability to image the surface showed the precise nanocrystal shape, which was generally obscured for all but the largest of the nanocrystals in conventional TEM images. The TOPO-only nanocrystals appeared elongated along the c-axis while the TOPO/HDA nanocrystals are

6.3 Semiconductor nanocrystals

217

Fig. 6.12 TOPO vs. TOPO/HDA surfactant. A and B are Z-STEM images of CdSe nanocrystals prepared with TOPO and a mixture of TOPO and HDA, respectively. The TOPO-only nanocrystals exhibit elongated facets (C). The relative intensity differences between atomic columns (D) seen in the images was used to identify the elongated facet as the (101) facet. A prime is used to denote an anion (Se)-rich surface. Adapted from McBride et al. (2004).

Fig. 6.13 Z-STEM images of TOPO/HDAcoated CdSe nanocrystals. A has one CdSe nanocrystal in the [001] orientation, marked with an arrow, showing the hexagonal crystal structure, B has a CdSe nanocrystal in the [100] orientation, marked with an arrow, which allows for imaging of the Cd and Se atomic columns. Adapted from McBride et al. (2004).

very round. The addition of HDA appears to control the growth rate along the c-axis, reducing the tendency for the nanocrystals to form ovoid shapes or rods. Figure 6.13 shows two TOPO/HDA CdSe nanocrystals in two different orientations. The atomic columns are well resolved in both images in addition to the appearance of an amorphous oxide on the surface. The presence of a surface oxide was initially detected using EELS (Kadavanich et al. 2001). The appearance of atomic dumbbells in the [100] image allows for identification of the various facets due to the polar crystal structure. The atomic dumbbell

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intensity contrast was used to help elucidate the growth mechanism of epitaxial shell coatings for core/shell nanocrystals.

6.3.3

Core/shell nanocrystals

In order to enhance the photostability and the fluorescence quantum yield of nanocrystals, shells of wider-bandgap materials are grown to passivate the surface trap sites of the nanocrystal and to aid in confining the electron and hole (Hines and Guyot-Sionnest 1996; Dabbousi et al. 1997; Manna et al. 2002). The determination of the effectiveness of the shell coverage is typically done by noting an increase in fluorescence, noticing an increase in average particle diameter using conventional TEM, and by XRD. Z-STEM imaging was used in an attempt to image the actual shell coverage and the core/shell interface (McBride et al. 2006). This work was done in collaboration with Quantum Dot Corp. (now a part of Invitrogen) to develop a commercial core/shell nanocrystal fluorescent probe. Figure 6.14 is the first Z-STEM image of a CdSe/ZnS core/shell nanocrystal (taken prior to aberration correction), synthesized using the standard literature preparation. The shell coating is clearly distinguishable from the core due its lower atomic number. The images obtained showed that the shell preferentially grows off one surface. This is a result of the different reactivities of the various facets of the core. In particular, the Se (001 ) surface is presumed the most reactive due to the lack of any surfactant coating. Due to the 12% lattice mismatch between CdSe and ZnS, CdS was used as an intermediate

Fig. 6.14 Z-STEM imaging of core/shell nanocrystals. A and B are images of CdSe/ZnS core/shell nanocrystals showing intensity contrast between core and shell (C). The black arrow in B and C indicates the interface between core and shell. D has cadmium doped into the shell, which improved shell coverage but reduced the amount of intensity contrast between core and shell. E shows a ZnS nanocrystal alongside a core/shell nanocrystal with the associated intensity profile (F). Since the ZnS nanocrystal does not have a shell, the intensity remains uniform over its entirety. The dashed arrows indicate the position where the intensity profiles (C and F) were obtained. Adapted from McBride et al. (2006).

6.3 Semiconductor nanocrystals

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Fig. 6.15 Z-STEM images of CdSe/CdS/ZnS core/shell rods. A and B are Z-STEM images of the 655-nm emitting commercial core/shell rods from Quantum Dot Corporation. The mass contrast between core and shell has been reduced, indicating that the shell material is pre-dominantly CdS. The atomic dumbbells in the image can be used to assign the flat end of the nanorod as the (001) surface. The difference between the dumbbell intensities (C) between the core/shell interface and the shell material indicates a change of material composition. D and E are Z-STEM images of 605-nm emitting commercial core/shell rods from Quantum Dot Corporation. They exhibit the bullet shape to a lesser extent. Both of these samples have quantum yields approaching 100%. Images B and E have been Fourier filtered to reduce background noise. The scale bars correspond to a length of 2 nm. Adapted from McBride et al. (2006).

shell coating to allow for a thicker shell and in an attempt to better coat the entire core surface. The initial attempts at graded shell structures are shown in Figs. 6.14(D–F). The image in Fig. 6.15 is a Z-STEM image of a QDot 655TM commercial core/shell nanocrystal, the end result of the graded-shell approach. The quantum yield of this material is near 100%. With the addition of Cd into the shell, the amount of Z contrast from the core and the shell is reduced. However, the high resolution afforded by aberration correction allows for clear identification of the atomic dumbbells and, as a result of the change in intensity between the Se anion and the S anion, allows for the identification of the core/shell interface. Both the QDot 655TM (Figs. 6.15(A–C)) and the QDot 605TM (Figs. 6.15(D–F)) imaged in this study exhibit a “nanobullet” shape. This is a result of the shell material still preferentially growing off specific facets. The atomic dumbbells in the images indicate that the thickest shell coverage is located on the anion-rich facets, as illustrated by Fig. 6.16. However, the improved total shell coverage seems to suffice due the high fluorescence

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Fig. 6.16 Preferential facet coverage. The high magnification view of a nanobullet (A) definitively reveals the sublattice polarity in the CdS and CdSe regions from the relative intensity, shown by the line trace C. A model structure commonly assumed for nanocrystals (B) can be directly compared to the image. Comparing the image with the model illustrates how the shell must be growing at a higher rate only on the Se-rich facets to form the bullet shape, as shown schematically in D. The white arrows indicate the caxis, while gray arrows indicate the Se-rich (101) facets. The scale bar in (A) corresponds to a length of 3 nm. Adapted from McBride et al. (2006).

quantum yield of this material. Future work to combat the facet-specific growth is needed as this material is still susceptible to photobleaching.

6.3.4

CdS x Se1−x alloy nanocrystals

A recent discovery by Bawendi et al. (Zimmer et al. 2006) demonstrated that the hydrodynamic radius of the quantum dot has an effect on in-vivo imaging. One approach to maintain the ability for multicolor fluorescence imaging and yet remove particle-size effects is to use an alloy core. Alloy nanocrystals allow for size-independent color tuning by modifying the ratios of the alloy’s constituents. In this work, homogeneous CdSx Se1−x core/shell nanocrystals synthesized where the sulfur to selenium ratio allowed for tuning over most of the visible spectrum (Swafford et al. 2006). In conjunction with Rutherford backscattering spectroscopy for elemental analysis, Z-STEM was used to ensure that the distribution of sulfur and selenium was uniform over the entire nanocrystal. Any sequestering of either would be indicated by a change in the image intensity, much like that of the core/shells. Figures 6.17(A and B) are Z-STEM images of CdS0.55 Se0.45 alloy nanocrystals. None of the alloys imaged showed signs of graded or core/shell structures. However, a large number of stacking faults were seen in all of the alloy samples. In order to determine if this is a result of the alloying process or the specific synthetic preparation, cores of pure CdS (not shown) and CdSe (Fig. 6.17(C)) were also imaged that were synthesized under the same conditions. The high number of stacking faults appears ubiquitous for cadmium-based nanocrystals prepared using oleic acid, as was used in this case. It is not known why the weaker binding oleic acid produces zinc-blende nanocrystals rich in stacking faults while stronger binding phosphonic acids

6.4 Semiconductor quantum wires 221

Fig. 6.17 Z-STEM of CdS0.55 Se0.45 alloyed nanocrystal. A and B are Z-STEM images of alloyed nanocrystals showing a large number of stacking faults but no contrast due to sequestering or core/shell structures. C is an image of CdSe nanocrystals prepared using oleic acid. In contrast to nanocrystals synthesized in TOPO/HDA, there is a large number of stacking faults and twins, the result of the oleic acid surfactant. Reproduced from McBride et al. (2008).

used in TOPO/HDA preparations produce wurtzite nanocrystals with very few stacking faults. In summary, aberration-corrected Z-STEM has improved our understanding of the nanocrystal’s surface and how it affects photoluminescence and growth. The first aberration-corrected images of CdSe cores, core/shells and alloys were obtained, yielding new insight into their structures. We now know that the reactive (001 ) surface dominates core growth as well as any epitaxial coating. Although, TOPO and HDA as a surfactant mixture seem to limit growth along the c-axis in cores, the same affect is not observed during shell growth. Future work will be needed to learn how to slow growth along this axis if uniform shell coverage is to be obtained.

6.4 6.4.1

Semiconductor quantum wires Nucleation of nanowires during epitaxial growth

Controlled growth of semiconductor quantum nano-object arrays is important for reproducible fabrication of optoelectronic devices, and therefore understanding the nature of the nucleation sites during growth, and the role of strain, is of critical need. In this example we show how the sublattice sensitivity of aberration-corrected Z-contrast imaging, combined with elasticity calculations, can give new and quantitative insights into these phenomena. Figure 6.18 shows a cross-sectional view of an InAsx P1−x nanowire sandwiched between InP layers, grown by molecular beam epitaxy (Molina et al. 2007). In this 110 projection, the cubic zinc-blende structure projects as dumbbells with the In column towards the left of the figure and the P or Asx P1−x column towards the right. The nanowire is visible because of the extra intensity from the As.

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Fig. 6.18 Z-contrast image of an InAsx P1−x nanowire between InP barrier layers, reproduced from Molina et al. (2007).

Fig. 6.19 Z-contrast images of the InAsx P1−x nanowires with increasing magnification, highlighting the contrast differences between the P columns of the InP layers and the Asx P1−x columns in the nanowires. It is clear that the transition from InP to InAsx P1−x nanowire is atomically abrupt, and an interface step is present under the nanowire. Adapted from Molina et al. (2007).

Close examination of the InP barrier layers does in fact show a weak intensity in the appropriate position that corresponds to the P column. The interface between the InP substrate and the nanowire can be located to atomic precision because of this contrast difference, as shown in Fig. 6.19. Atomic force microscopy examination of the nanowires deposited on an InP substrate showed the tendency for the nanowires to be located near step edges. While many of the steps were wavy, some wires nucleated near straight

6.4 Semiconductor quantum wires 223

segments of steps, and these gave a clear sharp image in cross-sectional Z-contrast images, as shown in Fig. 6.19. It is seen that the transition from the substrate InP layer to the nanowire is atomically abrupt. The location of the step is slightly off-center from the nanowire, indicating that the initial nucleation took place on the upper terrace of the step. Nucleation on the upper terrace is entirely in accord with expectations based on the 3.2% lattice mismatch between InAs and InP. Because of the existence of the step, a few unit cells next to the step on the upper terrace can relax sideways a little, reducing the lattice mismatch, and therefore making this area the preferential nucleation site. Finite-element elasticity calculations showed that the magnitude of the relaxation was indeed significant. With the nucleation taking place on the upper terrace the surface of the growing film will roughen, so this represents the onset of the Stranski–Krastanov transition from a two-dimensional to a three-dimensional growth mode (Molina et al. 2006). It can also be seen in Figs. 6.18 and 6.19 that there is a slight asymmetry in the As distribution within the wire, the region above the upper terrace showing brighter, and therefore containing more As. This is typical and was also confirmed by quantitative electron energy-loss spectroscopy, using the P edge intensity in the barrier layers as calibration. The resulting As concentration maps were used as input into a finite-element elasticity calculation (Molina et al. 2006, 2008). In the actual device structure, multiple layers of InP barrier layers and nanowire layers are grown, and cross-sectional TEM showed that they were not quite vertically aligned, but canted by a few degrees, as shown in Fig. 6.20. This observation could be accounted for quantitatively by the finite-element calculations. Modelling a buried nanowire showed that due to the microscopic asymmetry of the As distribution, an anisotropy also appeared in the stress on the surface of the subsequent InP barrier layer, with the peak stress emanating from the nanowire appearing slightly offset from the geometric center of the wire. Therefore, again the favorable nucleation site for the next nanowire is the position that minimizes the mismatch, which we find is slightly offset from the center of the underlying wire. With carefully

Fig. 6.20 Low-magnification TEM diffraction-contrast image showing the alignment of the nanowire stack is offset from the growth direction. Adapted from Fuster et al. (2004).

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controlled conditions all nanowires line up at a specific angle to the growth direction and the observed angle is in quantitative agreement with the finiteelement calculations.

6.4.2

Vertically aligned Si nanowires

Vertically aligned, free-standing, Si nanowire arrays are commonly grown using the vapor–liquid–solid mechanism employing a catalyst that forms a low-temperature eutectic with Si, for example, Au. When Si vapor comes into contact with a Au droplet, it dissolves into the droplet and when it becomes supersaturated it precipitates out onto the Si substrate, thereby growing a Si nanowire carrying a Au droplet on its tip, as shown in Fig. 6.21 (Oh et al. 2008). Significant surface diffusion of Au occurs, and the presence of small nanoparticles is evident in Fig. 6.21(a). As growth proceeds, these coarsen by Ostwald ripening, until the Au particle at the tip becomes depleted and growth stops (Fig. 6.21(b)). We have used the depth sensitivity of Z-contrast STEM to distinguish Au atoms embedded inside the volume of the nanowire from those decorating the surfaces. Starting with the beam focused before the specimen, a series of images was acquired successively with focus advancing into the sample in 2nm steps. The top surface of the thick region of the nanowire is seen in focus in Fig. 6.22(a), along with surface Au atoms (arrowed). Further decrease of defocus by 16 nm brings the edge of the nanowire into focus (Fig. 6.22(b)), while the Au atoms at the bottom surface come into focus only after decreasing the defocus by an additional ∼16 nm (Fig. 6.22(c)). Therefore, in the thick part of the nanowire, in the center of Fig. 6.22(b), the beam is definitely focused inside the nanowire, when atoms on the top and bottom surfaces are not in focus and are not visible in the image. We can now increase the magnification of the image and see individual Au atoms located inside the nanowire. We find several different configurations, one substitutional and three interstitial, as shown in Fig. 6.23.

Fig. 6.21 Low-magnification HAADF images of Si nanowires. (a) A Si nanowire capped with a solidified Au-Si droplet. (b) A faceted and tapered Si nanowire with no solidified droplet. The Z-contrast image clearly shows the surface decoration by Au covering the nanowires. Reproduced from Oh et al. (2008).

6.4 Semiconductor quantum wires 225

Fig. 6.22 Three frames selected from a through-focal series of HAADF images acquired in 2-nm defocus steps at the cross-sectional 112 orientation. Au (single atoms and nanoclusters) atoms existing at different surface locations can be identified. In each frame the Au nanoclusters in focus are indicated by arrows; (a) on the top sidewall at 0 nm, (b) at the edge sidewall at −16 nm and (c) on the bottom sidewall at −32 nm relative defocus. Reproduced from Oh et al. (2008).

Fig. 6.23 HAADF images of a Si nanowire in 110 zone-axis orientation (left panel). A slight image distortion caused during the scan was unwarped. Boxes show the regions used for intensity profiles, with Au atoms in various configurations arrowed; (a) substitutional; (b) tetrahedral; (c) hexagonal; (d) buckled Si–Au–Si chain configurations. The intensity profiles across the Si dumbbells correspond to a width of 18 pixels. Adapted from Oh et al. (2008).

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Fig. 6.24 Local views of defect configurations of Au in Si. (a) Substitutional; (b) tetrahedral; (c) hexagonal; (d) buckled Si–Au–Si chain configurations. The HAADF images (left panel) were processed by maximum entropy image deconvolution (HREM Inc.) using a Lorentziantype probe function. The corresponding atomic models (right panel) calculated by density-functional theory match the images closely. Adapted from Oh et al. (2008).

It is apparent that a very high concentration of Au atoms is seen inside the nanowire, but they are highly mobile, changing position from frame to frame and may be injected by the electron beam from the high concentration of Au atoms on the nanowire surface. In any case, it is interesting that the different configurations of the Au atoms can be reproducibly seen. The sites were investigated with density-functional theory (Fig. 6.24) and the substitutional site was found to be most stable, with the three interstitial sites being metastable. Furthermore, the number density of the sites was in accord with their relative formation energies, although indicating an effective specimen temperature of 1000 ◦ C, suggesting that the electron beam is likely to be inducing Au-atom migration.

6.5 6.5.1

Nanocatalysts Anomalous Pt–Pt distances in the Pt/alumina catalysts

The Pt/γ -alumina system is representative of many common heterogeneous catalysts, which consist of transition metals dispersed on a high surface area support. This system is often used for catalytic reduction and oxidation of automotive pollutants. Aberration correction has resulted in enormous gains in single-atom sensitivity. Before aberration correction, with a probe size of ˚ single Pt atoms were just detectable on a γ -alumina support around 1.3 A, (Nellist and Pennycook 1996), and this result in fact was the first to show that very small clusters or even single atoms could be important for catalysis. The improvement after aberration correction, when the probe size was reduced to ˚ was striking, however (Sohlberg et al. 2004). Now the signalaround 0.8 A, to-noise ratio is greatly enhanced, sufficient to allow accurate extraction of Pt atom positions, as shown in Fig. 6.25. It is apparent that the three Pt atoms do not form an equilateral triangle with regular Pt–Pt distances. The explanation for the distorted shape comes through density-functional total-energy calculations. Placing bare Pt trimers on a γ -Al2 O3 110 surface and relaxing the structure to equilibrium results in an almost equilateral

6.5 Nanocatalysts 227

Fig. 6.25 (Left) Z-contrast STEM image of Pt on the surface of γ -Al2 O3 close to the 110 orientation. Two Pt3 trimer structures are circled. (Right) Schematic of the configuration for the Pt3 OH unit on the (110) surface of γ -Al2 O3 , determined by first-principles calculations, Pt atoms shown circled. The circled V marks a vacant tetrahedral cation site; triangles mark possible H atom positions in the vacancy. Adapted from Sohlberg et al. (2004).

˚ close to the intertriangle with bond lengths of 2.59, 2.65, and 2.73 A, atomic spacings in metallic Pt. The longer bonds found experimentally can be explained by adding an OH group to the top of the trimer: Two of the ˚ in better agreement with the experimentally bonds lengthen to 3.1 and 3.6 A, ˚ The addition of the OH group also changes determined values (3.2 and 3.4 A). the electron density on the Pt atoms. There is a clear indication of depletion of electron density from the Pt–Pt bonds, which may explain the catalytic activity (Sohlberg et al. 2004). Although directly detecting OH groups on a single Pt trimer on a thick support would be extremely difficult, this example demonstrates that the enhanced sensitivity due to aberration correction provides vital information about the structure.

6.5.2

Rh on alumina

Aluminum-supported rhodium is an important heterogeneous catalyst, especially valuable for its ability to hydrogenate functionalized aryls while leaving the ring substituents intact (Freedman et al. 1955; Sokolskii et al. 1982; Hindle et al. 2006). There is significant incentive to understand the atomic-scale details of catalysts with this kind of specificity because such understanding could lay the foundations for predictive atomic-scale design of catalysts with high specificity. Atomic-resolution Z-contrast STEM images of a 1.2 wt. % Rh-on-alumina catalyst were obtained by Nellist and Pennycook (1996). These

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Fig. 6.26 (a) Z-STEM image of Rh on γ -alumina, reproduced from Nellist and Pennycook (1996). (b) Exploded view ∼1.0 × 1.0 nm. (c) Simulated image based on optimized Rh2 O3 -II structure on γ -alumina (100) showing the same area as (b), reproduced from Sohlberg et al. (2008).

images show the presence of very thin rafts of Rh atoms, a few nm2 in area, as shown in Fig. 6.26. The “raft” structures in the Rh/γ -alumina catalytic system can be better understood through the combined application of first-principles calculations and simulated Z-contrast imaging. The Z-STEM image in Fig. 6.26(a) shows a “raft” of Rh on the alumina surface. A particularly interesting feature is that the Rh atoms are resolved into rows in one direction, showing a separation of ˚ but do not resolve in the perpendicular direction, (along the rows). about 2.8 A, ˚ this indicates Given that the spatial resolution of the instrument was ∼1.3 A, ˚ that the separation of Rh atoms within the rows is less than 1.3 A. The presence or absence of oxygen within the Rh raft was not established experimentally since the instrument was not sufficiently sensitive to image oxygen. Even today with a next-generation instrument it would be difficult to distinguish oxygen in the raft from oxygen in the support. Studies of the Pt/γ -alumina (Sohlberg et al. 2004) and Cr/γ -alumina systems (Borisevich et al. 2007), however, suggest that the metal might be at least partially oxidized. (Both are reviewed herein.) This hypothesis is supported by first-principles calculations. While the Z-STEM imaging gave evidence that the Rh rafts are supported on the (100) face of γ -alumina, first-principles calculations found that the binding of Rh atoms to γ -alumina (100) is thermodynamically nonspontaneous (endothermic) (Sohlberg et al. 2008). It was found that binding to the (110) face is an exothermic process, but the most favorable binding occurs when the Rh atom is incorporated into an empty octahedral interstitial site in the first subsurface layer. Furthermore, it was found that clustering of Rh atoms on the surface is unfavorable, i.e. two isolated Rh adatoms on γ -alumina is a lower-energy configuration than a single Rh2 /γ -alumina structure. Larger clusters were found to spontaneously dissociate upon structural optimization.

6.5 Nanocatalysts 229

Fig. 6.27 Structure of Rh2 O3 -II. Note that in the orientation displayed, the separation of the rows of Rh atoms (light gray) is about ˚ but the projected separation of the 2.8 A, Rh atoms within the rows is much shorter. Reproduced from Sohlberg et al. (2008).

By contrast, Rh atoms are held in close proximity when in the form of an oxide. There are three known sesquioxides of Rh, denoted with Roman numerals I, II and III (Zhuo and Sohlberg 2006). Of these, the high-pressure phase II has a structure that is most consistent with that of the observed Rh rafts. In certain projections, as shown schematically in Fig. 6.27, it exhibits ˚ Structural optimization of a model rows of Rh atoms separated by about 2.8 A. consisting of a thin section of Rh2 O3 -II/γ -alumina(100) was found to yield a stable structure (Sohlberg et al. 2008). Z-STEM images were simulated based on the optimized theoretical structure using convolution simulation, an excellent approximation in the case of thin raft structures (Jesson and Pennycook 1995). As shown in Fig. 6.26(c), the simulated image captures the key structural features revealed by the Z-STEM image (Fig. 6.26(b)) rows ˚ without resolution of the individual species of intensity separated by 2.8 A along the rows. The presence of Rh2 O3 -II is also consistent with earlier TEM studies of γ -alumina-supported Rh catalysts where the formation of particles of the high-pressure Rh2 O3 -II phase was observed following high-temperature aging in air (Weng-Sieh et al. 1998). The growth of the high-pressure phase of Rh sesquioxide on γ -alumina presumably arises from the difference in atomic radius between Al and Rh. Rh is a larger atom than Al. The denser structure of the high-pressure Rh2 O3 phase gives a better “fit” to the Al2 O3 structure.

6.5.3

La stabilization of catalytic supports

γ -Alumina is one of many polytypes of Al2 O3 that are used as catalytic supports. A porous form of γ -alumina is used extensively because it has a large specific surface area. However, at temperatures in the range 1000–1200 ◦ C, γ alumina transforms rapidly into the thermodynamically stable α-alumina phase (corundum); the pores close and catalytic activity is degraded. A small amount of La doping increases the temperature at which this transition occurs. Many explanations have been proposed for this change, but until recently it was not clear which was correct. A combination of STEM, EXAFS and firstprinciples density-functional calculations was able to locate the La and explain this remarkable result (Wang et al. 2004). A demonstration that the La is

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Scanning transmission electron microscopy of nanostructures

Fig. 6.28 Z-contrast image of a sample of La-doped γ -Al2 O3 . Bright spots correspond to La atoms. (Left) An off-axis image suggests that the La atoms are clustered at edges. (Right) Close to the 110 axis, the support structure is revealed and the La atoms appear to be clustered on step edges. Adapted from Wang et al. (2004).

Fig. 6.29 (a) Z-contrast images of a La-doped γ -Al2 O3 sample obtained by changing the focus of the beam from the top surface of the sample to the bottom surface. (b) The La atom intensities peak at the top and bottom surfaces, showing the atoms to be located on the surfaces. Adapted from Wang et al. (2004).

present as single atoms was provided by Z-contrast imaging. In Fig. 6.28 single La atoms are seen superposed on the crystal lattice planes formed by the substrate. Furthermore, this image shows that they are found most often on step edges, which suggests that the La atoms are on the surfaces. Threedimensional microscopy provided an essential confirmation: By changing the focus of the beam through the sample as shown in Fig. 6.29, it was found that the La-atom intensities peak at only two planes (the top and bottom surfaces);

6.5 Nanocatalysts 231

Fig. 6.30 Schematics of the configurations for the (100) surface of γ -Al2 O3 , determined by first-principles calculations: (a) undoped, and (b) La-doped. The Al, O, and H atoms are shown in light gray, dark gray and white, respectively, while the La atom is circled in black. When a La atom is present on the surface, a surface Al atom relaxes from the surface into a cation vacancy as indicated by the arrows. The La atom occupies a site close to the initial location of the Al atom and binds to the nearest three oxygen atoms. Adapted from Wang et al. (2004).

they are not evenly distributed throughout the bulk. Experimentally, the La atoms are found only at the surfaces of the sample. Theory independently found that La atoms favor the surfaces over bulk sites by substantial energy differences (∼4 eV), with several possible configurations, having comparable energies. Figure 6.30 shows a schematic, indicating that a La atom on a (100) surface induces a significant local reconstruction event. In addition, theory demonstrated that it is not energetically favorable for La atoms to cluster on alumina surfaces; they remain as single ad-atoms. La atoms have a much larger binding energy on γ -alumina (7.5–9 eV) than on the α-alumina (4–5 eV), which is the final product of the undesirable phase transformation. Thus, the extra 3–5 eV binding energy required per La atom is the reason for the higher transition temperature. In simple terms, the La atoms prevent the phase change effectively by stabilizing the surface of the γ -alumina phase (Wang et al. 2004).

6.5.4

Cr on alumina

Chromia/alumina catalytic systems have been widely adopted by industry as catalysts for catalytic dehydrogenation of alkanes (Park and Ledford 1997; Flick and Huff 1999; Cherian et al. 2002). One of the biggest issues related with more commonly used chromia on γ -alumina is its rapid degradation rate, which necessitates replacing the catalyst every few weeks. However, it has recently been established that replacing the support with a very similar η-alumina greatly improves stability (Alerasool et al. 2006). Borisevich et al. (2007) used Z-contrast aberration-corrected STEM, together with EELS and extended X-ray absorption fine structure (EXAFS) and first-principles quantum-mechanical calculations to provide atomic-scale understanding of alumina/chromia systems, the oxidative dehydrogenation of alkanes, and the nanoparticle/substrate interactions that control catalytic activity. Z-contrast STEM studies demonstrate a marked difference in the Cr distribution on γ - and η-Al2 O3 . For γ -Al2 O3 at high loadings of Cr the distribution

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Scanning transmission electron microscopy of nanostructures

Fig. 6.31 (a) Z-contrast STEM image of Cr/γ -Al2 O3 , (b) EELS spectra acquired from different regions of Cr/γ -Al2 O3 , (c) Z-contrast STEM image of Cr/η-Al2 O3 with spots arrowed (see text), (d) EELS spectrum from Cr/η-Al2 O3 . Adapted from Borisevich et al. (2007).

is highly non-uniform, as illustrated by a Z-contrast image of the sample with ∼2.5 monolayer loading in Fig. 6.31(a). In the Z-contrast image Cr-rich regions (ZCr = 24) appear bright on the background of the light supporting alumina (ZAl = 13). Cr is visibly segregated into extended “patches” on the γ -alumina surface. The chemical nature of the contrast is confirmed by the EELS studies (Fig. 6.31(b)), which show a strong Cr signal on the “patches” and no detectable Cr signal in between. On η-Al2 O3 , on the other hand, no segregation was observed. Contrast in the STEM images of the sample with ∼0.9 monolayers of Cr was mostly uniform, with the thinnest, off-axis areas of the alumina flakes sometimes displaying faint spots (Fig. 6.31(c); spots arrowed). The EELS spectra, however, showed the clear presence of Cr and no discernible difference in Cr concentrations between different areas of the sample (Fig. 6.31(d)). The EXAFS data indicated a significant difference in Cr coordination numbers: 5.8 and 4.5 for γ - and η-alumina, respectively. It was also noted that while the Cr/η-Al2 O3 pattern showed a lack of long-range order, suggesting the existence of small monodispersed Cr clusters, the pattern for Cr γ -alumina surface corresponded ideally to the crystalline structure of Cr2 O3 .

6.5 Nanocatalysts 233

Fig. 6.32 Lowest-energy configurations for CrO3 on alumina surfaces (Al – light gray, O – dark gray, H – white). (a) On γ -Al2 O3 , (b) on η-Al2 O3 , (c) on η-Al2 O3 after interaction with C2 H6 molecule. Adapted from Borisevich et al. (2007).

To understand the difference in the behavior of CrOx clusters adsorbed on different transition aluminas, extensive first-principles calculations were performed. The resulting minimum-energy configurations for a CrO3 cluster are shown in Fig. 6.32. The bulk crystal structures of these two transition alumina polytypes are very similar: both are defect spinels in which 11% of the cation sites are vacant to satisfy the Al2 O3 stoichiometry. In γ -Al2 O3 most of the vacancies are located in tetrahedral sites, whereas in η-Al2 O3 they are distributed mainly in the octahedral sites (Zhou and Snyder 1991). This difference, however, makes for sufficiently distinct diffraction patterns (Zhou and Snyder 1991) and significantly affects the surface relaxation processes in these polytypes, which results in their different surface reactivity (Sohlberg et al. 1999). At the clean γ -Al2 O3 (110C) surface, all the oxygen atoms in the CrOx clusters are found to be located at the sites that match well with the oxygen sublattice of γ -Al2 O3 (Fig. 6.32(a)). In addition, the Cr atom binds to the surface oxygen atoms that occupy the site that corresponds to an octahedral site of bulk γ -Al2 O3 . Thus, Cr2 O3 particles can grow in a commensurate way on the γ -Al2 O3 surface. The η-alumina (110C) surface is markedly different from the corresponding γ -Al2 O3 surface in that it contains unsaturated three-coordinated Al atoms (Sohlberg et al. 2001; Rashkeev et al. 2003). The resulting configuration shows a mismatch of the oxygen sites of the CrO3 with the oxygen sublattice of the alumina substrate (Fig. 6.32(b)). It was therefore concluded that CrOx clusters on the η-alumina surface remain dispersed. Density-functional calculations were further used to explore the catalytic reaction of dehydrogenation of alkanes at the atomic level. As an example, Fig. 6.32(c) shows the key step of the dehydrogenation of a C2 H6 molecule at a CrO3 cluster supported on the η-Al2 O3 surface. When C2 H6 approaches the CrO3 cluster, two hydrogen atoms from the molecule get captured at the low-coordination oxygen sites and the remaining C2 H4 molecule is released. This process entails an energy gain of anywhere from 1.3 to 1.6 eV, depending on which O sites the hydrogens are attached to and on the value of x in CrOx . The key enabler for the process is the low coordination of the O

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Scanning transmission electron microscopy of nanostructures

atoms on the CrOx cluster. On the other hand, O atoms in the extended Cr2 O3 particles on the γ -Al2 O3 surface have saturated co-ordination spheres and cannot therefore be active in alkane dehydrogenation. The data thus indicates that the reconstruction on γ -alumina facilitates the formation of crystalline Cr2 O3 in registry with the substrate, while the reconstruction on η-alumina does not. The calculations further demonstrate that only the isolated CrOx clusters on the η-Al2 O3 surface are active in alkane dehydrogenation, with important implications for the longevity of catalysts.

6.5.5

CO oxidation by supported noble-metal nanoparticles

While gold (Au) in bulk form is not a very active catalyst, when prepared as supported nanoparticles it becomes one of the most active catalysts for several reactions including CO oxidation (Haruta et al. 1993). Low-temperature CO oxidation has some important practical applications and it is also interesting as a model reaction. Furthermore, the enhanced activity of nanosize Au is exciting because it contrasts with most other noble metals for this reaction. As a specific example, supported platinum (Pt) nanoparticles (Bamwenda et al. 1997) are less active than low-index Pt surfaces. There have been a wide variety of explanations for the high activity of small Au nanoparticles. First, it is apparent that as the Au nanoparticles become smaller, the number of particles per gram of Au will increase (scaling roughly as 1/d 3 for diameter d). The total surface area per gram will scale roughly as 1/d, while the perimeter (where a small reactant molecule can interact with both the support and the nanoparticle) will scale as 1/d 2 . Most authors therefore give a turnover frequency (a rate normalized by the number of sites) to help distinguish more exotic effects from this kind of simple scaling. Haruta and coworkers (Haruta 2004) have proposed that the perimeter must be important. In apparent contradiction of this perimeter model, Goodman and coworkers (Chen and Goodman 2004) have claimed that a bilayer structure, with effectively zero perimeter, is the most active structure and that negatively charged Au plays an important role. Landman and coworkers (Yoon et al. 2003) also found evidence to support the idea that the catalytic activity is related to negatively charged Au on MgO. However, Guzman and Gates (2004) have shown evidence that suggests the active site involves cationic Au. Norksov and coworkers (Lopez et al. 2004) have demonstrated that in calculations the coordination number appears key to the activity. Changes in coordination number seem to produce rather larger changes in the calculated reaction barriers than other factors (such as charge). In order to clarify this situation, we have examined an array of nanoparticles, prepared using both chemical (Yan et al. 2005) and physical (Veith et al. 2005) techniques and performed calculations to elucidate the reaction pathways and activation barriers. We constructed an ensemble of Au nanoparticles on TiO2 substrates and optimized their geometries using density-functional theory (Rashkeev et al. 2007). We found that at least one substrate O vacancy is needed to anchor

6.5 Nanocatalysts 235

Fig. 6.33 Adsorption of O2 and CO molecules on TiO2 -supported Au (a,b) and Pt (c,d) nanoparticles. Ti is shown as large, light gray sphers, O as smaller, dark spheres, Au in light gray, Pt in dark gray. In (b) and (d) a CO molecule is attached on the left of the nanocluster, with C bonding to the cluster, and an O2 molecule has been brought into contact with the surface. The relaxed configuration of the 11-atom supported Au nanoparticle ˚ (c,d) The corresponding figures for Pt before and after adsorption is shown in (a) and (b), where the O−O distance has increased to 1.39 A. ˚ Adapted from Rashkeev et al. (2007). nanoparticles, where the O−O distance is 1.48 A.

Au particles and previous work has shown that large Au particles are likely anchored by many such vacancies (Wahlstrom et al. 2003). To compare to other noble metals, in a subset of our ensemble of nanoparticles we replaced all Au atoms with Pt atoms and reoptimized geometries (Fig. 6.33). The adsorption of O2 and CO molecules on both types of nanoparticles was then optimized at various sites and the binding energies (or desorption energies) and reaction barriers were calculated. On virtually all Au particles, O2 adsorbs as a molecule. Adjacent O2 and CO molecules then react to produce CO2 , which desorbs, leaving a bonded O atom. On Pt surfaces and nanoparticles, O2 adsorbs strongly with a stretched O−O bond that dissociates easily, allowing adjacent O atoms and CO molecules to react. Figure 6.33 reveals a crucial difference between model Au and Pt nanoparticles. Upon adsorption, the neighboring Au−Au bonds are weakened, making the rotation and stretching needed for the CO + O2 reaction less energetically costly. Such weakening (akin to the structural fluxionality noted by Landmann and coworkers (Yoon et al. 2003)) does not occur at the corresponding Pt−Pt bonds. The Pt nanoparticles are more rigid, resulting in higher reaction barriers because it is harder for the strongly adsorbed molecules to move and interact. Figure 6.34 shows reaction barriers (E r ) from a nudged elastic band model for CO oxidation and O2 desorption energies (E d ). CO desorption energies were generally found to be larger and so are not shown. The most striking result for Au nanoparticles is the crossing of the E r and E d curves as a function

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Scanning transmission electron microscopy of nanostructures

Fig. 6.34 Reaction of O2 and CO molecules on TiO2 -supported Au (a,b) and Pt (c,d) nanoparticles. (a,c) The reaction energy profiles and schematics of the Au and Pt nanoparticles at the transition state (arrowed). (b,d) The desorption energy, E d , of an O2 molecule and the reaction barrier, E r , as a function of the average coordination number n of the two Au (or Pt) atoms to which the O2 molecule is attached. Points correspond to different adsorption sites and/or different nanoparticles located over oxygen vacancies in rutile (110) or anatase (101) surfaces. The average curves of E d (light gray triangles) and E r (dark gray triangles) are shown. The lines E d,p (light gray diamonds) and E r,p (black circles) correspond to the desorption energy and the reaction barrier for the bridge-bond perimeter sites of Au clusters. The Pt−Pt bonds are more rigid, resulting in a higher reaction barrier. Adapted from Rashkeev et al. (2007).

of the average co-ordination, n, of the two Au atoms to which O2 is bonded. Catalysis is favorable over desorption only at sites with average co-ordination n < 5, when E r < E d . At perimeter sites, O2 molecules bind more strongly and catalytic activity is favored even with n = 6 or 7 because the bridge bond to the substrate helps weaken the O−O bond. Reaction barriers on the perimeter also decrease with decreasing co-ordination. The behavior of Pt nanoparticles is distinctly different. Figure 6.30 shows that we always have E r < E d . In principle, catalysis is always favorable over desorption (for large nanoparticles, our E r values approach the known value at Pt surfaces ∼0.8 eV), but the absolute values become larger with decreasing co-ordination. In addition, perimeter sites for Pt particles were not catalytically active in the simulation as O2 molecules preferred other nanoparticle sites.

6.6 Magnetism in gold and silver nanoclusters

237

Fig. 6.35 High-magnification Z-contrast micrographs showing 10% wt-loaded Au on anatase after two stages of preparation. (a) In the precursor state following depositionprecipitation of Au, individual Au atoms are sharply resolved, suggesting stabilized nanoparticles. (b) In the most active form, after mild reduction in 12%-H2 at 423 K, individual Au atoms are not resolved, suggesting large structural fluxionality. Reproduced from Rashkeev et al. (2007).

The larger binding energies of CO and O2 to Pt nanoparticles relative to flat surfaces, imply longer residence times, effectively blocking active sites. Thus, from calculations it appears that coordination number is indeed the driving factor behind the change in activity with particle size in both Au and Pt nanoparticles. Images of supported Au nanoparticles provide some tantalizing evidence that the structural fluxionality (Yoon et al. 2003), necessary for high activity, occurs. Figure 6.35 shows atomic-resolution Z-contrast micrographs of assynthesized and reduced nanoparticles, prepared as part of an experimental investigation of CO oxidation by Au nanoparticles on TiO2 (Yan et al. 2005). In Fig. 6.35(a) Au atoms are sharply resolved, whereas in Fig. 6.35(b) Au nanoparticles appear blurred. The difference can be attributed to Au hydroxide present on the as-prepared samples, stabilizing the nanoparticle. The fuzziness of Fig. 6.35(b) provides evidence for large Au-atom motions, consistent with dynamic structural fluxionality and the low melting point of Au nanoparticles in this size range (Buffat and Borel 1976).

6.6

Magnetism in gold and silver nanoclusters

New and surprising mechanical, chemical, electronic and magnetic properties often arise when the size of a material is reduced to the nanoscale. The previous section showed, for example, that although gold is a noble metal and chemically inert in the bulk form, it becomes an effective catalyst on the nanoscale. The d orbitals of a gold atom are completely filled, and bulk gold shows diamagnetism. Recently, however, intriguing magnetic properties have been discovered in gold nanoparticles. Hori et al. (Hori et al. 1999; Yamamoto et al. 2004; Yamamoto and Hori 2006) reported ferromagnetism in gold nanoparticles protected by polymers and the presence of transitionmetal magnetic impurity was ruled out. Crespo et al. (2004) reported that gold nanoparticles stabilized by means of a surfactant are diamagnetic, similar to bulk gold. Because the interaction between the gold nanoparticles and the capping materials is believed to be weak, it becomes an interesting question

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Scanning transmission electron microscopy of nanostructures

whether “bare” gold nanoparticles can be ferromagnetic even though the d orbitals are completely full. First-principles density-functional theory (Hohenberg and Kohn 1964; Kohn and Sham 1965; Kresse and Furthmuller 1996) was used to study the electronic and magnetic properties of “bare” gold and silver nanoclusters (Luo et al. 2007b). Based on low-energy structures determined by Haberlen et al. (1997), gold clusters with Ih and Oh symmetry were considered. Three icosahedral (Ih ) clusters were investigated, Au13 , Au55 , and Au147 , consisting of one central atom and one, two, and three full shells of atoms, respectively. Six clusters with Oh symmetry are also considered: Au6 , Au13 , Au19 , Au38 , Au44 , and Au55 . The Oh clusters have the same local structure as the bulk face-centered cubic gold crystal. The Fermi level in bulk gold metal cuts through the broad 6s band, and the density-of-states at the Fermi energy is quite small. Therefore, bulk gold is non-magnetic (weakly diamagnetic due to core electrons). In contrast, the ground-state electronic structures of these high-symmetry clusters are found to be spin polarized. The highest-occupied molecular orbital (HOMO) is mostly made up of gold 6s states and is highly degenerate and partially filled. The whole cluster behaves like an isolated transition-metal atom such as Mn. Spin alignment within the cluster HOMO levels occurs, driven by Hund’s rule in the “superatom”. The spin density of the icosahedral Au55 cluster is shown as a contour plot in Fig. 6.36. Spin polarization occurs primarily on the outershell atoms. The mechanism of ferromagnetism due to Hund’s rule in a “superatom” can be explained more clearly by examining the electronic energy levels of the icosahedral Au13 cluster. When spin-polarization is not allowed, the HOMO level has a 5-fold degeneracy (for each spin), just like the d orbitals in an isolated atom. The HOMO level is half-occupied because only five electrons are available for both spins, as shown in Fig. 6.37(a). Because of the degenerate nature of the HOMO level, the total energy of the cluster can be lowered by even a weak exchange splitting, resulting in a

Fig. 6.36 Spin-density contour plot (unit: e/au3 ) of an icosahedral Au55 cluster. Adapted from Luo et al. (2007b).

6.6 Magnetism in gold and silver nanoclusters

239

Fig. 6.37 Degeneracy of energy levels in icosahedral Au13 cluster (a) no spinpolarization, 5-fold degenerate HOMO level is half-filled; spin-polarization allowed, (b) majority spin, 5-fold HOMO level, and (c) minority spin, 5-fold LUMO level. Adapted from Luo et al. (2007b).

spin-polarized ground state. This is exactly what happens in the icosahedral Au13 cluster when spin-polarization is allowed: the majority-spin HOMO level is filled with five electrons, while the corresponding minority-spin state is completely empty, thus becoming the lowest-unoccupied molecular orbital (LUMO), as shown in Figs. 6.37(b) and (c). The ground-state magnetic moment of icosahedral Au13 cluster is thus 5 μB . This is similar to the highspin state of an isolated Mn atom, with parallel spins among the five 3d electrons. The ground-state magnetic moments of different gold clusters have been determined from spin-polarized DFT calculations, and are listed in Table 6.1. The spin-polarization energies (E), defined as the total energy difference of each cluster between the spin-polarized and non-spin-polarized states, are Table 6.1 Ground-state magnetic moments (μ) of gold clusters and their spin-polarization energies (E) Clusters

μ(μB )

Au6 Oh Au13 Ih Au13 Oh Au19 Oh Au38 Oh Au44 Oh Au55 Ih Au55 Oh Au147 Ih

2 5 1 1 4 2 3 1 1

E (meV) −126.1 −417.1 −17.3 −19.5 −46.7 −23.2 −35.6 −4.5 −1.4

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Scanning transmission electron microscopy of nanostructures

Fig. 6.38 Hund’s exchange coupling strength of Au clusters as a function of their sizes, adapted from Luo et al. (2007b).

also shown. The magnetic moments depend sensitively on the symmetry of the clusters: for Au13 and Au55 clusters, the Ih clusters have higher degeneracy in the HOMO, thus larger magnetic moments than their corresponding Oh partners. We can extract the Hund’s exchange coupling strength J of each gold cluster from the spin-polarization energy (E) and the number of unpaired spins (n) by the following relation: E = −1/4 J n 2 . The extracted Hund’s exchange coupling strength is plotted in Fig. 6.38 as a function of the cluster size. The spin-exchange energy originates from Coulomb energies of the HOMO electrons with different spin configurations. It decreases rapidly as the size of the cluster increases, as expected. Silver clusters with the same structures have also been studied. Because silver is isoelectronic to gold, the magnetic properties of the silver clusters are very similar to gold clusters. In particular, their magnetic moments are the same as the corresponding gold clusters.

6.7

Charge ordering in manganites

Complex oxides such as manganites show a vast range of physical properties ranging from ferromagnetism, high-temperature superconductivity, colossal magnetoresistance to ferroelectricity. In addition, their properties can be further tuned through growth of superlattices or heterostructures. Key to the tuning of desired properties is the ability to maintain a mixed-valence state within the Mn sublattice through suitable doping. The structure is based on the perovskite lattice, Ax B1−x MnO3 , where A is a trivalent cation (La, Nd, Bi, Pr) and B a divalent cation (Sr, Ca, Ba), but typically will show reduced symmetry and the unit cell will be a multiple of the basic perovskite unit cell. For x = 0 the Mn would be in a +4 formal oxidation state, with a 3d3 electronic

6.7 Charge ordering in manganites

241

3 e 0 ), while for x = 1 the Mn would take a +3 formal oxidaconfiguration (t2g g 3 e 1 ). A fraction 1 − x of Mn ions per tion state with a 3d4 configuration (t2g g unit cell take the +4 oxidation state, while the rest take a +3 oxidation state. For some x values, these inequivalent Mn species can form ordered arrays at low temperatures. Evidence of this has come from the observation of superlattice reflections in diffraction that correlate with x, for example doubling or tripling of the Pnma unit cell of the parent CaMnO3 compound along the [100] or [101] directions for x = 0.5 or 0.33 in the Lax Ca1−x MnO3 system. Such observations imply a stable spatial ordering of the eg electrons, leading to the notion of “charge ordering” in these materials. However, the superstructures are accompanied by tilts and distortions of the oxygen octahedra (such as Jahn– Teller distortions) so that it was not at all obvious to what extent the superlattice spots represent structural changes, changes in electronic configuration or real charge disproportionation between inequivalent Mn sites. Figure 6.39(a) shows a Z-contrast image of a Bix Ca1−x MnO3 (BCMO) manganite along the pseudocubic projection with x = 0.37, which has an ordering temperature above room temperature. It is quite clear that the bright spots (representing Bi atoms within the Ca sublattice) are not showing any appreciable order. However, atomic-resolution EELS shows that the electronic structure is, in fact, ordered. EELS is a particularly powerful probe of the electronic configuration of first-row transition-metal oxides because the ratio

(a)

(b)

10 Å

Intensity

4

3 2

(c)

1

2.7 520

2.4

560

600 E loss (eV)

640

680

2.1 0

10

20

30

40

Fig. 6.39 (a) Z -contrast image of BCMO showing Bi dopant atoms as bright spots. (b) Electron-energy loss spectra typical of a Mn+3 O column (2) and a Mn+4 O column (3). Spectrum (4) comes from CaMnO3 , while (1) corresponds to bulk LaMnO3 . The oxygen K edge (around 530 eV) and the Mn L edge (around 644 eV) are shown. (c) L 2,3 ratio along the direction marked with an arrow in (a). Horizontal lines represent the L 2,3 ratio expected for a pure Mn+3 oxidation state (top) and a pure Mn+4 oxidation state (bottom). The positions of Mn planes along the scan have been marked with arrows. Reproduced from Varela et al. (2005).

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Scanning transmission electron microscopy of nanostructures

of the intensity of the L3 peak to the L2 peak, the L2,3 intensity ratio, correlates with the 3d-band occupation, i.e. the formal oxidation state (Leapman and Grunes 1980; Leapman et al. 1982; Rask et al. 1987; Krivanek and Paterson 1990; Paterson and Krivanek 1990; Kurata and Colliex 1993; Pearson et al. 1993). Figure 6.39(b) compares EELS spectra from the parent compounds CaMnO3 and LaMnO3 , with Mn in +4 and +3 oxidation states, respectively, to spectra obtained from selected columns in the BCMO. Figure 6.39(c) shows a line scan of the L2,3 intensity ratio along the 100 pseudocubic direction where the Mn oxidation state jumps from +4 to +3 along the directions of the arrow. The jumps occur every three or four Mn planes, consistent with the nominal doping level (Varela et al. 2005). Although the dopant is randomly distributed, the additional electrons have formed into an ordered stripe geometry of occupied eg orbitals on Mn sites, the periodicity of which is consistent with the macroscopically averaged periodicity observed in electron diffraction. Examination of the orbital occupation and total charge densities revealed clearly that there is primarily just orbital occupation ordering. The projected density of states on the two non-equivalent Mn sites is seen to be different, as shown in Fig. 6.40(a), and the calculated Mn 3d majority spin occupation and magnetic moment were also different, comparable although not exactly identical to that found in the parent LaMnO3 and CaMnO3 compounds with formal oxidation states of +3 and +4, respectively. In Fig. 6.40(b) the integrated charge density is compared around Mn atoms in

Fig. 6.40 (a) Projected density of states (PDOS) for the Mn 3d orbital of the two inequivalent Mn atoms [Mn(1) and Mn(2)] in 50%-doped CMO and the corresponding 3d PDOS in Mn metal. The Fermi energy is at E F = 0. (b) The integrated valence electron density within a sphere around Mn atoms in Mn metal and several Mn oxides as a function of the radius of the integration sphere. Reproduced from Luo et al. (2007a).

6.8 Summary 243

several oxides and in Mn metal and found to be practically indistinguishable. The charge associated with the additional eg electron is compensated by the outward relaxation of the two neighboring oxygen atoms, the Jahn–Teller distortion, which ensures the system remains essentially neutral. Thus, the charge ordering in this and most likely all perovskite-based manganites is really orbital-occupation ordering, the different patterns reflecting different 3D spatial arrangements of the Jahn–Teller distortions.

6.8

Summary

The revolution in nanoscience and that in electron microscopy, both predicted by Feynman almost a half century ago, are happening now, and are truly synergistic. The electron microscope is uniquely suited to the study of individual nanostructures, allowing differentiation of different structures and properties that is difficult or impossible to do with techniques that provide a spatial average. With the present generation of aberration correctors, which correct all aberrations up to 3rd order, it is possible to obtain sufficient sensitivity to image and spectroscopically analyze single atoms. In addition, such instruments provide sufficient depth sensitivity to locate individual atoms in three dimensions, ˚ with sub-Angstrom resolution laterally and a few nanometers resolution in ˚ depth, but with depth precision (for well-separated atoms) at the Angstrom level. The next generation of aberration correctors is also beginning to appear, correcting all geometric aberrations up to 5th order. Initial results with the 5thorder CEOS DCOR aberration corrector (M¨uller et al. 2006) in the ORNL FEI Titan 80-300 with a prototype high-brightness Schottky source have been presented above. At the time of writing, a Nion UltraSTEM was due to be delivered to ORNL, equipped with a Nion 5th-order aberration corrector and cold field-emission gun (Krivanek et al. 2008). These instruments should bring another jump in resolution, to the level of ˚ laterally and around 1 nm in depth, with a concomitant increase in single0.5 A atom sensitivity and depth resolution. Similarly, it will become possible to perform 3D EELS (D’Alfonso et al. 2007), to probe single impurity atoms at grain boundaries and dislocation cores, to link their electronic environment to macroscopic mechanical and electronic properties in a rigorous manner. With these new eyes it will become possible to see the ultimate atomic origins of materials properties with new clarity, whether it be for structural materials or electronic materials, for nanoscience or catalysis. We can also expect to see new probes appearing to measure properties of individual nanostructures in situ inside the STEM. For example, a cathodoluminescence detector would distinguish nanowires that were efficient light emitters from those that were not, for example because of the presence of a dislocation or defect (Pennycook 2008). Similarly, incorporation of a scanning tunnelling microscope inside the STEM would provide yet more insights into nanostructure functionality through local measure of electronic properties that could be correlated with EELS measurements, and also through the ability to applying bias to examine the nanostructure closer to its operating condition.

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There has perhaps never been a more exciting time to be involved in electron microscopy, since the human race crosses atomic resolution only once in its history, and that time is right now. Forevermore, we will be able to see atoms with clear vision.

Acknowledgments The authors would like to thank their collaborators in the work reviewed here, P. D. Nellist, O. L. Krivanek, N. Dellby, M. F. Murfitt, Z. S. Szilagyi, Y. Peng, H. M. Christen, W. Tian, R. Jin, B. Sales, D. G. Mandrus, T. Ben, D. L. Sales, J. Pizarro, P. L. Galindo, D. Fuster, Y. Gonzalez, L. Gonzalez, S. W. Wang, M. V. Glazoff, S. N. Rashkeev, S. H. Overbury, G. M. Veith, W. H. Sides and J. T. Luck, which was supported by the Division of Materials Sciences and Engineering, USDOE, in part by the Laboratory Directed Research and Development Program of ORNL, and by appointments (KvB, AYB, MPO) to the ORNL Postdoctoral Research Program administered jointly by ORNL and ORISE. Some of the instrumentation used in this research was provided as part of the TEAM project, funded by the Division of Scientific User Facilties, Office of Science, U.S. Department of Energy.

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Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications J.D. Taylor, B. Elliott, D. Dickel, G. Keskar, J. Gaillard, M.J. Skove, and A.M. Rao

7.1

Introduction

Micro- and nanocantilevers have the potential to revolutionize physical, chemical, and biological sensing. Their exceptionally small size allows for unprecedented sensitivities, improved dynamic performance and reliability, and low power consumption. Microcantilevers in particular are easily integrated into standard high-volume silicon manufacturing processes, making them relatively inexpensive and mass-producible (Beeby et al. 2004). They are sensitive to a variety of environmental parameters including temperature, pressure, humidity, and infra-red radiation (Sepaniak et al. 2002), and they can be made to respond selectively to specific chemical and biological species by means of functionalized surface treatments (Hierlemann 2005). These features make micro- and nanocantilevers ideal candidates for a wide variety of sensing applications and attractive alternatives to traditional sensing technologies. However, in order to fully realize cantilever-based sensing in micro- and nanoelectro-mechanical systems (MEMS and NEMS), an accurate and scalable detection method is required. This detection method must be capable of measuring changes in either the static or dynamic response that result from changes in environmental parameters. This chapter investigates a fully electrical (actuation and detection) scheme known as the harmonic detection of resonance (HDR) that meets these requirements and provides several unique advantages not present in other detection techniques.

7 7.1 Introduction

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7.2 Mechanical vs. electrical responses

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7.3 Analytic modelling

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7.5 Cantilevered multiwall carbon nanotubes (MWCNT)

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7.6 Conclusion

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Acknowledgments

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References

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Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

Microcantilevers are arguably the simplest microscale structures and certainly one of the most versatile. They may be considered as basic building blocks for more complicated microsystems (Lavrik et al. 2004). The singly clamped cantilever (“diving-board”) geometry will respond to much less force than other microstructures and is thus better suited for sensing applications in which deflections must be measured; however, doubly clamped (“bridge”) structures have on many occasions been used and are in general easier to manufacture (Howe and Muller 1986). Typical microcantilevers, like those used throughout this chapter, are singly clamped structures made of silicon using lithographic and surface micromachining processes. Nanocantilevers, however, are most commonly nanotubes or nanowires grown using chemical vapor deposition (CVD) (Gaillard et al. 2005). In addition to sensing, micro- and nanocantilevers have been used as both actuators (Minett et al. 2002) and microgenerators (Fang et al. 2006). In general, as the size of a sensor is reduced, sensitivity improves. It is therefore advantageous to investigate nanoscale sensors. Many measured environmental parameters depend on the surface to volume ratio, which is roughly a thousand times greater for nanoscale than for microscale structures. Also, intrinsic damping is generally less in nanostructures because of fewer defects. A full understanding of microcantilever-based sensing is necessary in order to extend this technology to the nanoscale. At the nanoscale the sensitivity to environmental parameters is usually much improved; however, the signal to background noise is generally much lower and the number of detection schemes applicable to the nanoscale is limited. For these reasons, this chapter focuses primarily on microcantilever-based sensing. After showing experiments and analysis at the microscale, we display initial results at the nanoscale.

7.1.1

Transduction mechanisms

There are several basic transduction mechanisms applicable to microand nanocantilever-based systems. These can broadly be classified as adsorbed/absorbed mass, induced stress, changes in pressure/damping, piezoeffects, and applied mechanical and electromagnetic forces. Systems may utilize these mechanisms for sensing (sensed parameter → mechanical response), detection (mechanical response → output signal), or actuation (input signal → mechanical response). Detection involves converting either the static deflection or the dynamic response of the cantilever into a useful output signal, usually electrical in nature. The dynamic response may include shifts in natural resonance frequency, changes in vibrational amplitude and phase, or changes in quality factor (Q-factor). 7.1.1.1 Adsorbed/absorbed mass Substances that adsorb on the surface or absorb into the bulk of a microcantilever increase its effective mass and decrease its resonance frequency. The sensitivity, S, to changes in mass of a microcantilever sensor with resonance

7.1 Introduction

frequency, ω0 , and effective mass, m, is given in eqn (7.1) assuming constant bending stiffness. From this expression it is obvious that structures with low effective mass and relatively high natural frequencies, e.g. micro- and nanocantilevers, are best suited to mass sensing because they offer higher sensitivities. Recently, adsorbed molecular masses as low as a few zeptograms (10−21 g) have been observed using nanocantilevers (Yang et al. 2006),   1/2 d k m ω −ω0 S=  = . (7.1) m dm 2m 7.1.1.2 Induced stresses Several mechanisms exist that can induce differential stresses in asymmetrically coated microcantilevers that may result in either static deflection (bending) of the cantilever or shifts in its resonance frequency. The effects of induced surface stresses are especially significant in micro- and nanostructures due to their large surface area to volume ratio. 7.1.1.2.1 Molecular adsorption/interfacial chemical reactions Cantilevers that have been coated on one side with a thin chemically selective receptor layer will bend as molecules adsorb on the surface. This adsorption may be of the low-energy Van der Waals type (physisorption) or the higher-energy covalent type (chemisorption). Spontaneous molecular adsorption causes a reduction of interfacial free energy and surface stress and a concomitant expansion of the material (Lavrik et al. 2004). The resulting stress gradient causes a static deformation of the cantilever. The adsorption process also tends to stiffen the cantilever, thereby increasing its resonance frequency, as opposed to mass loading, which lowers the frequency. A wide variety of highly selective chemical sensors have been developed utilizing these phenomena (Hierlemann 2005). 7.1.1.2.2 Analyte-induced expansion Cantilevers coated with a relatively thick analyte permeable receptor layer may bend due to analyte-induced swelling. Molecules may absorb into the bulk of the coating thereby changing either the internal stress or pressure depending on whether the coating is solid or gel-like (Sepaniak et al. 2002). This effect has been employed to measure humidity using polymeric hydrogel coatings (Lao et al. 2007). 7.1.1.2.3 Thermally induced stresses/calorimetry Thermally induced stresses arise due to unequal coefficients of thermal expansion in layered cantilevers. Typically, the cantilevers are coated with a thin metallic layer, e.g., gold. This mechanism is commonly referred to as the “bimetallic effect” and is frequently employed in home thermostats. The heat producing the thermal stresses may arise from several sources including embedded resistors (Hierlemann 2005) or IR radiation. Also, microcantilevers can be used as microcalorimeters to detect the heat produced during molecular adsorption or during subsequent associated exothermal reactions (Sepaniak et al. 2002). Typically, thermal actuation requires significantly more power than other transduction methods.

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7.1.1.2.4 Optical radiation Electromagnetic radiation of a variety of wavelengths from approximately 1 nm to 1 μm (UV–visible–near IR) gives rise to mechanical strains in microcantilevers, though the greatest deflections occur at IR frequencies. This can be attributed to both radiative heating and the generation of photoinduced free charge carriers. For silicon, these effects act in opposite directions (Datskos et al. 1998). 7.1.1.3 Pressure/damping Changes in the ambient pressure may affect the damping experienced by a microcantilever. Pressure changes affect the vibrational amplitude and Q-factor of the microcantilever resonator and thus must be measured dynamically. There are three basic pressure regimes: intrinsic, molecular, and viscous (see Section 7.4.2). Depending on the pressure regime, various physical parameters may be determined, such as defect density of the oscillator (Sullivan et al. 2007), molecular mass and gas composition (Weigert et al. 1996), or viscosity of the surrounding medium (Agoston et al. 2005; Quist et al. 2006). Variations in pressure due to acoustic waves can also be detected (Degertekin et al. 2001). 7.1.1.4 Piezo-electric/piezo-resistance Piezo-electric materials will generate a mechanical strain when subjected to an applied electric field. This phenomenon is extensively applied to actuate microcantilevers, notably in atomic force microscopes. Silicon is not intrinsically piezo-electric; therefore, a piezo-electric layer, e.g., lead zirconium titanate (PZT), must be deposited in postprocessing. This leads to a more complicated and costly production process as compared to capacitive designs. Piezo-electric materials also generate a voltage when mechanically strained. Therefore, piezo-electric coated microcantilevers may be used to detect both static and dynamic deflections arising from any of the other transduction mechanisms. Piezo-resistance is the change in resistivity of a material with applied stress. It is commonly used to detect the deflection of microcantilevers. Silicon is intrinsically piezo-resistive and this property can be enhanced by doping; thus piezo-resistive detection is highly compatible with standard CMOS processes. Piezo-resistive elements are typically placed at the base of the cantilever where the stresses from bending are greatest and are usually arranged in a Wheatstone-bridge configuration in order to negate common mode effects such as thermal variations (Hierlemann 2005). 7.1.1.5 Applied forces A variety of externally applied forces are capable of generating static and dynamic deflections in micro- and nanocantilevers. These forces may be either mechanical (applied directly or through inertial loading) or electromagnetic in origin. MEMS- and NEMS-based mechanical sensors offer the potential for improved sensitivities, lower power consumption, and wider bandwidths than conventional resistance strain gauges (Beeby et al. 2004).

7.1 Introduction

7.1.1.5.1 Mechanical force/torque Micro- and nanocantilevers are extremely sensitive to mechanically applied forces. Pico-Newton forces are routinely measured and even higher sensitivities have been achieved by cooling the cantilevers to millikelvin temperatures (Mamin and Rugar 2001). This mechanism forms the basis of contact mode atomic force microscopy (AFM), in which the surface topography of a sample is determined by scanning a microcantilever in the x y-plane and detecting its static deflection in the z-direction in response to surface contours (Binnig et al. 1986). Resonant strain gauges have also been developed using doubly clamped microbeams in which externally applied tensile strains stiffen the beam and increase its resonant frequency (Yan et al. 2003). 7.1.1.5.2 Gravitational/inertial Micromechanical accelerometers and gyroscopes, which measure linear and angular acceleration, respectively, are some of the most widespread MEMS devices. They are used in applications ranging from airbag-release systems to military inertial guidance (Beeby et al. 2004). These sensors function by measuring the static or dynamic deflection of a proof mass attached to a compliant support such as a microcantilever. The cantilever deflects in order to counteract the inertial loading of the proof mass due to the base acceleration. Accelerometers have also been developed based on resonant silicon structures that are more immune to environmental noise and better suited to sensing dynamic accelerations (Burns et al. 1996). 7.1.1.5.3 Electrostatic The electrostatic transduction mechanism is based on Coulomb’s law from which it follows that two oppositely charged elements will experience an attractive force. This mechanism is very common because it can be used for both actuation and detection and is quite straightforward to fabricate. If the elements can be modelled as a parallel-plate capacitor, the electrostatic force, FE , is given by eqn (7.2) (see Section 7.3.3). FE =

ε AV 2 , 2d2

(7.2)

where ε is the permittivity of the medium separating the electrodes, A is the plate area, V is the applied voltage, and d is the separation distance. The electrostatic force is a non-linear function of the separation distance and voltage. This non-linearity is essential to the HDR method; however, it is not always desirable, as in the case of non-contact mode AFM, in which a microcantilever is vibrated above the surface of a sample and shifts in its resonance due to variations in the electrostatic force that are measured. In these cases, feedback mechanisms are usually employed to keep the response sufficiently linear. 7.1.1.5.4 Magnetic A current-carrying element placed in a magnetic field experiences a Lorentz force in a direction perpendicular to both the current and magnetic field. This mechanism is the basis for magnetic force microscopy (MFM) and scanning Hall probe microscopy (SHPM) (Beeby et al. 2004). Also, a magnetic

253

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Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

microactuator has been developed that utilizes an electroplated permalloy that possesses a high magnetic permeability (Judy et al. 1995); however, because there are a limited number of magnetic materials compatible with current micromanufacturing processes and only planar coils are possible, it is very difficult to generate magnetic fields on chip, and thus magnetic transduction’s applicability to MEMS and NEMS has been somewhat limited.

7.1.2

Detection methods

Several detection schemes have been proposed to measure the static and/or dynamic response of micro- and nanocantilevers. The most common is laser reflectometry, in which a low-power laser is reflected off the cantilever and measured with a position-sensitive photodetector. This method is employed successfully in almost all AFMs. Other detection methods include various forms of microscopy (optical, scanning electron, or transmission electron), piezo-resistive or piezo-electric, interferometry, and diffraction methods. However, all of these detection schemes require complicated electronics that take up significant space and power and are not possible to integrate on a single chip. They are consequently not scalable to the micro- and nanoscale. For these purposes, the standard detection methods mentioned above are not suitable.

7.1.3

Electrostatic actuation and capacitive detection

Electrostatic actuation and capacitive detection presents an alternative that does meet these scaling criteria. In this method, a potential difference is applied between a conductive microcantilever and counterelectrode resulting in an attractive electro(quasi)static force. In response to this force, the cantilever deflects, and the capacitance of the arrangement varies, causing charge to move on and off the cantilever. If this charge or current can be measured, the mechanical vibration of the microcantilever can be deduced. However, electrostatic actuation and capacitive detection have traditionally proven difficult to implement. This difficulty can in large part be attributed to a parasitic signal that obscures the dynamic signal from the cantilever. This parasitic signal includes both the static capacitance of the microcantilever and counterelectrode and all the stray capacitance of nearby circuit elements. Several methods have been proposed to enhance the dynamic capacitance or lower the parasitic capacitance of the system, including single-electron transistors (Blencowe and Wybourne 2000), and comb drives (Bertz et al. 2001; Zhu 2008). Also, since the ratio of dynamic to parasitic signal depends on the ratio of cantilever deflection to total gap distance, some groups have attempted to minimize the parasitic effects by manufacturing cantilevers that are extremely close to the counterelectrodes. Each of these solutions increases the complexity, cost of production, and potential for malfunction.

7.1.4

Harmonic detection of resonance (HDR)

We have developed a capacitive detection method known as HDR that avoids the parasitic capacitance without significantly increasing the complexity of the

7.2 Mechanical vs. electrical responses 255

device. The electrical signal from a microcantilever when driven by a nearby counterelectrode has a “rich harmonic structure” (Gaillard et al. 2006), which can be attributed to non-linearities in the electrostatic force and mixing of the mechanical and electrical signals. The higher harmonic signals, at integer multiples of the driving frequency, do not suffer from significant parasitic effects. Consequently, by measuring the dynamic response of microcantilevers at these harmonic frequencies, significantly higher signal to background ratios (SBR) and Q-factors can be obtained, resulting in much improved sensitivity in HDR-based sensing devices. HDR presents several advantages over other detection schemes. It is an entirely electrical actuation and detection scheme, and consequently, it is directly scalable to micro- and nanodevices with straightforward integration into standard microlithographic processes. This allows for portable HDRbased sensing devices. HDR is also extremely simple. It requires no complicated components, such as lasers, magnets or piezo-electric elements, thereby reducing cost and potential for failure. HDR does require circuitry to detect the higher harmonics, but this should be possible to realize on a single chip. Finally, the gap distances in HDR can be relatively large, increasing available working distances and voltages and facilitating alignment. In the following sections of this chapter, a theoretical model is developed that explains many of the unique features of HDR, including the origin of the harmonics and superharmonic resonances and the conspicuous absence of the parasitic capacitance in the higher harmonics. Experimental results from HDR and corresponding mechanical measurements from an AFM are presented to differentiate between mechanical and electrical responses. Simulations are provided based on the theory that accurately reproduces the experimental results. An examination of non-linearities and Duffing behavior in microcantilevers is provided and their use in ultrasensitive sensors is discussed. Applications of HDR to gas and pressure sensing are then presented. Finally, cantilevered multiwall carbon nanotube (MWCNT) results are presented, demonstrating that HDR is not limited to the microscale.

7.2

Mechanical vs. electrical responses

There are distinct differences between the mechanical and electrical responses of an electrostatically actuated microcantilever. The mechanical response of a cantilever is simply its physical deflection as seen under a microscope, Fig. 7.6, or measured using AFM-based laser reflectometry. Like all oscillators, the mechanical response depends on both the amplitude and frequency of the applied force. In this chapter, the mechanical response is characterized by the tip deflection of the cantilever, z(t). As the cantilever deflects, charge moves on and off due to the variable capacitance. This current, which we call the electrical response, is measured in HDR. The current depends on the capacitance of the system that has the mechanical deflection as a parameter. In general, the electrical response exhibits more features than the mechanical response, such as parasitic capacitance and superharmonic resonances.

256

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

Photosensitive Position Detector

La

se

r

L Microcantilever z=d+z(t) x do Vdc

d

Counterelectrode

Vac

Lock-In Amplifier

Amplitude

Reference

2 34 1 5

Harmonic

Phase

A250

Input

Output

Fig. 7.1 A schematic of the AFM and HDR experiment.

Computer GPIB

7.2.1

Experimental apparatus

In order to examine the differences between the mechanical and electrical responses and understand the unique advantages of HDR as a sensing technology, we describe an experiment in which the mechanical motion of a cantilever and the electrical response are measured simultaneously using standard AFMbased laser reflectometry and the HDR technique, respectively. In this experiment, a cantilever is manipulated over an optical dark-field microscope. By cantilever, we mean either a silicon microcantilever or a cantilevered carbon nanotube. In most cases, this allows for simple positioning of the cantilever near the counterelectrode without the need for timeconsuming lithographic processes. The cantilever is placed parallel to and within 1–10 μm from the counterelectrode depending on its dimensions. An electrostatic force is generated by applying an ac voltage, Vac , with a dc offset, Vdc . This experiment was performed under ambient conditions, demonstrating that HDR does not require any elaborate apparatus to control temperature or pressure. The HDR system consists of an A250 pre-amplifier, a voltage oscillator, a dc power supply, and a lock-in amplifier, Fig. 7.1. It is useful to employ a Faraday cage that surrounds the metal contact for the counterelectrode and extends around the probe tip leaving about 2 mm of the tip exposed. This minimizes cross-talk between the metal contacts that hold the cantilever and the counterelectrode and helps to increase the signal-to-background ratio (SBR). These noise considerations are crucial when working at the nanoscale. The lock-in amplifier detects the output of the A250, which is proportional to the current, at a harmonic (integer multiple) of the oscillator driving frequency, Ω. As will be discussed later, harmonics of the applied ac voltage are essential to the HDR method.

7.2 Mechanical vs. electrical responses 257

It is worthwhile to briefly describe the operation of lock-in amplifiers since they are such an integral component of the HDR detection system. Lock-in amplifiers are electronic instruments capable of extracting extremely small signals of known frequency from otherwise noisy signals. For this reason they are ideally suited to measuring the higher-harmonic components of the cantilever electrical response that can be many orders of magnitude smaller than the noisy first harmonic. Lock-in amplifiers operate based on heterodyne-detection principles, in which a reference frequency signal is mixed (multiplied) with the input signal, V0 cos(Ωt), eqn (7.3). The result is a signal with a dc component that is proportional to the amplitude of the input signal, V0 , at the reference frequency and a component at twice the reference frequency. The dc component is isolated using a low-pass filter (integrator) with a time constant chosen such that the 2ω signal is strongly attenuated (Scofield 1994). Vmix = Vin (t) · Vref (t) = V0 cos( t) cos( t) =

1 V0 [1 + cos(2 t)] . (7.3) 2

The outputs of typical lock-in amplifiers are the inphase and quadrature (90◦ out of phase) components of the input signal at the reference frequency, from which the overall amplitude and phase shift of the measured signal can be determined. Most lock-in amplifiers multiply by a square-wave reference signal, which includes many higher harmonics. However, in HDR a “digital” lock-in is used (Stanford Research Systems Model SR830) that multiplies by a pure sinusoid, thereby providing more accurate harmonic measurements. The microcantilever assembly is then placed inside an atomic force microscope (Veeco CPII). The mechanical deflection of the cantilever can then be directly measured using the laser and photosensitive position detectors of the AFM. A lock-in is used in this case to separate out the harmonics of the mechanical signal from the photodiode.

7.2.2

Resonance

The standard steady-state mechanical response (amplitude and phase) of a single degree of freedom (SDOF) forced damped harmonic oscillator is given in eqn (7.2.2) (Pippard 1978). Of course at the resonance frequency, ω0 , the amplitude peaks and the phase difference changes by 180◦ . A=

F0 /m  1/ 2 2 ω02 − 2 + (2γ ω0 )2   −1 −2γ ω0 φ = tan , ω02 − 2

(7.4)

where A and φ are the amplitude and phase shift of the steady-state displacement. F0 and Ω are the magnitude and angular frequency of the applied force, and m, ω0 , and γ are the mass, resonance frequency, and dimensionless damping ratio of the oscillator, respectively.

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

1.6 0.3

60 0.2

Phase (Degrees)

40

Fig. 7.2 The amplitude and phase measured by our HDR system in the second harmonic of the current signal at ω0 for a silicon cantilever 110 μm long, 35 μm wide, and 2 μm thick. The inset shows the downshift in resonance frequency as the ac voltage is varied from 3 V (bottom) to 5 V (top) in 0.5-V increments. This shift is caused by the decrease in the effective spring constant as the ac voltage is increased. The black dotted line is a guide to the eye (Gaillard et al. 2006). Copyright American Institute of Physics.

0.1

20 0

Amplitude (mV)

80

237 238 239 240 Frequency (kHz)

1.4 1.2 1.0 0.8

–20

0.6

–40

0.4

–60 –80 232

Ampl. Phase

Amplitude (mV)

258

0.2

234 236 238 240 242 244 246 248 250 Frequency (kHz)

The micro-and nanocantilevers discussed in this chapter exhibit more complicated resonance phenomena than the SDOF oscillator described above. Specifically, the cantilevers possess several resonance peaks due to both higher modes of vibration and higher harmonics in the non-linear electrostatic driving force. The origin and nature of these peaks will be discussed thoroughly in the analysis section of this chapter.

7.2.3

Mechanical and electrical response spectra

The microcantilever exhibits a variety of resonance peaks, which are evident when the amplitudes of the mechanical (AFM) or electrical (HDR) signals are plotted over a wide range of frequencies. These frequency–response spectra are useful in differentiating between the mechanical and electrical responses and in examining the unique advantages of HDR. In general, each of the several peaks observed in both the mechanical and electrical responses resemble the SDOF resonance behavior governed by eqn (7.2.2), with the notable exception of the first harmonic of the current signal. As an example, the primary resonance peak, near ω0 , as observed in the second harmonic of the electrical (current) signal is presented in Fig. 7.2. This figure demonstrates that the resonance frequency and quality factor (Q-factor) of a micro- or nanocantilever can be accurately determined by examining only its electrical response. Furthermore, the inset of Fig. 7.2 shows that shifts in the resonance frequency can be observed electrically. In this case the shift is due to changes in the spring constant due to the applied voltage (see Section 7.3.4). Typical mechanical and electrical frequency responses from the experiments described above are presented in Figs. 3 and 4. Both the mechanical and the electrical signals contain several significant harmonic components. Each harmonic of the mechanical signal exhibits a single dominant resonance peak at a driving frequency of, ω0 /n, where n is the order of the harmonic.

7.2 Mechanical vs. electrical responses 259

Fig. 7.3 Mechanical response spectrum of a silicon microcantilever measured using AFM-based laser reflectometry. The largest peak is in the first harmonic at the primary resonance, ω0 ∼ 15.7 kHz. A superharmonic resonance is visible in the second harmonic at ω0 /2 and in the third harmonic at ω0 /3.

Fig. 7.4 Electrical (current) response spectrum of a silicon microcantilever measured using HDR. A parasitic capacitance exists in the first harmonic that increases linearly with frequency and obscures the resonance at ω0 ∼ 21 kHz. The higher harmonics do not exhibit significant parasitic effects. Note, the first harmonic has a different scale from the higher harmonics (Gaillard et al. 2006). Copyright American Institute of Physics.

260

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

Fig. 7.5 A polar plot of a silicon microcantilever (300 × 35 × 2 μm3 ) (diamonds) and a MWNT (triangles). The frequency is a parameter, with the beginning and ending frequencies indicated. For the silicon cantilever the frequency steps are 100 Hz, and for the MWNT the steps are 250 Hz. The plot for the silicon microcantilever, for which the amplitude is 20 times the indicated scale, illustrates the circle that a resonance displays on a polar plot. The much smaller signal from the MWNT shows the effect of a background signal of the same order of magnitude as the resonant signal. The double-headed arrow indicates the background signal amplitude and phase. One can see that the MWNT resonance shows a nearly complete circle on the polar plot with an offset due to the background signal (Gaillard et al. 2006). Copyright American Institute of Physics.

12.5 kHz 11 kHz

25 μV

12.5 μV

2.3 MHz

0

2.9 MHz

These are known as superharmonic resonances (Nayfeh and Mook 2001). The resonance peak of the first harmonic at ω0 , is well defined with a high signalto-background ratio (SBR). This is why laser-based detection of mechanical resonance has proven so successful in AFM applications. Parasitic capacitance does not affect mechanical responses. We can also see, in the second harmonic, a small peak driven at ω0 . This is due to non-linearities in the force on the cantilever, as will be examined analytically later. The electrical (current) response is noticeably different. The amplitude of the first harmonic increases nearly linearly with applied frequency until it approaches ω0 . This parasitic capacitance, linear in frequency, obscures the true resonant signal and dramatically reduces the SBR. This explains why capacitive detection proved so difficult before the advent of HDR.

7.2.4

Polar plots of resonance

The nature of the resonance peaks can often better be understood by examining their polar representations in which amplitude is plotted versus phase with driving frequency as the parameter. In the HDR polar plots, overlapping curves occur for each resonance peak (primary and superharmonic) existing in the harmonic spectrum. For instance, the second harmonics of the electrical signals for both a micro- and nanocantilever are presented in Fig. 7.5. The resonance frequency may be determined from the polar graph by noting where phase changes most rapidly, e.g. the top of the larger circle in Fig. 7.5. In some cases the polar representation shows that the resonance is no longer circular, but rather is closely approximated by a class of curves known as limac¸ons. It will later be shown that this results from highly non-linear systems where two separate terms contribute to the electrical signal, neither of which can be neglected.

7.3 Analytic modelling

7.3

Analytic modelling

The cantilever is a continuous system in which masses and forces are distributed along its length; however, it can be modelled more simply as a discrete multiple degree of freedom system (MDOF) using classical Euler–Bernoulli beam theory and a variational energy technique know as the assumed modes method (Tedesco et al. 1999). In this model each mode of vibration is governed by the typical second-order linear differential equation of motion (EOM) of a driven damped harmonic oscillator, eqn (7.5). This model applies equally for micro- and nanocantilevers assuming they are slender, homogeneous, and isotropic, [m z¨ (t) + b˙z (t) + kz(t) = F(t)]m ,

(7.5)

where z m (t) is the tip deflection of the mth mode. The effective modal parameters: mass, m m , damping, bm , stiffness, km , and force, Fm (t) are given in eqn (7.6) (Buchholdt 1997), L mm =

L ρ(x)A(x) [ψm (x)] dx 2

0

L km =

bm =

b(x) [ψm (x)]2 dx 0



d2 ψm (x) E I (x) dx 2

2

L dx

Fm (t) =

0

F(x, t)ψm (x)dx. 0

(7.6) The distributed parameters are the mass density, ρ(x), cross-sectional area, A(x), damping coefficient, b(x), Young’s modulus, E(x), area moment of inertia, I (x), and externally applied force, F(x, t). ψm (x) is the mth mode shape of the cantilever beam.

7.3.1

Modes of vibration

The mode shapes are the fundamental shapes that a vibrating structure can assume, or equivalently the eigenfunctions of its governing equation. All possible motions of a vibrating structure can be decomposed into a sum of these independent mode shapes. An SEM image showing the first two modes of vibration of a microcantilever beam is shown in Fig. 7.6. Each mode of vibration has a particular natural frequency and damping ratio. An equivalent representation of the cantilever EOM, eqn (7.5), is provided to emphasize the unique frequency response characteristics of each of the m modes of vibration,  z¨ (t) + 2γ ω0 z˙ (t) + ω02 z(t) =

F(t) m

 . m

(7.7)

261

262

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

Fig. 7.6 SEM images of the same cantilever vibrating at (a) the fundamental mode and (b) the second mode. The dimensions of this cantilever are w = 800 nm, t = 2 μm and l = 40 μm (Davis et al. 2003). Copyright Elsevier B.V.

The natural resonance frequency, ωm , and dimensionless damping ratio, γm , of the mth mode are   k/m ω0m ≡ m   b γm ≡ √ . (7.8) 2 km m Note the distinction between modes and harmonics. Often the term “harmonic”, which is defined as being an integer multiple of some fundamental frequency is confused with the modes of vibration. The confusion arises because for doubly clamped structures, e.g., violin strings, the frequencies of higher modes of vibration are all integer multiples of the first mode frequency. Thus, for doubly clamped systems, harmonic and modal frequencies are essentially interchangeable. For cantilevers, however, the frequencies of the higher modes are not integer multiples of the first, and thus harmonic and modal frequencies are not equivalent. The natural frequency of the mth mode for a general cantilever is given in (7.9). For rectangular cross-sections, e.g., the silicon microcantilevers exam ined in this report, the area moment of inertia is Irect = wh 3 12, and the crosssectional area is A = wh, where w and h are the width and thickness of the cantilever, respectively. For hollow cylindrical cantilevers, e.g., carbon nanotubes, we have ICNT = (π /64)(Do4 − Di4 ) and A = (π /4)(Do2 − Di2 ), where D0 and Di are the outer and inner diameters, respectively,  ω0m = (βm L)2

EI . ρ AL 4

(7.9)

7.3 Analytic modelling

263

Table 7.1 Modal natural frequencies for the free response of an undamped cantilever. βm L

Natural Frequency, ω0m

1.8751 . . . 4.6941 . . . 7.8548 . . .

ω01 ω02 = 6.3 ω01 ω03 = 17.5 ω01

Mode Number 1 2 3

The βm L values for the first several modes of a singly clamped (cantilever) beam and the corresponding natural frequencies are given in Table 7.1. As discussed previously, the higher mode frequencies are not integer multiples of the first (fundamental) mode and so should not be called harmonics. In this study, we are generally only concerned with the first mode of vibration (n = 1) for several reasons. The first mode has the greatest tip deflection, which facilitates both actuation and detection. Also, for the driving frequencies used, the amplitudes of the higher modes are negligible. Finally, the lock-in amplifier used has a limited frequency range; therefore, harmonics of higher modes could not be measured. However, the resonance of the second mode has been experimentally observed using HDR and is presented in Fig. 7.7.

7.3.2

Damping, Q-factor, and sensitivity

The Q-factor is a measure of the efficiency of an oscillator and is related to the sharpness of the resonance peak. The Q-factor can be defined, in general, as the ratio of the natural frequency, ω0 , and bandwidth, Δω, of the resonator. This definition is especially useful when determining the Q-factor of the superharmonic resonances. For a simple harmonic oscillator, the Q-factor is also related to the dimensionless damping ratio eqn (7.10) where ΔωFWHM is

2nd Harmonic

300 (a) 250

(b)

w02 = 5.86w01

Amplitude [μV]

20 200

w 02/2

20 150

1340

[kHz]

1440

10 660

50

230

10

15

100

0

15

[kHz]

740

w 01 240

750

1000 Freqeuncy [kHz]

1250

1500

Fig. 7.7 Second-harmonic electrical response of a microcantilever measured using the harmonic detection of resonance (HDR) technique. Visible in this response is the primary resonance of the first mode (a) ω01 ∼ 235 kHz, as well as the primary, ω02 ∼1377 kHz = 5.86ω01 , and first superharmonic resonances, ω02 /2 ∼ 688 kHz, of the second mode (b).

264

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

the bandwidth (i.e. the full width at half-maximum) of the resonance peak, Q=

1 ω0 . = ωFWHM 2γ

(7.10)

In cantilever sensing applications a high Q-factor is desirable. This is evident in the expression for the minimum measurable mass change, Δm min , of a cantilever sensor eqn (7.11), where the minimum observable frequency shift, Δωmin , is assumed to be proportional to the bandwidth, ΔωFWHM , given in eqn (7.10). Both a high Q-factor, Q, and sensitivity, S, are required to measure small changes in mass. m min =

7.3.3

ω0 ωmin ∝ . S SQ

(7.11)

Electrostatic force and the equations of motion (EOM)

The electrostatic force that drives the cantilever motion is found by taking the positive derivative of the energy stored in the electrostatic field eqn (7.12). Force is, in general, the negative derivative of energy with respect to distance. The positive sign here comes from the fact that the battery does work in moving charge off the cantilever in order to maintain a constant voltage (De Wolf 2001),   d 1 1 dC 2 2 CV V (t). (7.12) = FE (t) = dz 2 2 dz The applied voltage consists of an ac term at the driving frequency, Ω, and a dc bias. The electrostatic force, proportional to the voltage squared, may be expanded using trigonometric identities (7.13). In this form, it is evident that the force consists of a constant term, which shifts the average deflection of the cantilever towards the counterelectrode by a distance δ, and two harmonic terms at the driving frequency, Ω, and twice the driving frequency, 2Ω, 1 dC (Vdc + Vac cos ( t))2 2 dz   1 dC 1 2 1 2 2 = Vdc + Vac + 2Vdc Vac cos ( t) + Vac cos (2 t) . 2 dz 2 2

FE =

(7.13) From this we expect to see normal resonance phenomena when the applied frequency is equal to the natural frequency of the cantilever ( = ω0 ) and additionally, due to the second-harmonic term in the force, we expect a superharmonic resonance peak when the driving frequency is half the natural frequency ( = ω0 /2). If we now expand the capacitance of the cantilever and counterelectrode in a Taylor series about the average position, ζ = 0, where ζ (t) = z(t) − δ, see Fig. 7.1. 1 1 C(ζ ) = C0 + C1 ζ (t) + C2 ζ (t)2 + . . . + Cn ζ (t)n , 2 n!

(7.14)

7.3 Analytic modelling

where the capacitive coefficients, Cn , are given by  ∂ n C  . Cn = ∂ζ n ζ =0

(7.15)

Substituting this into the governing equation of motion for the first mode of the cantilever (ω01 = ω0 ) we arrive at ζ¨ (t) + 2γ ω0 ζ˙ (t) + ω02 ζ (t)

  1 2 1 = (C1 + C2 ζ (t) + . . .) 2Vdc Vac cos ( t) + Vac cos (2 t) . 2m 2 (7.16)

We may simplify the EOM eqn (7.16) by truncating the capacitance after the C2 term. This approximation is valid as long as the gap distance is large compared to the maximum cantilever deflection. We will see that this simplified model accurately reproduces the mechanical and electrical responses presented in the previous section. Non-linear effects from higher-order terms will be considered later in this chapter. The average displacement can be found using eqn (7.17). A model of the capacitance must be chosen in order to determine the capacitive coefficients, Cn ,  2 + 1V2 C Vdc 1 2 ac . (7.17) δ= 2k

7.3.4

Spring softening and pull-in

In general, if the force applied to a harmonic oscillator depends on the distance, then it may be expanded in a Taylor series m z¨ (t) + b˙z (t) + kz(t) = F0 +

∂F z + .... ∂z

(7.18)

Collecting the terms in z(t) yields m z¨ (t) + b˙z (t) + κz(t) = F0 ,

(7.19)

where the effective stiffness is given by κ=k−

∂F + .... ∂z

(7.20)

If the derivative of the force is positive, then the stiffness of the system is decreased by the applied force (spring softening). Likewise, if the derivative of the force is negative, then the stiffness increases (spring hardening). For the electrostatically actuated microcantilever, the attractive force decreases as the gap distance is increased, i.e. as the cantilever deflects away from the counterelectrode. The derivative of force with respect to z must then be positive. Therefore, we expect the microcantilever system to exhibit springsoftening effects. The equation of motion of the microcantilever is given in eqn (7.21). This spring softening due to an applied ac voltage is shown in the

265

266

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

inset of Fig. 7.2,

  1 1 C1 + C2 ζ (t) + C3 ζ (t)2 2 2   1 × 2Vdc Vac cos ( t) + Vac2 cos (2 t) , 2

m ζ¨ (t) + bζ˙ (t) + κζ (t) =

(7.21)

where the effective spring constant of the microcantilever due to the electrostatic force is 1 κ = k − (C1 + C2 ζ (t) + . . .) . (7.22) 2 We see in eqn (7.22) that the stiffness of the microcantilever system depends on the distance, ζ (t). The higher-order terms in ζ (t) cause the effective spring constant to increasingly soften as the cantilever deflects towards the counterelectrode. This spring softening results in a static pull-in phenomena. The cantilever will crash into the counterelectrode when the applied voltage, and consequently the deflection, exceed a threshold at which the effective spring constant goes to zero. Determination of dynamic instability is more complicated and is discussed in Hu et al. (2004), ζpull−in ≈

7.3.5

2k − C1 . C2

(7.23)

Capacitance model

In order to determine the response of the microcantilever, a model for the capacitance must be chosen. The simplest and most frequently used is the parallel-plate model εA εwL , (7.24) CPP = = d d0 − (δ + ζ (t)) where ε is the permittivity of the dielectric material (i.e. air), A = wL is the plate area of the capacitor, and d is the gap distance. The capacitive coefficients can then be calculated directly as εA C0 = CPP |ζ =0 = d0 − δ  dCPP  εA C1 = =  dζ ζ =0 (d0 − δ)2  d2 CPP  2ε A C2 = = (7.25)  dζ 2  (d0 − δ)3 ζ =0

The charge distribution has been solved (De and Aluru 2004) and is relatively uniform along the lengths of the cantilever and counterelectrode with only a moderate increase in charge density at the tip of the cantilever. Therefore, the electrostatic force is essentially constant and the parallel-plate model is justifiable. The electrostatic force along with its Taylor-series approximation is given in eqn (7.26). It has been shown that the series approximation is valid as long as

7.3 Analytic modelling

the maximum cantilever tip deflection is no greater than about half the initial gap distance. In this form it is obvious that for small tip deflections, ζ 1 are known as the nth higher harmonics. Substituting the Fourier expansion of the cantilever tip deflection eqn (7.27) into the electrostatic force eqn (7.26) and applying appropriate trigonometric identities demonstrates that the electrostatic force also assumes the form of a harmonic series,   N ε A (Vdc + Vac cos( t))2 2

j(n t+φn ) An e + ... 1+ Fe ≈ d 2d2 n=0

=

N

Fn e j(n t+φn ) .

(7.28)

n=0

From this we see that the superharmonic resonances are not limited to = ω0 /2, but in theory occur at all integer submultiples of the fundamental resonance ( = ω0 /n) at which the Fn component of the force excites the beam into resonance.

7.3.7

Harmonic balance analysis

A steady-state solution to the cantilever equation of motion can be found using the harmonic balance method. The harmonic balance method assumes a Fourier-series solution to the cantilever tip deflection eqn (7.27) with a

267

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Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

fundamental frequency equal to the applied frequency, Ω. Validation that this is the appropriate form of the solution is available in De and Aluru (2006). The harmonic series solution is then substituted into the microcantilever EOM eqn (7.21). Because the harmonic functions are linearly independent, the coefficients must independently equate to zero, yielding a set of N algebraic equations. The harmonic amplitudes of the tip deflection, An , may then be solved in terms of the applied frequency and other system parameters, yielding the harmonic frequency response spectrum for the displacement. Using the solution arrived at for ζ (t), the capacitance and current may also be calculated. Full solutions using this method almost always require a computer TM math routine, in this case MAPLE . However, we will next provide simple analytic solutions for low-order harmonics, which will provide insight into the complexity of the harmonic structure. The harmonic balance method was preferable to other perturbations techniques for several reasons. First, the harmonic balance method directly yields the amplitudes of the harmonics as a function of applied frequency, i.e. what is measured using HDR. Secondly, it provides a solution to the steady-state vibration of the cantilever before the onset of instability, which corresponds to the conditions in the experimental HDR results. Finally, the harmonic balance method is valid for all applied frequencies, not just those near a specific resonance, such as some other perturbation techniques (Nayfeh and Mook 2001).

7.3.8

Closed-form solutions

Using the method of harmonic balance, described above, it is possible to obtain analytic solutions for the mechanical and electrical response of the excited cantilever predicted by eqn (7.16). We will consider only the first two harmonics here, but the procedure for generating any higher-order solution should follow naturally, although with increasing mathematical complexity. We will also restrict ourselves to the case where only C1 and C2 are included in the expansion of C. It can be readily shown that this will not affect our results for the first and second harmonic, since, as can be seen from eqn (7.16), C3 will, at lowest order, produce a driving force of frequency 3ω. This term, as well as other high-order terms must be considered, however, in calculating the solution of any higher harmonics. Plugging the lowest-order solution into eqn (7.16), and neglecting (for the time being) terms of second order and greater, we find a result analogous to the damped, driven harmonic oscillator, b j A1 e j ( t+φ1 ) + ω02 A1 ej( t+φ1 ) m   1  C1 + C2 A1 e j( t+φ1 ) × 2Vdc Vac e j t + 12 V 2 ac e j2 t = m 1  2C1 Vdc Vac e j t + 2C2 A1 Vdc Vac e j(2 t+φ1 ) = m  1 1 (7.29) + C1 V 2 ac e j2 t + C2 A1 V 2 ac e j(3 t+φ1 ) . 2 2

− 2 A1 e j ( t+φ1 ) +

7.3 Analytic modelling

Keeping only the first term in the final expression, we can solve for the amplitude and phase of the first harmonic of the mechanical motion. These expressions closely resemble those of the linear SDOF oscillator, see eqn (7.2), A1 = tan φ1 =

2C1 Vdc Vac  2 m (ω 0 − 2 )2 + (2γ ω0 )2 2γ ω0 . 2 − ω2 0

(7.30)

For the second harmonic, we follow the same procedure as before, inserting a term from our harmonic balance into the driving equation and including only terms of the given order. We must remember to include previously omitted second-order terms. Following this procedure gives 2b j A2 e j(2 t+φ2 ) + ω2 0 A2 e j(2 t+φ2 ) m   1 1 = 2C2 A1 Vdc Vac e j(2 t+φ1 ) + C1 V 2 ac e j2 t + O(e j3 t ). m 2

−4 2 A2 e j(2 t+φ2 ) +

(7.31) Solving this equation for the amplitude and phase gives eqn (7.32), where we have defined an angle φ2  to simplify the notation. Notice that if C2 is zero, φ2 reduces to φ2  . From these equations, we can readily deduce the results shown earlier. The first harmonic will contain a single normal resonance peak at ω0 , while the second harmonic will have a dominant peak at ω0 /2, and a smaller peak, due to the contribution from A1 , at ω0 .  Vac (C2 A1 Vdc )2 + 14 (C1 Vac )2 + C1 C2 A1 Vdc Vac cos φ1 A2 = m (ω2 0 − 4 2 )2 + (2γ ω0 )2 tan φ2 = 4tan φ2  =

C1 Vac sin φ2  + 2C2 A1 Vdc 2sin(φ2  + φ1 ) C1 Vac cos φ2  + 2C2 A1 Vdc 2cos(φ2  + φ1 ) 4γ ω . 4 2 − ω2 0

(7.32)

For the electrical response, we will consider only the current through the cantilever, as the output of the A250 amplifier should be approximately proportional to this current. Again, we will only consider the first two harmonics, and therefore we may ignore higher-order contributions to the capacitance. The first harmonic (the first term in parentheses in eqn (7.33)) of the current signal has two parts. The first is proportional to the static capacitance, C0 , while the second is proportional to the mechanical resonance, A1 . The first term (the parasitic signal) obscures the second term (the dynamic signal). Notice that both the static and dynamic parts are linearly proportional to the

269

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Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

applied frequency, Ω, d

 C0 + xC1 +

x 2 C2 2



Vdc + Vac e j t



dq(t) = dt dt  j t C0 Vac + C1 A1 Vdc e jφ1 + 2 j e j2 t = j e   C2 2 A1 Vdc e j2φ1 + C1 A2 Vdc e jφ2 . (7.33) × C1 A1 Vac e jφ1 + 2

i(t) =

The second harmonic (the second term in eqn (7.33)) does not contain such a parasitic term. That is, all terms depend on the mechanical amplitudes, A1 and A2 . Thus, unlike the first harmonic, the higher harmonics do not increase linearly with applied frequency and do not suffer from the parasitic static capacitance, C0 . As should be evident from the above expression, higher-order harmonics differ greatly in the dominant contributions to their amplitude and Q-factor. For this reason, different harmonics, or different peaks in the same harmonic, will be affected more strongly by different phenomenon. This property may be utilized to design more robust HDR-based sensors that measure various properties of a system by simultaneously detecting shifts in multiple resonance peaks of the same cantilever.

7.3.9

Analytical results

The computed mechanical and electrical response spectra determined using TM are presented in Figs. 7.8 the harmonic balance method in MAPLE and 7.9, respectively. These theoretical responses agree excellently with the experimental results found using HDR, validating the analytical work.

1st Harmonic 2nd Harmonic 3rd Harmonic

1.8 Mechanical Amplitude [mm]

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 Fig. 7.8 Computed mechanical frequency response spectrum of a microcantilever. Compare with experimental AFM results, Fig. 7.3.

0.0 0

5

10

15

20

Driving Frequency, W [kHz]

25

7.4 Applications

1st Harmonic 2nd Harmonic 3rd Harmonic

Electrical Current Amplitude [mm]

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2

Fig. 7.9 Computed electrical (current) frequency response spectrum of a microcantilever. The first harmonic, showing the parasitic effects, has a different scale. Compare with experimental HDR results, Fig. 7.4.

0.0 0

5

10

15

20

25

Driving Frequency, W [kHz]

Specifically, the effects of parasitic capacitance are only significant in the first harmonic of the current signal, not in the mechanical response or higher harmonics of the current signal. Analytical predictions, such as these, that accurately predict amplitudes and resonance frequencies are extremely useful for the continued advancement of cantilever-based sensing.

7.4 7.4.1

271

Applications Non-linearity and Duffing

A characteristic feature of many MEMS devices that has recently received much attention is the non-linear response to ac driving signals (Isacsson et al. 2007). It has been postulated that non-linear spectral features may allow for a greater dynamic range and enhance the sensitivity (Zaitsev et al. 2005). While studying the behavior of electrostatically driven and measured cantilevers, it was noticed that it is possible to drive them hard enough to observe Duffinglike jumps in their amplitude–frequency behavior (Pippard 1978). Duffing-like behavior provides us with the ability to engineer the ultrahigh sensitivity of this bistability (Postma et al. 2005; Kozinsky et al. 2006). Typically in a Duffing resonator, above some critical driving amplitude, the response becomes a multivalued function of frequency in some finite frequency range. The presence of a bistable region results in a dramatic jump transition from a near-zero solution to one of high amplitude. This could be useful in sensing technologies (Almog et al. 2007). Non-linearity effects on resonance are often described using the classical Duffing equation (De and Aluru 2004), z¨ + ωo2 z = −2εγ z˙ − εαz 3 + FE (t),

(7.34)

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Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

A (a) No Duffing Excitation < critical Fig. 7.10 Steady-state solutions under different excitation amplitudes A showing three stages of Duffing behavior. The natural resonance frequency, ω0 , is shown by the central line that differs from the driving frequency. In region (III) the system has only one stable solution that with decreasing frequency bifurcates into a stable solution and an unstable (dashed curve) solution. This stable solution grows in amplitude higher than the stable solution accessible with increasing frequency as Ω continues to decrease (region II). When the slope becomes infinite, this stable solution drastically collapses to the lowamplitude solution that is stable again (shown by dashed arrows in region I). Fig. 7.10(c) shows what one would expect in the first harmonic for a large third-order non-linearity (Keskar et al. 2008b). Copyright IEEE.

A

(b)

w0

W Incipient Duffing Excitation = critical

(c)

w0

W

A Strong Duffing Excitation > critical

w0

W

the applied force is typically sinusoidal, FE (t) = K cos Ωt,

(7.35)

where Ω is the driving frequency and K is a constant. The equation of motion, (7.34), can be solved by either the method of multiple scales (Nayfeh and Mook 2001) or the method of harmonic balance. As we have more complex non-linearities than the classical case we do not see the typical Duffing behavior, but rather behavior that mimics it. Henceforward we will call these Duffing-like effects simply Duffing behavior. The response curves for our cantilevers have two stable states due to nonlinearities, Fig. 7.10, which lead to the so-called “jump” phenomena. When the driving frequency is slowly increased at constant amplitude, the response amplitude will jump up at a frequency less than the resonance frequency ω0 that is measured at low amplitude. The response amplitude will also jump down at a frequency less than ω0 when the frequency is decreased from well above ω0 . One speaks of “hard” and “soft” springs, in which the dynamic spring constant kd = dF/dz increases or decreases as z increases. For hard springs, the resonance curve bends toward higher frequency. For soft springs, such as those of our experiments, the resonance peak bends toward lower frequencies. In the frequency domain in which two stable steady-state solutions exist, the initial conditions determine which of these represents the actual response of the system. Thus, in contrast with linear systems, the steady-state solution of a non-linear system can depend on the initial conditions (Nayfeh and Mook 2001). Figure 7.10 shows a schematic of the stable steady-state solutions under different excitation amplitudes. Referring to this figure, we define some terminologies in the A vs. Ω curves that we use in the experimental data analysis.

7.4 Applications

273

600 for back 15.785 Amplitude (μV)

400

15.668

15.785 kHz 15.862 kHz μV 300 15.668 kHz

0°

500

180°

H2

600

300

200

15.862

100

0 15.6

15.7

15.8

15.9 16.0 Frequency (kHz)

16.1

16.2

In the first case where the excitation amplitude A (in our case, Vac ) is less than the critical amplitude, only one solution exists, and no bistability is possible, Fig. 7.10(a). For the case when the excitation amplitude equals the critical amplitude, the system is on the edge of the bistability, and there exists one point where A vs. Ω has an infinite slope showing incipient Duffing, Fig. 7.10(b). In the case when the excitation is greater than the critical value, the system is in the bistable regime having three possible solutions over some range of frequencies. Two of these solutions are stable. With increasing frequency the solution jumps from the low-amplitude stable solution (region I) to another high-amplitude stable solution (as shown by solid arrows) by-passing the unstable (experimentally unobservable) solution (shown dashed in region II). The large-amplitude solution is stable and decreases with increasing Ω and finally enters into region III. Interestingly, we have found that the Duffing behavior is more visible and prominent at the 3rd harmonic than at the second harmonic in air at room pressure (Keskar et al. 2008b). Hence, all the following experimental results were recorded at the third and higher harmonics of the ac voltage applied to the counterelectrode. The ac voltage applied to the counterelectrode was swept with both increasing and decreasing Ω. Figure 7.11 shows the response of a microcantilever at the third harmonic under hydrogen (760 torr) with a 10-μm gap distance (Keskar et al. 2008c). The microcantilever showed Duffing behavior with both increasing and decreasing Ω. To increase the amplitude, a hydrogen atmosphere was used to lower damping. For the hydrogen environment (Vac = 7 V, Vdc = 8.5 V) the signal amplitude jumped from 20 μV to 235 μV with increasing Ω (Fig. 7.11) and jumped down from 525 μV to 10 μV with decreasing Ω.

Fig. 7.11 Measured frequency spectra under 760 torr of hydrogen with increasing and decreasing Ω at the third harmonic (Vac = 7 V, Vdc = 8.5 V) with a 10-μm gap distance. The inset shows the polar plot mapped with increasing and decreasing Ω. The dotted line is an estimate of the unstable state. On the polar plot note that the stable states are largely circular arcs, with an odd turning in the decreasing Ω data near 15.785 kHz. The points that occur in the straight-line portions of the polar plot may be in part due to transient and or unstable states (Keskar et al. 2008c). Copyright IEEE.

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Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

120 Air

5th for 5th back

Ampltidue (μV)

100

14

80 60

ON 40

12.2

OFF

OFF

20 Fig. 7.12 Measured frequency spectra in air with increasing and decreasing Ω at 5th harmonic with a 4-μm gap distance showing ON/OFF characteristics (Vac = 3 V, Vdc = 8.5 V) (Keskar et al. 2008c). Copyright IEEE.

0 12

13 14 15 Driving Frequency W (kHz)

16

17

As expected, the spectra with increasing and decreasing Ω overlapped fairly well away from the two transition Ωs. Hence under hydrogen a strong Duffing behavior was observed in the backward direction at the jump-down frequency. To study the distance dependence of the Duffing behavior the gap distance between the cantilever and counterelectrode was decreased to 4 μm, lowering Vac and Vdc to keep the non-linearities from increasing too much. Since decreasing the gap should increase the non-linearities, Duffing jumps should be more prominent. The striking feature with a small gap distance was that there were often two peaks in the amplitude spectrum. The resonance peaks taken with decreasing Ω spread over a broader range from 12 to 16 kHz for all the harmonics making it difficult to read out the exact ω+ from the amplitude plot. At the fifth and sixth harmonics in the frequency region ∼14.5 kHz at ω+ and ∼13.8 kHz at ω– there is probably only one stable state but the instantaneous drop/jump in amplitude is so sharp that it mimics Duffing behavior. The highlighted “step”-like features observed in the frequency spectra at the sixth harmonic, Fig. 7.12, in both directions might have potential applications such as switches or frequency filters. Remarkable differences were identified in the amplitude and the phase during the transition at the third and fourth harmonics as compared to the fifth and sixth harmonics at lower gap distance. This shows that the Duffing jumps are exaggerated when observed in the higher harmonics of the driving signal.

7.4.2

Pressure sensing

Silicon microcantilevers have also been extensively studied as sensors for pressure, temperature, mass and viscosity measurements. Here, we focus on the resonance response as a function of pressure in different regimes—the intrinsic regime, the molecular-flow regime, the viscous regime, and transition

7.4 Applications

275

Pressure Gauge

Gas out Cantilever DC supply

Lock-in Amplifier Harmonic 2 Input 1 3

Ref

A250 Chargesensitive pre-amplifier

GPIB

Signal Generator Sync

Gas In

Counterelectrode

Out GPIB

Fig. 7.13 Pressure-sensing experimental HDR setup (Keskar et al. 2008a). Copyright Elsevier B.V.

regimes in between. In the intrinsic regime (10−8 − 10−6 torr), air damping is insignificant compared to the intrinsic damping of the vibrating cantilever itself. Hence, the resonant frequency ω0 and the quality factor, Q, are nearly independent of air pressure, p. In the molecular region (10−6 − 10−1 torr), the collisions of air molecules with the vibrating cantilever cause the damping. For the viscous region ( p > 10−1 torr), the velocity of the cantilever is always much smaller than the speed of sound in the medium and hence we can consider air as a viscous fluid. However, there can be turbulence, in which case the damping is roughly proportional to the square of velocity (Landau and Lifshitz 1987). In this investigation, laminar flow with negligible turbulence is assumed. The dependence of Q on p can be explained by the Christian model (Christian 1966), which emphasizes that the Q is proportional to 1/ p in the molecular region. We conducted a study of a resonating cantilever as a function of chamber pressure using the setup shown in Fig. 7.13 (Keskar et al. 2008a). Here, we used HDR to measure the response of a silicon microcantilever mounted on a chip (Gaillard et al. 2006). The experimental setup consisted of an A250 charge amplifier, a signal generator, a dc power supply, and a lock-in amplifier as shown in Fig. 7.13. A silicon microcantilever was aligned parallel to a tungsten counterelectrode and epoxied to a small DIP chip. This chip was then plugged into a board mounted inside a glass chamber, which was subjected to changes in pressure and environment. Using a turbomolecular pump, the chamber pressure was varied from 10−3 torr to room pressure (760 torr) under a variety of gaseous environments. The pressures were measured at the gas inlet side using the (KJLC BDG Series) Bourdon dial gauges. As the pressure is increased, the damping increases, and consequently, the Q-factor and vibrational amplitude decrease and the resonance frequency shifts. Figure 7.14 shows the frequency-response spectra for a variety of pressures in a helium environment. The Q-factor is inversely proportional to the

276

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

Fig. 7.14 Change in quality factor, Q, and resonance frequency, ω0 , as pressure is varied from 4E − 3 to 4E + 2 torr. The inset shows the inverse relationship between pressure and Q (Keskar et al. 2008a). Copyright Elsevier B.V.

Fig. 7.15 Change in Q factor with gas (molecular regime) (Keskar et al. 2008a). Copyright Elsevier B.V.

pressure, as shown in the inset. It can be seen in Fig. 7.15 that the resonant frequency, amplitude, and thus the quality factor changed with the variations in environment at room pressure (760 torr). The response of a different silicon microcantilever of the same dimensions at a 12-μm gap distance to the change in the gas environment was greatly enhanced in a hydrogen atmosphere with a sharper resonant peak implying a better Q. In an H2 environment the signal amplitude is about three times that in air. As per the theoretical prediction, in the H2 environment (M = 2, lightest), the amplitude and Q are the highest, whereas the lowest amplitude and Q are

7.4 Applications

found in the heaviest gas, Ar (M = 40). The ω0 first increased with a decrease in the mass of the gas molecules from Ar to He, but for H2 ω0 decreased. The decrease in resonant frequency as the mass of the gas atom decreased from He to H2 may be due to a weakly observed Duffing behavior in the H2 environment, in which the cantilever approached the counterelectrode more closely, due to the much larger amplitude, which, as mentioned above, would lead to a lowered effective spring constant (Keskar et al. 2008a). In any case, the differing results demonstrate the ability to differentiate between gases surrounding the cantilever. This can be regarded as an advantage or disadvantage. As with a thermocouple gauge, this would have to be calibrated for different gases to give the correct pressure. On the other hand, it would be possible to distinguish between different average molecular masses at constant pressure.

7.4.3

Active gas sensing

The HDR system is not only sensitive to environmental conditions such as damping, it also may respond to changes in dielectric constant or polarity index due to the nature of the capacitive measurement. Modifying the experimental setup slightly allowed us to introduce different vapors into the chamber housing the HDR chip (Keskar et al. 2008b). All experiments were carried out by (i) maintaining the microcantilever near its resonant frequency, and (ii) recording the changes in the cantilever response (change in the amplitude ΔA and phase signals ΔΦ) due to vapors of polar and non-polar solvents. Solvent vapors (water, hexane, benzene, methanol and iso-propanol) were transported from the bubbler using air as the carrier gas. The amplitude and phase signals for a cantilever resonating in air and exposed to 100 sccm of air bubbled through methanol are compared in Fig. 7.16(a). The peak amplitude increased from 150 μV in air to 170 μV. Likewise, the phase in air that resembles that of a damped simple harmonic oscillator, is also sensitive to the environment around the microcantilever and it decreased upon exposure to methanol. Once we determined ω0 using the phase and peak amplitude, we set the lock-in amplifier close to ω0 to detect the ΔA and ΔΦ induced by the change in the environment in the vicinity of the cantilever. ΔΦ is defined as ΦA − ΦM , where ΦA and ΦM correspond, respectively, to the magnitude of phase change in air and methanol (see Fig. 7.16(a)). Thus, ΔA and ΔΦ signals can serve as sensitive indicators for the presence or absence of specific gases near the resonating cantilever. In Fig. 7.16(b), we plot ΔA and ΔΦ for a microcantilever resonating in air and intermittently exposed to 100 sccm of methanol. While ΔA increased by ∼6 μV, ΔΦ decreased by ∼5 deg, indicating a slight decrease in ω0 . The greater increase in peak amplitude (∼20 μV) observed in Fig. 7.16(a) as compared to the increase in ΔA(∼6 μV) seen in Fig. 7.16(b) can be attributed to the greater exposure time in the former case as the cantilever was continuously exposed to methanol vapors. The sensing of various polar and non-polar solvents (n-hexane, benzene, methanol, iso-propanol and water) showed an increase in amplitude and

277

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

160

40

140

20

120

0

100

(a)

–20

80

–40

60

–60

40 20 17.0

Amlitude in air Amlitude in 100 sccm of air bubbled through methanol Phase in air Phase in 100 sccm of air bubbled through methanol

17.2 17.4 17.6 17.8 18.0 Driving Frequency W (kHz)

–12 180 100 sccm of air

Methanol

–16

176

–18 172

DF DA

(b)

168

–20 –22 –24 –26

164

–80

–14

DF(Deg.)

60

DA(mV)

180

Phase (deg.)

Amlitude (mV)

278

–28

–100 18.2

0

20

40

60

80 100 120 140 160 Time (s)

Fig. 7.16 (a) Response spectrum showing amplitude (solid squares) and phase (solid circles) upon exposure to 100 sccm of air continuously bubbled through methanol. The reference spectrum showing amplitude (hollow squares) and phase (hollow circles) in pure air is shown for comparison. (b) Dependence of ΔA and  (measured at the selected ω0 = 17.71 kHz) on the amount concentration of protium (H2 ) molecules present in the vicinity of the cantilever (Keskar et al. 2008b). Copyright International Frequency Sensor Association.

n-hexane 0.049

DA/A

benzene 0.033 methanol 0.030 Fig. 7.17 Normalized amplitude changes upon exposure of the cantilever vibrating near ω0 to 100 sccm of air bubbled through water, iso-propanol, methanol, benzene and n-hexane. The numbers on the left axis indicate ΔA/A for each individual solvent vapor (Keskar et al. 2008b). Copyright International Frequency Sensor Association.

isopropanol

0.012

water 0.010 0

20

40

60

80 100 120 140 160 Time (s)

decrease in phase upon exposure to 100 sccm of air bubbled through all these solvents. It shows that the normalized amplitude A/A is highest for nhexane and decreases uniformly with decreasing molecular mass of the solvent vapors under study. The vapor pressures at room temperature for water, isopropanol, methanol, benzene and n-hexane are 15, 44, 128, 100 and 127 mm of Hg, respectively. Hence, due to the lower vapor pressures for water and isopropanol we get very small changes in amplitude for these two solvents, as seen in Fig. 7.17. We can correlate the peculiar trends in the normalized amplitude responses for these solvents to their different dependences on mass, dielectric constant

7.5 Cantilevered multiwall carbon nanotubes (MWCNT) 279

and polarity index. The normalized amplitude showed a decreasing trend with increase in polarity index, dielectric constant and inverse of mass. Although it was not possible to determine exactly which property determined the result, the previous work would favor the molecular mass of the solvent. As opposed to solvent vapors, while sensing gases the normalized amplitude decreased with increasing molecular mass of the gas excepting methane.

7.5

Cantilevered multiwall carbon nanotubes (MWCNT)

While detection of mechanical oscillations using the optical detection method has proven useful for microcantilevers, the reflected beam intensity is insufficient for the accurate determination of oscillations in nanosized cantilevers. Importantly, MEMS and NEMS systems involve electrical detection of mechanical motion at a very small scale. Therefore, from a commercial viewpoint, electrical detection of motion such as the capacitive readout of the mechanical motion is highly desirable since it can be readily integrated with NEMS devices that are fully compliant with standard CMOS technologies. Recently, a doubly clamped single-walled carbon nanotube (SWCNT) was electrically actuated and its resonant frequency was detected using a mixer technique (Sazonova et al. 2004). Although this technique is valuable for a beam in a guitar-like configuration, it cannot be applied to a cantilevered nanostructure. Alternatively, a technique for detecting nanoscale displacements has been demonstrated using a single-electron transistor (Knobel and Cleland 2003). However, this device operates at low temperatures (30 mK) and a relatively high magnetic field (8 T). Electrically induced mechanical oscillations in MWCNTs have also been recorded using non-electrical-detection methods that involved the use of a transmission electron microscope (Poncharal et al. 1999), scanning electron microscope (Yu et al. 2000) field emission microscope (Wang et al. 2002) or an optical microscope (Gaillard et al. 2005). For realistic applications, the resonating system must be portable and therefore a capacitive readout of the mechanical motion as described above for microcantilevers is ideal. In this section we show that the HDR method in which two specific geometries are used may be more useful than any other of the above-referenced method for detecting oscillations in nanosized cantilevers. Using a dark-field microscope equipped with a XYZ micropositioning stage, a single MWCNT can be readily manipulated into a cantilevered geometry (“diving board”) at the end of a sharpened probe tip (inset figure in Fig. 7.18). The cantilevered MWCNT is placed parallel to and within 1–10 μm of another sharpened tungsten probe tip that serves as the counterelectrode. The MWCNT is forced into resonance by applying an ac voltage with a dc offset on the counterelectrode under ambient conditions. The resulting current on the nanotube is shown in Fig. 7.18 (Gaillard et al. 2006). Even for such a small cantilevers vibrating in air, the amplitude signal is well resolved with a Q ∼ 30, Fig. 7.18. As expected, in the absence of the

280

Harmonic detection of resonance methods for micro- and nanocantilevers: Theory and selected applications

30

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Fig. 7.19 Amplitude and phase spectra for a single MWNT driven into resonance using the HDR method in the tip-to-tip geometry. The schematic shows the mechanical oscillations that are induced in the MWNT by the electric force Fe when the MWNT and the counterelectrode are separated by a finite distance (Ciocan et al. 2005). Copyright American Chemical Society.

2.8

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1.50E – 06

–40 –60

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1.00E – 06 5.00E – 07

–80 –100 2.25E + 06

Amplitude (V)

Fig. 7.18 Electrical data for the amplitude (symmetric peak, solid circles) and phase (open circles) of a MWCNT 7 μm long and 50 nm in diameter near resonance measured under ambient conditions. The background is indicated by the nearly horizontal data. Even for such a small cantilever vibrating in air, the signal is well resolved with a Q ∼ 30. The inset shows the MWCNT cantilever positioned over the tungsten probe tip counterelectrode (Gaillard et al. 2006). Copyright American Institute of Physics.

Amplitude (μV)

26

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2.35E + 06 2.45E + 06 Frequency (Hz)

0.00E + 00 2.55E + 06

MWCNT a flat baseline is measured for the amplitude signal. As in the case of the microcantilever, the effective spring constant can be tuned by the applied an ac voltage in nanocantilevers (Gaillard et al. 2006). Because of the large signal-to-background ratio present in our HDR method, tuning of the resonant frequency is easily observable even in nanotube-based cantilevers. Mechanical oscillations in a cantilevered MWCNT can also be detected using the so-called tip-to-tip geometry (Ciocan et al. 2005). A resonance spectra for a CVD-grown MWCNT with L = 10 μm, inner tube diameter Di = 17 nm, and outer tube diameter Do = 57 nm is depicted in Fig. 7.16. For a MWCNT clamped at one end, the frequency of the mth mode of vibration is given by eqn (7.36) (Poncharal et al. 1999). This equation was also presented

7.6 Conclusion 281

in Section 7.3.1 for a cantilever of arbitrary geometry,   Do2 + Di2 E b βm2 1 , fm = 8π L 2 ρ

(7.36)

where L is the tube length, Do and Di are the outer and inner tube diameters respectively, ρ the density of the MWCNTs, and the βm s are determined from the boundary conditions to be β1 ≈ 1.875; β2 ≈ 4.694; β3 ≈ 7.855. We measured the density ρ of MWCNTs to be ρ = 2100 kg/m3 . Note that we measure the bending modulus of the nanotube. As long as the nanotube does not change its geometry by buckling or any other such deformation, the bending modulus is equal to Young’s modulus. Micro- and nanocantilevers are an especially promising platform for a wide variety of sensing applications. Numerous transduction mechanisms may be utilized for sensing, detection, or actuation, including adsorbed/absorbed mass, induced stresses, changes in pressure/damping, and externally applied forces. In this chapter a novel detection method for micro- and nanocantilevers has been introduced known as the harmonic detection of resonance (HDR). The HDR method has the advantage of being entirely electrical, both in actuation and detection, and thus is straightforward to integrate into standard CMOS technology. Unlike most standard detection schemes, such as laser reflectometry, HDR requires no complicated or bulky electronics; therefore, it is scalable and inexpensive. The HDR method is extremely capable of measuring the oscillations of microcantilevers and individual MWCNTs in ambient conditions. As a first application of the HDR technique applied to nanosized cantilevers, we have measured the bending modulus of a MWCNT to be 29.6 GPa from three experimentally observed resonances using an iterative algorithm (Ciocan et al. 2005). This bending modulus is in excellent agreement with those reported in the literature from stress–strain measurements of MWCNTs with comparable dimensions. This fully electrical approach for exciting and measuring the resonance frequencies is convenient and advantageous in devices that use MWCNTs as the circuit elements or detectors.

7.6

Conclusion

HDR overcomes the problem of parasitic capacitance, which significantly limited the performance of previous capacitive sensors, by measuring resonances at higher harmonics (integer multiples) of the driving frequency. A theoretical model of the microcantilever was developed that explained the harmonic resonances in terms of the non-linear electrostatic force. A distinction was drawn between the mechanical and electrical (current) responses of the microcantilever and experimental frequency response spectra of each (mechanical and HDR) were presented. The electrical response in particular was shown to have a “rich harmonic structure” in which each harmonic exhibits both primary and superharmonic resonances. In general, the various resonant peaks respond in unique ways to the physical properties of the system thus offering

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the possibility for a new class of sensors in which a range of parameters are measured simultaneously on a single cantilever. An examination of non-linearities in microcantilevers was provided. Duffing-like behavior was demonstrated experimentally including the characteristic folding of the resonance peaks and introduction of bistable regions. The application of these non-linear phenomena to ultrasensitive sensors was also discussed. Applications of HDR to gas and pressure sensing were then presented. Changes in pressure were shown to affect both the quality factor and resonance frequency of micromechanical oscillators. Also, microcantilevers were utilized as sensitive indicators for the presence or absence of specific gases. Finally, cantilevered multiwall carbon nanotube (MWCNT) results were presented, demonstrating that HDR is not limited to the microscale. HDR is one of the few detection methods applicable to the nanoscale, and it is certainly the simplest to implement. The HDR method was used to determine the bending modulus of a MWCNT, 29.6 GPa, which was in excellent agreement with standard results, further justifying HDR’s effectiveness as a sensing technology. The vast potential of nanotechnology is only beginning to be realized. Due to their versatility and simplicity, micro- and nanocantilevers will almost certainly be an integral component of many future MEMS and NEMS devices. As such, there will always be a need for a simple and effective sensing and actuation method, and for this there is perhaps no better choice than HDR.

Acknowledgments The authors gratefully acknowledge support through NSF NIRT grants 0210559 and 0304019, and Clemson University’s Center for Optical Materials Science and Engineering Technologies. The authors thank Mr. Jason Reppert of Clemson University for his help with the preparation of the manuscript.

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References Ciocan, R., Gaillard, J., Skove, M.J., Rao, A.M. Nano. Lett. 5 (12), 2389 (2005). Datskos, P.G., Rajic, S., Datskou, I. Appl. Phys. Lett. 73 (16), 2319 (1998). Davis, Z.J., et al. Sens. Actuators A: Phys. 105 (3), 311 (2003). De, S.K., Aluru, N.R. J. Microelectromech. Syst. 13 (5), 737 (2004). De, S.K., Aluru, N.R. J. Microelectromech. Syst. 15 (2), 355 (2006). De Wolf, D.A. Essentials of Electromagnetics for Engineers (Cambridge University Press, 2001). Degertekin, F.L., Hadimioglu, B., Sulchek, T., Quate, C.F. Appl. Phys. Lett. 78 (11), 1628 (2001). Fang, H-B., Liu, J.-Q., Xu, Z.-Y., Dong, L., Wang, L., Chen, D., Cai, B.-C., Liu, Y. Microelectron. J. 37 (11), 1280 (2006). Gaillard, J., Skove, M., Rao, A.M. Appl. Phys. Lett. 86 (23), 233109 (2005). Gaillard, J., Skove, M.J., Ciocan, R., Rao, A.M. Rev. Sci. Instrum. 77, 073907 (2006). Hierlemann, A., (ed.) H. Baltes, H. Fujita, D. Liepmann, Microtechnology and MEMS (Springer-Verlag Berlin, 2005). Howe, R.T., Muller, R.S. IEEE Trans. Elect. Dev. 33 (4), 499 (1986). Hu, Y.C., Chang, C.M., Huang, S.C. Sens. Actuators: A. Phys. 112 (1), 155 (2004). Isacsson, A., Kinaret, J.M., Kaunisto, R. Nanotechnol. 18, 195203 (2007). Judy, J.W., Muller, R.S., Zappe, H.H. J. Microelectromech. Syst. 4 (4), 162 (1995). Keskar, G., Elliott, B., Gaillard, J., Taylor, J.D., Skove, M.J., Rao, A.M. Sens. Actuators: A. Phys. 147, 203 (2008a). Keskar, G., Elliott, B., Skove, M.J., Gaillard, J., Rao. A.M. Sens. Transd. 91 (1), (2008b). Keskar, G., Elliott, B., Skove, M.J., Rao, A.M. IEEE Senrs J. 8, 1948 (2008c). Knobel, R.G., Cleland, A.N. Nature 424 (6946), 291 (2003). Kozinsky, I., Postma, H.W.C., Bargatin, I., Roukes, M.L. Appl. Phys. Lett. 88, 253101 (2006). Landau, L.D., Lifshitz, E.M. Fluid Mechanics, 10 vols. (2nd English edn., 6: Pergamon Press, 1987). Lao, C.S., Kuang, Q., Wang, Z.L., Park, M.-C., Deng, Y. Appl. Phys. Lett. 90 (26), 262107-3 (2007). Lavrik, N.V., Sepaniak, M.J., Datskos, P.G. Rev. Sci. Instrum. 75 (7), 2229 (2004). Mamin, H.J., Rugar, D. Appl. Phys. Lett. 79 (20), 3358 (2001). Minett, A., Fr`aysse, J., Gang, G., Kim, G., Roth, S. Curr. Appl. Phys. 2 (1), 61 (2002). Nayfeh, A.H., Mook, D.T. Pure and Applied Mathematics (Wiley-Interscience, New York, 2001). Pippard, A.B. The Physics of Vibration, 2 vols. (1: Cambridge University Press, 1978). Poncharal, P., Wang, Z.L., Ugarte, D., de Heer, W.A. Science 283 (5407), 1513 (1999). Postma, H.W.C., Kozinsky, I., Husain, A., Roukes, M.L. Appl. Phys. Lett. 86, 223105 (2005). Quist, A., Chand, A., Ramachandran, S., Cohen, D., Lal, R. Lab on a Chip 6 (11), 1450 (2006). Sazonova, V., Yaish, Y., Uestuenel, H., Roundy, D., Arias, T.A., McEuen, P.L. Nature 431 (2006), 284 (2004).

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research Francis L. Martin and Hubert M. Pollock

8.1

Introduction

In any biomedical research programme, three questions are almost certain to arise at some stage: 1. What can we see? 2. What are the components? 3. What are the interactions and functionality in specific situations? At the cellular or subcellular level, the first of these will be answered with the help of a microscope, of a type that will depend on how fine is the detail that needs to be imaged. The second question may well require some specialized technique of analytical instrumentation, such as thermal analysis or optical spectroscopy. The third requires knowledge of the underlying cell biology, combined with the know-how to apply the aforementioned analytical instrumentation to address a specific situation, for example, whether there is a DNA conformational change associated with cancer. There is clearly an advantage in having techniques that allow both extremely high-resolution microscopy and infrared (IR) or Raman spectroscopy to be performed in one and the same instrument. One such newly developed technique, known as photothermal microspectroscopy (PTMS), has already enabled a number of findings of biomedical value in this field to be published, and examples of these in particular will be discussed in this chapter. Cellular biomolecules absorb the mid-IR (λ = 2.5–25 μm) via vibrational transitions that are derived from individual chemical bonds. For some years, methodologies such as synchrotron Fourier-transform infrared (FTIR) microspectroscopy have been employed for cell-by-cell characterization. Thus,

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from an IR spectrum of a received sample, a “biochemical-cell fingerprint” may be generated; such applications have immense potential in disease diagnosis and characterization. Meanwhile, there have been enormous recent developments in the area of “near-field” technology and in its application to the imaging or localized spectroscopy of solid-state, organic and biological samples. Novel approaches that exploit the photothermal effect have now been successfully used to achieve chemical resolution combined with subwavelength spatial imaging, giving localized PTMS (alternatively termed photothermal temperature fluctuation, PTTF) (Hammiche et al. 2004a). Combined with multivariate analyses such as principal component analysis (PCA) that allow for the reduction of large spectroscopic datasets towards cluster analysis, so as to allow for the identification of chemical entities that might contribute to differences between different sample categories, there is now the possibility to derive from a cell, or a subcellular compartment, a biomolecular signature (Martin et al. 2007). In the case of cancer diagnostics, one might exploit this approach to rapidly discriminate IR spectra that deviate from a “normal” state, and in particular to identify the wavenumbers and thus the responsible chemical alterations. A major obstacle to the early diagnosis of cancers has been the difficulty in differentiating between molecular species within large populations of cells (i.e. of the order of 1 × 106 cells per sample). In early, often insidious, disease, only a very small proportion of cells in a biopsy or cytology sample might be (pre-)cancerous and the visual identification of these will be a “needle in a haystack” problem. Moreover, spectral differences between normal and (pre-)cancerous may be confined to extremely subtle shifts or changes in the heights of certain peaks. To attack this clinical problem, a novel approach involves chemical characterization of such cells, using IR spectroscopy interfaced with an information-processing algorithm (see, for example, Walsh et al. 2007a,b). Such an emerging technology platform is being applied to other biological problems such as stem-cell identification through biomarker development and their in-situ tissue localization (German et al. 2006b; Bentley et al. 2007; Walsh et al. 2008a). This chapter focuses in particular on developments in instrument technology, in the field of biomedical IR spectroscopy, that have led to the possibility of non-destructively interrogating nanomolecular cellular alterations. This raises the prospect of generating susceptibility-to-adenocarcinoma spectral signatures (German et al. 2006a), high-throughput screening platforms (Hammiche et al. 2007) or an in-situ stem-cell characterization procedure (Grude et al. 2007, 2008; Walsh et al. 2008a).

8.2 8.2.1

Some existing mid-IR microspectroscopy techniques FTIR microspectroscopy

The basic instrumental method that is playing an increasingly prominent role in solving challenges within microbiological, clinical and pharmaceutical

8.2 Some existing mid-IR microspectroscopy techniques

laboratories is FTIR microspectroscopy. Useful reviews are those by Naumann (2000, 2001); Petrich (2001); Dukor (2002); Maquelin et al. (2002); McIntosh and Jackson (2002); Naumann and Puppels (2002); Kazarian and Chan (2006); Srinivasan and Bhargava (2007); and Wang and Mizaikoff (2008). As additional examples of papers describing applications in the biomedical field, we mention those by Diem et al. (1999); Gazi et al. (2004); Hammiche et al. (2004a); Fernandez et al. (2005); Podshyvalov et al. (2005); and Tfayli et al. (2005); see also the comment by Martin and Fullwood (2007). As emphasized by Bird et al. (2008), such work has shown that the technique has reached the point at which unsupervised multivariate analysis of the observed spectral patterns can reveal distinct spectral classes that correlate well with visual cytology. The potential medical impact of FTIR microspectroscopy has been compared with that of magnetic resonance imaging (MRI) (McIntosh et al. 1998): the latter allows physicians to image macroscopic tissue structures, while IR microspectroscopy characterizes biochemical changes at the cellular level. The term IR microscope covers two main types of diffraction-limited microscopy. The first provides optical visualization plus IR spectroscopic data collection. The second (more recent and more advanced) technique employs focal plane array (FPA) detection for IR chemical imaging, where the image contrast is determined by the response of individual sample regions to particular IR wavelengths selected by the user (Lewis et al. 1995; Kidder et al. 2002; Hammond and Wooten 2005; Levin and Bhargava 2005; and Kazarian and Chan 2006). It is also useful to distinguish between two types of operation: tunedradiation or “dispersive” and broadband, the first using radiation that is tuned to one particular wavelength, while broadband (white) light includes the entire spectral region of interest. In tuned-radiation microspectroscopy, a series of measurements at different wavelengths gives a spectrum. With broadband radiation, an interferometer allows the signal to be measured as a function of time; these data are then converted to intensity as a function of frequency (i.e. the spectrum) by means of the Fourier-transform method. Likewise, there are two principal modes of operation: mapping, where we view the whole spectrum measured at a particular spatial location, for purposes of chemical identification; and imaging at one particular waveband, chosen in order to obtain image contrast between different components of the sample. The same terminology applies to other types of instrumentation, such as Raman microspectroscopy. Spectra may be measured one point at a time using a single-element detector (single-point mapping), or as a two-dimensional FPA image, typically 256 to 16 384 pixels. Until the early 1990s, a significant barrier to commercialization of IR imaging was that the FPAs needed to read IR images were not readily available as commercial items. FTIR microspectroscopy and Raman microscopy are widely used analytical methods for the identification and characterization of molecular species in biological systems. Variants of FTIR microspectroscopy include attenuated total reflection-FTIR (ATR) microspectroscopy (Romeo et al. 2003), Raman microspectroscopy (Huang et al. 2003; Baena and Lendl 2004; Haka et al. 2005; Short et al. 2005) and PTMS (Hammiche et al. 2004a, 2007). As with their bulk spectroscopy counterparts, each imaging technique has particular

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strengths and weaknesses, and is best suited to fulfil different needs. Among the micro-optical techniques, FTIR microspectroscopy has a prime position within many industrial laboratories. Photons that interact with a sample are either absorbed at specific energies, or scattered. Conventionally, the IR beam is detected optically in the far field after it has interacted with the sample. In a transmission measurement, where the radiation goes through a sample (mounted on an IR-transparent window such as BaF2 or ZnS) and is measured by a detector placed on the far side of the sample, the energy transferred from the incoming radiation to the molecule(s) is calculated as the difference between the emitted and detected energies. In a “transflectance” measurement, the same energy-difference measurement is made, but a thin layer of a sample is placed onto a reflecting substrate such as a gold mirror: the source and detector are located on the same side of the sample. The radiation passes through the sample and is reflected back by the mirror surface, so that the radiation has effectively passed through the sample twice. Transflectance spectra have superimposed on them a “nuisance” weaker specular reflectance spectrum, which has the appearance of a first-derivative spectrum. Water is a very strong absorber of mid-IR radiation and wet samples often require advanced analytical approaches such as ATR spectroscopy, where the effects of water are mitigated to a great extent. Here, the surface of a sample is placed into intimate contact with a higher refractive index, IR-transparent internal reflection element, generally either ZnSe, type II diamond, or germanium. Its spectrum is then recorded with IR radiation that is incident through the reflection element at an angle greater than the critical angle. There is also a fiber-optic version of ATR spectroscopy known as FEFA (fiber-based evanescent-field analysis), in which the light is applied to the sample through an IR-transparent fiber (Lambrecht et al. 2006). In FTIR transmission microscopy, for a continuous sample such as an organic polymer film, the thickness requirements are typically that the sample is at most 10 μm thick: this should enable a satisfactory fingerprint spectrum showing all the peak maxima in the spectrum to be recorded, in which the strongest bands have transmission minima in the region of 10% to 20%. The sample should also be of uniform thickness, and preferably non-scattering or at least minimally scattering. In order to record transmission spectra that are free from other optical artefacts and in order to avoid the formation of interference fringes appearing as a superimposed sinusoidal pattern in the spectrum, a common practice is to flatten samples such as fibers or powder particles, so that the material being examined is of a uniform thickness. This is often done in a compression cell. Such treatments may, of course, alter the property of interest, for example changing its crystallinity, polymorphic form, or molecular orientation. In the ATR mode, the relative intensity of bands in ATR spectra increases with increasing wavelength. Anomalous dispersion can sometimes give rise to distorted band shapes, with bands appearing to have a somewhat firstderivative-like appearance. The effect is related to the change of refractive index through an absorption band. Distortions are most apparent for strong bands and when operating at lower angles of incidence. A recently developed and convenient sampling technique, single-bounce micro-ATR, has become

8.2 Some existing mid-IR microspectroscopy techniques

very widely used, in which a small region of the sample is located and pressed onto the face of a small ATR element, commonly a diamond. Spectra may also be recorded in specular (front-surface) reflection mode from suitable samples (i.e. smooth, optically thick materials) but the resultant spectra are heavily distorted and must be treated (usually with a Kramers–Kronig transform) to separate the refractive index and absorption index components of the spectra. In classical FTIR microspectroscopy, the spatial resolution is limited by the diffraction limit. The blackbody (globar) source of white radiation conventionally used for FTIR microspectroscopy is relatively dim. To achieve the highest possible spatial resolution, the beam cross-section is restricted by means of an aperture to, say, 15 μm × 15 μm. Depending on the microscope objective used, this can give an effective single pixel size of less than 2 μm × 2 μm, small enough not to significantly degrade the resolution below the diffraction limit of the mid-IR radiation employed, i.e. half of a wavelength ranging from 1.25 μm to 12.5 μm (Kazarian and Chan 2006). Even so, this resolution will normally preclude single-cell measurements, and with a very small aperture the signal-to-noise ratio (SNR) is seriously degraded. A synchrotron IR radiation source (SRS) generates a highly collimated beam of photons of a much higher brilliance than can be used in conventional FTIR microspectroscopy, and may be delivered through an even smaller sampling aperture, in order to improve the spatial resolution up to the point at which it becomes diffraction-limited (Holman et al. 2000; Sahu et al. 2005). This allows for the acquisition of spectra at a spatial resolution of 5 μm and a SNR ∼1000 times greater than conventional FTIR microspectroscopy (Tobin et al. 2004a; Gazi et al. 2005). Through the small aperture the strong beam may be focused at the single-cell level, so that an individual ≈10 μm × 10 μm epithelial or stromal cell may be interrogated. Synchrotron FTIR microspectroscopy holds great promise for the interrogation of intracellular dynamics and molecular changes. It may also be applied to acquire spectra of single, living cells in media (Holman et al. 2000; Moss et al. 2005). It has recently been used with some success to investigate different cell types in prostate (German et al. 2006a) and corneal (German et al. 2006b; Bentley et al. 2007) tissue (for reviews of wider biomedical applications, see Tobin et al. 2004a,b; Dumas et al. 2007). However, use of this technique is restricted due to the fact that there is only a handful of synchrotron IR beamlines worldwide and the costs associated with maintaining and using these facilities is enormous. To be able to characterize individual intracellular components would be of very great value but the spatial resolution of synchrotron FTIR microspectroscopy is close to its practical limits, and is unlikely to be able to significantly improve on its current performance in that respect.

8.2.2

Raman microspectroscopy

Whereas FTIR microspectroscopy measures the absorbance of light by a sample, Raman microspectroscopy is a vibrational spectroscopic technique that measures an inelastic light-scattering process in which photons incident on

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an interrogated sample transfer energy to or from molecular vibrational modes (Raman and Krishnan 1928; Everall 2002). The x-scale of the Raman spectrum denotes energy shift (in cm−1 ) relative to the frequency of the laser used as the source of radiation. Briefly, the Raman spectrum arises from inelastic scattering of incident photons, which requires a change in polarizability with vibration, as opposed to IR absorption, which requires a change in dipole moment with vibration; the selection rules in the two cases are different. The end result is spectral information that is in many cases complementary to that given by the mid-IR. Because such frequency shifts (i.e. energy transfers) are unique for each molecule, Raman spectra provide detailed types of information concerning the structure and dynamics of materials that are not provided by absorption measurements (Baena and Lendl 2004; Haka et al. 2005). In Raman microscopy, either dispersive technology (for the highest sensitivity) or Fourier-transform technology (for maximum wavelength accuracy) may be employed. One advantage of the Raman scattering technique is that operation is possible in aqueous solutions and as such, it has potential in-vivo applications such as endoscopy or skin-surface analysis (Short et al. 2005).

8.2.3

Summary

In summary, if we compare Raman with FTIR microspectroscopy, a fair generalization would be that IR spectra are easy to interpret and yield information on cellular molecular and macromolecular components (proteins, lipids, nucleic acids); furthermore, it constitutes a valuable tool for the identification of microorganisms. FTIR spectroscopy has a great future for use on tissue samples that contain little or no water, such as de-waxed microtomed sections or partially dehydrated tissue samples (Martin and Fullwood 2007). Raman shifts characterize particular bonds and bond angles, and the spectrum, with its multiple sharp lines, gives an excellent fingerprint. Raman spectroscopy is useful for analyzing molecules without a permanent dipole moment that would not show up on an IR spectrum. A useful “exclusion rule” states that for molecules with an inversion center, no modes can be both IR- and Ramanactive. The main difficulty of Raman spectroscopy is separating the weak inelastically scattered light from the intense Rayleigh-scattered laser light. The technique is experimentally “more complicated”, as different tissue types and samples have different needs for choice of optimum excitation wavelength (to avoid obscuring fluorescence or tissue damage) (Dukor 2002). As an example of the biochemical applications of Raman spectroscopy, we note that it has in many hollow organs diagnozed conditions such as dysplasia in Barrett’s oesophagus, adenomas in the colon and squamous dysplasia in the larynx (Stone et al. 2000). Raman spectroscopy does not suffer from the “water problem”, and accordingly, has more potential in the supremely important area of in-vivo diagnosis (Perelman 2006): its sensitivity is comparable with that of FTIR spectroscopy (Chowdary et al. 2006) although it is significantly slower in collecting high-resolution data (Wang and Mizaikoff 2008). Much effort is currently being devoted to the possibility of using Raman spectroscopy

8.3 Development of near-field techniques

as a clinical tool, involving, for example, fiber-optic systems based on diode laser illumination. Recent technological developments now make it possible to differentially select Raman spectra below several millimeters of turbid media such as tissue. In the long term, it may prove possible to use Raman spectroscopy to provide automatic diagnosis for a number of conditions. Later, we outline current possibilities for exploiting the benefits of Raman spectroscopy for imaging at the sub-μm level.

8.3

Development of near-field techniques

Despite the achievements of these microspectroscopic methods, there remain the restrictions imposed by sample preparation requirements, as well as the major problem of diffraction, which effectively imposes limits on the lateral spatial resolution. New far-field optical techniques such as super-resolution fluorescence microscopy have recently achieved lateral resolutions of below 100 nm, but extending IR investigations to below the diffraction limit requires the use of more specialized approaches, using, for example, one of the various types of near-field scanning probe. A number of analytical techniques discussed earlier may, in principle, be incorporated into the design of a scanning probe microscope, such as an atomic force microscope (AFM) or a scanning near-field optical microscope (SNOM or NSOM). Until recently, most attempts to apply scanning probe technology to localized spectroscopy have employed UV/visible absorption or photoluminescence to provide the signal that determines the contrast in the microscope image. The last few years have seen efforts to harness the analytical power of IR spectroscopy and Raman scattering (for reviews see Dragnea et al. 1999; Dragnea and Leone 2001; Oesterschulze et al. 1994, 1996; Pollock and Smith 2002).

8.3.1

Scanning near-field optical microscopy

With SNOM, in order to achieve the near-field criterion, the size of the light source must be less than the optical wavelength. Examples of such a source are (i) an aperture at the tip of a tapered optical fiber or at the hollow apex of a modified AFM cantilever, or (ii) a minute object that acts as a light-scattering source (“apertureless SNOM”). Fiber-based IR-SNOM, reviewed by Pollock and Smith (2002), tends to suffer from the generic constraints of low signal levels and difficulties in preparing high-throughput fibers, especially at wavelengths greater than ∼4 μm. IR absorption cross-sections of most materials are small. Moreover, typical photoconductive detectors for IR wavelengths are several orders of magnitude less sensitive than the photomultiplier tubes or avalanche photodiodes used for visible SNOM. A promising apertured SNOM technique uses a modified AFM cantilever at whose hollow apex an aperture has been defined by direct-write electron-beam lithography (Zhou et al. 1999). A popular version of apertureless SNOM employs the near-field of a

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Fig. 8.1 (left) Phase s-SNOM image of 18-nm diameter tobacco mosaic virus particle at 1858 cm−1 . The image contrast results from the amide I vibrational resonance of the virus shell proteins. (right) IR s-SNOM phase spectrum of a single virus particle. Each data point is the average, over the virus interior, of data from many images, and the curve represents a model predication (reprinted in part, with permission, from Brehm, M., Taubner, T., Hillenbrand, R., Keilmann, F. (2006) c 2006 American ChemNano Lett. 6, 1307,  ical Society).

back-illuminated subwavelength aperture to provide highly localized excitation of a sample, i.e. the use of a subwavelength scattering center to permit the far-field detection of the evanescent near-field of a sample. At present, the most promising approach appears to be IR apertureless scattering SNOM (s-SNOM, see Keilmann and Hillenbrand 2004), since the resolution is comparable to or better than fiber-based systems, throughput is not an issue and, probably most important in terms of SNOM, it is gaining wider acceptance as an analytical technique because the difficulty in preparing high-quality apertured probes is eliminated. The two approaches, sometimes respectively referred to as illumination and collection mode SNOM, can achieve spatial resolution in the range 10–100 nm. The local optical response measured by elastic light scattering from a tip relates to the complex dielectric value of the sample. Hence, s-SNOM can assess the sample’s optical dispersion, from which one can recognize the IR fingerprint signature (Brehm et al. 2006; Cvitcovic et al. 2007). The theoretical link between the s-SNOM signal and the spectral signature involves some advanced physicochemical modelling: the IR field concentration at the probe tip excites a collective oscillation of phonons in the sample, and if the sample material and crystal structure in a particular locality fulfils certain conditions, this switches the dielectric constant of that locality to a negative value. This then creates new electromagnetic modes that are confined to the surface (“surface polaritons”); the IR light is confined likewise, a proportion thereof is scattered and can be detected interferometrically. Brehm and coworkers (2006) obtained the local fingerprint IR s-SNOM spectra of 30 nm to 70 nm PMMA beads and 18 nm cylindrical tobacco mosaic viruses (Fig. 8.1). It has since been shown that s-SNOM can generate materialspecific image maps of sub-10 nm gold and polymer particles adsorbed on a Si substrate (Cvitcovic et al. 2007). In practice, IR s-SNOM fingerprint signatures such as this contain much less detail than does a true IR absorption spectrum. Nevertheless, chemical recognition by IR s-SNOM could prove to be a robust, label-free analytical tool for nanoscience, as claimed by researchers in the Nano-Photonics group at CeNS Martinried. Their pioneering work (see, for example, Keilmann and Hillenbrand 2004) included a demonstration of spatial resolution close to 30 nm and detailed chemical contrast in phase-separated polymer films with better than 100 nm resolution.

8.3 Development of near-field techniques

8.3.2

Near-field Raman microscopy

A similar principle of employing a subwavelength scattering center is used in near-field Raman microspectroscopy. It is possible to achieve mapping by surface-enhanced Raman scattering (SERS) at nanometer spatial resolution (Smith et al. 1995; Webster et al. 1998; Pettinger et al. 2004). This offers the prospect of operation in aqueous solutions where the strong IR absorption of water causes problems for IR measurements. The signal upon which it relies is up to six times weaker than when using IR microspectroscopy, but the efficiency of the effect depends strongly on the roughness of the AFM tip surface. A single particle (∼100 nm diameter) of metal at the apex of a tip can act as a “hot spot”, giving rise to highly intense signals (Wang et al. 2005). In this form, the technique has alternatively been termed tip-enhanced Raman spectroscopy (TERS). In near-field optical sensing, as reviewed by Vo-Dinh (2008), a bioreceptor such as an antibody, enzyme, protein, or nucleic acid uses a biochemical mechanism to give a “molecular recognition” signal, which may then be sensed using, for example, an optical fiber (or indeed, any of a large variety of other types of transducer). Nanobiosensors have been described that can use either of two distinct types of biological-sensing molecule: antibodies, or synthetic peptide substrates coupled to a fluorophore. The optical nanofiber is given a biosensitive layer that can either contain biological recognition elements or be made of biological recognition elements covalently attached to the transducer. The interaction between the target analyte and the bioreceptor produces a physicochemical perturbation that can be converted into a measurable effect. Nanosensors have been developed and used to detect biochemical targets inside single cells, and recently, the combination of s-SNOM and SERS has been demonstrated to detect biochemicals on solid substrates with subwavelength, 100-nm spatial resolution (Vo-Dinh et al. 2006). These authors reported the use of the SERS technique as a tool for detecting specific nucleic acid sequences, demonstrated the possibility of using Raman and/or SERS labels as gene probes, and cited SERS enhancements from 1013 to 1015 , thus anticipating the possibility for single-molecule detection with SERS. Figure 8.2 shows a SERS image and spectrum of Chinese hamster ovary (CHO) cells incubated with silver SERS nanoparticles functionalized with DNBA and antiepidermal growth factor (EGF) antibody. The SERS image was acquired at the major SERS peak at 1333 cm−1 using a 633nm HeNe laser for excitation. Although the difficulty of low signal levels in near-field spectroscopic imaging is compounded when we attempt to use Raman scattering or IR absorption as the contrast mechanism, the potential benefits of high-spatial resolution chemical mapping will continue to be a potent driving force in the development of vibrational near-field spectroscopy. The exciting developments of scanned metal particle probes that may permit high degrees of surface enhancement to increase Raman scattering (in some samples) will probably attract a great deal of attention in the near future and the use of metallized apertureless probes seems likely to become the method of choice, provided that ease and reliability of fabrication can be achieved. For further discussion of IR and Raman

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SERS intensity (A.U.)

(c)

Fig. 8.2 Global SERS images and spectra of CHO cells incubated with nanoparticles functionalized with DNBA and antibody to EGF. The bright-field image (a), the total SERS image (b), and the spectrum (c) of a CHO cell are shown. Using a 633-nm HeNe laser for excitation, the image (b) was acquired at a SERS intensity signal of 1333 cm−1 , which showed correlation with the DNBA characteristic peak in (c) (reprinted from Vo-Dinh, T., Kasili, P., Wabuyele, M. (2006) Nanomedicine: NBM 2 c 2006 with permission from Elsevier). 22, 

45

30

15

400

1200

2000

Raman shift / cm–1

surface-enhanced vibrational spectroscopy (SEVS) techniques, see Chapter 15 by Kodali and Bhargava in volume III of this Handbook.

8.3.3

Photothermal microspectroscopy (PTMS): nanosampling

We next describe the use of near-field techniques in which temperature rather than light intensity is measured, using a thermal AFM probe. Various versions of microscopy that exploit the propagation of thermal waves have been described, such as photothermal reflectance microscopy/thermoreflectance microscopy (see, for example, Dietzel et al. 2003; Tessier et al. 2003). For microscale and submicroscale imaging, as with a standard AFM, the height of the probe above the surface being scanned is controlled by a feedback system that maintains constant the force between probe and surface: thus the probe height is used to create image contrast. The thermal type of probe allowed scanning thermal microscopy (SThM) to be developed and exploited to detect subsurface structure (as reviewed by Majumdar 1999). In the form of micro- or nanothermal analysis (nano-TA; Pollock and Hammiche 2001), the technique was extended to include chemical characterization and fingerprinting at the scale of a few μm, and has been used in combination with conventional Raman microscopy (Ye et al. 2007). By exploiting the photothermal effect, the same thermal probe has made possible the development of PTMS (Hammiche et al. 1999), which shares with photoacoustic spectroscopy the ability to be

8.3 Development of near-field techniques

used on a broad range of material types including thin films and powders: the requirements for sample preparation are far less strict in photothermal, as compared to most conventional methods of spectroscopic analysis (Hammiche et al. 2004a). In contrast to IR or Raman s-SNOM, the spectra given by PTMS correspond closely to ATR IR spectra (Bozec et al. 2001) and may readily be interpreted using established methods of data analysis. As will now be described, PTMS is opening up new avenues in biomedical research where localized IR spectroscopic data at subwavelength spatial resolution are required. As in SThM, the conventional near-field tip is replaced by a miniature thermometer. This sensor is integrated in a cantilevered structure, so that spatial scanning yields topographic images as usual, although in practice the spatial resolution will normally be lower than that of a standard topographic AFM image. Areas of interest may then be selected for sampling by IR spectroscopy. If a sample absorbs IR radiation to which it is exposed, it heats up. IR absorption causes photon-induced vibrations of specific molecular bonds, such as stretching and bending. As these molecular vibrations relax, evanescent thermal waves are emitted via a non-radiative process (Williams and Fleming 1997). If the IR beam is intensity modulated, as in a normal FTIR spectrometer, the measured temperature, and hence the output of the thermal probe, fluctuates accordingly: moreover, variation of the modulation frequency may be used to control the depth below the surface that is being sampled. In the FTIR version of PTMS, this time-varying output signal is amplified and fed into the external input of the same spectrometer, thus providing an interferogram that replaces the interferogram normally obtained by means of direct detection of the IR radiation transmitted by the sample. The Fourier-transform algorithm is performed on this interferogram after digitization, in order to convert it to a photothermal spectrum by transformation from the time domain to the frequency domain to give a spectrum (Fig. 8.3). So that this may be performed as a function of location on the sample surface, the probe is mounted on a simple positioning system, or, if images are required, in an AFM. In either case, an appropriate optical interface is required, to direct the spectrometer’s external IR beam onto the sample and to increase the flux at the sample surface in order to increase the SNR. The IR beam is focused to a spot, which is 2 mm in diameter. Probe and sample surface are brought into contact at the focal point.

8.3.4

True absorbance spectra from PTMS

Even in applications where spatial discrimination is not required, PTMS has certain other advantages over the various other optical detection methods. As with Raman chemical imaging, PTMS requires little or no sample preparation. With Raman, physical sample sectioning may be required to expose the surface of interest, with care taken to obtain a surface that is as flat as possible. The conditions required for a particular measurement dictate the level of invasiveness of the technique, and samples that are sensitive to high-power laser radiation may be damaged during analysis. PTMS does not suffer from these imitations. However, unlike PTMS, Raman is relatively insensitive to the

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Fig. 8.3 Composite polymer sample: industrially prepared thin film of polymer X sandwiched between two slabs of polymer Y (from Hammiche, A., Bozec, L., German, M.J., Chalmers, J.M., Everall, N.J., Poulter, G., Reading, M., Grandy, D.B., Martin, F.L., Pollock H.M. c (2004) Spectroscopy 192, 20 2004 Spectroscopy Magazine, reproduced with permission). The junction is shown in cross-section. (Top left) Thermal conductivity image (75 μm × 75 μm) obtained with the bowtie thermal probe. (Top right) Tapping mode AFM phase image, different area (50 μm × 50 μm) of same interface, acquired with a 1.5-μm diameter Nanonics thermal probe (Grandy, D.B. and Reading, M. (2004), reproduced with permission). The corresponding PTMS spectra labelled “sample + background” are derived from data obtained at points marked on the top left image, where the left- and right-hand sets of spectra correspond to upper and lower crosses, respectively. As discussed later, the probe itself gives a background spectrum that is subtracted to give the spectra that characterize these regions of the sample.

presence of water in the sample, and is therefore useful for imaging samples that contain it. Optical-detection FTIR microspectroscopy may not be suitable for IR-absorbing sample materials. Samples that are opaque to IR radiation, such as condensed chromatin, may be strongly IR-absorbing and, thus, difficult to analyze in transmission mode. PTMS permits the direct measurement of heat generated as a result of sample material absorbing radiation, and hence generates true absorption spectra. In the absence of a brilliant light source such as SRS, PTMS allows individual types of suspicious cells to be selected at the bench-top. Thus, any diluting effects derived from an IR signature of normal adjacent cells are eliminated, and differences in spectral abnormalities are greatly enhanced. Application of multivariate data analyses to derived spectral datasets enables the identification of the molecular groups responsible for differences between different categories of cell. This approach has been employed to generate a spectroscopic

8.4 Towards a brilliant benchtop IR source 297

fingerprint for susceptibility-to-adenocarcinoma in the prostate (German et al. 2006a); to monitor cell-cycle distributions in a mammary cell line (Hammiche et al. 2005); and, to discriminate different categories of exfoliative cervical cytology (Walsh et al. 2007a, 2008b). The same techniques may be used to identify a new type of marker for stem cells (Bentley et al. 2007; Grude et al. 2007, 2008; Walsh et al. 2008a). Standard reflection/absorption FTIR methodologies require that samples be deposited on specially coated glass slides (Cohenford and Rigas 1998; Argov et al. 2002; Tobin et al. 2004a). In contrast, PTMS generally requires little or no sample preparation. Although another spectroscopic method, ATR spectroscopy, allows for spectrum collection regardless of sample thickness, it requires that the sample be sufficiently smooth or soft, so that the surface of the measuring crystal can be brought within range of the evanescent wave (Kazarian and Chan 2006). Current techniques often rely on the interrogation of a dried monolayer sample; spectra of samples in their original aqueous state often exhibit less-intense amide II and phosphate bands, which suggest that dehydration affects the secondary structure of proteins (Dukor 2002). Finally, we mention an additional use of thermal probes, known as thermally assisted nanosampling (Price et al. 2000; Reading et al. 2002; Harding et al. 2008). Here, material is extracted from the surface of a specimen by using the heated tip to soften a region and then pulling the tip away as it is cooling, in order to leave a small quantity adhered to the probe. This may then be qualitatively and quantitatively analyzed by a number of alternative techniques, including micro-thermogravimetric analysis, NMR spectroscopy, size-exclusion chromatography or spectroscopy. As required, the analysis is performed either in situ or after the material had been removed or dissolved from the tip (on a later page, Fig. 8.10 gives an example). This could be an attractive proposition in a case where non-destructive imaging is not required, and as regards simplicity, sensitivity and speed of data acquisition, the technique can compare favorably with microspectroscopy. A recent sophisticated use of thermally assisted nanosampling has been described by Park et al. (2008), who used it with Raman spectroscopy to identify paraffin at the femtogram level. Of course, use of the technique in the biomedical field will be limited by the fact that it is applicable only if the sample material’s surface tension and viscosity allow it to be mounted on the tip in this way.

8.4

Towards a brilliant benchtop IR source

As outlined above, an FTIR setup has been employed in most of the PTMS studies performed so far. They have, to some extent, suffered from the limitation imposed by the weakness of conventional “Globar” thermal sources of white radiation, resulting in long acquisition times, measured typically in minutes per spectrum. For this reason, to our knowledge no work has yet been reported in which the AFM was used to obtain PTMS image contrast. Promising alternative light sources (Table 8.1) are (a) the SRS (as discussed above), and (b) high-power monochromatic but tuneable sources of mid-IR radiation, which recent advances in laser technology have made available.

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research Table 8.1 Types of IR light source. Type of source

Flux (watts per square millimetre, per wavenumber in cm−1 taken at 2000 cm−1 )

Thermal (Globar)

1.7 × 10−5 (estimated from an assumed value of 0.1 W/mm2 for the whole of the wavelength range) 10−3 to 10−2

Synchrotron beam focused to 15 μm × 20 μm (Miller et al. 1997) OPA (unfocused beam)

10−3 averaged over time

Work in progress aims to incorporate a PTMS facility into a SRS setup (Bozec et al. 2002; Hammiche 2006). In contrast, it is highly likely that for routine laboratory use where near-field microscopy with spectroscopic image contrast is required, a laser source will in due course become standard. For biomedical studies, typically one would like a laser that is tunable over the wavelength range 5 μm to 10 μm, costing no more than a few thousand dollars and with modest space requirements, given the space occupied by the scanning probe equipment and spectrometer. Such ideal midIR light sources do not (yet) exist. However, often it will not be necessary to obtain data over this entire spectral range: in situations where the principal constituents are known a priori, and where the image contrast is to be determined by the distribution and concentration of, say, one or two of these constituents, then the only light-source wavelengths needed will be those at which these particular constituents absorb strongly. The two most promising types of source are the quantum cascade laser and the optical parametric amplifier (OPA). Quantum cascade lasers (QC devices) use transitions between subbands created by quantum confinement in a semiconductor heterostructure (Faist 2006). In the so-called distributed-feedback-type, a grating is etched in the active region, to force the operation of the laser at a very specific wavelength determined by the grating periodicity. As a consequence, the laser operates at a single frequency that may be adjusted slightly by changing the temperature of the active region. QC devices have been shown to be capable of operating at wavelengths ranging from below 4 μm up to tens of μm. Tuning is performed by varying the temperature of the active region, but each laser has a limited tuneability, the tuning range being a few per cent (from 1% to 8%) of peak emission wavelength. However, Maulini and coworkers (2005) report the use of a combination of two tuning methods, coarse (achieved by rotating the grating) and fine (changing the cavity length and the laser chip temperature), to obtain a wavenumber range of 175 cm−1 around a center frequency of 1850 cm−1 . In any case, in principle several such lasers could be custommanufactured, each delivering a different wavelength chosen to coincide with a particular spectral band of relevance to the intended experiment, and driven in sequence by a common power supply. Thus, a QC device could be an excellent choice for applications where there is no need to cover the whole mid-IR region, but only a few specific wavelengths, in cases where the requirement

8.5 Possible advantages of using normal AFM probes 299

is to pick out the spatial distribution of one or two substances whose chemical composition is already known beforehand. The OPA type of bright tuneable source employs an optical parametric oscillator (OPO) (Vodopyanov et al. 2000, 2004), i.e. an oscillator operating at optical frequencies whose resonance frequency and damping parameters vary with time in a defined way. Non-linear optical interaction is used to convert a near-IR monochromatic input (“pump”) laser wave into two output waves of lower frequency, via a three-wave mixing process. This process is known as optical parametric generation (OPG). The heart of an OPO is a non-linearoptical crystal, consisting, for example, of epitaxially grown GaAs, AgGaS2 , lithium niobate, or ZnGeP2 . In this crystal, the pump photon decays into two less-energetic photons (signal and idler). Tuneability is achieved by using a crystal of periodically modulated orientation, which thereby behaves as an artificially created grating of optical non-linearity: the period of this grating determines the output wavelengths of the OPO. The resultant OPA source of laser radiation can thus emit light of variable wavelengths. Tuning either the pump wavelength or the temperature of the crystal allows the mid-IR output to be varied over a wide range, e.g. from 0.5 μm to as much as 10 μm if provision is made to allow a choice of OPO crystal. Typical specifications include a pulse of typical length in the tens of ps range, delivered at a repetition rate of 30 Hz. Bozec and coworkers (2001) first demonstrated the use of an OPO for PTMS, and resolved the C–H band in polypropylene. One may make order-of-magnitude comparisons of the flux obtained using different types of light source (Table 8.1). Two other types of source have in the past attracted attention in this context, namely lead-salt and tuneable dye lasers. The leadsalt type is flawed as regards reliability and ease of manufacture, and has a relatively weak output. Tuneable dye lasers are tuneable over a wide range but to date, only in the near-IR (maximum wavelength less than 2 μm if the technique of difference frequency mixing of Q-switched Nd:YAG laser with the dye laser radiation is used (Chatterjee et al. 2001).

8.5

Possible advantages of using normal AFM probes

Later, we describe various alternative types of near-field thermal probe. One disadvantage is that they tend to require a larger tip than that of a normal AFM probe, and to have a relatively high cost. Accordingly, the possibility of using the sharper and cheaper normal type of probe to obtain an IR spectrum is attractive. Indeed, AFM-type microcantilevers have been used as highly sensitive and selective chemical sensors. One approach is to coat the surface of the lever with a material that selectively adsorbs or binds a given target substance, e.g. cellular material. When the cantilever comes into contact with the target substance, the interaction will bend it and will shift its resonance frequency. For a variety of reasons, so far the use of cantilevers as chemical sensors has involved physical contact with the analyte, thus severely limiting the possibility of an automated, controlled and localized analysis of tissue samples or cellular material.

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The possibility of using standard AFM cantilevers to obtain IR spectra has been explored by a number of research teams. For example, Hammiche and coworkers (2007) have used a photothermomechanical recording method, employing a standard AFM cantilever probe of low spring constant. The operation is simple: the probe is placed a few μm above the surface of the sample that is illuminated by IR radiation. As the sample absorbs radiation it heats up, and a heavily damped heat wave is generated in the layer of air in contact with the sample surface. The cantilever, being located within this layer, vibrates. These cantilever mechanical vibrations are sensed as in a standard AFM, and the recorded signal is fed to the external input of an FTIR spectrometer as in a PTMS experiment. Fourier transformation then generates a true absorption spectrum. A significant advantage of this method, as compared with other microcantilever techniques, is its ability to obtain spectra in non-contact mode, so that damage to delicate samples is minimized, and the probe may be used many times with no danger of contamination by sample material. When this is not an issue, the setup will also operate in contact mode (Hammiche et al. 2000a,b, 2004b). Here, the heat generated from IR absorption causes the sample surface to expand, so that the cantilever is deflected upwards (“scanning thermal expansion microscopy”). In either mode, the direct measurement of IR radiation absorption means that optically opaque samples that cannot be analyzed by means of transmission methodologies can thus be analyzed. In an analysis of different categories of exfoliative cervical cytology, normal, low-grade, high-grade and severe dyskaryosis (? carcinoma) were successfully distinguished using this approach (Hammiche et al. 2007). Dazzi (2008) has described a promising technique, known as AFM-IR or photothermally induced resonance (PTIR), which likewise uses a normal AFM probe to measure the thermal expansion. This photoacoustic method differs from the photothermomechanical method described above, in that it employs irradiation from a pulsed source, in this case a free-electron laser. Typically, each macropulse contains about 500 micropulses of ≈2 ps long. The sample is deposited on the upper surface of an IR-transparent prism of ZnSe, silicon or germanium, and the pulsed IR laser illuminates it from below at the total internal reflection angle (30◦ ). The pulse produces a very rapid transient local absorption-induced heating of the sample, giving an acoustic wave that propagates up the AFM cantilever. Detection of the transient resonances of the cantilever vibration is a much more sensitive way of quantifying the surface thermal expansion of the sample than measuring this directly. Depending on the Q of the cantilever, the duration of the signal is a few hundred μs. Because the time response of the AFM feedback loop is too slow to compensate for the transient deformation induced by the laser burst, whereas the detection of the cantilever resonances is very rapid, separate images of the topography and local IR absorption can be generated. Moreover, the effect of thermal diffusion, which could limit spatial resolution, is minimized. The wavelength spectra are obtained by recording the FFT power density spectra of the cantilever deflection as a function of wavelength, and it can be shown that the maximum amplitude of the cantilever oscillation is proportional to the rate of absorption of thermal energy.

Recent publications have described the successful application of the AFMIR technique to a variety of samples such as DNA, viruses and bacteria (Dazzi et al. 2006, 2007). In the work published to date, the samples, illuminated from below, were sufficiently thin for the probe at the upper surface to detect the signal that was generated by total internal reflection. By selecting the principal frequency component (66 kHz) of the cantilever vibration, the AFMIR spectrum of the bacterium is obtained. As can be seen in Fig. 8.4, as measured on a small quantity dried from a concentrated solution of bacteria the AFM-IR spectrum fits very well the standard FTIR spectra of Escherichia coli. The absorption bands (amide I, II, III and the 1080 cm−1 DNA line) and their relative amplitude are well reproduced. Figure 8.5 shows an example of a chemical mapping image of a single bacterium, obtained at the amide II wavenumber. Individual isolated hyphae (filaments) of a fungal pathogen, Candida albicans, have been mapped at two different wavelengths, showing, for example, that the polysaccharide known as mannan is distributed nonuniformly within a hypha.

8.6

Experimental procedures for PTMS

The spatial resolution achieved with the PTIR technique is impressive, but there are two principal limitations. Scattering reduces the amount of light transmitted through, and absorbed by, an isolated object of subwavelength size, and to allow for this effect is not straightforward. In addition, artefacts in the form of variations in image intensity can appear if the distribution of light through the thickness of the sample is not uniform: this requirement sets an upper thickness limit of the order of λ/15 (e.g. 400 nm at a wavelength of 6 μm). The s-SNOM technique of the CeNS Martinried group could require a significant new theory to interpret the spectra generated, especially if the spectra generated are to be compared with those stored in current libraries listing the spectra available from bulk IR techniques with a view to interpretation and identification of materials. In contrast, one benefit of the PTMS approach is that the spectra generated to date have reliably and reproducible correlated well with those generated by photoacoustic IR spectroscopy (Bozec et al. 2001, 2003; Hammiche et al.

301

Absorbance/A.u.

8.6 Experimental procedures for PTMS

Wavenumber/cm–1

Fig. 8.4 E. coli bacteria: comparison of the FTIR spectrum and the AFM-IR spectrum. The black curve plots the AFM-IR amplitude of the 66-kHz vibration component of the cantilever. The gray curve shows the corresponding FTIR spectrum (reprinted from Dazzi, A., Prazeres, R., Glotin, F., Ortega, J.M. (2006) Infrared Phys. Technol. 49, 113 c 2006, with permission from Elsevier). 

Fig. 8.5 (Left): Standard mode topographic AFM image of E. coli bacterium topography (1.25 μm × 1.25 μm). (Right): The corresponding AFM-IR chemical mapping at 1550 cm−1 , i.e. on the amide II absorption band. The IR absorption is seen to have a non-uniform distribution within the bacterium (from Dazzi, A., Prazeres, R., Glotin, F., Ortega, J.M. (2006) Infrared Phys. c 2006 Elsevier, reproTechnol. 49, 113  duced with permission).

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2004a). As we have seen, in general, use of the PTMS technique does not impose severe requirements for special methods of sample preparation: the sample is illuminated from above, so that there is no restriction on sample thickness (in principle, such “top-down” illumination could be used for AFMIR likewise). Of course, top-down illumination means that the probe system as well as the sample will give a thermal signal, which then must be taken into account as exemplified in Fig. 8.3. However, the issue of fixation of tissues when applying such techniques to investigate biological investigations calls for comment. Many disease states require close monitoring over time and an archived tissue resource would be useful for intraexperimental comparative purposes. The first priority for such a diagnostic tool is preparation to prevent self-destruction or autolysis of tissue. This is required to preserve tissue integrity, and should not be confused with sample preparation for analysis. Fixation of samples prior to analysis, itself an artefact, is necessary to archive tissues prior to transport to diagnostic laboratories. Studies using other methods have adopted elaborate methodologies, such as tissues from paraffinembedded blocks that were sectioned and mounted on ZnSe or BaF2 windows prior to dewaxing (Argov et al. 2002; Gazi et al. 2004), tissues frozen on resection and cryomicrotomed onto such windows prior to air drying (Tobin et al. 2004a), exfoliative cytology in preservative prior to application to windows (Cohenford and Rigas 1998), or even the biomonitoring of viable cells (Holman et al. 2000) or cells air dried (Salman et al. 2003) on ZnSe windows. In contrast, recent PTMS studies simply required 70% ethanol to fix cells in situ. Based on staining quality and preservation of tissue architecture, ethanol- or formalin-fixed tissues have been noted to retain their architectural features in a superior fashion to frozen tissues (Gillespie et al. 2002). More importantly for this study, these authors noted that ethanol fixation, when compared to formalin fixation, consistently allowed for better visual identification, using microscopy, of prostate-epithelial-cell nuclear detail. This permitted more accurate grading of hyperplastic, pre-malignant, and tumour cell nuclei. Ethanol fixation through reversible precipitation appears not to induce denaturing effects (Sainte-Marie 1962) and will rapidly dehydrate and preserve the integrity of cellular material. To date, analysis of cellular material using PTMS suggested that the ethanol-fixed samples studied were stable over long periods of time (∼6 months). For PTMS there are three types of thermal probe in general use. Whether a resistive or a thermocouple type of probe is used, the probe is incorporated into a cantilever structure, so that when mounted in an AFM, force feedback may be employed in the usual way, for positioning and topographic imaging. Most of the research published to date has employed the resistive “Wollaston” probe, which consists of a 5-μm diameter, 90% platinum-10% rhodium looped filament of Wollaston wire, approximately 200 μm in length (Dinwiddie et al. 1994). This type of probe was originally developed for purposes of SThM and micro-TA, and accordingly can be used as a heater at the same time as it acts as a resistance thermometer. For PTMS, the heating facility is not strictly essential, although the ability to clean the probe by heating is often a very great advantage. The temperature is sensed via changes in the electrical resistance of the apex of the V-shaped platinum/rhodium wire. Resistance

8.6 Experimental procedures for PTMS

thermometry is based on the temperature dependence of the resistance of a suitable sensing element. The element resistance is determined by passing a bias current through it and measuring the voltage induced across it. Sensitivity can be improved by increasing the length of the filament and/or by increasing bias current. If spatial resolution is not of importance, the size of the filament can be increased (although Johnson noise has to be taken into account). Bias current can be increased up to the point where Joule heating does not affect the sample material. Given the scale of this device, the ultimate spatial resolution is one or two μm at best, and the cost is relatively high since the construction does not lend itself to large-scale automated production. However, for many applications it has proved to be very reliable, especially as it is robust both thermally and mechanically. Microfabricated types of resistive thermal probe have recently been manufactured, allowing measurements to be performed at a still finer scale. The probes used for nano-TA are silicon MEMS-fabricated probes, similar to standard AFM probes but incorporating a resistive heater / sensing element at the end of the cantilever (Kim et al. 2007). Another type of microfabricated probe, a thermocouple type in which the temperature-sensing element is closer to the tip / sample contact interface and known as the “bowtie” probe, is commercially available and is better suited to PTMS at the highest possible spatial resolution. Device materials used include silicon nitride, silicon oxide, gold, nickel, and palladium (Zhou et al. 1998; Hammiche et al. 2000b). Electronbeam lithography is used to write the devices onto the square tops of AFM tips in the form of blunt wet-etched pyramids (240 probes at a time on a 3-inch silicon wafer). After the substrate has been etched to the pyramid-definition level, plasma-enhanced chemical vapor deposition is used to deposit the layer required to form the cantilever. The simplest version of probe consists of a simple “cross” junction formed by running two narrow (∼80 nm) wires of gold and palladium across the probe apex, to form a thermocouple junction where they meet. The size of the active region (a factor in determining maximum spatial resolution) is the size of the junction, and not the area of the blunt apex of the tip. However, the size of this apex may prevent the probe from sensing regions within narrow pits or holes in the sample. To remedy this, the electron-beam lithographic process is adjusted so as to give a cantilever shape that ends in a sharp point near the top of the pyramid. The sensing wire is then written over the end of the sharp point in such a way that the metal is half on and half off the tip. Probes of this type have been demonstrated to give a lateral resolution better than 100 nm and a temperature resolution better than 0.1 ◦ C. This fabrication procedure can readily be modified so as to produce probes of the resistive type, when the ability to clean the probe by heating is required (Dobson et al. 2005). A third type is the glass-sleeved platinum probe supplied by Nanonics Imaging Ltd (Jerusalem, Israel). The sensing element of the Nanonics probe (Dekhter et al. 2000) is intermediate in size between that of the Wollaston and bowtie probes, and is fabricated from a glass bar containing two platinum wires, using a special pipette-making procedure. Given its ability to provide spectral data (as explained below), the instrument can in principle be operated in either of two modes: the mapping mode, in which IR spectra are obtained from individual regions selected from the

303

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research

SThM image; and the IR imaging mode, in which an image is obtained whose contrast is determined by the local concentration of material that absorbs in a given band of the spectrum. To date, data from the IR imaging mode are as yet unpublished. With the resistive type of probe, typically the probe tip is cleaned by Joule heating (to ∼600 ◦ C when the Wollaston type is used). Photothermal interferograms are averaged over time (typically 2000 co-additions) for enhancement of signal-to-noise characteristics, followed by conversion by Fourier transformation to spectra displaying photothermal amplitude expressed as a function of wavenumber (cm−1 ). Atmospheric background spectra are taken in non-contact mode and inspected to ensure that the tip is free from biological contamination. With top-down illumination, a spectroscopic measurement consists of two recordings, one with the probe in contact with the sample and the other with the probe alone in the path of the IR beam (see Fig. 8.3). The latter results in a background spectrum that, in the case of the metallic Wollaston probe, is just the emission spectrum of the source, with superimposed bands arising from absorption by CO2 and water vapor as the beam travels through the air. The micromachined probe, with its nonmetallic constituents, gives a background spectrum that includes in addition a strong absorption band between 1200 cm−1 and 700 cm−1 , with weaker bands between 2200 cm−1 and 2000 cm−1 and between 3200 cm−1 and 3000 cm−1 . The glass component of the Nanonics probe also gives a background spectrum. In each case, the absorption spectrum of the sample is obtained by normalization. A first normalization is then performed, by dividing the sample spectrum by the background. After baseline subtraction and normalization to a prominent peak such as the amide II peak (≈1540 cm−1 ), the data are then ready for processing, as discussed below. It is likely that a still more definitive spectrum would result if the improved method of spectrum subtraction of Mongeau and coworkers (1986) were to be used.

8.7

Prospects for high spatial resolution in near-field FTIR spectroscopy

There are a number of factors determining the spatial discrimination of the PTMS technique. Two levels of requirement can be considered, depending on the application: a “strong” requirement, when no spectral contamination is allowed, and a “weak” requirement in which some spectral contamination is accepted provided that sufficient discrimination is achieved for mapping over a sample. The spatial resolution is limited principally by the size of the probe and the temperature distributions that result from the absorption of IR energy by the inhomogeneities that are to be identified. Consider a phase-separated sample containing domains of two types, only one of which absorbs the radiation and heats up accordingly. Four principal factors will be involved: • the size of the probe, and hence the probe/sample contact area; • the sharpness of the temperature distributions that result from the absorption of IR energy by the inhomogeneities that are to be identified;

8.7 Prospects for high spatial resolution in near-field FTIR spectroscopy

• the finite heat capacity of the probe, and the consequent perturbation of the temperature distributions to be detected; and, • the magnitude of the thermal diffusion length, which depends on the frequency of the temperature fluctuations resulting from the intensity modulation of the FTIR beam. An approximate method of quantifying the effect of modulation uses the concept of thermal diffusion length (L) (Rosencwaig 1980; Almond and Patel 1996; Mandelis 2000). The key property of thermal waves is their evanescent behavior, leading to the possibility of having spectral contamination across boundaries. The damping of a thermal wave is frequency dependent, and for one-dimensional propagation, the length L 1 is defined as the distance at which the amplitude has decayed to 1/e of its original value:  L1 =

μ , πf

(8.1)

where μ is thermal diffusivity and f is frequency. It is well known that in photoacoustic spectroscopy, increasing the modulation frequency may be used to reduce the thermal diffusion length, and thus to reduce the depth of material that is being sampled. With PTMS, for a given wavenumber the modulation frequency is determined by the choice of oscillation frequency of the interferometer mirror. Accordingly, the relation between mirror oscillation frequency and thermal diffusion length allowed Bozec et al. (2001) to explain data on spectroscopic detection of thin layers and subsurface material. The actual modulation frequency is of course not constant, but depends on the wavelength of the radiation as well as the mirror oscillation frequency, leading to a range of thermal diffusion lengths within the sample. Taking into account the frequency content of the interferogram obtained in a measurement on a typical polymer at an oscillation frequency of 2 kHz, L 1 ranges from 18 μm at an optical wavelength λ = 20 μm, to 5.5 μm at λ = 2 μm. Of particular relevance to spatial resolution is the fact that at higher mirror frequencies the effective diffusion length is reduced laterally as well as in depth. Figure 8.6 shows some theoretical examples of heat leakage. In an outstandingly lucid and informative review, Mandelis (2000) has described how the concept of a characteristic diffusion length is valid in the threedimensional case also. It is possible to characterize the situation that involves three-dimensional heat flow as well as temperature modulation, as in the case of a thermal source of finite size, such as a small domain that is photothermally heated. In the limiting case as the modulation frequency is reduced to zero, to use the quantity L 1 that applies for one-dimensional propagation would clearly be invalid. This case was discussed by Smallwood and coworkers (2002), who derived values of an effective thermal wavelength L f . Their three-dimensional finite-element analysis leads to the following simple relation: L f2

×



L 0 −4 + 4 L 1 −4

1/2

+ L 0 −2

 = 2,

(8.2)

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research

(a) temperature Z

Fig. 8.6 Lateral “leakage” of heat from a domain heated by modulated IR absorption: (a) Zero modulation frequency: effect of absorbed IR energy within near-surface region (a disc in the x y-plane), incident flux is 1400 W m−2 , uniform absorbed heat flux (17 W m−2 within a 200-nm band), absorption coefficient 0.4 μm−1 , thermal conductivity 0.15 W m−1 K−1 . The resulting excess temperature distribution is shown in 3 dimensions as a function of radial distance ρ and depth z as given by the formula:

r

X

Z

heated surface layer: disc, 0.3 mm radius, 30 nm thick (b)

T (temperature) y x

T (ρ, z) =

T=

∞ Q s ρ0 /(2ke ) e−oz J0 (σρ)J1 (σρ0 )dσ/σ,

m Kc

e jw

t

-∞

t–t

x

erfc

d

m (t – t)

2

w = 100 Hz

0

where σ is spatial frequency in the radial direction (from Hammiche, A., Pollock, H.M., Reading, M., Claybourn, M., Turner, P.H., Jewkes, K. (1999) Appl. Spectrosc. c 1999 The Society for Applied 53, 810  Spectroscopy, reproduced with permission); (b) If the heated domain is a semi-infinite half-plane (left-hand side, edge is the yT -plane): the relevant equation is as shown. Plots of temperature–amplitude show leakage at two modulation frequencies ω (μ and kc are diffusivity and thermal conductivity); and, (c) Time–distance– temperature plot: the heated material takes the form of a strip of half-width 0.3 μm, and of infinite length in the y-direction. At t = 0 it is sharply defined, and the temperature after a time 1/ω is plotted. (Function used involves error functions and exponential integral functions; see Carslaw and Jaeger 1947.)

w = 1000 Hz

–10

–5

0

10 5 Distance from edge / mm

(c) w = 1 MHz T (temperature) y x

Temperature after time 1/w w = 10 MHz 1 mm

which expresses L f in terms of two other characteristic lengths, one of which is L 1 as defined above. The other is a “zero-frequency thermal wavelength”, L 0 , whose value is of the order of the source dimension (the exact value also depends on values of thermal conductivity and heat-loss coefficient, but is almost independent of the thermal properties of the medium, for a wide range of practical cases). Thus, even with no modulation, the temperature in the medium falls rapidly with distance from the source, with L 0 being governed by its size. Note that this description assumes that the effect of lateral spread of the thermal wave in the adjacent air gap (see, for example, Mieszkowski et al. 1989) may be neglected. Figure 8.7 (top) shows how thermal diffusion length

8.7 Prospects for high spatial resolution in near-field FTIR spectroscopy

307

y] 1

nc

fre

e qu

g[ Lo 5

3 L1 /mm 6

4

2 0 6

2

4

1000

silver, 170 silicon, 73

100 L1 /mm

0 source size /mm

air, 19

10

polymer, 0.1

1

0.1 10

100

1k 10k 100k 1M

Frequency / Hz L f varies with source size and modulation frequency, for a typical polymer of thermal diffusivity 10−7 . At the limit of very large source size and high frequency, the relation reduces to that given by eqn (8.1) (L ≈ L 1 , plane-wave case). This is shown for four different materials in Fig. 8.7 (bottom panel). In practice, the frequency may be varied over a range of 1 kHz to 30 kHz, by varying the mirror oscillation frequency. Bozec (2003) performed PTMS experiments in order to characterize the effect of thermal diffusion on the recorded spectra, using a discontinuous monolayer of 5-μm diameter polystyrene spheres deposited from an aqueous suspension onto a polyethylene terephthalate (PET) substrate. Spectra were recorded at two mirror oscillation frequencies, 2.2 kHz and 10 kHz. The characteristic polystyrene 700 cm−1 ring out-of-plane vibration

Fig. 8.7 Thermal diffusion lengths: (top) variation of effective thermal diffusion length L f with source size and modulation frequency for a typical polymer of thermal diffusivity 10−7 ; and, (bottom) variation of thermal diffusion length L 1 for onedimensional propagation (plane-wave case, very large source size) with modulation frequency, for four different materials (thermal diffusivities are shown in units of −1 10−6 m2 s ). For air and for a typical polymer, thermal diffusion lengths range from 115 μm at a wavenumber of 4000 cm−1 , to 360 μm at 400 cm−1 .

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research

0

Area of the CH2 2940 cm–1 PS peak / A.U.

Fig. 8.8 Effect of increased mirror peak speed on decay of C−H stretch amplitude of PTMS signal, across a polystyrene/rocksalt boundary (Grandy and Reading 2004): (top) Cluster of 5-μm polystyrene particles on rocksalt, which is non-absorbing at these wavelengths. (bottom) Lateral resolution diagram, at two frequencies as indicated (Wollaston probe, 15-min acquisition time). The stars in the top panel shows successive locations at which the spectra were obtained. At the faster mirror peak speed (giving a faster modulation frequency), the decay of the thermal wave as a function of position is sharper, and the signal decays to 50% over about 2.5 μm, in reasonable agreement with theory.

50mm

100mm

1.6 kHz

4.0 kHz

–20

–15

–10

–5

0

5

10

Distance from interface / mm

peak was resolved at a mirror oscillation frequency of 10 kHz, whereas at 2.2 kHz (greater value of thermal diffusion length) it was barely seen as a shoulder on the adjacent strong PET peak. In a separate experiment with a similar layer deposited onto rocksalt, Grandy and Reading (2004) confirmed that at a faster mirror peak speed (giving a faster modulation frequency), the decay of the thermal wave as a function of position is sharper, as shown in Fig. 8.8. In a more precise experiment (Bozec 2003), a bilayer sample of polyethylene (PE) against PET, prepared by embedding a portion of the sample into an acrylic resin followed by microtoming to give a flat surface, the chief feature of interest was the variation of intensity of two PE C−Hstretch vibrations, in a spectral region where no features are seen in the background spectrum of either type of probe. Each of two types of thermal probe was used in turn. First, single-beam spectra of both PET and PE were recorded by placing the probe at 10 μm from the interface on successive sides. Further spectra were taken at successively smaller separations until the distance between two consecutive points became too small for the measured spectra to be distinguished. In Fig. 8.9, we see that there is a variation in the intensity of the C−Hstretch of PE that follows the distance of the probe from the interface (nominal position “0”). This intensity slowly decreases, as the probe moves towards the PET. These signals appear to be well- distinguished provided that the successive measurements are made at positions that are separated by a minimum value, which appears to be 2.5 ± 0.5 μm for the Wollaston probe

8.7 Prospects for high spatial resolution in near-field FTIR spectroscopy

309

1 2

1 2 3 4 5

3 4 5

6

3000

2900 Wavenumber/cm–1

6

3000

3100

2800

2900

2800

Wavenumber/cm–1 10 PE

1 3

4

5 0

2

5

5 PET 10 mm

6

Fig. 8.9 C−H stretch peaks (2960 cm−1 ) from −CH2 group in PE, measured by PTMS at positions close to a PE/PET interface as indicated (spectral resolution −8 cm−1 ; mirror peak speed 0.13 cm s−1 ; 1024 co-additions: (left) with Wollaston probe: spatial resolution 2.5 ± 0.5 μm (estimated error arising from uncertainty in the accuracy with which it was possible to position the probe); (right) with bowtie probe: spatial resolution ≤ 1.2 ± 0.5 μm (from Bozec 2003).

and ≤ 1.2 ± 0.5 μm for the micromachined probe: see in particular traces 4 and 5. We see that the concept of thermal diffusion length can be a useful guide to spatial discrimination: the value of thermal diffusion length and, hence the spectral discrimination, may be varied by increasing the peak speed of the spectrometer mirror, or by using an additional source of optical modulation, such as a chopper. If the conditions are such that there is contamination of the spectra of domain A by that of domain B, but a “pure” spectrum of A is nevertheless required, there are two possible approaches: • Either the data may subsequently be processed by a weighted subtraction of the B spectrum, if this is already known, in order to isolate the spectrum of A. This method requires that the operator can identify spectral signatures from either of the phases in order to perform an accurate weighted subtraction.

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research

• Alternatively, PTMS may be used in combination with the nanosampling technique already mentioned, which avoids the thermal contamination problem when the B spectrum is obtained. A further goal for the technique is that of true IR imaging, with spatial resolution at the tens of nm level. This will primarily involve optimizing the behavior as well as requiring one of the available types of powerful nonbroadband sources, and would provide a useful tool in situations where data covering the whole spectrum are not needed. Moreover, spectral contamination could be acceptable (“weak criterion”), if the requirement is simply for qualitative detection of changes in composition to be detected spectroscopically across small distances to yield image contrast. Finally, it should be remembered that the photothermal-induced resonance technique mentioned above uses normal AFM probes, which tend to give smaller probe / sample contact areas than can be achieved with the more complex thermal probes. Consequently, this approach may prove to have advantages when the highest possible spatial resolution is required, although its potential has not yet been fully explored.

8.8

Spectroscopic detection of small particles

Of obvious interest is the question of the smallest quantity of a given material that will yield useful spectral data. As outlined above, nanosized particles will give recognizable s-SNOM spectra. It has also been shown that a recognizable PTMS spectrum of polystyrene may be obtained from a cluster of as few as two polystyrene spheres of diameter 5 μm, attached to a rocksalt substrate (Fig. 8.10). This was discussed by Hammiche and coworkers (2004b), who also showed that spectra are obtained from samples of even smaller volume if they are picked up by the probe and removed from the substrate using the thermally assisted nanosampling method described earlier. As shown in Fig. 8.10, this method can give significantly greater SNR, since there is no solid substrate to act as a heat sink, thereby lowering the temperature of the sample. We conclude that if a polymer particle is either isolated or surrounded by IR-transparent material, such a sample having a mass of a few tens of femtograms can be analyzed. We conclude that much has already been achieved with the PTMS approach, even at the currently obtained level of spatial resolution. We have seen that the technique provides a novel, non-destructive microprobe approach to midIR spectroscopy, for a wide range of sample geometries and physical forms, and with little or no need for sample preparation. The value of the spatial resolution is not yet comparable with those obtained by s-SNOM (Keilmann and Hillenbrand 2004), which can achieve a resolution in the region of 30 nm. However, unlike PTMS data, s-SNOM data are obtained over only a very small spectral range (typically a few wavenumbers). The ability to combine AFM imaging with localized spectroscopic analysis within a single instrumental procedure such as PTMS may represent an important step forward in the analysis of complex inhomogeneous samples. Subcellular resolution is potentially

8.9 Data analysis

311

Fig. 8.10 Polystyrene microspheres (nominal diameter 5 μm) embedded into the surface of rocksalt (Hammiche, A., Bozec, L., Pollock, H.M., German, M.J. Reading, c 2004 M.J. (2004b) J. Microsc. 213, 129  Wiley-Blackwell, reproduced with permission). (Top left) as imaged with a Wollaston wire probe; (top right) photograph of a few polystyrene spheres picked up by the Wollaston wire probe; (bottom) photothermal spectra obtained at positions shown at top left as (i) a pair of spheres, (ii) a larger cluster, and (iii) from the spheres shown at top right; and (iv) an absorbance spectrum of polystyrene taken from a database.

achievable in the future with the use of new types of micromachined near-field probes under development, and it is likely that fine-scale PTMS measurements will provide key data for the study of a far larger range of samples than can be outlined here.

8.9

Data analysis

It is possible that FTIR microspectroscopy will in due course be incorporated into the battery of tools used routinely in the typical pathology or cell biology laboratory. IR spectra can be acquired very rapidly (minutes or less) using general laboratory methodologies. Problems arise when groupings are to be identified from patterns obtained by the processing of the high-dimensional data, where the variables (such as wavenumbers) may number several thousand. Not surprisingly, as Fig. 8.11 shows, the large number of variables and the superposition of spectral features when fingerprinting whole cells, even if measured one at a time, make it difficult to identify the significant underlying variance. Accordingly, data usually cannot be quantitatively analyzed by univariate methods such as single-peak or peak-ratio evaluation, and the choice of an efficient method of multivariate data reduction and classification (Cox 2005) is supremely important. Such methods may include chemometric analysis

312

Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research

Invasive

2.0 Absorbance / au

Normal

Fig. 8.11 IR spectra acquired from exfoliative cervical cytology. A single spectrum derived from cytology with a histological characterization as normal is compared to one derived from a case of severe dyskaryosis (? invasive) carcinoma (for more detail, see Walsh et al. 2007a,b, 2008b).

1.5 1.0 0.5 0.0 1700

1500 1300 Wavenumber /cm–1

1100

900

(Mark 2002), linear discriminant analysis (LDA) (Fearn 2002), probabilistic or advanced neural networks (Yang 2002; Wenning et al. 2008), and fuzzy logic. The allocation of datasets from “unknown” samples to the correct class(es) (i.e. “normal” vs. “suspicious”) could typically involve reference to a database of fingerprint spectra. Ideally, it should be able to cope with the different spectral ranges suited to different methods of preparing a given type of sample. What might be needed in each case may include procedures for filtering, normalization, classification, validation, estimation of confidence limits, etc., and in general will probably need to be tailored to the particular situation. For this to succeed, one approach will be to employ some validated and accessible statistical package that delivers a yes/no answer (or a red light/green light). Fortunately, a number of appropriate statistical techniques exist. Consider an experiment in which microspectroscopic data have been derived from many samples, separated by conventional methods, such as histology, into distinct a priori groups or classes of tissue (e.g. normal vs. cancerous) or patient identity. Statistical models such as “ANOVA” (analysis of variance) may be used to test for significant differences between, for example, independent groups that are assumed to be normally distributed. More generally, one can envisage various alternative objects of such an experiment, such as answering one or more of the following questions: • Does the data analysis classify the spectra, by grouping them into posterior clusters, or otherwise? • If so, to what extent does this classification agree with the list of a priori class labels? • When spectra are obtained from “unknown” samples (i.e. samples having no a priori class labels), can each spectrum be classified, using, for example, data from a training or validation set previously and successfully analyzed, and with what level of confidence?

8.9 Data analysis

• Once any groupings (clusters) have appeared, can we obtain classspecific information on which variables give rise to the observed separation of spectra into clusters? Most of the relevant available statistical techniques may be divided into “unsupervised classification” and “supervised” (classification or regression). The supervised methods make use of some a priori knowledge (such as allocation to normal or malignant classes) of the spectra to be analyzed. Regression describes continuous dependent-variable data, e.g. of values of constituent concentrations, or class variables, in terms of the independent variables, in this case wavenumbers. It finds “factors” analogous to the principal components of PCA (discussed below), but that can better relate to the constituent compositions. The links between some of the most widely used methods are schematically displayed in Fig. 8.12. For details, and a discussion of the complex issue of how to choose the appropriate multivariate data-analysis technique in IR spectroscopy-based diagnostics, we refer the reader to the impressively lucid review by Wang and Mizaikoff (2008). In the biomedical field, cell characteristics will fluctuate markedly within physiological reference ranges, so that in order to identify samples that deviate from a normal state, large numbers of individual samples are needed for robust studies. A popular method of exploratory cluster analysis is PCA (see Fig. 8.12), which captures as much variability as is contained in the original data in a much smaller number of entries (an example of data compression). For each wavenumber variable, we may tabulate, along a single axis, the value of that variable measured for each of the samples. This gives us a single column vector. Each principal component (PC) is a vector consisting of a linear combination of the many hundreds of these raw data vectors, calculated in such a way as to reveal maximal amounts of variation, which plotting the original vectors fails to enable us to achieve. The PCs appear in order, starting with PC1, which reveals the biggest spread between spectra. Each spectrum becomes a single point, or score, in n-dimensional hyperspace and using selected PCs as axes, the data may be analyzed for clustering when viewed in a particular direction. Typically, the first 8 to 10 or even fewer PCs need be used. Nearness in hyperspace implies pattern recognition, and the separation of sample clusters in the plots signifies structurally dissimilar groups. Thus, PCA is a powerful technique for determining the clustering of the data themselves (see Fig. 8.13), but is unable to distinguish within-group from between-group variances. However, “scores plots” such as those given by PCA that present the results of clustering are somewhat subjective, being dependent on the choice of axis rotation and of which PCs to employ. There is no real guarantee that PCA’s maximum variance directions will necessarily correspond to the best segregation directions in the data: it can be other, non-maximum variance directions in the data that carry the best class-separating information. For these reasons, PCA is in general unlikely to lead to robust validation procedures such as are needed in good laboratory practice. Moreover, as pointed out earlier, there is one potential disadvantage of using PCA alone; namely, it does not unambiguously give the optimum grouping into clusters.

313

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research

Fig. 8.12 Some multivariate data-analysis techniques. Note: (i). Classifies spectra into groups (classes, categories, clusters). (ii). Supervised; data compression included. Training set is required, to build calibration model. Can also be used as a classification tool. (iii). For exploratory revelation of data structure, including any unexpected grouping: no training set or a priori class membership allocation is used. (iv). For more precise classification (makes use of pre-defined classes). When performing classification modelling, spectra are divided between training (calibration) set and validation set or “target”). (v). Spectra are grouped by similarity between datasets. (vi). The principal components (PCs) capture as much variability as is contained in the original data in a much smaller number of entries (an example of data compression). Each spectrum becomes a single point, or score, in n-dimensional hyperspace. Nearness in hyperspace implies pattern recognition. (vii). Yields dendrograms that reveal “similarity” (nearness in hyperspace) between spectra. Requires no assumptions about the number of groups required. (viii). Attempts to find the centers of natural clusters in the data, by minimizing total intracluster variance. The number of clusters is pre-defined. (ix). Also gives detail of how each spectrum is related to each cluster. Each point has a degree of belonging to clusters, rather than belonging completely to just one cluster. (x). Uses a separate PC model for each class. Training data set used, consisting of samples (or objects) with a set of well-clustered class membership. An unknown is classified as belonging to a specific group if it lies “closest” to the group and within an allowed threshold (see Fig. 8.14). A sample can be identified as belonging to more than one class (or to none of the classes). KNN is a simple alternative method that assigns each unknown to just one category defined in the training set. (xi). Closely related to ANOVA (analysis of variance), but the variable is a category label instead of a numerical quantity. Groups are separated by maximizing the ratio of between-group variance to within-group variance. Preliminary data compression (e.g. by PCA) is required. (xii). Useful if no clear linear relationship between variables. Training to cover all variations expected. (xiii). Can include data compression, taking account of class differences and allowing for variances of dependent-variable data (e.g. values of constituent concentrations), as well as of wavenumber data. (xiv). Employs probabilistic (Bayesian) separation of classes. (xv). Uses PCA for data compression, without considering class differences. (xvi). Exploits the fact that each peak extends over several wavenumbers (Martin et al. 2007). Classifies via two wavenumbers: checks predictions’ stability across nearby wavenumbers; includes compression.

One remedy is to make use of LDA. Unlike PCA on its own, the addition of LDA allows for a choice of any pre-determined classes to be taken into account. LDA is a powerful and much less subjective method for finding clusters, but applying it directly to high-dimensional data such as IR spectra allows

8.9 Data analysis

315

Fig. 8.13 SRS FTIR microspectroscopy facilitates the interrogation of cells of a large bowel crypt on a cell-by-cell basis. This allows the generation of individual “biochemical-cell fingerprints” in the form of vibrational IR spectra. Combined with PCA, this allows for the reduction of each individual IR spectrum into a score (each symbol) towards cluster analysis. Subsequent generation of a loadings plot allows identification of the wavenumbers that are responsible for cluster separation of the tissue-sample categories i.e. normal vs. pre-cancerous.

too much scope for discrimination to be achieved by chance, in directions that represent mainly noise. However, PCA may be used to reduce the dimensions of the dataset, and LDA then will find new variables (linear discriminants) by maximizing the ratio of between-class variance to within-class variance (the “PCA-LDA” procedure; Fearn 2002). Alternatively, stepwise LDA may be used (Martin et al. 2007). Figure 8.15 shows two examples of PCA-LDA clustering. A significant limitation of dimension-reduction techniques such as PCA is the lack of interpretability of the resulting classifiers. In microspectroscopy, we generally need to know which variables (wavenumbers), and hence which chemical bonds, were primarily responsible for any observed significant differences between spectra. As emphasized by Wang and Mizaikoff (2008), a detailed chemical interpretation of the molecular signatures obtained will always enhance the accuracy and reliability of classification techniques that are based merely on pattern recognition. The Mann–Whitney U test has traditionally been employed to compare absorption spectra at each wavenumber (Greene and D’Oliveira 1989). This test is a non-parametric test for statistical difference between unpaired groups, and does not assume that the data are normally distributed. It calculates the probability that the two groups of values being compared are drawn from the same distribution by chance. Probability diagrams are constructed for each pair of absorption spectra to be compared, and the P-value given is the probability of this chance occurrence. Regions

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Microspectroscopy as a tool to discriminate nanomolecular cellular alterations in biomedical research

Fig. 8.14 SIMCA scores plot output projection of PC2 vs. PC3 from pre-processed absorption spectra taken from two regions of an oral tissue section using single-cell synchrotron FTIR microscopy (sampling areas 10 μm × 10 μm) (from Fisher, S.E., Harris, A.T., Chalmers, J.M., Tobin, M.J. Vibrational spectroscopy for medical diagnosis (eds. Diem, M., Chalmers, J.M., Griffiths, c P.R.) 2008 John Wiley & Sons Limited, reproduced with permission).

Fig. 8.15 PCA-LDA scores plots of spectral data. Each point represents one spectrum, stars show cluster means assuming a normal distribution, and confidence ellipses show probability contours for each of the three clusters. Points lying outside a contour drawn at a particular value of p, as labelled, have a probability of only (1− p), or less, of belonging to that cluster. (Left) spectra acquired from epithelial cells in particular regions of prostate from one patient, as described in German et al. (2006a): peripheral zone, gray circles; transition zone, black squares; and cancerous zone, gray triangles–complete cluster separation is seen; (right) PTMS spectra from human intestinal crypts: putative stem-cell regions (gray circles), transit-amplifying regions (black squares) and differentiated regions (gray triangles) (from Walsh et al. 2008a). The confidence ellipses indicate a degree of separation but with some overlap.

8.9 Data analysis

317

Fig. 8.16 Cluster vector loadings plots. (top) PCA-LDA-processed data, as for Fig. 8.15 (left) but obtained from six patients, analyzed two tissue types and one patient at a time. Each loadings plot corresponds to one patient as indicated. (bottom) As at top, showing detail of the low-wave number region. For all six patients, there is fair agreement on which spectral regions have been primarily responsible for clustering revealed by scores plots (Martin, F.L., German, M.J., Wit, E., Fearn, T., Ragavan, N., Pollock, H.M. (2007) c 2007 Mary Ann J. Comput. Biol. 14, 1176  Liebert Inc., reproduced with permission).

of spectra with a P 0 is the electron charge, E F the Fermi energy of the free tube and f E (μ) the Fermi distribution. We define Vb = VL − VR to be the bias voltage. The same tight-binding Hamiltonian is used both in the conductivity calculation as well as for performing molecular-dynamics simulations for structural relaxation ensuring consistency in the calculations. We consider both the defect-free Rh-C60 as well as the defected one containing one Cv per C60 molecule. In the inset of Fig. 21.12 we show the calculated transmission function T (E) for the two cases. As can be observed, there are no conduction channels above the Fermi energy (set to zero) in the defect-free system. The presence of Cv s introduces more transport channels, some of which extend above the Fermi energy. The main part of Fig. 21.12 shows the corresponding I–V curves for the two cases. As is apparent from this figure, the defects (Cv s) lead to an increase in the current due mainly to the creation of new conduction channels below and above the Fermi energy as well as to the narrowing of the electron gap.

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Structural, electronic, magnetic, and transport properties of carbon-fullerene-based polymers

21.5

Magnetic coupling among magnetic moments

As stated above, it appears that a consensus has been reached regarding the origin of the magnetic moments in C-based materials; that this is induced by structural, topological and impurity defects. The unresolved issue, however, pertains to how these magnetic moments are coupled ferromagnetically. The ferromagnetic coupling among the defect-induced magnetic moments is not only a local effect due to the point-group symmetry of the defect site, the band splitting, and the band-filling factors of the atomic orbitals of the surrounding defect (and/or the impurity) atoms (and/or ligands) (Mpourmpakis et al. 2003, 2005). These features, while playing a primary role in the formation of the magnetic moments, also contribute to the development of the magnetic coupling either in a direct or indirect way. However, the ferromagnetic coupling among neighboring magnetic moments depends crucially on their mutual interaction that, in turn, is established by the induced charge transfers. That is, it depends strongly on the electron–electron (e-e) correlations and as previously discussed (see details in Sections 21.2 and 21.3), the way e-e correlations promote the magnetic coupling can be viewed from different angles. McConnell and Welch (1967) attributed the ferromagnetic coupling to the kinetic exchange, i.e. a contribution to CI (see Section 21.5.1). Similarly, other researchers rely on the CI approach in order to explain it. On the other hand, most of the efforts concerning this investigation are based on SRMs because (as we have already discussed) of their simpler and computationally tractable formalism. In the following, we make a more detailed reference to the proposed CIMs and discuss the way we mapped our SRM results onto a generalization of the McConnell model. As will be shown, this generalization is not only applicable to the C-based materials but also to the doped NTIMs as well.

21.5.1

Kinetic exchange

Strictly speaking, the kinetic exchange interaction is a configuration interaction (CI) term that plays an important role in chemical bonding. In a simplistic scenario it can be viewed as the outcome of quantum interference effects among the basic set functions (for example, atomic orbitals (AOs)) which are used to describe the molecular orbitals (MOs) (for a recent discussion see, for example, Yamaguchi et al. 2006 and references therein). This is played out in Anderson’s Hamiltonian, namely in the description of a magnetic (impurity) atom embedded into a non-magnetic metallic host (Anderson 1961) in which a single electron state is described in terms of Bloch functions of the host lattice and d-type AOs of the magnetic impurity. This model formed the basis for the study of the diluted magnetic semiconductors (DMSs). As shown by Schrieffer and Wolff (1966), Anderson’s Hamiltonian (Anderson 1961) can be transformed into the form of an s −d exchange model in which two fundamental interaction mechanisms (between a valence electron and the magnetic impurity) prevail (Larson et al. 1988; Blinowski

21.5 Magnetic coupling among magnetic moments 765

and Kacman 1992). One of them is an energy-dependent exchange interaction, Jkk  , of negative sign (i.e. of antiferromagnetic (AF) order). The other is a spinindependent direct s−d exchange interaction (Schrieffer and Wolff 1966). The former is known as the kinetic exchange interaction and within the k-space perturbation theory it can be viewed as originating from the virtual transitions of a conduction electron between the conduction sp-bands and the d or f bands of the magnetic impurity. Later considerations have shown that depending on the band filling of the d-band and the spin multiplicity of the ground state of the magnetic ion, the kinetic exchange can be of AF or FM Kondo-like or exhibiting a more complicated form (Blinowski et al. 1996). In the presence of more than one magnetic ions embedded in a non-magnetic host the kinetic exchange interaction is expected to play a key role in their mutual interaction. That is, a significant part of the polarization experienced by the conduction electron in the neighborhood of the magnetic ion is attributed to the kinetic exchange that, in turn, affects the indirect interaction between the magnetic ions. In view of this picture the kinetic exchange interaction is expected to play a crucial role in the description of mediated exchange interactions as, for example, the superexchange and the double-exchange interactions. It is apparent that the perturbative computation of the kinetic exchange requires a knowledge of the whole spectrum (and the corresponding wavefunctions) of the Hamiltonian of concern. The knowledge of the excited states is a pre-requisite for the calculation of the kinetic exchange; it is a consequence of the single-configuration approximation (and the single-electron approximation) of the solution that is looked upon. It may be possible, however, to avoid the single-electron perturbative approach to the kinetic exchange and limit the number of the excited single-electron states needed for its calculation. This may be achieved by attempting a multiconfigurational description of the electronic states of the DMSs, while keeping only a few Slater determinants (configurations), namely those that can guarantee the capture of the major part of the correlation effects. In such a procedure, the kinetic exchange will be the result of interference terms among the various molecular configurations used in the construction of the many-electron wavefunctions and is inherently incorporated within the calculated total energy. As characteristic examples of multiconfigurational approaches we point out the following: • The organic charge-transfer salts consisting of alternating donor (D) and acceptor (A) sites viewed as alternating D–A chains extending all over the material. In these, it may be possible for the triplet state of a neutral D–A dimer (D0 A0 , participating in a D–A chain) to induce a triplet state of their charge-transfer configuration (D+ A− ) through a CI, with the latter facilitated by the electric fields developed by the charge-transfer processes. This triplet state is then propagated along the D–A chain as suggested by McConnell (McConnell and Welch 1967). Therefore, in such an approach two multielectron configurations, namely 0 D0 A and D+ A− , are involved in the CI formulation of the problem (see Fig. 21.13) and associated with them the kinetic exchange results from their interference.

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Structural, electronic, magnetic, and transport properties of carbon-fullerene-based polymers

Fig. 21.13 Schematic representation of McConnell’s CI approach (McConnell and Welch 1967).

• The superexchange model of Kramers and Anderson for antiferromagnetism as applied to MnO (Kramers 1934; Anderson 1950). In this, the wavefunction of the system is written in terms of two MOs, namely the ground-state MO and an excited MO. In the former, the O atom mediating the interaction of two Mn atoms that are its first nearest neighbors has two antiparallel electrons. The latter is a charge-transfer state in which one electron from the oxygen atom moves onto one of the Mn atoms. • The double-exchange model of Anderson and Hasegawa (1955) proposed for explaining the origin of ferromagnetism in the mixed-valence manganites of perovskite structure (Anderson and Hasegawa 1955). In this, three MO configurations are constructed by combining the most probable atomic (electronic) configurations of one oxygen atom and its two neighboring Mn ions by taking into account the spin multiplicity of each constructed MO. Here, once again, the charge-transfer states are essential and are included in the CI approach. • The consideration of van Vleck of Ni ferromagnetism (van Vleck 1953).

21.5.2

Generalized McConnell model

In Section 21.1, it was pointed out that features similar to those found in the magnetic C60 -based polymers were also identified in some of the magnetic doped-NTIMs. These observations led us to propose that these two different classes of materials belong to the same class of magnetic materials whose magnetism is due to defects (Andriotis and Menon 2005; Andriotis et al. 2003, 2005a,b, 2006a, 2009). The similarities between these two classes of materials pertain mainly to the development of the ferromagnetic coupling among the magnetic moments that in the C-based materials are induced by the defects, while in the class of the doped-NTIMs are provided by the impurity atoms (either magnetic or non-magnetic). In both classes, however, the magnetic moments are induced by the defects (either of intrinsic or extrinsic type). The model we proposed in order to explain the origin of this defectinduced magnetism is a generalization of McConnell’s theory (McConnell and Welch 1967). According to our proposal, the presence of two kinds of defects (of donor and acceptor type, respectively) is necessary in order to promote positive- and negative-charge and spin-density localizations, self-sustained by the development of strong electric fields. A schematic view of our proposal is shown in Fig. 21.14. According to this, the defects provide locations for localizing separately the induced positive and negative charge transfers shown as light and dark shaded neighborhoods in the system. In any or both of these localization regions, electron spin density is also localized, forming the

21.5 Magnetic coupling among magnetic moments 767

Fig. 21.14 Pictorial view of a magnetic system described in terms of electron charge transfers leading to locally positive and negative regions (as shown by light- and dark-shaded regions, respectively) as well as spin-density transfers leading to the formation of magnetic moments (indicated by arrows). The charge localizations are organized in periodic arrays of donor (D) and acceptor (A) regions.

magnetic moments (shown with the short arrows). This settlement (which is not necessary to be periodic as in Fig. 21.14), leads to the development of the electric fields (shown by E in the figure) which, in turn, establish the ferromagnetic coupling (shown as J AB ) among the magnetic moments. It is understood that the details for the formation of the magnetic moments is mostly system specific, depending on the defect types, the point group symmetry of their location in the crystal, the band splitting and fillings of the AOs associated with the atoms in the neighborhood of the defects, etc. However, it is strongly affected by the electronic structure of the (host) material (bandgap, spin polarization and position of spin states relative to the gap, impurity levels, etc.). The exchange coupling among the magnetic moments appears to be associated with the remote delocalization and overlap of the MOs both of which are induced by the Cv s as this was sufficiently discussed in Section 21.4.2. In this, it was shown that the delocalization is remote and selective, establishing, thus, remote overlap among the MOs. It is this remote overlap that contributes to the exchange coupling among the magnetic moments. A characteristic example appears to be the case of the C60 trimer (shown in Fig. 21.9). In this view, and at least on the nanoscale, doping concentrations smaller than the percolation threshold can lead to ferromagnetism. Therefore, metallicity seems not to be a pre-requisite for the establishment of magnetism. Nevertheless, as we have seen, defects eliminate the electron energy gap of the two-dimensional Rh-C60 polymer as demonstrated in Fig. 21.3.

21.5.3

Similarities with magnetic non-traditional inorganic materials

The similarities that we identified between the magnetic features of the C60 based polymers and those of the NTIMs made us check our model proposal in

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Structural, electronic, magnetic, and transport properties of carbon-fullerene-based polymers

the case of the ZnO codoped with two kinds of substitutional impurities that can act as donor and acceptor sites. In particular, we studied the ZnO codoped with Co2+ and Cu+ ions (to be denoted as Zn(Co,Cu)O). Our preliminary results have indicated that, in agreement with Fig. 21.14, one of the defects (Co) provides mainly the magnetic moments, while the other acts as a ferromagnetic coupling mediator even at distances as long as 3rd nearest neighbors in the Zn sublattice. Interestingly, it was found that a ferromagnetic coupling between the Co ions is established if the Cu atom intervening between is spin polarized, a picture reminiscent of the RKKY interaction (Ruderman and Kittel 1954; Kasuya 1956; Yosida 1957) between magnetic ions embedded in a metal (Lathiotakis et al. 2008).

21.6

Conclusion

The defect-induced ferromagnetism in C-based materials is an exciting and challenging phenomenon. It has attracted intense efforts in recent years and many models have been proposed soon after its observation. From the results of the present work and other reported ones, it has become clear that two interrelated but, in many aspects, mutually independent processes are responsible for its development. One of these processes has mainly local character. It is associated with the development of the magnetic moments. It depends on the type of the defect, its point-group symmetry, its ligand fields and the type and band filling of the ligand AOs. These factors specify the electron charge and spin transfers towards or away from the defect region; they are affected by the electronic structure of the host material. The second process establishes the ferromagnetic coupling among the induced magnetic moments. This is mainly based on the strength of the electrostatic fields that are developed by the charge transfers. As our calculations have shown, in the C60 -based polymers, the Cv s and the 2 + 2 cycloaddition bonds are indeed responsible for the development of large electrostatic dipole moments. These are responsible for large electric fields that, in turn, enable the defect sites to retain the charge and spin localization in their region. This picture is reminiscent of McConnell’s model for ferromagnetism in the magnetic organic salts and allows us to propose a generalization of it as a possible mechanism for the defect-induced carbon ferromagnetism. Our generalization considers the presence of two kinds of defects, which can act as donor and acceptor sites, respectively, as a necessary pre-requisite for the occurrence of magnetism. This defect combination ensures the presence of charge neutrality and can lead to the necessary development of the electric fields that will promote the charge localization. The proposed generalization is not limited solely to the role of the kinetic exchange interaction. All CI contributions have their share in this phenomenon. We have arrived at this conclusion based on our numerical results that revealed large defect-induced charge transfers and, associated with them, large electric fields, both of which were found to promote the triplet state configuration as the energetically most favorable one for the ground state. Finally, it is worth noting that further investigations led us to the observation that the proposed explanation of the carbon magnetism is equally well applicable in the case

References

of doped-NTIMs that, as demonstrated, do indeed exhibit those characteristic features that underline the magnetism of the C-based materials.

Acknowledgments The present work is supported through grants by DOE (DE-FG02-00ER45817 and DE-FG02-07ER46375) and US-ARO (W911NF-05-1-0372).

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Magnetic nanowires: Fabrication and characterization

22 22.1 Introduction

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22.2 Metallic nanowire fabrication, the state-of-the-art

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22.3 Structural characterization

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22.4 Magnetic reversal process: Single nanowire

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22.5 Magnetic anisotropy and interactions: Role of geometrical arrangement

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22.6 Transport measurements

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22.7 Temperature-driven effects

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22.8 Dynamic properties of magnetization

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Kleber Roberto Pirota, Marcelo Knobel, Manuel Hernandez-Velez, Kornelius Nielsch, and Manuel V´azquez

22.1

22.9 Future perspectives

815

References

817

Introduction

Magnetic nanowires offer a number of very interesting opportunities in many areas of advanced technology, including patterned magnetic media, magnetic sensor devices and microwave technology. The controlled production of metallic nanowires with unusual properties attracts increasing interest owing to their potential applications in emerging technologies requiring nanoscale devices, such as, high-density magnetic storage, magnetotransport phenomena, catalysis, controlled optomagnetic response, field emitter devices, nanomechanical resonators, spin-transfer systems and multifunctional sensors (Li et al. 1999; Cui et al. 2001; Husain et al. 2003; Mancoff et al. 2005; Yan et al. 2005; Lee et al. 2007; Piraux et al. 2007). In particular, metallic nanowires have emerged as elemental building units for nanocircuit fabrication (e.g. interconnecting elements) (Duan 2003). Thus, the expected relevant role of the increased packing density of integrated circuits and functional nanodevices has called for advanced nanowire research in the coming years. On the other hand, magnetic properties of low-dimensional materials are of significant importance in novel research fields, such as spintronics, magnetoelectronics and magnetic recording. Therefore, the development of novel alternatives to enhance the magnetic data storage in perpendicular recording, based on patterned highly ordered nanowires, turns out to be a priority area. Theoretically, nanowire arrays oriented perpendicular to the substrate and with reduced diameter and mutual separation could give rise to data-storage densities as high as 700 Gbits/inch2 . The research effort on nanostructured materials and particularly on one-dimensional (1D) systems has allowed deeper studies of physical and chemical principles at the nanoscopic level. From the experimental point of view, properties and effects such as: enhanced storage and transference of information, the absorption-edge blue shift in confined systems, conductance quantization, enhanced mechanical properties and other features (Krans et al. 1995; Alivisatos et al. 1996; D´ıaz

22.2 Metallic nanowire fabrication, the state-of-the-art 773

et al. 2004) have been widely verified. In addition, the increasing theoretical and computational developments have led to a deeper understanding of these systems for the new generation of nanoelectronic and optomagnetic devices (Saib et al. 2003; Dobrzynski 2004). The magnetic behavior of the nanowires has been investigated, by micromagnetic simulations and, experimentally, by magnetometry and magnetic force microscopy. Moreover, transport properties of single nanowires were also used to infer the magnetization process and magnetic structure of such systems. Also, magnetostatic interactions among nanowires within a patterned template (mainly anodized-alumina membranes) have been extensively investigated. In this chapter, we will mainly focus on the magnetic properties of patterned arrays of metallic magnetic nanowires electrodeposited into the pores of anodized-alumina membranes (AAM). The complex magnetization processes, both in isolated nanowires and in collectively patterned arrays, will be discussed in detail. The magnetic anisotropy depends on the nanowire composition and preparation conditions. Typically, there is a competition between shape and crystalline anisotropy that determines the magnetic behavior. The most recent developments concerning micromagnetic simulations and the spinwave configurations within a nanowire, will be re-examined and analyzed. Further, the projected potential of future studies will be assessed. This chapter is organized as follows: Section 22.2 presents the state-of-theart on fabrication techniques of nanowires, Section 22.3 is devoted to analyzing the microstructure of magnetic nanowires, paying attention to the properties of nanowires electrodeposited within the pores of alumina (AAM) templates. In Section 22.4, a detailed discussion about the magnetic properties of such systems is given, focusing on the properties of single nanowires. The collective behavior of arrays where the interactions among the magnetic entities play an important role is discussed in Section 22.5. The transport properties of magnetic nanowires are reviewed in Section 22.6. Section 22.7 is devoted to the temperature-dependent effects (such as magnetoelastic-induced anisotropy) while Section 22.8 describes the dynamic properties of the magnetization such as ferromagnetic resonance characteristics and spin-wave excitations in ferromagnetic nanowires. Finally, an overview on the perspectives of future research is given in Section 22.9.

22.2

Metallic nanowire fabrication, the state-of-the-art

In this section several synthesis routes for metallic nanowire fabrication will be described. We will pay particular attention to self-assembled bottom-up techniques and their combination with traditional physical techniques for growing bulk materials and thin films, such as physical vapor deposition or sputtering. As previously mentioned, we will focus on electrochemical growth of nanowires based on template-assisted methods using anodic alumina templates (V´azquez et al. 2004a, 2004c). Such nanoscale arrays can be prepared by traditional and sophisticated top-down techniques that, in general, need ultrahigh-vacuum requirements,

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Magnetic nanowires: Fabrication and characterization

and by highly successful bottom-up strategies for synthesizing submicrometer materials. The combination of both strategies has also been successfully used to obtain nanowire arrays and different nanostructured materials with different functionalities. One of the most relevant routes for nanoscale fabrication is based on lithographic techniques. Highly sophisticated lithographic routes, developed in the quest of achieving higher resolution of integrated circuit features on nanometric scale, have brought tremendous advances in microelectronics. Most of these achievements can be extended to the nanotechnology (MIT 2006). Nevertheless, the smaller the dimensions, the more expensive the lithographic processes, which is strongly related to the wavelengths used to define the smallest features, e.g. connecting metal wires on a wafer surface (Krauss and Chou 1997). In recent years, the high-quality process of mask production constitutes one of the elements giving rise to increasing cost of products based on lithographic methods. Nowadays, much effort is centered on the production of maskless lithography and nanoimprint lithography (Chou et al. 1996; Li et al. 2001), which is expected to allow, in the near future, the replication of patterns with features down to 10 nm. The most extensively studied bottom-up technique to obtain nanowires and nanowire arrays are: (i) self-assembly following or profiting by the characteristics of a solid surface as its relief or anisotropic crystallographic structure, (ii) direct self-organization within the channels and cavities of porous material, (iii) the formation of inorganic mesoscopic material by self-assembly of surfactant organic molecules, (iv) the introduction of solid/liquid interface, (v) supersaturation control to modify the usual growth of a seed and (vi) selfassembly of zero-dimensional nanostructures (Rao et al. 2003; Xia et al. 2003). In particular, metallic nanowires have been produced by chemical vapor deposition (CVD) (Fang et al. 2005), solvothermal and hydrothermal techniques (Sun and Xia 2002; Yao et al. 2005) and template-assisted methods (Cheng et al. 2006; Yamamoto et al. 2006). Anodic alumina membranes (AAM) (Masuda and Fukuda 1995; Martin 1996; Choi et al. 2003a; Masuda et al. 2003), nanochannels array glass (Nguyen et al. 1998) and track-etched polycarbonate membranes (Sch¨onenberger et al. 1997) have been the most used templates for metallic nanowires fabrication. However, it is extremely difficult to obtain homogeneous pore diameters below 10 nm in these templates. In this regard, one interesting approach is the use of metal organic chemical vapor deposition (MOCVD) to prepare nickel nanowires within the pores of AAM previously coated with carbon, with diameter as small as 4 nm (Pradhan et al. 1999). In this method the porous film is subjected to carbon deposition by thermal decomposition of propene resulting in a uniform coating of the AAM walls. Then, that nanocomposite is subjected to MOCVD using the necessary precursors. Taking advantage of the natural or artificial marks created on solid surfaces, the nanowire growth can be performed on the solid surfaces by means of different techniques such as, molecular beam epitaxy (MBE), electron beam evaporation (EBE), phase–shift optical lithography and sputtering (M¨uller et al. 2001; Sun et al. 2005). By following this strategy, insulated continuous wires can be obtained, e.g. gold nanowire fabrication has been achieved on

22.2 Metallic nanowire fabrication, the state-of-the-art 775

a previously prepared Si 557 surface (Crain et al. 2003). The theoretical explanation of the obtained results predicts the formation of bands on the metallic surface with gaps similar to those described for bulk semiconductors. Other surface manipulation for nanowire growth is based on the stress sources created by misfit dislocations at interphases (Xie et al. 1995; Shchukin and Bimberg 1999). This stress field can give rise to nanowire formation. Silicon surfaces have been used as templates to fabricate ordered indium nanowire arrays by means of molecular beam epitaxy (MBE) (Liang et al. 2004). The results bring out the importance of controlling the kinetic parameters to obtain such arrays and the growth mechanism of nanowires through nanocluster accumulation process. Noble-metal nanowire arrays formed by induced planes on silicon surfaces have been reported (Song et al. 1999). The process can be addressed so that vapor or liquid surrounding the solid surface acts as a substrate. This compensates dangling bonds or minimizes the surface stress and therefore gives rise to thermodynamically stable surface morphology. This strategy has opened up a potential way for the growth of functional metal nanowire arrays on polar surfaces of different materials. On the other hand, short metal nanowires have been fabricated using scanning tunnelling microscopy (STM) in combination with transmission electron microscopy (TEM), especially for quantum conductance studies (Costa-Kramer et al. 1997; Ohnishi et al. 1998; Kondo and Takayanagi 2000). Electrical conductance quantization in gold nanowires has been reported elsewhere (Pascual et al. 1993). Since metal nanowires exhibiting conductance quantization are usually very short, they have often been identified both as nanocontacts and quantum wires. The fabrication of stable nanowires with features comparable to Fermi wavelengths is actually a very hard task (Rodr´ıgues and Ugarte 2001). Different studies on this aspect have been reported in transition-metal nanowires such as Fe, Ni and Co (Untiedt et al. 2004). Mechanical and electrochemical methods have been used to fabricate such small nanowires. In the former, a metal quantum wire is formed by means of the mechanical separation of two electrodes using a STM tip (Smith 1995; Landman et al. 1996; Pascual et al. 2005). On the other hand, the electrochemical methods already reported essentially consist of (a) metal anodization based on AFM (Snow et al. 1996), where the AFM tip is used as a cathode and the water from the ambient humidity is used as an electrolyte, (b) selective plating on the STM tip that, after slow dissolution, gives rise to the formation of a quantum wire between the tip and the other electrode (Li and Tao 1998), (c) electrochemical deposition and etching to obtain Cu, Ag, Ni, Pd and Pb quantum nanowires between two gold electrodes (Li et al. 2001). Conductance quantization has also been reported in Ni wires that have been electrodeposited in the nanopores of track-etched polymer membranes (Elhoussine et al. 2002). The Hall effect in platinum nanowires has been recently studied in the situation where the size responsible for the spin accumulation is either smaller or larger than the spin diffusion length (Vila et al. 2007). In these studies, a Pt wire, 100 nm wide and 5 to 20 nm thick, is in ohmic contact with a Cu wire, 160 nm in width and 100 nm in thickness, running between two Permalloy wires, 100 nm in width and 30 nm in thickness, were prepared by means

776

Magnetic nanowires: Fabrication and characterization

of conventional e-beam lithography, e-beam evaporation and lift-off process. The formation of metallic nanowires and nanochannels through guided self— assembly has been reported (Alaca et al. 2004). In this method, the initiation and termination points of the nanostructure are pre-designed, so that, the pattern evolution is dictated by stress—assisted cracking on a dielectric film previously grown by plasma-enhanced chemical vapor deposition (PECVD) and attached to a silicon substrate in which cracks are used like molds for the nanowire growth by filling them with a desired material. Metal nanowires have been also obtained by selective electrodeposition of conductive metal oxide at the step edges of highly oriented pyrolytic graphite (HOPG), starting with the metal-oxide reduction to form metal nanowires. This strategy is labelled as electrochemical step–edges decoration (ESED) and it has been used for Mo nanowire fabrication (Zach et al. 2000), although it could be used for Cu and Fe and other metal nanowire fabrication as well. Very recently, this technique has been further used for gold nanowire fabrication combined with physical vapor deposition (PVD) where gold nanoparticles are used as nucleation templates to catalyze the electrodeposition process (Cross et al. 2007). This method cannot be applied for metal nanowire growth where oxide compounds are neither stable nor conductive. In fact, discontinuous nanowires consisting of individual clusters are formed (Liu and Penner 2000; He and Tao 2003). Crystalline zinc nanowires fabrication is acquiring increasing importance due to their broad field of applications. Zn nanowires with diameters less than 15 nm possess interesting thermal properties related to their magnetoresistance (Heremans et al. 2003). Further, they can exibit superconductor behaviours (Wang et al. 2005) and are also used as precursors of ZnO nanowire arrays with attractive applications in the fields of optoelectronic and magneto-optics (Sekar et al. 2006; Zhang et al. 2006). Recently, Zn nanowires have also been produced on alumina substrates by using ceramic methods of milling, e.g. boron and zinc oxide powders are mixed together and heated in a nitrogen atmosphere (Chen et al. 2007). Metallic nanowire growth based on patterned nanometer lines over large areas using nanowires as masks has been recently developed (Whang et al. 2003). Whang et al. used core-shell Si−SiO2 nanowires uniaxially compressed on a Langmuir–Blodgett trough aligned with a controlled pitch and then transferred to a silicon substrate. Reactive ion etching (RIE) is used to remove the SiO2 shell from the sides and the top of the nanowires. After the deposition of metals (in this reference, chromium), the nanowires mask is removed to obtain parallel metallic lines over the whole substrate surface. Recently, the same group using a similar strategy, produced assemblies of nanowires into integrated device arrays (Whang et al. 2004). Some advantages of this approach are: the formation of ordered monolayers over large areas, the facility for transferring the organized monolayers to substrates and the possibility of obtaining multilayers by repeating this process. This has allowed the growth of aligned nanowires with strict control on several important parameters, such as, the nanowire pitch, their orientation, and the array size. Also, this approach enables the room-temperature growth of the arrays and multilayered devices that make it compatible with the required low costs and the use of flexible

22.2 Metallic nanowire fabrication, the state-of-the-art 777

substrates for applications in integrated functional nanosystems (Cui et al. 2001; DeHon 2003). The use of DNA for nanostructure fabrication (Alivisatos 1996) has already been proposed as a way for building up nanoelectronic devices. In particular, the fabrication of metallic nanowires based on the self-assembly of complementary DNA used as localized templates on specific surface sites has been reported (Braun et al. 1998; Martin et al. 1999). Anodization techniques are increasingly being used to create highly ordered functional nanostructures, mainly nanowire arrays grown within AAM by means of electroplating techniques. Till now, aluminum metal constitutes the only element that allows fabrication of highly ordered porous membranes with hexagonal symmetry in a centered densely packed array of pores (Jessensky et al. 1998). The template–synthesis strategy for nanofabrication has been described in detail by Hulteen and coauthors (Hulteen et al. 1997). The anodization followed by electroplating processes for tailoring functional 1D nanostructures somehow fulfils some of the demanding requirements of new nanotechnologies. By means of this strategy it is possible to get a better control of: (i) ordering degree, i.e. the size of crystalline single domains (up to several square micrometers), (ii) the single 1D structure diameter (from a few nanometers to more than 200 nm) and length (from tens to thousands of nanometers), (iii) the lattice parameter of ordered arrays (from around 60 to 500 nm). A schematic view of AAM is presented in Fig. 22.1. The parallelism among the arrangement of pores with a symmetry axis perpendicularly oriented to the substrate surfaces can be considered the most interesting and distinctive characteristic of AAM. Electrochemical growth, either galvanostatic or potentiostatic, or a combination of both, has been the

Al2O3

Porous self-assembled nanostructure. Centered hexagonal symmetry Diameter size range: 20–240 nm Spacings: 50–500 nm

Adjustable thickness: from tens of nm to 100 micrometers Al

Patterned aluminum substrate

Fig. 22.1 Schematic view of an anodized-alumina membrane (AAM) showing its principal geometrical parameters.

778

Magnetic nanowires: Fabrication and characterization

commonly used procedure to obtain magnetic nanowires based on AAM (Choi et al. 2003a; Masuda et al. 2003). Thus, AAM have became a relevant material in nanotechnology and in nanobiotechnologies, where the particular attention is given to the use of AAM as templates for a self-assembled large variety of systems (Masuda et al. 2001; Bae et al. 2002). The details of the anodization and electrodeposition processes to fabricate the metallic magnetic nanowires inside the porous AAM will be given in the next subsection.

22.2.1

Anodic alumina membranes (AAM) and the electrodeoposition of the metallic magnetic nanowires

Commonly, AAM are fabricated using the well-known two-step anodization process introduced in Masuda and Fukuda (1995) that we describe in the following and as pursued in our laboratory. The details of the aluminum anodization process, alumina phase formation and the influence of impurities may be found elsewhere (Thompson 1997). Aluminum foils with high purity (99.999%) are used as the starting material. After being annealed for recrystallization and to enhance the foils texture, they are degreased and electropolished in a mixture of perchloric acid (HClO4 ) and ethanol (C2 H5 OH) under a constant voltage of 20 V. This process is carried out under magnetic stirring in order to achieve higher homogeneous etching to reduce the surface roughness. In general, the pore diameter and the distance between them (d and D, respectively) directly depend on the anodization voltage and temperature at the process is performed, and they are found to range between 15 to 200 nm and 30 to 500 nm, respectively. Despite this rather loose control of the geometrical parameters, highly ordered pores with narrow size distribution are obtained by controlling the temperature and voltage corresponding to the specific acid used. In a typical AAM fabrication, different acidic aqueous solutions can be used as electrolytes, the most common being the oxalic, sulfuric and phosphoric acids. The optimum parameters and the geometrical characteristics of the AAM obtained with each acid are collected in Table 22.1. The degree of order is basically determined by the duration of anodization. Highly ordered pore arrays are observed only after a long anodization time, and as a result it is observed only on the bottom of the films, at the Al2 O3 /Al interface. This is the principal reason why a second anodization process is required. The electrochemical reactions that take place during the anodic oxidation of Table 22.1 Optimum conditions and resultant geometrical paramenters for optimum well-ordered pores in AAMs. Acid oxalic phosphoric sulfuric

Concentration

T (◦ C)

Voltage (V)

d (nm)

D (nm)

0.5 M 1% weight 0.3 M

3–5 0–1 1–2

40 200 25

35 180 25

105 500 65

22.2 Metallic nanowire fabrication, the state-of-the-art 779

aluminum are the following: Al (s) + 3/2 H2 O (l) → Al2 O3 (s) + 3H+ (l) + 3e− Al(s) → Al3+ (l) + 3e− , in the anode and 3H+ (l) + 3e− → 3H2 (g), in the cathode, where s: solid; l: liquid and g: gas phases. The first reaction is verified in the oxide barrier formation. A combination of this reaction and the second one occur in the formation of porous oxide as a function of additional electrochemical parameters such as the applied voltage, temperature, pH, etc. After a long anodization time (typically more than 24 h), the formed porous oxide is removed from the Al foil by a wet chemical etching with a mixture of phosphoric and chromic acid at 45 ◦ C. The Al surface in contact with the bottom of the alumina layer presents the same spatial ordering displayed by the AAM bottom surface. This self-assembled patterned Al acts as a template for the second anodization step using the same electrochemical parameters as for the first one. The second anodization, with the order degree already defined by the first one, is achieved in a much shorter time (e.g. 2 h typically gives about 4 μm of pore length for oxalic acid under the optimum conditions of Table 22.1). Typical HRSEM images from the surfaces of AAM fabricated in our group are shown in Fig. 22.2. In this figure, one can see the evolution of the pore diameter when the AAM (anodized with oxalic acid) is subjected to phosphoric acid etching (5% vol. at 30 ◦ C) for different times. According to our experience the pore diameter increases around 2 nm per min. under these conditions. The explanation of why pores grow at all and why their size distribution is quite narrow can be found in the models based on the electric-field distribution at the pore tips (Thompson and Wood 1981; Parkhutik and Shershulsky 1982). On the other hand, there is not a complete understanding and agreement about the well-aligned and highly ordered pore formation in these nanostructures. Insights into the self-organizing process can be found in Jessensky et al. (1998). Recently, great advances have been achieved in the fabrication of singlecrystalline nanopore arrays of AAM by using a combination of nanoimprint and lithographic techniques with a subsequent anodization process of aluminum metal and other metallic or semiconductor substrates (Asoh et al. 2003; Choi et al. 2003b). According to this novel strategy, nanostructured titanium nitride (TiN) molds have recently been developed (Navas et al. 2007). TiN thin films were grown by reactive magnetron sputtering using a titanium target (99.995% purity) and different AAM as substrates. The influence of AAM pore diameter and spacing, their thicknesses and sputtering parameters on the TiN molds, has been reported in the work. In all cases, the mean thickness of TiN films is 0.7 μm, estimated from the deposition rate. To facilitate the manipulation of such thin nanostructured films, Ti layers around 1 μm thick were sputtered onto the TiN films and then, the resulting sandwich (Ti/TiN) was attached to a silicon wafer. A schematic view of the fabrication processes is shown in Fig. 22.3.

780

Magnetic nanowires: Fabrication and characterization

(a)

(b)

(c)

Fig. 22.2 Typical HRSEM images from the surfaces of AAM fabricated in our group. (a) AAM as prepared, with oxalic acid. (b) After 10 min of phosphoric acid etching and (c) after 30 min of phosphoric etching.

TiN Al2O3 Pre-patterned Al

TiN mold Silicon

Pressure

AI electropolished

AI electropolished

Fig. 22.3 Schematic representation of the fabrication and subsequent inprint of an Al foil of a TiN nanoimprint tool (see text for description). From Figure 1 of Navas et al. (2007). Copyright 2007 IOP.

AI2O3

22.2 Metallic nanowire fabrication, the state-of-the-art 781

m 1.0 μ

Z (nm)

(a)

16 14 12 10 8 6 4 2 0 –2 0

2

4

6

8

10

X (μm) (b)

Aluminum foils and anodic alumina were detached from the conformed device by means of two different selective chemical etchings described elsewhere (Pirota et al. 2004). A relevant property of these molds is their hardness, which is higher than those previously fabricated on silicon carbide and silicon nitride. Pre-patterned aluminum foils are obtained by applying different pressure on the aluminum foil surface. The use of these pre-patterned foils enables the use of a single anodization process to obtain highly ordered AAM. In this way, AAM replicas with high quality can be fabricated using one anodization process. Due to their mechanical properties, particularly the hardness, the TiN molds can be used many times with minimum degradation and corrosion. An atomic force microscope (AFM) image of typical pre-patterned Al foils is shown in Fig. 22.4. The electrodeposition of metals can be performed by different techniques such as constant current (galvanostatic), constant voltage (potentiostatic), current–voltage mixture pulses (Nielsch et al. 2000) and by alternating pulses (Metzger et al. 2000). A brief discussion of the reasons for the selection of one or other method is given in the following paragraphs. As was already mentioned, the bottom of the AAM presents a thick oxide (insulating) barrier of the order of 1 nm per volt of anodization potential. The

Fig. 22.4 (a) AFM image of a pre-patterned Al foil after a nanoimprint done with a TiT tool fabricated as described in the text. (b) Depth profile of the dashed line in (a) showing the homogeneity of the inprint process.

782

Magnetic nanowires: Fabrication and characterization

electrodeposition of metallic material through this barrier is not straightforward. With the AAM attached to the aluminum substrate (used as a cathode in the electrodeposition process), high potential is required for tunnelling the electrons through the barrier layer. In this situation, dc electrodeposition is very unstable and a uniform filling of the pores cannot be properly achieved. In this case, a cathode reaction in an acid media can lead to a partial removal of the barrier oxide and the formation of pores in the barrier layer. To reach uniform and complete filling of pores using dc electrodeposition, it is necessary to detach the AAM from the aluminum substrate by a chemical etching process that can be done with a supersaturated Hg chloride solution. Subsequently, the barrier layer can be removed from the nanostructured matrix by a H2 PO4 or NaOH solution. Then, a metallic contact is deposited on one side of the freestanding AAM by sputtering or other deposition method (Masuda et al. 1998). It is important to mention that this procedure is practical for thick AAM, as it is easier to handle. The ac electrodeposition process is also used for filling the pores of the AAM. In this case, AAM remains attached to the aluminum substrate and the metal is deposited by using an alternating potential (AlMawlawi et al. 1991; Li 1997). Here, the electrodeposition is not limited by the oxide barrier thickness and the rectifying contact allows the metal deposition. Till now, the most efficient route for filling the AAM pores is agreed to be the pulsed electrodeposition method (PED). This procedure, initially used for electrodeposition of thin metallic films (Tang 1995), was firstly applied in combination with a highly ordered nanostructured template material by Nielsch et al. (2000). This method is reliable for the deposition onto high aspect materials and can compensate for the slow diffusion-driven transport in the pores. Using this method, the AAM also remains attached to the aluminum substrate and rectifying contact also allows the metal deposition. In comparison to ac deposition, an additional advantage of the PED method is the better control of the deposition parameters, such as deposition rate and ion concentration at the deposition interface. The thinning of the barrier layer that can be performed via chemical pore widening (e.g. in a phosphoric acid solution), current-limited anodization process or by a combination of both, significantly improving the quality and homogeneity of the grown nanowires. The current-limited anodization method for thinning the barrier layer can be done in different ways, as described in Nielsch et al. (2000). The magnetic nanowires of transition metals are commonly electrodeposited in an aqueous solution utilizing the well-known Watts baths as electrolyte (Watts 1916). In our laboratory, homogeneous magnetic nanowires were electroplated using a modulated pulse signal consisting of a 2-ms pulse constant negative current of 70 mA/cm2 followed by a constant positive voltage pulse of 5 V (2 ms) (see Fig. 22.5). After 1 s without any applied pulse for an ion concentration recovery, the electroplating again follows with a negative current pulse and so on. The polarities here are defined as the positive electrode being the aluminum substrate. The deposition can be controlled by measuring the voltage during the negative current pulse. Figure 22.6 shows the negative current pulse voltage as a function of the pulse number in the case of Ni nanowires. In this

22.2 Metallic nanowire fabrication, the state-of-the-art 783

V (V)

5

0

I (mA)

–5

–10

–15

–20

–25 2.0×100 4.0×100 6.0×100

0.0

1×103 1×103 1×103 1×103 1×103 1×103 1×103

time (ms)

Fig. 22.5 Voltage–current mixed pulses (PED method) for magnetic nanowires deposition.

14

Voltage (V)

12

10

Zone 3a

Zone 2a

Zone 1a 8

6

4 0

50

100

150

200

250

300

Pulse number

figure we can identify three different zones. Zone 1 is identified as the initial deposition inside the dendrites (that is formed at the bottom of the pores during the thinning of the barrier layer by the current-limited anodization method) at the bottom of the AAM pores. Zone 2 represents the homogeneous growth of the nanowires and zone 3 characterizes the beginning of the Ni deposition on the top of the AAM matrix, denoting the overfilling of the pores.

Fig. 22.6 Evolution of the voltage measured during the negative curent pulse of the PED method for magnetic nanowires (in this case, Ni) electrodeposition. This type of graphic helps in the determination of the quality of the electrodeposition. From Figure 3 of Pirota et al. (2004). Copyright 2004 Elsevier B. V.

784

Magnetic nanowires: Fabrication and characterization

The fabrication of multilayered nanowires can be performed via two different procedures. The first one consists of a pulse-plating method in which two or more metals can be deposited from a single solution by switching between the electrodeposition potentials of the two constituents. Contrary to the deposition of single homogeneous nanowires described above, this method requires the use of a three-electrode cell: the working electrode, the counterelectrode and the reference electrode. This single-bath method has been used by several groups to fabricate different multilayered nanowires such as Co/Cu, NiFe/Cu, CoNi/Cu, Ni/Cu or Fe/Ni (Blondel et al. 1994; Piraux et al. 1994; Liu et al. 1995; Wang et al. 1996; Dubois et al. 1997a). The second way to fabricate multilayered nanowires makes use of two separate electrolytes. According to this method, it is necessary to change the electrolyte for each desired layer. To avoid possible contaminations from the baths, cell cleaning before each layer deposition is very important. The work of Blondel et al. (1997) compares the electronic transport properties of Co/Cu multilayered nanowires fabricated from single-or dual-bath techniques. More recently, Pirota et al. also fabricated Co/Cu multilayered nanowires using this second method and realized that, for thinner layers the Co grows in a fcc phase instead of the common hcp one (Pirota et al. 2005) (see Section 22.3). Multilayers of ferromagnetic–insulator–ferromagnetic multilayer film for magnetoresistive tunnel junctions have also been successfully grown (see, for example, Liu et al. 2005 for the case of Ni/NiO/Co case).

22.3

Structural characterization

X-ray diffraction (XRD) is generally used to investigate the structural characteristics of magnetic nanowires. The crystallographic phase of the grown nanowires can be determined and, in addition, information about lattice parameter, texture, overall crystalline quality and coherence length can be obtained (Maurice et al. 1998). Transmission electron microscopy (TEM) also allows one to investigate the overall crystalline state of the nanowires. In most cases, the membrane matrix is dissolved by some chemical etching and the nanowires are deposited onto a proper TEM grid. In addition to the structural measurements, energy-dispersive spectroscopy of X-rays (EDX) and electron energyloss spectroscopy (EELS) techniques give information about the local average composition of the samples and their band structure. In the case of multilayered nanowires, the composition determined by EDX combined with the lattice period obtained through TEM, allows one to measure the layer thickness with high precision. This cannot be achieved by TEM measurements, especially when different atoms have similar atomic weights. It is agreed that homogeneous copper, cobalt and permalloy (Ni80 Fe20 alloy usually labelled as Py) nanowires present a close-packed structure as in the corresponding bulk phases: face-centered cubic (fcc) for Cu and Py and hexagonal close packed (hcp) for Co. In the case of NiFe, the structure strongly depends on its composition, as has been studied by Liu et al. (2005), where Fe1−x Nix is shown to have a bcc structure for x < 0.35, a mixed phase of bcc and fcc for 0.35 < x < 0.5 and only a fcc phase for x > 0.5.

22.3 Structural characterization 785

20 nm

a

5 nm

Fig. 22.7 TEM images of Fe nanowires electrodeposited into the pores of oxalic anodized AAM. (b) Zoom of (a).

Information about the structure of Ni and Fe nanowires is not abundant. Recent studies (M¨onch et al. 2007) have been reported on Fe nanowires or nanorods encapsulated into C nanotubes of great interest for biomedical sensors. In these studies the structural and Fe phase composition studies have been performed using M¨ossbauer spectroscopy and high-resolution transmission electron microscopy (HRTEM). To date, this information constitutes a controversial issue strongly dependent on the synthesis method used for the nanowire growth. Regarding Ni nanowires grown in AAM by electroplating, structural studies have concluded that its polycrystalline structure and texture are insensitive to the depositing parameters (Tian et al. 2003). Figure 22.7 shows high-resolution TEM (HRTEM) images of a free-standing Ni nanowire of 35 nm diameter, fabricated in our laboratory, under two different magnifications. These figures clearly show the atomic planes oriented in different directions and clear evidence of the polycrystalline structure of the nanowires. Tian and coauthors (2003) showed that Ni wires are always polycrystalline, although they did not exclude the possibility of single-crystal growth using other techniques under pulse or ultrasonic conditions. The grazing-angle Xray diffraction spectra (XRD), for an array of Ni nanowires grown by a pulsedcurrent method on AAM, have confirmed the appearence of a single-crystalline phase (Pirota et al. 2008). Taking into account the very small mass of metal elements grown in nanowire arrays, the most efficient techniques for determining structural properties are based on synchrotron radiation. Iron nanowires grown within the pores of AAM by an ac plating procedure were characterized by high-energy X-ray diffraction (HED) (Benfield et al. 2001). It was demonstrated that Fe nanowires were made of crystallites with a strongly preferred orientation perpendicular to the AAM pore axis. This technique also allows one a high sensitivity to determine the phase grown and/or the preferred orientations of the

b

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Magnetic nanowires: Fabrication and characterization

nanowires to the pore sizes that constitutes key information for understanding their magnetic behavior. As an example of the application of XRD to nanowire arrays, the results for homogeneous Co nanowires are consistent with the stable hcp lattice with an additional peak assigned to the Al present in the samples. Although the nanowires are polycrystalline in all samples, as can be determined by quantitative measurements, there is a clear correlation between the nanowire size and crystallite size, which increases with the nanowire length as well as the volume fraction of oriented material (Pirota et al. 2008). In this case, synchrotron X-ray diffraction (XRD) was used to investigate the crystallographic phase and the texture of the Co nanowires. Nevertheless, the relative intensities of the peaks do not agree with the expected intensities for an hcp lattice. The intensity ratio [1 0 −1 0/1 0 −1 1] experimentally obtained is larger than the expected value, which denotes the presence of a strongly preferred orientation (texture) in the [10–10] direction. The experiments were performed in the LNLS (Campinas, Brasil) XRD2 beamline using reflection geometry with ˚ For each sample, a ˚ radiation and for Cu Kα radiation (1.54056 A). 1.6314 A conventional θ − 2θ curve was obtained together with rocking curves (detector angle fixed and sample scanned) for two peaks. Moreover, it is known that the crystalline structure of the electrodeposited Co thin film strongly depends on a large number of parameters, such as electrolytic bath composition, pH, deposition current density, temperature, agitation and electrodeposition dynamics (Budendorff et al. 2000). The most common (hcp) structure is usually obtained by electrodeposition at room temperature, while the fcc phase is stable at temperatures above 400 ◦ C. Both the c-axis of the hcp phase as well as the phase itself depends on the pH of the solution. Higher pH (>5) favors a hcp structure with the c-axis parallel to the nanowire length. For lower pH (