Applied Scanning Probe Methods II: Scanning Probe Microscopy Techniques (NanoScience and Technology) (v. 2)

NanoScience and Technology NanoScience and Technology Series Editors: P. Avouris B. Bhushan K. von Klitzing H. Sakaki...

Author: Bharat Bhushan | Harald Fuchs

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NanoScience and Technology

NanoScience and Technology Series Editors: P. Avouris B. Bhushan K. von Klitzing H. Sakaki R. Wiesendanger The series NanoScience and Technology is focused on the fascinating nano-world, mesoscopic physics, analysis with atomic resolution, nano and quantum-effect devices, nanomechanics and atomic-scale processes. All the basic aspects and technology-oriented developments in this emerging discipline are covered by comprehensive and timely books. The series constitutes a survey of the relevant special topics, which are presented by leading experts in the field. These books will appeal to researchers, engineers, and advanced students.

Semiconductor Spintronics and Quantum Computation Editors: D.D. Awschalom, N. Samarth, D. Loss Nano-Optoelectonics Concepts, Physics and Devices Editor: M. Grundmann Noncontact Atomic Force Microscopy Editors: S. Morita, R. Wiesendanger, E. Meyer Nanoelectrodynamics Electrons and Electromagnetic Fields in Nanometer-Scale Structures Editor: H. Nejo Single Organic Nanoparticles Editors: H. Masuhara, H. Nakanishi, K. Sasaki Epitaxy of Nanostructures By V.A. Shchukin, N.N. Ledentsov and D. Bimberg Applied Scanning Probe Methods I Editors: B. Bhushan, H. Fuchs, S. Hosaka Nanostructures Theory and Modeling By C. Delerue and M. Lannoo Nanoscale Characterisation of Ferroelectric Materials Scanning Probe Microscopy Approach Editors: M. Alexe and A. Gruverman

Magnetic Microscopy of Nanostructures Editors: H. Hopster and H.P. Oepen Silicon Quantum Integrated Circuits Silicon-Germanium Heterostructure Devices: Basics and Realisations By E. Kasper, D.J. Paul The Physics of Nanotubes Fundamentals of Theory, Optics and Transport Devices Editors: S.V. Rotkin and S. Subramoney Single Molecule Chemistry and Physics An Introduction By C. Wang, C. Bai Atomic Force Microscopy, Scanning Nearfield Optical Microscopy and Nanoscratching Application to Rough and Natural Surfaces By G. Kaupp Applied Scanning Probe Methods II Scanning Probe Microscopy Techniques Editors: B. Bhushan, H. Fuchs Applied Scanning Probe Methods III Characterization Editors: B. Bhushan, H. Fuchs Applied Scanning Probe Methods IV Industrial Applications Editors: B. Bhushan, H. Fuchs

Bharat Bhushan Harald Fuchs (Eds.)

Applied Scanning Probe Methods II Scanning Probe Microscopy Techniques

With 263 Figures Including 7 Color Figures

123

Editors: Professor Bharat Bhushan Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM) 650 Ackerman Road, Suite 255, The Ohio State University Columbus, OH 43202-1107, USA e-mail: [email protected]

Professor Dr. Harald Fuchs Center for Nanotechnology (CeNTech) and Institute of Physics University of Münster, Gievenbecker Weg 11, 48149 Münster, Germany e-mail: [email protected]

Series Editors: Professor Dr. Phaedon Avouris IBM Research Division, Nanometer Scale Science & Technology Thomas J. Watson Research Center, P.O. Box 218 Yorktown Heights, NY 10598, USA

Professor Bharat Bhushan Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM) 650 Ackerman Road, Suite 255, The Ohio State University Columbus, OH 43202-1107, USA

Professor Dr., Dres. h. c. Klaus von Klitzing Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1 70569 Stuttgart, Germany

Professor Hiroyuki Sakaki University of Tokyo, Institute of Industrial Science, 4-6-1 Komaba, Meguro-ku Tokyo 153-8505, Japan

Professor Dr. Roland Wiesendanger Institut für Angewandte Physik, Universität Hamburg, Jungiusstrasse 11 20355 Hamburg, Germany DOI 10.1007/b98404 ISSN 1434-4904 ISBN-10 3-540-26242-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26242-8 Springer Berlin Heidelberg New York Library of Congress Control Number: 2003059049 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product liability: The publishers cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Typesetting and production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design : design & production, Heidelberg Printed on acid-free paper 2/3100/YL - 5 4 3 2 1 0

Foreword

The Nobel Prize of 1986 on Scanning Tunneling Microscopy signaled a new era in imaging. The scanning probes emerged as a new instrument for imaging with a precision sufficient to delineate single atoms. At first there were two – the Scanning Tunneling Microscope, or STM, and the Atomic Force Microscope, or AFM. The STM relies on electrons tunneling between tip and sample whereas the AFM depends on the force acting on the tip when it was placed near the sample. These were quickly followed by the Magnetic Force Microscope, MFM, and the Electrostatic Force Microscope, EFM. The MFM will image a single magnetic bit with features as small as 10 nm. With the EFM one can monitor the charge of a single electron. Prof. Paul Hansma at Santa Barbara opened the door even wider when he was able to image biological objects in aqueous environments. At this point the sluice gates were opened and a multitude of different instruments appeared. There are significant differences between the Scanning Probe Microscopes or SPM, and others such as the Scanning Electron Microscope or SEM. The probe microscopes do not require preparation of the sample and they operate in ambient atmosphere, whereas, the SEM must operate in a vacuum environment and the sample must be cross-sectioned to expose the proper surface. However, the SEM can record 3D image and movies, features that are not available with the scanning probes. The Near Field Optical Microscope or NSOM is also member of this family. At this time the instrument suffers from two limitations; 1) most of the optical energy is lost as it traverses the cut-off region of the tapered fiber and 2) the resolution is insufficient for many purposes. We are confident that NSOM’s with a reasonable optical throughput and a resolution of 10 nm will soon appear. The SNOM will then enter the mainstream of scanning probes.

VI

Foreword

In the Harmonic Force Microscope or HFM, the cantilever is driven at the resonant frequency with the amplitude adjusted so that the tip impacts the sample on each cycle. The forces between tip and sample generate multiple harmonics in the motion of the cantilever. The strength of these harmonics can be used to characterize the physical properties of the surface. It is interesting to note that this technology has spawned devices of a different kind. In one instance, the tip is functionalized in a way that allows the attachment of a single protein. Withdrawing the tip from a surface stretches the attached molecule and measures the elastic properties of single protein molecules. In another the surface tension on the surface of the cantilever is modified with a self-assembled monolayer of molecules such as thiols. The slight bending of the beam is easily detected with the components developed for use in the scanning probes. This system is used to detect the presence not only of the monomolecular layers but also of single molecules attached to the initial self-assembled monolayer. The extensive material in this field means that the variety of topics is larger than can be accommodated in four volumes. The Editors, Profs. Bhushan and Fuchs, must have great powers of persuasion for they have done a remarkable job in collecting this set of paper in a relatively short period of time. The collection will become a milestone in the field of scanning probes. c. f. quate Leland T. Edwards Professor (Research) of Engineering Stanford University Stanford, California Co-inventer of AFM in 1985

Preface

The rapidly increasing activities in nanoscience and nanotechnology supported by sizable national programs has led to a variety of efforts in the development and understanding of scanning probe techniques as well as their applications to industrial and medical environments. Beyond imaging, scanning probe techniques representing the eyes of nanotechnology allows us to investigate surfaces and interfaces close to surfaces at the nanometer scale and below, thus providing information about structure, mechanical, electronic, and magnetic properties. It became apparent during the collection phase of Vol. I in 2003 that many more activities exist which deserve presentation. Therefore, this three volume set was prepared in order to display the wide breadth of this field and also to provide an excellent compendium for recent developments in this area. The response of colleagues and research groups being asked to contribute has been very positive, such that we decided, together with the publisher, to rapidly move on in these areas. It became possible to collect excellent contributions displaying first hand information from leading laboratories worldwide. The present volumes II–IV cover three main areas: scanning probe microscopy (SPM) techniques (Vol. II); characterization (basic aspects, research, Vol. III); and industrial applications (Vol. IV). Volume II includes overviews on sensor technology based on SPM probes, high harmonic dynamic force microscopy, scanning ion conduction microscopy, spin polarized STM, dynamic force microscopy and spectroscopy, quantitative nanomechanical measurements in biology, scanning micro deformation microscopy, electrostatic force and force gradient microscopy and nearfield optical microscopy. This volume also includes a contribution on nearfield probe methods such as the scanning focus ion beam technique which is an extremely valuable tool for nanofabrication including scanning probes. Volume III includes the application of scanning probe methods for the characterization of different materials, mainly in the research stage, such as applications of SPM on living cells at high resolution, macromolecular dynamics, organic supramolecular structures under UHV conditions, STS on organic and inorganic low dimensional systems, and ferroelectric materials, morphological and tribological characterization of rough surfaces, AFM for contact and wear simulation, analysis of fullerene like nanoparticles and applications in the magnetic tape industry. The more relevant industrial applications are described in Vol. IV, which deals with scanning probe lithography for chemical, biological and engineering applications, nanofabrication with self-assembled monolayers by scanned probe lithography, fabrication of nanometer scale structures by local oxidation, template effects of

VIII

Preface

molecular assemblies, microfabricated cantilever arrays, nanothermomechanics and applications of heated atomic force microscope cantilevers. Certainly, the distinction between basic research fields of scanning probe techniques and the applications in industry are not sharp, as becomes apparent in the distribution of the individual articles in the different parts of these volumes. On the other hand, this clearly reflects an extremely active research field which strengthens the cooperation between nanotechnology and nanoscience. The success of the series is solely based on the efforts and the huge amount of work done by the authors. We gratefully acknowledge their excellent contributions in a timely manner which helps to inform scientists in research and industry about latest achievements in scanning probe methods. We also would like to thank Dr. Marion Hertel, Senior Editor Chemistry, and Mrs. Beate Siek of Springer Verlag for their continuous support, without which this volume could never make it efficiently to market. January, 2006

Prof. Bharat Bhushan, USA Prof. Harald Fuchs, Germany

Contents – Volume II

1

Higher Harmonics in Dynamic Atomic Force Microscopy Robert W. Stark, Martin Stark . . . . . . . . . . . . . . . . . . . .

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8

Multimodal Model of the Microcantilever . . . Overview . . . . . . . . . . . . . . . . . . . . . Modal Analysis . . . . . . . . . . . . . . . . . Tip–Sample Interaction . . . . . . . . . . . . . State Space Formulation . . . . . . . . . . . . Dynamics: Linearized Tip–Sample Interaction Poles and Zeros . . . . . . . . . . . . . . . . . Dynamics: Nonlinear Interaction . . . . . . . . Optical Readout . . . . . . . . . . . . . . . . .

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4 4 5 7 9 11 13 16 20

1.3

Higher Harmonic Imaging . . . . . . . . . . . . . . . . . . . . . .

23

1.4 1.4.1 1.4.2 1.4.3

Spectroscopy: Distinguishing Two Polymers Overview . . . . . . . . . . . . . . . . . . . . Experimental Details . . . . . . . . . . . . . Signal Analysis . . . . . . . . . . . . . . . .

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27 27 28 28

1.5

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

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33

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Atomic Force Acoustic Microscopy Ute Rabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.1 2.1.1 2.1.2 2.1.3

Introduction . . . . . . . . . . . . . . . . . . . . . Near-field Acoustic Microscopy . . . . . . . . . . Scanning Probe Techniques and Nanoindentation . Vibration Modes of AFM Cantilevers . . . . . . .

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38 39 40 41

2.2 2.2.1 2.2.2 2.2.3

Linear Contact-resonance Spectroscopy Using Flexural Modes Flexural Vibrations of Clamped-free Beams . . . . . . . . . . . The Point-mass Model . . . . . . . . . . . . . . . . . . . . . . Experiments with Clamped-free Beams . . . . . . . . . . . . .

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42 44 47 48

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2.3

Contact Forces as Linear Springs and Dashpots . . . . . . . . . . .

51

2.4 2.4.1 2.4.2 2.4.3

Characteristic Equation of the Surface-coupled Beam Discussion of the Characteristic Equation . . . . . . . Influence of an Additional Mass . . . . . . . . . . . . Roots of the Characteristic Equation with Damping . .

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55 58 61 63

2.5

Forced Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.6

Imaging and Contrast Inversion . . . . . . . . . . . . . . . . . . .

70

2.7

Sensitivity of the Flexural Modes . . . . . . . . . . . . . . . . . . .

73

2.8 2.8.1

Quantitative Evaluation . . . . . . . . . . . . . . . . . . . . . . . . Experiments for Quantitative Evaluation . . . . . . . . . . . . . . .

76 78

2.9

Nonlinear Forces . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

2.10

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

A A.1 A.2 A.3

Appendix . . . Definitions . . UAFM-mode . AFAM-mode .

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84 84 85 86

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

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Scanning Ion Conductance Microscopy Tilman E. Schäffer, Boris Anczykowski, Harald Fuchs . . . . . . .

91

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3.2 3.2.1 3.2.2 3.2.3

Fundamental Principles Basic Setup . . . . . . Nanopipettes . . . . . . Electrodes . . . . . . .

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92 92 95 96

3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5

Ion Currents Through Nanopipettes . . Background Theory . . . . . . . . . . . Simple Analytical Model . . . . . . . . Finite Element Modeling . . . . . . . . Experimental Current-Distance Curves Imaging with Ion Current Feedback . .

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97 97 97 99 101 102

3.4 3.4.1 3.4.2

Advanced Techniques . . . . . . . . . . . . . . . . . . . . . . . . . Modulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . Applications in Bioscience . . . . . . . . . . . . . . . . . . . . . .

103 104 106

3.5 3.5.1 3.5.2 3.5.3

Combination with Other Scanning Techniques Combination with Atomic Force Microscopy . Application in Material Science . . . . . . . . Combination with Shear Force Microscopy . .

107 108 108 111

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Application in Bioscience . . . . . . . . . . . . . . . . . . . . . . .

114

3.6

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

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116

4

Spin-Polarized Scanning Tunneling Microscopy Wulf Wulfhekel, Uta Schlickum, Jürgen Kirschner . . . . . . . . . .

121

4.1 4.1.1 4.1.2 4.1.3

Introduction . . . . . . . . . . . . . . . . . . . The Resolution Problem in Magnetic Imaging Magnetism and Spin . . . . . . . . . . . . . . . The Tunneling Magnetoresistance Effect . . .

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121 121 122 122

4.2 4.2.1 4.2.2 4.2.3

The Principle of Spin-polarized Scanning Tunneling Microscopy The Constant Current Mode . . . . . . . . . . . . . . . . . . . . The Spectroscopic Mode . . . . . . . . . . . . . . . . . . . . . . Differential Magnetic Imaging Mode . . . . . . . . . . . . . . .

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124 125 125 126

4.3

Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . .

127

4.4 4.4.1 4.4.2

Ferromagnetic Domains and Domain Walls . . . . . . . . . . . . . Ultra-sharp Domain Walls in Co(0001) . . . . . . . . . . . . . . . Asymmetric Néel Caps in Fe(001) . . . . . . . . . . . . . . . . . .

128 129 131

4.5 4.5.1 4.5.2

Antiferromagnets in Contact with Ferromagnets . . . . . . . . . . Mn on Fe(001) and Topologically Induced Frustrations . . . . . . The Layered Antiferromagnet Cr on Fe(001) . . . . . . . . . . . .

133 133 136

4.6 4.6.1 4.6.2

Bulk Versus Surface: Which Electronic States Cause the Spin Contrast? . . . . . . . . . Voltage Dependence of the TMR Effect in Co(0001) . . . . . . . . Voltage Dependence of the TMR Effect in Cr/Fe(001) . . . . . . .

137 137 139

4.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140

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140

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Dynamic Force Microscopy and Spectroscopy Ferry Kienberger, Hermann Gruber, Peter Hinterdorfer . . . . . .

143

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

144

5.2

Scanning Probe Microscopy . . . . . . . . . . . . . . . . . . . . .

145

5.3

Dynamic Force Microscopy Imaging . . . . . . . . . . . . . . . . .

146

5.4 5.4.1 5.4.2 5.4.3

Force Spectroscopy Principles . . . . . Theory . . . . . . . Applications . . . .

149 149 151 153

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Combined Imaging and Spectroscopy . . . . . . . . . . . . . . . .

158

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Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . .

161

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161

6

Sensor Technology for Scanning Probe Microscopy and New Applications Egbert Oesterschulze, Leon Abelmann, Arnout van den Bos, Rainer Kassing, Nicole Lawrence, Gunther Wittstock, Christiane Ziegler . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.1

Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . .

165

6.2 6.2.1

Material Aspects of Probe Fabrication . . . . . . . . . . . . . . . . Mechanical Properties of Cantilever Probes . . . . . . . . . . . . .

166 167

6.3 6.3.1 6.3.2

Scanning Near-Field Optical Microscopy . . . . . . . . . . . . . . Principle of Near-Field Optics . . . . . . . . . . . . . . . . . . . . Probes for Scanning Near-Field Optical Microscopy (SNOM) . . .

174 174 175

6.4 6.4.1

Probes for Ultrafast Scanning Probe Microscopy . . . . . . . . . . Improved Sampling Technique . . . . . . . . . . . . . . . . . . . .

179 181

6.5 6.5.1 6.5.2

Functionalized Tips . . . . . . . . . . . . . . . . . . . . . . . . . . Tip Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182 182 183

6.6 6.6.1 6.6.2

Scanning Electrochemical Microscopy . . . . . . . . . . . . . . . . Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

186 186 189

6.7 6.7.1 6.7.2 6.7.3 6.7.4

Tips for Magnetic Force Microscopy . Ideal Tip Shape . . . . . . . . . . . . Hand-Made Tips . . . . . . . . . . . . Coating AFM Tips . . . . . . . . . . Tip Planes: The CantiClever Concept

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192 192 193 194 195

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Quantitative Nanomechanical Measurements in Biology Małgorzata Lekka, Andrzej J. Kulik . . . . . . . . . . . . . . . . . Stiffness of Biological Samples . . . . . . . Cell Structure . . . . . . . . . . . . . . . . Determination of Young’s Modulus . . . . Brief Overview of the Application of AFM to Studies of Living Cells . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . .

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237

8

Friction Force Microscopy . . . . . . . . Friction and Chemical Force Microscopy Applications of FFM/CFM . . . . . . . . Summary . . . . . . . . . . . . . . . . . .

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Scanning Microdeformation Microscopy: Subsurface Imaging and Measurement of Elastic Constants at Mesoscopic Scale Pascal Vairac, Bernard Cretin . . . . . . . . . . . . . . . . . . . .

241

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241

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Review and Physical Background of Near-Field Acoustic Microscopes . . . . . . . . . . . . . . . . . . . . . . Review of Near-Field Microscopes . . . . . . . . . . . . . . . Physical Basis for Near-Field Acoustics and the Scale Effect Mechanical Approach . . . . . . . . . . . . . . . . . . . . . . Models of Subsurface Sensing Using Acoustic Waves and Surface Bending . . . . . . . . . . . . . . . . . . . . . .

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242 242 244 247

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252

8.3.1 8.3.2 8.3.3

Imaging and Measurement with Scanning Microdeformation Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . Application to Subsurface Imaging . . . . . . . . . . . . . . . Characterization of Local Mechanical Constants . . . . . . .

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254 254 256 259

8.4 8.4.1 8.4.2 8.4.3

Specific Application . . . Thin Film Measurements Shape Memory Alloy . . Viscosimetry . . . . . . .

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260 260 264 267

8.5

Ultimate Metrology: Measurements at the Mechanical Noise Level

274

8.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

278

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279

8.2.1 8.2.2 8.2.3 8.2.4 8.3

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Electrostatic Force and Force Gradient Microscopy: Principles, Points of Interest and Application to Characterisation of Semiconductor Materials and Devices Paul Girard, Alexander Nikolaevitch Titkov . . . . . . . . . . . . .

283

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285

9.2 9.2.1 9.2.2

Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principles of Surface-voltage Measurements . . . . . . . . . . . .

285 286 287

XIV

9.2.3


9.2.4

Detection of Strong Local Electrical Effect via the “Topographic” Data . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

294 296

9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.3.5

Observation and Interpretation DC Observations . . . . . . . Ω Observations . . . . . . . . 2Ω Observations . . . . . . . Surface Voltage Observations . Guidelines for Interpretation .

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297 299 300 300 302 302

9.4 9.4.1 9.4.2 9.4.3

Future Opportunities . . . . . . . . . . . . . . . Interest in the KFGM Method . . . . . . . . . . Spatially Resolved Observations . . . . . . . . . Another Way to Estimate the Maximum Possible Spatial Resolution . . . . . . . . . . . . . . . . .

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304 304 309

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311

9.5 9.5.1 9.5.2

Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . Applications Under Ambient Conditions . . . . . . . . . . . . . . Vacuum or UHV Applications . . . . . . . . . . . . . . . . . . . .

313 314 316

9.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

316

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

318

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10

Polarization-Modulation Techniques in Near-Field Optical Microscopy for Imaging of Polarization Anisotropy in Photonic Nanostructures Pietro Giuseppe Gucciardi, Ruggero Micheletto, Yoichi Kawakami, Maria Allegrini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

10.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321

10.2 10.2.1

Polarimetric Imaging . . . . . . . . . . . . . . . . . . . . . . . . . The Jones Formalism . . . . . . . . . . . . . . . . . . . . . . . . .

322 325

10.3

Electromagnetic Field Diffracted by a SNOM Aperture . . . . . .

327

10.4 10.4.1 10.4.2 10.4.3

Experimental Implementations . . . . . . . . . . . . Static Polarization SNOM . . . . . . . . . . . . . . Polarization-Modulation SNOM: Illumination Mode Polarization-Modulation SNOM: Collection Mode .

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333 333 337 342

10.5 10.5.1 10.5.2 10.5.3 10.5.4 10.5.5

Applications of SNOM Polarimetry . . . . . . . . . . . . . Polarization Responses of Photonic Waveguides . . . . . . Measuring Stress-Induced Birefringence . . . . . . . . . . . Polarization Anisotropy in Mesoscale-Structured Materials Polarization Anisotropy in Polymers . . . . . . . . . . . . . Polarization Anisotropy in Photoluminescence Emission . .

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344 345 348 349 351 354

10.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

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11

XV

Focused Ion Beam as a Scanning Probe: Methods and Applications Vittoria Raffa, Piero Castrataro, Arianna Menciassi, Paolo Dario .

361

11.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361

11.2 11.2.1 11.2.2 11.2.3 11.2.4

Description of the System . . . . . System Overview . . . . . . . . . Liquid Metal Ion Source (LMIS) . Ion Optics . . . . . . . . . . . . . Dual Beam Systems . . . . . . . .

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362 362 363 364 365

11.3 11.3.1 11.3.2 11.3.3 11.3.4 11.3.5

FIB Processes . . . . . . Imaging . . . . . . . . . Milling . . . . . . . . . . Gas-Assisted Etching . . Gas-Assisted Deposition Ion Beam Lithography .

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367 367 372 376 377 379

11.4 11.4.1 11.4.2 11.4.3

Main Applications . . . . . . . . . . FIB as an Analytical Technique . . FIB in the Semiconductor Industry . Micromachining . . . . . . . . . . .

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380 381 389 401

11.5

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . .

408

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409

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Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Contents – Volume III

12

Atomic Force Microscopy in Nanomedicine Dessy Nikova, Tobias Lange, Hans Oberleithner, Hermann Schillers, Andreas Ebner, Peter Hinterdorfer . . . . . . .

1

12.1

AFM in Biological Sciences . . . . . . . . . . . . . . . . . . . . .

1

12.2 12.2.1 12.2.2 12.2.3 12.2.4 12.2.5

Plasma Membrane Preparation for AFM Imaging . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Plasma Membrane Preparation . . . . . . . . . . . . . . . Atomic Force Microscopy . . . . . . . . . . . . . . . . . Molecular Volume Measurements of Membrane Proteins . AFM Imaging . . . . . . . . . . . . . . . . . . . . . . . .

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4 4 5 7 7 7

12.3 12.3.1 12.3.2 12.3.3

AFM Imaging of CFTR in Oocyte Membranes Introduction . . . . . . . . . . . . . . . . . . . Does the CFTR Form Functional Assemblies? Two CFTRs are Better Than One . . . . . . . .

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10 11 11 13

12.4 12.4.1 12.4.2 12.4.3 12.4.4

Single Antibody–CFTR Recognition Imaging . Introduction . . . . . . . . . . . . . . . . . . . Tethering of Antibodies to AFM Tips . . . . . AFM Imaging and Recognition . . . . . . . . . A Single Antibody Sees a Single CFTR . . . .

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16 16 17 17 17

12.5 12.5.1 12.5.2 12.5.3

Single Cell Elasticity: Probing for Diseases Introduction . . . . . . . . . . . . . . . . . Force–Mapping AFM . . . . . . . . . . . . Can One Protein Change Cell Elasticity? .

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19 19 20 21

12.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

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XVIII

13


Scanning Probe Microscopy: From Living Cells to the Subatomic Range Ille C. Gebeshuber, Manfred Drack, Friedrich Aumayr, Hannspeter Winter, Friedrich Franek . . . . . . . . . . . . . . . . .

27

13.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

13.2 13.2.1 13.2.2

Cells In Vivo as Exemplified by Diatoms . . . . . . . . . . . . . . Introduction to Diatoms . . . . . . . . . . . . . . . . . . . . . . . . SPM of Diatoms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28 28 30

13.3

Interaction of Large Organic Molecules . . . . . . . . . . . . . . .

33

13.4 13.4.1 13.4.2

Nanodefects on Atomically Flat Surfaces . . . . . . . . . . . . . . Ion Bombardment of Highly Oriented Pyrolytic Graphite (HOPG) Bombardment of Single Crystal Insulators with Multicharged Ions

37 38 42

13.5 13.5.1 13.5.2

Subatomic Features . . . . . . . . . . . . . . . . . . . . . . . . . . Atom Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Electron Spin Detection with AFM and STM . . . . . . . .

45 45 47

13.6

Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . .

50

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

14

Surface Characterization and Adhesion and Friction Properties of Hydrophobic Leaf Surfaces and Nanopatterned Polymers for Superhydrophobic Surfaces Zachary Burton, Bharat Bhushan . . . . . . . . . . . . . . . . . .

55

14.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

14.2 14.2.1 14.2.2 14.2.3 14.2.4

Experimental Details . . . . . . . . . . . Instrumentation . . . . . . . . . . . . . . Samples . . . . . . . . . . . . . . . . . . Roughness Factor . . . . . . . . . . . . . Test Matrix for Nanopatterned Polymers .

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58 58 59 61 62

14.3 14.3.1 14.3.2

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . Hydrophobic Leaf Surfaces . . . . . . . . . . . . . . . . . . . . . . Nanopatterned Polymers . . . . . . . . . . . . . . . . . . . . . . .

63 64 74

14.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

15

15.1

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Probing Macromolecular Dynamics and the Influence of Finite Size Effects Scott Sills, René M. Overney . . . . . . . . . . . . . . . . . . . . .

83

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84


XIX

15.2

The Glass Transition and Molecular Mobility . . . . . . . . . . . .

85

15.3 15.3.1 15.3.2 15.3.3 15.3.4 15.3.5 15.3.6

Macromolecular Probing Techniques . . . . . . . . . . . . . Static Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . Modulated Contacts . . . . . . . . . . . . . . . . . . . . . . . Calibration of Lateral Forces in Scanning Probe Microscopy . Shear Modulation Force Microscopy (SM-FM) . . . . . . . . Friction Force Microscopy (FFM) . . . . . . . . . . . . . . . Tribological Models for FFM . . . . . . . . . . . . . . . . . .

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90 90 92 93 97 98 99

15.4 15.4.1 15.4.2 15.4.3

Internal Friction and Dynamics near the Glass Transition . . . . . Molecular Relaxations . . . . . . . . . . . . . . . . . . . . . . . . Structural Heterogeneity and Relaxation near the Glass Transition Cooperative Molecular Motion During the Glass Transition . . . .

103 103 105 107

15.5 15.5.1 15.5.2 15.5.3 15.5.4

Constraints and Structural Modifications near Interfaces Interfacial Plasticization . . . . . . . . . . . . . . . . . . Dewetting Kinetics . . . . . . . . . . . . . . . . . . . . Disentanglement Barriers . . . . . . . . . . . . . . . . . Interfacial Glass Transition Profiles . . . . . . . . . . .

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109 109 110 111 113

15.6 15.6.1 15.6.2 15.6.3

Mechanical Operations in Nanoscopic Polymer Systems Indentation Contact Mechanics . . . . . . . . . . . . . . Rim Formation During Indentation . . . . . . . . . . . . Strain Shielding and Confined Plasticity in Nanoscopic Polymer Systems . . . . . . . . . . . . .

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115 116 120

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122

Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

15.7

16

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Investigation of Organic Supramolecules by Scanning Probe Microscopy in Ultra-High Vacuum Laurent Nony, Enrico Gnecco, Ernst Meyer . . . . . . . . . . . . .

131

16.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

16.2 16.2.1 16.2.2 16.2.3

Methods . . . . . . . . . . . . . . . . . . . Organic Molecular Beam Epitaxy (OMBE) Scanning Tunneling Microscopy (STM) . . Atomic Force Microscopy (AFM) . . . . .

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132 132 134 137

16.3 16.3.1 16.3.2 16.3.3 16.3.4 16.3.5 16.3.6 16.3.7

Molecules . . . . . . . . . . Fullerenes . . . . . . . . . . Porphyrins . . . . . . . . . . Phthalocyanines . . . . . . . Perylene Derivatives . . . . . Lander Molecules . . . . . . PVBA Molecules . . . . . . Decacyclene and Derivatives

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142 142 142 142 144 144 144 144

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XX


16.4 16.4.1 16.4.2

Molecules on Metals . . . . . . . . . . . . . . . . . . . . . . . . . STM Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Contact AFM Investigations . . . . . . . . . . . . . . . . . .

145 145 157

16.5 16.5.1 16.5.2

Molecules on Semiconductor Surfaces . . . . . . . . . . . . . . . . STM Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . Non-Contact AFM Investigations . . . . . . . . . . . . . . . . . .

162 162 165

16.6 16.6.1 16.6.2

Molecules on Insulating Surfaces . . . . . . . . . . . . . . . . . . . STM Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . Non-contact AFM Investigations . . . . . . . . . . . . . . . . . . .

167 168 169

16.7 16.7.1 16.7.2

Manipulation of Single Molecules . . . . . . . . . . . . . . . . . . STM Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . Non-contact AFM Investigations . . . . . . . . . . . . . . . . . . .

171 171 175

16.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

176

17

One- and Two-Dimensional Systems: Scanning Tunneling Microscopy and Spectroscopy of Organic and Inorganic Structures Luca Gavioli, Massimo Sancrotti . . . . . . . . . . . . . . . . . . .

183

17.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

17.2

Basic Principles of STM and STS . . . . . . . . . . . . . . . . . .

185

17.3 17.3.1 17.3.2

Inorganic Overlayers . . . . . . . . . . . . . . . . . . . . . . . . . 1D Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2D Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188 188 196

17.4 17.4.1 17.4.2

Molecular Overlayers . . . . . . . . . . . . . . . . . . . . . . . . . 1D Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2D Overlayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 202 208

17.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

212

18

Scanning Probe Microscopy Applied to Ferroelectric Materials Oleg Tikhomirov, Massimiliano Labardi, Maria Allegrini . . . . . .

217

18.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

217

18.2

Development of Scanning Probe Techniques for Ferroelectrics . .

217

18.3 18.3.1 18.3.2 18.3.3

Scanning Force Microscopy Non-Contact Mode . . . . . Contact Mode . . . . . . . . Voltage-Modulated SFM . .

220 220 221 222

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XXI

18.3.4 18.3.5 18.3.6 18.3.7 18.3.8

Resonance Modes of EFM Lateral Force . . . . . . . . Frontal Force . . . . . . . . Second Harmonic . . . . . Tapping Mode . . . . . . .

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224 228 232 233 234

18.4 18.4.1 18.4.2 18.4.3 18.4.4 18.4.5

Scanning Optical Microscopy . . . Pure Optical Microscopy . . . . . . Scanning Electrooptic Microscopy . Near-Field Electrooptic Microscopy Micro-Spectroscopic Techniques . . Second Harmonic Microscopy . . .

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235 235 237 242 244 245

18.5 18.5.1 18.5.2 18.5.3 18.5.4 18.5.5 18.5.6 18.5.7

Applications to Ferroelectrics . . . . . . Imaging of Domains and Domain Walls Writing Patterns . . . . . . . . . . . . . Phase Transitions . . . . . . . . . . . . Morphotropic Phase Boundary . . . . . Relaxors . . . . . . . . . . . . . . . . . Thin Films . . . . . . . . . . . . . . . . Artificial Nanostructures . . . . . . . .

. . . . . . . .

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247 247 248 249 250 251 251 252

18.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

19

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. . . . .

. . . . .

Morphological and Tribological Characterization of Rough Surfaces by Atomic Force Microscopy Renato Buzio, Ugo Valbusa . . . . . . . . . . . . . . . . . . . . . .

261

19.1.1 19.1.2 19.1.3

Characterization of Surface Roughness by Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . Statistical Methods for Stationary Random Surfaces . . . . . . . . Statistical Methods for Fractal Surfaces . . . . . . . . . . . . . . . Estimation of Morphological Parameters from AFM Topographies

263 264 266 270

19.2 19.2.1 19.2.2 19.2.3

Modeling Contact Mechanics for Rough Surfaces . . Early Phenomenological Contact Theories . . . . . Contact Mechanics Theories for Fractal Roughness . On the Molecular Origins of Amontons’ Law . . . .

. . . .

272 273 277 284

19.3 19.3.1

. . . .

286

19.3.2 19.3.3

Investigations of Multi-Asperity Contacts by AFM . . . . . AFM Characterization of Surface Roughness for Tribological Purposes . . . . . . . . . . . . . . . . . . . Contact Mechanics Investigations at the Nanometer Scale . Contact Mechanics Investigations on the Micrometer Scale

. . . . . . . . . . . .

286 288 291

19.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

294

19.1

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. . . .

. . . .

. . . .

XXII

20


AFM Applications for Contact and Wear Simulation Nikolai K. Myshkin, Mark I. Petrokovets, Alexander V. Kovalev . .

299

20.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299

20.2

Scale Factor in Tribology . . . . . . . . . . . . . . . . . . . . . . .

299

20.3 20.3.1 20.3.2 20.3.3 20.3.4 20.3.5

AFM as a Tool of Contact Simulation . . . Contact of Rough Surfaces . . . . . . . . . Rough Contact with Adhesion . . . . . . . Multilevel Contact Models . . . . . . . . . Simulation of Contact Using AFM Images Nanomechanical Probing of Soft Layers . .

. . . . . .

300 300 303 307 309 312

20.4 20.4.1 20.4.2

AFM in Wear Simulation . . . . . . . . . . . . . . . . . . . . . . . Nanoscratching and Nanowear with AFM Tip . . . . . . . . . . . . Wear Simulation in AFM Contact Mode . . . . . . . . . . . . . . .

316 317 320

20.5

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

323

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

324

21

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AFM Applications for Analysis of Fullerene-Like Nanoparticles Lev Rapoport, Armen Verdyan . . . . . . . . . . . . . . . . . . . .

327

21.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

21.2 21.2.1 21.2.2

Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friction Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . AFM Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . .

328 328 329

21.3

Characterization of Fullerene-Like Solid Lubricant Nanoparticles .

330

21.4

Friction of Solid Lubricant Films . . . . . . . . . . . . . . . . . . .

331

21.5

Friction and Wear of the Surfaces Lubricated with Oil + IF Nanoparticles . . . . . . . . . . . . . . . . . . . . . .

333

21.6

Friction of IF Nanoparticles Under Severe Contact Conditions . .

336

21.7

Mechanisms of Friction of the IF Nanoparticles . . . . . . . . . .

339

21.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341

22

Scanning Probe Methods in the Magnetic Tape Industry James K. Knudsen . . . . . . . . . . . . . . . . . . . . . . . . . . .

343

22.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

343

22.2

Atomic Force Microscopy . . . . . . . . . . . . . . . . . . . . . .

345


XXIII

22.2.1 22.2.2 22.2.3

Topographic Characterization of the Magnetic Tape . . . . . . . . Topographic Characterization of Heads . . . . . . . . . . . . . . . Tape Roughness Analysis . . . . . . . . . . . . . . . . . . . . . . .

345 349 351

22.3 22.3.1 22.3.2 22.3.3

Magnetic Force Microscopy . . . . . . . . . . . . Methodology . . . . . . . . . . . . . . . . . . . . . Characterization of the Magnetic Tape with MFM Characterization of Heads with MFM . . . . . . .

. . . .

358 358 359 364

22.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367

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Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

Contents – Volume IV

23

Scanning Probe Lithography for Chemical, Biological and Engineering Applications Joseph M. Kinsella, Albena Ivanisevic . . . . . . . . . . . . . . . .

1

23.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

23.2

Modeling of the DPN Process . . . . . . . . . . . . . . . . . . . .

4

23.3 23.3.1 23.3.2 23.3.3 23.3.4

Patterning of Biological and Biologically Active Molecules . DNA Patterning . . . . . . . . . . . . . . . . . . . . . . . . . Protein Patterning . . . . . . . . . . . . . . . . . . . . . . . . Peptide Patterning . . . . . . . . . . . . . . . . . . . . . . . . Patterning of Templates for Biological Bottom-Up Assembly

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7 8 10 13 15

23.4 23.4.1 23.4.2 23.4.3 23.4.4 23.4.5 23.4.6 23.4.7 23.4.8 23.4.9 23.4.10 23.4.11

Chemical Patterning . . . . . . . . . . . . Thiols . . . . . . . . . . . . . . . . . . . ω-Substituted Thiols . . . . . . . . . . . Silanes and Silazanes . . . . . . . . . . . Deposition of Solid Organic Inks . . . . . Polymers . . . . . . . . . . . . . . . . . . Polyelectrolytes . . . . . . . . . . . . . . Dendrimers . . . . . . . . . . . . . . . . Deposition of Supramolecular Materials . Deposition of Metals . . . . . . . . . . . Deposition of Solid-State Materials . . . Deposition of Magnetic Materials . . . .

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17 17 18 19 20 21 23 23 24 25 26 27

23.5

Engineering Applications of DPN . . . . . . . . . . . . . . . . . .

28

23.6

Future Challenges and Applications . . . . . . . . . . . . . . . . .

30

23.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

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XXVI

24


Nanotribological Characterization of Human Hair and Skin Using Atomic Force Microscopy (AFM) Bharat Bhushan, Carmen LaTorre . . . . . . . . . . . . . . . . . .

35

24.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

24.2 24.2.1 24.2.2

Human Hair, Skin, and Hair Care Products . . . . . . . . . . . . . Human Hair and Skin . . . . . . . . . . . . . . . . . . . . . . . . . Hair Care: Cleaning and Conditioning Treatments, and Damaging Processes . . . . . . . . . . . . . . . . . . . . . . .

39 39

24.3 24.3.1 24.3.2

Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . Hair and Skin Samples . . . . . . . . . . . . . . . . . . . . . . . .

51 53 57

24.4 24.4.1

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59

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59

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70

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78

24.4.5

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . Surface Roughness, Friction, and Adhesion for Various Ethnicities of Hair . . . . . . . . . . . . . . . . . . Surface Roughness, Friction, and Adhesion for Virgin and Chemically Damaged Caucasian Hair (with and without Commercial Conditioner Treatment) . . . . . Surface Roughness, Friction, and Adhesion for Hair Treated with Various Combinations of Conditioner Ingredients Investigation of Directionality Dependence and Scale Effects on Friction and Adhesion of Hair . . . . . . . . . . . . . . . . . Surface Roughness and Friction of Skin . . . . . . . . . . . . .

. . . .

85 98

24.5

Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

24.4.2

24.4.3 24.4.4

25

46

Nanofabrication with Self-Assembled Monolayers by Scanning Probe Lithography Jayne C. Garno, James D. Batteas . . . . . . . . . . . . . . . . . .

105

25.1 25.1.1 25.1.2 25.1.3 25.1.4

SPM-Based Methods of Lithography . . Bias-Induced Nanofabrication . . . . . . Force-Induced Nanofabrication of SAMs Dip-Pen Nanolithography (DPN) . . . . . Automated Scanning Probe Lithography .

. . . . .

105 107 108 110 111

25.2 25.2.1 25.2.2 25.2.3

Patterning with Self-Assembled Monolayers . . . . . . . . . . . . Structure of SAMs . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of SAM Nanopatterns Generated by Force-Induced SPL Nanofabrication of SAMs by DPN and Bias-Induced SPL . . . . .

112 112 114 118

25.3

Directed Fabrication of Polymeric Structures . . . . . . . . . . . .

120

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XXVII

25.4

Fabrication of Metallic Structures . . . . . . . . . . . . . . . . . .

122

25.5 25.5.1 25.5.2 25.5.3

Nanoscale Patterning of Proteins . . . . . . . . . . . . Protein Arrays Generated by DPN . . . . . . . . . . . Applying Bias-Induced SPL for Protein Nanopatterns Protein Immobilization on SAMs Generated by Force-Induced SPL . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

126 127 128

. . . . . . .

129

Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . .

130

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

25.6

26

Fabrication of Nanometer-Scale Structures by Local Oxidation Nanolithography Marta Tello, Fernando García, Ricardo García . . . . . . . . . . .

137

26.1

Introduction to AFM Nanolithographies . . . . . . . . . . . . . . .

137

26.2

Basic Local Oxidation Aspects . . . . . . . . . . . . . . . . . . . .

138

26.3

Mechanism and Kinetics . . . . . . . . . . . . . . . . . . . . . . .

141

26.4

Feature Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

26.5

Applications I: Patterning, Data Storage and Template Growth . .

146

26.6

Applications II: Nanoelectronic Devices . . . . . . . . . . . . . . .

151

26.7

Parallel Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . .

154

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

27

Template Effects of Molecular Assemblies Studied by Scanning Tunneling Microscopy (STM) Chen Wang, Chunli Bai . . . . . . . . . . . . . . . . . . . . . . . .

159

27.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

27.2

27.2.2 27.2.3 27.2.4

Single Guest Molecule Immobilization with Assembled Molecular Networks . . . . . Hydrogen Bonded Supramolecular Networks and Single Molecule Inclusions . . . . . . . . Van der Waals Interaction Stabilized Networks Metal-Organic Coordination Networks . . . . Covalently Bonded Molecular Grids . . . . . .

. . . . . . . . . . .

160

. . . .

. . . .

160 163 165 166

27.3 27.3.1 27.3.2

Intralayer Heterogeneous Molecular Arrays . . . . . . . . . . . . . Hydrogen Bond Stabilized Heterogeneous Lamellae . . . . . . . . Van der Waals Interaction Stabilized Intralayer Arrays . . . . . . .

166 167 168

27.4 27.4.1

Interlayer Effect on Molecular Adsorption and Assemblies . . . . Site Selective Adsorption . . . . . . . . . . . . . . . . . . . . . . .

171 172

27.2.1

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XXVIII


27.4.2 27.4.3

Molecular Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . Directional Assembly of Nanoparticle Arrays . . . . . . . . . . . .

176 177

27.5

Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . .

179

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

28

Microfabricated Cantilever Array Sensors for (Bio-)Chemical Detection Hans Peter Lang, Martin Hegner, Christoph Gerber . . . . . . . .

183

28.1 28.1.1 28.1.2 28.1.3 28.1.4

Introduction . . . . . . . . . Sensors . . . . . . . . . . . . Cantilevers . . . . . . . . . . Cantilever Operating Modes Cantilever Arrays . . . . . .

. . . . .

183 183 184 186 192

28.2 28.2.1 28.2.2

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement Chamber . . . . . . . . . . . . . . . . . . . . . . . . Cantilever Functionalization . . . . . . . . . . . . . . . . . . . . .

196 196 198

28.3 28.3.1 28.3.2

Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Artificial Nose for Detection of Perfume Essences . . . . . . . . . Label-Free DNA Hybridization Detection . . . . . . . . . . . . . .

203 204 206

28.4

Applications and Outlook . . . . . . . . . . . . . . . . . . . . . . .

209

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

210

29

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Nano-Thermomechanics: Fundamentals and Application in Data Storage Devices B. Gotsmann, U. Dürig . . . . . . . . . . . . . . . . . . . . . . . .

215

29.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

215

29.2 29.2.1 29.2.2 29.2.3 29.2.4

Heat Transfer Mechanisms . . . . . . . . . . . . . . Heat Generation in Microcantilevers . . . . . . . . . Heat Transfer Through Air and Silicon . . . . . . . Heat Transfer Through Radiation . . . . . . . . . . . Heat Transfer Through a Tip-Surface Point Contact

. . . . .

215 216 217 222 224

29.3

Momentum Transfer Through Air . . . . . . . . . . . . . . . . . .

227

29.4 29.4.1 29.4.2 29.4.3 29.4.4 29.4.5 29.4.6 29.4.7

Thermomechanical Nanoindentation of Polymers . . . General Considerations . . . . . . . . . . . . . . . . . Indentation Experiments . . . . . . . . . . . . . . . . Interlude: Carbon Nanotube Tips . . . . . . . . . . . . Interlude: Thermal Force and Indentation Formation . Interlude: Rim Formation on Polymer Samples . . . . Indentation Kinetics and the Indentation Mechanism . Interlude: Thermo-Nano-Mechanics Without a Heater

229 229 230 232 234 234 236 239

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XXIX

29.5

Thermomechanical Nanowear Testing . . . . . . . . . . . . . . . .

241

29.6 29.6.1 29.6.2

Application to Data-Storage Devices . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaling Challenges for Nanoindentation of Polymers . . . . . . . .

243 243 245

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

248

30

Applications of Heated Atomic Force Microscope Cantilevers Brent A. Nelson, William P. King . . . . . . . . . . . . . . . . . . .

251

30.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

251

30.2 30.2.1 30.2.2 30.2.3

Physical and Environmental Sensing . . . . . . . . . . . . Pressure Sensing . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity Mapping and Subsurface Imaging . Topographical Detection . . . . . . . . . . . . . . . . . .

. . . .

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252 252 253 258

30.3 30.3.1 30.3.2 30.3.3 30.3.4

Chemical Sensing Applications . . . . . . . Calorimetry . . . . . . . . . . . . . . . . . Mass Detection . . . . . . . . . . . . . . . Time-of-Flight Scanning Force Microscopy Explosives Detection . . . . . . . . . . . .

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261 261 262 263 264

30.4 30.4.1 30.4.2

Data Storage and Lithography . . . . . . . . . . . . . . . . . . . . Data Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

264 265 269

30.5

Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . .

272

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

272

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Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Contents – Volume I

Part I

Scanning Probe Microscopy

1

Dynamic Force Microscopy André Schirmeisen, Boris Anczykowski, Harald Fuchs . . . . . . .

3

Interfacial Force Microscopy: Selected Applications Jack E. Houston . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Atomic Force Microscopy with Lateral Modulation Volker Scherer, Michael Reinstädtler, Walter Arnold . . . . . . . .

75

Sensor Technology for Scanning Probe Microscopy Egbert Oesterschulze, Rainer Kassing . . . . . . . . . . . . . . . .

117

Tip Characterization for Dimensional Nanometrology John S. Villarrubia . . . . . . . . . . . . . . . . . . . . . . . . . .

147

2 3 4 5

Part II

Characterization

6

Micro/Nanotribology Studies Using Scanning Probe Microscopy Bharat Bhushan . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171

Visualization of Polymer Structures with Atomic Force Microscopy Sergei Magonov . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207

Displacement and Strain Field Measurements from SPM Images Jürgen Keller, Dietmar Vogel, Andreas Schubert, Bernd Michel . .

253

7 8 9

AFM Characterization of Semiconductor Line Edge Roughness Ndubuisi G. Orji, Martha I. Sanchez, Jay Raja, Theodore V. Vorburger . . . . . . . . . . . . . . . . . . . . . . . . . 277

10

Mechanical Properties of Self-Assembled Organic Monolayers: Experimental Techniques and Modeling Approaches Redhouane Henda . . . . . . . . . . . . . . . . . . . . . . . . . . .

303

XXXII

11 12

Contents – Volume I

Micro-Nano Scale Thermal Imaging Using Scanning Probe Microscopy Li Shi, Arun Majumdar . . . . . . . . . . . . . . . . . . . . . . . .

327

The Science of Beauty on a Small Scale. Nanotechnologies Applied to Cosmetic Science Gustavo Luengo, Frédéric Leroy . . . . . . . . . . . . . . . . . . .

363

Part III Industrial Applications 13 14 15

16

SPM Manipulation and Modifications and Their Storage Applications Sumio Hosaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

389

Super Density Optical Data Storage by Near-Field Optics Jun Tominaga . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429

Capacitance Storage Using a Ferroelectric Medium and a Scanning Capacitance Microscope (SCM) Ryoichi Yamamoto . . . . . . . . . . . . . . . . . . . . . . . . . . .

439

Room-Temperature Single-Electron Devices formed by AFM Nano-Oxidation Process Kazuhiko Matsumoto . . . . . . . . . . . . . . . . . . . . . . . . .

459

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

List of Contributors – Volume II

Leon Abelmann Systems and Materials for Information storage group MESA + Research Institute P.O. Box 217, 7500 AE Enschede, The Netherlands e-mail: [email protected] Maria Allegrini INFM and Dipartimento di Fisica “Entrico Fermi”, Università di Pisa Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy e-mail: [email protected] Boris Anczykowski nanoAnalytics GmbH, Gievenbecker Weg 11, 48149 Münster, Germany e-mail: [email protected] Piero Castrataro Scuola Superiore Sant’Anna, Polo Sant’Anna Valdera – CRIM Lab Viale Rinaldo Piaggio, 34, 56025 Pontedera (PI), Italy e-mail: [email protected] Bernard Cretin Department LPMO, FEMTO-ST Institute, UMR CNRS 6174 32 avenue de l’Observatoire, 25044 Besançon Cedex, France e-mail: [email protected] Paolo Dario Scuola Superiore Sant’Anna, Polo Sant’Anna Valdera – CRIM Lab Viale Rinaldo Piaggio, 34, 56025 Pontedera (PI), Italy e-mail: [email protected] Harald Fuchs Center for Nanotechnology (CeNTech) and Institute of Physics University of Münster, Gievenbecker Weg 11, 48149 Münster, Germany e-mail: [email protected] Paul Girard LAIN, UMR CNRS 5011, CC 082, Université de Montpellier II Place E. Bataillon, 34095 Montpellier Cedex 5, France e-mail: [email protected]

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Hermann Gruber Institute for Biophysics, J. Kepler University Altenbergerstr. 69, A-4040 Linz, Austria e-mail: [email protected] Pietro Giuseppe Gucciardi CNR-Istituto per i Processi Chimico-Fisici Via La Farina 237, I-98123 Messina, Italy e-mail: [email protected] Peter Hinterdorfer Institute for Biophysics, J. Kepler University Altenbergerstrasse 69, A-4040 Linz, Austria e-mail: [email protected] Rainer Kassing University of Kassel Institute for Microstructure Technologies and Analytics, IMA Technological Physics Heinrich-Plett-Str. 40, D-34132 Kassel, Germany e-mail: [email protected] Yoichi Kawakami Department of Electronic Science, Graduate School of Engineering Kyoto University, Nishikyo-ku, Katsura, 615-8510 Kyoto, Japan e-mail: [email protected] Ferry Kienberger Institute for Biophysics, J. Kepler University Altenbergerstr. 69, A-4040 Linz, Austria e-mail: [email protected] Jürgen Kirschner Max-Planck-Institut für Mikrostrukturphysik Weinberg 2, 06120 Halle, Germany e-mail: [email protected] Andrzej J. Kulik Ecole Polytechnique Fédérale de Lausanne, EPFL – IPMC – NN 1015 Lausanne, Switzerland e-mail: Andrzej.Kulik@epfl.ch Nicole Lawrence (geb. Schwendler) Technische Universität Kaiserslautern Erwin-Schrödinger Strasse, 67663 Kaiserslautern, Germany e-mail: [email protected]


Małgorzata Lekka The Henryk Niewodniczañski Institute of Nuclear Physics Polish Academy of Sciences Radzikowskiego 152, 31-342 Kraków, Poland e-mail: [email protected] Arianna Menciassi Scuola Superiore Sant’Anna, Polo Sant’Anna Valdera – CRIM Lab Viale Rinaldo Piaggio, 34, 56025 Pontedera (PI), Italy e-mail: [email protected] Ruggero Micheletto Department of Electronic Science, Graduate School of Engineering Kyoto University, Nishikyo-ku, Katsura, 615-8510 Kyoto, Japan e-mail: [email protected] Egbert Oesterschulze Universität Kaiserslautern, Fachbereich Physik Physik und Technologie der Nanostrukturen Erwin-Schrödinger Straße, 67653 Kaiserslautern, Germany e-mail: [email protected] Ute Rabe Fraunhofer Institute for Nondestructive Testing, IZFP, Bldg. 37 D-66123 Saarbrücken, Germany e-mail: [email protected] Vittoria Raffa Scuola Superiore Sant’Anna, Polo Sant’Anna Valdera – CRIM Lab Viale Rinaldo Piaggio, 34, 56025 Pontedera (PI), Italy e-mail: [email protected] Tilman E. Schäffer Center for Nanotechnology (CeNTech) and Institute of Physics University of Münster, Gievenbecker Weg 11, 48149 Münster, Germany e-mail: [email protected] Uta Schlickum Max-Planck-Institut für Mikrostrukturphysik Weinberg 2, 06120 Halle, Germany e-mail: uta.schlickum@epfl.ch Martin Stark Ecole Polytechnique Fédérale de Lausanne Institut des Sciences et Ingénierie Chimiques Laboratory of Ultrafast Laser Spectroscopy 1015 Lausanne, Switzerland e-mail: Martin.Stark@epfl.ch

XXXV

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Robert W. Stark Ludwig-Maximilians-Universität München and Center for NanoScience (CeNS) Dept. Earth and Environmental Sciences, Section Crystallography Theresienstr. 41, 80333 München, Germany e-mail: [email protected] Alexander N. Titkov Ioffe Physico-Technical Institute, 26 Polytecknicheskaya 194021 St Petersburg, Russia e-mail: [email protected] Pascal Vairac Department LPMO, FEMTO-ST Institute, UMR CNRS 6174 32 avenue de l’Observatoire, 25044 Besançon Cedex, France e-mail: [email protected] Arnout van den Bos Systems and Materials for Information storage group MESA + Research Institute P.O. Box 217, 7500 AE Enschede, The Netherlands e-mail: [email protected] Gunther Wittstock Carl von Ossietzky Universität Oldenburg Carl von Ossietzky Str. 9–11, 26129 Oldenburg, Germany e-mail: [email protected] Wulf Wulfhekel Max-Planck-Institut für Mikrostrukturphysik Weinberg 2, 06120 Halle, Germany e-mail: [email protected] Christiane Ziegler Technische Universität Kaiserslautern Erwin-Schrödinger Strasse, 67663 Kaiserslautern, Germany e-mail: [email protected]

List of Contributors – Volume III

Maria Allegrini INFM and Dipartimento di Fisica “Enrico Fermi”, Università di Pisa Largo Bruno Pontecorvo 3, 56127 Pisa, Italy e-mail: [email protected] Friedrich Aumayr Institut für Allgemeine Physik, Technische Universität Wien Wiedner Hauptstraße 8-10/134, A 1040 Wien, Austria e-mail: [email protected] Bharat Bhushan Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM) 650 Ackerman Road, Suite 255, The Ohio State University Columbus, OH 43202-1107, USA e-mail: [email protected] Zachary Burton Shell Global Solutions (US) Inc. 3333 Highway 6 South, Houston, TX 77082-3101, USA e-mail: [email protected] Renato Buzio National Institute for Physics of Matter INFM Via Dodecaneso 33, 16146 Genova, Italy e-mail: buzio@fisica.unige.it Manfred Drack GrAT Center for Appropriate Technology, Technische Universität Wien Wiedner Hauptstraße 8-10/965, A 1040 Wien, Austria e-mail: [email protected] Andreas Ebner Institute for Biophysics, J. Kepler University Altenbergerstr. 69, A-4040 Linz, Austria e-mail: [email protected]

XXXVIII


Friedrich Franek Austrian Center of Competence for Tribology Viktor Kaplan-Straße 2, A 2700 Wiener Neustadt, Austria Institut für Sensor- und Aktuatorsysteme, Technische Universität Wien Floragasse 7/2, A 1040 Wien, Austria e-mail: [email protected] Luca Gavioli INFM and Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore via dei Musei 41, I-25121 Brescia, Italy e-mail: [email protected] Ille C. Gebeshuber Austrian Center of Competence for Tribology Viktor Kaplan-Straße 2, A 2700 Wiener Neustadt, Austria Institut für Allgemeine Physik, Technische Universität Wien Wiedner Hauptstraße 8-10/134, A 1040 Wien, Austria e-mail: [email protected] Enrico Gnecco National Center of Competence in Research in Nanoscale Science University of Basel 4056 Basel, Switzerland e-mail: [email protected] Peter Hinterdorfer Institute for Biophysics, J. Kepler University Altenbergerstr. 69, A-4040 Linz, Austria e-mail: [email protected] James K. Knudsen 3328 York Bay, Woodbury, MN 55125 e-mail: [email protected] Alexander V. Kovalev Tribology Department Metal-Polymer Research Institute of Belarus National Academy of Sciences Kirov st. 32A, Gomel, 246652, Belarus e-mail: [email protected] Massimiliano Labardi INFM, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy e-mail: [email protected] Tobias Lange Institute of Physiology II Robert-Koch Str. 27b, 48149 Muenster, Germany e-mail: [email protected]


XXXIX

Ernst Meyer National Center of Competence in Research in Nanoscale Science University of Basel 4056 Basel, Switzerland e-mail: [email protected] Nikolai K. Myshkin Tribology Department Metal-Polymer Research Institute of Belarus National Academy of Sciences Kirov st. 32A, Gomel, 246652, Belarus e-mail address: [email protected] Dessy Nikova Institute of Physiology II Robert-Koch Str. 27b, 48149 Muenster, Germany e-mail: [email protected] Laurent Nony L2MP, Equipe Nanostructuration, Faculté des Sciences de Saint-Jérôme 13397 Marseille, France e-mail: [email protected] Hans Oberleithner Institute of Physiology II Robert-Koch Str. 27b, 48149 Muenster, Germany e-mail: [email protected] René M. Overney Department of Chemical Engineering, University of Washington Box 351750, Seattle, WA 98195-1750, USA e-mail: [email protected] Mark I. Petrokovets Tribology Department Metal-Polymer Research Institute of Belarus National Academy of Sciences Kirov st. 32A, Gomel, 246652, Belarus e-mail: [email protected] Lev Rapoport Department of Science, Holon Academic Institute of Technology 52 Golomb St., Holon 58102, Israel e-mail: [email protected] Massimo Sancrotti INFM and Dipartimento di Matematica e Fisica Università Cattolica del Sacro Cuore via dei Musei 41, I-25121 Brescia, Italy Laboratorio TASC-INFM Strada Statale 14, km. 163.5, Basovizza, I-34012 Trieste, Italy e-mail: [email protected]

XL


Hermann Schillers Institute of Physiology II Robert-Koch Str. 27b, 48149 Muenster, Germany e-mail: [email protected] Scott Sills Department of Chemical Engineering, University of Washington Box 351750, Seattle, WA 98195-1750, USA e-mail: [email protected] Oleg Tikhomirov Institute of Solid State Physics, Chernogolovka 142432, Russia INFM, Largo Bruno Pontecorvo 3, 56127 Pisa, Italy e-mail: [email protected] Ugo Valbusa National Institute for Physics of Matter INFM and Dipartimento di Fisica dell’Università degli Studi di Genova Via Dodecaneso 33, 16146 Genova, Italy e-mail: valbusa@fisica.unige.it Armen Verdyan Department of Science, Holon Academic Institute of Technology 52 Golomb St., Holon 58102, Israel e-mail: [email protected] Hannspeter Winter Institut für Allgemeine Physik, Technische Universität Wien Wiedner Hauptstraße 8-10/134, A 1040 Wien, Austria e-mail: [email protected]

List of Contributors – Volume IV

Chunli Bai National Center for Nanoscience and Technology Beijing 100080, P.R. China e-mail: [email protected] James D. Batteas National Institute of Standards and Technology Surface and Microanalysis Science Division 100 Bureau Drive, Stop 8372, Gaithersburg, MD 20899 e-mail: [email protected] Bharat Bhushan Nanotribology Laboratory for Information Storage and MEMS/NEMS (NLIM) Ohio State University, Columbus, OH 43210, USA e-mail: [email protected] U. Dürig IBM Research GmbH, Zurich Research Laboratory Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland e-mail: [email protected] Fernando García Instituto de Microelectrónica de Madrid, CSIC C/ Isaac Newton 8, 28760, Tres Cantos, Madrid, Spain e-mail: [email protected] Ricardo García Instituto de Microelectrónica de Madrid, CSIC C/ Isaac Newton 8, 28760, Tres Cantos, Madrid, Spain e-mail: [email protected] Jayne C. Garno Department of Chemistry, Louisiana State University 232 Choppin Hall, Baton Rouge, LA 70803 e-mail: [email protected]

XLII


Christoph Gerber National Competence Center of Research in Nanoscale Science, Basel Klingelbergstrasse 82, CH-4056 Basel, Switzerland e-mail: [email protected] B. Gotsmann IBM Research GmbH, Zurich Research Laboratory Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland e-mail: [email protected] Martin Hegner National Competence Center of Research in Nanoscale Science, Basel Klingelbergstrasse 82, CH-4056 Basel, Switzerland e-mail: [email protected] Albena Ivanisevic Purdue University, Weldon School of Biomedical Engineering 500 Central Drive, West Lafayette, Indiana 47907-2022 e-mail: [email protected] Joseph M. Kinsella Purdue University, Department of Biomedical Engineering 500 Central Drive, West Lafayette, Indiana 47907-2022 e-mail: [email protected] Hans Peter Lang National Competence Center of Research in Nanoscale Science, Basel Klingelbergstrasse 82, CH-4056 Basel, Switzerland IBM Zurich Research Laboratory, Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland e-mail: [email protected] Carmen LaTorre Owens Corning, Insulating Systems Business 2790 Columbus Road, Route 16 (Bldg 20-1), Granville, OH 43023, USA e-mail: [email protected] William P. King Woodruff School of Mechanical Engineering, Georgia Institute of Technology 771 Ferst Drive N.W., Atlanta, GA 30332-0405 e-mail: [email protected] Brent A. Nelson Woodruff School of Mechanical Engineering, Georgia Institute of Technology 771 Ferst Drive N.W., Atlanta, GA 30332-0405 e-mail: [email protected]


Marta Tello Instituto de Microelectrónica de Madrid, CSIC C/ Isaac Newton 8, 28760, Tres Cantos, Madrid, Spain e-mail: [email protected] Chen Wang National Center for Nanoscience and Technology Beijing 100080, P.R. China e-mail: [email protected]

XLIII

1 Higher Harmonics in Dynamic Atomic Force Microscopy Robert W. Stark · Martin Stark

1.1 Introduction Aside from its ability to image and visualize the specimen, atomic force microscopy (AFM) offers the potential to perform a wide variety of quantitative measurements. For example, indenting the surface of a polymer with the tip allows us to characterize its viscoelastic properties, while long-range electrostatic forces can be used to detect the number of charges stored in a semiconductor nanostructure. In AFM, a force (from the sample) acting on the sensing tip of the atomic force microscope will influence the motion of the cantilever that the tip is attached to. The motion of this cantilever is observed and quantified by a detection system, and eventually parameterized into the quantities of interest. Fundamentally, all AFM is dynamic: the motion of the tip varies with time under the influence of the forces acting upon it. However, parameterization of the measurements is required to establish a link between the measurements and the physical quantities of interest. It is precisely this task – the establishment of a link between the detected motion of the lever and the physical property of the sample – that gives rise to the wide variety of AFM modes currently employed. Amplitude modulation (also known as tapping mode or intermittent contact mode) has become a standard technique used to perform surface imaging by dynamic AFM under ambient conditions [1]. In this imaging mode, the cantilever is excited at its fundamental resonance to an oscillation amplitude of typically 10 nm to 100 nm. The interaction between the tip and the sample limits the oscillatory amplitude. In order to track the surface profile, feedback circuitry controls the relative position of the tip to the sample during scanning, keeping the oscillation amplitude constant. Phase imaging [2] (the acquisition of the phase lag between the driving force and the cantilever response) has proven to be a powerful method of characterizing mechanical and chemical surface properties. Specific chemical information is obtained by employing functionalized tips [3]. Apart from the topography, the average deflection is recorded together with the amplitude and the phase of the cantilever motion during conventional phase imaging. This additional data corresponds to the first two Fourier coefficients of the AFM signal: the zeroth harmonic (DC component) and the amplitude and phase of the first harmonic. However, since the amplitude signal is used for feedback in order to track the surface topography, this signal is equivalent to the control error and does not contain novel information. The phase signal carries information on the local energy dissipation [4], which

2

R.W. Stark · M. Stark

is determined by chemical [3] and mechanical surface properties [2, 5]. A precise interpretation of the phase signal remains difficult, however, since it includes contributions from the local topography [6]. The slowly varying part of the signal spectrum (DC) is also referred to as the “tapping mode deflection”. This DC-signal encodes the time-averaged interaction force [7–10]. More information is available from analyzing further signal components or the entire spectral response. Although the free cantilever can be approximated by a onedimensional harmonic oscillator for many applications [11], the broadband signal formation can only be understood by taking higher eigenmodes into account (see Fig. 1.1). When approaching the surface, the oscillation amplitude is confined by the sample surface on one side, creating an analogy to the “impact oscillators” studied in structural mechanics [12–15]. The confinement induced by the sample surface acts as a barrier, and it this is defined by the attractive and repulsive surface forces (by the chemical and mechanical properties of the sample) [16–18]. Despite the nonlinear interaction, tip oscillation remains periodic under stationary measurement conditions [19]. As a consequence, any influences from nonlinearities will appear as higher harmonics of the fundamental frequency [4, 20–27]. Deviations from the purely sinusoidal behavior of the oscillation can also arise from dissipative effects. The tip approaches and retracts from the surface during a single oscillation cycle. Energy dissipation causes hysteresis of the tip motion and introduces higher Fourier components [22, 26, 28]. The signal amplitudes of the higher harmonics of a periodic signal generated by a nonlinear system can therefore be used to infer the characteristics of the nonlinearity by balancing the harmonics of the system input and the output [29–32]. Furthermore, it is important to note that the microcantilever is an extended structure with a complex dynamical behavior [33–36]. The cantilever can undergo a variety of oscillatory motions that are related to the flexural, torsional and bending modes of the structure [37–42]. Multimodal AFM techniques take advantage of the resonance structure of the cantilever [43–45]. The

Fig. 1.1. Scheme of higher harmonics generation in atomic force microscopy. There are two aspects that are important for the spectral characteristics of the AFM signal: the nonlinearity of the tip–sample interaction, and the multiple resonances of the cantilever. Due to the nonlinear tip–sample interaction, the force signal is distorted. Higher harmonics of the driving frequency prevail in the spectrum, which is modified by the transfer characteristics of the cantilever. The spectral characteristics are therefore closely related to the tip–sample forces, such as the mechanical, electrostatic and van der Waals forces

1 Higher Harmonics in Dynamic Atomic Force Microscopy

3

resonant frequencies of the higher eigenmodes of commercial AFM cantilevers are usually not integer multiples of the fundamental resonance frequency [34, 41], although harmonic cantilevers modified to purpose have recently been introduced [46]. Two key elements can be identified that contribute to a deviation of the cantilever dynamics from a simple harmonic oscillator: firstly, the non-linearity introduces harmonics (integer multiples of the fundamental oscillation frequency) into the signal spectrum; secondly, higher eigenmodes of the cantilever can be excited (1.1). One important source of higher harmonic generation – electrostatic force – was exploited very early in the development of Kelvin probe force microscopy [47–50]. The attractive force due to a contact potential difference ∆φ between a tip and a sample separated by a distance D is given by Fel (D) =

1 dC ∆φ2 , 2 dD

(1.1)

where C is the capacitance of the tip–sample system. An AC voltage UAC with a frequency ω and a DC (ω = w) offset UDC is applied between tip and sample for potential imaging in Kelvin probe force microscopy. The electrostatic force is then Fel (D) =

1 dC [∆φ − UDC − UAC sin(ωt)]2 . 2 dD

(1.2)

This leads to spectral force components at the harmonic frequencies DC, ω and 2ω: 1 dC 1 2 2 (1.3) FDC (D) = (∆φ − UDC ) + UAC , 2 dD 2 dC Fω (D) = − (1.4) [(∆φ − UDC ) UAC sin(ωt)] dD and F2ω (D) =

1 dC 2 U cos(2ωt) . 4 dD AC

(1.5)

This 2ω component can be used to compensate for the DC potential offset caused by UAC in (1.3) [51]. In an electrolyte, the electrostatic force between tip and sample separated by a distance D follows an exponential law: Fel (D) = Fmax e−D/λ ,

(1.6)

with a characteristic decay length (Debeye length) λ that depends upon the electrolyte concentration. This nonlinear electrostatic force is thought to contribute to the generation of higher harmonics [52]. The following chapter emphasizes the nature of the micromechanical cantilever as a distributed parameter system. The assumption herein is that the cantilever is a geometrically extended structure with an infinite number of eigenfrequencies with corresponding modal shapes (eigenvectors). For theoretical investigations of dynamic AFM, often only the fundamental mode is taken into account, neglecting higher modes. This single degree of freedom (SDOF) approximation permits the


4

prediction of basic features of nonlinear dynamics, such as the existence of different oscillatory states [53, 54], the phase lag between driving force and system response [4], or the pressure at the contact area [55]. In an SDOF model with an external force F(t) and state variables of tip displacement x and velocity x˙ x(t) ¨ +

ω0 x(t) ˙ + ω20 x(t) = ω20 F(t) k , Q

(1.7)

the cantilever is fully characterized by its angular resonant frequency ω0 , the spring constant k, and the quality factor Q. The effective mass m is determined by m = k/ω20 . The quality factor is related to the damping γ by Q = 1/(2γ) and can be understood as the amplification factor of an excitation Ad at a resonance ω = ω0 . In this case the oscillation amplitude is given by A0 = Q Ad . In the state space form (1.7) can be rewritten as 2 0 1 0 ω0 x˙ = F(t) , (1.8) x+ 1 k −ω20 −ω0 Q A

b

introducing the state vector x = [x, x] ˙ T , the system matrix A and the input vector b. Generally speaking, the SDOF approximation holds as long as there are only small contributions to the system dynamics at frequencies higher than the fundamental resonance frequency [56]. However, the approximation completely fails to predict the system characteristics at higher frequencies such as those of higher harmonics or of a chaotic response. Obviously, the SDOF approximation does not account for higher resonances of the cantilever. Additionally, there is a second feature of the distributed parameter system that is not present in the SDOF model: a distributed system can exhibit transmission minima (the cantilever can act as a notch filter that strongly suppresses any dynamics at specific frequencies). This effect has been described as a “dynamically frozen cantilever” [57]. In the following, a multiple degree of freedom (MDOF) state space model of an AFM cantilever interacting with the specimen surface is developed. This approach allows us to use an identical model to investigate the characteristics of both freely vibrating and surface-coupled cantilevers. Additionally, this model provides a direct theoretical reason for the generation of higher harmonics from the nonlinearity of the contact between the tip and the sample in tapping mode AFM. The theoretical results are corroborated by experimental data. Analysis of the higher harmonics of the excitations enables us to differentiate between different materials.

1.2 Multimodal Model of the Microcantilever 1.2.1 Overview There are several strategies that could be used to account for the higher order modes of the cantilever. A straightforward approach can be obtained through finite element [41,


5

58–61] or finite difference analysis [62] of the oscillating cantilever. In this case, the full geometry of the cantilever must be modeled using numerical simulations. Unfortunately, such simulations are highly intensive. However, the computational effort required can be reduced by taking into account only the vibration modes that significantly contribute to the dynamics. This leads to a truncated model consisting of only a limited number of eigenmodes. This simple mode truncation, however, has an important drawback: the exact frequencies of the transmission minima also depend upon the contributions from higher eigenmodes. Thus, a truncation of the model causes the frequencies of the transmission minima to be shifted to higher frequencies [63,64]. However, since the exact frequency of the transmission minima is of minor importance to the understanding of the generation of higher harmonics, we will neglect this effect in the following. The interaction of a silicon tip with a glass surface was modeled for the analytical and numerical calculations. A rectangular silicon cantilever with an idealized optical detection scheme was assumed. A model with N = 3 modes was used in the numerical simulation of nonlinear dynamics, and a model with N = 5 modes was used for linearized analysis. The parameters used are indicated in Table 1.1. Table 1.1. Parameters used to simulate the generation of higher harmonics in atomic force microscopy. There are two aspects that are important for the spectral characteristics of the AFM signal: the nonlinearity of the tip–sample interaction, and the multiple resonances of the force sensor. The interplay between the various tip–sample forces and the spectral characteristics is currently being researched Resonant Spring Modal Young’s Poisson Young’s Poisson Tip Hamaker frequency constant quality Modu- ratio Modulus ratio for radius constant factor lus tip for tip sample sample

Interatomic distance parameter

ω0

a0

normalized

k

Qn

10 N/m 200

Et

νt

129 GPa 0.28

Es 70 GPa

νs 0.17

R

H

20 nm 6.4

×10−20

J 0.166 nm

1.2.2 Modal Analysis As a first step, a modal analysis provides the geometrical shape of the eigenmodes and the corresponding resonant frequencies of the cantilever. These are the most important input parameters for the numerical simulation of the temporal evolution of the system. The advantage of such a modal analysis is that the approach is feasible for arbitrary structures, and it is common engineering practice [65]. For more complicated cantilever geometries, the eigenvalues and eigenvectors can be obtained from a finite element analysis [65, 66]. As a generic and instructive example, higher harmonic generation resulting from the tip–sample interaction will be discussed for the case of a rectangular


6

AFM cantilever. Together with further simplifications, such as modal damping and external forcing at the tip, an explicit analytical solution can be obtained for the partial differential equation. These solutions will be used as parameters in the subsequent numerical analysis. The equation of motion for the flexural vibrations of a freely vibrating and undamped cantilever beam can be approximated by the Euler–Bernoulli equation [67] EI

∂ 2 z(ξ, t) ∂ 4 z(ξ, t) + m =0, ∂ξ 4 ∂t 2

(1.9)

with vertical displacement z(ξ, t), normalized position ξ ∈ [0, 1] along the cantilever, time t, flexural stiffness E I, and constant mass per unit length m. The resonant frequency of the nth eigenmode ωn is related to the respective eigenvalues kn by ω2n = (kn )4 E I/m .

(1.10)

The eigenvectors (modal shapes) ϕ(ξ) are determined by the boundary conditions. The cantilever is clamped at ξ = 0. The boundary conditions are ϕ(w) = 0 for the displacement and ϕ (w) = 0 for the deflection slope. At ξ = 1 the cantilever is free, there are no external torques or shear forces; ϕ (x) = 0 and ϕ (1) = 0, respectively. The eigenvalues can be obtained from the characteristic equation cos kn cosh kn = −1 .

(1.11)

Here, the asymptotic solution kn(a) = (n − 12 )π of (1.11) is a useful approximation for higher modes (n > 3). If the tip is in contact with the surface but can slide laterally (clamped–pinned beam: ϕ(x) = 0 and ϕ (x) = 0), one correspondingly obtains cos kn sinh kn + sin kn cosh kn = 0 .

(1.12)

The eigenvectors of the free cantilever are given by: ϕn (ξ) = cos kn ξ − cosh kn ξ −

cos kn + cosh kn (sin kn ξ − sinh kn ξ) . sin kn + sinh kn

(1.13)

The first four flexural eigenmodes are illustrated in Fig. 1.2. By assuming nonclassical boundary conditions through the addition of translational and rotational springs at the clamped end instead of the rigid constraints ϕ(w) = ϕ (w) = 0, the accuracy of the theoretical calculations can be increased still further [68]. To simplify the problem, we assume that the massless tip is located at the free end at ξ = 1. Additionally, we assume that the laser light used to detect cantilever movement is focused on the free end, and that all forces, such as magnetic actuation and tip–sample forces, act on the free end. With these assumptions, the driving and the tip–sample forces are collocated with the deflection sensing system (ξdrive = ξsens = ξtip ). This avoids difficulties such as nonminimum phase response or time delays that can occur in noncollocated systems [69]. A more realistic analysis of optical detection is discussed in Sect. 1.2.8. From the modal bending shape we obtain the modal displacement ϕn (ξ) and the modal deflection angle ϕn (ξ), which is an idealized light lever readout with an


7

Fig. 1.2. Illustration of the first four flexural eigenmodes of a freely vibrating rectangular cantilever beam

infinitely small laser spot. These modal parameters can be understood as coupling coefficients. The modal displacement describes the coupling efficiency of a tip–sample force to the corresponding mode, and the modal deflection describes the coupling of the respective mode to the sensing system. As we will see in Sect. 1.2.4, there are subtle differences between the transfer characteristics of a deflection angle readout (light lever) as compared to a position readout (tip displacement and interferometer). 1.2.3 Tip–Sample Interaction DMT Model The interaction between tip and sample is determined by the attractive and repulsive surface potentials. The distance between tip and sample is D = z s + z. The scalar z is the tip deflection and the scalar z s is the distance between the undeflected cantilever and the sample. Van der Waals forces dominate the interaction in the attractive regime (D ≥ a0 ). In the repulsive regime (D < a0 ), the tip–sample forces are calculated from the Derjaguin–Muller–Toporov (DMT) model [70]. The DMT model describes a situation where the energy dissipation due to tip–sample contact is negligible. To avoid an unphysical divergence, the parameter a0 is introduced, corresponding to the interatomic distance [54]. The tip sample forces are given by ⎧ 2 D ≥ a0 ⎨ −HR/ 6 (z s + z) Fts (z) = , (1.14) √ ⎩ −HR/6a2 + 4 E ∗ R (a0 − z s − z)3/2 D < a0 0 3 where H is the Hamaker constant and R the radius of the tip. The effective contact −1

stiffness is calculated from E ∗ = 1 − νt2 /E t + 1 − νs2 /E s , where E t and E s are the respective elastic moduli and νt and νs the Poisson ratios of the tip and the sample, respectively. It should be mentioned that the choice of bulk material parameters for the nanoscale simulations can only be a crude estimation. It has been shown that hard-


8

ness of a material at the nanometer scale can differ significantly from its microhardness [71]. Additionally, one should bear in mind that silicon tips are typically covered by a silicon oxide layer [72]. Nevertheless, numerical simulations based on these bulk parameters provide useful predictions of the system dynamics. For very small oscillations around an equilibrium position z 0 , (1.14) can be linearized [34], leading to D ≥ a0 −HR/ 3 (z s + z 0 )3 ∂ ∗ √ = . (1.15) kts = − Fts (z) ∗ 1/2 ∂z 2E R (a0 − z s − z 0 ) D < a0 z=z 0 In the following sections, the contact stiffness kts∗ is normalized to the cantilever spring constant k by kˆ ts = kts∗ /k. Electrostatic Forces Aside from the attractive van der Waals forces, the electrostatic interaction between the tip and the sample is another source of higher harmonics generation. From (1.1), it is clear that a nonlinear electrostatic interaction is also present during imaging due to a contact potential difference, since dC(D)/ dD is not linear. For smooth surfaces the electrical capacitance between a sphere with radius R and a plane is given for D/R 1 by R θ , (1.16) C(D) ≈ 2πε0 R ln + 1.843 + D 63 with the parameter θ ∈ [0, 1], and ε0 being the electrical permittivity of the medium [73]. Thus, within the limit of small oscillation amplitudes, the electrostatic force (1.1) acting at the tip is reciprocally proportional to D [74] with Fel (D) = −

πε0 R ∆φ2 . D

(1.17)

A slight roughness of tip or sample can prevent divergence in (1.17) [73]. For large distances D/R 1 [74], the electrostatic force is Fel (D) = −

πε0 R2 ∆φ2 . D2

(1.18)

The attractive van der Waals force for the plane sphere configuration exhibits a similar distance relationship FvdW (D) = −

HR . 6D2

(1.19)

In the limit D/R 1 the attractive electrostatic and van der Waals forces exhibit the same force vs. distance relationship [75]. Thus, control of the potential ∆φ between tip and sample is essential in order to distinguish between electrostatic and van der Waals forces in a mechanical AFM experiment.


9

1.2.4 State Space Formulation In the following, a multiple degree of freedom (MDOF) model of the cantilever will be developed, which employs the state space formalism. The cantilever is conceived as a linear time-invariant system that is subject to a nonlinear output feedback caused by tip–sample interaction, as illustrated in Fig. 1.3. This feedback perspective of dynamic AFM [30,76] allows for a numerically efficient investigation of the dynamics of the system [9, 77]. The input (1) accounts for external forcing at the free end of the cantilever as well as for the tip–sample forces. External forces at input (1) can be experimentally realized by a magnetic actuation of the cantilever, for example. Further system inputs such as the displacement of the sample (input(2)) or the cantilever base (input (3)) will not be discussed. Output (1) is the deflection of the free end of the cantilever (the physical displacement of the tip) and output (2) corresponds to the light lever readout of the system (the signal that is usually measured). The state space form of the equations of motion of the cantilever with N degrees of freedom (eigenmodes) is x˙ = Ax + bu , y = Cx

(1.20)

analogous to the SDOF formulation introduced in (1.8). The time-dependent state vector x = (x1 , x2 , . . . ) = (xn=1 , ∂t xn=1 , . . . ) contains the generalized displacements and velocities of the individual eigenmodes. The 2N × 2N matrix A is the system matrix, b is the input vector, and the scalar u is the force input. In the case of

Fig. 1.3. (a) Schematics of dynamic atomic force microscopy. Commonly used inputs and outputs of the dynamical system are indicated. The system can be accessed by applying forces u(t) directly onto the free end of the cantilever (input 1), such as the tip–sample interaction or external actuation. Alternative inputs include base excitation of the sample (input 2) or excitation of the cantilever support (input 3). The tip deflection z tip (t) (output 1) determines the tip–sample interaction force. However, often this output cannot be observed directly. Most commonly, the angle of the cantilever deflection is read out using an optical lever detection scheme determining the difference signal S− (t) (output 2). (b) Graphical representation of the dynamic system. The cantilever is represented by a linear and time-invariant (LTI) submodel. The tip–sample interaction force is a nonlinear output feedback that depends on the tip position


10

multiple inputs, the vector b and the scalar u transform into a matrix and a vector, respectively. The output vector y consists of the tip displacement output y1 and the photodiode signal output y2 , which are linear combinations of the system states xi . The corresponding weights are given by the output matrix C. Note that, in comparison to the structure of (1.8), it is now necessary to introduce an output matrix in order to distinguish between different outputs. There is no feed-through between the input and the outputs. The modal quality factors are usually larger than 100 under ambient conditions. This means that there is only a very weak intrinsic damping and a weak damping due to the surrounding gas. Thus, we can assume a modal damping which leads to a decoupling of eigenmodes that implies normal modes: the modal contributions oscillate in phase. The system matrix A has a very simple structure and consists of the 2 × 2 submatrices of the eigenmodes along the main diagonal. In other words, the cantilever bending line is stationary for normal modes. This is in contrast to nonmodal damping, which leads to complex mode shapes where the phase angle varies along the structure [66]. Such “traveling” or complex modes can occur in liquids due to viscous damping in the fluid (see, for example, Fig. 8 in [78]). The submatrices are constructed from the eigenfrequency ω n = ωn /ω1 and the modal quality factor Q n , which is related to the modal damping γn = 1/(2Q n ). The eigenfrequencies are normalized to the fundamental resonant frequency of the cantilever. The dynamics of the nth mode can be described by the modal state vector xn , the system matrix An , and the input vector bn , using x xn = n , x˙n An =

0 1 ωn /Q n − ωn2 −

0 bn = ϕn (1)/Mn

,

(1.21)

.

The modal state vector simply consists of the modal displacement and velocity. The modal system matrix corresponds to that of a single degree of freedom harmonic oscillator. The first component of the input vector bn is zero, while the second component describes the coupling of an input force to the eigenmode. It is given by the respective modal displacement ϕn at the tip position ξtip = 1, weighted with the 1 respective generalized modal masses Mn = 0 mϕn (ξ)2 dξ. Without restriction the mass density is normalized to Mi = m = 1. The dynamics of the N degrees-of-freedom system is thus given by ⎤ ⎡ A1 x˙1 ⎢ x˙2 ⎥ ⎢ 0 ⎢ . ⎥=⎢ . ⎣ . ⎦ ⎣ . . . ⎡

x˙ N

⎤⎡ ⎤ ⎡ ⎤ x1 b1 ··· 0 · · · 0 ⎥ ⎢ x2 ⎥ ⎢ b2 ⎥ ⎥⎢ . ⎥ + ⎢ . ⎥u . .. ⎦⎣ . ⎦ ⎣ . ⎦ . . . 0 0 AN xN bN 0 A2 .. .

(1.22)


11

The output matrix C describes the combination of the system states yielding the output vector with the tip displacement y1 and the deflection readout y2 . For the individual eigenmodes, the contribution to the total output is given by ϕn (ξtip ) 0 . (1.23) Cn = ϕn (ξsens ) 0 Here, ξtip = 1 and ξsens = 1 are the positions of the tip and the detection laser along the cantilever, respectively. The modal bending shapes are given by (1.13). This leads to the system output

y1 y2

⎡

⎤ x1 ⎢ x2 ⎥ ⎥ = C1 C2 · · · C N ⎢ ⎣ .. ⎦ . .

(1.24)

xN It is advantageous to normalize the output channels to the respective frequency response at ω = 0 (DC) by yˆ1 y /yDC = 1 1DC . (1.25) yˆ2 y2 /y2 The normalized deflection output yˆ1 leads to a normalized quasi-static spring constant for the cantilever, kˆ cant = 1. Normalization of the light lever sensor, yˆ2 , reflects the usual quasi-static calibration of the photodiode response from a force vs. displacement curve on a hard sample. The nonlinear interaction between tip and sample can be modeled as an output feedback. The tip displacement is fed back to input (1) through the interaction force Fts ( yˆ1 ). Modeling of the so-called active Q-control is straightforward in this model. For the standard time-delay type of Qcontrol, output yˆ2 is fed back to input (1) with a time delay ∆t and amplified by a gain factor K . Finally, it is worth mentioning that the modes n > N are neglected although they also contribute to the low-frequency response [79]. Additionally, process and measurement noise such as thermomechanical noise, electronic 1/ f noise or shot noise will also be present in the system. 1.2.5 Dynamics: Linearized Tip–Sample Interaction In order to understand the frequency shifts due to attractive or repulsive forces in the light of a MDOF model, it is useful to discuss the case of very small oscillations first. In this situation, we can linearize the tip–sample interaction (1.15). The output feedback is directly proportional to the system’s position output (1). Thus, the elastic surface properties can be conceived as a proportional feedback with a gain parameter kˆ ts . Correspondingly, a dissipative interaction due to viscous sample properties could be represented by a differential controller. For brevity, dissipative tip sample interaction or other time-dependent interaction forces will not be discussed.

12


The influence of the tip–sample interaction on the amplitude and the phase response of the system described by (1.22) to (1.24) is illustrated in the Bode plots in Fig. 1.4a. The linearized van der Waals interaction corresponds to a spring with a negative force constant. This leads to a softening of the dynamic system and a corresponding redshift of the resonant frequencies (dash-dotted) in comparison to the freely oscillating cantilever (solid). In contrast, a repulsive tip–sample interaction corresponds to a spring with a positive spring constant. This interaction hardens the system, which results in a blueshift of the resonant frequencies (dashed) [11]. It is important to note that the resonant frequencies depend on the gain factor of the output feedback. However, they are not affected by the choice of the output channel. In contrast, compared to the light lever output (2), the frequencies of the transmission minima are different for the position output (1). The relative frequency shift of the higher eigenmodes is of course smaller than that of the fundamental mode because the frequency shift depends upon the stiffness ratio kˆ ts /m ωˆ 2n , which rapidly decreases with an increasing mode number n. For a very stiff tip–sample contact with kˆ ts = 105 , the system approximately corresponds to a cantilever with its end pinned to the sample surface (Fig. 1.4). The resonant frequencies of the stiff system – seen in the light lever output (2) – correspond to the frequencies of the transmission minima of the displacement output of the free system. This is not surprising, since the poles (resonances) of a constrained system generally correspond to the zeros (antiresonances) of the free system [80]. It is remarkable that the displacement output (1) exhibits a flat amplitude response, whereas in output (2) the resonances remain detectable. In the constrained system,

Fig. 1.4. Bode plot (frequency response) of the cantilever. (a) Position output and (b) idealized light lever readout. The frequency response of the free cantilever is indicated by the solid line. Moderately attractive and repulsive interaction forces shift the resonances to lower and higher frequencies respectively (dashed, dash-dotted). Note that the respective resonance frequencies are the same for both outputs, whereas the transmission minima depend upon the output channel. A strong surface coupling (kts = 105 ) leads to a flat transfer function with a small gain in the position output (a). The system resonances are still visible in the photodiode signal


13

poles and zeros cancel for output (1), but not for output (2). A physical interpretation is straightforward: the cantilever was modeled as a beam that is actuated by a force at the free end. However, there is now an additional stiff spring attached to the free end which directly counteracts an external driving force. These examples clearly illustrate that a very precise definition of the system inputs and outputs is essential. With the actuation directly at the tip position, there are no measurable dynamics of the tip displacement, whereas a rich structure of resonances and transmission minima can be observed in the light lever output in the case of a surface-coupled cantilever. Physically, this means that there are also different system transfer characteristics, depending on the measurement system. The employment of a light lever readout will provide dynamic characteristics that are different to those from interferometric detection. In order to tailor the system dynamics to specific experimental or control engineering needs, it is also possible to combine a light lever detection with an interferometric detection. For feedback purposes, a linear combination of the signals obtained from a light lever and an interferometer can be advantageous. In robotics, such linear combinations of position and deflection angle detection are used in order to avoid undesired system dynamics such as an unwanted nonminimum phase response of single flexible effectors [69,81]. 1.2.6 Poles and Zeros ∞ The Laplace transformation G(s) = 0 e−st g (t) dt is a convenient tool for the analysis of dynamic systems. The main advantage of the Laplace transformation is the simple calculation of integrals and derivatives with respect to time, which become multiplications and divisions. Thus, differential equations transform into polynomial equations, which are easier to solve. The system output is calculated by convolving the system’s impulse response with the input signal. System characteristics such as resonances, transmission zeros, damping and system stability are conveniently illustrated in the Laplace domain in the so-called pole zero and root locus maps. In the following we will discuss the influence of tip–sample interactions on the system dynamics in the Laplace plane. For the position output (1) of the dynamic system given by (1.22) and (1.24), the state space model can be rewritten as a rational transfer function G(s) =

n

N ϕn ξtip ϕn ξtip = . s2 + ωn s/Q n + ω2n n=1

(1.26)

Thus, the transfer function has the form G(s) =

Z(s) , P(s)

(1.27)

where the denominator P(s) and the numerator Z(s) are polynomials of the complex frequency variable s = iω. This means that the numerator and denominator can be factorized. The roots of P(s) and Z(s) are referred to as poles and zeros, respectively.

14


The transfer function can be understood as a complex landscape in the Laplace plane, introducing a real part s = σ +iω. Upon plotting the magnitude of the Laplace transform function |G(s)|, the landscape consists of sharp peaks (poles) and valleys (zeros), as illustrated in Fig. 1.5a. The complex Laplace transform function diverges at the poles and equals zero at the zeros. On a logarithmic scale, the zeros appear as sharp peaks in the negative direction. The amplitude response of the system can be determined from a cross-section through this landscape along the imaginary axis Fig. 1.5b. A projection of the landscape onto the Laplace plane is shown in Fig. 1.5c. Note that, for the cantilever model, the poles and zeros are located in the left half plane, very close to the imaginary axis. Usually, poles and zeros are indicated in the so-called pole zero map as illustrated in Fig. 1.5d. Here, poles are usually marked by crosses, zeros are indicated by open circles. The circles of equal frequency and the lines of equal damping are also indicated. Poles in the left half of the plane have

Fig. 1.5. Illustration of the transfer function G(s) for the dynamics of a microcantilever in the Laplace plane. (a) The magnitude log10 (|G(s)|) as a surface plot on a logarithmic scale. The transfer function diverges at the poles. (b) The frequency response is obtained by a cross-section through the landscape along the imaginary axis. (c) Contour plot of the landscape (a). (d) The pole zero map is obtained from the contour plot by omitting the contour lines. Poles are visualized with crosses, zeros with circles. The dashed lines specify the lines of equal damping as indicated, while the dashed semi circles indicate the locations of equal frequency


15

a positive damping and are thus stable. In contrast, poles in the right half of the plane exhibit negative damping and are instable. Similarly, the location of the zeros in the Laplace plane is important to the system’s dynamics. Zeros in the left half of the plane can translate into minima in the amplitude response, whereas zeros in the right half of the plane indicate a nonminimum phase response. An additional feedback can manipulate the location of the poles in the complex plane. It also can add or suppress poles. This means that by adding a feedback circuitry such as Q-control, the damping of the system can be reduced or increased and the resonant frequency can be changed. The system zeros are not affected by the feedback, as they depend on the actuators and the sensors of the system. For atomic force microscopy, this means that the system dynamics critically depend on the actuation design of the cantilever and on the deflection sensor system. When a feedback with time delay is added, as for example for the time-delay Q-control [82, 83], the transfer function G(s) ceases to be a rational function, and the mathematical analysis becomes more complicated. A root locus map in the Laplace plane enables a very compact representation of the migration of the system’s poles caused by an additional feedback circuit imposed by the tip–sample forces. The map in Fig. 1.6 visualizes the variations in eigenfrequency and damping in response to a varying effective tip–sample stiffness kˆˆ ts . Here, poles (crosses) and zeros (circles) are shown that belong to the subsystem for the signal flow from input (1) to tip position output (1). For a free cantilever, all of the poles are located in the left half of the plane, and the dynamic system is stable. For the truncated model of a free cantilever, the number of zeros n zero is determined by the number of poles n poles = 2N with n zero = n poles − 2. For the collocated actuation and sensing system, the poles and zeros alternate in the frequency response. Such a collocated configuration of tip and light lever readout is generally desirable in AFM. For an in-depth discussion of the roles of poles and zeros and the collocation of the sensor and the actuator in mechanical systems, see [69, 80, 81]. The zeros are independent from kˆ ts and remain stationary. In the single degree of freedom approximation (N = 1), the dynamic system in (1.8) does not have any zeros. In this case, the resonant frequency diverges to infinity with increasing tip–sample stiffness. This is the reason why the single degree of freedom approximation performs only poorly describes contact mode AFM and completely fails to describe atomic force acoustic microscopy. The migration path of the poles towards the zeros is indicated in Fig. 1.6a for a tip–sample stiffness kˆ ts > 0. For an infinite contact stiffness, the poles and zeros cancel, except for the conjugate complex highest order poles, which diverge. The arrows indicate the direction of migration for an increasing kˆ ts . As can be seen in Fig. 1.6, the poles migrate along bent lines and not along the lines of equal damping. This means that the modal damping can vary substantially depending upon the tip–sample contact. In Fig. 1.6b, the migration of the poles is illustrated for kˆ ts < 0. The arrow indicates the direction of migration for increasing kˆ ts . The poles now migrate towards zeros with lower frequencies. For kˆ ts < −mω21 = −1, one of the dominant poles is real and positive and the system becomes unstable. Experimentally, this instability is well known as “snap-in” [84]. When a soft cantilever closely approaches the surface,

16


Fig. 1.6. Root locus map of the linearized closed loop system from input (1) to output (1). The real axis was stretched significantly in order to emphasize the pole migration. The locations of equal damping are indicated by the radial lines, the circles of equal frequency are strongly stretched and appear as horizontal lines. Poles and zeros are indicated by crosses and circles respectively. (a) The arrows give the direction of the pole migration for increasing kˆ ts > 0. For a repulsive tip–sample interaction, the poles migrate towards the next zero with increasing sample stiffness. The highest order poles of the system diverge to infinity. The zeros remain stationary, as they are independent from the feedback. (b) Pole migration for increasing attractive interaction (decreasing kˆ ts < 0). Here, the dominant pole pair (fundamental resonance) migrates towards the real axis. For kˆ ts < −1 one pole is in the right half of the plane and the system is unstable

the cantilever suddenly bends at a certain distance from the surface due to attractive van der Waals forces. Within the framework of this model, the “snap-in” can be understood as an instability of the closed loop system. 1.2.7 Dynamics: Nonlinear Interaction Nonlinearities in the tip–sample contact lead to the generation of higher harmonics. In this section, the theoretical background is illustrated by numerical simulation results. For the numerical simulation, a rectangular cantilever interacting with the surface as described by the nonlinear force law (1.14) was assumed [77]. The simulations were implemented in Matlab Release 13 and Simulink (The Mathworks Inc., Natick, MA, USA). The parameters are summarized in Table 1.1. Since the AFM signal


17

strongly depends on the alignment of the detection system (see Sect. 1.2.8), only the position output will be considered in this section. Figure 1.7a shows the simulated evolution of the amplitude and phase of the first harmonic, together with the total harmonic distortion THD and the calculated average force f in relation to the size of the tip–sample gap. The amplitude, phase, and average force exhibit the features predicted by the SDOF approximation as being due to the transition from the low amplitude state to the high amplitude state [11]. The anharmonic contributions in the system output (1) were also calculated. The total −1/2 ∞ −1/2

∞ , where harmonic distortion is defined by THD = n=2 |cn | n=1 |cn | |cn | is the amplitude of the nth harmonic, as obtained by the fast Fourier transform (FFT) algorithm. Thus, the THD is the fraction of power that is contained in the higher harmonics. Far away from the sample, the amplitude of the fundamental oscillation is 2|c1 | = 20 nm and the phase is φ = −90◦ . There is only a very small average attractive force and a very small total harmonic distortion. Approaching z s = 20 nm, the system is still in the net attractive (low amplitude) regime, as can be seen by the net negative interaction force. As the strength of the attractive interaction increases, the THD of the output signal also increases. Between z s = 18.5 nm and z s = 18 nm, the system moves to the high amplitude state (arrows). This transition occurs in the temporal evolution of the phase as well as in the average interaction force. It is also visible in the THD, which increases by 50%. Upon further approach, the dynamics of the system change at z s = 2.8 nm. The THD decreases significantly and recovers

Fig. 1.7. The amplitude and phase of the first harmonic, the total harmonic distortion, and the average tip– sample forces, as obtained from the numeric simulation. The transition from the low amplitude state to the high amplitude state can be identified by the phase jump (arrow). The transition between both states is accompanied by a step in the total harmonic distortion of the position output (THD, arrow). Subharmonics are generated in a small gap z s , which lead to a reduction in the THD (asterisk). The transition from net repulsive to net attractive forces is evident in the average force f (arrow)

18


at z s = 1 nm, before dropping to zero. This behavior is either explained by the generation of subharmonics, where spectral power is transferred into subharmonics, or by nonperiodic dynamics, where the power is distributed over the entire spectrum. The characteristic features discussed for the THD of simulated data compare well to those seen experimentally [23, 24, 85]. The evolution of the harmonic signal provides additional information on the tip sample interaction. The results from the numerical simulation are shown in Fig. 1.8a. The harmonics are grouped according to the eigenmode that contributes most to the respective signal. In Fig. 1.8, the DC value |c0 |, also known as the “tapping mode deflection”, is shown together with the FFT amplitudes |c1 | to |c4 |, which are grouped around the fundamental frequency. The absolute value of the static deflection due to the average force |∆z| = | f avg |/k perfectly agrees with the DC value |c0 | of the photodiode signal. Thus, the DC value, which is easy to measure in a standard AFM by low-pass filtering the photodiode signal, is a measure of the average tip–sample force. The group below shows the amplitudes |c5 | to |c13 | around the second eigenmode. The amplitudes of these harmonics drop with the onset of period doubling and recover again before the amplitudes drop to zero. The harmonics grouped around the third mode exhibit a significant jump in the transition from the low amplitude to the high amplitude solution (arrow). The different characteristics of the harmonics associated with a particular eigenmode lead to the idea of parameterizing the anharmonic signal in terms of a modal harmonic distortion, as discussed in Sect. 1.4.3. Complex

Fig. 1.8. Simulated harmonics obtained in the position output. They are grouped together according to the eigenmode that contributes most to the signal. (a) The DC contribution (dashed) is a measure of the average tip–sample interaction force. b The harmonics 5–13 around the second and (c) 13–23 around the third eigenmode. (d) Subharmonics are generated for a small separation between cantilever and sample


19

dynamics appear for small tip–sample gaps of (z s < 5 nm). The subharmonics ω/2, 3/2ω and 11/2ω are shown in Fig. 1.8a. To further investigate the dynamics, it is instructive to analyze the evolution of the spectrum during an approach. Selected spectra obtained by numerical simulation are displayed in Fig. 1.9. The evolution of the spectrum in the range 0 and 2.5ω0 is illustrated by the spectrogram in Fig. 1.10. Away from the sample surface at z s = 21 nm there is only a weak anharmonic contribution. Closer to the surface, higher harmonics prevail over the entire spectrum (see for example z s = 5.3 nm). At smaller separations, more complicated dynamics can be observed. At z s = 2.4 nm the subharmonics are clearly visible, which indicates period doubling. At z s = 1.38 nm the energy is distributed over the entire spectrum, although the harmonics are pronounced. Finally, at z s = 0.4 nm the tip snaps onto the surface and is not then released. Here again we stress that the results from simulation are consistent with those from experiment for a large variety of phenomena, such as the abrupt increase in the contributions from higher harmonics with the onset of repulsive forces, the generation of subharmonics, as well as the overall evolution of the spectrum when approaching a surface [24, 76]. From the numerical data it is clear that the nonlinear tip sample interaction induces a harmonic distortion of the AFM signal. The higher harmonics are experimentally accessible. Increasing the influence of the nonlinearity further, through either a closer approach to the specimen or increased driving amplitude, causes higher periodicity and even chaos [86–88]. The higher harmonics can be used on-line to quantify the interaction forces, and may even be used as input

Fig. 1.9. Fast Fourier transform (FFT) of the position output at various tip–sample gaps z s . Note the period doubling at z s = 2.4 nm, the irregular spectrum at z s = 1.38 nm, the periodic motion again at z s = 1.3 nm, and the nonperiodic system behavior at z s = 1.1 nm

20


Fig. 1.10. (color) Spectrogram of the simulated approach (for decreasing z s ). The color depicts the magnitude of the respective frequency component. The frequency is normalized to the fundamental resonance. Initially, only the fundamental and the second harmonics prevail. At z s ≈ 5 nm, complicated dynamics set in and the amplitudes of intermediate frequencies increase. After a periodic motion (z s = 2.5 nm), period doubling occurs, followed by nonperiodic motion. Within the parameter range for nonperiodic motion there are windows of periodic behavior (see Fig. 1.9 with z s = 1.3 nm for example)

to the feedback loop [89]. Furthermore, the experimental results show that higher eigenmode excitation is an important feature in dynamic AFM. 1.2.8 Optical Readout After discussing the dynamics of the system, we will now address the question how the detection of higher eigenmodes can be optimized. Today, most commercially available atomic force microscopes are equipped with light lever detection (Fig. 1.11). There are two parameters related to this that are relevant to system performance: the diameter and the position of the laser spot [90, 91]. Thus, we will evaluate the influence of adjustments made to the laser spot on the transfer characteristics of the sensing system. In Sect. 1.2.4 we showed that the output vector y is a linear combination of the weighted system states (1.23) and (1.24). From (1.13), we know that ϕn (x) = 2(−1)n . In the general case, the parameters for idealized light lever readout ϕn (x) are replaced by the modal coupling factors cn , which account for the coupling of the individual eigenmodes to the light lever readout. These considerations lead to the output matrix −2 0 2 0 · · · 2(−1) N 0 . (1.28) C= c1 0 c2 0 · · · c N 0 Here, output (1), represented by the first row, is still the tip deflection output as defined in Sect. 1.2.4. The second row accounts for light lever readout with a laser spot of a finite size and an arbitrary adjustment of the laser spot on the cantilever.


21

Fig. 1.11. Scheme of light lever detection in atomic force microscopy. The inset illustrates illumination by a rectangular laser beam. Reprinted with permission from [96]. Copyright (2004), American Institue of Physics

In the following we will calculate the coefficients ci . To this end, we must consider the reflection of the laser light on the curved cantilever. As we have seen, the geometrical shape of a vibrating cantilever beam can be described by (1.13). A Gaussian beam profile is assumed for the laser beam. The irradiance of the laser beam is (ξ − ξ0 )2 IC (ξ) = I0 exp −8 , (1.29) w2 √ with I0 = 8/π P0 /w [92]. Here, P0 is the total light power, w is the 1/ e2 -width along the cantilever axis, and x0 is the position of the spot center on the cantilever beam, as illustrated in Fig. 1.11. The spot size w is normalized to the cantilever length. The corresponding real scalar field is (2 (ξ − ξ0 ))2 E c (ξ, ξ0 ) = I0 exp − . (1.30) w2 For small deflections Aϕ(x) λ (A is the oscillation amplitude), the modal output coupling factor is ci (ξ0 ) = Si− /A, where the difference signal Si− is obtained by integration via S− (ξ0 ) = β [0,1]

Φ(ξ) − Φ(ξ ) dξ dξ E c (ξ, ξ0 ) E c ξ , ξ0 ξ − ξ

(1.31)

[Equation (10) in [93]]. The parameter β is a prefactor that corrects for losses in the optics and also includes the wavelength of the laser. The cantilever shape Φ(ξ) is a linear combination of the orthogonal modal shapes ϕi (ξ). Thus, the modal output coupling factors can be calculated from ci (ξ0 ) = n Sig [0,1]

ϕi (ξ) − ϕi (ξ ) dξ dξ E c (ξ, ξ0 ) E c ξ , ξ0 . ξ − ξ

(1.32)

22


The prefactor β has been included in the normalization factor n sig . Equation (1.32) implies that the components of the output matrix C depend on both the position of the laser spot ξ0 and its spot size w. The detection optics are not changed. In static applications such as contact mode and force spectroscopy, it has been shown that maximum detection sensitivity can be obtained with a spot diameter of about w = 0.9 [92]. It is not immediately clear how it is possible to achieve a good sensitivity for the detection of higher eigenmodes. To compare the detection performance of different spot sizes, the following experimental laser alignment procedure was assumed. In the first step, the laser is adjusted to the free end of the cantilever to obtain the maximum sum signal. Then, in the second step the laser spot is moved to the free end of the cantilever until the sum signal is reduced to 90% of its maximum value. Obviously, this alignment procedure does not provide optimum sensitivity since 10% of the total laser power is lost. Nevertheless, such a procedure allows the user to adjust the laser spot to the free end without the need for high-resolution viewing optics. Bode plots of the corresponding transfer functions are illustrated in Fig. 1.12. Transfer functions G w (s) were calculated for the spot diameters w = 0.1, w = 0.3, and w = 0.9. For comparison they are normalized to the maximum static sensitivity for a spot diameter of w = 0.9 at the optimum beam alignment for quasi-static AFM measurements (ξ0 = 0.59, sum signal 99.9%). The static gains at ω = 0 were |G 0.1 | = 0.13, |G 0.3 | = 0.38, and |G 0.9 | = 0.89. This is in line with the observation that, for low frequency applications, a 1/ e2 -spot diameter of about the length of the cantilever provides the optimum sensitivity [93–95]. However, the amplitude response of G 0.3 outperformed that of G 0.9 for the eigenmodes n ≥ 3 (arrow), and G 0.1 outperformed G 0.3 for n ≤ 6 (arrow). The reason for this is that the root of ϕ3 (x) = 0 is ξ ≈ 0.7 (ϕ6 (0.9) = 0). The 1/ e2 -width of the laser spot is then of the same size as the part of the modal shape with the same slope (Fig. 1.11). Additionally, the third eigenmode is nearly canceled out in G 0.9 . Physically, this can be explained by noting that the extended laser spot integrates the slope over the illuminated area of the respective eigenmode. In the difference signal, contributions from areas with negative and positive signals cancel each other out, leading to a reduced sensitivity in this frequency range. This example shows that it is possible for higher eigenmodes to be undetectable if the laser spot is not adjusted appropriately. Thus, Fig. 1.12 illustrates the trade-off encountered during the design of the light lever detection system. A system optimized for contact mode and force spectroscopy (w ≈ 0.9) does not transmit the response of the higher eigenmodes appropriately. This can be overcome by reducing the spot diameter. The price of doing so is reduced sensitivity in the low frequency range. In a commercial AFM system, the experimentalist can usually change the position of the laser spot on the cantilever, but he cannot change its size. There are two further points that also have to be considered for the laser alignment in an existing instrument. (i) Signal artifacts can be caused by interference between the laser light reflected from the cantilever and light reflected from the sample surface. These effects are minimized by maximizing the light reflected from the cantilever (by maximizing the sum signal). This also leads to a maximized low frequency sensitivity, as required in contact mode and standard dynamic AFM. (ii) In contrast, in high frequency


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Fig. 1.12. Bode plot of the transfer functions G w (s) for w = 0.9 (short dash), w = 0.3 (dash), and w = 0.1 (solid). The sum signal is 90% for all curves. The insets (not to scale) highlight details of the transfer functions where |G 0.1 | < |G 0.3 | < |G 0.9 |, |G 0.9 | < |G 0.1 | < |G 0.3 | and |G 0.9 | < |G 0.3 | < |G 0.1 |. Note that the third eigenmode is virtually undetectable for w = 0.9 due to an almost pole-zero cancellation (asterisk). Reprinted with permission from [96]. Copyright (2004), American Institue of Physics

experiments the laser spot should be moved to the free end of the cantilever, accepting a loss of sum signal in order to increase the high frequency sensitivity [96]. Thus, in practice the operator must account for the trade-off between high and low frequency sensitivity. Most importantly for quantitative AFM analysis, Fig. 1.12 tells us that the transfer function of an AFM critically depends on the laser beam adjustment and the laser spot size. The gain factors of the individual modes may vary as well as the frequencies of the transmission minima. Thus, in any quantitative AFM experiment involving the detection of higher eigenmodes, the transfer characteristics of the system must be measured in order to avoid large experimental errors [97, 98].

1.3 Higher Harmonic Imaging As we have seen in the preceding sections, it is the nonlinear interaction between tip and sample that gives rise to higher harmonics. By recording the amplitudes of higher harmonics of the driving frequency, images can be obtained, enabling materials to be differentiated [23, 99, 100]. The signal-to-noise ratio can be improved by employing cantilevers with tailor-made resonance characteristics [46, 101]. Under UHV conditions it was possible to achieve higher harmonic imaging with subAngstrom resolution using a light-atom probe (carbon) on a tungsten sample in frequency modulation AFM [27]. In the following, we demonstrate higher harmonic imaging in amplitude modulation AFM, employing a secondary lock-in amplifier that demodulates the AFM signal at a multiple of the driving frequency. A commercial atomic force microscope (TopoMetrix Explorer, Veeco/TM Microscopes, Sunnyvale, CA, USA) was modified to detect the anharmonic signals. In this system, the bandwidth was limited to f c ≈ 50 kHz by the photodiode pream-

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plifier. Thus, signals are damped by a factor of ten at f = 500 kHz due to the low pass characteristics of the electronic circuit. An external function generator (DS 345, Stanford Research Systems) was used to generate the AFM driving signal. The signals were demodulated by a lock-in amplifier (EG&G 5302, EG&G Inc., Princeton, NJ, USA) with a time constant of 100 µs. The lock-in amplifier output (x, y) was transformed into polar coordinates (r, φ) by dedicated converter electronics. The signal amplitude r was used for AFM feedback. A second lock-in amplifier (SR 844, Stanford Research Systems) was used to detect the higher harmonic signals (Fig. 1.13). The reference signal was generated with a second function generator of the same type. To maintain phase stability, both function generators were coupled by a 10 MHz frequency standard. This separate detection of fundamental and harmonic frequencies ensured that only higher harmonics of the AFM signal were detected, avoiding possible harmonics generation caused by signal mixing in the lock-in amplifier. In order to obtain higher order harmonics within the limited bandwidth of the photodiode preamplifier, a cantilever with a low resonant frequency was employed (silicon cantilever NSC 11, lever A, v-shaped, nominal spring constant 3.0 N/m, Mikromasch, Tallinn, Estonia). The resonant frequencies of the transverse modes were f 1 = 52.2 kHz, f 3 = 264.5 kHz, and f 5 = 417.6 kHz. The eigenfrequencies f 2 and f 4 belong to torsional vibration modes of the cantilever that do not contribute the force signal. The actual spring constant c = 3.1 N/m was determined by analysis of the thermomechanical noise of the cantilever in combination with a finite element analysis [41]. To demonstrate that higher harmonic imaging allows a clear differentiation between distinct materials, a shadow evaporation mask of a 4 nm-thick platinum carbon (Pt-C) layer on a fused silica coverslip was imaged in tapping mode AFM. The cantilever was excited at its fundamental resonance frequency of f 1 = 52.2 kHz. The amplitude of the freely vibrating tip was set to A0 = 50 nm and the setpoint was adjusted to an amplitude of 60% of the free amplitude. The topographical image and

Fig. 1.13. Experimental set-up for the detection of anharmonic signals for imaging purposes. A commercial AFM is equipped with a second lock-in amplifier for the detection of anharmonic signals. Reprinted with permission from [99]. Copyright (2003), American Institue of Physics


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the control error (amplitude) image were acquired simultaneously together with one higher harmonic image. In order to compare the images of several harmonics, higher harmonic images were obtained consecutively. Figure 1.14 shows the topography (a), control error (b) and higher harmonics (c–e). In the higher harmonic images in Fig. 1.14c–e, the material contrast between the Pt-C film and the glass substrate is prominent. The images at 3 f 1 and 8 f 1 (c,e) show that the Pt-C layer appears darker than the glass substrate. However, compared to the harmonics in Fig. 1.14c,e, a contrast reversal was observed at 5 f 1 in Fig. 1.14d. From cross-sectional analysis, it can be seen that the signal-to-noise ratio of the higher order harmonic image at 8 f 1 was increased by an order of magnitude compared to the image at acquired at 3 f 1 . The harmonics that are close to a resonance frequency of the cantilever are enhanced, thus improving the ratio between signal and instrument noise. The fifth harmonic is close to a resonance, resulting in an increased signal amplitude compared to the third harmonic. This enhancement can be seen in the cross-sectional analysis in Fig. 1.14c–e. Although simulations based on the simplified model of the tip sample contact in (1.14) explain the signal enhancement of the various modes,

Fig.1.14.(a) Topographic (b) control error and (c-e) higher order harmonic images of a 4 nm-thick Pt-C test structure on a fused silica cover slip. The driving frequency was f = 52.2 kHz, and the detection frequencies were (c) 3 f = 156.6 kHz, (d) 5 f = 261.0 kHz, and (e) 8 f = 417.6 kHz. Reprinted with permission from [99]. Copyright (2003), American Institue of Physics

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they do not reproduce the contrast inversion observed in Fig. 1.14 (data not shown, see [99]). Here, more sophisticated contact mechanical models are needed that account for details of tip–sample interactions such as adhesion, capillary condensation and viscous damping. The practical use of higher harmonic signals for qualitative characterization of the imaging forces is illustrated in Fig. 1.15. A test specimen with a 100 nm-high silicon oxide grating on a silicon substrate was imaged in tapping mode. The amplitude of the freely vibrating tip was set to A0 = 45 nm, and the amplitude setpoint was adjusted to 80% of the free amplitude. The structure of the grating was obtained in the topography (a) and control error (b) images. However, in the higher harmonic image (c), a clear contrast was apparent. This can be explained by nonlinear tip sample interactions in the imaging process, that lead to the coexistence of two distinct stable states: the so-called high- and low-amplitude states [54]. A cross-sectional analysis reveals transitions of the dynamic system from the low-amplitude state in the dark regions to the high-amplitude state in the bright regions. The sharp step in the higher harmonic image (arrow, Fig. 1.4c) was accompanied by a sharp line in the control error image (b) and a small pseudo-topographical step of about 4 nm (a). Since this step was induced by the transition from the low amplitude to the high amplitude state, it does not reflect a real variation in the topography. Thus, the higher harmonic image can be used along with the phase image to assess the reliability of topographical images. Even with the limited bandwidth of this set-up, it is possible to obtain images at higher harmonics. Since the images of higher harmonics are directly related to the tip–sample forces, they can be used as monitor signals in order to optimize imaging conditions. Basically, large amplitudes in the higher harmonics indicate large repulsive tip–sample forces, whereas small amplitudes are indicative of gentle imaging conditions. In order to obtain reliable topographic information, the variations in the higher harmonic signal should also be minimized, as this corresponds to a uniform imaging state. Thus, in conclusion, there are two main benefits from the acquisition of higher harmonic data. The higher

Fig. 1.15. Detail of a silicon test structure imaged in tapping mode (scan direction right to left). (a) Topography, (b) control error and (c) eighth harmonic. The instabilities arising from the bistable behavior of the system are difficult to see in the conventional images (a) and (b). However, in the harmonic image (c), a strong contrast is apparent, indicating both operating regimes. Reprinted with permission from [99]. Copyright (2003), American Institue of Physics


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harmonic image is useful for optimizing the imaging conditions in tapping mode, and it enables a qualitative differentiation between dissimilar materials, as we will discuss in more detail in the following section.

1.4 Spectroscopy: Distinguishing Two Polymers 1.4.1 Overview Imaging at specific higher harmonics was discussed in the previous section. Here, in turn, we focus on the spectral broadband behavior during the scan. This example describes possible strategies for exploiting the broadband information. The sample used consists of the two polymers PMMA and PS, that have similar Young’s moduli of about 3.3 GPa [102]. The local sample composition is known from the preparation process. During imaging across the boundary between both materials, the photodiode signal was recorded along different scan-lines. Figure 1.16 shows the topography and the corresponding phase image. We will focus on basic aspects that are suited to introducing concepts associated with signal analysis of dynamic AFM. More advanced methods would undoubtedly provide even more detailed results. For the sake of clarity, however, we will restrict ourselves to straightforward analysis. The first step is to find the representative spectrum of each of the materials. In a second step we use this information for material identification, in order to verify the method. In the third step the spectra are parameterized in terms of harmonic distortion, offering a direct experimental implementation.

Fig.1.16.Polymer mixture. (a) Topography and (b) phase image of the boundary region of PMMA (left) and PS (right). A significant step of approximately 40 nm separates the two polymers, as is apparent in the topography. PS appears brighter in the phase image. One scan-line is highlighted (see text). Note the groove structure traversing the PMMA in the lower half of the images. Scales (black to white): 62 nm (topography), 30◦ (phase). The scale bar equals 500 nm

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1.4.2 Experimental Details The sample (courtesy of H. Sturm, BAM, Braunschweig, Germany) was prepared by dropping melts of poly(methylmethacrylate) (PMMA) poly-(methylmethacrylate), i.e. PMMA and poly(styrol) (PS) onto a freshly cleaved salt crystal. The structure was embedded in epoxy resin. After dissolving the salt, the fairly smooth structure shown in Fig. 1.16 is left. During the preparation process, the sample was precharacterized to clearly locate the domains of PMMA and PS. Imaging was carried out on a commercial AFM (Nanoscope IIIa, Digital Instruments, Santa Barbara, CA, USA) with purpose-built electronics for the extraction of the photodiode signal. A high-pass filter was used to eliminate drift in the photodiode signal. After filtering and pre-amplification (SR560, Stanford Research Systems, Sunnyvale, CA), the signal was recorded at 5 M sample/s, at 12 bit resolution (DACQ-board PCI-6110 E controlled by LabView software, both National Instruments, Austin, TX), and analyzed off-line. The v-shaped cantilever (NSC 11, lever A, v-shaped, nominal spring constant 3.0 N/m, Mikromasch, Tallinn, Estonia) was driven at its fundamental resonance frequency of 44.3 kHz. At an imaging scan-rate of 2.03 scan lines per second and 512 pixels per line, this frequency corresponds to approximately 21.3 oscillations per pixel. The image was acquired at a 67% set-point of 100 nm free amplitude. During scanning, the conditioned photodiode signal and the topographic signal were recorded simultaneously. The data acquisition was not synchronized to the scanning, in order to capture trace, retrace and possible transients. One time-trace of 1.5 × 106 samples (300 ms of data) covers 623.6 image pixels, which is more than one scan-line. Time-traces were recorded during several images. As drift demanded a readjustment of the set-point, several curves exhibited different fundamental frequency amplitudes. These curves were rejected. The remaining six time-traces were acquired under constant conditions within two subsequent images. Five of them were used to construct an ensemble for determining the average spectra, while the sixth trace was used as an independent validation trace. 1.4.3 Signal Analysis Figure 1.17 depicts the raw data of the scan-line highlighted in Fig. 1.16. To emphasize the digital nature of the data and to emphasize the character of a time-trace we give the raw units – least significant bit (LSB) and time. A step of approximately 40 nm visible at the boundary of the two materials gives a clear landmark. The material boundary is shown in Fig. 1.17, as well as the reversal of scan-direction. Below the overview, a few oscillation periods recorded on the two materials are shown as examples of the typical signal shape. Note that the RMS-signal amplitude was used as the input for the feedback loop. Important variations in amplitude occur after the step in topography. At the step, the tip is slightly withdrawn from the sample, increasing the amplitude of the fundamental oscillation while at the same time decreasing the strength of the


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Fig. 1.17. Raw data associated with the scan-line highlighted in Fig. 1.16: (a) topography, (b) time-trace of the detector signal, (c) examples of the signal recorded on the two materials. To stress the character of the raw data, we kept the units of “least significant bits” (LSB). The step in topography measures approximately 40 nm and separates the two polymers as indicated. While the time-trace was recorded, the scan-direction was reversed (“Trace” to “Retrace”, as indicated). As the differences between the cantilever motions on PMMA and PS cannot be identified by eye, signal analysis methods are necessary

tip–sample impact. As will be shown later, this topographically-induced transient response leaves a clear fingerprint. A visual comparison of the cantilever response on both materials (Fig. 1.17c) does not allow us to estimate signal distortion, nor to distinguish between the materials. The differences are too small, partly because of the scanning conditions, and partly because they have similar material properties. Averaging techniques are necessary to extract characteristic differences. For this purpose, two respective sub-ensembles were created from all of the acquired scan-lines containing data taken on PMMA and PS. As the material boundary is known, sorting the data is straightforward. To exclude boundary effects, data close to the material step were not considered. Furthermore, one scan line was completely excluded from the sub-ensembles and will serve as an independent data set for testing. A Fourier transformation of nonoverlapping sections of 4096 points (0.8192 ms or approximately 36.3 oscillation periods) led to ensembles of spectra containing 714 members in the case of PS and 567 members in the case of PMMA. Using iterated correlation averaging, the characteristic amplitude spectra on PMMA and PS (Fig. 1.18) were obtained from a final average of 200 spectra each. At the fundamental frequency (44.32 kHz), the magnitudes of both spectra are equal, due to feedback regulation, so the spectra were normalized to the amplitude at that frequency. Clusters of peaks at multiples of the fundamental frequency are clearly visible, as well as the modal structure imposed by the spectral response of the cantilever. The DC contribution does not contain valuable information, because the signal is high-pass filtered at 14.5 kHz. Briefly comparing the spectra, no particular feature is striking enough to separate both materials. However, a detailed comparison (Fig. 1.19) reveals important differences, which are discussed in the following.

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Fig. 1.18. Characteristic amplitude spectrum of the AFM signal, as obtained on (a) PMMA and (b) PS. Both spectra are normalized to the amplitude at the first harmonic (in other words the driving frequency, which is set to the fundamental resonance of the free lever). Each spectrum represents the average of 200 raw spectra (see text)

A systematic difference between the spectra is present in the frequency group assigned to the third oscillatory eigenmode of the lever around harmonic 13. In this frequency band, the PMMA amplitude is always larger to that for PS. Figure 1.19b gives a summary of the differences between the two spectra relative to the average

An,PMMA + An,PS . At this point, one value for each harmonic, An,PMMA − An,PS important observation should be stressed. The difference between the spectra of the two materials is by no means uniform. The time-course of the interaction on the two materials varies not only in its strength but also in its shape, which points to dissipative effects. Qualitatively, the impact of the tip on the sample associated with the spectrum is faster and/or harder on PMMA compared to PS. Rather than changing the elastic behavior, we suspect differences in the viscoelastic, adhesion or wetting properties at the origin. There is also a significant contrast in the second harmonic. It is known from careful inspection of the instrument that the second harmonic is

Fig. 1.19. (a) Harmonics spectra measured on PMMA and PS. (b) Comparison of the higher harmonics measured on PMMA and PS. Positive values indicate that their is a higher relative contribution for PMMA at that frequency than for PS


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not necessarily a good spectral indicator, as it is already present in the excitation. It might also be corrupted by nonlinearities of the detection. Are the differences in the spectra, as detailed above, really significant enough to distinguish both materials? This question is addressed using cross-correlation of the characteristic spectra with spectra along the scan-lines. We define the identification parameter ID as the ratio between the cross-correlation of the spectrum AScan-line (ω) of the scan-line with the characteristic spectrum APMMA (ω) of PMMA and the crosscorrelation of the spectrum of the scan-line with the characteristic spectrum APS (ω) of PS. Both cross-correlations around time t are only evaluated on the frequency band of the third eigenmode of the lever. Thus: APMMA (ω) × AScan-line (ω) − 1 (1.33) ID(t) = APS (ω) × AScan-line (ω) ω ∈ Band of third eigenmode For ID (t) < 0, the material around time t on the time-trace is identified as PS, for ID (t) > 0 the material is identified as PMMA. We first apply this test to the marked scan-line (Fig. 1.17). Figure 1.20 shows the smoothed identification parameter ID(t) along the scan-line together with the raw-data of the topography signal. The inset recalls the topographic image. Indeed, the topographic step marks the material boundary, which is again found by the identification parameter. Remarkably, a dip in the topographic scan-line at around 20 ms is identified as PS (marked by arrows). This position matches a pronounced groove in the topographic image (arrow). From the identification parameter ID(t), as well as from careful inspection of the topographic and phase images, we assume the groove to be a fracture penetrating the covering layer of PMMA into the underlying PS. Previously, the spectral details were used to analyze the time traces. Now, in fact, we conceive the scan line as a variation of a quantity with the position on the sample. Again we focus on the scan-line highlighted in Fig. 1.16. The topography, the

Fig.1.20.Material identification for the highlighted scan-line (inset: topographic image) using the identification parameter defined in (1.33). The topography along the scan-line is given together with the identification parameter ID. The material in the groove (arrows) is identified as PS (see text)


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relative phase lag, and the harmonic signal distortion are all considered, as illustrated in Fig. 1.21. For the phase lag, the average of all lines in the image is also given (thick line). Fast variations linked to features in the topography are smoothed out, and thus the average scan line is representative of the phase difference on the two materials. In the definition of the phase, we use positive values for a signal delayed with respect to the average signal, and vice versa for negative values. Thus, the more positive the phase lag, the higher the dissipation. Regarding the scan-line, we state a phase difference of 1.5◦ , indicating a higher dissipation on PS. A calculation of the total harmonic signal distortion according to A2 (nω0 ) n=2,3... THD = A2 (nω0 ) n=1,2,3...

yields an overall measure of the signal content in higher harmonics. For the case presented, this overall quantity distilled from the full spectrum does not show a significant difference for the two materials. The THD remains at about 5%, and only drops at the topographical step, which is the clear fingerprint of the control error. By defining a modal harmonic distortion where the distortion is only evaluated in a frequency band around a distinct eigenmode of the lever, we have found a material-sensitive parameterization. Figure 1.21d shows the modal harmonic distortion evaluated for the third flexural eigenmode of the lever (including harmonics 9–17). The modal harmonic distortion in the high-frequency band is more pronounced on PMMA than on PS. This example shows how to define a suitable parameterization that may provide a material specificity that is similar to that yielded by the approach using the characteristic spectra. However, the full spectra enables a verification of whether the chosen parameterization is sharp enough to separate materials or material properties in the intended way.

Fig. 1.21. (a) Topography, (b) phase, (c) total and (d) modal harmonic distortion. Both materials can be distinguished in the phase signal and in the modal harmonic distortion. The total harmonic distortion does not provide a sufficient contrast


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1.5 Outlook This chapter has demonstrated that a deeper understanding of the cantilever dynamics permits subtle experiments probing surface forces that can be used to discriminate dissimilar materials. The relation between higher harmonics and different sample properties such as viscoelasticity, adhesion, wetting properties, surface potential, polarizability (van der Waals interaction) or specific chemical properties is not yet completely understood. However, there are several lines of development based on a full spectral analysis of the AFM signal that will provide deeper insights into short-range surface forces in the future. In order to correlate the cantilever dynamics with surface properties, a reconstruction of the forces acting at the tip over time can provide further information. Although a similar inverse problem for the simpler case of a periodically-forced harmonic oscillator was solved more than a century ago [103], a full spectral inversion is difficult due to the limited signal-to-noise-ratio encountered in AFM measurements [104, 105]. However, by taking advantage of the periodicity of the tip motion in dynamic AFM, it is possible to reconstruct the tip–sample forces [76]. Progress in the determination of surface forces can also be expected from analyzing the system’s response to a parametric excitation. By driving the cantilever at a subharmonic of a mechanical resonance, small signals can be enhanced, permitting identification of materials [106, 107]. Feedback schemes based on the detection of higher harmonics will further enhance the performance and force sensitivity of AFM measurements [27, 51, 89]. The quantification of surface forces demands a comprehensive theory of the dynamics of AFM which must cover (among others) the full range of AFM techniques, from contact mode to ultrasonic AFM and tapping mode to noncontact AFM. We did not, however, aim to solve this task by discussing the generation of higher harmonics in atomic force microscopy. Rather, we restricted ourselves to aspects of the AFM dynamics, where the motion remains periodic. Still, we hope that the concepts on which we base our considerations may help to advance the theoretical understanding of dynamic AFM. Acknowledgements. We are indebted to Wolfgang Heckl (LMU Munich, Germany), Andreas Stemmer (ETH Zurich, Switzerland), Reinhard Guckenberger and Fritz Keilmann (both MPI for Biochemistry, Martinsried, Germany), who gave us the opportunity to pursue our investigations and provided valuable discussions. Without the many fruitful discussions with Georg Schitter (UCSB, USA) and Reiner Hillenbrand (MPI for Biochemistry, Martinsried, Germany), much of the work discussed here would not have been possible. RWS gratefully acknowledges financial support from the German Federal Ministry for Education and Research (BMBF) within the programme “Young Scientists Competition Nanotechnology” (Grant 03N8706) and the Swiss Commission for Technology and Innovation (KTI, TopNano21, Grant 6351.1).


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46. Sahin O, Yaralioglu G, Grow R, Zappe SF, Atalar A, Quate C, Solgaard O (2004) Sensor Actuat A 114:183 47. Mackel R, Baumgartner H, Ren J (1993) Rev Sci Instrum 64:694 48. Kikukawa A, Hosaka S, Imura R (1995) Appl Phys Lett 66:3510 49. Nonnenmacher M, O’Boyle MP, Wickramasinghe HK (1991) Appl Phys Lett 58:2921 50. Jacobs HO, Knapp HF, Muller S, Stemmer A (1997) Ultramicroscopy 69:39 51. Okamoto K, Sugawara Y, Morita S (2002) Appl Surf Sci 188:381 52. van Noort SJT, Willemsen OH, van der Werf KO, de Grooth BG, Greve J (1999) Langmuir 15:7101 53. Gleyzes P, Kuo PK, Boccara AC (1991) Appl Phys Lett 58:2989 54. García R, San Paulo A (2000) Phys Rev B 61:R13381 55. Behrend OP, Oulevey F, Gourdon D, Dupas E, Kulik AJ, Gremaud G, Burnham NA (1998) Appl Phys A 66:S219 56. Rodríguez TR, García R (2002) Appl Phys Lett 80:1646 57. Kolosov OV, Castell MR, Marsh CD, Brix A (1998) Phys Rev Lett 81:1046 58. Hirsekorn S (1998) Appl Phys A 66:1 59. Yaralioglu GG, Atalar A (1999) Rev Sci Instrum 70:2379 60. Arinero R, Leveque G (2003) Rev Sci Instrum 74:104 61. Chudoba T, Schwarzer N, Richter F (1999) Thin Solid Films 355–356:284 62. Turner JA, Hirsekorn S, Rabe U, Arnold W (1997) J Appl Phys 82:966 63. Clark RL (1997) J Dyn Syst–T ASME 119:390 64. Reza Moheimani SO (2000) J Dyn Syst–T ASME 122:237 65. Hatch MR (2001) Vibration simulation using MATLAB and ANSYS. Chapman & Hall/CRC, Boca Raton, FL 66. Argyris J, Mlejnek H-P (1988) Die Methode der finiten Elemente. Vieweg, Braunschweig/Wiesbaden 67. Clough RW, Penzien J (1993) Dynamics of structures. McGraw-Hill, Singapore 68. Rinaldi G, Packirisamy M, Stiharu I (2004) Int J Nanotechnol 1:292 69. Spector VA, Flashner H (1990) J Dyn Syst–T ASME 112: 186 70. Derjaguin BV, Muller VM, Toporov Yu P (1975) J Colloid Interf Sci 53:314 71. Gannepalli A, Mallapragada SK (2001) Nanotechnology 12:250 72. Haefliger D, Plitzko JM, Hillenbrand R (2004) Appl Phys Lett 85:4466 73. Boyer L, Houze F, Tonck A, Loubet JL, Georges JM (1994) J Phys D Appl Phys 27: 1504 74. Hao HW, Baro AM, Saenz JJ (1991) J Vac Sci Technol B 9(2):1323 75. Dianoux R, Martins F, Marchi F, Alandi C, Comin F, Chevrier J (2003) Phys Rev B 68:45403 76. Stark M, Stark RW, Heckl WM, Guckenberger R (2002) Proc Natl Acad Sci USA 99:8473 77. Stark RW, Schitter G, Stark M, Guckenberger R, Stemmer A (2004) Phys Rev B 69:085412 78. Sader JE (1998) J Appl Phys 84:64 79. Fleming AJ, Reza Moheimani SO (2003) IEEE T Contr Syst T 11:726 80. Miu DK (1993) Mechatronics. Springer, Berlin Heidelberg New York 81. Yuan K, Liu LY (2003) J Robotic Syst 20:581 82. Hölscher H (2002) Surf Sci 515:517 83. Rodríguez TR, García R (2003) Appl Phys Lett 28:4821 84. Hao HO, Baró AM, Saenz JJ (1991) J Vac Sci Technol B 9:1323 85. Stark RW (2003) Proc 1st Int Meeting on Applied Physics 2003 (aphys2003), 13–18 October 2003, Badajoz, Spain, arXiv:physics/0501061 86. Sasaki N, Tsukada M, Tamura M, Tamura R, Abe K, Sato N (1998) Appl Phys A 66:S287 87. Basso M, Giarré L, Dahleh M, Mezic I (2000) J Dyn Syst–T ASME 122:240 88. Rützel S, Lee SI, Raman A (2003) Proc R Soc Lond A 459:1925 89. Schiener J, Witt S, Stark M, Guckenberger R (2004) Rev Sci Instrum 75:2564

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90. Proksch R, Schäffer TE, Cleveland JP, Callahan RC, Viani MB (2004) Nanotechnology 15:1344 91. Schäffer TE, Fuchs H (2005) J Appl Phys 97:083524 92. Schäffer TE (2002) J Appl Phys 91:4739 93. Schäffer TE, Hansma PK (1998) J Appl Phys 84:4661 94. Putman CAJ, de Grooth BG, van Hulst NF, Greve J (1992) J Appl Phys 72:6 95. Gustafsson MGL, Clarke J (1994) J Appl Phys 76:172 96. Stark RW (2004) Rev Sci Instrum 75:5053 97. Scherer MP, Frank G, Gummer AW (2000) J Appl Phys 88:2912 98. Stark M, Guckenberger R, Stemmer A, Stark RW (2004) Proc IEEE Nano 2004, 17–19 August 2004, Munich, Germany 99. Stark RW, Heckl WM (2003) Rev Sci Instrum 74:5111 100. Crittenden S, Raman A, Reifenberger R (2004) Workshop on Advanced Dynamic AFM Methods, 8–10 October 2004, Tres Cantos, Madrid, Spain 101. Sahin O, Quate CF, Solgaard O, Atalar A (2004) Phys Rev B 69:165416 102. Bandrup J, Immergood EH (eds) (1989) Polymer handbook. Wiley, New York 103. Franklin WS (1894) Phys Rev 1:442 104. Todd BA, Eppell SJ, Zypman FR (2001) Appl Phys Lett 79:1888 105. Todd BA, Eppell SJ (2003) J Appl Phys 94:3563 106. Patil S, Dharmadhikari CV (2003) Appl Surf Sci 217:7 107. Abdel-Rahman EM, Nayfeh AH (2005) Nanotechnology 16:199

2 Atomic Force Acoustic Microscopy Ute Rabe

Abbreviations a1 , a2 , a3 , a4 and A1 , A2 , A3 , A4 aC α A b cP dY 0 , d X0 δn , δL E E∗ ε ET , ES G∗ G T, G S Fn , FL f ϕ φ, φLat Φ γ , γLat h ηAir I L L1 L2 ∗ k∗ , kLat kC

Constants in the shape function y(x) Contact radius Wave number of the flexural waves Cross-section area of the cantilever Thickness of the cantilever Ratio φLat /φ Normalized amplitude of the sensor tip in y- and x-direction, respectively Normal, lateral contact deflection (in the coordinate system of the sample surface) Young’s modulus of the cantilever Reduced Young’s modulus of the contact Displacement of the additional mass m L from the center of the beam Young’s modulus of the tip, the surface Reduced shear modulus of the contact Shear modulus of the tip, the surface Normal, lateral forces (in the coordinate system of the sample surface) Frequency Tilt angle of the cantilever Dimensionless normal, lateral contact function Phase Normal, lateral contact damping Height of the sensor tip Damping constant describing losses by air Area moment of inertia Length of the cantilever Distance from the fixed end of the cantilever to the position of the tip Distance from the free end of the beam to the tip Normal, lateral contact stiffness Static spring constant of the cantilever

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m∗, m mL M MT , MS N(α), N0 (α) p, pLat Q R ρ S S0 , S1 , S2 , S3 , S4 σ t u0 w x, x2 y, y2 X, T , U ω Ω ψ AFAM AFM FMM SAFM SAM SLAM SMM UAFM UFM

U. Rabe

Effective mass, real mass of the cantilever Additional mass Moment Indentation modulus of the tip, the sample surface Denominators in the formulas for forced vibration of the beam Dimensionless normal, lateral contact damping Quality factor Radius of the sensor tip Mass density of the cantilever Contact area Terms in the characteristic equation Sensitivity Time Amplitude of excitation Width of the cantilever Coordinate in length direction of the cantilever Deflection of the cantilever in its thickness direction Auxiliary functions in the boundary conditions Circular frequency Characteristic function ψ = αL dimensionless wave number Atomic force acoustic microscopy Atomic force microscopy Force modulation microscopy Scanning acoustic force microscopy Scanning acoustic microscopy Scanning local acceleration microscopy Scanning microdeformation microscopy Ultrasonic atomic force microscopy Ultrasonic force microscopy

2.1 Introduction Materials with an artificial nanostructure such as nanocrystalline metals and ceramics or matrix embedded nanowires or nanoparticles are advancing into application. Polymer blends or piezoelectric ceramics are examples for materials of high technical importance possessing a natural nanostructure which determines their macroscopic behavior. Structures such as thin films or adhesion layers are of nm dimensions in only one direction, but micromechanical devices and even nano-devices are feasible which are of nm size in three dimensions. In the nm range the mechanical properties such as hardness or elastic constants and the symmetry of the crystal lattice can vary with the size of the object. Furthermore, the properties of structures with nm-dimensions depend strongly on the surrounding material and on the boundary conditions in general. Therefore there is a need to measure mechanical properties on a nano-scale.

2 Atomic Force Acoustic Microscopy

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2.1.1 Near-field Acoustic Microscopy Ultrasonic imaging based on transmission and reflection of ultrasonic waves has been used for a long time for elasticity measurements and flaw detection in different areas such as physics, nondestructive testing and medicine. By measuring the dispersion of laser generated surface acoustic waves in the frequency range of several 10 MHz up to several 100 MHz, the elastic constants of thin films can be determined [1]. The distance covered by the acoustic wave between the generating laser spot and the receiver (about 10 mm) defines the lateral resolution of this technique. In the classical scanning acoustic microscopy (SAM), which was developed in the 1970th, an acoustic wave is focused onto the sample surface by a sapphire lens, and the reflected acoustic waves are detected [2]. The imaging contrast depends on the acoustic impedance ρv(ρ = mass density of the sample, v = sound velocity) and consequently on the elastic constants of the sample. Defects like cracks, inclusions or mechanical stresses influence the acoustic impedance and can also be imaged if they are in a certain penetration depth inside the sample. According to Abbe’s principle, techniques using focused waves are restricted in their lateral resolution to about half a wavelength. A focused acoustic beam in water yields a spot diameter of about 1 µm at 1 GHz frequency. The local resolution can be improved if an acoustic wave is guided towards the sample by a structure which is smaller and closer to the surface than the acoustic wavelength. Different acoustic microscopes based on this near-field principle were proposed [3–6]. One drawback of this type of microscope is a low signal level because the traveling acoustic waves have to pass through an nm-scaled structure. Alternatively, the near-field sensor can be constituted by a small object which vibrates at one of its resonant frequencies. Güthner et al. used a resonating tuning fork [7] vibrating at 33 kHz in air. One corner of the tuning fork served as a sensor tip. Hydrodynamic interaction between the senor tip and the sample surface damps the vibration and shifts the resonant frequency of the tuning fork. As the damping depends mainly on the thickness of the air layer between sensor and sample, there is almost no contrast by the elasticity of the sample, but the technique can be used to image topography. More recently, tuning forks or length extension quartz resonators are used as force sensors for high-resolution atomic force microscopy in air and in vacuum [8]. After the invention of the Atomic Force Microscope (AFM) [9] near field microscopy was strongly promoted and various operation modes and related techniques emerged. Since 1993, several microscopes combining AFM with ultrasonic imaging have been developed, named for example ultrasonic force microscopy (UFM) [10], scanning acoustic force microscopy (SAFM) [11], atomic force acoustic microscopy (AFAM) [12], and ultrasonic atomic force microscopy (UAFM) [13]. These techniques can be considered as special types of dynamic force microscopy or as nearfield techniques from the point of view of ultrasonic imaging. The sensor tip of the AFM has a radius of only several nm up to several 100 nm. The tip–sample contact radius, which is orders of magnitude smaller than the acoustic wavelength, defines the local resolution. To reach high local resolution, nanoscaled probes can either be used as emitters or as detectors of ultrasound. In the latter case acoustic waves are generated with

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conventional ultrasonic transducers, and a scanning probe microscope is used to detect the acoustic wave fields at the sample surface. In atomic force microscopy the tip–sample forces are a nonlinear function of tip–sample distance. The nonlinear forces cause frequency mixing, if an ultrasonic excitation signal is applied to a transducer below the sample and another vibration with a slightly different frequency is excited in the cantilever and its sensor tip [11, 14]. In the scanning acoustic force microscopy [14] mixing of two surface waves which propagate in different direction is exploited to image surface acoustic wave fields with submicron lateral resolution. Interference phenomena caused by scattering of a plane wave by a disk-shaped structure were observed in this way [15]. The nonlinearity of the tip–sample forces has a rectifying effect which is exploited in ultrasonic force microscopy [10, 16]. In UFM, an ultrasonic transducer generating longitudinal waves is placed below the sample. The amplitude of the sinusoidal excitation applied to the transducer is modulated with a saw-tooth signal. The acoustic wave causes a high frequency out-of-plane surface vibration with a low-frequency amplitude modulation. The sensor tip of the AFM is in contact with the vibrating sample surface and when the threshold amplitude is reached, the sensor tip lifts off from the surface. A lock-in amplifier which operates at the modulation frequency detects the envelope of the high frequency signal. This rectifying property was called “mechanical diode effect”. A qualitative image of elastic sample properties and contrast from subsurface objects can be obtained [16–18]. A rectifying effect due to the nonlinear forces is also observed when the amplitude modulated vibration is excited at the fixed end of the cantilever (“waveguide UFM”) [19]. The contrast in UFM was examined by different research groups [20, 21]. The advantage of the mixing technique and the UFM is the low bandwidth which is required for the position detector in the AFM. Because the modulation frequency can be chosen in the kHz range, a direct detection of signals at MHz or even GHz frequencies is not necessary. As both, elasticity and adhesion, contribute to the image contrast [22], it is difficult to separate surface elasticity quantitatively from adhesion. 2.1.2 Scanning Probe Techniques and Nanoindentation Different dynamic operation modes of the scanning force microscope were suggested to measure elasticity on an nm scale. In the force-modulation mode (FMM) the sensor tip is in contact with the probed surface, and the surface is vibrated normally or laterally at a frequency below the first resonance of the cantilever [23]. The amplitude or phase of the vibration of the cantilever is evaluated. Force modulation microscopy provides elasticity contrast of softer samples like for example polymers. If stiffer materials like metals and ceramics are to be examined, the contact stiffness between tip and surface becomes much higher than the spring constant of the cantilever (0.1 N/m up to several 10 N/m, depending on the type of beam) and the contrast decreases. Instead of applying the force indirectly by varying the distance between the surface and the fixed end of the cantilever, other research groups applied a magnetic force directly to the cantilever [24,25]. Some authors extend force-modulation microscopy to the higher modes of the cantilever [25–27]. Here, this type of operation is called contact-resonance spectroscopy and will be treated in detail in the next paragraphs.


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In the pulsed force mode [28] the distance between the fixed end of the sensor and the sample surface is also modulated at a frequency below the resonant frequencies of the sensor. The amplitude is so high (10–500 nm) that the tip loses contact with the sample surface during its vibration cycle. Characteristic points in the vibration signal of the cantilever are evaluated to image elasticity and adhesion. In scanning local acceleration microscopy (SLAM) the sensor tip is in contact with a sample surface that is vibrated out-of-plane with a frequency slightly above the first flexural resonance of the cantilever [29]. Using a temperature controlled SLAM instrument, Oulevey et al. observed martensitic phase transformation in NiTi-alloy [30]. Force-modulation microscopy can be considered as direct detection of low frequency acoustical vibrations by an AFM. A sensor tip which touches the sample surface during its vibration cycle for example in FMM or tapping mode [31] radiates sound into the sample, but the amplitudes are usually below the detection limit of commercial transducers [32]. If the sensor of the AFM is magnified by only one or two orders of magnitude like in scanning microdeformation microscopy (SMM), the acoustical amplitudes transmitted through the sample become detectable [33]. Subsurface features were imaged by SMM in transmission mode [34]. The vibration of the cantilever was measured with a piezoelectric element [35] or with an optical interferometer pointing onto the cantilever [36]. The AFM sensor and the SMM cantilever are so similar to each other that many aspects of the equation of motion and the contact mechanics models are identical. As the radius of the SMM sensor tip is larger than the radius of an AFM tip the tip–sample interaction is easier to control and macroscopic contact models are easier to apply. Because the contact area scales with the size of the sensor tip, the lateral resolution is lower in SMM than in the AFM based techniques. Nanoindentation, which was originally developed to measure hardness, can also be used for elasticity measurements [37]. In dynamic nanoindentation a low amplitude sinusoidal force modulation is superimposed to the quasi static load applied to the indenter. Amplitude and phase of the vibration of the electromechanical system, constituted by the indenter and the force detection unit, are evaluated to measure contact stiffness as a function of load [38]. The indenter tips are of Berkovich type made of diamond or spherical with a radius of 100 µm. The local resolution of the nanoindenter is typically 100 nm to 200 nm because of the mechanical stress field in the sample at a minimum penetration depth of about 20 nm. Seyed Asif et al. imaged the real and imaginary parts of the Young’s modulus of carbon fibers in epoxy matrix by dynamic nanoindentation [39]. Especially for examination of softer materials like polymers, dynamic nanoindentation became very popular during the last years. An overview of commercial instruments can be found for example in a publication by Bushan and Li [40]. 2.1.3 Vibration Modes of AFM Cantilevers The typical dimensions of cantilevers for atomic force microscopy are several 100 µm in length, several 10 µm in width and several 100 nm up to several µm in thickness. The sensors are small plates or beams having distributed mass, and they can be excited to different modes of vibration such as flexural or torsional modes. While

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most of the dynamic scanning probe applications relied on the fundamental modes of the beams, for noise analysis the importance of the higher eigenmodes was recognized by different authors [41, 42]. In atomic force acoustic microscopy and ultrasonic atomic force microscopy the flexural resonant frequencies of atomic force microscope cantilevers are measured [13, 43, 44]. The sensor of the AFM can be considered as a cantilever beam which is clamped at one end and free at the other end. If the sensor tip is in contact with a sample surface tip–sample interaction forces change the end conditions and all resonant frequencies of the cantilever are shifted. Furthermore, tip–sample interaction causes damping and changes the width of the resonance curves. When the cantilever is in flexural vibration, a large component of the amplitude of the tip apex is normal to the sample surface. Therefore flexural modes are used to measure normal tip–sample contact stiffness, which in turn depends on the Young’s modulus of tip and sample, on the shear modulus of tip and sample, on the contact area, and on the adhesion forces. In most cases, the flexural modes cause an angular oscillation of the cantilever beam at the position where the tip is fixed, which in turn leads to an oscillation of the tip apex parallel to the surface. This means that the flexural modes are also influenced by lateral tip–sample contact stiffness and friction. Torsion of AFM cantilevers is used to measure lateral forces and friction [45,46] and a variety of dynamic operation modes exploiting torsional vibration is known. Similar to AFAM and UAFM the torsional resonant frequency and the width of the resonant peaks can be determined to measure in-plane surface properties [47,48]. At low amplitudes the sensor tip sticks to the sample surface, and at higher amplitudes sliding sets in. The stick-slip events cause a characteristic plateau in the shape of the contact-resonance curves [49]. Torsional modes are not the only type of vibration exhibiting a strong component of lateral tip–sample amplitude. Bending of the cantilever in its thickness direction leads to non-negligible lateral tip–sample displacement [50], and lateral bending modes of the cantilever can be used for imaging and spectroscopy [51]. In the so-called “overtone atomic force microscopy” Drobeck et al. exploited the torsional vibration modes of V -shaped cantilevers to measure in-plane surface stiffness [52, 53]. Elasticity contrast was obtained on Al−Ni−Fe quasicrystal samples [52], and the lateral stiffness of Si, Al, and CdTe surfaces was evaluated quantitatively [53]. The results were obtained in ambient conditions by analysis of the thermal noise of the cantilevers.

2.2 Linear Contact-resonance Spectroscopy Using Flexural Modes Contact-resonance spectroscopy techniques which exploit the flexural modes of the cantilevers can be organized according to the method of excitation. In the AFAMtechnique a transducer below the sample excites longitudinal waves which cause outof-plane vibrations of the investigated surface (transducer 1 in Fig. 2.1) [43, 44]. Via the tip–sample contact forces the vibrations are transmitted into the cantilever. The flexural vibrations can also be excited by a transducer which generates oscillations of the fixed end of the beam (transducer 2 in Fig. 2.1). This technique (UAFM) was first


43

Fig. 2.1. In the AFAM-mode the flexural vibrations of the cantilever are excited by a transducer below the sample (transducer 1). The vibrations can also be excited by a transducer which excites the fixed end of the cantilever (transducer 2, UAFM-mode). The low frequency components of the beam deflection signal are used to control the static deflection of the cantilever. The high frequency components of the signal are evaluated for example by a lock-in-amplifier for ultrasonic spectroscopy or imaging

published by Yamanaka and Nakano and continuously developed afterwards [13,54– 56]. Alternatively, a ZnO transducer can be fabricated directly on the sensor beam to excite the vibrations in the UAFM-mode [57–59]. Dupas et al. fixed one end of the cantilever to a piezoelectric bimorph [60]. For simplicity the technique is designated here by AFAM or UAFM, when the vibrations are excited by the sample surface or by the fixed end of the beam, respectively. Excitation by a concentrated harmonic force acting directly on the cantilever [25] or spectral analysis of thermomechanical noise of the sensor [52, 53] is also reported for contact-resonance spectroscopy. In the first AFAM experiments the ultrasonic transducer below the sample was excited with electrical spikes [61]. The resulting out-of-plane deflection of the investigated surface is a short pulse with a spectrum depending on the resonant frequency of the transducer and the transmission of the sample. The pulse at the surface excites a vibration in the cantilever which decays according to the time constants of contact damping and air damping. Fourier transformation of the cantilever vibrations showed that the spectrum contained contact resonances. However, with the pulse excitation the senor tip loses contact with the sample surface very easily and the signals are strongly influenced by adhesion. The current state of the art in AFAM and UAFM is to excite the vibration with sinusoidal signals and to sweep the frequency to measure spectra. Frequency sweeps are also common in other dynamic AFM operation modes, but in the techniques discussed here the sensor tip is in contact with the

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sample surface while the spectra are measured. This means that the static deflection of the cantilever must be controlled or monitored during the sweep. Furthermore, the techniques discussed here do not only use the fundamental resonant frequency of the cantilever but consider the higher modes, too. The necessary bandwidth of excitation and detection depends on the type of sensor beam which is used. Softer samples like polymers can be examined with sensors having a spring constant of only a few N/m or less, while stiffer samples require stiffer beams with several 10 N/m spring constant. As will be shown in the following, for example the first contact-resonance frequency of the types of beams listed in Table 2.2 is always lower than 1 MHz. This means that ultrasonic imaging is possible with a bandwidth of 1 MHz. For quantitative evaluation, the bandwidth should be large enough to cover two contact resonances at a minimum. The continuous wave signals are excited by a function generator. A fast lockin amplifier can be used to determine amplitude and phase of the vibration of the cantilever. Alternatively, the ultrasonic signal and reference can be down converted to a lower frequency and reference (20 kHz) and subsequently fed to a lock-inamplifier [62]. In this case the lock-in-amplifier works at a fixed lower frequency and needs not to follow the swept frequency. A network analyzer can also be used to excite and to detect the signals in spectral measurements [57]. Dupas et al. used a stroboscopic technique and increased the bandwidth of detection to 4 MHz by modulating the light intensity of the laser diode of the beam-deflection detector [60]. There are different possibilities of acoustical imaging using contact resonances which are discussed in Chap. 6. 2.2.1 Flexural Vibrations of Clamped-free Beams The most widespread commercial cantilevers are either of triangular or of approximately rectangular shape. The vibrational modes for example of V-shaped beams or dagger-shaped beams can be calculated numerically [63, 64] or using analytical approximations [65]. However, analytical formulas can be evaluated extremely fast and they may provide a deeper understanding of how the different parameters influence the vibration. This analysis will therefore be restricted to cantilevers with approximately rectangular shape because they can be described with satisfactory precision by analytical models. Under the precondition that its length is much larger than its width and thickness, the micro fabricated sensor beam can be regarded as an elastic beam. If x is the coordinate in length direction of the beam and y the coordinate in thickness direction, the equation of motion for damped flexural vibrations becomes [66]: EI

∂y ∂2 y ∂4 y + ηAir ρA + ρA 2 = 0 4 ∂x ∂t ∂t

(2.1)

where E is the Young’s modulus of the cantilever, ρ is its mass density, A is the area of its cross section, I is the area moment of inertia and ηAir is a damping constant expressing the dissipation caused by the air. y(x) is the deflection at position x, ∂y/∂x is the slope of the beam at position x, E I∂ 2 y/∂x 2 is the torsional moment, and


45

E I∂ 3 y/∂x 3 is the shear force. One seeks a harmonic solution in time with angular frequency ω = 2π f . The solution of the differential equation of motion may be written:

y(x, t) = y(x) · y(t) = a1 eαx + a2 e−αx + a3 eiαx + a4 e−iαx eiωt . (2.2) The mode shape y(x) can also be expressed as: y(x) = A1 (cos αx + cosh αx) + A2 (cos αx − cosh αx) + A3 (sin αx + sinh αx) + A4 (sin αx − sinh αx) ,

(2.3)

where a1 , a2 , a3 , a4 and A1 , A2 , A3 , A4 are constants. By substituting the solution (2.2) in the equation of motion (2.1) one obtains the dispersion relation for a flexural wave with complex wave number α: E Iα4 + iρAηAir ω − ρAω2 = 0 ! 4 ρA α± = ± (ω2 ∓ iηAir ω) . EI In absence of damping the dispersion equation simplifies to: " (αL)2 1 E I f = . 2π L 2 ρA

(2.4) (2.5)

(2.6)

Boundary conditions must be fulfilled if the beam is of finite length L. By substituting the boundary conditions in the general solution one obtains a characteristic equation. For a beam with one clamped end and one free end one finds in textbooks [66, 67]: cos αL cosh αL + 1 = 0

(2.7)

The roots αn L of this equation can be calculated numerically, where n = {1, 2, 3, . . . } is the mode number. Examples are listed in Table 2.1. Using the dispersion relation one obtains the resonant frequencies of the beam. The quality factor Q of the resonances is given by: Q=

ωn ωn = ∆ω ηAir

(2.8)

According to (2.8), the quality factor Q increases with the mode number. Experimentally one often observes an increase in Q for the first modes up to 1 MHz followed by a decrease at still higher frequencies [44]. This means that the damping ηAir is in fact a function of the frequency. Quality factors of the first flexural resonances of commercial sensors in air are typically between Q = 200 and Q = 900 for sensors made of single crystal silicon. With most commercial atomic force microscopes the cantilever can be excited to forced vibration when its one end is free. The fundamental mode of the clamped-free beam and the quality factor of the resonance can be measured in this way. On the

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Table 2.1. Dimensionless wave number αn L and frequency ratio f n / f 1,free for different boundary conditions of the beam: clamped-free ( f n,free ), clamped-pinned ( f n,pin ), and clamped-clamped ( f n,clamp ) n

(αn L)free

f n,free / f 1,free

(αn L)pin

f n,pin / f 1,free

(αn L)clamp

f n,clamp / f 1,free

1 2 3 4 5 6 7

1.87510 4.69409 7.85476 10.99554 14.13717 17.27876 20.42035 ≈ 2n−1 2 π

1.00 6.27 17.55 34.39 56.84 84.91 118.60

3.92660 7.06858 10.21018 13.35177 16.49336 19.63495 22.77655 ≈ 4n+1 4 π

4.39 14.21 29.65 50.70 77.37 109.65 147.55

4.73004 7.85320 10.99561 14.13717 17.27876 20.42035 23.56194 ≈ 2n+1 2 π

6.36 17.54 34.39 56.84 84.91 118.60 157.90

other hand, the constants ρ, A, E, I, and ηAir are often unknown. It is therefore better to rewrite the dispersion relations (2.5) and (2.6) in terms of measurable quantities: " ω2 ηAir ω α± L = ±α1,free L · 4 2 ∓i 2 ω1,free ω1,free " ≈ ±1.8751 · f f 1,free

4

f f 1,free

(αL)2 = 2 . α1,free L

2 ∓i

1

f

Q 1,free f 1,free

(2.9)

(2.10)

Due to the high quality factors Q of the flexural modes in air the shift of the resonant frequencies caused by air damping is negligible. Here the resonant frequencies of the clamped-free beam are sometimes called “free” resonances. In this case “free vibration” is not meant as the opposite of “forced vibration” but relates to the end condition of the beam. The theory predicts a certain ratio of the free flexural resonant frequencies regardless of the cross section and the length of the beam, for example: f 2,free (4.6941)2 = = 6.27 . f 1,free (1.8751)2

(2.11)

The first ten solutions αn L for the clamped-free beam without damping (ηAir = 0) are listed in Table 2.1. Note that the bending modes are not equidistant in frequency. According to the dispersion equation, the frequency ω is not proportional to the wave number α, but ω ∼ α2 . For the higher modes the frequency interval to the next higher mode increases with the square of the frequency. The higher bending modes are no harmonics of the fundamental frequency.


47

2.2.2 The Point-mass Model It is very common in atomic force microscopy to approximate the fundamental flexural mode of the clamped-free beam by a harmonic oscillator (Fig. 2.2). The constants of a point-mass model are chosen such that the resonant frequency of the harmonic oscillator ω0 equals the first resonant frequency of the beam ω1,free : ! kC ≡ ω0 (2.12) ω1,free = m∗ The static spring constant kC of the cantilever for forces acting at x = L is: kC =

3E I Eb3 w = L3 4L 3

(2.13)

The second expression holds for rectangular beams, where w and b are the width and thickness, respectively. The effective mass m ∗ is now calculated according to m∗ =

kC 2 ω1,free

=

3ρLwb 1 ≈ m 4 (α1,free L) 4

(2.14)

Here m is the real mass of the cantilever. The motion of the beam at the free end (x = L) can now be described by the equation of motion of a point-mass m ∗ : m ∗ y(L, t) + m ∗ ηAir y(L, t) + kC y(L, t) = 0 ¨ ˙

(2.15)

Interaction between the sensor tip located at x = L and a sample surface gives rise to forces, which can be represented by a second spring with the stiffness k∗ . The second spring leads to a change of the resonant frequency of the system according to [44, 68, 69]: " ! k∗ k ∗ + kC ω= = ω 1 + (2.16) 0 m∗ kC It has been shown that the point-mass model fails to predict quantitatively correct amplitudes and resonant frequencies when the contact stiffness k∗ is of the same order of magnitude or larger than the static spring constant kC of the sensor [70]. Nevertheless, the fundamental frequency of the clamped-free beam ω0 or f 0 = ω0 /2π, the static spring constant kC , and the effective mass m ∗ are very useful to characterize a sensor.

Fig. 2.2. Optical micrograph of an AFM cantilever and point-mass model

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2.2.3 Experiments with Clamped-free Beams With respect to application it is important to examine how well real sensors correspond to the flexural beam model. It is helpful to use a calibrated optical interferometer with a bandwidth of several MHz to examine as many of the higher modes of the cantilever as possible [44, 71]. For example, Hoummady et al. examined higher flexural modes of AFM cantilevers interferometrically in order to use them for imaging [72] and Cretin and Vairac used an optical interferometer to measure the vibration of the sensor in SMM [36]. Figure 2.3 shows a set-up with a Michelson-heterodyne interferometer [73]. The chip at which the cantilever is fixed is glued to an ultrasonic transducer (shear wave transducer V155, 5 MHz center frequency, Panametrics-NDT, Waltham, MA, USA). The transducer is excited with a function generator (model 33120A, Agilent Technologies). Due to the high quality factors Q of its modes, peak-to-peak voltages of several volts are sufficient to excite vibration amplitudes of the beam of several nm or even several 10 nm. Though a longitudinal wave transducer excites flexural waves more effectively, the shear wave transducer was chosen because it excites as many types of modes as possible. Torsional resonances and lateral modes could be examined in this way, too. Depending on the dimensions and material constants of the beam, it can happen that for example a torsional mode and a flexural mode have almost the same frequency. Such a situation leads to mode coupling [71, 74] and consequently to difficulties in quantitative evaluation. The He − Ne Laser beam of the interferometer is focused onto the cantilever with a microscope objective. The resulting spot diameter is only a few µm. The cantilevers are mounted on a motor-driven x-y table to scan their surface with the help of two stepper motors.

Fig. 2.3. Set-up of the optical interferometer to measure the amplitudes of the clamped-free cantilever vibration modes. The chip of the cantilever is glued to an ultrasonic transducer which provides an excitation of vibration up to a few MHz. The laser beam of the interferometer is focused onto the cantilever to a spot size of a few µm. Step motors controlled by a computer scan the surface of the cantilever


49

Knowledge of the two-dimensional mode shape helps to identify modes which appear as unknown peaks in the resonance spectrum. The interferometer signal is read with a digital oscilloscope, amplitude and phase of the vibration are obtained by fast Fourier transformation. A three dimensional visualization of such a measurement is shown in Fig. 2.4. A spectral analysis showed that the beams made of silicon single crystal behaved linearly at vibration amplitudes of a few ten nm, i.e. they vibrated only at the frequency of excitation. After measurement of the fundamental mode the higher resonant frequencies of the clamped-free beam can be predicted with the frequency ratios in Table 2.1. Usually the values calculated in this way are close enough to the experimental frequencies to detect the higher modes easily. In detail, however, there are systematic deviations from the theory which depend on the type of beam. The deviations from theory can be visualized by dividing the frequencies of the higher modes by the theoretical ratio f n,free / f 1,free and plotting them as a function of mode number. In case of perfect correspondence of the sensor with the model such a plot would provide a horizontal line. Figures 2.5a and b show the resonant frequencies of a number of silicon beams. Typical dimensions of the beams examined here are listed in Table 2.2 The absolute resonant frequencies of the beams vary because of thickness variations of the

Fig. 2.4. Bending modes of a cantilever made of single crystal silicon measured with an optical interferometer in air. The frequency of the first mode was below detection limits of the interferometer. Mode n = 10 was not detected for unknown reasons. The dimensions of the beam were 1.61 × 45 × 440 µm3 (thickness × width × length) and the static spring constant was kC = 0.094 N/m according to the manufacturer

50

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Fig. 2.5. (a) Resonant frequencies of the first four flexural modes of 23 cantilevers from 6 different wafers (type FM, Nanosensors). (b) Resonant frequencies of the first three flexural modes of 19 cantilevers from two different wafers (type NCL, Nanosensors). The frequencies of the higher modes were divided by the theoretical frequency ratio f n / f 0 . Modes of each individual beam are connected by a line. The same symbol and line type was used for beams from the same wafer. In case of perfect correspondence to the flexural beam model, the frequencies would be connected by horizontal lines. The strong deviations in (b) were caused by multiple peaks Table 2.2. Typical data of single crystal silicon cantilevers used here as provided by the manufacturer (Nanosensors, NanoWorld AG, Neuchatel, Switzerland). The cantilevers are of approximately rectangular shape. The cross-section of the beams is trapezoidal Type of beam

Length L [µm]

Width w [µm]

NCL FM CONT

215–235 215–235 440–460

20–45 20–35 42.5–57.5

Thickness b Spring constant Resonant frequency [kHz] [µm] kC [N/m] 6–8 2–4 1–3

21–98 0.5–9.5 0.02–0.77

146–236 45–115 6–21

silicon wafers used for manufacturing. Furthermore, for one type of beam (FM) the frequencies increase more strongly, while for another type (NCL) they increase less with the mode number than predicted by the model. These systematic deviations are caused by details which are not considered in the simple analytical model. For example the beams are made of mechanically anisotropic material, the clamping of the fixed end is not infinitely stiff and symmetric, and the cross-section of the beams is not constant. The influence of these details on the resonant frequencies can be examined theoretically by finite-element calculations [75]. For example Hurley et al. examined the deviations caused by a triangular end of the beam (dagger shape) [64]. Another practical problem which often occurs when resonant frequencies are measured in an AFM is double peaks or even multiple peaks. Mode coupling can only happen if two modes have frequencies close to each other. Therefore it is likely that most of the unwanted maxima in the spectrum are caused by maxima which are already present in the spectrum of the vibration which excites the beam. The spectrum of the forced linear vibration is a product of the vibration spectrum of the chip and the spectrum of the cantilever. The chip of the cantilever has typical dimensions of 4 × 2 × 0.5 mm3 (length × width × thickness) and is therefore a mechanical structure with numerous resonant frequencies. The exact spectrum depends on how


51

the chip is clamped [75]. In the AFAM-mode ultrasonic waves transmitted through a sample generate the surface vibration, which means that in this case the spectrum of the exciting signal depends on the material, the dimensions, and the clamping of the sample.

2.3 Contact Forces as Linear Springs and Dashpots When the sensor tip is in contact with a sample surface, the tip–sample forces change the end conditions of the beam. Figure 2.6 shows the situation schematically. The tip–sample forces are a nonlinear function of the tip–sample separation and they can also be hysteretic. Here it is assumed that the vibration amplitudes are so small that the forces can be represented by a system of linear springs and dashpots. In case of bending modes one can treat the problem in only two dimensions. There are forces normal to the surface represented by the normal contact stiffness k∗ and the contact damping γ and forces lateral to the surface represented by the lateral contact ∗ stiffness kLat and a lateral contact damping γLat . For technical reasons the cantilever is tilted with respect to the surface by an angle ϕ (11◦ to 15◦ ), i.e. the coordinate system of the surface is rotated by ϕ with respect to the coordinate system of the cantilever. The normal and lateral contact stiffness can be calculated from contact mechanics models [74,76,77]. Depending on the material of which tip and surface are made, all sorts of physical forces, including magnetic and electrostatic forces, can contribute to the contact stiffness. The different force models will not be discussed in detail here, but it will be assumed that the surface preparation was such that elastic forces prevail. Though in principle all types of forces can be measured by AFAM, predominance of elasticity can be ensured by choosing high enough static loads applied by the cantilever (several 100 nN).

Fig. 2.6. The sensor tip is in contact with the sample surface. A contact area of radius aC forms between the tip and the surface and the tip senses forces lateral and normal to the sample surface which are modeled by linear springs and dashpots

52

U. Rabe

The Hertzian model describes the contact between two nonconforming elastic bodies of general anisotropy [78]. In the simplest case the bodies are mechanically isotropic, the sample is considered as flat and the sensor tip is represented by a sphere with a radius R. If a normal force Fn acts onto the sphere, a contact radius aC forms: (2.17) aC = 3 3Fn R/4E ∗ . If the adhesion forces are so small that they can be neglected, the normal force Fn is given by the static deflection of the cantilever multiplied with the spring constant of the cantilever. The sum of the indentation in the contacting bodies, δn , i.e. the amount the two bodies approach is given by: ! 9Fn2 3 δn = , (2.18) 16RE ∗2 The normal contact stiffness k∗ is: 3 k∗ = 2aC E ∗ = 6E ∗2 RFn . Here, E ∗ is the reduced Young’s modulus of the contact which is given by

1 − νS2 1 − νT2 1 = + , E∗ ES ET

(2.19)

(2.20)

where E S , E T , νS , νT , are Young’s modulus and Poisson’s ratio of the surface and the tip, respectively. After AFAM experiments, the tip shape often deviates from that of a sphere [79]. In this case the shape of the tip can be described more generally by a body of revolution. It has been shown that for axisymmetric indenters on elastically isotropic half spaces the contact stiffness k∗ , i.e. the derivative of the applied load Fn with respect to the indention depth, δn , generally obeys the equation [80]: k∗ =

dFn 2 √ ∗ =√ SE dδn π

(2.21)

Here S = π × aC2 is the contact area. Therefore the relation E ∗ = k∗ /(2aC ) ,

(2.22)

which can be derived from the Hertzian model, still holds in this more general case. These formulas are derived on the assumptions of a frictionless contact. The Hertzian model is only valid if the contact area is small compared with the dimensions of the contacting bodies and their radii of curvature [78], which means that the contact radius must be smaller than the tip radius aC R. Linear elastic theory is only valid if the mechanical stresses remain small enough. In AFM these two conditions are easily violated when a sensor tip with a radius of a few nm contacts a metallic or ceramic sample surface.


53

The contact radius defines the lateral resolution in contact-resonance spectroscopy. A typical contact radius in AFM ranges from several nm up to several tens of nanometers, depending on the tip radius and the elasticity of the tip and the sample. Many polycrystalline materials like metals and ceramics, which appear mechanically homogeneous on a macroscopic scale, show a local variation in elastic constants for a scanning probe microscope because the tip senses the individual grains within the polycrystalline aggregate. Each grain represents a small single crystal. This means that imaging by AFAM can only be explained when the sample surface is no longer treated as an isotropic material. Furthermore AFM sensor tips made of single crystalline silicon are not elastically isotropic, and this holds for other tip materials as well. Mechanically anisotropic materials are described by more than two elastic constants. In the most general case of two non-conforming bodies of general shape and anisotropy, the contact area is elliptical [78]. The reduced Young’s modulus of the contact is a function of the indentation δn , contains combinations of the elastic constants of tip and sample and cannot be separated into a sum of a contribution from tip and surface like in (2.20). Vlassak and Nix examined the indentation of a rigid parabolic punch in an anisotropic surface [81]. They showed that the contact area remains spherical if a three- or fourfold rotational symmetry axis perpendicular to the boundary exists. In this case (2.19) and (2.21) remain valid if the isotropic reduced elastic modulus E/(1 − ν2 ) is replaced by an indentation modulus that is calculated numerically from single crystal elastic constants [82]. Equation (2.20) is replaced by: 1 1 1 = + E∗ MS MT

(2.23)

where MS and MT are the indentation modulus of the sample and the tip, respectively. The required symmetry holds for silicon sensor tips which are oriented in (001) crystallographic direction. Because of its fourfold symmetry the tip does not alter the rotational symmetry if it is in vertical contact with an isotropic body or with a sample which also has a fourfold rotational symmetry axis along the tip and indentation axis. Even for bodies which do not have a three- or fourfold symmetry axis (2.23) can be used as a first approximation. The error made by application of this equation depends on the anisotropy and can be estimated [83]. The tilt angle ϕ of the cantilever causes tip–sample forces tangential to the surface when the surface is moved in its normal direction. Additionally, the flexural vibrations cause an angular deflection ∂y/∂x at the sensor-tip position and consequently a lateral deflection h∂y/∂x of the sensor tip apex. Tangential forces were treated by Mindlin [78]. The lateral contact stiffness depends on the effective shear stiffness G ∗ of the contact: dFL ∗ kLat = = 8aC G ∗ (2.24) dδL 1 2 − νS 2 − νT = + (2.25) ∗ G GS GT where δL is the lateral contact deflection, and G S and G T are the shear modulus of the sample and the tip, respectively. For isotropic bodies the ratio between the ∗ normal contact stiffness k∗ and the lateral contact stiffness kLat is independent of the

54

U. Rabe

normal force Fn : ∗ 8aC G ∗ 4G ∗ kLat = = . k∗ 2aC E ∗ E∗

(2.26)

As Mazeran and Loubet pointed out [25], for E T E S ∗ 2(1 − νS ) kLat ≈ . ∗ k 2 − νS

(2.27)

Taking a range of Poisson’s ratio from 0.1 for diamond to 0.5 for rubber, the ratio ∗ /k∗ varies from 2/3 to 18/19 with an of lateral and normal contact stiffness kLat average value of 0.85 [25]. Like in the case of the normal contact stiffness, it will be necessary in future to calculate shear stiffness, taking into account the mechanical anisotropy of the contacting bodies. An important question for nondestructive testing is whether techniques like AFAM or UAFM are able to detect sub-surface features. The typical frequency f of ultrasonic excitation is 100 kHz to 10 MHz. The velocity of longitudinal acoustic waves in the solids under examination ranges typically from v = 1 mm/µs (polymers) to v = 10 mm/µs (ceramics). In this case the acoustic wavelength λ = v/ f ranges from 100 µm to 10 cm. Consequently the acoustic wavelength is larger than the scan width of the AFM and orders of magnitude larger than the contact radius. This confirms that the vibrating tip can be seen like a dynamic indenter and the penetration depth of the ultrasonic techniques is given by the decay of the mechanical stress field in the sample. According to the Hertzian contact model the decay length is several multiples of the contact radius (see Fig. 2.7). Therefore it is possible to measure the film thickness with AFAM or UAFM [57], provided the films are thin enough. Yaralioglu et al. calculated the contact stiffness of layered materials using the impedance of a mechanical radiator [58]. They examined thin films of photo resist, W, Al, and Cu on silicon single crystal. They were able to show that in the low frequency limit ( f → 0) the impedance method provides the same results for the contact stiffness k∗ as the Hertzian contact model and that this method is well suited to

Fig. 2.7. Mechanical stress field in a Hertzian contact as a function of penetration depth z into the surface. The average normal pressure pn in the contact area is given by pn = Fn /πaC2 , where Fn is the normal force and aC is the contact radius. The compressional stresses σz and σr have their maximum at the surface and the principal shear stress τ1 reaches its maximum in the sample [78]


55

calculate the influence of subsurface defects on the contact stiffness [84]. Tsuji et al. observed subsurface dislocation movement in graphite with UAFM. They calculated the influence of a subsurface layer with lower Young’s modulus on the contact stiffness using finite elements [85–88]. The penetration depth of the mechanical stress field can be enhanced by increasing the contact radius, however, this causes a loss in lateral resolution. According to Fig. 2.7 the compressional stress has its maximum at the sample surface, while the shear stress reaches its maximum below the surface. Therefore techniques which exploit lateral vibrations like torsional contact-resonance spectroscopy should be very sensitive to subsurface defects.

2.4 Characteristic Equation of the Surface-coupled Beam In the linear model shown in Fig. 2.8 the forces between the sensor tip and the surface are represented by a system of springs and dashpots. The length of the cantilever from the clamped end to the free end is L. The sensor tip is located at position L 1 and L 2 = L − L 1 is the distance between the tip position and the free end. Again, one seeks a harmonic solution in time. Two sections of the cantilever with individual solutions for the mode shapes, y1 (x1 ) and y2 (x2 ), respectively, are defined. The function y1 (x1 ) describes the amplitude of the section which begins at the clamped end at x = 0 and ends at the tip position at x = L 1 . For simplicity one can use y(x) instead of y1 (x1 ), because the two functions are identical on the first section of the beam. The function y2 (x2 ) begins at the free end of the beam and ends at the tip position at x2 = L 2 . At the clamped end of the beam the deflection and the slope of

Fig. 2.8. Linear model of the cantilever when the sensor tip is in contact with a sample surface. The tip–sample interaction forces are represented by springs and dashpots. The cantilever is tilted by an angle ϕ with respect to the surface. The sensor tip of height h is located at a position L 1 on the length axis of the cantilever. The distance of the tip to the free end of the cantilever is L 2

56

U. Rabe

the beam must be zero, while at the free end of the beam the forces and moments have to be zero. The end conditions for y(x) and y2 (x2 ) are therefore: ⎧ 2 ⎧ ∂ y ⎪ ⎪ ⎨ y(x) = 0 ⎨ 2 =0 . (2.28) x = 0 : ∂y(x) y = L : ∂x3 ⎩ ⎪ ∂ y =0 ⎪ ⎩ = 0 ∂x ∂x 3 The solution (2.3) and its derivatives together with the foregoing end conditions yield A1 = A3 = 0 for y(x) and A2 = A4 = 0 for y2 (x2 ). The shape functions therefore have the form: y(x) = A2 (cos αx − cosh αx) + A4 (sin αx − sinh αx) y2 (x2 ) = A1 (cos αx2 + cosh αx2 ) + A3 (sin αx2 + sinh αx2 ) .

(2.29)

The partial solutions y(x) and y2 (x2 ) must be coupled continuously at the tip position at x = L 1 i.e. at x2 = L 2 . ⎧ ⎨ y(x) = y2 (x2 ) (2.30) x = L 1 or x2 = L 2 : ∂y(x) ∂y2 (x2 ) ⎩ =− ∂x ∂x2 The negative sign in the equation for the derivatives appears because the x2 -axis is defined in the negative direction of the x-axis. This direction was only chosen for convenience of calculation. Note that the x-axis (and the x2 -axis) of the cantilever is not parallel to the surface when the cantilever is tilted by an angle ϕ as shown in Fig. 2.8. In this text the terms “y-axis” and “x-axis” always correspond to the coordinate system of the cantilever. The moments and the forces on the sensor tip lead to further boundary conditions at the coupling position. One can first consider a simplified case where the x-axis of the cantilever is parallel to the sample surface ∗ = 0, and γ = 0). The and only forces normal to the surface are acting (ϕ = 0, kLat boundary condition for the shear forces at x = L 1 is in this case: EI

∂3 y ∂ 3 y2 ∂y(L 1 , t) . + E I 3 = k∗ y(L 1 , t) + γ 3 ∂x ∂t ∂x2

(2.31)

The solution looked for is a harmonic wave of the form y(x, t) = y(x) exp(iωt). The time derivatives can therefore be calculated and reformulated using the dispersion relation (2.6) neglecting the air damping. This leads to the boundary condition 1 ∗ ∂ 3 y ∂ 3 y2 + = · k y(L 1 , t) + γ iωy(L 1 , t) 3 3 ∂x EI ∂x2 " $ % k∗ 1 2 = y(L 1 , t) + γ iα . EI E IρA

(2.32)

A contact function φ(α) is defined, which contains contact stiffness and contact damping:


φ(α) = 3

57

k∗ + i(αL 1 )2 p . kC

(2.33)

The spring constant kC of the cantilever (2.13) was used. The dimensionless damping constant p is defined as [70]: L 1γ 3γ L1 L 1 (1.875)2 γ p= √ = = . L (1.875)2 m ∗ ω0 L mω0 E IρA

(2.34)

Substituting the contact function φ(α) in (2.31), the boundary condition now becomes φ(α) ∂ 3 y ∂ 3 y2 + = 3 y(L 1 ) . 3 ∂x 3 ∂x2 L1

(2.35)

In the same way a lateral contact function φ(α)Lat is defined: φLat (α) = 3

∗ kLat + i(αL 1 )2 pLat kc

L 1 γLat pLat = √ . E IρA

(2.36)

The angle ϕ of the cantilever with relation to the surface causes cross-coupling between lateral and normal signals. Forces Fx in length-direction of the beam acting on the sensor tip give rise to a moment M = h Fx at the end of the beam. The angular deflection ∂y/∂x of the beam at the tip position causes a deflection in x-direction h∂y/∂x of the tip apex. In summary, this leads to the following boundary conditions for the forces and the moments at the tip position: ⎧ 2 ∂y(x) ∂ y(x) ∂ 2 y2 (x2 ) ⎪ ⎪ ⎪ − X(α)y(x) − = −T(α) ⎪ 2 2 ⎪ ∂x ∂x ∂x ⎨ 2 x = L 1 or x2 = L 2 : . ⎪ 3 3 ⎪ ⎪ ∂ y(x) ∂ y2 (x2 ) ∂y(x) ⎪ ⎪ + = U(α) · y(x) + X(α) ⎩ ∂x 3 ∂x ∂x23 (2.37) The auxiliary functions T , X and U are defined as follows: T(α) =

h2 h2 φ(α) sin2 ϕ + 3 φLat (α) cos2 ϕ 3 L1 L1

X(α) =

h sin ϕ · cos ϕ [φLat (α) − φ(α)] L 31

U(α) =

1 1 φ(α) cos2 ϕ + 3 φLat (α) sin2 ϕ . 3 L1 L1

(2.38)

The four boundary conditions at the coupling position are used to determine the four unknown constants in the shape functions y(x) and y2 (x2 ). The solutions of this eigenvalue problem define an infinite set of discrete wave numbers αn . After some

58

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pages of calculation one obtains the following characteristic equation: Ω(α) ≡ S4 + S3 T(α) + S2 X(α) + S1 U(α) + S0 T(α)U(α) − X 2 (α) = 0 (2.39) where S0 , S1 , S2 , S3 , and S4 stand for the following terms: S0 = (1 − cos αL 1 cosh αL 1 )(1 + cos αL 2 cosh αL 2 ) S1 = α [−(1 − cos αL 1 cosh αL 1 )(sin αL 2 cosh αL 2 − sinh αL 2 cos αL 2 ) + (1 + cos αL 2 cosh αL 2 )(sin αL 1 cosh αL 1 − sinh αL 1 cos αL 1 )] S2 = 2α2 [sin αL 1 sinh αL 1 (1 + cos αL 2 cosh αL 2 ) + sin αL 2 sinh αL 2 (1 − cos αL 1 cosh αL 1 )] S3 = α3 [(sin αL 1 cosh αL 1 + sinh αL 1 cos αL 1 )(1 + cos αL 2 cosh αL 2 ) − (sin αL 2 cosh αL 2 + sinh αL 2 cos αL 2 )(1 − cos αL 1 cosh αL 1 )] S4 = 2α4 (1 + cos αL cosh αL)

(2.40)

The last term in the characteristic equation (2.39) can be further simplified by substituting the definitions of the auxiliary functions: T(α)U(α) − X 2 (α) =

h2 φ(α)φLat (α) L 61

(2.41)

2.4.1 Discussion of the Characteristic Equation In order to understand the characteristic equation of the surface-coupled beam, it is helpful to consider simple cases. If for example all the spring constants and the dashpot constants are set to zero, the auxiliary functions T(α), X(α), and U(α) become zero too. This means that all terms except S4 vanish in Ω(α) and the characteristic equation reduces to (2.7) which is the equation of the clamped-free beam. As a next step one can consider only a normal spring k∗ , the lateral spring constant and the contact damping are still considered to be zero. This means that the contact function and the lateral contact function become φ(α) = 3k∗ /kC and φLat (α) = 0, respectively. If in addition the beam is not tilted (ϕ = 0), and if the tip is located at the end of the beam (L 1 = L and L 2 = 0), the auxiliary functions X(α) and T(α) vanish and U(α) simplifies to U(α) = φ(α) = 3k∗ /kC . The characteristic equation for a clamped-spring-coupled beam [13, 44] then follows:


(αL)3 (1 + cos αL cosh αL) + 3

59

k∗ (sin αL cosh αL − sinh αL cos αL) = 0 kC (2.42)

Two examples of the shape function y(x) of this case are shown in Fig. 2.9. Note that (2.42) contains the characteristic equation of the clamped-free beam and the characteristic equation of the clamped-pinned beam, combined by the factor (αL)3 kC /3k∗ . The clamped-pinned case is reached when the normal contact stiffness k∗ goes to infinity. Starting from the clamped-pinned case one can now add a lateral spring which causes feedback forces proportional to the angle of the end of the beam. A lateral spring fixed to the sensor tip is equivalent to a torsional spring which is fixed directly to the end of the beam [25]. The characteristic equation of this case can be obtained by dividing (2.39) by U(α) and subsequently considering the case U(α) → ∞. From the remaining terms S1 + S0 T(α) = 0 one obtains the following characteristic equation: 3

∗ h 2 kLat (1 − cos αL cosh αL) + αL (sin αL cosh αL − sinh αL cos αL) = 0 L 2 kC (2.43)

Finally, when the lateral spring constant too goes to infinity, one obtains a cantilever which is clamped at both ends. The simplified cases of the characteristic equation discussed above are shown in Fig. 2.9. Furthermore, the shapes of the first (n = 1) and the third (n = 3) mode are shown. In the spring-coupled cases, the mode shapes change continuously with the contact stiffness, Fig. 2.9 shows one example of normal contact stiffness and one of lateral contact stiffness. Other special cases of the characteristic equation (2.39) were published in literature [44, 60, 89]. The characteristic equation for a beam with normal and lateral springs at the end at x = L was first published by Wright and Nishiguchi. [89]. The continuous change in mode shape as a function of contact stiffness is accompanied by a continuous change in resonant frequency. The lines labeled “A” in Fig. 2.10 show the resonant frequencies of the first four flexural modes as a function of normalized contact stiffness k∗ /kC for the clamped-spring-coupled beam (2.42). In this case the contact-resonance frequencies of the n’th mode are always lower than the free resonant frequency of the subsequent mode (n + 1). The gaps in the spectrum between the clamped-pinned mode and the next clamped-free mode are shown as grey areas. If the sensor tip is moved away from the end of the beam (L 2 > 0), the maximum possible frequency shift increases. The case of a relative tip position L 2 /L = 5% is shown as lines “B” in Fig. 2.10. If it happens for a mode that the distance of its last vibration node to the end of the beam becomes equal to or larger than the distance of the tip to the end, L 2 , this mode merges with the next higher mode and the situation becomes difficult to survey. As the wavelength decreases with increasing mode number, there are always higher modes for which this limit is surmounted. In the experiments it is therefore better to evaluate only modes for which the wavelength is greater than L 2 . A torsional spring at the end of the beam (lines “C” in Fig. 2.10) also increases the shift of the resonant frequencies to higher values, such that the gaps in the spectrum disappear. The approximations for large n in Table 2.1 show that

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Fig. 2.9. Characteristic equations for some selected simple cases of the cantilever in flexural vibration. The shape functions y(x) for the first flexural mode (n = 1) and the third mode (n = 3) are shown

the frequency of the (n + 1)’th clamped-free mode equals the frequency of the n’th clamped-clamped mode. In a real experiment there will always be normal and lateral forces at the same time. However, even if the lateral contact stiffness is of the same size as the normal contact stiffness, its influence on the cantilever vibration is smaller than the influence of the normal stiffness, due to the factor h 2 /L 2 which appears in the characteristic equation. The lateral forces are transformed into a moment by the lever-arm h of the sensor tip which is shorter than the cantilever length L. Typical sensor tip heights h are 5–15 µm, typical cantilever lengths L are 100–400 µm, i.e. h 2 /L 2 ranges from 0.15 to 0.0125. As a consequence, the lateral contact stiffness influences the frequency little if the normal contact stiffness k∗ is low compared to the spring constant of the cantilever kC . At higher contact stiffness k∗ the frequency shift comes closer to its upper limit, and ∗ becomes remarkable. It was first the influence of the lateral contact stiffness kLat pointed out by Mazeran and Loubet [25] that the influence of the lateral contact ∗ stiffness kLat is high when the ratio of the normal contact stiffness to the spring constant of the cantilever k∗ /kC is high. As a consequence, lateral forces may not be neglected if contact resonances are measured with soft sensor beams on stiff surfaces.


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Fig. 2.10. Shift of the normalized resonant frequencies f n / f 0 of the first four flexural modes as a function of normalized contact stiffness k∗ /kC . The lines labeled “A” show the case when only a spring normal to the sample surface is fixed to the cantilever. In case “B” the normal spring is fixed slightly away from the free end of the cantilever. In case “C” lateral tip–sample forces act additionally to the normal forces. A constant ratio between lateral and normal contact stiffness ∗ /k ∗ = 0.85) was assumed (kLat

Dupas et al. [60, 90] calculated the characteristic equation for the complete model shown in Fig. 2.8 with an additional spring at the clamped end of the cantilever. They examined the influence of the position and of the length of the sensor tip on the frequency shift of the first flexural mode. Arinero and Lévêque [26] calculated the contact vibrations of a beam with finite elements and examined the influence of the tip among other parameters. Wu et al. examined the influence of the length of the sensor tip on the frequency shift of the vibration modes [91]. 2.4.2 Influence of an Additional Mass An additional point-mass m L on the cantilever shown schematically in Fig. 2.11 causes an inertial force proportional to −ω2 m L . The influence of the mass can be found out easily if it is located at the tip position x = L 1 on the length axis of the beam [44, 92]. In this case the auxiliary function U(α) (2.38) in the boundary conditions (2.37) changes. If the center of gravity of the mass is displaced by ε from the middle plane of the beam, a moment proportional to −ε2 ω2 m L [92] has to be added in the auxiliary function T(α). The form of the characteristic equation as well as the auxiliary function X(α) remain unchanged. With the help of the dispersion relation (2.6) one obtains: ε2 m L h2 2 2 (φ(α) sin ϕ + φ (α) cos ϕ) − (αL 1 )4 Lat L 31 L 31 m 1 1 mL U(α) = 3 (φ(α) cos2 ϕ + φ Lat (α) sin2 ϕ) − 3 (αL 1 )4 L1 L1 m T(α) =

(2.44)

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Fig. 2.11. An additional point mass m L is located on the cantilever at x = L 1 . The mass is displaced by ε from the middle plane of the beam indicated by the chain dotted line

It was assumed that ρAL 1 ≈ m, where m is the mass of the cantilever. This is a good approximation if the sensor tip is located close to the end of the beam. Without lateral forces, with a tip at the end of the beam (L 2 = 0), without tilt (ϕ = 0), and without damping one obtains the characteristic equation derived by Muraoka [92]: (αL)3 [(1 + cos αL cosh αL) − Lε 2 mmL (αL)3 ∗ (sin αL cosh αL + sinh αL cos αL)] mL k 3 − = 0 (2.45) (αL)4 + kC m sin αL cosh αL − sinh αL cos αL 2 − Lε 2 mmL (αL)3 (1 − cos αL cosh αL) 2

This equation was used to calculate the frequency shift in Fig. 2.12a. With increasing additional mass the cantilever behaves more and more like a point-mass oscillator. The frequency shift of the point-mass oscillator is according to (2.16): " k∗ f = 1+ (2.46) f0 kc This behavior is shown in Fig. 2.12 as a dashed line (p1). In case of an additional mass at x = L the point-mass model predicts the following frequency shift: " ! −1 f k∗ mL = 1+ · 1+4 (2.47) f0 kc m An additional mass increases the effective mass of the beam and lowers its first free resonant frequency. The frequency shift according to the point-mass oscillator with additional mass (2.47) is labeled (p2) in Fig. 2.12. If the additional mass is high enough (m L /m = 5) there are sections of the dispersion curves of the beam which coincide with the point-mass model. If contact-resonance frequencies are measured


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Fig. 2.12. Influence of an additional mass m L on the frequency shift of the contact-resonance frequencies (a) without lateral forces ( displacement of the mass ε = 0, m L /m = 5, position of the mass L 1 /L = 1) [92], (b) with lateral forces (position L 1 /L = 0.9, ratio of lateral contact stiffness to normal contact stiffness k∗Lat /k∗ = 0.85, beam angle ϕ = 11◦ , cantilever length divided by tip height L/h = 15, mass displacement ε = 0, m L /m = 5). The dashed lines show the frequency shift of the equivalent point-mass oscillator without additional mass (p1) and with additional mass m L (p2)

and evaluated which lie on these sections, the simple point-mass formulas can be used to calculate the normalized contact stiffness k∗ /kC from the contact-resonance frequencies [92]. Muraoka observed an improved contrast when he used a cantilever with a W-particle glued onto its end for imaging in AFAM [93]. The characteristic equation with the additional mass can also be used to calculate the influence of the sensor tip on the vibration modes. For most commercial cantilevers, however, the influence of the mass of the tip is negligible. 2.4.3 Roots of the Characteristic Equation with Damping In the discussion of the characteristic equation the influence of the contact damping has not been considered so far. In a simplified case without lateral forces and tilt of the beam, the characteristic equation becomes [70]: ∗ k 3 2 (αL) (1 + cos αL cosh αL) + 3 + i(αL) p kC × (sin αL cosh αL − sinh αL cos αL) = 0 (2.48) The solutions αn L of this equation are complex numbers. The influence of the contact damping on the real part and the complex part of the wave number is shown in Fig. 2.13 [70]. The shapes of the first three modes in the case of a ratio of the normal contact stiffness to the spring constant of the cantilever k∗ /kC = 10 and without contact damping ( p = 0) are shown as insets. With increasing contact damping p mode 1 shows stiffening behavior, i.e. the frequency increases, while

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Fig. 2.13. Influence of the dimensionless contact damping p on the first three modes in the case k∗ /kC = 10. The real part (filled symbols) and the imaginary part (open symbols) of the eigenvalues αn L are plotted as a function of p

mode 2 shows softening behavior. Both modes have a maximum of the imaginary part of the wave number in the frequency interval where the real part of αn L changes strongly. Note that the horizontal scale in Fig. 2.13 is logarithmic. The stiffness of mode 2 decreases strongly, the imaginary part of αn L increases at first, then it decreases again and in this part of the curve the absolute values of the real and the imaginary parts are identical. A closer experimental and theoretical study of how the vibration modes behave as a function of interaction damping could lead to new techniques to measure tip–sample interaction damping or even internal friction at sample surfaces.

2.5 Forced Vibration Three types of excitation shown in Table 2.3 play an important role in AFM. (a) It is possible to vibrate the clamped end of the cantilever harmonically in its thickness direction with an amplitude u c (t) = u 0 eiωt (UAFM). (b) The sample surface is set into normal vibration with an amplitude u S (t) = u 0 eiωt (AFAM). (c) A concentrated harmonic force F(t) = F0 eiωt – for example a magnetic force – acts directly on the cantilever. The excitation signal is added to the boundary conditions. Afterwards the general solution and its derivatives are substituted in the boundary conditions and the constants A1 , A2 , A3 and A4 are calculated as a function of excitation frequency. Table 2.3 shows a simplified situation with only a spring and a damper at the end of the beam. The boundary conditions are in the AFAM mode, case (b): ⎧ ∂2 y ⎪ ⎪ ⎨ 2 =0 y = a0 (2.49) x = 0 : ∂y x = L : ∂x3 =0 ⎪ φ(α) ∂ y ⎪ ⎩ ∂x = 3 (y − u 0 ) ∂x 3 L

Table 2.3. Boundary conditions and solutions for the forced flexural vibration of a cantilever beam. Three techniques of excitation are treated: (a) Vibration of the clamped end of the cantilever, (b) sample surface vibration, (c) modulated concentrated force acting at x = L

2 Atomic Force Acoustic Microscopy 65

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From the first two boundary conditions one obtains A1 = A3 = 0, and with the third boundary condition the shape function becomes: y(x) = A2 (cos αx − cosh αx) + A4 (sin αx − sinh αx)

(2.50)

From the last boundary condition the constants A2 and A4 are calculated: A2 = −

u 0 φ(α) (sin αL + sinh αL) 2N(α)

A4 =

u 0 φ(α) (cos αL + cosh αL) (2.51) 2N(α)

The denominator of the two constants in the foregoing equations is the same: N(α) = (αL)3 (1 + cos αL cosh αL) + φ(α)(cosh αL sin αL − sinh αL cos αL) (2.52) The complex amplitude of vibration is obtained by substituting (2.51) and (2.52) in (2.50): y(x) =

u 0 φ(α) [−(sin αL + sinh αL)(cos αx − cosh αx) 2N(α) + (cos αL + cosh αL)(sin αx − sinh αx)]

(2.53)

The detector, for example the laser spot of an interferometer, is usually located at the end of the beam. In this case one only needs the amplitude at x = L, and (2.53) simplifies to: y(L) = u 0

sin αL cosh αL − sinh αL cos αL N(α)

(2.54)

Commercial atomic force microscopes are generally equipped with beam-deflection sensors. Their signal is proportional to the angle of the cantilever, given by the derivative of y(x). At the end of the beam at x = L the derivative is: αLu 0 sin(αL) sinh(αL) ∂y = ∂x x=L L N(α)

(2.55)

The boundary conditions and shape functions for the three mentioned cases of excitation are shown in Table 2.3. The amplitudes and slopes in (2.54) and in (2.55) are complex numbers. Real amplitude and phase Φ of the signals are calculated here using: |y(x)| =

Im[y(x)]2 + Re[y(x)]2

Φ(x) = − arctan

Im[y(x)] Re[y(x)]

(2.56)

The normalized vertical amplitude of the sensor tip dY 0 and the normalized lateral amplitude d X0 of the sensor tip are defined according to: ∂y (2.57) dY 0 = |y(L)| /u 0 d X0 = h (L) /u 0 ∂x


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The lateral amplitude is the amplitude in x-direction, parallel to the length axis of the cantilever. Note that the vertical and lateral amplitudes dY 0 and d X0 , respectively, are only normal and lateral to the sample surface if the cantilever is not tilted (ϕ = 0). Figure 2.14 shows examples of forced vibration calculated with the formulas in Table 2.3 (a) and (b). The normalized vibration amplitude dY 0 and the normalized lateral vibration amplitude d X0 are shown for three different values of normalized contact stiffness k∗ /kC . The lateral amplitude, which is proportional to the beamdeflection signal, becomes very small in the case k∗ /kC = 10 for the first mode and very high in the case k∗ /kC = 100. The maxima are marked as (1) and (2) in Fig. 2.14(c) and (d). Figure 2.15 shows the corresponding mode shapes y(x). The shape of the first mode is similar to the shape of a pinned mode in the case k∗ /kC = 100. This leads to high lateral amplitude at the end of the beam and a strong beam-deflection signal. In the case k∗ /kC = 10 the lateral deflection becomes almost zero at the end of the beam, and the first mode is hardly visible if a beamdeflection sensor is used and if the laser spot is focused to the end of the beam. The beam-deflection detector is most sensitive when the laser spot is located in one of the nodes of the vibration mode. The first mode will therefore be detectable if the

Fig. 2.14. Forced vibration of a cantilever calculated with the formulas shown in Table 2.3 for excitation at the clamped end and excitation by the sample surface. The vertical and lateral amplitudes of the sensor tip d Y 0 and d X0 , respectively, are shown for three different values of contact stiffness. In Figs. (a) and (b) the vibration was excited at the clamped end (UAFM) and in Figs. (c) and (d) the vibration was excited by sample surface vibration (AFAM). The damping constants used for the calculation were Q 1,free = 300 and γ/(m ∗ ω0 ) = 0.4

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Fig. 2.15. Shape functions y(x) of the modes corresponding to the peaks marked as (1) and (2) in Figs. 5.1 (c) and (d). The spring and the dashpot representing the contact forces are not shown here

laser spot is moved towards the middle of the beam. However, this decreases the sensitivity of the detection system to static forces and there is the risk that a position more in the middle of the beam corresponds to an antinode of one of the higher modes. All solutions in Table 2.3 have the same denominator N(α). The denominator contains the characteristic equation (here the characteristic equation of the simplified case without tilt of the cantilever and without lateral forces). In the case of low damping the zeros of the denominator are the resonant frequencies of the system, and the resonant frequencies are independent of the method of vibration excitation, as can be seen in Fig. 2.16. The measured amplitudes, however, depend strongly on the type of excitation, on the method of detection (amplitude or slope) and on the position of the detecting laser spot on the length axis of the cantilever. In the AFAM-mode one observes anti-resonances between the resonant frequencies where

Fig. 2.16. Vertical amplitude dY 0 and lateral amplitude d X0 in the UAFM and AFAM mode. The case of a normalized contact stiffness of k∗ /kC = 10 was chosen. A measurement at the end of the beam was assumed (x = L). The vertical lines show that the detected resonant frequencies are independent of the type of excitation and also independent of whether one detects the amplitude of the beam (equal to the vertical tip amplitude) or the slope of the beam (proportional to the lateral tip amplitude) if the damping is low. The constants used for the simulation were: Resonant frequency of the clamped-free beam f 1,free = 150 kHz, Q 1,free = 300, contact damping γ/(m ∗ ω0 ) = 0.4


69

Fig. 2.17. Amplitude (dashed line) and phase (continuous line) in the AFAM-mode (a) and in the UAFM-mode (b). In the AFAM-mode (a) anti-resonances accompanied by a phase shift of −π are observed. The data for the calculation were the same as in Fig. 2.16

Fig. 2.18. Forced vibration in the AFAM-mode. The lateral amplitude d X0 is shown. This signal is proportional to the beam-deflection signal. (a) Influence of the position of the sensor tip (b), influence of the lateral forces in the case of normalized contact stiffness k∗ /kC = 100, contact damping γ/(m ∗ ω0 ) = 0.4. (c) Influence of the position of the laser spot on the detected signal, (d) influence of the lateral forces in the case k∗ /kC = 1000, γ/(m ∗ ω0 ) = 2. For the lateral forces in (b) and (d) a ratio of lateral to normal contact function φLat (α)/φ(α) = 0.85, an angle ϕ = 11◦ , and a ratio cantilever-length to tip-height of L/h = 15 was assumed

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the amplitude becomes zero and the phase shifts by −π (Fig. 2.17). This holds also for the slope of the cantilever (not shown here). The solutions for the forced vibration of the complete system according to Fig. 2.8 including lateral forces and tip position can be found in the appendix. A few examples of forced vibration are plotted in Fig. 2.18. The tip position influences the resonant frequencies strongly in all cases, as can be seen in Fig. 2.18a. A shift of the tip from the end position (L 1 /L = 1) to a position only 10% of the length of the beam closer to the fixed end (L 1 /L = 0.9) leads to a dramatic change in the spectrum. The lateral forces influence the spectrum less when the normal contact stiffness divided by the spring constant of the cantilever is lower (k∗ /kC = 100) (b), and they influence the spectrum more when the normal contact stiffness is high (k∗ /kC = 1000) (d). The detector position influences the amplitudes of the measured peaks (c). The first mode, which is not well visible with the beam-deflection sensor at the end of the beam, is much better visible if the detector is located at x = 0.8L. The resonance frequencies do not depend on the detector position, but the frequencies of the anti-resonances depend strongly on the position of the detector on the length axis of the beam. This was also pointed out by Arinero and Lévêque [26].

2.6 Imaging and Contrast Inversion There are different possibilities to evaluate the contact resonances for imaging, as for example amplitude or phase imaging. The cantilever-sample system is excited with a frequency close to a contact resonance [62]. The contact-resonance frequency must be higher than the upper cut-off of the feedback loop. The feedback loop of the AFM works with the quasi static and low frequency components of the cantilever signal and keeps the static deflection of the cantilever at a predefined value while the surface is scanned. Parallel to the topography image the ultrasonic amplitude or phase is measured for example with a lock-in-amplifier and displayed as a color coded image. An example of a topography image and an ultrasonic amplitude image of a piezoelectric ceramic is shown in Fig. 2.19. In the topography image the grains of the polished sample surface can be seen. In the ultrasonic image substructures are visible within the grains which are the ferroelectric domains. The contact stiffness in AFAM and UAFM depends on the orientation of the domains [83, 94]. The contrast of the amplitude images depends strongly on the frequency of operation [95,96] as shown in Fig. 2.20. If the frequency of operation is far from any contact resonance, the contrast vanishes. If the frequency of excitation is lower than the average contact-resonance frequency of the sample, regions with high stiffness will cause lower amplitude of vibration than regions with low stiffness. And if the frequency is higher than the average contact-resonance frequency the contrast inverts. Experimental observation of contrast inversion is an indicator for a contactresonance frequency. It is recommended to operate above the contact-resonance frequency if possible, because the images are more intuitive if surface areas with higher stiffness show higher amplitude of vibration. Imaging of samples with a high variation of Young’s modulus as for example polymer samples with ceramic inclusions will lead to a shift of the contact-resonance


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Fig. 2.19. Topography-image (a) and AFAM amplitude-image (b) of a piezoelectric ceramic [82]. The size of the image is 10 × 10 µm2 . The grey scale in the topography image covers 20 nm. The spring constant of the cantilever was 41 N/m. The first two flexural resonant frequencies of the clamped-free beam were 162 kHz and 1004 kHz. The ultrasonic amplitude image was taken above the first contact-resonance frequency at 698 kHz. The white arrows point to a pore in the ceramic surface

Fig. 2.20. Contrast inversion depending on the ultrasonic excitation frequency [95]. A smaller section of the same surface as in Fig. 2.19 was imaged (a) at a frequency below the contactresonance frequency (695 kHz) and (b) above the contact-resonance frequency (699 kHz). The large triangular domain in the middle of the image changes contrast when the frequency of imaging is changed. The principle of contrast inversion is explained in the simulated spectra below. Areas with higher contact stiffness lead to a higher contact-resonance frequency. The dotted lines indicate schematically the frequency of operation below and above the average resonant frequency. The white arrows point to the same pore as seen in Fig. 2.19. The ultrasonic amplitude is always low and contrast inversion is not observed when the tip is on the pore. The topography of the pore leads to a strong variation of contact stiffness and consequently to a contact-resonance frequency much different from the flat areas of the sample

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frequency within one image amounting to several multiples of the half-width of the resonant peaks. In this case it is no more possible to find a frequency which is close to the contact-resonance maximum on both phases. Several contrast inversions may happen within one image and the qualitative correspondence between amplitude and stiffness is lost. Strong variations of contact stiffness are also caused by topographical features. One such example can be seen in Fig. 2.20. If the tip is above the small pore in the ceramic which is indicated by white arrows in Fig. 2.19 and Fig. 2.20, the amplitude of vibration is low in both cases of excitation frequency. Probably, the contact-resonance frequency of this area is much different from the contactresonance frequency of the flat surface regions. The contrast in the amplitude images is caused by the shift of the contactresonance frequency with contact stiffness. Furthermore, the amplitude at contact resonance varies with contact stiffness. The theoretical amplitudes at contact resonance of the first three modes are shown in Fig. 2.21. Amplitude or phase imaging is a convenient method to obtain qualitative images of contact-stiffness variation. For quantitative evaluation, however, it is necessary to measure the contact-resonance frequency in every pixel of the image. Kobayashi

Fig. 2.21. Vertical amplitude of the sensor tip dY 0 and lateral amplitude d X0 at the contactresonance frequency of the first three flexural modes as a function of normalized contact stiffness. A detector position at the end of the beam (x = L) was assumed and a low contact damping (γ/(m ∗ ω0 ) = 0.4) was used for the calculation. The curves were obtained by taking the amplitude maxima of simulated contact-resonance curves. All modes have local minima in the lateral amplitude for contact stiffness values at which the slope of the shape function y(x) has local minima at x = L


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Fig. 2.22. Topography, height scale 10 nm (a) and contact resonance AFAM-image (b) of nanocrystalline nickel with an average grain size of 167 nm fabricated by pulsed electrodeposition. The sample was polished with diamond paste and alumina solution before imaging [119]. The grey scale of the contact-resonance image ranges from 730 kHz to 750 kHz. The size of the images is 1.5 × 1.5 µm2

et al. developed a phased-locked-loop for the UAFM in order to follow the contactresonance frequency during imaging [97]. The technique was enhanced by amplitude detection by Yamanaka et al. [98]. Fukuda et al. showed that the Q-control technique [99] which was originally developed for tapping mode AFM can also be applied in UAFM [100]. Fast computers and high storage capacities make it possible to sweep spectra in every image point to track the contact-resonance frequency [101, 102] or to save full vibration spectra in every image point (Solver and Ntegra, NT-MDT, Moscow, Russia). The velocity of imaging is only limited by the frequency of vibration and by the contact damping. If contact resonances between 100 kHz and 1 MHz are measured, an image of 128 × 128 pixels takes about half an hour. Topography and contact-resonance frequency images of a nanocrystalline nickel sample are shown in Fig. 2.22. The sample was polished before imaging. The average grain size of the material was 167 nm.

2.7 Sensitivity of the Flexural Modes The influence of the forces which act upon the sensor tip is different for different modes because the mode shape and the wavelength depend on the mode number. In case of quantitative evaluation, the frequency shift of the contact resonances is relevant. The derivative ∂ f/∂k∗ was calculated analytically by Turner and Wiehn [65]. Normalized quantities marked with the symbol “∧” are used here to shorten the calculations: kˆ =

k∗ , kc

kˆ Lat =

∗ kLat , kc

fˆ =

f f 1,free

,

(2.58)

and abbreviations for combinations of the trigonometric and hyperbolic functions which appear frequently are introduced:

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ssh(x) = sin(x) sinh(x) cch(x) = cos(x) cosh(x) cch ± (x) = 1 ± cos(x) cosh(x) mix ± (x) = sin(x) cosh(x) ± cos(x) sinh(x)

(2.59)

The dimensionless wave number ψ is defined as ψ = αL. The characteristic equation is a function of the wave number, the contact stiffness and the contact damping. The characteristic equation (2.39) is rewritten in normalized quantities, and only the dependency on the normalized contact stiffness kˆ ∗ is considered: & ' Ω ψ(kˆ ∗ ), kˆ ∗ = 0 (2.60) The derivative of this equation with respect to kˆ ∗ yields [65]: dψ ∂Ω =− ∗ ˆ dk ∂ kˆ ∗

∂Ω ∂ψ

−1 (2.61)

If this formula is applied for example to the characteristic equation of the springcoupled cantilever (2.42) one obtains: −3mix − (ψ) dψ = dkˆ ∗ 3ψ 2 cch + (ψ) − ψ 3 mix − (ψ) + 6kˆ ∗ ssh(ψ)

(2.62)

As the sensitivity is given by the shift of the resonant frequency and not by the shift of the dimensionless wave number, the chain rule and the dispersion relation (2.10) are used: ∂ fˆ ∂ fˆ ∂ψ = ∂ψ ∂ kˆ ∗ ∂ kˆ ∗

∂ fˆ ψ =2 ∂ψ (1.8751)2

(2.63)

Now, the sensitivity σ can be calculated: σ=

−6ψmix − (ψ) d fˆ = dkˆ ∗ (1.8751)2 [3ψ 2 cch + (ψ) − ψ 3 mix − (ψ) + 6kˆ ∗ ssh(ψ)]

(2.64)

To evaluate the sensitivity, the normalized wave number or the contact-resonance frequency is first determined using the characteristic equation (2.42). Afterwards, (2.64) is applied. The sensitivity of the first four flexural modes of the spring-coupled cantilever is shown in Fig. 2.23. In the limit of vanishing contact stiffness (kˆ ∗ = 0) the normalized wave number is ψ = 1.8751 and cch + (1.8751) = 0. Therefore the sensitivity of the first flexural mode is in the limit kˆ ∗ = 0: d fˆ 6 ∗ n = 1 : σ(kˆ = 0) = = ≈ 0.5 (2.65) 4 ∗ ˆ (1.8761) dk ∗ k =0


75

In general one obtains for the n’th mode in the limit kˆ ∗ = 0: 6 d fˆ = 2 ∗ ˆ (1.8761) (ψn,free )2 dk ∗

(2.66)

k =0

For each value of the contact stiffness one can determine a flexural mode with highest sensitivity. The approximate crossing points of the sensitivity curves in Fig. 2.23 are listed in Table 2.4. The higher the contact stiffness, the higher is the most sensitive mode. The acoustic impedance at the sensor-tip position increases with the mode number [70]. One can also say that the dynamic load which is imposed by the cantilever onto the sample increases with increasing frequency proportional m ∗ ω2 y(L) [27, 29]. However, the maximal value of the sensitivity σ ≈ 0.5 is only reached by the first mode for small contact stiffness (k∗ → 0). This means that the sensitivity can be improved by using higher modes, but the sensitivity of the higher modes of a soft beam will be smaller than the sensitivity of the first mode

Fig. 2.23. Sensitivity σ of the first four flexural modes as a function of normalized contact stiffness [65]. The dashed line shows the sensitivity according to the equivalent point-mass model Table 2.4. Linear model of the cantilever when the sensor tip is in contact with a sample surface. The tip–sample interaction forces are represented by springs and dashpots. The cantilever is tilted by an angle ϕ with respect to the surface. The sensor tip of height h is located at a position L 1 on the length axis of the cantilever. The distance of the tip to the free end of the cantilever is L 2 Mode number n Slope at x = L becomes zero Mode (n + 1) becomes more sensitive than mode n Sensitivity of n’th flexural mode for k∗ → 0

1 k∗ /kC = 9.5 k∗ /kC ≈ 9.9 σ ≈ 0.09 σ = 0.485

2 k∗ /kC = 83 k∗ /kC ≈ 69 σ ≈ 0.04 σ = 0.077

3 k∗ /kC = 279 k∗ /kC ≈ 216 σ ≈ 0.02 σ = 0.028

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of a stiffer beam. Unfortunately, a stiffer beam makes the control of the static force more difficult. As the static force is also an important parameter, the choice of the optimal cantilever is always a compromise. Using the Rayleigh–Ritz method, Turner and Wiehn calculated an approximate formula for the sensitivity of the first five flexural and torsional modes [65]. This solution can also be applied to cantilevers with variable cross-section as for example V-shaped beams. The analytical formula of Turner and Wiehn was extended to contact damping by Chang et al. [103]. They found that the contact damping reduces the sensitivity of the modes for small values of contact stiffness k∗ . Furthermore the influence of lateral forces and tilt [104] and the length of the sensor tip [91] were examined. As the contact stiffness depends on the radius of the sensor tip and the Young’s moduli of tip and surface, it is not possible to calculate general formulas for the sensitivity as a function of Young’s modulus of the sample. For such a calculation one must assume a certain tip radius, elastic modulus and static force [96].

2.8 Quantitative Evaluation The characteristic equation derived in Chap. 4 can be solved for an analytical expression which describes the normalized contact stiffness k∗ /kC as a function of contact resonance. The ratio cP between lateral and normal contact function is defined as: cP =

φLat (α) φ(α)

(2.67)

Substituting the auxiliary functions X, T , U (2.38) in the characteristic equation (2.39, 2.40), one obtains a quadratic equation in the contact function φ(α): h2 h φ(α) sin2 ϕ + cP cos2 ϕ + L 21 S2 φ(α) sin ϕ cos ϕ [cP − 1] 2 L1 L1 2 2 h (2.68) + L 1 S1 φ(α) cos ϕ + cP sin2 ϕ + S0 2 φ2 (α)cP = 0 L1

L 41 S4 + L 31 S3

The quadratic term in φ(α) vanishes if there are no lateral forces (cP = 0). If the cantilever is parallel to the sample surface (ϕ = 0), one obtains the following solution for the contact function φ(α): φ(α) = −

L 41 S4 2(αL 1 )3 cch + (αL) = L 1 S1 cch − (αL 1 )mix − (αL 2 ) − cch + (αL 2 )mix − (αL 1 )

(2.69)

In the experiment a contact-resonance frequency f Cont is measured. If the resonant frequencies of the free cantilever are known, the dispersion relation (2.10) containing the measurable quantities can be used to calculate the wave numbers from the measured contact-resonance frequencies: αn,free L f Cont = cn f Cont αL = f n,free

αn,free L with cn := f n,free

(2.70)


77

In case of small contact damping, its influence on the contact-resonance frequency can be neglected, and the contact function reduces to φ(α) ≈ 3k∗ /kC . One obtains an analytical formula to calculate the normalized contact stiffness k∗ /kC directly from the contact-resonance frequencies [105]. √ √ k∗ (cn f n )3 cch + (cn f n ) (2.71) =3 √ √ kC cch − (cn √f n L 1 /L)mix − (cn √f n L 2 /L) −cch + (cn f n L 2 /L)mix − (cn f n L 1 /L) However, the position of the sensor-tip is usually not known. If one measures the contact-resonance frequencies of two flexural modes and assumes that the contact stiffness is the same for both modes, one can calculate the sensor-tip position numerically [105]. If lateral forces are considered, the quadratic equation must be solved: " L 2 L 4 S4 φ(α) = g ± g2 − 21 1 h S0 cP L 1 S1 (cos2 ϕ + cP sin2 ϕ) + L 21 S2 Lh1 (cP − 1) sin ϕ cos ϕ 2

g=−

+L 31 S3 Lh 2 (sin2 ϕ + cP cos2 ϕ) 1

2

2S0 Lh 2 cP

(2.72)

1

Here g is only an auxiliary quantity. In principle the normalized contact stiffness k∗ /kC can again be calculated if the contact damping is low. However, the tilt angle ϕ of the cantilever, the height h of the sensor tip with relation to the ∗ /k∗ of lateral to normal cantilever length L, and the ratio cP = φLat /φ ≈ kLat contact constants must be known. The tilt angle ϕ is technically defined by the cantilever holder in the AFM. The height of the sensor tip is usually specified by the manufacturer of the commercial probes. The length of the cantilever beam can be measured with an optical microscope. A more precise measurement of the tip height is only possible with a scanning electron microscope. Instead of calculating the contact stiffness from the contact-resonance frequencies, one can also use the formulas for the forced vibration and fit the experimental resonances to the theoretical curves by parameter variation [60, 90]. Because of the various unknown parameters, however, the error in the obtained contact stiffness values can be high. In contrast to the other parameters, the ratio between the lateral and the normal stiffness cP is not a parameter of the instrument, but depends on the elastic constants of the sensor tip, on the local elastic constants of the sample, and the other tip–sample interaction forces. If cP is unknown, care should be taken to work in the range of the dispersion curves where the influence of the lateral forces is low (see Chap. 4.1). The lateral forces shift the resonant frequencies of the modes in a different way than the normal forces. Therefore it should be possible to determine normal and lateral contact stiffness by fitting the measured spectra to the simulated spectra. However, up to now, as discussed in Chap. 2.3, real cantilever beams do not agree perfectly with the analytical model. On the other hand, the lateral contact stiffness can in

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principle be measured if torsional or lateral contact-resonance experiments are made at the same location of the sample. 2.8.1 Experiments for Quantitative Evaluation For quantitative evaluation the resonant frequencies of the clamped-free beam are needed. As far as the bandwidth is sufficient the free resonant frequencies can be measured with the internal sweep programs of the commercial instruments. Alternatively an optical interferometer as described in Chap. 2.3 can be used. Two flexural modes are necessary at minimum. However, it is better to determine the resonant frequencies of more flexural modes in order to test how much the cantilever deviates from the model of the clamped-free beam. Then the sensor tip is brought into contact with a sample and a predefined static deflection is set by the operator. The contact-resonance frequencies of at least two modes are measured by sweeping the frequency with the help of the AFAM or UAFM set-up described in Chap. 2. The amplitude of excitation should be kept as small as possible in order to avoid nonlinearities in the spectrum. As contact-resonance frequencies use to shift as a function of applied load, variation of the applied static load helps to distinguish contact resonances from other signals. An example of contact-resonance spectra of three flexural modes and with three different static deflections of the cantilever is shown in Fig. 2.24.

Fig. 2.24. Contact-resonance spectra measured with a silicon cantilever of 41 N/m spring constant on a silicon single crystal [103]. The resonances of the clamped-free beam and contact resonances of the first three flexural modes are shown. The static load for the contact resonances was 410 nN, 820 nN, and 1230 nN. It can be seen that all contact-resonance frequencies are higher than the corresponding free resonance and that the contact-resonance frequencies increase with increasing static load


79

The sensor-tip position L 1 on the length axis of the cantilever is then calculated numerically. It is assumed that the normalized contact stiffness which is calculated using (2.71) or (2.72) from the measured contact-resonance frequencies must be the same for all contact modes if the static load is identical. If more than two modes are measured, the obtained tip position will in most cases be different for each pair of modes. These differences yield the systematic error in the normalized contact stiffness k∗ /kC which typically amounts to 20% up to 40% at present. The systematic error is caused by the deviation of the real cantilever from the analytical model. After having determined the normalized contact stiffness one has to apply a model for the tip–sample force interaction to obtain an elastic constant of the surface. An elastic constant of the sample surface can be calculated from k∗ /kC using the contact mechanical formulas (2.19) and (2.20) or (2.23), if the spring constant of the cantilever, the normal load, and the elastic constants of the sensor tip and its radius are known. Usually these constants are not known precisely enough, and reference measurements on samples with known elastic constants are necessary. Figure 2.25 shows a sequence of measurements on two silicon single crystal surfaces with different orientation [82]. A reproducible difference between the contact-resonance frequencies of the two surfaces is visible, but also an average increase in the contactresonance frequencies and in the contact stiffness. The increase in contact stiffness is caused by tip wear [62, 79, 106]. Often the contact-resonance frequency increases strongly during the first measurements. A more stable situation is reached after the tip has been rounded so much that the pressure in the contact area is reduced sufficiently to stop further tip damage. The different symbols in Fig. 2.25 belong to different static cantilever loads. As can be seen, the static force has a strong influence

Fig. 2.25. Sequence of contact-resonance measurements on two different silicon single crystal surfaces of different orientation [82]. The contact-resonance frequencies of the first two flexural modes are shown (a) and (b). Four different static loads of the cantilever, 420 nN, 840 nN, 1260 nN, and 1680 nN were chosen. The contact stiffness k∗ calculated from (a) and (b) is shown (c). The cantilever made of silicon single crystal with the dimensions 120 × 30 × 2.8 µm3 (length × width × thickness) had a spring constant of 42 N/m. The first two clamped-free resonant frequencies of the sensor were 287.96 kHz and 1596.37 kHz

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on the contact-resonance frequency. The shift of the contact stiffness as a function of applied load can be used to estimate the shape of the sensor tip [56]. If the sensor tip is broken, so that the end of the tip is flat, the contact-resonance frequencies do not shift any more as a function of applied load. A flat tip could be an advantage, because in this case the contact area would be constant and known. On the other hand it is difficult to align a flat tip parallel to the sample surface. By comparison of the normalized contact stiffness on the two samples, the difference of the elastic constants can be calculated. When the normal load Fn and the tip radius R are identical for a measurement on a sample of interest and on a reference sample, one can conclude from the Hertzian model, (2.19): " k∗3 ∗ ∗ E = E Ref · (2.73) ∗ 3 kRef ∗ ∗ and kRef are the reduced Young’s modulus and the contact stiffness of the Here E Ref ∗ ∗ · k∗ /kRef , holds in reference sample, respectively. A simpler expression, E ∗ = E Ref the case of a flat tip. The indentation modulus of the sample MS can be calculated by solving (2.23): MS−1 = E ∗−1 − MT−1 . In case of an isotropic sample with known Poisson’s ratio the Young’s modulus follows directly from MS using (2.20). The indentation modulus of the tip can be calculated from the elastic single crystal constants of the tip material if they are known, or it can be measured using two reference samples: ( M1 M2 1 − k1∗3 /k2∗3 ( (2.74) MT = M2 k1∗3 k2∗3 − M1

Here M1 , M2 , k1∗ , and k2∗ are the indentation modulus and the contact stiffness of reference samples 1 and 2, respectively. The steps for quantitative measurements are summarized schematically in Fig. 2.26. By comparison of the contact stiffness on two samples the difference in the elastic constants can be determined. The elastic contact generates shear stresses and compressional stresses in the tip and the sample. The anisotropy in the indentation

Fig. 2.26. Schematic list showing the measurements and steps of evaluation which are necessary to obtain quantitative elastic constants of a sample surface


81

modulus is therefore lower than the anisotropy in the Young’s modulus. It was shown for silicon single crystal surfaces with two different orientations that the measured difference in contact stiffness k∗ agreed with the difference in k∗ predicted by the indentation modulus calculated from the elastic single crystal constants of silicon [82]. Nanocrystalline ferrites [105], diamond like carbon thin films [79], and clay minerals [107] were examined by contact-resonance spectroscopy. Figure 2.27 shows contact-resonance spectra on polystyrene, clay, mica and fused silica [107]. Polystyrene and fused silica were chosen as reference materials. Clay minerals fill the pores between fused silica grains in sandstone. They influence the average elastic constants of sand stone strongly, but measurement of their elastic constants is not easy. With the AFAM-measurements it was possible to estimate the Young’s modulus of the clay minerals. Dupas et al. imaged WC and Co-Phases in WC-Co Cermets and examined NiTi-epoxy composites by contact-resonance spectroscopy. They obtained quantitative results by fitting the measured spectra to simulations [60]. Hurley et al. examined the elastic constant of a niobium film quantitatively by AFAM [64], and Hurley and Turner investigated the influence of ambient humidity on quantitative AFAM results [108]. Piezoelectric ceramics like BaTiO3 and PZT were investigated by AFAM and the domain contrast was examined [83]. The indentation modulus of BaTiO3 ceramics and PZT ceramics was calculated and compared to experiments [83]. Tsuji et al. evaluated the stiffness of domain boundaries in piezoelectric material by UAFM [109]. Up to the present, commercially available cantilevers have large tolerances in their dimensions and they do not behave perfectly like a clamped beam. Therefore reliable quantitative measurements are only possible if one chooses reference samples with elastic constants close to the constants of the sample of interest. In addition, the error in the obtained values for the elastic constants depends on the control of the static load. It is essential for the calibration by reference sample to compare measurements that are made with the same normal load. In most commercial instruments, however, there is no possibility to apply a calibrated force to the cantilever.

Fig. 2.27. Contact-resonance spectra on polystyrene, clay, mica and fused silica [104]. The spectra of the first and the second contact-resonance frequency of a cantilever with a spring constant of 1.5 N/m are shown

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Fig. 2.28. Contact stiffness image of a nanocrystalline nickel sample. The contact stiffness values were calculated from contact-resonance frequencies. The size of the image is 1.5 × 1.4 µm2 . The spring constant of the employed cantilever was 48 N/m and its free resonant frequencies were 166 kHz and 1031 kHz for the first and the second flexural mode, repectively

The force is applied indirectly by approaching the sample and the fixed end of the cantilever. Due to the tilt of the cantilever with respect to the surface, the cantilever buckles and the static bending depends on the normalized contact stiffness k∗ /kC . Because of the different amount of buckling identical static beam-deflection signals do not correspond to identical normal loads Fn if the elastic constants of the samples differ strongly. With the new imaging techniques discussed in Chap. 6 it is possible to obtain contact-resonance images instead of only amplitude images. From two resonant frequency images of two different modes one can calculate a contact stiffness image as shown in Fig. 2.28 and an indentation modulus image if reference measurements are available [101, 102].

2.9 Nonlinear Forces A simple force curve contains an attractive adhesive part and a repulsive elastic part [110,111]. In AFM in air, adhesion forces caused by a water meniscus dominate and drop almost to zero when the water meniscus breaks. When the amplitude of vibration is slowly increased, the nonlinearity of the force is more and more sensed by the AFM. The resonance curves develop a steep edge at the left side of the spectrum, the maximum shifts to lower frequencies and an amplitude jump is noticed with hysteresis when the direction of the frequency scan is inverted [62]. An example of contact-resonance spectra measured with different amplitudes of excitation is shown in Fig. 2.29. It has been shown that the contribution of the local elasticity and the adhesion to the contact forces can be separated by exploiting the amplitude dependence of the resonant frequency, the so-called backbone curve [112]. In the regime of small nonlinearity, when the tip remains in contact with the sample surface during its vibration cycle, the cantilever vibration behavior resembles that of a macroscopic Hertzian contact oscillator [113]. The harmonic balance method [112] or the multiple scales technique [74] can be applied to solve the equation of motion approximately. Amplitude jumps caused by bistability are typical for a nonlinear system with softening behavior [114]. If the amplitude of surface vibration is increased, one can observe that the high frequency amplitude of the cantilever first increases linearly. After a certain thresh-


83

Fig. 2.29. Contact-resonance spectra showing the influence of the nonlinear forces [62]. (a) When the excitation amplitude of the transducer is increased, the spectra become asymmetric and develop a steep edge. (b) Hysteresis depending on the direction of the frequency scan

old, however, the minimum of the force curve is passed through during a vibration cycle and the amplitude of vibration of the cantilever remains constant or even decreases [115]. At the same time a DC offset of the cantilever develops, i.e. the sensor tip lifts off the sample surface. In this amplitude region UFM and SAFM are operating. At very high amplitudes the tip jumps so far away from the sample surface that the water meniscus breaks and the interaction forces drop almost to zero during a vibration cycle. The contact with the sample can now be considered as an impact where the impact time is short compared to the period of oscillation. In this regime subharmonics have been observed [116] and the ultrasonic mode resembles the intermittent contact or tapping mode where the vibration is excited by the transducer in the cantilever holder. Different authors have examined the influence of the nonlinear forces on the spectrum of the first flexural resonance in the tapping mode and in the non-contact mode [117–119]. Recently Stark et al. applied a state-space model to examine flexural vibration spectra in the tapping mode including the higher modes [120].

2.10 Conclusions Analytical models to describe flexural cantilever vibration for quantitative measurement of elastic constants have been summarized in this article. Examples of application by different research groups have been given. The challenge in quantitative AFAM will be to manufacture and model sensors in such a way that a characterization of every individual sensor by electron microscopy is not necessary. Furthermore, sensor tips with a well defined radius, a high stiffness and a high resistance to breaking and wear must be developed. Imaging techniques which acquire the contact-resonance frequency or full spectra in every point of the surface are more favorable than mere amplitude or phase imaging. Such techniques will make it possible to map local surface stiffness and contact damping quantitatively. In cases where high local resolution is not required, the precision of nanoindentation with regard to quantitative Young’s modulus will probably be higher, because nanoindentation provides calibrated force measurements and forces

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normal to the surface. On the other hand, the development of contact mechanics models for dynamic indentation is an active area of research, where nanoindentation and the AFM-techniques can stimulate each other. The potential of the high frequency AFM techniques like AFAM and UAFM is local resolution in the nm-range which can in principle be achieved, the possibility of fast image acquisition and the possibility to examine frequency-dependent phenomena. Furthermore, the measurement of frequencies is an advantage, as it avoids difficult calibration of amplitude. By combination of the results from different bending, lateral bending and torsional modes, elastic constants in three directions relative to the sample surface can be measured by AFM. Nonlinear tip–sample forces such as friction and adhesion change the shape of the resonance curves in a characteristic way. Quantitative evaluation of nonlinear resonance curves is a still larger challenge than elasticity measurement. Numerical modeling and absolute measurement of the vibration amplitudes of tip and sample will be required in this case. Acknowledgements. The author would like to thank W. Arnold, all colleagues, coworkers, and students for support and collaboration. Critical proofreading of this manuscript by S. Hirsekorn is gratefully acknowledged. Part of this work was supported by the German ministry of education and research (BMBF) in the frame of the research program “Nanoanalytik” under the contract No. FKZ13N8326.

A Appendix A.1 Definitions In order to calculate the forced vibration for the model shown in Fig. 2.8, including the angle ϕ of the cantilever, lateral forces, and tip position, abbreviations are defined for combinations of the trigonometric and hyperbolic functions which are needed frequently: s1± = sin αL 1 ± sinh αL 1

s2± = sin αL 2 ± sinh αL 2

± c± 1 = cos αL 1 ± cosh αL 1 c2 = cos αL 2 ± cosh αL 2 ssh 1 = sin αL 1 sinh αL 1 ssh 2 = sin αL 2 sinh αL 2 cch 1 = cos αL 1 cosh αL 1 cch 2 = cos αL 2 cosh αL 2

cch ± 1 = 1 ± cos αL 1 cosh αL 1

cch ± 2 = 1 ± cos αL cosh αL

cch + 1+2 = 1 + cos α(L 1 + L 2 ) cosh α(L 1 + L 2 ) mix1± = sin αL 1 cosh αL 1 ± cos αL 1 sinh hαL 1 mix2± = sin αL 2 cosh αL 2 ± cos αL 2 sinh hαL 2

(A.1)

As can be seen in Table 2.3, the formulas for the forced vibration excited by a concentrated force acting at the free end of the beam are very similar to the solutions in


85

Table 2.3 case (b), excitation by surface vibration. Therefore only the UAFM-mode and AFAM-mode are calculated here. In both cases, the calculation is based on the equation of motion (2.1), the dispersion relation (2.5) and the solution (2.3). The definitions of the contact functions φ(α) and φLat (α), the dimensionless contact damping p and the auxiliary functions X, T , and U in (2.33)–(2.38) are not repeated here. They can be taken from the main body of the text. As discussed in Chap. 4, two functions y(x) and y2 (x2 ) are defined for the two sections of the beam from the clamped end to the tip position and from the free end to the tip position, respectively. A.2 UAFM-mode In case of a vertical vibration of the clamped end of the beam the boundary conditions are: ⎧ 2 ∂ y2 (x2 ) ⎪ ⎪ =0 ⎨ (x) = u 0 ∂x22 x = 0 : ∂y(x) x2 = 0 : 3 ∂ y2 (x2 ) ⎪ =0 ⎪ ⎩ =0 ∂x ∂x23 ⎧ ⎨ y(x) = y2 (x2 ) ∂y2 (x2 ) x = L 1 or x2 = L 2 : ∂y(x) =− ⎩ ∂x ∂x2 ⎧ 2 2 ∂y(x) ∂ y(x) ∂ y2 (x2 ) ⎪ ⎪ − = −T(α) − X(α)y(x) ⎨ ∂x 2 ∂x ∂x22 x = L 1 or x2 = L 2 : ∂y(x) ∂ 3 y(x) ∂ 3 y2 (x2 ) ⎪ ⎪ ⎩ + = U(α) · y(x) + X(α) 3 3 ∂x ∂x ∂x2 (A.2) By combining the functions y(x) andy2 (x2 ) and the first four boundary conditions one obtains: u0 y(x) = (cos αx + cosh αx) + A2 (cos αx − cosh αx) + A4 (sin αx − sinh αx) 2 (A.3) y2 (x2 ) = A1 (cos αx2 + cosh αx2 ) + A3 (sin αx2 + sinh αx2 ) The remaining four boundary conditions together with these solutions lead to a linear system of equations with four unknown constants A1 , A2 , A3 and A4 . Solving the linear system for the constants leads to: *

+ + u0 ) + − + − 4 + + 3 2 −2α5 c− 1+2 − α Tc2 s1 + α X s2 s1 + c2 c1 − α Us2 c1 N(α)

u0 ) 5 2α ssh 1+2 + α4 T ssh 1 mix2+ + mix1− cch + A2 = 2 N(α)

− + + 2 −2α3 X ssh 1 ssh 2 + cch 1 cch + 2 + α U ssh 1 mix 2 − mix 1 cch 2 * −α(TU − X 2 )ssh 1 cch + 2 A1 =

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*

+ + u0 ) − − + − 4 − + 3 2 −2α5 c− 1+2 − α Ts2 s1 + α X −c2 s1 + s2 c1 + α Uc2 c1 N(α)

u0 ) + + + − α4 T 2ssh 1 cch + A4 = −2α5 mix1+2 2 + mix 1 mix 2 N(α) *

+ (A.4) −2α3 X mix1− cch + 2 − mix 1 ssh 2 A3 =

With the common denominator N(α): + − + + 4 N(α) = 2{−2α5 cch + 1+2 + α T(mix 2 cch 1 − cch 2 mix 1 ) − − − + − 2 − 2α3 X(cch + 2 ssh 1 + ssh 2 cch 1 ) + α U(mix 2 cch 1 − cch 2 mix 1 ) − − α(TU − X 2 )cch + 2 cch 1 }

(A.5)

If the sensor tip is located at the end of the beam (L 2 = 0), the constants and the denominator simplify to: * u0 ) 4 −α ssh − α3 Tmix − + 2α2 Xcch + αUmix + + (TU − X 2 )ssh 2N0 (α) u0 {α4 mix + + 2α3 Tssh + 2α2 Xmix − − 2αUcch A4 = 2N0 (α) − (TU − X 2 )mix + }N0 (α) = α4 cch + + α3 Tmix + + 2α2 Xssh A2 =

+ αUmix − + (TU − X 2 )cch −

(A.6)

The amplitude and the slope at the end of the beam (x = L) are in this case (L 2 = 0): u0 {α4 c+ + α3 Ts+ − α2 Xc− } N0 (α) u0 ∂y (L, α) = α {α4 s− − α2 Xs+ + αUc− } ∂x N0 (α)

y(L, α) =

(A.7)

The forced vibration in the UAFM-mode with lateral forces and tilt of the cantilever, but without variable tip position, was discussed in a publication [121]. Unfortunately, there are several typewriting errors in the formulas in this publication. The correct boundary conditions were published for example in [104]. A.3 AFAM-mode In the case of vibration of the surface in its normal direction, one obtains components of excitation in vertical direction (y-direction) and in lateral direction (x-direction) in the coordinate system of the cantilever. The formulas derived here can also be used for excitation of flexural modes by lateral surface vibration. The components of excitation in the coordinate system of the cantilever need only to be changed in this case. In case of normal surface vibration the boundary conditions are:


87

⎧ ∂2 y ⎪ ⎪ ⎨ y(x) = 0 =0 2 x = 0 : ∂y(x) y = L : ∂x3 ⎪ =0 ∂ y ⎪ ⎩ ∂x =0 ∂x 3 ⎧ ⎨ y(x) = y2 (x2 ) ∂y2 (x2 ) x = L 1 or x2 = L 2 : ∂y(x) =− ⎩ ∂x ∂x2 ⎧ 2 2 ∂y(x) ∂ y(x) ∂ y2 (x2 ) ⎪ ⎪ − X(α)y(x) − = a0 − T(α) ⎨ ∂x 2 ∂x ∂x22 x = L 1 or x2 = L 2 : ∂ 3 y(x) ∂ 3 y2 (x2 ) ∂y(x) ⎪ ⎪ ⎩ + = −b0 + U(α) · y(x) + X(α) 3 ∂x 3 ∂x ∂x2 (A.8)

With the components of excitation in the coordinate system of the cantilever: a0 = −u 0

h φLat (α) sin ϕ L 31

b0 = u 0

1 φ(α) cos ϕ L 31

(A.9)

Using the first four boundary conditions one obtains the following solutions for the two sections of the cantilever: y(x) = A2 (cos αx − cosh αx) + A4 (sin αx − sinh αx) y2 (x2 ) = A1 (cos αx2 + cosh αx2 ) + A3 (sin αx2 + sinh αx2 )

(A.10)

The constants are again calculated using the four boundary conditions at the tip position and solving the linear system of equations for the constants A1 , A2 , A3 , and A4 : + ) + + − − − + 2 A1 = −α3 a0 c+ 2 ssh 1 + s2 mix 1 + c2 cch 1 − α b0 c2 mix 1 + s2 ssh 1 * + − + − −s2− cch − 1 + α (Xa0 − Tb0 ) c2 cch 1 + (Xb0 − Ua0 ) s2 cch 1 /N(α) ) + + − − − 2 A2 = −α3 a0 mix2+ s1− + ssh 2 c− 1 − cch 2 c1 + α b0 mix 2 c1 + ssh 2 s1 * + + − + − +cch + 2 s1 + α (Xa0 − Tb0 ) cch 2 c1 − (Xb0 − Ua0 ) cch 2 s1 /N(α) + ) + − − − − − 2 A3 = α3 a0 c+ 2 mix 1 − s2 ssh 1 − s2 cch 1 − α b0 −c2 ssh 1 + s2 mix 1 * − − − + − +c− 2 cch 1 + α (Xa0 − Tb0 ) s2 cch 1 − (Xb0 − Ua0 ) c2 cch 1 /N(α) ) + − A4 = α3 a0 −ssh 2 s1+ + mix2+ c− 1 − cch 2 s1 − + + + +α2 b0 −ssh 2 c− 1 + mix 2 s1 − cch 2 c1 * + + − (A.11) +α (Xa0 − Tb0 ) cch + 2 s1 + (Xb0 − Ua0 ) cch 2 c1 /N(α) The denominator N(α) is the same as in the UAFM-mode (A.5). If the position detector is located at the end of the beam (x2 = 0), the solution simplifies to: y2 (0) = 2A1

∂y2 (0) = −2αA3 ∂x2

(A.12)

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References 1. Hurley DC, Tewary VK, and Richards AJ (2001) Meas Sci Technol 12:1486 2. Every AG (2002) Meas Sci Technol 13:R21 3. Zieniuk JK and Latuszek A (1989) in: Shimizu H, Chubachi N, Kushibiki J (eds) Acoust Img 17. Plenum Press, New York, p 219 4. Khuri-Yakub BT, Cinbis C, Chou CH, Reinholdtsen PA (1989) in: McAvoy BR (ed) Proc. IEEE Ultrasonics Symp, p 805 5. Kulik A, Attal J, Gremaud G (1993) in: Wei Y and Gu B (eds) Acoustical Imaging 20. Plenum Press, New York, p 241 6. Quate CF, Khuri-Yakub BT, Akamine S, Hadimioglu BB (1994) United States Patent 5,319,977 7. Günther P, Fischer UCh, Dransfeld K (1998) Appl Phys B 48:89 8. Giessibl FJ (1998) Appl Phys Lett 73:3956 9. Binnig G, Quate CF, Gerber Ch (1986) Phys Rev Lett 56:930 10. Kolosov O and Yamanaka K (1993) Jpn J Appl Phys 32:22 11. Chilla E, Hesjedahl T, Fröhlich H-J (1997) Phys Rev B 55:15852 12. Rabe U and Arnold W (1994) Appl Phys Lett 64:1493 13. Yamanaka K and Nakano S (1996) Jpn J Appl Phys 35:3787 14. Rohrbeck W and Chilla E (1992) Phys stat sol (a) 131:69 15. Hesjedahl T and Behme G (2001) Europhys Lett 54:154 16. Yamanaka K, Ogiso H, Kolosov O (1994) Jpn J Appl Phys 33:3197 17. Kolosov OV, Castell MR, Marsh CD, Briggs GAD, Kamins TI, Williams RS (1998) Phys Rev Lett 81:1046 18. Yamanaka K (1996) Thin Solid Films 273:116 19. Inagaki K, Kolosov OV, Briggs GAD, Wright OB (2000) Appl Phys Lett 76:1836 20. Inagaki K, Matsuda O, Wright OB (2002) Appl Phys Lett 80:2386 21. Huey BD (2002) in: Maev RG (ed) Acoust Img 26. Kluver Academic/Plenum Publishers, p 183 22. Dinelli F, Castell MR, Ritchie DA, Mason NJ, Briggs GAD, Kolosov OV (2000) Phil Mag A 80:2299 23. Maivald P, Butt H-J, Gould SAC, Prater CB, Drake B, Gurley JA, Elings VB, Hansma PK (1991) Nanotechnology 2:103 24. Florin E-L, Radmacher M, Fleck B, Gaub HE (1994) Rev Sci Instr 65:639 25. Mazeran P-E and Loubet J-L (1997) Tribol Lett 3:125 26. Arinero R and Lévêque G (2003) Rev Sci Instr 74:104 27. Troyon M, Wang Z, Pastre D, Lei HN, Hazotte A (1997) Nanotechnology 8:163 28. Rosa A, Weilandt E, Hild S, Marti O (1997) Meas Sci Technol 8:1 29. Burnham NA, Kulik AJ, Gremaud G, Gallo PJ, Oulevey F (1996) J Vac Sci Technol B14:794 30. Oulevey F, Gremaud G, Mari D, Kulik AJ, Burnham NA, Benoit W (2000) Scripta mater 42:31 31. Elings VB and Gurley JA (1993) US Patent No. 5,266,801 32. Hirsekorn S, Rabe U, Arnold W (2001) Appl Phys A 72 [Suppl.]:S87 33. Cretin B and Sthal F (1993) Appl Phys Lett 62:829 34. Robert L and Cretin B (1999) Surf Interface Anal 27:568 35. Vairac P and Cretin B (1996) Appl Phys Lett 68:461 36. Cretin B and Vairac P (1997) Appl Phys Lett 71:2082 37. Oliver WC and Pharr GM (1992) J Mater Res 7:1564 38. Lorenz D, Fränzel W, Einax M, Grau P, Berg G (2001) J Mater Res 16:1776 39. Seyed Asif SA, Wahl KJ, Colton RJ, Warren OL (2001) J Appl Phys 90:1192 40. Bushan B and Li X (2003) Int Mater Rev 48:125


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41. Butt H-J and Jaschke M (1995) Nanotechnology 6:1 42. Salapaka MV, Bergh HS, Lai J, Majumdar A, McFarland E (1997) J Appl Phys 81:2480 43. Rabe U, Janser K, Arnold W (1996) in: Tortoli P and Masotti L (eds) Acoustical Imaging 22. Plenum Press, New York, p 669 44. Rabe U, Janser K, Arnold W (1996) Rev Sci Instr 67:3281 45. Meyer G and Amer NM (1990) Appl Phys Lett 57:2089 46. Marti O, Colchero J, Mlynek J (1990) Nanotechnology 1:141 47. Kawagishi T, Kato A, Hoshi Y, Kawakatsu H (2002) Ultramicroscopy 91:37 48. Pfeiffer O, Bennewitz R, Baratoff A, Meyer E (2002) Phys Rev B 65:161403 49. Reinstädtler M, Rabe U, Scherer V, Hartmann U, Goldade A, Bushan B, Arnold W (2003) Appl Phys Lett 82:2604 50. Krotil HU, Weilandt E, Stifter Th, Marti O, Hild S (1999) Surf Interface Anal 27:341 51. Caron A, Rabe U, Reinstädtler M, Turner JA, Arnold W (2004) Appl Phys Lett 85:6398 52. Drobeck T, Stark RW, Gräber M, Heckl WM (1999) New J Phys 1:15.1 53. Drobeck T, Stark RW, Heckl W (2001) Phys Rev B 64:045401 54. Yamanaka K and Nakano S (1998) Appl Phys A 66:S313 55. Yamanaka K, Noguchi A, Tsuji T, Koike T, Goto T (1999) Surf Interface Anal 27:600 56. Yamanaka K, Tsuji T, Noguchi A, Koike T, Mihara T (2000) Rev Sci Instr 71:2403 57. Crozier KB, Yaralioglu GG, Degertekin FL, Adams JD, Minne SC, Quate CF (2000) Appl Phys Lett 76:1950 58. Yaralioglu GG, Degertekin FL, Crozier KB, Quate CF (2000) J Appl Phys 87:7491 59. Crozier KB, Yaralioglu GG, Degertekin FL, Adams JD, Minne SC, Quate CF (2000) Proc SPIE 4099:48 60. Dupas E, Gremaud G, Kulik A, Loubet J-L (2001) Rev Sci Instr 72:3891 61. Rabe U and Arnold W (1994) Ann Phys 3:589 62. Rabe U, Kester E, Arnold W (1999) Surf Interface Anal 27:386 63. Stark RW, Drobeck T, Heckl WM (2001) Ultramicroscopy 86:207 64. Hurley DC, Shen K, Jennett NM, Turner JA (2003) J Appl Phys 94:2347 65. Turner JA and Wiehn JS (2001) Nanotechnology 12:322 66. Weaver W, Timoshenko SP, Young DH (1990) Vibration Problems in Engineering. John Wiley, New York 67. Stokey FW (1976) Vibration of Systems Having Distributed Mass and Elasticity. In: Harris CM, Crede CE (eds) Shock and Vibration Handbook, McGraw-Hill, New York, Chapter 7 68. Radmacher M, Tillmann RW, Gaub HE (1993) Biophys J 64:735 69. Burnham NA, Gremaud G, Kulik AJ, Gallo P-J, Oulevey F (1996) J Vac Sci Technol B 14:1308 70. Turner JA, Hirsekorn S, Rabe U, Arnold W (1997) J Appl Phys 82:966 71. Reinstädtler M, Rabe U, Scherer V, Turner JA, Arnold W (2003) Surf Sci 532-535:1152 72. Hoummady M, Farnault E, Yahiro T, Kawakatsu H (1997) J Vac Sci Technol B 15:1539 73. Monchalin J-P (1986) IEEE Transactions UFFC-33:485 74. Turner JA and Hurley DC (2003) Instrumentation, Mesure, Métrologie 3:117 75. Reinstädtler M (2004) Elastische und tribologische Oberflächencharakterisierung auf der Nanoskala mittels Torsions- und Lateralmoden von Sensorbalken der Rasterkraftmikroskopie. PhD Thesis, Naturwissenschaftlich-Technische Fakultät III der Universität des Saarlandes, unpublished 76. Pietrement O and Troyon M (2000) J Colloid and Interface Science 226:166 77. Schwarz UD (2003) J Colloid and Interface Science 261:99 78. Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge 79. Amelio S, Goldade AV, Rabe U, Scherer V, Bushan B, Arnold W (2001) Thin solid films 392:75 80. Pharr GM, Oliver WC, Brotzen F (1992) J Mat Res 7:613

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81. Vlassak JJ and Nix WD (1993) Phil Mag A67:1045 82. Rabe U, Amelio S, Kopycinska M, Hirsekorn S, Kempf M, Göken M, Arnold W (2002) Surf Interface Anal 33:65 83. Rabe U, Kopycinska M, Hirsekorn S, Muñoz Saldaña J, Schneider GA, Arnold W (2002) J Phys D: Appl Phys 35:2621 84. Sarioglu AF, Atalar A, Degertekin FL (2004) Appl Phys Lett 84:5368 85. Tsuji T and Yamanaka K (2001) Nanotechnology 12:301 86. Tsuji T, Irihama H, Yamanaka K (2002) JSME Int J A 45:561 87. Tsuji T, Irihama H, Yamanaka K (2002) Jpn J Appl Phys 41:832 88. Yamanaka K, Tsuji T, Irihama H, Mihara T (2003) Proc SPIE 5045:104 89. Wright OB and Nishiguchi N (1997) Appl Phys Lett 71:626 90. Dupas E (2000) Cartographie et spectroscopie des propriétés méchaniques à l’échelle du nanomètre par spectrométrie acoustique locale à fréquence variable: Thèse no 2278, EPFL Lausanne, unpublished 91. Wu T-S, Chang W-J, Hsu J-Ch (2004) Microelectronic Engineering 71:15 92. Muraoka M (2002) JSME Int J A 45:567 93. Muraoka M (2005) Nanotechnology 16:542 94. Karapetian E, Kachanov M, Kalinin SV (2005) Phil Mag 85:1017 95. Rabe U, Amelio S, Hirsekorn S, Arnold W (2000) in: Halliwell M and Wells NT (eds) Acoust Img 25. Kluwer Academic/Plenum, New York, p 253 96. Rabe U, Amelio S, Kester E, Scherer V, Hirsekorn S, Arnold W (2000) Ultrasonics 38:430 97. Kobayashi K, Yamada H, Matsushige K (2002) Surf Interface Anal 33:89 98. Yamanaka K, Maruyama Y, Tsuji T, Nakamoto K (2001) Appl Phys Lett 78:1939 99. Anczykowski B, Cleveland JP, Krüger D, Elings V, Fuchs H (1998) Appl Phys A 66:S885 100. Fukuda K, Irihama H, Tsuji T, Nakamoto K, Yamanaka K (2003) Surf Sci 532-535:1145 101. Hurley DC, Kos AB, Rice P (2005) Mater Res Soc Proc 838E:08.2.1 102. Hurley DC, Kopycinska-Müller M, Kos AB, Geiss RH (2005) Adv Eng Mat 7:713 103. Chang W-J, Fang T-H, Chou H-M (2003) Phys Lett A 312:158 104. Chang W-J (2002) Nanotechnology 13:510 105. Kester E, Rabe U, Presmanes L, Tailhades Ph, Arnold W (2000) J Phys Chem Solids 61:1275 106. Kopycinska-Müller M, Geiss RH, Rice P, Hurley DC (2005) Mater Res Soc Proc 838E:010.16.1 107. Prasad M, Kopycinska M, Rabe U, Arnold W (2002) Geophys Research Lett 29:13-1 108. Hurley DC and Turner J (2004) J Appl Phys 95:2403 109. Tsuji T, Ogiso H, Akedo J, Saito S, Fukuda K, Yamanaka K (2004) Jpn J Appl Phys 43:2907 110. Hirsekorn S, Rabe U, Arnold W (1996) Europhysics News 27:93 111. Hirsekorn S, Rabe U, Arnold W (1997) Nanotechnology 8:57 112. Muraoka M and Arnold W (2001) JSME Int J A 44:396 113. Turner JA (2004) J Sound and Vibration 275:177 114. Nayfeh AN and Mook DT (1979) Nonlinear Oscillations. Wiley, New York 115. Hirsekorn S, Rabe U, Arnold W (1997) Nanotechnology 8:57 116. Burnham NA, Kulik AJ, Gremaud G (1995) Phys Rev Lett 74:5092 117. Wang L (1998) Appl Phys Lett 73:3781 118. Sasaki N and Tsukuda M (1999) Appl Surf Sci 140:339 119. Lee SI, Howell SW, Raman A, Reifenberger R (2002) Phys Rev B 66:115409 120. Stark RW, Schitter G, Stark M, Guckenberger R, Stemmer A (2004) Phys Rev B 69:085412 121. Rabe U, Turner J, Arnold W (1998) Appl Phys A 66:S277 122. Kopycinska-Müller, M (2005) PhD-Thesis, Naturwissenschaftlich-Technische Fakultät III der Universität des Saarlandes, unpublished

3 Scanning Ion Conductance Microscopy Tilman E. Schäffer · Boris Anczykowski · Harald Fuchs

3.1 Introduction The era of scanning probe microscopy was launched in 1981 with the construction of the first scanning tunneling microscope (STM) by Binnig, Rohrer and coworkers [1]. This microscope is based on the measurement of tiny tunneling currents between a sharp metal probe and a flat metal sample. For the first time, individual atoms could be laterally resolved on conducting surfaces. But more than that, the STM has triggered the development of a large family of new microscopes, the so-called scanning probe microscopes (SPM), that are all based on a small, locally confined probe which is sensitive to various kinds of physical quantities. In 1984, Pohl et al. invented the near-field scanning probe microscope (NSOM) that enables optical imaging of surfaces with sub-wavelength resolution [2]. In 1986, in the same year when they received the Nobel Price for their development of the STM, Binnig et al. invented the atomic force microscope (AFM) [3]. This microscope is based on the measurement of minute forces. One of its main applications is the creation of highresolution topographic images of surfaces, where it has the advantage over STM of not requiring conducting probes and samples. The additional benefit of the AFM that it can image in different environmental conditions including aqueous buffer solution makes it well suited for biological applications [4]. Several related scanning probe microscopes have been developed so far, including the magnetic force microscope (MFM) [5, 6], the electric force microscope [7] and the scanning electrochemical microscope (SECM) [8]. In this chapter, we focus on the scanning ion conductance microscope (SICM), which was invented by Hansma et al. in 1989 [9, 10]. The SICM records the ionic current through a small aperture as this aperture is scanned over the surface of a sample immersed in an electrolyte solution. The aperture is usually formed by a nanopipette. Two electrodes are placed in the electrolyte: one inside the pipette (pipette electrode), and one outside the pipette in the bath over or under the sample (bath electrode). By applying a voltage between both electrodes and recording the ion current through the pipette, locally resolved images of the ion current over the sample surface can be generated. Despite the many possible applications of such a microscope, the SICM is one of the least developed scanning probe microscopy technique so far. Only few setups are described in literature [9–17]. Most researchers have used drawn-out glass capillaries as pipettes similar to those used in intracellular recording and patch clamp experiments [18,19], but some also used microfabricated probes [20]. Typical opening diameters of the tips of the

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glass nanopipettes are 30–100 nm, and lateral scan resolutions in the same range are obtained. There is a strong dependence of the measured current on the tipsample distance (“current squeezing effect”). The tip-sample distance can therefore be kept constant during scanning by using the current as an input to a feedback loop, thereby generating images of sample topography. This particularly works for non-porous samples where the bath electrode is positioned on the same side of the sample as the pipette. Pipette and sample do not come into mechanical contact with each other, and soft and delicate samples such as living cells can be imaged in this “non-contact” configuration [14]. To improve the imaging capabilities of the SICM, several methods have been devised that are mostly based on the principle of an ac measurement. By modulating either the pipette current or the tip-sample distance and by using lock-in amplifiers, these methods exhibit several advantages over the regular dc method [16, 17, 21]. All these methods use the measurement of the local ion conductance to generate images of sample topography. For many applications, however, it becomes important to measure complementary quantities, most prominently such that allow a determination of sample topography independent of the ion current. For this purpose, two different techniques have been developed so far: (a) SICM with a complementary AFM distance control [12] and (b) SICM with a complementary shear-force control [15]. The first technique uses a bent nanopipette that is coated with a reflective metal layer and that is scanned over the sample surface like an AFM tip. Different AFM imaging modes can be used this way, including contact mode and tapping mode imaging. The second technique uses a nanopipette that is positioned perpendicularly to the sample surface and that is set into transverse vibrations. Arising mechanical shear-forces between tip and sample provide an independent measure of sample height. Both techniques allow the simultaneous and complementary imaging of sample topography and local ion conductance. We organize this chapter in the following way: in Sect. 3.2, we cover fundamental principles concerning the measurement of the local ion conductance. In Sect. 3.3, we present theoretical approaches and experimental implementations for measuring ion currents through nanopipettes close to surfaces. In Sect. 3.4, we move on to advanced techniques for the imaging of ion currents that are based on the local measurement of conductance only. In Sect. 3.5, we finally move our attention to the combination of SICM with other scanning techniques. Methods that provide an independent measurement of sample topography are especially important as they enable high-resolution measurements of the local permeability of a sample to ions.

3.2 Fundamental Principles 3.2.1 Basic Setup The basic principle of SICM is shown in Fig. 3.1a. A nanopipette with a small opening diameter is positioned close to a sample surface. A voltage is applied accross the electrolyte via two electrodes, one inside the pipette and the other one in the bath

3 Scanning Ion Conductance Microscopy

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Fig. 3.1. (a) Schematic of a basic SICM setup for current measurements over a non-porous sample. The tip of a pulled nanopipette is brought into close proximity to the sample surface. A voltage, U, is applied between the electrode in the pipette and that in the bath, causing an ionic current to flow through the opening of the pipette tip. With decreasing tip-sample distance, this current decreases as a result of the “squeezing” of the current into a narrowing conductance path (gap) between pipette and sample. Since no mechanical contact occurs during this process until “touch-down”, the measured leakage current, IL , provides a contact-free measure of tip-sample distance. By scanning the sample with respect to the pipette tip, two-dimensional images of sample topography can be recorded. (b) Equivalent circuit for this setup. The resistance of the pipette, RP , is in series with the resistance of the leakage path, RL

solution. Usually, silver/silver chloride electrodes are used. The induced current flows from the pipette electrode through the pipette, leaves the pipette through the narrow gap between pipette and sample surface, and is collected by the bath electrode (the direction of the current can also be reversed). The closer the pipette tip is to the sample, the more this leakage current is restricted (“squeezed”) by this narrow gap [9,22]. The detected current therefore directly gives a measure of the tip-sample distance. Especially important is the fact that this measurement is achieved without actual mechanical contact between tip and sample. It is therefore possible to record non-contact images of sample topography by scanning the sample relative to the tip and by using a feedback loop for controlling the tip-sample distance. We will see below that this property is especially valuable for the imaging of soft biological samples such as living cells. To further characterize this basic SICM measurement setup, we consider the equivalent circuit (Fig. 3.1b). The applied voltage, U, drives a current between the electrodes. This current needs to pass through the narrow neck of the pipette tip that constitutes the resistance RP , and, in series, through the leakage resistance of the thin gap between pipette and sample, RL . Since it is this leakage resistance that is usually of interest as it provides a measure of the tip-sample distance, it must not be much smaller than the pipette resistance. Otherwise, the overall resistance and therefore the measured current, I, would be dominated by the pipette resistance. This makes it difficult to extract the leakage resistance which becomes increasingly

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larger for smaller tip-sample distances. We assume here that other resistances in the system such as resistances in the wires, in the bulk electrolyte and in the solid-liquid junctions are small. Furthermore, we neglect electrochemically induced potentials that can be compensated for by applying an offset voltage. We also note that, when fast changes of the current are to be measured, the capacitances in the system need to be considered. In many cases it will be interesting to measure the current through conductive paths in the sample, such as pores or channels. For this purpose, we consider a slightly different setup (Fig. 3.2a). Characteristic for this type of measurement is that there is a separate compartment on each side of the sample. Each compartment is filled with electrolyte, and there are one or more conductive paths through the sample between the two compartments. Similarly to the setup described in Fig. 3.1, the thin end of a pulled pipette is brought into close proximity to the sample surface. One electrode is inside the pipette, and the other electrode is in the compartment below the sample. Again, a voltage is applied between the electrodes. Now there are different paths for the current. If the pipette would form an ideal, tight seal with the sample, then the only possibility for the current would be to travel through a conductive path (pore or channel) in the sample directly below the tip (IC ). In a realistic case, however, the seal is never perfect. Some current “escapes” through the thin gap between pipette and sample, and subsequently flows through the sample via other conductive paths, making up the leakage current, IL , that acts as a background current.

Fig. 3.2. (a) Schematic of a basic SICM setup for current measurements through a porous sample. There are two separate compartments with the sample at their interface. The pipette with the pipette electrode is placed above the sample, and the bath electrode is placed below the sample. The current can leave the pipette opening either through a conductive path in the sample directly below the tip (IC ) or can “escape” sideways through the thin gap between tip and sample as leakage current, IL . The closer the tip is to the sample surface, the smaller this leakage current is (due to the current squeezing effect), and the better the conductance through the sample can be measured. (b) Equivalent circuit for this setup. The resistance of the conductive path through the pore directly below the tip, RC , and the resistance of the leakage path, RL , are in parallel with respect to each other, and in series to the pipette resistance, RP


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We again consider the equivalent circuit to further characterize this setup (Fig. 3.2b). The current first passes through the narrow neck of the pipette tip that constitutes the resistance RP . The following two paths are symbolized by the channel resistance, RC , and by the leakage resistance, RL , in parallel to each other. Since we are usually interested in the current IC that flows through the channel below the tip, we need to measure RC . Therefore, the pipette resistance RP should be as small as possible, and the leakage resistance RL should be as large as possible, requiring the gap between pipette and sample to be small. 3.2.2 Nanopipettes Pipettes for SICM face similar requirements as those used for intracellular recording. Most importantly, they need to be quite sharp at the tip, with opening diameters of a few tens of nanometers. This minimizes the risk of cell damage during cell puncturing in intracellular recording, and maximizes the resolution in SICM. Pipettes are usually drawn from glass (e.g., borosilicate) capillaries with an initial outer diameter of 1–2 mm. Pipette pullers based on a heated coil can be used, but laserbased pipette pullers are preferable. In these pipette pullers an infrared laser beam locally heats up the glass capillary which is subsequently drawn apart by force. This creates sharp tips with opening diameters down to a few tens of nanometers (Fig. 3.3). A small opening diameter results in a relatively large pipette resistance RP . To minimize this resistance, the pipette could be filled with a highly conductive solution (e.g., 2–3 M KCl). The diffusion of this solution into the bath solution

Fig. 3.3. Scanning electron microscope (SEM) image of the tapered end of a nanopipette as it is typically used as an SICM probe (sputter coated with a thin layer of platinum to improve SEM imaging). The size of the opening depends on the pulling parameters such as pulling force, velocity, and heat. Inner opening diameters in the range of 30–100 nm are routinely obtained


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would be small due to the small tip diameter, and could be further reduced with the help of an agar salt bridge, for example. Physiological electrolyte solution of lower concentration, however, also works, resulting in typical pipette resistances on the order of 100 MΩ. 3.2.3 Electrodes The electrodes that are typically used in SICM are silver/silver chloride electrodes. This type of electrode is also used as reference electrodes in potentiometry and voltammetry experiments. Silver/silver chloride electrodes have electrochemical properties that make them well suited for application in SICM. Furthermore, they are easily fabricated, for example by electrolytic deposition of silver chloride on a layer of metallic silver. The silver/silver chloride (Ag/AgCl) electrode consists of a silver (Ag) wire that is coated with solid silver chloride (AgCl) and immersed in an electrolyte containing chloride ions (e.g., a solution of KCl or NaCl). In thermodynamic equilibrium, the solution is saturated with silver chloride: AgCl(solid) Ag+ (aq) + Cl− (aq)

(3.1)

The equilibrium constant for this reaction is K = 1.76 × 10−10 (T = 25 ◦ C), so the concentration of Ag+ ions in solution is very small. The Ag/AgCl electrode is described by a reversible redox reaction, in which the chloride atoms in the solid silver chloride receive an electron and go into solution as chloride ions, leaving metallic silver, and vice versa: AgCl(solid) + e− Ag(solid) + Cl− (aq)

(3.2)

This reaction occurs close to the electrode surface (< 1 nm distance). The potential that will build up at the electrode in thermodynamic equilibrium is given by the Nernst equation. The Nernst equation relates the electrode potential, E, to the concentration (more precisely: the activity) of the chloride ions in solution [23]. RT 1 ln F [Cl− ] ∼ = +0.222 V − 0.0591 V × lg[Cl− ]

E = E0 +

(3.3)

E 0 is the standard potential of the Ag/AgCl electrode (with respect to the Standard Hydrogen Electrode, at T = 25 ◦ C), R ∼ = 8.315 J mol−1 K−1 is the molar gas constant, T is the absolute temperature (here: T = 298 K), F ∼ = 96,485 C mol−1 is the Faraday constant, and the square brackets denote the concentration. In SICM, two Ag/AgCl electrodes are used as anode and cathode, respectively. This setup is simpler than that for voltammetric experiments, where three electrodes (working electrode, auxiliary electrode and reference electrode) are used and controlled by a potentiostat. For the SICM setup, simply an external voltage is applied between the two electrodes inducing a faradaic ionic current in the electrolyte which is measured with a low-current amplifier.


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3.3 Ion Currents Through Nanopipettes 3.3.1 Background Theory The measured current, I, that passes through the pipette depends on the effective ohmic resistance in the current path of the electrolyte, Reff : I=

U . Reff

(3.4)

U is the applied voltage. The resistance of the electrolyte is mainly determined by the narrow regions it passes through. Considering the equivalent circuit of Fig. 3.2b, the effective resistance becomes Reff = RP +

RC RL . RC + RL

(3.5)

We are often interested in the channel resistance RC . However, the channel resistance depends on the particular configuration in which it is measured. For configurations with simple boundary conditions, the channel resistance can easily be calculated. For example, for a cylindrical channel of length L and radius r that is capped by a (perfectly conducting) electrode at each opening, the ohmic channel resistance is given by capped

RC

=

Lρ , πr 2

(3.6)

where ρ is the specific electrical resistivity of the electrolyte. Such an ideal configuration, however, never occurs in SICM. A more realistic configuration is when the electrodes are far removed from either end of the conductive channel. The resistance then increases, because the current additionally traverses the distance from the bulk medium to the disk-like mouths of the channel. This gives rise to an “access” ∞ that acts in series with the channel resistance [18]: resistance Racc & πr ' ρ capped ∞ + Racc = L+ . (3.7) RC∞ ≈ RC 2 πr 2 But also this configuration does not precisely apply to SICM, mainly because one electrode is always confined to the interior of the pipette. The precise meaning of RP , RC and RL in (3.5) therefore depends on the particular boundary conditions (for example, the tip-sample distance). Nevertheless, (3.5) is useful in many cases, since it provides a simple allocation of the resistances in the current path. 3.3.2 Simple Analytical Model At the heart of SICM lies the dependence of the measured current on the tipsample distance that comes about in so-called current-distance curves. Far away


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from the sample surface, the current through the pipette is determined by the pipette resistance RP . Speaking in terms of (3.5), the leakage resistance is zero. In closer proximity to the sample surface, the leakage resistance increases due to the current squeezing effect. Therefore, the effective resistance increases and the measured current drops. In the ideal case of a perfect seal between pipette and surface at zero tip-sample distance, the current becomes zero at contact. In the case of a conically tapered pipette tip with an opening angle that is not too large, the pipette resistance can be approximated by [15] RP ≈

ρ LP , πr0 ri

(3.8)

where r0 is the inner diameter of the thick end, ri is the inner diameter of the thin end (at the pipette tip), and L P is the length of the conical pipette (Fig. 3.4, inset). This formula can be derived by sectioning the cone into successive disks of infinitesimal thickness, and summing the resistances of all disks (this assumes that disk-shaped electrodes are placed in the openings of the thin and the thick end). Nitz et al. [15] constructed a simple analytical model for the distance-dependent leakage resistance RL (in the absence of channels in the sample, i.e. RC → ∞). They obtained RL (z) ≈

3ρ ln(ra /ri ) , 2πz

(3.9)

where ra is the outer diameter of the thin end (pipette tip) and z is the tip-sample distance. The measured current thus becomes & z 0 '−1 I(z) ≈ I0 1 + , (3.10) z where I0 = U/RP is the measured current in the absence of the sample (or with a large tip-sample distance) and z0 =

3r0 ri ra ln . 2L P ri

Fig. 3.4. Calculated current vs. distance (IZ) curve, based on a simple analytical model [inset and (3.10)]. The parameters used for calculating the curve are r0 = 0.3 mm, ri = 50 nm, ra = 85 nm, L P = 10 mm, ρ = 1.09 Ω m and U = 100 mV

(3.11)


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Equation (3.10) describes a current that drops to zero for z = 0 and that asymptotically approaches I0 for z → ∞. At z = z 0 , the current is half of that asymptotic value. For typical parameters (r0 = 0.3 mm, ri = 50 nm, ra = 85 nm, L P = 10 mm, ρ = 1.09 Ω m, U = 100 mV), we obtain I0 = 432 pA and z 0 = 0.52 nm. It becomes apparent that the current starts to significantly drop only at very small tip-sample distances (Fig. 3.4). 3.3.3 Finite Element Modeling In order to further characterize and understand current flow when the pipette tip is close to a sample, we modeled the system using finite elements [24]. Due to the numerical nature of this modeling, more complex configurations can be analyzed. For example, it becomes straightforward to include a channel in the sample. We chose a conical pipette with an inner and outer tip opening diameter of 100 nm and 200 nm, respectively, and with an inner opening half-angle of 6◦ . We placed the pipette electrode in the plane 1000 nm into the pipette, and the bath electrode in the plane 320 nm below the tip opening. A cylindrical sample disk with a diameter and thickness of 700 nm and 30 nm, respectively, was placed concentrically below the pipette. Optionally, a thin concentric channel of 40 nm diameter was inserted in this sample disk. The electrolyte was assumed to be a 0.1 M NaCl solution, with a specific electrical resistivity of ρ = 1.09 Ω m [25]. The pipette walls and the sample were assumed electrical insulators. The governing differential equation of this system is the Poisson equation for the scalar electrical potential φ, ∆φ = 0 ,

(3.12)

where ∆ is the Laplace operator. The problem was then solved self-consistently with finite element modeling. Dirichlet boundary conditions apply at the pipette and bath electrodes, between which the voltage of 0.1 V was applied. Homogenous Neumann boundary conditions apply at the pipette walls and the sample, as well as at the system boundaries. The dimensionality of the system in the modeling process was reduced from three to two spatial dimensions by utilizing the cylindrical symmetry. The current density vector j directly derives from the electric field vector E, which in turn directly derives from the scalar electric potential: j=

1 1 E = − ∇φ . ρ ρ

(3.13)

The current flowing through the pipette was then simply obtained by integrating the current density over an area A spanning fully across the pipette interior, where only components of the current density that are normal to the local area element were considered: j · dA .

I=

(3.14)

A

The dot denotes the scalar product. We modeled three different experimental configurations. First, we removed the sample from the system, leaving only the pipette

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in the bath solution. The results of the finite element modeling process for this configuration are displayed in Fig. 3.5a. The local current density vectors in the axial symmetry plane are symbolized as arrows. Additionally, the absolute value of the current density is encoded in the grayscale, where darker areas denote larger current densities. It is apparent that the current density inside the pipette close to its tip is large (∼ = 1 − 2 × 105 A/m2 ), and that it drops to almost zero soon after leaving the pipette opening (within a distance of about half of the tip opening diameter). The total current flowing through the pipette is ∼ = 1.41 nA (corresponding to Reff ∼ = 70.9 MΩ). In Fig. 3.5b, we consider a different configuration by positioning an insulating sample surface 20 nm away from the pipette tip. This case demonstrates the current squeezing effect, as the current now has to escape sideways through the narrow gap between pipette tip and sample surface. But this relatively small tip-sample distance (20% of the tip diameter) does not lower the total current through the pipette (∼ = 77.5 MΩ) = 1.29 nA, corresponding to Reff ∼ by a large amount, compared to the case in Fig. 3.5a. This behavior will be discussed more below. In Fig. 3.5c, we add a small concentric channel with a diameter of 40 nm to the sample. Some of the current can now escape through this channel, thereby decreasing the effective resistance of the system. The total current through the pipette increases again slightly to ∼ = 1.33 nA (corresponding to Reff ∼ = 75.2 MΩ). It becomes apparent that even at these relatively small tip-sample distances, the influence of the sample, with or without channels, is relatively small. For larger effects to occur, the pipette tip needs to be positioned even closer to the sample, as we will see in the following paragraph. We repeated these modeling steps for a series of tip-sample distances, and plotted the total current as a function of tip-sample distance (Fig. 3.6). In the absence of the conductive channel through the sample (dashed line), the current starts to drop significantly at tip-sample distances below 20 nm (about half of the tip radius). At zero distance, the current is zero. In the presence of the conductive channel (solid

Fig. 3.5. Finite element analysis of the current squeezing effect at the tip of the pipette. (a) Pipette tip in the absence of a sample. The current density is large inside the pipette close to its end, and drops to almost zero soon below the pipette tip. (b) Pipette tip in the presence of a solid insulating sample. This added constraint causes the current to “leak” through the gap between tip and sample. (c) Pipette tip in the presence of a sample with a concentric channel below the tip. The current takes two alternative paths: it “leaks” through the gap between the tip and sample, and it flows through the channel in the sample. With a decreasing tip-sample distance, an increasing amount of current flows through the channel in the sample. For all three images, the arrows symbolize local current density vectors and the grayscale quantizes the absolute value of the current density. The parameters of the model are listed in the main text. Grayscale range (white to black): 0 − 2 × 105 A/m2


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Fig. 3.6. Calculated current vs. distance (IZ) curve, based on a finite element analysis. Two cases are considered: a sample without a central conductive channel (dashed line) and a sample with a central conductive channel (solid line). In both cases, the current starts to drop significantly only at tip-sample distances below half of the pipette tip radius. In the presence of the channel, the current reaches a finite value at zero distance, while it drops to zero in the absence of the channel

line), the current behaves similarly, but does not drop below a minimum value (here: ≈ 0.95 nA). In this case, the leakage current is zero and all the current flows through the channel in the sample. 3.3.4 Experimental Current-Distance Curves Current-distance curves manifest the basic principles of SICM, and provide useful information about the tip-sample interaction. In experimental current-distance curves, the pipette tip is ramped towards the sample surface with the help of a piezoelectric actuator (in z-direction). Simultaneously, the pipette current is recorded. A typical curve is shown in Fig. 3.7. When the pipette tip is far away from the sample surface, the current is about constant and independent of the z-piezo position (right hand side of graph). When the tip-sample distance gets in the range of the opening radius of the pipette tip, the cross section of the current path will decrease due to the thin gap forming. The resistance therefore increases and the measured current decreases. In

Fig. 3.7. Measured current vs. distance (IZ) curve on a glass surface. The current significantly decreases from a constant value to almost zero when approaching the surface. The inner tip radius of the pipette was 150 nm [15]

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the case of soft samples, often there is a seal that forms between the tip and sample, causing the measured current to decrease to almost zero. This is in contrast to hard samples such as glass coverslips, where the current often does not reach zero at any tip-sample distance. The reason for this is that neither the tip nor the sample is deformable to even out a slightly tilted or rough pipette opening, so a gap of finite size will be maintained. In many cases, the current starts to increase again when the pipette is further lowered into the sample. This effect has been attributed to a bending of the pipette that increases the leakage current [15]. 3.3.5 Imaging with Ion Current Feedback The dependence of the measured ion current on the tip-sample distance can be utilized to control this distance. This allows scanning the tip laterally over a sample at a preset vertical distance. Generally, scanning is achieved with the help of piezoelectric actuators that move the tip relative to the sample in each of the three spatial dimensions. A feedback loop keeps the tip-sample distance constant by controlling the vertical piezoelectric actuator so that the measured ion current stays constant. Simultaneous scanning in lateral direction then generates a topographic image of the sample surface (Fig. 3.8). One characteristic of this ion current feedback is that topographic data is obtained without mechanical contact between tip and sample. Gentle imaging of very soft samples such as living cells is therefore possible. Korchev et al., for example, imaged the topography of living cells with SICM to monitor dynamic changes in cell volume at a resolution of 2.5 × 10−20 liter [26] and to localize single active ion channels on the cell surface [27]. The lateral resolution of SICM topography images, however, is typically limited to several dozen nanometers, set by the inner opening diameter of the pipette tip. In the next section, we will discuss improvements of the current feedback technique that allow an even better and more reliable tracking of the sample surface.

Fig. 3.8. Topography of a monolayer of MDCK cells on a glass support, imaged with SICM in ion current feedback. Vertical range: 8.2 µm. Image size: 26.4 × 26.4 µm2 . Experimental data by P. Heidenreich, University of Münster


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3.4 Advanced Techniques The original SICM technique utilized the distance-dependence of the ion current to control the tip-sample distance with the help of a feedback loop. Provided that no local or temporal variations in the conductivity of the electrolyte occur, the feedbackcontrolled scanner will keep the distance between the probe and the sample constant while scanning the surface. Although this approach has successfully been applied to image the topography even of delicate biological samples, some inherent issues and weak points remain. Some of the problems of the original SICM setup will be discussed in the following paragraph. Since the SICM relies on a measurement of the dc ion current flowing between the pipette and the bath electrode, its performance strongly depends on the long-time stability of the overall electrochemical conditions. Various factors can affect in particular the dc voltage and current between the electrodes. For instance, temperature drifts as they can easily occur during scanning for hours can cause significant dc offsets, which in turn have a direct effect on the current-based feedback system. Furthermore, drifts due to a gradual contamination of the electrodes over time can also degrade the performance of the scanning system, ultimately making it impossible to track the sample topography for a long time. If the SICM is to be combined with other electrophysiological investigations further problems may arise. For instance the application of drugs or additional voltages or currents can also have significant effects on the electrochemical potential between the SICM electrodes. Using the dc ion current as a feedback signal might therefore be impractical. Another inherent weak point of a current-based feedback system results from the geometric shape of the pipette’s tapered end. In order to obtain a significant current squeezing effect when the probe is brought into close proximity to the sample surface, the tapered end should be even and flat. Such a leveled probe typically shows a strong dependency of the ion current on the vertical tip-sample distance. At the same time, however, this geometry makes the probe practically insensitive to protrusions on the sample surface if they are approached from the side during lateral scanning. As a consequence, the probe runs into these protrusions and then bends due to the lateral movement of the scanner. Such a bending will result in a small tilt of the opening of the pipette with respect to the sample surface, leading to a reduction of the squeezing effect and thereby to an increase of the ion current. The feedback loop of the SICM will then force the probe even further toward the sample, ultimately causing a tip crash and damaging both sample and probe. For the same geometric reasons the probe can easily get trapped in pits or pores that might be present on the sample surface. In general, rough surface morphologies are therefore difficult to scan in a stable manner with a simple current-based feedback system. As a first remedy the average distance between the probe and the sample could be kept rather large by simply adjusting the current setpoint of the feedback loop accordingly. Such a workaround, however, would degrade the sensitivity and the resolution of the SICM. Therefore, advanced techniques are required to overcome the inherent technical limitations of a simple current-based feedback system. A number of different approaches have been conceived to achieve this goal. They can be subdivided into two classes: SICM systems of the first class still rely on using the

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ion current as feedback signal for distance control, but apply special modulation techniques to minimize effects of the afore mentioned issues. We will discuss those techniques in the following paragraphs. In the second class of advanced SICM systems, the original concept of a current-based feedback system is abandoned. Instead, the SICM is provided with a current-independent feedback system to control the tip-sample distance. The local ion current is no longer used for distance control but recorded simultaneously with the surface topography as it is measured by a separate feedback system. Typical implementations are based on concepts well-known from other scanning probe techniques such as dynamic mode atomic force microscopy or shear force concepts. We will discuss them in detail in Sect. 3.5. 3.4.1 Modulation Methods Different methods for improving the performance of an SICM while retaining the original idea of using the local ion current as feedback signal for controlling the tip-sample distance have been introduced. In particular, modulation techniques have proved to be effective for minimizing the inherent problems of a current-based feedback loop. They share the common idea that the risk of hitting a surface feature during lateral scanning can be effectively reduced by modulating the tip-sample distance. Furthermore, modulation techniques can be useful to overcome limitations of the original SICM concept regarding long-term stability as they are caused by dc offsets and drift effects. Gitter et al. [28], Mann et al. [29] and Happel et al. [30] employ a type of pointspectroscopy technique as it is know from other scanning probe methods such as atomic force microscopy [31]. Here the probe is approached to and then retracted from the sample surface by typically several micrometers in each data point of the scanning area. Thus the lateral movement of the scanner can be performed while the probe is far away from the surface. Consequently, the risk of the probe laterally colliding with a protrusion on the surface or it getting trapped in a depression is reduced. However, this comes at the price of slow scanning speed, since the probe is not tracking the surface under continuous control of a feedback loop. Instead in each data point the probe is approached toward the sample until a predefined trigger point is reached, i.e. until a decrease of the ion current below a set threshold value signals that the probe is in close proximity to the surface. The z-position of this trigger point is stored and then the vertical movement is reversed and the probe retracted from the surface. This type of point-spectroscopy can practically be based on any type of trigger signal, for instance the “squeezing” effect of the dc ion current as it occurs typically a few tens of nanometers above the surface. While the point-spectroscopy approach provides an effective workaround for overcoming the inherently low lateral sensitivity of the probe, it does not address problems caused by time-varying dc offsets. To minimize such long term drift effects additional measures have to be taken. One option is applying either short voltage or current pulses [29, 30] instead of applying a constant voltage between the pipette and bath electrodes. This reduces unwanted effects caused by slowly changing dc potentials at the electrodes. Another method to improve the performance of a SICM with a current-based feedback


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Fig. 3.9. Setup of a distance-modulated SICM. The tapered pipette is attached to a piezo-electric element which modulates the distance between the probe and the sample surface. Typically a modulation frequency in the range of a few hundred Hertz and a modulation amplitude of a few tens of nanometers is applied. If the tapered end of the probe is in close proximity to the surface the resulting leakage current IL comprises a dc and an ac component. The amplitude of the latter depends on the average distance between the probe and the sample an serves as the input signal of a feedback loop

system has been proposed by the group of Shao [17, 21] and by the group of Korchev [16]. They keep the voltage between the two electrodes constant, but modulate the z-position of either the sample or the probe with an amplitude of a few tens of nanometers by means of a piezo-electric element (Fig. 3.9). If the probe is far away from the sample surface such a small distance modulation with a typical frequency of a few hundred Hertz does not significantly affect the ion current. If the tapered end of the pipette, however, is brought into close proximity to the surface so that the “squeezing effect” sets in, such a distance modulation leads to a modulation of the ion current. The amplitude of the resulting ac component in the ion current can then be detected with a lock-in amplifier. The latter allows recovering even noisy, low-level ion current signals by using the voltage signal which modulates the tip-sample distance as a reference signal. The detected amplitude of the modulated ion current is then used as feedback signal to control the average tip-sample distance, i.e. the feedback system tries to maintain a constant amplitude of the ion current while scanning the surface. This method of distance control has advantages over the conventional dc current-based approach because it makes the measurement less sensitive to changes in ionic strength or other dc drift effects.

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3.4.2 Applications in Bioscience Advanced modulation techniques as they were discussed in the previous section allow to scan delicate biological surfaces such as living cells more gently. The improved performance of a distance-modulated SICM allowed Gorelik et al. to obtain well resolved images of fine surface structures such as microvilli on living cells as depicted in Fig. 3.10 [32]. Since the introduction of the distance-modulation technique reduces the risk of uncontrolled contact between the probe and the sample and also improves long-term stability, it becomes possible to continuously image specific surface areas for several hours and thereby to study dynamic processes [33, 34].

Fig. 3.10. Monolayer of A6 epithelial cells imaged by distancemodulated SICM. The fine structure on the cell surface is well resolved and shows numerous microvilli. Reprinted with permission from Gorelik et al. [32]

Fig. 3.11. SICM scan of an A6 cell surface (left) and ion channel recording on top of a single microvillus (right). The pipette’s position is marked by an arrow in the line profile. The channel was identified as a K+ channel on the basis of the reversal potential of the current-voltage curve. Reprinted with permission from Gorelik et al. [32]


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Another type of application is to employ the SICM for performing patch-clamp experiments at specific surface sites [32, 34]. In such experiments the SICM is first used to scan the sample surface. The spatially resolved images of the surface topography allow identifying regions or structures of interest. The scan unit is then used to position the tapered SICM probe over a user-selected spot. Finally, the probe is engaged to the sample surface to form a gigaohm seal for subsequent patch-clamp recording. This combination of high-resolution SICM scanning and patch-clamping allows obtaining ion channel recordings on selective surface spots with lateral position control in the range of a few tens of nanometers. Such high precision is out of reach with conventional positioning methods such as those using light microscopy. With the SICM it becomes possible to investigate the activity of single ion-channels on nanometer-sized structures on native biological samples such as living cells. For instance, the tapered SICM pipette can be deliberately positioned on top of a single microvillus on an epithelial cell and used for subsequent patchclamp recording (Fig. 3.11).

3.5 Combination with Other Scanning Techniques The ability of the SICM to image biological samples in vivo with a resolution in the range of a few tens of nanometers makes it an interesting candidate for to combine with other scanning microscopy techniques. In doing so different strategies can be pursued: the first category of techniques sticks to the idea of using the ion current signal to maintain a constant distance between the probe and the sample while scanning. In addition a complementary data channel, for instance local optical data, is simultaneously recorded. Combinations of SICM with scanning nearfield optical microscopy (SNOM) or scanning confocal microscopy (SCM) fall in this category. The groups of Korchev and Shao modified the original SICM setup in such a way that the end of the tapered pipette also serves as a near-field light source for SNOM [21, 35, 36]. This can be achieved by coupling laser light into the pipette via an optical fiber. Coating the outside of the pipette with a reflective metal layer helps to confine the laser light to the aperture, i.e. the tapered pipette’s end. Provided that the sample and the substrate are transparent the SNOM signal can then be collected through an objective and detected by a photomultiplier located underneath the SICM head. Living cells were successfully investigated with such a combined SICM/SNOM setup by Korchev et al. An alternative way to utilize the SICM probe as a confined light source for SNOM was suggested by Bruckbauer et al. [37,38]. The method is based on fluorescence as it occurs when a calcium indicator, with which the pipette is filled, binds with calcium in the sample solution and is illuminated with a laser. The mixing zone where the fluorescent complex forms serves as a localized light source. SICM has also been successfully combined with SCM [39]. The setup comprises an inverted light microscope fully configured for scanning confocal microscopy, on which the SICM head is placed. During lateral scanning the vertical position of the sample is controlled by a standard SICM, i.e. ion current based feedback loop. As a consequence the optical confocal volume, which is located just below the pipette’s

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end, follows the topography of the sample. This allows capturing fluorescence images of a surface simultaneously with topographic data. Thereby interaction of fluorescent nanoparticles, for instance virus-like particles, with the surface of fixed or living cells can be studied. An alternative strategy for combining SICM with other scanning microscopy techniques abandons the original idea of an ion current based feedback system. Instead, other distance control schemes are employed to keep the scanned probe at a fixed, predefined distance to the sample surface and then to record local variations in the ion current as they might be caused by variations in local ion-conductivity or sample-permeability for instance. So far, we have considered only distance-control methods that are exclusively based on the measurement of the ion conductance. This works well for samples with a homogenous distribution of ion conductance. In many cases, however, it is of interest to study inhomogeneities in the local ion conductance, such as those that arise due to the presence of channels through a thin porous membrane, for example. In these cases, a mechanism independent from the ion conductance is required to keep the tip-sample distance constant. In the following sections we discuss different methods that solve this problem and present applications using these methods. 3.5.1 Combination with Atomic Force Microscopy One possibility for a simultaneous measurement of ion current and sample topography is based on the use of a bent nanopipette as a cantilever [12]. With such a configuration, the sample topography can be imaged using a standard AFM measurement setup. Simultaneously, the ion current can be monitored and recorded as a complementary measurement signal. The invention of the tapping mode in liquid [40, 41], which simplified imaging of soft samples in solution, initiated the development of a novel microscope: Tapping-mode AFM combined with a SICM [12]. In this design, a bent nanopipette [42, 43] is used as both force sensor and ion conductance probe (Fig. 3.12). This bent nanopipette is vibrated perpendicular to the sample with the help of a piezo stack. Typical vibration amplitudes at the tip are ≈ 10 nm rms. The measured vibration amplitude of the pipette serves as input signal to a feedback loop that controls the pipette-sample distance, thereby generating the topography signal when scanning. Simultaneously, the ion current is recorded and used to generate a complementary image of the ion conductance. While such a microscope can also be operated in contact mode (using the dc deflection of the pipette), better resolution (both in topography and ion conductance) is generally obtained in tapping mode due to the absence of lateral imaging forces. 3.5.2 Application in Material Science In this section, we will present an application of a tapping mode SICM in the field of biomineralization. This field deals with the processes by which living organisms synthesize organic-inorganic composite materials. These materials often exhibit elaborate micro- and nanostructures. They usually contain just a small


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Fig. 3.12. Combined scanning ion conductance microscope and tapping-mode atomic force microscope. A bent nanopipette is used as both an ion conductance probe and an atomic force microscope cantilever. Tapping mode operation enables topographic imaging of soft samples. The tip-sample distance is thereby kept constant by mechanically vibrating the pipette and using feedback to keep the vibration amplitude constant. The vibration amplitude of the cantilever is detected with nanometer precision using the optical beam deflection method. The pipette is coated with a layer of gold (thickness ≈ 30 nm) to increase its reflectivity. Simultaneously to the topography, the ion current through the pipette is recorded, thereby generating complementary images of topography and local ion conductance

amount of biopolymers, but those biopolymers play a critical role in the assembly of the mineral. Furthermore, these organic-inorganic composite materials often exhibit extraordinary mechanical, electrical, magnetic and chemical properties. By improving the knowledge about the mechanisms of biomineralization, new models might be created that aid in the artificial synthesis of new, high-performance materials. The example we will consider here is the formation of nacre (“mother of pearl”) by the marine mollusc Haliotis rufescens (red abalone). There has been a great interest in this material, and significant progress in the elucidation of its structure and function has been made over the last years. Nacre is a micro-laminate composite of highly oriented calcium carbonate (CaCO3 ) crystals in the aragonite polymorph and an organic matrix in the form of thin organic sheets, probably consisting of mostly proteins and polysaccharides. This composite material exhibits an intricate micro- and nanostructure of high regularity: it consists of an interdigitating brickwork array of crystal tablets and organic matrix. This composite structure is one of the reasons for the exceptionally large fracture toughness of nacre, which is ≈ 3000× larger than that of pure mineral. Various models have been set up that explain the fabrication process of nacre. With the use of electron diffraction microscopy and atomic force microscopy, it has

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Fig. 3.13. Schematic of a simple model of nacre growth, in which aragonite crystal tablets grow through pores in organic interlamellar sheets. SICM was used to proof the presence of such pores in the organic sheets, and to image their distribution and conductivity. (a) Side view. (b) Top view. Reprinted with permission from Schäffer et al. [13]. Copyright (1997) American Chemical Society

been found that the tablets are single crystals, and that the crystallographic axes of these crystals are nearly identical within several layers of subsequent tablets. This observation has resulted in a prominent model that explains the formation of nacre by a process called “hetero-epitaxial nucleation”. In this process, the growth of the crystal is periodically capped by an interlamellar organic layer, onto which a new crystal nucleates in a hetero-epitaxial fashion. Alternatively, abalone nacre may be formed by a continuous crystal growth of the mineral tablets through pores (channels) in the interlamellar organic sheets. Significant indirect and direct evidence has been favoring this alternate model that is based on growth through “mineral bridges” (Fig. 3.13). SICM was one of the techniques that generated strong evidence for this alternate model. For that purpose, demineralized interlamellar organic sheets were prepared from abalone nacre by decalcification. These organic sheets were imaged with a SICM in tapping mode operation without drying, fixing or coating the sample. The tapping mode operation ensured that a constant tip-sample distance was maintained. There are areas of high and low ion conductance on the organic sheet (Fig. 3.14a). In some areas with a predominantly low ion conductance, there are small features of locally enhanced ion conductance (Fig. 3.14b). With the help of a statistical analysis, these features were attributed to individual pores in the organic sheet. Several observations support this conclusion. First, the average ion conductance of these features is 28 ± 11 nS (after subtracting a conductivity background of 50 nS that is caused by leakage). From this quantity, the diameter of the pores in the organic sheet with a thickness of ≈ 30 nm calculates to 15–40 nm, which is in agreement with observations using atomic force or scanning electron microscopy. Second, an estimate of the pore density in the organic sheet can be found from the integrated conductance over the scan area. This estimate, 58 ± 12 pores/µm2 , is in the same range as the pore density seen in AFM images. Third, calcified features


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Fig. 3.14. SICM images of demineralized interlamellar organic sheets of abalone nacre. (a) Regions of small (dark) and large (bright) ion conductance can be seen, proving that the sheets are permeable for ions. Scalebar: 1 µm. Grayscale range (black to white): 50–200 nS. (b) Expansion of the box in (a). Some areas with predominantly low ion conductance show small, distinct features that might represent individual pores (arrows). Scalebar: 400 nm. Grayscale range (black to white): 50–200 nS. Reprinted with permission from Schäffer et al. [13]. Copyright (1997) American Chemical Society

that could be mineral bridges were observed using transmission electron microscopy (TEM). The realization that there are pores in the interlamellar organic sheets has several important consequences. On one hand, these pores can transport ions across those sheets, and possibly even soluble proteins that are necessary for growth control of the crystal tablets. But most importantly, there is the possibility that the crystal itself grows through those pores, thereby forming mineral bridges through the interlamellar organic sheets. This alternate model of nacre growth readily explains several experimental observations, among those the mentioned coherence of the crystallographic axes between consecutive tablets. 3.5.3 Combination with Shear Force Microscopy Another possibility for a simultaneous measurement of ion current and sample topography lies in the use of the nanopipette as shear force sensor. Shear force sensors are well known from SNOM setups, where they serve to keep an optical fiber at constant distance to the sample during scanning [44, 45]. In a shear force configuration, the nanopipette is vertically oriented with respect to the sample surface (Fig. 3.15). A small piezo-electric stack that is attached to the nanopipette excites transverse mechanical vibrations in the drawn-out end of the nanopipette (parallel to the sample surface). The vibration amplitude strongly depends on the tip-sample distance due to shear forces. The shear forces increase sharply at small tip-sample distances, thereby decreasing the vibration amplitude. The vibration amplitude is therefore well suited as a measure for the sample topography. The technical challenge is finding a method of detecting this vibration amplitude on the nanometer scale. Several methods have been established so far, including optical detection [44] and detection using a piezo-electric tuning fork sensor [46]. Piezo-electric detection

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Fig. 3.15. Schematic illustration of the setup of a combined scanning ion conductance and shear force microscope. The lateral oscillatory motion of the probe’s tapered end is induced by a piezo element to which the micropipette is attached. By focusing a laser beam on the tapered probe and detecting the diffracted light with a segmented photodiode the mechanical oscillation is converted into an electrical signal. This is analyzed by means of a lock-in amplifier whose reference channel is synchronized with the driving signal of the dither piezo. The amplitude of the probe’s oscillation serves as the input signal of a feedback loop. The latter controls the vertical position of the scanner in such a way that a constant, predefined oscillation amplitude is maintained during xy-scanning. As a consequence the SICM probe follows precisely the sample topography. Simultaneously the local ion current flowing between the electrodes in the micropipette and under the porous sample is recorded. Thus possible variations in the local ion conductivity can be analyzed along with the surface topography

methods face some principal problems when they are used for imaging in liquids, since electrical shortcuts may occur in the electrolyte medium. Solutions include coating the tuning fork with an electrical insulating layer [47], applying a custom piezo-electrical detection [48], or using a diving bell concept, where a water-air interface is forced close to the sample surface [49]. On the other hand, optical detection of shear forces principally works in liquid just as well as in air [50]. In the optical detection method, a laser beam is focused onto the vibrating section of the nanopipette near its tip (Fig. 3.15). The incident laser beam is scattered by the vibrating tip, thereby modulating the intensity distribution of the beam. An optoelectronic detector, usually based on a split photodiode, detects these intensity modulations of the scattered beam and generates a signal that is proportional to the vibration amplitude of the nanopipette. This signal is then fed to the scan controller that attempts to keep the signal constant with the help of a feedback loop, thereby generating a topographic image of the sample surface when scanning. Simultaneously, the ion current from an electrode below the sample to the electrode in the nanopipette is recorded and used to generate a complementary image of the ion conductance through the sample. One advantage of the shear force method over the tapping-mode method (Sect. 3.5.1) for measuring the tip-sample distance is that the pipettes need not go


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Fig. 3.16. Shear force topography image of a layer of fixed MDCK cells in buffer solution. Vertical range: 7.4 µm. Image size: 40 × 40 µm2 . Sample by S. Schrot, experimental data by M. Böcker, University of Münster

through the tricky process of being mechanically bent at their tip. One could think that the relatively large effective vertical spring constant of a vertically oriented pipette could impose a possible thread to soft samples. On the other hand, the non-contact nature of shear force imaging allows scanning of surfaces with small tip-sample interaction forces. It is possible to image soft samples as delicate as cells in liquid with a shear force setup (Fig. 3.16). The successful implementation of a combined scanning ion conductance and shear force microscope was demonstrated by Nitz et al. [15]. Using such a setup (Fig. 3.15), they acquired simultaneous images of topography and ion current on a polycarbonate membrane (Fig. 3.17). The locations of the pores can clearly be seen in the topography image as local depressions. The complementary ion current image shows an increased ion current at those locations. As further test samples, these researchers also used porous mica surfaces. Porous mica surfaces with defined pore sizes can be produced by a track-etching process [51]. In this process, a mica sample is irradiated with high energy ions, thereby inducing

Fig. 3.17. Complementary (a) shear force topography and (b) ion current images of a polycarbonate membrane with pores of 0.4 µm diameter. Image size: 6 × 6 µm2 . Reprinted with permission from Nitz et al. [15]

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Fig. 3.18. Complementary (a) shear force topography and (b) ion current images of a Muscovite mica surface in which rhombic pores with an opening area of 200×400 nm2 were inserted by ion radiation and subsequent etching with hydrofluoric acid. Image size: 3.2 × 3.2 µm2 . Reprinted with permission from Nitz et al. [15]

metastable states in the crystal lattice within a radius of 100 to 1000 nm from the ion trace. Subsequent etching with hydrofluoric acid produces rhombic pores with a diameter that can be influenced by the acid temperature and by the etching time. Simultaneous images of topography and ion current on such a porous mica surface show an increased ion current over the pores in the sample (Fig. 3.18), again demonstrating the feasibility of the shear force approach. 3.5.4 Application in Bioscience The combined scanning ion conductance and shear force microscope is an ideal tool for investigating local variations in the ion conductance of biological specimen. This is of special interest for research in the field of the barrier-forming structures such as endothelial or epithelial cell layers. In multicellular organisms such structures form the interface between different fluid compartments and play an important role for inter- and transcellular processes [52, 53]. Gaining insight into the complex barriercrossing transport mechanisms is a common interest of cell biology, medicine, and pharmacology, as malfunctioning of these barriers leads to pathological implications. In particular, knowledge about the permeability of barriers for substances such as drugs is of great relevance. Hence special electrochemical and microscopic methods are required to study the ion-permeability of barrier-forming cell structures. For instance, experimental techniques such as the measurement of transepithelial electrical resistances (TER) provide valuable information about the barrier properties of cell layers [54, 55]. In addition to such integrating measurements of the total cell layer impedance, the combined scanning ion conductance and shear force microscope can provide further insight into cellular transport mechanisms, because it allows the simultaneous recording of topographic data and local ion-conductance with high lateral resolution.


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Fig. 3.19. Complementary (a) shear-force topography and (b) ion current images of a layer of live MDCK-II cells. Some but not all of the cell-cell contacts are visible in topography. The simultaneously recorded current image shows an increased ion-permeability over tight-junctions, independent of the local surface corrugations. Image size: 22 × 22 µm2 . Sample provided by J. Seebach, experimental data by J. Kamp, University of Münster [66]

As depicted in Fig. 3.19, the shear-force-combined SICM has successfully been used to investigate intercellular structures, namely tight-junctions between living cells. The results show the benefits of having two independent information channels: while the topographic shear-force data alone does not permit to differentiate between a permeable ion-channel and a closed cavity in the cell layer, the ion current image provides this important piece of information. The possibility of this type of combined microscope to recognize ionic transport channels in hard as well as soft samples opens perspectives for further applications in fields as diverse as biochemistry/pharmacology, corrosion research, quality control of coatings or artificial membranes.

3.6 Outlook While the SICM itself already has a broad range of possible applications, further techniques which are based on similar components can be envisioned. For instance the tapered micropipette of the SICM can be used not only as a high-resolution scanning or patch-clamping probe for in vivo characterization of biological surfaces, but also as a source for localized material delivery. The material flux from the probe to the sample substrate can either be driven by electric fields, capillary forces, hydrostatic pressure, or by ultrasonic excitation. Such type of nanolithography has been successfully applied to deposit metals [56–59], chemicals [60, 61], and also biological material such as DNA, proteins or other biomolecules [62–65]. A possible field of application is the construction of nanostructured biosensors. Furthermore, having the ability to deliver material in a controlled manner through the micropipette of a SICM opens the perspective to study the effects of local drug delivery on

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biological specimen, providing a nanoscopic tool for drug-induced manipulation of individual cells. Acknowledgements. We thank M. Böcker, P. Heidenreich and J. Kamp for contributing to this chapter with their experimental results. We thank H.-J. Galla, A. Janshoff, M. Madeja, S. Schrot, M. Schwarz, J. Seebach, E.-J. Speckmann, C. Steinem, J. Wegener for support and discussions. This research was supported in part by the German Federal Ministry of Education and Research (BMBF), and by the Deutsche Forschungsgemeinschaft (DFG) by the Gemeinnützige HertieStiftung/Stifterverband für die Deutsche Wissenschaft (SCHA 1264).

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4 Spin-Polarized Scanning Tunneling Microscopy Wulf Wulfhekel · Uta Schlickum · Jürgen Kirschner

Summary. We give an introduction to spin-polarized scanning tunneling microscopy (Sp-STM), a magnetic imaging technique with nanometer lateral resolution. Sp-STM allows to record constant current images which represent the electron density near the sample surface and simultaneously the spin polarization of the electron density which is related to the magnetic moment. It is shown how magnetic and topographic information can be separated using a modulation technique of the magnetization of bulk ferromagnetic tips. It is demonstrated that the out-ofplane component as well as one well-defined in-plane component of the spin polarization can be recorded. Finally, it is shown that with Sp-STM valuable information on the spin-resolved electronic structure and on the fundamental processes of tunneling spins may be obtained.

4.1 Introduction The imaging of magnetic domains in small structures is one of the important basis of modern micro-magnetism. While the existence of magnetic domains was initially not directly shown with an imaging technique, but was deduced from the observation of discontinuous jumps in the hysteresis loops [1], magnetic imaging has become the method of choice when investigating magnetic structures. Since the early works of Barkhausen, several magnetic imaging techniques, i.e. techniques that map one or more components of the magnetization or related quantities, were developed. The measured quantity can be e.g. the magnetic vector potential, the magnetization, the magnetic stray field or the spin polarization. We here focus on spin-polarized scanning tunneling microscopy (Sp-STM), which maps the spin polarization of a conductive sample surface. 4.1.1 The Resolution Problem in Magnetic Imaging In modern applications, e.g. magnetic recording media or magnetic random access memory, the size of the magnetic structures or domains is in the range of several 10 nm and the trend of miniaturization is still holding on. This poses some severe resolution problems to most established magnetic imaging techniques. For example, magnetic force microscopy (MFM) is intrinsically limited in lateral resolution by the physical effect used to obtain magnetic contrast. MFM relies on the long range magnetic dipolar interaction of a magnetic tip and the magnetic charges of the sample. To obtain reasonable forces, the magnetic volume of the tip has to be of sufficient

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size. Further, the magnetic tip has to be lifted several nm above the surface to avoid other interactions like the van-der-Waals force to cover up the small magnetostatic forces. These constrains in localization of the interaction lead to a limit of the MFM resolution around several 10 nm [2]. Resolution limits of similar size exists for most other magnetic imaging techniques. More details on this can be found in [2]. To further develop new magnetic recording media, alternative magnetic imaging techniques are required. In fundamental studies of magnetism, there exists a similar need for high resolution magnetic imaging. Many fundamental questions in magnetism are linked to processes that operate on the atomic scale and are beyond the lateral resolution of the established techniques. At the forefront of nano-science, high resolution imaging techniques are in general of high interest and impact. Sp-STM may contribute substantially in the study of magnetism on the nanometer scale, as its resolution is given by the lateral extension of the tip area that contributes to the tunneling process. Under favorable conditions, this may lead to sub-nanometer lateral resolution [3, 4]. We illustrate the capabilities of this new technique and compare it to MFM. 4.1.2 Magnetism and Spin The key to Sp-STM lies in the relation of magnetism and the electronic structure of matter. In ferromagnets, the spin sensitive exchange interaction between localized electrons (Heisenberg model) or electrons in a delocalized electron gas (Stoner model) splits the electronic density of states into minority and majority densities (see Fig. 4.1a). This is in contrast to paramagnetic substances, where the distributions of spin-up and spin-down electrons are identical. In ferromagnets, the spin splitting leads to an imbalance of the total occupation of the electrons of different spins and as a consequence to the magnetic moment of the atom. The direction of magnetization of the ferromagnet is in general collinear to the direction of the spin polarization, which relates the magnetization to the spin polarization. A measurement of one or more of the components of the spin polarization can give information on one or more components of the direction of magnetization, respectively. 4.1.3 The Tunneling Magnetoresistance Effect The splitting of the density of states has immediate consequences on the tunneling current. Pioneering field-emission experiments with ferromagnetic tips have shown [5, 6] that during the tunneling process from the tip into the vacuum, the spin polarization of the density of states of the tip is partly transferred to the emitted electrons. A spin-polarized current was observed. This can easily be understood on the basis of Fermi’s golden rule. The tunneling process itself, i.e. the transmission probability through the potential step of the work function in front of the tip, is not spin dependent. The observed polarization is just a consequence of the different number of states for minority and majority electrons: the more states are allowed to tunnel, the higher the resulting tunneling current. In this simplified picture, the tun-

4 Spin-Polarized Scanning Tunneling Microscopy

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neling current from a spin-polarized tip is spin-polarized according to the imbalance of electrons with spin-up and spin-down. Jullière discovered that when electrons tunnel between two ferromagnets, not only the current is spin-polarized but also the size of the current is influenced [7]. In his experiment, two magnetic films were separated by an insulator film to form a planar tunnel junction. The two magnetic films were chosen to have different coercive fields. This permitted to align their magnetization parallel or antiparallel as a function of an applied magnetic field. Jullière found that the tunneling conductance G (and by this the resistance) depends on the relative orientation of the magnetization of the two layers. This effect is called the tunneling magnetoresistance (TMR) effect. For parallel orientation G was higher than for antiparallel orientation. This finding can be explained on basis of a simple model. As above, we neglect any spin dependence in the transmission through the barrier and focus solely on the density of states of the two electrodes. Under the assumption of a small bias voltage across the junction and in the absence of spin-flip scattering, the density of states in the ferromagnets near the Fermi energy determine the tunneling conductance. For parallel orientation, the majority/minority electrons of the first electrode tunnel into the majority/minority states in the second electrode, respectively, as sketched in Fig. 4.1b. Using Fermi’s golden rule, the conductance G is proportional to the density N of initial (i) and final (f ) states at the Femi edge. Combining both spin channels, the conductance for parallel oriented magnetizations is given by: f

f

G ↑↑ ∝ N↑i N↑ + N↓i N↓

(4.1)

Fig. 4.1. (a) Spin-split density of states N of a ferromagnet. ↑/↓ indicate majority/minority states. (b), (c) Tunneling between two ferromagnetic electrodes. In (b) respectively (c) the magnetization of the two electrodes is parallel respectively antiparallel. The conductivities G for tunneling from the left to the right electrode are indicated by arrows


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For antiparallel orientation (see Fig. 4.1c), electrons of majority character in one electrode tunnel into states of minority character in the other electrode and the conductance is given by a mixed product: f

f

G ↑↓ ∝ N↑i N↓ + N↓i N↑

(4.2)

These two conductivities in general differ leading to a variation of the tunneling current with the relative orientation of magnetization of the electrodes. Later, Slonczewski treated the problem of spin-polarized tunneling more rigorously [8]. Neglecting higher order spin effects like spin accumulation, he calculated the dependence of G on the angle θ between the magnetization of the two electrodes. With the spin polarization P = (N↑ − N↓ )/(N↑ + N↓ ), and neglecting barrier effects, the conductance is given by: G = G 0 (1 + P i P f cos θ)

(4.3)

Slonczewski’s calculations for the angular dependence of the TMR effect was later experimentally confirmed [9]. In general, if a finite bias is applied, all states between the two Fermi levels are involved in tunneling. They have to be weighted according to their tunneling probability, which is among other things energy dependent. This scenario is more complex but G can be expressed using effective, i.e. correctly weighted, densities or polarizations. More details on this will be given in Sect. 4.6.1.

4.2 The Principle of Spin-polarized Scanning Tunneling Microscopy With the invention of STM, Binning et al. realized an imaging technique with a lateral resolution that is capable to resolve single atoms [10, 11]. In STM, the apex of a conductive tip is placed near the surface of a conductive sample. Between sample and tip, a bias voltage is applied and a small tunneling current flows, that exponentially decays with tip sample separation. In the constant current mode of STM, a feed back mechanism adjusts the tip sample distance such that a constant tunneling current is obtained. When the tip is scanned over the surface, the tip apex moves on lines of constant current. In the simplest model, these lines are related to lines of constant density of states, i.e. reflect the sample topography. Already in 1988, Pierce suggested to set up a STM that uses the TMR effect to image simultaneously the sample magnetization [12]. All that is needed in addition to convential STM is a spin-polarized tunneling current. He suggested two different approaches to SpSTM. Besides the obvious use of ferromagnetic, i.e. spin polarized tips, he discussed the possibility to photo-excite spin-polarized carriers in GaAs tips. The latter was realized by Suzuki et al. [13]. In their approach, circularly polarized light was used to excite spin-polarized carriers into the conduction band of the tip that then tunnel into the sample. The spin polarization of the electrons can be selected by the helicity of the light. By modulating the helicity, modulations in the tunneling current were induced due to spin dependent tunneling. The modulations were detected with a lock-in amplifier to separate spin information from topographic information. The success of


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this approach was limited. It suffered from a rather low contrast and an unintended additional magneto-optical contrast of low lateral resolution. In this chapter, we focus on Sp-STM experiments with ferromagnetic tips. This approach was more successful. In the pioneering publication of Pierce, three different imaging modes of Sp-STM were suggested. All have been realized experimentally and are briefly explained below. We will, however, focus on only one of the three imaging modes. A detailed description of all imaging modes was given recently, elsewhere [14]. 4.2.1 The Constant Current Mode In the constant current mode, STM images are taken with non magnetic and ferromagnetic tips and the results are carefully compared. Wiesendanger et al. were the first to report results obtained with this mode [15] on the layer-wise antiferromagnetic Cr(001) surface [16]. Using tungsten tips, topographic constant current line-scans revealed atomic steps on Cr(001) of the expected step height of 0.14 nm while using a ferromagnetic CrO2 tip, alternating step heights of 0.16 and 0.12 nm were observed. This was attributed to the TMR effect between the ferromagnetic tip and the ferromagnetically ordered Cr atoms on the terraces. When the spin polarizations of the tip and the Cr terrace atoms are parallel, the tunneling current is enhanced due to the TMR effect (see (4.3)) and in the constant current mode of the STM, the tip is retracted by a small amount (0.02 nm). On the adjacent atomic terrace on Cr, the spin polarization of the terrace atoms is opposite due to the topological antiferromagnetic order of Cr(001) [16]. Therefore, on this terrace the TMR effect leads to a reduction of the current and the STM tip approaches slightly. This mechanism results in alternating step heights seen with a spin-polarized tip. However, no separation of topography and spin information could be obtained in this imaging mode and reference measurements had to be acquired with non-magnetic tips. 4.2.2 The Spectroscopic Mode The spectroscopic mode of Sp-STM allows under certain circumstances the separation of topographic information from spin information. This mode is also named spin-polarized scanning tunneling spectroscopy (Sp-STS). It was initially suggested by Pierce [12] and Stroscio et al. [17] and was first realized by Bode et al. [18]. It uses the fact that the spin polarization of the tunneling current is a function of the energy, i.e. the spin polarization of the states that contribute to tunneling depends on the sample bias. For example, when a finite negative sample bias U is applied between tip and sample, in principle the occupied sample states in the range of width U below the Fermi level of the sample contribute to the tunneling. In the tunneling process, the electrons tunnel into the unoccupied tip states of the range U above the Fermi level of the tip. The spin polarization of both, the tip and the sample states, contribute to the tunneling. Therefore the spin polarization of the tunneling current in general varies with sample bias. To illustrate the mode of operation, let us assume that the spin polarization increases with U. When the magnetic surface is

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imaged, the topographic image contains both information on the electronic density and on the spin, as mentioned above. Any variations of the conductance G due to the TMR effect are compensated by changes in the tip sample distance. In Sp-STS, the feed back loop of the STM is switched of and U is increased. As a consequence, the tunneling current changes. As we have assumed a rising spin polarization, the tunneling current for parallel oriented tip and sample magnetization increases more than for antiparralel orientation. Pierce et al. suggested to vary U significantly. Nevertheless, the imaging mode also works for small modulation of U. In this case, the variations in the tunneling current are proportional to the differential conductance dI/dU. In the pioneering experiment by Bode et al., the observed peak height of the spin-split surface state of Gd(0001) in laterally resolved dI/dU spectra was used to obtain magnetic information [18]. The imaging mode is now widely used. It has the advantage that tips coated with a thin film of a ferromagnet or antiferromagnet can be used so that the magnetic stray field of the tips can be minimized or completely be avoided [19, 20]. The obvious disadvantage is that the dI/dU signal in Sp-STS only weakly depends on magnetism. It much stronger depends on general variations of the density of states caused by e.g. compositional, structural or morphological changes of the sample. A more detailed overview on this imaging mode can be found in [2]. 4.2.3 Differential Magnetic Imaging Mode In the differential magnetic imaging mode, a bulk ferromagnetic tip is used whose magnetization is modulated. Due to the TMR effect, these modulations lead to modulations of the tunneling current that are related to the spin polarization of the sample. The basic concept of this mode is directly related to (4.3). In the experiment, a magnetically bistable tip is used. The tip magnetization is periodically switched between the two stable configurations of opposite magnetization. This is equivalent to changing the sign of the spin polarization of the tip apex. In the experimental set-up, the magnetization of the tip is reversed by an alternating current through a small coil that is fixed to the tip. The frequency of the alternating current lies above the cut-off frequency of the feed back loop of the STM [21]. Thus, the feed back loop only detects the averaged tunneling current for the two spin polarizations (positive and negative) of the tip apex. As can easily be seen from (4.3), in the averaged tunneling current I = I0 all spin-dependent currents cancel out such that the constant current image contains no magnetic information. With a phasesensitive lock-in amplifier, the alternating part of the tunneling current ∆I is detected which is proportional to P i P f cos Θ. It contains all the spin information. This way, topographic and spin information are strictly separated and an image of the spin component along the magnetization axis of the tip can be recorded simultaneously with the topography [21]. The alternating magnetic field induced within the coil has to be large enough to fully reverse the magnetization of the tip. The alternating field, however, also creates induction currents in the tunneling loop. For this reason, in an experiment only soft magnetic materials may be used as tips. Moreover, magnetostriction of the tip during the reversal must be avoided. The early experiments of Johnson et al. suffered from large magnetostriction of the Ni tip, so that no stable magnetic or topographic imaging was possible [22].


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4.3 Experimental Set-up In our experiment, the magnetization of a ferromagnetic electrode is switched by applying a small alternating current to a coil wound around the electrode. The direction of magnetization of the STM electrode is mostly determined by its shape. For imaging the out-of-plane component of the spin polarization, a sharp and pointed tip is used [23]. This is schematically shown in Fig. 4.2a. The alternating magnetic field induced by the coil is large enough to fully reverse the magnetization of the tip. In the tip, the magnetization direction lies always along the tip axis due to the large shape anisotropy. Thus, at the tip apex where the tunneling occurs, the magnetization is perpendicular to the sample surface. To image an in-plane component of the sample spin polarization, a ring is used as a STM electrode [24] as schematically shown in Fig. 4.2b. The magnetic field of the coil switches the ring between the two stable circular magnetic configurations. At the bottom of the ring, the magnetization lies tangential to the ring, i.e., in the surface plane. By choosing the plane in which the ring is oriented, the magnetization direction of the ring is defined and thus the direction of the sensitivity in the surface plane for the measured spin signal is known. Figure 4.3 shows images of the STM electrodes experimental realization. STM tips for recording the out-of-plane componen of the spin polarization (see Fig. 4.3a) were prepared by electrochemical etching from thin CoFeSiB wires of 130 µm diameter. As etching agent, a dilute mixture of HCl and HF was used that was suspended by surface tension as a thin liquid membrane in a Pt ring during etching. The pH value was tuned such that the formation of silica from the Si in the amorphous wire was prevented. Using low etching currents of the order of 250 µA, pointed tips of cone angles typically between 8 and 15◦ were created (see bottom of Fig. 4.3a). Due to the large shape anisotropy, the tip magnetization is along the tip axis [25]. The magnetic tips were fixed in a non magnetic shaft around which the coil is wound. The chosen tip material offers extremely low coercivities in the range of 50 µT and negligible magnetostriction (smaller than 10−8 ). This ensures that during switching of the tip, magnetization, the magnetostriction in the created domain walls is small enough that virtually all mechanical vibrations are suppressed [25]. With detailed micromagnetic simulations of the tip switching process, the expected magnetostriction could be estimated to be below 10−14 m, i.e. is irrelevant in the experiment. For imaging one in-plane component of the spin polarization, ring shaped STM electrodes of the same CoFeSiB material were used [24]. Figure 4.3b shows an

Fig. 4.2. Schematic representation of Sp-STM electrodes for measuring (a) the out-of-plane component and (b) the in-plane component of the sample spin polarization

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Fig. 4.3. Sp-STM electrodes for measuring (a) the out-of-plane component and (b) the inplane component of the sample spin polarization. The zoomed images show scanning electron microscopy images of the pointed end and of the bottom of the ring, respectively

image of a ring of about 2 mm outer diameter and a thickness of 25 µm. The rings were electrochemically etched from a thin CoFeSiB foil. The coil wound around the ring is clearly visible in Fig. 4.3b. By choosing the ring orientation, the direction of sensitivity in the surface plane is defined. The outer perimeter of the rings were polished to even out the roughness caused by the etching. After polishing, the perimeter is smooth with some minor polishing traces (see bottom of Fig. 4.3b). Although the rings used as STM electrodes are not sharp, a lateral resolution below 1 nm could be achieved [24, 26]. Most likely nano tips exist at the apex which give a high lateral resolution. To obtain spin contrast, the STM electrodes need to be cleaned in situ by Ar sputtering. The contrast can be increased by coating the CoFeSiB electrodes with several monolayers (ML) of Fe.

4.4 Ferromagnetic Domains and Domain Walls In this section we give several examples on magnetic domains and domain walls in ferromagnets to illustrate the magnetic imaging capabilities of Sp-STM for the in and out-of-plane components of the magnetization. All studies were carried out in ultra high vacuum and at room temperature. The samples were cleaned by Ar ion etching followed by thermal annealing. The sample surfaces were checked for impurities with Auger electron spectroscopy (AES) and no contaminations could be found within the sensitivity limit of our spectrometer. The crystal structure of the surfaces were check with low energy electron diffraction (LEED). All surfaces showed sharp


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LEED spots and a low background intensity indicating a single crystalline surface with low concentration of defects. 4.4.1 Ultra-sharp Domain Walls in Co(0001) When using tips that are magnetized perpendicular to the sample plane, the out-ofplane component of the spin polarization may be imaged [21]. While magnetic stray fields of the tip cannot be avoided in this configuration, hard magnetic samples like hcp Co could be imaged without problems. As an example, the closure domain pattern of Co(0001) is presented. Hcp Cobalt has a uniaxial magetocrystalline anisotropy with an easy direction along the c-axis, i.e. perpendicular to the selected (0001) surface. Due to the minimization of the stray field energy, the single domain state is unstable and splits up into a Lifshitz closure domain pattern. Since for Co the magnetic anisotropy and the dipolar energy are of the same order of magnitude, no perfect and simple closure domain structure occurs on the (0001) surface. Instead, the magnetization of most areas of the surface is strongly rotated away from the surface normal and a dendritic pattern is formed [27]. In the MFM image (see Fig. 4.4a), the complex domain pattern of Co(0001) is visible. The limited lateral resolution of MFM becomes apparent at the magnification of the image. In contrast to MFM, Sp-STM reveals the full detail of the fractal structure as depicted in Fig. 4.4b. When zooming into the ends of the fractal branches sharp features in the otherwise smooth contrast can be observed as shown in Fig. 4.5a. The contrast across these sharp features resembles domain walls. The observed contrast corresponds, however, not to a 180◦ domain wall but to an angle of rotation of only 20◦ . The width of the transition region between the two domains is only of the order of 1 nm. At first sight, these small widths seem to contradict common knowledge about domain walls. Bloch walls in bulk Co have a width of ≈ 11 nm. To exclude instrumental reasons for the observation of such sharp walls, we take the following consideration. A mechanism that often causes artificially sharp walls is a magnetostatic pick-up mechanism in which the domain wall is dragged along with the magnetic tip during

Fig. 4.4. (a) MFM and (b) Sp-STM image of the fractal domain pattern of Co(0001). The scans were performed on the same sample but not on the same area

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Fig. 4.5. (a), (b) Sp-STM images of a sharp domain wall on Co(0001). (b) No shift in the wall position is observed when inverting the scan direction excluding dragging of the wall

scanning until it snaps off. In that case, a sharp transition would be observed at the point of snapping off. To test for this mechanism, we recorded the wall while scanning from the right to the left and from the left to the right (see Fig. 4.5b). If the wall was dragged along and snapped off, opposite displacements of the wall for scanning in the two directions should be seen. The domain wall, however, appears at exactly the same position for both directions ruling out any significant dragging. To understand the origin of the specific type of 20◦ wall and to calculate its expected width, we focus on details of the domain pattern of Co(0001). In Co(0001) the magnetocrystalline anisotropy favors a magnetization along the surface normal. To reduce the stray field energy of the sample, domains of opposite magnetization along the normal separated by 180◦ domain walls are formed in the bulk of the crystal. This magnetization configuration reduces the overall stray field, but still produces a large number of surface charges, since the flux is not kept inside the crystal. As Hubert et al. suggested, the system can reduce the amount of surface charges by a partial flux closure with tilted surface domains [28]. From the ratio between magnetocrystalline and dipolar energy one may calculate the angle θ, the magnetization is tilted from the surface normal. Minimizing the free energy results in a large angle of θ ≈ 80◦ [23], i.e. the flux closure is obtained by almost in-plane magnetized surface domains. Hence, one expects to find 20◦ domain walls on the surface in agreement with our Sp-STM observations.

Fig. 4.6. (a) Calculated energy of a 20◦ domain wall in Co(0001) as function of its width. (b) Experimental wall profile (points) and calculated wall profile (solid line) of a 20◦ domain wall


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To estimate the width of such a 20◦ domain wall we use a one dimensional model of the domain wall including the magnetic exchange energy and the anisotropy energy of Co [23]. Figure 4.6a shows the wall energy as a function of the wall width w. The minimum energy is found at a wall width of only 1.5 nm. The resulting theoretical wall profile agrees well with the experimentally observed wall profile as depicted in Fig. 4.6b. Especially, the experimental wall profile does not appear blurred when compared to the calculated one indicating a lateral resolution better than 1 nm. The experiments illustrate the possibility, that a domain wall may be narrower than the √ magnetic exchange length of A/K without violation of micromagnetic rules (A and K are the magnetic exchange constant and the magnetic anisotropy, respectively). This holds since the angle of rotation across the wall is small [23]. 4.4.2 Asymmetric Néel Caps in Fe(001) Using ring electrodes, the bottom of the ring is magnetized tangential such that the in-plane component of the spin polarization along the ring plane is imaged [24]. In case the ring is perfect, the magnetic flux is closed and no magnetic stray field exits. This allows the imaging of extremely soft magnetic samples. As an example, the Nèel caps in Fe-whiskers were investigated. Fe-whiskers are needle-like Fe single crystals that are terminated by (001) surfaces. The magnetic ground state is the so called Landau state, in which the flux inside the Fe-whisker is closed. The whisker splits up into two elongated domains that are magnetized along the long side of the whisker. The magnetization at the surface of an Fe-whisker lies in-plane. The domains are separated by a 180◦ domain wall. At the two ends of the whiskers, closure domains are formed. Figure 4.7a shows a magneto optic Kerr effect (MOKE) image of an Fe-whisker taken after sample surface preparation in the UHV chamber. The 180◦ domain wall along 100 can be seen immediately. The MOKE images were recorded to find the position of the wall such that the ring electrode of the Sp-STM can be placed close to the wall.

Fig. 4.7. (a) Magneto optic Kerr image of an Fe-whisker. (b) Topographic and (c) spin signal recorded simultaneously. The 180◦ domain wall seen in (a) is also clearly visible in (c)

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Figure 4.7b,c show the topographic and the magnetic Sp-STM images taken at the same time. In the topographic image, flat terraces of several 100 nm width separated by atomic steps are visible. Fe-whisker surfaces are rather flat which is a necessary condition for imaging with a ring as a dull STM electrode. Nevertheless, the ring electrode allows topographic imaging with nm resolution which can be explained by the existance of nano-tips on the outer ring perimeter. In the image of the spin signal, clearly two domains separated by a 180◦ domain wall running along the 100 direction can be seen. The wall is rather wide. This is not caused by the lateral resolution of the instrument but is due to the magnetic properties of Fe. In the interior of the Fe-whisker, the two domains are separated by a charge free Bloch wall, i.e. the magnetization in the wall rotates in the plane of the wall. At the surface of the whisker, this would lead to an orientation of the wall magnetization perpendicular to the sample surface creating magnetic surface charges. This is energetically unfavorable so that the magnetization of the wall at the surface rotates in the surface plane, i.e. a Néel cap is formed. As has been shown in detailed calculations, the curling of the Bloch wall to a surface Néel cap is asymmetric [29] resulting in an asymmetric wall profile at the surface. In Fig. 4.8, the solid line represents the micromagnetically simulated line profile across the 180◦ domain wall. It shows the in-plane component of the magnetization pointing along the domain wall. The measured wall profile (squares) perfectly agrees with the calculated data within the lateral calibration error of the scanner (≈ 10%). The agreement between theory and experiment indicates not only that with Sp-STM one may image a well-defined in-plane component of the spin polarization. It also shows that the ring electrode is indeed free of stray fields. Even small fields of the order of 100 µT are sufficient to move the domain walls in whiskers by several µm. As we see practically no blurring in the experimental wall profile when compared to the theoretical profile, the local stray field of the ring have to be significantly below the above mentioned value.

Fig. 4.8. Calculated (solid ) and measured (squares) wall profile across the Néel cap on an Fewhisker. The calculations were taken from [29]


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4.5 Antiferromagnets in Contact with Ferromagnets Antiferromagnetic surfaces can be investigated as well, as in Sp-STM the spin polarization at the surface is mapped. We here focus on topological antiferromagnets. In these, the magnetic moments within one atomic layer of the crystal couple ferromagnetically while the adjacent layers couple antiferromagentically. In other words: the material is a layered antiferromagnet. We show two examples for these systems which both were grown on Fe(001) as a substrate and both show an in-plane spin polarization. 4.5.1 Mn on Fe(001) and Topologically Induced Frustrations As a model system for the topological antiferromagnets, we focus on Mn grown on Fe(001). Mn can be stabilized at room temperature in a body centered tetragonal (bct) structure on Fe(001) [30]. The bct structure is stable up to about 20 ML [31–33]. Mn films have an out-of-plane lattice constant of 0.323 nm, i.e. a little larger than the 0.287 nm of Fe. Mn grows in a layer-by-layer mode up to 10 to 20 ML [34]. The critical thickness strongly depends on the substrate quality and the growth temperature. When Mn was deposited at room temperature, no intermixing was found. The onset of intermixing was observed by AES for substrate temperatures above 420 K [33, 38]. In STM studies, interdiffusion of Fe into the first ML Mn was found at substrate temperatures above 370 K [39]. The intermixing was observed until the fourth Mn layer [37]. At the Mn surface of films between 4 to 10 ML, small regions with rectangular cross-shaped patterns start to form [36,37]. It was speculated that these small rectangular islands are local reconstructions and a precursor to threedimensional growth. The first evidence that Mn on Fe(001) is a layer-wise antiferromagnet was reported by Walker and Hopster [38]. This was confirmed with scanning electron microscopy with polarization analysis (SEMPA) [34] and later with spin-polarized STM in the spectroscopic mode [20].

Fig. 4.9. (a) Topographic STM image and (b) line scan of 7 ML Mn on Fe(001). (c) Magnetic Sp-STM image of the same area and (d) line scan revealing the layer-wise antiferromagnetic order of bct Mn. The direction of sensitivity for the spin is parallel to the whisker axis and is indicated in (c)

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Figure 4.9 shows the example of a thin Mn film grown on Fe(001). The topographic image of the Mn film indicates an imperfect layer-by-layer growth as several layers are exposed at the surface. The terraces are separated by single atomic steps as can be deduced from the line scan in Fig. 4.9b. The Fe substrate was homogenously magnetized in one direction over the whole imaged area, as determined by Kerr-microscopy. The direction of sensitivity of the ring was chosen collinear to the magnetization of the Fe substrate. Thus, the imaged spin signal shows the projection of the spin component collinear to the Fe magnetization. The Sp-STM images taken simultaneously to the topographic images show an alternating contrast on the atomic terraces of Mn (see Fig. 4.9c) confirming a layer-wise antiferromagnetic order. In agreement to that, the spin signal is identical on all terraces but the sign alters (see the line scan in Fig. 4.9d). This ideal antiferromagnetic order was found on most areas of the Mn film. The unperturbed layer-wise antiferromagnetic order is, however, disturbed in case a step of the underlaying Fe substrate is present. Figure 4.10a presents schematically the topological and magnetic situation of Mn layers overgrowing a step edge of the Fe substrate underneath. The thickness of the Mn layers on both sides of a monatomic Fe step differs by one ML. Due to the vertical lattice mismatch, subatomic steps are formed at the Mn film surface at the position of Fe step edges. The situation of the magnetic order above such step edges is more complicated. An undisturbed layer-wise antiferromagnetic order within the Mn film is not possible when the Mn moments at the interface on both sides of the step edge are aligned in the same direction by the Fe substrate. Instead, Mn layers which meet at the position of the Fe step edge are magnetized oppositely. This leads to a magnetic frustration [40, 41]. Figure 4.10b shows a Sp-STM image of the topography of a Mn film grown over a monatomic step of the Fe substrate. In the topography, a buried Fe step edge is running almost vertically through the center of the imaged area (black arrows as guideline). The line profile in Fig. 4.10c taken along the black line in Fig. 4.10b shows a step of monatomic height between two different Mn terraces (≈ 0.16 nm) and a step of subatomic height (≈ 0.018 nm) at the position of a buried Fe step edge due to the different lattice constants of Fe and Mn. In Fig. 4.10d clearly the layer-wise antiferromagnetic order between the Mn islands and the Mn layer underneath is visible. Following the way of the buried Fe step edge, a magnetically frustrated region is present. Along the buried Fe step edge, a reversal of the spin contrast appears. In this region the spin polarization of the Mn rotates by 180◦ . The observation of magnetically frustrated regions at the surface of thin Mn films at the position of buried Fe step edges indicates that the magnetic frustrations are extended throughout the whole Mn film down to the interface, as schematically shown in Fig. 4.10a. This implies that the coupling energy at the interface between Fe and Mn is higher than the domain wall energy in the Mn film which is likely for thin films. Figure 4.10e presents an averaged line profile across the topologically enforced magnetic frustration at the position of the box in Fig. 4.10d. The measurement indicates that the magnetic frustration has a certain lateral extension. To estimate the wall width at the surface, the experimental profile is fitted with a tanh-function and is plotted as a solid line. It reproduces the shape of the transition region well. Due to the good agreement, this function is used to determine the wall width which is in this case 4.6 ± 0.2 nm.


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The width of the frustrated regions was studied as function of the Mn film thicknesses (see Fig. 4.11). The smallest width of 1.2 nm was imaged between the second and third ML Mn and the widest one of 6.9 ± 0.3 nm between 18 and 19 ML. Thicker Mn films could not be investigated due to the phase transition to α-Mn resulting in a three-dimensional growth and rough surfaces. The widening is a consequence of minimizing the exchange energy. At the interface, an atomically sharp frustration is enforced at the buried Fe step edge. This costs a maximum of magnetic exchange energy. In the layers above, the wall widens which reduces the exchange. Eventually, the frustration reaches the bulk wall width of a 180◦ domain wall in Mn, which represents the energetic minimum. No further widening is expected. This limit was, however, not reached in our measurements, as we observed no sign of saturation of the width but a linear increase. We can therefore conclude, that the bulk wall width must be much larger than 7 nm [26].

Fig. 4.10. (a) Schematic sketch of the topographic and magnetic situation of a buried Fe step. Sp-STM image of (b) the topography and (d) the corresponding spin signal of 11.9 ML Mn on Fe(001). One buried Fe step edge is running almost vertically through the center of the images, indicated by arrows. (c) Line profile taken along the black line in (b) showing a monatomic Mn step and a step of subatomic height formed by a buried Fe step. (e) Line profile (averaged over 70 lines) across the magnetically frustrated region in the Mn over-layer at the position of the box in (d). The solid line represents a fit to the wall profile

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Fig. 4.11. The width of magnetically frustrated regions of Mn surface layers as a function of the Mn film thickness in ML (bottom scale) and equivalent in nm (top scale). The solid line is a linear fit to the experimental data points

4.5.2 The Layered Antiferromagnet Cr on Fe(001) Cr(001) is a topological antiferromagnet, as already discussed in the introduction in Sect. 4.2.1. Cr can be grown pseudomorphically on Fe(001) and shows an in-plane spin polarization collinear to that of the underlying substrate [42]. We used ring electrondes aligned along the whisker axis to image the antiferromagnetic surface. Figure 4.12a shows the topography of the Cr film. Two atomic layers are exposed. As expected, the Sp-STM image recorded simultaneously shows the antiferromagetic order of the terraces (see Fig. 4.12b). The observed spin contrast is, however, only a few percent of the total tunneling current, indicating that the electronic states that are involved in the tunneling process are only weakly spin-polarized. This holds especially for tunneling conditions, where the surface state is not involved [17]. More details on the the tunneling states are discussed in Sect. 4.6.2.

Fig. 4.12. (a) Topographic STM image and (b) magnetic Sp-STM image of the same area of a 12.8 ML Cr on Fe(001)


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4.6 Bulk Versus Surface: Which Electronic States Cause the Spin Contrast? Analogous to STM, Sp-STM can be used to gain information of the density of states and the processes during tunneling. In contrast to STM, Sp-STM gives spin resolved information. This additional information may be used to learn more about the electronic structure of the sample and the tunneling process between the tip and the sample, itself. The capabilities of Sp-STM are illustrated with two model systems, ferromagnetic hcp Co and antiferromagnetic bcc Cr. 4.6.1 Voltage Dependence of the TMR Effect in Co(0001) When biasing a tunneling junction by the voltage U, not only the states at the Fermi energy contribute to the tunneling current but all states between the Fermi levels of the tip and the sample. More precisely, the tunneling current I is given by [43]: I∝

E F +eU

− 2d ~

ρS (E) e

! & ' 2m Vb −E+ 12 eU

ρT (E − eU )dE

(4.4)

EF

where ρS and ρT are the density of states of the sample and the tip. Equation (4.4) is valid for both spin channels. With this generalized Jullière model it becomes obvious that the size of the TMR should depend on the bias voltage in a way that reflects the sample and tip spin polarizations. In planar junctions, the TMR often decreases with increasing bias voltage. Due to its technological importance, many studies have been devoted to this aspect. In the early work of Jullière, a bias as small as 3 mV was needed to halve the value of the TMR. With increased control of the preparation, this value increased up to 700 mV over the years [44]. Besides the above mentioned density of states effect, several other models have been proposed to explain this behavior. Hot electrons from the positive electrode might be scattered in a spin-dependent way at defects in the amorphous barriers [45] or might create magnons [46]. These mechanisms reduce the spin polarization and, consequently, the TMR. To pinpoint the mechanism, difficulties partly related to the complex structure of planar tunnel junctions (polycrystalline electrodes, poorly characterized amorphous barriers) have to be overcome. Here, Sp-STM measurements of the voltage dependence of the TMR across the vacuum barrier are helpful. Obviously, no effects due to impurities in the spacer are present. Magnon creation and the density-of-states effect are still present. Measurements of the TMR were carried out in ultra-high vacuum using Sp-STM with an amorphous tip and a Co(0001) sample. During the measurement, the bias voltage was varied while keeping the tip at a fixed position and measuring the averaged tunneling current It and the modulated current ∆I. The TMR, defined as the asymmetry δ of the tunneling currents observed for parallel and antiparallel magnetization alignment, was obtained from It and ∆I. The measured TMR (Fig. 4.13a) obtained with the tip stabilized at 1 V, 1 nA is almost constant with bias voltage.

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Fig. 4.13. Tunnel magneto-resistance δ of a clean Co(0001) surface versus bias voltage, obtained with a magnetic tip stabilized at 1 V, 1 nA (a) and at 100 mV, 1 nA (b)

This is in contrast to the case of planar tunnel junctions with amorphous spacers. If spin-dependent scattering at magnons was the dominant mechanism for the drop of the TMR, a similar decrease of the TMR with bias voltage should also be present in our case. Its absence, however, indicates that this mechanism is not dominant. Most interestingly, the density of states effect seems to be absent as well. Both, the spin polarization of tip and sample vary strongly with energy and a variation of the TMR should be expected. This discrepancy may be lifted easily, when having a closer look at the tunneling process. At typical tunneling conditions, the tip and sample are separated by a vacuum barrier of several atomic distances. Under these conditions, electrons with perpendicular momentum dominate the transport [8] and states with a momentum parallel to the sample plane contribute only marginal. Taking into account this effect, one may explain the constant TMR with the band structure of hcp Co along the surface normal. Figure 4.14a reveals that around the Fermi edge there is only one minority band. When assuming perpendicular tunneling, only a single band of the sample contributes to the tunneling current and no bias dependence of the TMR is expected. This interpretation was further confirmed by ab-initio calculations [47]. For small barrier widths, tunneling via surface states could become important in STM experiments [17]. This might lead to pronounced changes in the TMR. Figure 4.13b presents the experimental TMR versus bias voltage obtained at a smaller tip-sample separation (feed back conditions: 100 mV, 1 nA). For negative bias voltages, a constant TMR is still observed. For positive bias voltages, however, a dip at 200 mV is found. Most likely the dip is related to majority state which reduce the minority-dominated spin polarization at this energy. Indeed, inverse photoemission measurements revealed a majority surface state of Co(0001) at ≈ 0.2 eV [48] which is also present in the calculated layer-resolved density of states (see Fig. 4.14b). At small tip-sample separations, the tunneling probability through this surface state can be enhanced and thus would decrease significantly the TMR.


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Fig. 4.14. (a) Theoretical spin-resolved band structure of Co(0001) along the Γ –A direction. (b) Theoretical spin-resolved density of states as function of distance from the surface

4.6.2 Voltage Dependence of the TMR Effect in Cr/Fe(001) Analogous to the studies on Co(0001), the TMR as function of the bias voltage was measured for thin Cr films grown on Fe(001). Figure 4.15 shows the result for a 12 ML thick Cr film at room temperature. The overall observed spin contrast is small. This is in agreement with the fact that Cr is a layered antiferromagnet. Due to the symmetry of the magnetic order, a translation of a Cr crystal by one layer is equivalent with an inversion of the spin. This symmetry implies that the bulk states in Cr are spin polarized but spin-up and spin-down bands are degenerate. Therefore, in a tunneling experiment, no spin polarization should be caused by tunneling into the bulk states. In agreement to that, we only observe a small size of the TMR effect of about 0.65%. The non vanishing TMR is most probably related to the finite thickness of the Cr film lifting the degeneration slightly. Around the well known surface state of Cr located close to the Fermi level [49], the TMR is increased. This can be explained by tunneling into the spin polarized surface state, for which the above mentioned symmetry rules do not apply. We can therefore conclude that similar to the case of Co(0001), the tunneling current and its spin polarization is given by both bulk states and surface states. Depending on the symmetry of the states and the tunneling conditions, the balance between the two contributions to the spin polarization varies, i.e. the surface sensitivity of Sp-STM is a function of the tunneling parameters. This effect is of only minor importance in ferromagnetic films, but may become of high importance in layered systems of different electronic structure.

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Fig. 4.15. Tunnel magneto-resistance δ of ≈ 12 ML Cr on Fe(001) versus bias voltage

4.7 Conclusion With the successful operation of Sp-STM in the differential magnetic imaging mode for out-of-plane and in-plane spin polarization, Sp-STM has become an established technique. Nevertheless, it is a young techniques and future studies on ferromagnetic and antiferromagnetic systems will give a deeper understanding of magnetism on the nanometer scale. For antiferromagnets, the real space information gained is complementary to the information already available from magnetic scattering techniques like neutron scattering. Sp-STM allows to tackle the new class of problems in antiferromagnets: frustrations and aperiodic structures. Finally, Sp-STM may be used to investigate the density of states and its spin polarization via the processes of spin-polarized tunneling. By this, it has the potential to become a valuable tool for electron spectroscopy.

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15. Wiesendanger R, Güntherodt HJ, Güntherodt G, Gambino RJ, Ruf R (1990) Phys Rev Lett 65:247 16. Blügel S, Pescia D, Dederichs PH (1989) Phys Rev B 39:1392 17. Stroscio JA, Pierce DT, Davies A, Celotta RJ, Weinert M (1995) Phys Rev Lett 75:2960 18. Bode M, Getzlaff M, Wiesendanger R (1998) Phys Rev Lett 81:4256 19. Kubetzka A, Bode M, Pietzsch O, Wiesendanger R (2002) Phys Rev Lett 88:057201 20. Yamada TK, Bischoff MMJ, Heijnen GMM, Mizoguchi T, van Kempen H (2003) Phys Rev Lett 90:056803 21. Wulfhekel W, Kirschner J (1999) Appl Phys Lett 75:1944 22. Johnson M, Clarke J (1990) J Appl Phys 67:6141 23. Ding HF, Wulfhekel W, Kirschner J (2002) Europhys Lett 57:100 24. Schlickum U, Wulfhekel W, Kirschner J (2003) Appl Phys Lett 83:2016 25. Wulfhekel W, Hertel R, Ding HF, Steierl G, Kirschner J (2002) J Magn Magn Mater 249:368 26. Schlickum U, Janke-Gilman N, Wulfhekel W, Kirschner J (2004) Phys Rev Lett 92:107203 27. Unguris J, Scheinfein MR, Celotta RC, Pierce DT (1989) Appl Phys Lett 55:2553 28. Hubert A, Schäfer R (1998) Magnetic Domains, Springer-Verlag, Berlin, pp 315 29. Scheinfein MR, Unguris J, Blue JL, Coakley KJ, Pierce DT, Celotta RJ (1991) Phys Rev B 43:3395 30. Heinrich B, Arrott AS, Liu C, Purcell ST (1987) J Vac Sci Technol A 5:1935 31. Purcell ST, Johnson MT, McGee NWE, Coehoorn R, Hoving W (1992) Phys Rev B 45:13064 32. Kim SK, Tian Y, Montesano M, Jona F, Marcus PM (1996) Phys Rev B 54:5081 33. Andrieu S, Foy E, Fischer H, Alnot M, Chevrier F, Krill G, Piecuch M (1998) Phys Rev B 58:8210 34. Tulchinsky DA, Unguris J, Celotta RJ (2000) J Magn Magn Mater 212:91 35. Pfandzelter R, Igel T, Winter H (1997) Surf Sci 389:317 36. Pierce DT, Davies AD, Stroscio JA, Tulchinsky DA, Unguris J, Celotta RJ (2000) J Magn Magn Mater 222:13 37. Yamada TK, Bischoff MMJ, Mizoguchi T, van Kempen H (2002) Surf Sci 516:179 38. Walker TG, Hopster H (1993) Phys Rev B 48:3563 39. Bischoff MMJ, Yamada T, Quinn AJ, van Kempen H (2002) Surf Sci 501:155 40. Berger A, Hopster H (1994) Phys Rev Lett 73:193 41. Berger A, Fullerton EE (1997) J Magn Magn Mater 165:471 42. Pierce DT, Unguris J, Celotta RJ, Stiles MD (1999) J Magn Magn Mater 200:290 43. Lang ND (1986) Phys Rev B 34:5947 44. Boeve H, Girgis E, Schelten J, De Boeck J, Borghs G (2000) Appl Phys Lett 76:1048 45. Zhang J, White R (1998) J Appl Phys 83:6512 46. Moodera JS, Nowak J, van de Veerdonk RJM (1998) Phys Rev Lett 80:2941 47. Ding HF, Wulfhekel W, Henk J, Bruno P, Kirschner J (2003) Phys Rev Lett 90:116603 48. Math C et al (2001) Surf Sci 482-485:556 49. Kleiber M, Bode M, Ravlic R, Wiesendanger R (2000) Phys Rev Lett 85:4606

5 Dynamic Force Microscopy and Spectroscopy Ferry Kienberger · Hermann Gruber · Peter Hinterdorfer

In recent years, considerable attention has focused on biological applications of the atomic force microscope (AFM), in particular on high-resolution imaging of individual biological molecules and complexes. The AFM is unique since it not only allows to image individual molecules under near-physiological conditions, but it can also monitor and visualize dynamic processes at the single molecule level. In dynamic force microscopy (DFM), the cantilever vertically oscillates during the lateral scan and touches the sample therefore only intermittently at the end of its downward movement. This imaging mode significantly reduces frictional forces, thus it yields high-resolution images of single molecules which are only weakly immobilized on a support. Furthermore, the capability of AFM to measure forces in the pico-Newton range has opened the possibility to investigate inter- and intra-molecular forces at the single molecule level. In particular, the interaction between tip-bound ligands and surface-bound receptor molecules can be analyzed in terms of affinity and rate constants. Moreover, force spectroscopy experiments yield details on structural parameters of the binding pocket, on the molecular dynamics of the recognition process and on the energy-landscape of the interaction. Similar experiments were also used to drive single proteins from the native state through their complex energy landscape into the unfolded state, leading to a detailed picture of the mechanical unfolding pathway for a range of biomolecules. By combining DFM imaging with force spectroscopy, receptor sites on a surface can be localized with nanometer positional accuracy, rendering possible to acquire topographical images simultaneously with recognition images.

Abbreviation AFM DFM PEG BR SPM NHS PSGL-1 GFP Q-factor

atomic force microscope dynamic force microscopy poly(ethylene glycol) bacteriorhodopsin scanning probe microscopy, N-hydroxysuccinimide P-selectin glycoprotein ligand-1 green fluorescent protein quality factor

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F. Kienberger · H. Gruber · P. Hinterdorfer

5.1 Introduction Since its invention in the late eighties, the atomic force microscope (AFM) [1] has increasingly been used for the visualization of biomolecules and complex biological structures [2]. Its great advantage compared to other high-resolution methods like electron microscopy and X-ray diffraction is that the measurements can be carried out in a fluid environment, including physiological medium. In addition, it does not alter the natural state of the specimen, i.e. there is no need to stain, freeze or crystallize biological samples. Furthermore, it spans a range of dimensions from nanometer up to hundreds of microns, which is an additional advantage over most other techniques. The exceptional signal-to-noise ratio of the AFM allows individual biomolecules to be imaged under physiologically relevant conditions at sub-nanometer resolution [3], making possible to observe single biomolecules at work [4] and to study the complex pathways of their interactions. In addition to high-resolution imaging of bio-molecular assemblies, the measurement of mechanical forces at the molecular level has provided detailed insights into the function and structure of many biological systems [5]. Intra- and intermolecular interactions can be studied directly at the molecular level, as exemplified by detailed analysis of the binding potentials of receptor-ligand pairs involved in cell adhesion [6, 7], polysaccaride elasticity [8, 9], DNA mechanics [10, 11], and the function of molecular motors [12, 13]. AFM also allows to select individuals from a large population of molecules. Thus, time-dependent processes can be followed without synchronizing large population of molecules [14], rendering possible to study transient intermediate states and individual reaction pathways, thus yielding valuable information on the complex and dynamic interactions underlying many biological processes. In conclusion, single molecule force spectroscopy has become a versatile tool for the structural and functional investigation of single biomolecules. In the last decade, the resolution of topographical imaging, the force sensitivity, and the acquisition speed have been much improved by optimisation of sample preparation [15], new image acquisition modes [16–18], and by continuous developments of AFM instrumentation [14, 19]. In particular, the development of dynamic force microscopy (DFM) methods has minimized the problem of lateral forces when imaging delicate and weakly adhering bio-molecular assemblies [20]. Using DFM, soft biological samples are significantly less deformed and molecular objects weakly adhered to surfaces are not easily displaced by the forces applied [21]. Furthermore, the combination of DFM with force spectroscopy allows for localization of binding sites on surfaces with nanometer positional accuracy, providing for single-molecule maps of specific types of molecules in a compositionally complex sample while simultaneously carrying out high-resolution topographic imaging [22, 23]. Because it can identify specific components, the combination of DFM imaging and force spectroscopy can be used to map composition and to simultaneously detect compositional changes during biological processes.

5 Dynamic Force Microscopy and Spectroscopy

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5.2 Scanning Probe Microscopy Twenty years ago, the first scanning probe instrument, i.e. the scanning tunnelling microscope, opened a new avenue for the investigation of the micro- and nano-world. Scanning probe microscopy (SPM) measures the near-field physical interactions between the scanning probe tip and the atoms on a surface [24]. Atomic force microscopy (AFM), a member of the SPM family, operates by moving a sharp tip supported on a micro-cantilever horizontally over a surface, thereby recording a topographical map of the surface [26]. Deflections of the cantilever caused by the tip–sample interaction are detected by a split photodiode. The magnitude of the deflection is registered by the change in the direction of a laser beam reflected at the end of the cantilever. The deflection signal of the cantilever is used to keep the force applied to the sample constant by moving either the sample or the cantilever up and down via a servo-system. The surface topography is then reconstructed from the vertical movements of the sample or the cantilever, respectively. In this imaging mode, called contact-mode AFM, the probe is incessantly in contact with the sample. To prevent deformation or even destruction of the soft biological sample, the vertical force applied by the cantilever is typically adjusted to small values (∼ 100 pN) [27]. The adjustment of the indentation force is done in a force-distance cycle. To construct a force curve, the cantilever is approached towards the surface and the deflection signal is registered by the photo-diode (Fig. 5.1a). As soon as tip and substrate make contact during the downward approaching phase (red trace in Fig. 5.1a), the cantilever is deflected upwards resulting in a linear rise of the deflection signal according to Hooks law. In the subsequent withdrawal phase (black retrace in Fig. 5.1a), the tip is then raised (or the sample is lowered) and the probe and substrate are drawn apart. The cantilever can now return to its original resting position (at 0 nm extension). By adjustment of the force via the force-distance cycle, the vertical force applied during imaging can be reduced to smaller values. Although the vertical forces can be reduced as described above, the typical lateral forces are still hampering the investigation of relatively delicate and loosely attached biological specimens [28]. The problem of such unfavourable tip-surface interactions, however, has largely been overcome by the development of DFM methods such as tapping-mode AFM [16, 17] and MACmode AFM [18, 29]. In DFM, the cantilever is oscillated close to its resonance frequency as it scans over the surface and it touches the sample only at the end of its downward movement. When the cantilever and sample come into intermittent contact, the oscillation amplitude of the cantilever is lowered. These changes in amplitude are exploited to produce an image of the sample topography. In tapping-mode AFM, the excitation of the cantilever oscillation is due to an acoustic vibration of the overall liquid cell, whereas an alternating magnetic field is used in MACmode AFM to excite the magnetically coated cantilever sinusoidally. In contrast to tapping-mode, the amplitude-frequency curve in the MACmode closely follows the characteristics of a damped harmonic oscillator (Fig. 5.1b) [30–32]. Therefore amplitude and frequency of the cantilever oscillation can be adjusted more accurately in MACmode AFM. The amplitude and the corresponding amplitude reduction are adjusted in an amplitude-distance cy-

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Fig. 5.1. Principles of scanning probe microscopy. (a) Force-distance cycle of a bare silicon nitride tip acquired on a hard surface under buffer conditions. The approach curve (red line) and the retracting curve (black line) are shown. (b) Resonance curve of an oscillating cantilever. For imaging, the excitation frequency is adjusted to a value given by the resonance peak (∼ 9 kHz). (c) Amplitude-distance cycle acquired on a hard surface. The amplitude of oscillation is shown for both the approach (trace, red line) and the retraction (retrace, blue line). The free oscillation amplitude was set to 5 nm at an oscillation frequency of 8 kHz

cle [21]. Figure 5.1c shows the amplitude-distance cycle of an oscillating cantilever on a hard surface in buffer solution. The amplitude of the oscillation is shown for both the approach (trace, red) and the retraction (retrace, black) of the cantilever as a function of the cantilever-surface distance. Far away from the surface, the oscillation amplitude is not influenced by the sample and is the same during both approach and retraction. As the tip is pushed further towards the surface, the amplitude rapidly decreases until it is damped to zero at full contact (trace, red line). Upon retraction of the cantilever from the surface (retrace, black line), the amplitude starts to increase until it reaches the free amplitude again. The free oscillation amplitude is set to a low value (∼ 5 nm). This is important for a minimal energy input during imaging and prevents disruption of the sample, thereby allowing for stable imaging. Both contact- and dynamic force microscopy modes have been employed for high-resolution imaging of biomolecules. In fact, some of the highest resolution images reported in the literature were obtained using contact mode. In these studies, mostly two-dimensional arrays of membrane proteins were imaged to subnanometer resolution using conventional AFM cantilevers in aqueous buffered conditions [33, 34]. The high image-resolution was made possible by the close-packing of the biomolecules which makes them more resistant to the deformation by the forces of the cantilever during imaging. In contrast, DFM imaging has mainly been applied to single, isolated biomolecules on surfaces and to supra-molecular complexes [35] because it overcomes the strong lateral imaging forces associated with contact mode AFM [36].

5.3 Dynamic Force Microscopy Imaging The critical parameters to adjust in DFM are the oscillation frequency, the oscillation amplitude, and the amplitude reduction for the feedback during imaging (i.e. the set point). In general, small oscillation amplitudes of ∼ 5 nm are necessary to prevent


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displacement of the biological specimens [21]. Prior to imaging, a proper setpoint with ∼ 20% amplitude reduction (e.g. 1 nm at 5 nm free amplitude) is determined in an amplitude-distance cycle. This results in an oscillation amplitude of 80% of the free amplitude during imaging (i.e. 4 nm at 5 nm free amplitude). A reduction of 1 nm in amplitude correlates with an imaging force of F ∼ 50 pN for a cantilever spring constant of k = 100 pN/nm and a Q-factor of 1.8 (F ∼ k∗ a/Q) [37]. Since the estimated force of 50 pN is only transiently exerted at the lower part of the oscillation, an average force of much less than 50 pN is applied during scanning. Consequently, no lateral indentation of the sample takes place during imaging in dynamic force microscopy. Moeller et al. have recently demonstrated that DFM allows imaging of native protein surfaces with similar resolution as in contact-mode AFM [38]. Topographical imaging of native purple membranes with tapping-mode AFM revealed single polypeptide loops of bacteriorhodopsin (BR) molecules at a lateral resolution of ∼ 1 nm and a vertical resolution of ∼ 0.1 nm. Figure 5.2a shows the extra-cellular face of the purple membrane which exhibits trimeric structures protruding ∼ 0.4 nm out of the lipid bilayer. The trimers are arranged in a trigonal lattice of ∼ 6 nm side length. While contact-mode AFM gave a somewhat better resolution on isolated purple membrane (0.5 nm) [33], it was not possible to resolve single membrane-bound molecules [39]. In contrast, both the trimeric arrangement of BR molecules and morphological details of membrane-bound antibodies were well resolved in a recent DFM study [40]. Figure 5.2b shows the surface of purple membrane to which an antibody is bound; the trimeric lattice of BR-molecules and the arrangement of the two Fab fragments of the antibody are clearly visible. A three-dimensional representation shows that the antibody with its two Fab fragments appears 2–3 nm in height (Fig. 5.2c). The maximum distance of two Fab fragments of the same antibody was observed to be ∼ 18 nm, indicating a strong overall intrinsic flexibility of the antibody-hinge region. In addition, membrane-bound antibodies showed many different conformations that ranged from globular to V- and to Y-like conformational shapes. Imaging under near-physiological conditions is crucial for the study of the native structure and function of bio-molecular complexes. However, the oscillation of a cantilever in a liquid is strikingly different from the oscillation in air or vacuum. First of all, the cantilever motion drags the surrounding liquid, leading to an increase of the effective mass and a corresponding decrease of the resonant frequency. Second, the strong hydrodynamic interaction between the cantilever and the liquid produces a low quality factor (Q-factor) which is about two orders of magnitude lower than in air [19]. The low quality factor of the cantilever in a liquid implies much higher forces between an oscillating tip and the sample than in air. To overcome this, a new dynamic force microscopy technique has been introduced [37]. The low Q-factor of the cantilever is increased by up to three orders of magnitude with the help of a positive feedback control, coupled to a phase-locked loop unit used to track the resonance of the cantilever. Moreover, the phase shift and the resonant frequency can be used to investigate the energy dissipation and conservative interactions between tip and sample, respectively. High-Q dynamic force microscopy, as it is called, has been applied to living cells [41] and polymeric surfaces [42]. A resonant frequency shift image acquired with an active Q control, combined with a phase locked-loop

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Fig. 5.2. Imaging of antibodies attached to purple membranes. Since the contact time and friction forces are significantly reduced in dynamic force microscopy, the imaging of soft and weakly attached biological samples is greatly facilitated. (a) Topographical image of the extra-cellular purple membrane surface using tapping-mode AFM exhibits a trimeric arrangement of bacteriorhodopsin (BR) molecules at high lateral resolution. The scan size was 50 nm2 . The full gray-level scale ranges from 0 to 1.5 nm. (b) Topographical image of the cytoplasmatic side of purple membrane to which an antibody is bound using MACmode. The trimeric lattice structure of BR and the arrangement of the two Fab fragments of the antibody are clearly resolved. The scan size was 20 nm2 . The full gray-level scale ranges from 0 to 3 nm. (c) Three-dimensional representation of (b) showing the heights of the structures. The BR-trimers (indicated with dotted lines) protrude 0.8 nm out of the membrane, whereas the two Fab -fragments of the antibody (see labelling) are ∼ 2 nm in height

unit, is shown in Fig. 5.3a. The effective Q factor was increased to ∼ 300 in liquid, resulting in a remarkable contrast between the cell and the substrate, as well as between different intracellular structures. Figure 5.3b (left panel) shows a Qcontrolled resonance-frequency shift image (dashed square in Fig. 5.3a) revealing a well-defined arrangement of fibers. The corresponding phase-shift image without

Fig. 5.3. Dynamic force microscopy imaging of a living cell using Q-enhancement. For imaging in liquid, the low quality factor (Q) of the cantilever was increased up to three orders of magnitude, reducing the tip–sample forces significantly. (a) Resonant frequency shift image of a living rat kidney cell using Q-enhancement and tracking the resonant frequency. The effective Q was 300 at a scan size of 68 µm2 and a frequency range of 200 Hz. The square marked was imaged at higher resolution in (b). (b) Left panel. Resonant frequency image using Q-enhancement and corresponding line profile. Right panel. Phase shift image of the same region with conventional DFM. The profiles are shown for the same scan line marked with a dashed line. The scan size was 10 µm2 . The phase shift image without Q-control shows lower resolution and lower signal-to-noise ratio of the same region. The enhancement of the spatial resolution can be also seen in the corresponding profiles along the same scan line


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Q-control of the same region shows much lower resolution and poorer signal-to-noise ratio (Fig. 5.3b, right panel). Moreover, the distribution of the fibers is distorted in the latter image, probably due to high forces between the tip and the sample. The enhancement of the spatial resolution can also be seen in the corresponding crosssection profiles along one same scan line (Fig. 5.3b, lower panels).

5.4 Force Spectroscopy A number of techniques are presently available to investigate intra- and intermolecular forces acting within or between biomolecules. The most prominent tools are optical tweezers [43, 44], the bio-membrane force probe [45], and the atomic force microscope. These techniques span a measurable force window ranging from entropic forces at several femto-Newtons up to the rupture of covalent bonds at several nano-Newtons [46]. Using AFM, both receptor-ligand interactions (i.e., inter-molecular forces) and the unfolding patterns of single biomolecules (i.e., intra-molecular forces) can be studied. For studying specific binding between biomolecules, an increasing force is exerted on a receptor-ligand complex, and the dissociation process is followed over time. Dynamic aspects of recognition are addressed in force spectroscopy experiments, in which the time scale is systematically varied in order to analyze the changes in conformations and states during receptor-ligand dissociation [47]. Such experiments allow for estimation of affinity, rate constants, and structural data of the binding pocket [48]. For studying intramolecular forces, single molecules are stretched between the cantilever tip and the substrate, yielding insight into the molecular determinants of mechanical stability and the role of force-induced conformational changes in the regulation of physiological function [49]. 5.4.1 Principles In molecular recognition force spectroscopy experiments, the binding of ligands immobilized on AFM tips towards surface-bound receptors (or vice versa) is studied by applying a force to the receptor-ligand complex until the bond breaks at a measurable unbinding force [50]. Such experiments require that one or few ligand molecules are permanently tethered to the apex of the AFM tip, usually by covalent bonding via a flexible linker molecule [51]. As an example, the covalent coupling of ligands to gold-coated tip-surfaces via a free SH-group of the ligand guarantees a sufficiently stable attachment, because these bonds are about ten times stronger than typical ligand-receptor interactions [52]. This chemistry has been used for various types of receptor-ligand studies, but it requires a prior deposition of a gold layer onto the silicon nitride or silicon oxide tip surface [53]. A viable alternative to functionalized gold tips is the direct tethering of ligand molecules to the surface of a silicon or silicon nitride tip (Fig. 5.4a). In the first step, amino groups are generated on the tip surface, either with ethanolamine hydrochloride or with gas phase silanization [48, 54, 55]. These two procedures do not cause stickiness and yield a very low

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number of amino groups on the apex of the tip [56], as is desired for single molecule experiments. In the second step of the anchoring protocol, a distensible and flexible linker is often used to space the ligand molecule from the amino-functionalized tip surface by several nanometers. The ligand on the spacer molecule can freely orient and diffuse within a certain volume provided by the length of the tether, thereby achieving unconstrained binding to its receptor. Poly(ethylene glycol) (PEG), an inert water soluble polymer has been used often as flexible linker [57]. Crosslinker lengths have been varied whereby ∼ 6 nm seem to give a good compromise between high mobility of the ligand and narrow lateral resolution of the target site. The heterobifunctional crosslinkers used for tip-ligand coupling carry two different reactive groups at their ends [58]. One of these is usually a carboxyl group, activated in the form of a N-hydroxysuccinimide (NHS) ester. This group couples to the amino group on the tip, yielding a stable amide bond. In the third step, the outer end of the crosslinker (e.g. a thiol-reactive group) reacts with a suitable function on the ligand molecule (e.g. a thiol group), resulting in the flexible attachment of the ligand to the tip (Fig. 5.4a). Interaction forces of single ligand-receptor pairs are measured in force-distance cycles using a ligand-carrying tip mounted on a cantilever and a target surface with

Fig. 5.4. Specific immobilization of biomolecules to the AFM-tip for studying single-molecular interaction forces. (a) A distensible and flexible polymer linker was used to bind a single antibody covalently to the tip, enabling the ligand to freely reorient and to achieve unconstrained binding to its receptor immobilized on the surface. (b) In force-distance cycles, an AFM-tip carrying ligands is brought into contact with a surface that contains the respective cognate receptors, so that a ligand-receptor bond is formed. During a subsequent retraction of the tip, an increasing force is applied to the ligand-receptor bond until the interaction breaks at a critical force, the unbinding force


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firmly attached receptor molecules. A typical force-distance cycle with a distensible PEG tether between tip and ligand is shown in Fig. 5.4b. At a fixed lateral position, the tip vertically approaches the surface and is subsequently retracted. During this cycle, the cantilever deflection (i.e., force) is continuously measured and plotted versus tip-surface separation (i.e., distance). At the beginning of the tip-surface approach (Fig. 5.4b, ‘approach’ curve), the cantilever deflection remains zero. Upon tip-surface contact (i.e. at a distance of 0 nm), the cantilever bends upward, consistent with a repulsive force that linearly increases with the distance. Subsequent tip-surface retraction (Fig. 5.4b, ‘retract’ curve) first leads to relaxation of the cantilever bending until the repulsive force drops to zero. Upon further retraction (i.e. at a distance of > 0 nm), the cantilever progressively bends downwards, reflecting an attractive force that increases with increasing tip-surface separation. The shape of this non-linear force-distance profile is determined by the elastic properties of the flexible PEGcrosslinker and shows parabolic-like characteristics, which mirrors the increase of the spring constant of the polymer chain during extension [59]. The physical connection between the tip and surface exerts the increasing force until the ligandreceptor complex finally dissociates at a certain critical force, termed unbinding force, whereupon the cantilever jumps back to the resting position. If the ligand on the tip does not form a specific bond with the receptor on the surface, the recognition event (i.e., the parabolic shaped curve) is missing and the retrace looks like the trace (as shown in Fig. 5.1a). In addition, the specificity of ligand-receptor binding is usually demonstrated by blocking experiments with free ligands which are injected into the solution so as to block the receptor sites on the surface. As a consequence, all specific recognition signals completely disappear and only occasional adhesion events are observed [48, 56]. 5.4.2 Theory The binding energy of a ligand-receptor interaction E B , given by the free energy difference between the bound and the free state, is the common parameter to describe the strength of a bond. It determines the ratio of bound complexes to the product of free reactants at equilibrium in solution and is related to the equilibrium dissociation constant K D , given by the kinetic rate constants kon and koff , K D = koff /kon , through K D = exp(−E B /kB T ). In order to get an estimate for the interaction force f from binding energy E B , the dimension of the binding pocket may be described by the characteristic length dimension x, f = E B /x. Typical values of E B = 20 kB T and x = 0.5 nm yield f ∼ 170 pN as a crude estimate for the strength of a single ligand-receptor bond. Classical mechanics describes the force required to separate interacting molecules as the gradient in energy along the interaction potential. The molecules would therefore dissociate when the force exceeds the steepest gradient in energy. However, activation barriers, temperature, time scales, and the detailed characteristics of the energy landscape, all known to determine ligand-receptor dissociation, cannot be interpreted with this purely mechanical picture [47]. Viewing ligand-receptor binding on the single-molecule level, the average lifetime of a ligand-receptor bond, τ(0), is given by the inverse of the kinetic off-rate constant, τ(0) = 1/koff . Therefore, ligands will dissociate from receptors even with-

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out any force applied to the bond at times larger than τ(0). In contrast, if molecules are pulled faster than τ(0), the bond will resist and require a force for detachment [60]. Accordingly, unbinding forces do not resemble unitary values, but are depending on the dynamics of the unbinding experiments. At the millisecond to second time scale of AFM experiments, thermal impulses govern the unbinding process. In the thermal activation model, the lifetime of a complex in solution is described by a Boltzmann ansatz, τ(0) = τosc exp(E b /kB T ) [61], where τosc is the inverse of the natural oscillation frequency and E b the energy barrier for dissociation. Hence, due to the thermal energy, there is a finite probability of overcoming the energy barrier E b , which leads to the separation of the ligand-receptor complex. A force acting on a binding complex deforms the interaction energy landscape and lowers the activation energy barrier (Fig. 5.5). The lifetime τ( f ) of a bond loaded with a constant force f is given by τ( f ) = τosc exp(E b − fx/kB T ) [62], x being interpreted as the distance of the energy barrier from the energy minimum along the direction of the applied force. The lifetime τ( f ) under constant force compares therefore to the lifetime at zero force, τ(0), according to τ( f ) = τ(0) exp(− fx/kB T ) for a single sharp energy barrier for which a mono-exponential dependence is characteristic. Using AFM, ligand-receptor unbinding is commonly measured in force-distance cycles, where an effective force increase or loading rate r can be deduced from r = d f/dt, equal to pulling velocity times effective spring constant [63]. The combination of the Boltzmann ansatz with the stochastic description of the unbinding process predicts different unbinding force distributions at different loading rates r (Fig. 5.6a) [64]. The maximum of each force distribution, f ∗ (r), reflects the most probable unbinding force for the respective loading rate r. f ∗ is related to r through f ∗ (r) = kB T/x ln(r.x/kB T.koff ). Apparently, the unbinding force f ∗ scales linearly with the logarithm of the loading rate (Fig. 5.6b). For a single barrier, this would give rise to a simple, linear dependence of the force on the logarithm of the loading rate. In cases where more barriers are involved along the escape path (Fig. 5.6c), the curve will follow a sequence of linear regimes, each of which marks a particular barrier (Fig. 5.6b) [65]. In force spectroscopy experiments, the variation in the pulling speed applied to specific ligand-receptor bonds therefore will lead to detailed structural and kinetic information of the interaction. Length scales of energy barriers are obtained from the slope of the spectroscopy plot (i.e., force versus loading rate) and extrapolation to zero forces yields the kinetic off-rate for the dissociation of the complex in solution.

Fig. 5.5. Energy profile for dissociation over a single energy barrier. Under a constant force, the barrier is linearly decreased by the applied force F, giving rise to a characteristic length scale x that is interpreted as the distance of the energy barrier from the energy minimum along the projection of the force


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Fig.5.6.Unbinding force distributions, force spectroscopy plot and energy landscape of a complex bio-molecular bond. (a) Histograms of unbinding forces of the biotin-streptavidin interaction recorded in force-distance cycles at different loading rates. The maximum of each histogram depicts the most probable unbinding force at a given loading rate. (b) Plotting the most probable unbinding force against the loading rate results in the force-spectroscopy plot for biotin-streptavidin (open circles) and biotin-avidin (closed triangles). (c) The slopes of the linear regimes in the force-spectroscopy plot mark distinct activation barriers along the direction of force (force f under an angle θ to the molecular coordinate) and yield the energy landscape along the reaction coordinate. The external force adds a mechanical potential that tilts the energy landscape and lowers the barriers (dotted line)

5.4.3 Applications The avidin-biotin complex is often regarded as the prototype of a receptor-ligand pair, due to its enormously high affinity (K D = 10−13 M) and long bond lifetime (τ(0) = 80 days). First realizations of single-molecule recognition force detections were therefore made with biotin and its cognate receptors, streptavidin [53], the succinilated form of avidin, or avidin itself [66, 67]. Unbinding forces of 250 pN and 160 pN have been reported for streptavidin and avidin, respectively. Recent force-spectroscopy experiments of the avidin-biotin interaction revealed an energy landscape with more than one prominent energy barrier (Fig. 5.6c) [65]. Using bio-force probe measurements, the loading rate was varied over eight orders of magnitude, yielding a detailed picture of the force-spectroscopy curve, with unbinding forces from 5 pN (at smallest loading rates) to 200 pN (at highest loading rates) (Fig. 5.6b). Distinct linear regimes that demonstrate the thermally activated nature of the bond breakage are visible and abrupt changes in slope imply a number of sharp energetic barriers along the dissociation pathway (Fig. 5.6c). In a different study, single-molecule force spectroscopy was applied to get insights into the binding of Ran, a molecule that regulates assembly and disassembly of the receptor-cargo complexes in the nucleic cellular membrane, to the nuclear import-receptor importin β1 [68]. By use of recognition force spectroscopy it was found that the complex of RanGTP and importin β1 alternates between two distinct conformational states of different interaction strength. The force distributions shifted to higher forces by increasing the loading rate, a behaviour that fits to the abovementioned models (Fig. 5.7a). For Ran-GTP, these distributions also had a unique

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Fig. 5.7. Force spectroscopy investigations revealed a molecular switch between conformational states of proteins from the nuclear pore complex. (a) Unbinding force distributions with one (RanGDP-importin) and two populations (RanGTP-importin) were obtained. Increasing the loading rate resulted in a shift of their most probable unbinding forces to higher values. (b) Force-spectroscopy plot of the two different interactions together with the thermal noise level. The bi-modal unbinding force distribution of the RanGTP-importin interaction clearly deviates from the classical picture for a single-path rupture topology (as shown in Fig. 5.5) and indicates that two distinct conformational isomers are present. (c) Conceptual two-dimensional energy landscape of the complex (RanGTP-importin) under applied mechanical force. The application of force lowers the barriers for dissociation, as well as the barrier separating the two states, allowing molecules trapped in the low-strength state to cross over to the high-strength state

bi-modal appearance (two black Gaussian curves at each loading rate in Fig. 5.7a), which could not be described by the escape from a single potential well. In contrast, a model involving the independent dissociation of two distinct molecular complexes, each leading to a force distribution of its own, accurately fitted the data. The two force populations were shifted to higher forces at increasing loading rates (Fig. 5.7b) and also relatively changed in size, with the second (higher-force) population representing a greater fraction at higher loading rates. The results thus indicate that the interaction between RanGTP and importin β1 can lead to two distinctly different bound states, each associated with an individual dissociation pathway. Implicit in this model is that the two states are separated by a high activation energy barrier which hampers interstate transitions (Fig. 5.7c). The application of external force decreases this barrier, allowing molecules trapped in the lower-strength state to pass the barrier to the higher-strength state. In all abovementioned examples of force-spectroscopy, the loading rate dependence of the unbinding force was measured by changing the pulling velocity of the force exerted to the interaction. The lifetime of the molecular bond can then be calculated for different forces using the Boltzmann ansatz [69]. Direct measurements of lifetimes are only possible using a force clamp, where a constant and adjustable force is applied to the complex and the time duration of bond survival is detected such that the lifetime of the interaction is directly measured at the corresponding force. It has been suggested that for selectin-ligand bonds which mediate the tethering and rolling of flowing leukocytes on vascular surfaces, the application of force could prolong bond lifetimes by deforming the molecules such that they lock more


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tightly (catch bonds) [70]. For catch bonds, the lifetime increases with force; this is opposite to slip bonds, where the application of force lowers the energy barrier between the bound and free states. Slip bonds have been widely observed (see all abovementioned examples), whereas catch bonds have been demonstrated for the first time [70]. It was shown that increasing force first prolonged, and then shortened the lifetimes of P-selectin complexes with P-selectin glycoprotein ligand-1 (PSGL-1), revealing both catch and slip bond behaviour (Fig. 5.8). Using PSGL-1 modified AFM-tips and P-selectin immobilized onto a bilayer-support (Fig. 5.8a), the interaction was measured in a pre-set force-time profile (Fig. 5.8b). In contrast to conventional force-distance cycles, the cantilever was only retracted up to a predetermined distance and then stopped, in order to apply a constant tensile force to the bond. The lifetime at the applied constant force was then simply determined by the duration of the adhesion (Fig. 5.8b). Plotting measured lifetime against applied force revealed a biphasic curve (Fig. 5.8c), corresponding to catch-bond behaviour in the low force range and conventional slip bond characteristics in the high force regime. This indicates that a molecular interaction can change from a catch bond in one force regime to a slip bond in another force regime, a phenomenon which existing models had not predicted before. Dynamic force spectroscopy not only allows to investigate the interaction between receptors and ligands but also shows great performance in measuring intra-

Fig. 5.8. Catch bonds investigated by force-spectroscopy experiments. The application of force can either shorten bond lifetimes (slip bonds, Figs. 5.5 and 5.6), or prolong bond lifetimes by deforming the molecules such that they lock more tightly (catch bonds). (a) Functionalization of the AFM-tip with cell adhesion molecules. PSGL-1 molecules were immobilized on the tip (as monomers and dimers), whereas P-selectin was bound to a bilayer. (b) Force-distance cycle between the PSGL-1 modified tip and P-selectin on the surface without a binding event (upper panel), and with corresponding binding and lifetime measurement (lower panel). (c) Lifetime measurements of the P-selectin PSGL-1 bond in dependence of the applied force. A biphasic relationship between lifetime and force was found, indicating the presence of catch bonds (i.e., bond lifetimes increased with force), succeeded by slip bonds (lifetimes decreased with force)

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molecular force profiles of single molecules. The application of mechanical force to biological polymers like proteins, polysaccarides and DNA produces conformations that are different than those previously investigated by chemical or thermal denaturation [71]. The force-induced domain unfolding in proteins [72], length transitions caused by conformational changes in the sugar rings [73] or in the secondary structure of polysaccarides [74], and alterations of the secondary structure of DNA molecules were studied in great detail [75], elucidating the molecular determinants of mechanical stability and the role of force-induced conformational changes in the regulation of physiological function. In these experiments, the molecule is held between the tip and the support and its visco-elastic properties are studied in force-distance cycles. Similar to molecular recognition force-spectroscopy experiments, a detailed picture of the complex mechanical unfolding pathway through a rough energy landscape can be gained by varying the dynamics of pulling. In a recent example, single green fluorescent proteins (GFPs) were driven from the native state through their complex energy landscape into the completely unfolded state (Fig. 5.9) [76]. The GFP was stretched between a gold support and a gold-coated cantilever (Fig. 5.9a), and conventional force-distance cycles were applied to measure GFP-unfolding and stretching (Fig. 5.9b). In force spectroscopy and lifetime-measurements it was found that mechanical GFP unfolding proceeds via two successive intermediate states (Fig. 5.9c). The transition from the native state to the first intermediate state occurred near thermal equilibrium at ∼ 35 pN, corresponding to an equilibrium free energy cost of 22kB T . A second meta-stable mechanical intermediate was found associated with one complete β-strand detached from the GFP-barrel. Quantitative analysis of force distributions and lifetimes therefore led to a detailed picture of the unfolding energy landscape.

Fig. 5.9. Force-spectroscopy of GFP unfolding. Single-molecule pulling experiments can be used to investigate mechanical unfolding pathways through complex energy landscapes. (a) Unfolding experiments of the green fluorescence protein (GFP). GFP was sandwiched between IgG molecules and mechanically elongated, leading to a characteristic and reproducible sawtoothshaped mechanical unfolding force pattern. (b) Force extension curves of the GFP-IgG construct. In all traces at extensions larger than 80 nm, the well-studied sawtooth pattern of IgG-domain unfolding is visible (coloured in blue). At shorter extensions and lower forces, an additional peak appeared (coloured in green) reflecting GFP unfolding. (c) Cartoon of the multidimensional energy landscape of GFP unfolding. The red arrows indicate the course of the mechanical unfolding pathway


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Force-clamp methods used for measuring the lifetime of specific ligand-receptor bonds (see Fig. 5.8b) have also been applied to directly measure the end-to-end length of single proteins during its folding reaction at the single-molecule level (Fig. 5.10) [77]. The end-to-end distance of a poly-ubiquitin molecule was monitored during reversible unfolding-folding cycles. Figure 5.10a shows the changes in length of a single ubiquitin polyprotein in response to the stretching force displayed in Fig. 5.10b. Ubiquitin was first unfolded and extended at a high force, then the stretching force was clamped and protein folding was recorded. The folding trajectories were continuous and marked by several distinct stages. These results demonstrated that folding reactions occurred continuously and not between well-defined discrete states. Similar force-clamp experiments of polysaccaride molecules [78]

Fig. 5.10. Force-clamp spectroscopy monitors the folding trajectory of a single protein. The end-to-end length of the protein ubiquitin is plotted as a function of time in (a), whereas the corresponding applied force-profile is shown in (b). Molecular unfolding events, labelled 1 to 5, that occur during the stretch-relaxation cycle are shown in the inset of (a). Stretching the poly-ubiquitin chain at 120 pN using force-clamp (b), it triggers a series of unfolding events that appear as a staircase of 20 nm steps (a), marking the unfolding of the individual ubiquitins in the chain. After 4 s, the force was relaxed to 15 pN, and the protein spontaneously contracted in stages until it reached its folded length. To confirm that the polyprotein had folded, the stretching force was raised back to 122 pN at 14 s, with the effect that the ubiquitin chain extended in steps of 20 nm back to its fully unfolded length

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or the muscle protein titin [79] have also demonstrated that single-molecule force spectroscopy experiments give new insights into the physics of molecular unfolding. Methodical extensions of conventional force spectroscopy experiments allow visco-elastic properties to be measured during unfolding processes [80]. By adding small oscillation amplitudes to the vertical displacement of the cantilever in a forcedistance cycle, the phase and amplitude response reveal both the elastic (i.e., conservative) and viscous (i.e., dissipative) contributions to the unfolding process of single polymers [59] and complex biomolecules [81]. Using Q-factor enhancement, it was shown that the tracking of the resonant frequency helped to separate elastic from viscous forces during unfolding of single biomolecules [37].

5.5 Combined Imaging and Spectroscopy Combination of high-resolution imaging with single-molecule force spectroscopy provides unprecedented possibilities for analysing inter- and intra-molecular forces [22, 23]. Individual bio-molecular complexes observed in topographical images and forces studied in force spectroscopy experiments yield insights into the assembly of supra-molecular structures at the level of single molecules [82]. In one of the first studies, high-resolution imaging was combined with force spectroscopy to investigate the interaction forces between the individual protomers of the hexagonally packed intermediate layer (HPI-layer) [83]. After imaging the HPI layer the AFM stylus was attached to individual protomers, allowing for force-spectroscopy experiments. Structural alterations of individual protomers were attributed to intra-molecular unfolding events recorded in force-distance cycles, showing that single protomers were removed at pulling forces of ∼ 300 pN. Furthermore, it was possible to sequentially unzip entire bacterial pores formed by six HPI protomers. In a similar study, the combination of imaging and unfolding was used to extract single bacteriorhodopsin molecules (BR) from purple membrane patches (Fig. 5.11) [84]. Individual BR-molecules were first localized (Fig. 5.11a), extracted from the membrane (Fig. 5.11b), and the remaining vacancies were imaged again (Fig. 5.11c). Other studies revealed new and exciting insights into the energy landscape of BR-unfolding and their different unfoldingpathways [85, 86]. Besides studying intra-molecular force patterns in unfolding experiments, the identification and localization of specific ligand-receptor interactions on bio-surfaces such as cells and membranes are of particular interest. For this, high-resolution imaging must be combined with specific ligand-receptor detection, so that binding sites can be assigned to bio-molecular structures. By combining dynamic force microscopy imaging with molecular recognition force spectroscopy, an imaging method for the detection and localization of binding sites was recently developed (Fig. 5.12) [23,87,88]. In the first realization, an antibody tethered to an AFM-tip via a ∼ 8 nm long PEG-tether (Fig. 5.12a) was used in the dynamic force-microscopy mode to scan along a surface loosely covered with antigens [23]. Since the oscillation amplitude of the cantilever was only 5 nm, the antibody on the tip always had a chance for recognition when passing an antigenic binding site. Imaging with


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Fig. 5.11. Imaging and unfolding of single bacteriorohodopsin (BR) molecules. Individual BRmolecules were first localized by topographical imaging and then extracted from the membrane. (a) Topographical image of the cytoplasmatic purple membrane surface. BR assembles in trimers (outlined trimer) that arrange in a hexagonal lattice. An individual protein was selected (circled area) and unfolded by keeping contact with the protein surface for ∼ 1 s. (b) Force-distance cycles subsequently recorded by retracting the AFM-tip from the purple membrane. A characteristic pattern of force peaks corresponding to the unfolding of single BR-molecules was observed. (c) After acquisition of the force-distance cycles, the same surface area was imaged again. The topographical image shows that one single BR monomer is missing, the one which was unfolded in (b)

the antibody-functionalized AFM tip gave strikingly different images than use of a bare silicon-nitride tip. Antigenic molecules imaged with the antibody-modified tip differed significantly both in height and diameter. The profiles of single antigenic molecules appeared ∼ 1 nm higher and 10 nm broader than profiles from the purely topographical image (i.e., acquired with a bare silicon-nitride tip). Increased heights reflect the amplitude reduction owing to antibody-antigen recognition, whereas increased diameters are due to the additional length of the antibody plus crosslinkermolecule. The antigenic positions were therefore determined by the maxima of the cross-section profiles obtained at a resolution of ∼ 3 nm. The specific nature of the antibody-antigen interaction was tested by injecting free antibody into the liquid cell, so as to block the antigenic sites on the surface. Subsequent images showed the corresponding reduction in apparent height and diameter. With a refined methodology, a new imaging mode was developed and tested on the same antibody-antigen interaction system rendering possible to separate the topography signal from the simultaneously recorded recognition signal (Fig. 5.12b) [87]. Topography and recognition images were acquired using tips that were magnetically oscillated during scanning and contained antibodies directed against the antigenic molecule immobilized onto the surface. For cantilevers with low Q-factors driven at frequencies below resonance, the surface contact only affected the downward deflections (i.e., the minima) of the oscillations, whereas binding of the antibody on the tip to the antigenic molecule on the surface only affected the upward deflections (i.e., the maxima) of the oscillations. The recognition sig-

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Fig. 5.12. Topographical imaging and simultaneous detection of specific bio-molecular interaction. By combining dynamic force microscopy with single-molecule recognition force spectroscopy, topographical images can be recorded together with recognition images at the same time (a). (b) The raw cantilever deflection signal of the oscillating cantilever is split via the TRECbox into upper parts (corresponding to the recognition image) and lower parts (corresponding to the topographical signal). The topographical image of chromatin and the corresponding recognition image are shown in (c) and (d), respectively. These images were obtained simultaneously from one scan of the sample by using an antibody tethered to the AFM-tip

nals were therefore well separated from the topographic signals. Topography and recognition images were simultaneously recorded using a special electronic circuit with which the maxima and the minima of each sinusoidal cantilever deflection period were depicted. The oscillation-minima were used to record the topography image, and the oscillation-maxima provided the data for the recognition image. In Figs. 5.12c and 12d, respectively, the topography image and the simultaneously acquired recognition image are shown. In this study, a specific type of molecule (histone H3) in a complex sample of chromatin adsorbed onto a surface was recognized by an anti-histone H3 functionalized tip while simultaneously recording a high-resolution image of the same sample [88]. The recognition spots in the recognition image (Fig. 5.12d) revealed the locations of histone H3 in the topographical image (Fig. 5.12c) with high lateral resolution and high efficiency. The specificity of the recognition process was tested by perfusing histone H3 into the liquid cell, resulting in the complete disappearance of the recognition signals. In addition to the localization of histone H3 in the complex biological sample of chromatin, the com-


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positional changes during histone remodelling were also recorded in the topography and recognition images. The simultaneous investigation of both topography and recognition will open a wide field of applications for investigating biological structure-function relationships in native environments on the nanometer scale, because the technique can map composition on top of a topographical image and can detect compositional changes occurring during biological processes.

5.6 Concluding Remarks Atomic force microscopy has evolved into an imaging method that yields fine structural details on live, biological samples like proteins, nucleotides, membranes and cells in their physiological environment and at ambient conditions. Due to its high lateral resolution and sensitive force detection capability, the exciting option of measuring inter- and intra-molecular forces of biomolecules on the single-molecule level has also become possible. The proof-of-principle stage of the pioneering experiments has already evolved into high-end analyses methods for exploring kinetic and structural details of interactions underlying protein folding and molecular recognition processes. Data obtained from force spectroscopy include physical parameters not measurable by other methods and opens new perspectives in exploring the regulation of the dynamics of biological processes. New instrumental developments allow to investigate the chemical composition of the sample. Together with improvements of the sensitivity and acquisition speed this has paved the way to exciting fields in nano-bioscience and nano-biotechnology. Acknowledgements. This work was supported by Austrian Science Foundation projects P14549/ P15295 and the GEN-AU initiative of the Austrian Ministry of Education, Science and Culture. We thank Dr R. Zhu for carefully reading the manuscript.

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6 Sensor Technology for Scanning Probe Microscopy and New Applications Egbert Oesterschulze · Leon Abelmann · Arnout van den Bos · Rainer Kassing · Nicole Lawrence · Gunther Wittstock · Christiane Ziegler

6.1 Introductory Remarks The invention of scanning tunneling microscopy (STM) by Binnig and Rohrer [1–3] and atomic force microscopy (AFM) by Binnig et al. [4] has unleashed the development of a new class of analytical tools that have revolutionized surface science over the past two decades. The ability to study various surface properties on an atomic scale [5] combined with the ability to manipulate or arrange molecular structures [6, 7] has blossomed into what recently has recently been termed nanoscience technology. The heart of any scanning probe microscope tool is the sensing element, simply termed the probe. It consists in almost every case of a sharpened tip with an extremely small radius of curvature that determines the interaction volume with the local sample surface and thus the lateral resolution during the scanning process. We will only concentrate in this chapter on AFM-based probes (cantilever probes consisting of a mechanical beam with or without an integrated sharp tip), because they have proven to be the most flexible arrangement for future developments in SPM. The various AFM operation modes and their operational limits are not the scope of this report, and details on them may be found elsewhere ( [8–11] and references therein). The same holds for the various forces involved in the scanning process (e.g. [12–15]). Giving an overview of recent probe developments resembles a Sisyphus work, because scanning probe microscopy (SPM) has found widespread use not only in

Fig. 6.1. Set-up of a multifunctional standalone SPM cantilever probe

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physics but also in biology, chemistry, mechanical, and electrical engineering [5]. Therefore, we will only focus on the developments in SPM probe technology that will hopefully culminate in the multifunctional stand-alone probe sketched in Fig. 6.1 in the near future. In this review we have tried to highlight probes used for very different applications, including near-field optics, semiconductor device characterization and nanofabrication, ultrafast electrical field sampling, functionalized probes for biological applications, electrochemical microscopy, and magnetic microscopy.

6.2 Material Aspects of Probe Fabrication The development of SPM probes adapted to certain applications has to be separated into at least two different steps. The first covers the selection of an appropriate sensor concept for the desired application. The second involves the choice of a proper material that not only allows probes of an appropriate sensitivity to be obtained but that also meets the boundary conditions of the experimental environment, in terms of chemical inertness, mechanical hardness, low abrasion, and so on. Before embarking on examples of probes for various applications in accordance with step 1, we would like to emphasize the importance of choosing the proper material. In the early stages of AFM cantilever probes (a mechanical beam with an integrated sharp tip at its very end), they were fabricated manually [4]. However, Table 6.1. Properties of some important materials used for MEMS fabrication of SPM probes. SI denotes semi-insulating material (001) Si [16]

Material (001) GaAs [17]

diamond [18]

2.329 168.4 129.2 8,503.3 80 1,150 (25)

5.317 121.5 85.5 4,780.3 60 680 (100)

3.515 1,163.6 1,050.3 18,194.5 – 18,000 (100)

Thermal heat capacity cp (J/kg K) Heat conductivity k (W/mK) Melting point Tm (K) Thermal expansion coeff. (µV/K) Seebeck coefficient (µV/K)

700 150 1,415 2.6 –1,600...1,500

500 45.5 1,513 6.86 –680...130

516 2,000 graphite formation 0.8

Optical and electrical refractive index n (λ = 633 nm) Static dielectric constant Gap energy E G (eV) Electron mobility µn (cm2 /V s) Hole mobility µp (cm2 /V s) Electrical resistivity ρ (Ω cm)

3.4 11.8 1.12 (indirect) 120–1,300 70–500 105

3.878 13.18 1.42 9,200 (SI) 400 (SI) 109 (SI)

2.41 5.40 5.45 1,900 1,600 1016

Properties Mechanical properties Mechanical density ρm (g/cm3 ) Youngs modulus E [110] (GPa) Youngs modulus E [100] (GPa) E [110] /ρ (m/s) Torsional modulus G (GPa) Hardness (load (g))

6 Sensor Technology for Scanning Probe Microscopy and New Applications

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it soon became clear that reliable experiments demand reproducible fabrication of probes with well defined material properties. Some noteworthy physical properties of the materials used most often in the fabrication of microelectromechanical systems (MEMS) – silicon, gallium arsenide, and diamond – are summarized in Table 6.1. 6.2.1 Mechanical Properties of Cantilever Probes In order to explain the influence of the material and the geometrical properties of a cantilever beam in an AFM experiment, we now give a very brief theoretical introduction to the mechanics of cantilever beams. A more detailed description may be found in [9] and references therein. The displacement z(y) of a cantilever directed in the y-direction is governed by the following differential equation in frequency space: ! " I E d4 z (y) − κn4 z (y) = 0 with κn defined as ωn = κn2 . (6.1) dy4 A ρ The eigenfrequency ωn of the nth eigenmode thus depends on the geometrical cantilever properties, which are given by the moment of inertia I of the beam and its cross-sectional area A, and on the mechanical properties: the Young’s modulus E and the mechanical density ρ. In the simple case of a beam of length l and rectangular cross-section, the solution to the associated eigenvalue problem results in the following eigenvalues for the first five flexural modes: κn l = 1875; 4694; 7855; 10,996; 14,137 .

(6.2)

Taking the compliance of the beam, defined as: k = 3E

I , l3

(6.3)

into account, we obtain the well known relations for ω1 and k of the first flexural mode: " d E 1 bd 3 (6.4) and k = E 3 . ω1 = 1.0149 2 l ρ 4 l A more detailed description of the mechanics of a beam made of isotropic and homogeneous material is given elsewhere [19]. This theoretical approach was extended to study the mechanical behavior of cantilever probes fabricated from layered materials [20–22]. The influence of the mechanical properties of the material on the frequency of √ the first flexural mode of a beam in (6.1) is given in terms of the ratio E/ρ. In accordance with Table 6.1, single crystalline silicon obviously offers quite adequate properties. This explains its widespread use as standard material in the fabrication of MEMS and in particular SPM probes [23]. However, it is worth noting that the values

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√ for E/ρ mentioned in Table 6.1 differ only by a factor of 3.81 at the most. The same variation in ω can be easily achieved, for instance, by increasing the cantilever length l by a factor of 1.95 while keeping all other parameters constant. This means that the material chosen for the cantilever beam becomes decisive only for cantilever beams with extreme geometrical dimensions (where the technological fabrication processes constitute the limiting case). For more information on the influence of the properties of crystalline materials, see [24]. 6.2.1.1 Silicon The enormous popularity of silicon or thin films of silicon nitride or silicon dioxide as probe material is due not only to their extraordinary properties but also to the huge number of technological processes and the vast amount of experience that can be drawn upon during processing [25,26]. In the case of silicon, the fabrication of probes relies in most cases on bulk micromachining. In particular, sharp tips are obtained by underetching masking pads during wet or plasma etching processes [27]. The typical radius of curvature of a silicon tip, 10–20 nm, can be further improved by repeated thermal oxidation at reduced temperatures and subsequent removal of the oxide layer [28, 29]. This sharpening process exploits the unique rheological behavior of thermally grown silicon dioxide layers, as will be discussed later in some detail. Tips with a radius of curvature of less than 5 nm are available (see [24, 30]). 6.2.1.1.1 Homogeneity of Cantilever Probes Total thickness variations in silicon wafers of typically 1–5 µm limit the reproducible fabrication of cantilevers with exactly the same mechanical properties, as is obvious from (6.4). Etch stop layers were successfully introduced to obtain silicon membranes of homogeneous thickness, which reduces the scatter in mechanical cantilever properties substantially [32]. Thin films of etch-resistant material as well as p/n junctions obtained by doping have been proposed and used for this purpose [33, 34]. Recently, surface micromachining utilizing SOI (silicon on insulator) wafer has become very popular because the thickness of the cantilever is well controlled by the thin cap silicon layer (see [35,36]). If serial fabrication is not imperative, a focused ion beam (FIB) is a useful tool, particularly if extremely small and precise geometries are of interest, in other words probes with ultrasmall compliance [37]. 6.2.1.1.2 Probes Made of Thin Deposited Layers Another option for solving this problem is to fabricate the cantilevers from thin deposited layers, such as silicon nitride [38]. In this case, tip fabrication relies on molding from anisotropically etched pyramidal shaped etch grooves [38] or the fabrication of Spindt-type tips made by evaporation through an orifice [39]. One advantage of this with respect to bulk micromachining is the ability to control the


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thickness of the cantilever precisely by the deposition or oxidation process. However, the fragility of the thin cantilever membrane requires a mechanically stable holder to be mounted in a parallel process to the cantilever. Various elaborate and complex processes have been considered for this purpose, such as anodic bonding [38], silicon-to-silicon bonding [40], soldering [41], and glueing [42]. 6.2.1.1.3 Standalone Cantilever Probes and Nanometric Oscillators Using silicon as the substrate material results in additional benefits, due to the huge number of technological processes previously established for the VSLI fabrication of electronic devices [23]. Various approaches to the integration of electrical sensors in cantilever probes have been used. In this section we discuss sensors and actuators that are intended for the standalone operation of silicon-based AFM probes. Piezoresistive transducers consisting of a single resistor or of four resistors that form a Wheatstone bridge were integrated into the base of a silicon cantilever probe to detect both its bending and twisting [43–49]. The compact probe design makes it perfect for experiments performed under UHV conditions or with strongly restricted geometry. It avoids any kind of alignments that are necessary in conventional optical [50–54] or electrical deflection methods [55–57]. A typical example of an array of silicon cantilevers with integrated piezoresistive readout is shown in Fig. 6.2. It also constitutes an important prerequisite for AFM imaging with increased bandwidth utilizing an array of cantilever probes. Quate et al. have demonstrated in their pioneering work the individual control of at least 50 cantilevers, integrating both a piezoresistive readout and a microactuator based on thin ZnO films [44]. Highspeed imaging [47, 58], parallel lithography [59–61], as well as parallel data storage with high data densities [62] have since been presented. In this context, the top notch millipede project of IBM must be mentioned as one of the first routinely working ultrahigh SPM data storage devices [63, 64]. A further increase in temporal bandwidth is achieved by utilizing nanometric oscillators with resonance frequencies of typically a few GHz [65–69]. A parallel array of millions of nanometric oscillators was proposed and presented by Kawakatsu et al. [70, 71].

Fig. 6.2. (a) Array of silicon cantilever probes with integrated piezoresistive Wheatstone bridge. (b) Each cantilever is perforated at the center of the Wheatstone bridge to improve the sensitivity by increasing the mechanical strain in the remaining material [31] (reproduced courtesy of NaScaTec GmbH, Kassel, Germany)

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6.2.1.1.4 Sensors Based on a Cantilever Set-up The unintentional temperature sensitivity of piezoresistors integrated into cantilevers was exploited to detect heat emerging from the chemical reactions of very small amounts of chemicals on the cantilever beam, and this led to the development of so-called chemical noses. Meanwhile, various approaches have been discussed in the literature [72–78]. The first exploited the bimetal effect of layered cantilevers [79,80]. 6.2.1.1.5 Sensors Integrated Into the Probe Tip So far our discussion has concentrated on the cantilever as a mechanical tool that transforms forces affecting the tip into a bending of the beam. However, the tip itself offers a realm to implement ultrasmall sensors. This entails some difficulty: integrating sensors into a three-dimensional structure only microns across is a real challenge. However, doing so results in the fascinating prospect of being able to investigate different surface properties simultaneously. Owing to the rich variety of sensors that have been integrated into the tip (see [5, 24]), we can only discuss a few examples in the second part of this chapter. 6.2.1.2 Carbon Carbon has been used in various chemical modifications for probe fabrication. Aside from electron beam deposited (EBD) carbon tips, sp2 - and sp3 -bonded carbon (nanotubes and polycrystalline diamond) has attracted enormous attention as a tip material owing to its extraordinary properties. 6.2.1.2.1 Electron Beam Deposited Tips Electron beam assisted gas phase deposition of carbon-containing material was developed in 1990 by Akama et al. [81] (see also [82]) to fabricate carbon tips with a high aspect ratio, very small radius of curvature, and an appropriate hardness for most common applications [83, 84]. However, a few years later, Dai et al. provided evidence that carbon nanotubes are a much better choice due to their excellent physical properties [85]. 6.2.1.2.2 Carbon Nanotube Tips The synthesis of carbon nanotubes was reported by Iijima for the first time in 1991 [86]. Carbon nanotubes consist of single or multiple seamless hollow cylinders made from folded graphene that is composed of sp2 -bonded carbon hexagons [87]. The diameters of single- or multiwalled nanotubes are in the nanometer range,


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whereas typical nanotube lengths are in the micron range [88]. This enormous aspect ratio, together with a very small and reproducible tip radius of curvature make them perfect AFM tips for high-resolution imaging [89–91] as well as imaging in deep structures [85]. As can be conjectured from the similar structure of graphite in its basal plane, nanotubes exhibit a tremendous value of Young’s modulus of about 0.4–1.3 TPa for typical rope diameters of 20–2 nm [92–94]. Even more fascinating, nanotubes are both stiff and gentle because buckling occurs if the load exceeds a critical force, the so-called Euler force [85]. The enormous strain resilience of carbon nanotubes avoids any damage to both the nanotube as well as the sample during buckling in contrast to almost any conventional silicon or silicon nitride tip [95–98]. Furthermore, Wong et al [99] and Terrones et al. [100] successfully demonstrated the potential of nanotubes for sensitive chemical imaging by attaching molecules that serve as molecular sensors to the carbon nanotube tip. Although carbon nanotubes have attracted exceptional attention in the SPM community, no convincing batch fabrication process has yet been established. Almost all fabrication schemes that have been presented so far rely on manual attachment to conventional AFM cantilever tips [85, 101–103]. 6.2.1.2.3 Diamond Probes In applications that require probes of high compliance, tip wear becomes an important issue. Table 6.1 indicates that sp3 -bonded carbon in the form of diamond, with its unique mechanical properties, is the best candidate to fulfill the requirements of mechanical hardness and low abrasion. However, it also has other extraordinary properties, such as a very high thermal conductivity and chemical inertness. Namely, the large electrical band gap of 5.45 eV and the ability to control the carrier density over more than ten decades by doping with boron and nitrogen means that diamond excels as a material for high-temperature semiconducting devices [104]. Initially, grains of single-crystalline diamond were utilized to manually fabricate probe tips by fracture [108, 109] and grinding [110, 111]. With the introduction of CVD diamond growth, the first probes made of sharpened metal wires with statistically distributed CVD-grown diamond grains were introduced [112, 113]. But the latter suffer among other things from their insufficient reproducibility. The first approaches based on the molding of diamond tips were introduced by two groups [114–117]. Some typical examples are shown in Fig. 6.3. Their usefulness was demonstrated by utilizing them for the electrical characterization of electronic devices with high-resolution nanopotentiometry, spreading resistance microscopy, and capacitance microscopy [118]. Improved processes, namely the tip-on-tip approach [107] and the double molding process [119], were introduced later to improve the quality of molded diamond probes. However, the polycrystalline texture of the CVD-deposited diamond film causes tremendous difficulties during batch-processing of AFM probes, due to, for example, incomplete filling of the pyramidal molds owing to the small nucleation density, insufficient control over the selective growth of CVD diamond on the silicon substrate, low quality of the diamond film at the tip, and the intricate process needed to attach a holder to the diamond membrane [120, 121]. The application of molded

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Fig. 6.3. (a)–(c) SEM images of AFM cantilever probes made of polycrystalline diamond utilizing the conventional molding technique [105, 106]. (d) An example of a tip-on-tip diamond probe [107]

diamond probes to an AFM set-up was hindered due to the strong optical scattering of the laser beam used for beam deflection measurement from the very roughly grown surface of the polycrystalline diamond surface, which constituted of (111) oriented grains. Coating the commercial silicon tips was one suggestion for overcoming these problems, but this comes at the expense of an increased radius of curvature [116, 118, 122]. Recently, the projection mask technique was introduced in order to overcome most of the abovementioned problems [120]. This method utilizes an already structured substrate to define the vertical cantilever geometry as shown in Fig. 6.4. The lateral geometry of the cantilever is defined consecutively as a two-dimensional pattern on a conventional optical projection mask. This pattern is transferred via a proximity lithography and etching process onto the already structured substrate.

Fig. 6.4. Scheme of the projection mask technique. The outline of the cantilever is defined via a proximity lithography and etching process, while the vertical geometry is given by the prestructured substrate. The probe can be fabricated from both the substrate or a thin deposited film [120]


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Finally, a mechanical holder necessary for ease of handling is fabricated from the substrate used to create the cantilever probe. The projection mask technique is capable of probe fabrication from the substrate as well as from thin films deposited onto the substrate. The feasibility of this method was demonstrated for all-diamond cantilever probes. SEM images of typical diamond probes are given in Fig. 6.5. These probes were successfully applied for nanofabrication of ultrasmall single electron in-plane gate transistors fabricated by mechanical scribing with the diamond tip while the depth of scribing was monitored electrically [121].

Fig. 6.5. SEM images of all-diamond AFM cantilever probes made by the projection mask technique: (a) diamond membrane with a v-shaped cantilever probe mounted on a silicon holder, (b) single cantilever with a pointed tip, (c) bottom view of the cantilever in (a) revealing the rough growth surface of HFCVD diamond, (d) sharpening of the diamond grains by plasma treatment [120]

6.2.1.3 Gallium Arsenide Gallium arsenide (GaAs) is highly attractive due to its unique electronic and optical properties [123]. In this context it is considered to be a substitute for the rich class of compounds of III/V- and II/VI-semiconductors. In particular, its direct band gap predestines GaAs for a variety of optoelectronic and low-dimensional electronic devices. Section 6.3.2 will focus, for instance, on the application of a very compact surface-emitting quantum well laser for an active optical near-field probe. However, in the investigation of ultrafast phenomena, GaAs is also one of the predominant substrate materials for the establishment of photoconductive switches, as will be

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discussed in Sect. 6.4. Another noteworthy application of GaAs is its use as an efficient photon-excited source of spin polarized electrons [124]. The first GaAs sensors were made by cleaving single crystalline GaAs wafers twice in order to obtain very sharp tips at the intersection of the two edges [125]. Meanwhile, it is common practice to batch-fabricate GaAs cantilever probes with integrated tips, and the process works in general analogously to that based on silicon [126]. However, the covalent/ionic bonding between the Ga and As atoms in GaAs leads to an intricate etching behavior [123]. On the one hand, this exacerbates the fabrication of MEMS structures by wet chemical etching, but on the other hand it can also be exploited to obtain extreme tip shapes that are not feasible when using silicon or diamond materials [127]. In the following, probes for different applications will be discussed in more detail.

6.3 Scanning Near-Field Optical Microscopy 6.3.1 Principle of Near-Field Optics The optical resolving power is restricted by two spatial low pass filter processes. The first is inherent to the nature of light, because the wavenumber k = 2π/λ of monochromatic light of a given(wavelength λ limits the size of the object spatial frequency component kObject = k2x + k2y in the object plane. However, propagation in free space requires a positive and real component k z in the propagation direction: ( (

2 (6.5) k z = k02 − kObject = k02 − k2x + k2y . Hence, higher object spatial frequencies than k give rise to an imaginary k z and thus to evanescent waves – waves that are exponentially damped in free space and are not accessible by far-field optics. The second low pass filter stems from the geometry of the lens itself; it is due to the diffraction at the lens barrel. In terms of Fourier optics it is apparent from

Fig. 6.6. Imaging an object with transmission function T(x) with a lens of focal length f and diameter D results in the Fourier transform F {T(x)} = T (k x ) in its back focal plane [128]


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the simplifying scheme in Fig. 6.6 that only Fourier components with k x ≤ kmax are x capable of passing through a lens of focal length f and diameter D. Assuming this,

D kmax x = sin ϑ max ≈ tan ϑ max = = NA , k0 2b

(6.6)

where NA is the numerical aperture of the lens. This limits the largest spatial and thus the lowest spatial wavelength λmax that can pass through frequency kmax x x the lens: = k0 NA , kmax x

and λmax = x

λ0 . NA

(6.7)

The latter expression is in accordance with the arbitrarily defined Rayleigh’s criterion for the resolving power of a lens [129]. Overcoming the limitations of far-field optics obviously requires the detection of the evanescent (the high spatial frequency) near-field components. This is achieved with an optical link that picks up part of the near-field component in the vicinity of the sample surface and converts it into propagating waves. Optical links can be grouped into aperture carrying wave guiding probes, scattering (apertureless) probes, and immersion lens probes [130]. In the following we restrict our discussion to the first type of probes. For a more extensive discussion, refer to [24,130] and references therein. 6.3.2 Probes for Scanning Near-Field Optical Microscopy (SNOM) 6.3.2.1 Aperture-Based Near-Field Optical Probes An aperture probe consists of a subwavelength-sized aperture in an optical opaque screen. If the latter is brought in close vicinity to the sample surface to pick up nearfield optical information, the diffraction is (to a good approximation) negligible, and the aperture size rather than the wavelength determines the lateral resolution. For the detection of near-field components, the aperture is, in almost every case, integrated into the very tip of a pointed tapered metal-coated waveguide. Owing to the cut-off effect, the latter shows unfavorable waveguide properties; tapering the waveguide below half the wavelength gives rise to strong damping due to the lack of any propagating waveguide mode [131–134]. From this brief discussion, we can draw the conclusion that an appropriate aperture probe should offer at the very least: a pointed tip with a subwavelength aperture, a very small taper region to reduce transmission losses, a proper mechanical compliance to keep the aperture mechanically in track with the sample surface during scanning (image formation), and most importantly, reproducible optical and mechanical probe properties for reliable imaging. Most of the requirements discussed can easily be fulfilled if near-field optics is combined with scanning force microscopy integrating an aperture tip into a conventional SFM cantilever probe fabricated by MEMS technology [40, 136–138].

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Hence, we introduced an modified technology for integrating a hollow silicon dioxide aperture tip into a conventional silicon cantilever probe, as can be seen from the schematic cross-section of this tip in Fig. 6.7a [135]. The finished aperture SNOM/SFM probe is shown in Fig. 6.7b, whereas c and d show zoom images of the aluminum-coated tip, forming a 50 nm sized aperture in the opaque metal layer at the tip’s apex, respectively. Two different technological routes are available for aperture fabrication. One exploits the pronounced oxide retardation at concave (and convex) silicon structures during the thermal oxidation of silicon, whereas the other is based on the unique properties of plasma etching of structured silicon surfaces. Owing to the self-adjusting nature of both methods for top-to-bottom nanostructure fabrication, we discuss them in some detail. For planar silicon substrates, oxidation follows, to a good approximation, the simplifying Deal/Grove model that predicts diffusion-limited (reaction rate-limited) growth for the oxide layer if the grown silicon dioxide layer exceeds (is less than) about 20–40 nm or so [140]. However, the Deal/Grove model does not account for the intrinsic mechanical stress originating from the volume expansion (by a factor of 2.25) upon oxidizing a single Si atom to obtain a SiO2 molecule [141–143]. Thus, Kao et al. emphasized the influence of the intrinsic stress affecting both the temperature-dependent diffusion coefficient and the reaction rate, that gives rise to remarkable oxide growth retardation at concave and convex surface structures in comparison to planar ones [144, 145]. In the refined model discussed by Senez et al., silicon dioxide was treated as a viscoelastic material with a nonlinear stressdependent viscosity [146]. Form this unique material behavior, we inferred that pronounced oxide retardation should occur at high oxidation temperature [135]. This was proven both experimentally and theoretically, as can be seen from Fig. 6.8.

Fig. 6.7. (a) Cross-section of a SNOM aperture probe carrying a hollow tip made of a thin silicon dioxide layer. The latter is pierced at its apex and coated with an opaque aluminum film in order to obtain aperture SNOM tips. (b) The complete SFM/SNOM cantilever probe, (c)–(d) SEM images of the pyramidal hollow tip and a 50 nm aperture, respectively [135]


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Fig. 6.8. (a) Calculated stress distribution and outline of the oxide layer on a silicon v-groove structure that was wet oxidized at a temperature of 900 ◦ C. (b) Theoretical predictions are experimentally confirmed by SEM imaging of cross-sections of v-groove structures [135]. Simulations were performed with the numerical code FLOOPS from M. E. Law and A. Tasch [139]

To accomplish apertures in the silicon dioxide layer, the structures shown in Fig. 6.8b are opened from the bottom by removing first the silicon substrate and subsequently part of the oxide layer by wet etching with BHF. Due to the retarded oxide thickness at the apex, it opens reproducibly during the etching process at the apex to form an aperture. For SNOM aperture tips, pyramidal structures are used instead of grooves. The second technological route to accomplishing aperture probes also starts also with an inhomogeneous silicon dioxide layer grown by steam oxidation on octahedral etch pits. However, perforation of the silicon dioxide layer at the pyramidal apex forming apertures is accomplished this time by reactive ion plasma etching with CHF3 [147]. As can be seen from Fig. 6.9a and b, this simple method allows us to define 90 nm openings at the concave apex without any lithography process. It should be emphasized that due to the large opening angle of 70.5◦ , the taper of our aperture tip is rather short and thus the transmission is rather high in comparison to tapered fiber probes. Nevertheless, it is not appropriate for some interesting applications like Raman spectroscopy or single molecule detection. For this purpose coaxial arrangements [149–151] as well as antenna structures like the bow-tie antenna were proposed [152, 153]. The latter consists, as shown in Fig. 6.10a, of two triangular metal electrodes that face each other and are separated by a tiny gap. Simulations underline that the bow-tie antenna has the ability to focus electromagnetic waves, even at optical wavelengths, into the gap of the bow-tie antenna (see Fig. 6.10b) with almost total efficiency [154]. However, the realization of gaps of less than 50 nm size on a three-dimensional structure is currently a real challenge.

Fig. 6.9. Silicon dioxidecoated v-grooves in silicon (a) before and (b) after reactive ion plasma etching with CHF3 [147]

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Fig. 6.10. (a) Scheme of a bow-tie antenna probe consisting of two triangular electrodes facing each other and leaving a tiny gap between them [148]. (b) Electric field energy distribution in the plane of the bow tie antenna, showing its focusing capability (lighter color indicates high density) [24]

Fig. 6.11. SEM images of a hollow silicon dioxide tip coated with aluminum. The latter was partly removed by focused ion beam milling in order to establish a pointed dipolar bow-tie antenna (in collaboration with H. U. Danzebrink and Ch. Lehrer) [153]

Figure 6.11 shows first results for a microfabricated bow-tie antenna tip based on our silicon dioxide aperture probe [153]. The two arms of the antenna were accomplished by focused ion beam removal of triangular parts of the metal layer that was first deposited onto the whole aperture tip. Figure 6.11b proves that gaps of about 50 nm width can be realized in this serial process. 6.3.2.2 Active Near-Field Optical Probes A completely different approach giving an active (light emitting) active near-field probe for SNOM and SFM applications is presented in Fig. 6.12. It comprises a conventional monolithic GaAs cantilever probe with a sharpened tip and an integrated laser diode that illuminates the tip from the rear. Light emission is restricted by a subwavelength-sized aperture in the 100 nm thin metal layer at the tip’s apex. The high refractive index of GaAs (n = 3.8 at a wavelength of 980 nm) is advantageous because it reduces the transmission intensity losses of the tapered tip due to the cut-off effect [127]. The compact geometry of VCSEL allows near-field cantilever probes to


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Fig. 6.12. (a) Schematic view and (b) SEM image of the first active GaAs near-field probe with an integrated VCSEL operating at a wavelength of about 977 nm [126, 127]

be produced that are similar in dimensions to conventional SFM cantilevers. Furthermore, our probe design is capable of exploiting the beam of the VCSEL emitted in the opposite direction for beam deflection measurement in the SFM mode, omitting the need for any external light source. Thus, these kind of probes are destined to be operated in a parallel arrangement that is advantageous for, say, a very compact set-up in a high-density near-field optical data storage device [155] or for near-field optical lithography.

6.4 Probes for Ultrafast Scanning Probe Microscopy In ultrafast scanning probe microscopy (USPM), the voltage contrast in an electronic device is detected with a pump/probe experiment [156, 157]. An optical train of ultrashort pulses emitted from a Ti sapphire laser is converted into an electrical pulse using a photoconductive switch [158–160]. This pulse is sent to the device under test (DUT). Simultaneously, the same optical pulses are used to turn on and off a photoconductive switch [161, 162] integrated into a GaAs SFM cantilever in order to measure a high speed electrical signal using a photoconductive sampling technique [163]. A SEM image of an appropriate sampling probe is shown in the lower part of Fig. 6.13. It carries a microwave strip line on top of a thin low temperature grown (LT) GaAs layer that was deposited onto a cantilever made from an insulating GaAs substrate [164]. Illuminating the gap between two of the adjacent microwave electrodes with ultrashort optical pulses allows the voltage, which is picked up with the strip line on the GaAs cantilever from the DUT, to be sampled. The time resolution is achieved by varying the time delay between the synchronized pump and probe beam [165]. A typical result from an investigation of a microwave strip line is given in Fig. 6.14. The experiment was conducted as shown in Fig. 6.14a. The temporal signal shown in Fig. 6.14b was picked up at a fixed position on the ground electrode and reveals a pulse width for the strip line mode of about 1.2 ps [164, 166]. Figure 6.14 c and d show similar results from a spatial/temporal investigation where the cantilever was scanned across the strip line in the contact and noncontact mode to obtain the complete pulse shape. Obviously, the differentiated signal is obtained

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Fig. 6.13. Scheme of a scanning microwave microscope based on a GaAs cantilever with two adjacent photoconductive switches used for launching and subsequent detection of ultrafast signals that interact locally with the sample surface. SEM image of a realized USPM cantilever with microstrip line [164]

Fig.6.14.(a) Spatial/temporal sampling of the field distribution of pulses launched on a microwave strip line with a photoconductive switch integrated onto a GaAs cantilever. (b) Time-resolved electric field signal on the grounded electrode of the transmission line: in contact mode Ic , in non-contact mode Inc , and the numerically differentiated contact mode signal. (c) and (d) Spatial/temporal measurements of the field distribution of the strip line mode in contact and noncontact mode, respectively [164, 166, 167]


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in noncontact mode, owing to the differentiating behavior of the gap capacitance between the cantilever tip and sample surface [166]. Operating the cantilever probe in the noncontact mode should therefore permit a high-resolution investigation of the local dielectric function in a microwave microscope, as schematically shown in the upper part of Fig. 6.13. 6.4.1 Improved Sampling Technique It should be pointed out that, for the conventional photoconductive sampling technique described above, the carrier lifetime limits the temporal resolution of the system to 1–2 ps [168, 169]. Usually materials with reduced carrier lifetimes such as low temperature grown GaAs (LT GaAs) or ion implanted semiconducting layers are utilized [170–174]. However, introducing a high defect density using such techniques usually results in low carrier mobility, poor stability, and a lack of fabrication processes for other devices. To get rid of these material restrictions, we proposed an alternative sampling method. The idea of the improved sampling concept relies on the mathematical theorem that differentiation of the Heaviside function Θ(t − t0 ) results in Dirac’s delta function: δ (t − t0 ) =

∂Θ (t − t0 ) . ∂t

(6.8)

That is, if the photoconductive switch is formed using a material with a long carrier lifetime, such as semi-insulating GaAs (SI-GaAs), then the transfer function of the photoconductive switch over a time-scale of a few picoseconds corresponds to a step-like function. If this photoconductive switch is integrated into the GaAs cantilever and the latter is operated in the noncontact mode, sampled signals will be differentiated and thus the time resolution should be improved substantially. First theoretical simulations confirm the working principle, as can be seen in Fig. 6.15 [167]. To demonstrate the high efficiency of this improved technique,

Fig. 6.15. The original signal Vin on the strip line is a superposition of two 500 fs-long pulses delayed by 300 fs. Additionally shown are the signals obtained for conventional sampling with fast LT-GaAs material in the contact mode (dashed line) and for the novel technique with long carrier lifetime material in the noncontact mode (dotted line) [167]

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a superposition of two pulses delayed by 300 fs was sampled. The signal obtained in the contact mode for the conventional sampling technique with LT GaAs and the noncontact signal for the novel technique with SI-GaAs are plotted together in Fig. 6.15. As can be seen, the high temporal resolution of the new technique allows us to resolve the two superimposed pulses, whereas the conventional method, even with a switch response as short as 600 fs, is not capable of recovering the double peak. This concept may be used in future for ultrafast pulse generation and sampling.

6.5 Functionalized Tips SFM is mainly known and used as an imaging tool. Functionalization of the SFM tip has opened up a new field of nanochemical applications. For a defined tip–surface interaction, specific imaging of the surface – so-called chemical force mapping – is possible, and it is possible to determine adhesion forces between biomolecules and substrate surfaces on a molecular scale. In the following, different types of tip modifications and their applications will be described. 6.5.1 Tip Modification Tip modifications can be grouped into specific and nonspecific functionalizations. 6.5.1.1 Nonspecific Tip Modification The advantage of nonspecific functionalization is its simple preparation procedure and the low cost of binding molecules. However, the disadvantage is the uncertainty over which part of the protein is fixed and which part is free for binding. Therefore, measurements can only be statistically analyzed and interpreted. To achieve a good statistical background, many time-consuming measurements have to be performed. An example of nonspecific modification is the fixation of a protein on the tip surface by means of glutardialdehyde (C5 H8 O2 ). Glutardialdehyde is a small protein which cross-links proteins to form a closed proteinlayer (see Fig. 6.16).

Fig. 6.16. Nonspecific tip functionalization


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6.5.1.2 Specific Tip Modification Specific binding of biomolecules permits defined and reproducible measurement. Depending on the functional head group, a chemical or a biological property can be analyzed [175]. To modify the tip, a linker molecule (spacer 1; thiols or silanes) with a functional head group (hydrophobic, OH, COOH groups) is be fixed onto the tip surface (see Fig. 6.17 step 1). This chemical functionalization allows chemical force microscopy (CFM) or scanning force spectroscopy (SFS) measurements to be taken (see Sect. 6.5.1.1). To perform force measurements of single proteins or biomolecules, a second spacer can be introduced between the protein and the modified tip surface (see Fig. 6.17 step 2 and 3). Spacer 2 avoids the need to measure interactions between the linker molecules (instead of protein) and the sample surface.

Fig. 6.17. Specific tip functionalization (Nanocraft, Renton, WA, USA)

6.5.2 Applications 6.5.2.1 Chemical Force Microscopy Chemical force mapping ([176], and refer to Nanocraft, Renton, WA, USA) combines lateral force microscopy (LFM) with modified tips. As the cantilever is scanned over the substrate surface, variations in interaction between the modified tip and the sample surface will cause the tip to stick or slip during the scan, resulting

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in twisting of the cantilever. The torsion angle is proportional to the interaction force. This torsion can be detected by a four-section photodiode, which monitors the vertical as well as the lateral deflection of the cantilever. By scanning the same area with different chemically functionalized tips, a chemical force map can be built up (compare Fig. 6.18).

Fig. 6.18. Principle of chemical mapping (Nanocraft, Renton, WA, USA)

6.5.2.2 Scanning Force Spectroscopy Surface property changes due to protein adsorption play an important role in many fields of medicine and engineering. Scanning force spectroscopy [177, 178] can be used to study the interaction forces between a molecule – here a biomolecule – and either a surface or another biomolecule. In this method a protein is immobilized on the tip of the cantilever. Then the cantilever is moved towards the surface,

Fig. 6.19. (a) Principle of force measurement. (b) A typical force–distance curve of a biofuntionalized tip


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where it resides for a short time before being retracted (see Fig. 6.19). The bending of the cantilever can be monitored in relation to the distance from the tip to the surface in a so-called force–distance curve. If the force constant of the tip is known, the bending can be used to measure the interaction force between the protein on the tip and the substrate surface. A typical force–distance curve is shown in Fig. 6.19. 6.5.2.3 Miniaturized Sensor Technology Another development is the use of cantilevers as chemical and/or biochemical sensors [179, 180]. In this application the cantilevers are not brought into contact with a surface. Instead, they are used to detect chemicals in the gas or liquid phase. For this purpose, the cantilever is again functionalized with a coating to which the analyte has a specific affinity. If the analyte binds to the cantilever coating, the cantilever changes its physical properties. This reflects the surface tension. If only one side of the cantilever is coated, this results in a bending of the cantilever which can be detected. Another principle is based on using the cantilever as harmonic oscillator. The cantilever is made to perform harmonic oscillations near to its resonance frequency. Due to the attached analyte, the mass changes, and so does the resonance frequency (see Fig. 6.20). In the gas phase the frequency change is directly proportional to the mass loading, and hence a quantitative signal is achieved [181]. In the liquid phase, viscosity changes also influence the resonance frequency and, in particular, the resonance width [182].

Fig. 6.20. Principle of a resonant cantilever sensor

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6.6 Scanning Electrochemical Microscopy 6.6.1 Principles Scanning ElectroChemical Microscopy (SECM) is a scanning probe technique that relies on measuring faradaic currents at a microdisk electrode surrounded by an insulating sheath. The ultramicroelectrode (UME) is immersed in an electrolyte solution and scanned over the specimen surface at a distance of a few electrode radii (Fig. 6.21). As typical electrode radii are ≤ 12 µm, the working distance and the lateral resolution are typically similar. In 1986 Engstrom et al. [183] used an UME to measure local concentrations of reactants and products converted or generated at a macroscopic specimen electrode – the first SECM experiment. Later Bard and coworkers supplied a magnificent contribution that laid the theoretical foundations for SECM [184–186]. One of the major differences of this technique from other scanning probe techniques is that the origin of the signals is quantitatively understood and can be used on a routine basis to extract physicochemical parameters about interfacial reactions and transport phenomena from the measured data. This facilitates the functional analysis of various important solid/liquid interfaces, like passivated metals, coating materials, conducting polymers, fuel cell and biosensor components, the metabolism of biological tissue, and adherently growing cells. The theory and application of SECM has been compiled into an authoritative book that is warmly recommended to all who want to learn more about it or who consider entering the field [187]. Furthermore, a number of more condensed reviews are available [188], that cannot be cited in completeness. However, a comprehensive survey of the literature is maintained and updated [189]. SECM can also be used for local wet-chemical surface modification [190].

Fig. 6.21. Schematic of a basic SECM apparatus. The current flows between the probe electrode (UME) and an auxiliary electrode (Aux). If needed, the sample can be connected as a second working electrode (WE2) to the bipotentiostat. The sample current then flows between the sample and the auxiliary electrode


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SECM can be used in three working modes: the so-called feedback mode, the generation-collection mode, and the direct mode. Each of the working modes has its particular advantages and that make each suitable for treating specific problems. Among the different modes, the feedback experiments are the best understood theoretically. The experiment is carried out in a solution that contains one form of a quasireversible redox couple R O + ne− . For simplicity it is assumed for this explanation that only the reduced form R is initially present in the working solution. This compound is called the “mediator”, because it is used to shuttle electrons between the sample and the probe. The mediator R is oxidized at the UME (probe) according to R −→ O + ne− when the working electrode has a sufficiently positive potential with respect to the reference electrode. The electrochemical cell is completed by an auxiliary electrode through which the current flows (Fig. 6.21). A steady-state diffusion-limited reaction current is produced in the solution bulk, and this can be calculated according to (6.9) (see Fig. 6.21) [191]: i T∞ = 4n FDc∗rT ,

(6.9)

where i T∞ is the probe current measured at a large distance from any surface (T for tip, ∞ for quasi-infinite distance), n is the number of electrons transferred per molecule R, F is the Faraday constant, D is the diffusion coefficient of R, c∗ is the bulk concentration of R, and rT is the radius of the active electrode area of the probe electrode. If the probe electrode approaches a smooth insulating and inert surface, such as glass, the presence of the surface will block the diffusion of R to the UME. Therefore i T (d) decreases with decreasing d (Fig. 6.22, curve 1). The current response is given as the normalized current (with respect to i T∞ ) versus the distance d in units of rT The decrease can be calculated precisely using different numeric techniques [192–194]. Individual geometric arrangements are assumed and the diffusion equation is then solved. Many calculated data points can be approximated by analytical functions, for which a number of options are available (for more details about these, see [193]). Such functions are used to determine d for particular

Fig.6.22.(a) Normalized experimental currents as a function of normalized UME–sample distance and schematic representation of the diffusion at the UME: 1) hindered diffusion 2) mediator regeneration by diffusion-controlled mediator recycling at the sample (◦); (–) theoretical values from [193] for RG = 15.2, rT = 12.85 µm, mediator is 1 mM ferrocene methanol. Printed with permission from [195]. Copyright (2001) Springer-Verlag; (b) to scale representation of the UME at a common working distance of about 1 rT . Courtesy of G. Wittstock

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experiments. It should be noted that in most sketches, the probe–sample distance is enlarged to insert reaction equations into the space. Figure 6.22b provides a realistic representation of the situation, that illustrates the diffusional shielding of the active electrode area, which is a prerequisite for this working mode. If the UME approaches a conducting surface, an electrochemical recycling process occurs at the sample: O + ne− −→ R. R will diffuse back to the UME and cause an increase in the probe current. The current–distance curve is plotted in Fig. 6.22. Curve 2 shows the situation when the reactions at both the UME and at the sample are diffusion-controlled. The reaction at the sample can proceed with finite reaction rate. In this case, for each heterogeneous rate constant at the sample an individual approach curve results. Recycling of the mediator is not only caused by electrochemical reactions at the sample but also by catalytic reactions (such as enzymatic reactions) from the local etching of the sample. Figure 6.23 suggests two types of applications. In the first, the surface is approached at fixed lateral coordinates (x, y) and a normalized rate constant is extracted with the help of analytical approximations for the curves in Fig. 6.23. As one can see, the z travel range used to determine i T∞ must be more than 10rT . Therefore, SECM is not usually performed with piezoelectric scanners known from STM or SFM, but instead with dedicated instrumentation [188]. On the other hand, one can scan the surface at fixed d. In this case, the image reflects a reaction rate image of the surface. It is this attractive feature that makes SECM so attractive for chemical applications despite the limited lateral resolution. The generation-collection mode is simpler. In this case the probe detects a compound that is not present in the bulk solution and is generated or released only at the sample surface (Fig. 6.24). This mode has found widespread use in the analysis of biological samples, passive layers of valve metals, and transport through pores in membranes and skin.

Fig. 6.23. Family of approach curves for different normalized rate constants for the reaction at the sample κ = ksample rT /DR , calculated according to [196]; ksample , pseudo first-order rate constant for the reaction at the sample surface; DR , diffusion coefficient of R


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Fig. 6.24. Schematic of the generation-collection mode

The direct mode uses the probe as a microscopic counter electrode for the sample. In this case, the reactions at the probe and at the sample are not necessarily opposite to each other; for example solvent electrolysis at the probe and material conversion at the sample. This mode has mainly been used in the field of surface modification. 6.6.2 Applications A comprehensive survey of possible applications can be found in [187, 188]. Understandably for such a chemically sensitive imaging technique, the main application areas are where interfacial reactions are crucial to understanding or optimization. Topographical imaging, though possible, should not be regarded as the main strength of the technique. Applications include investigations of biochemically active surfaces, such as biosensors and biological specimens, and studies of corrosion phenomena and electrocatalysis related to fuel cell research. We give one example here, which nicely illustrates how bot, the functional imaging capabilities of SECM and its local surface modification capabilities can be used together. Patterned self-assembled monolayers are intensively studied interfacial supramolecular systems. One of the advantages of such a monolayer is the passivation it can provide to a metal surface. SECM is one of only a few techniques that can probe the passivating properties of such layers without the need to apply extreme potentials to the sample, which may alter the monolayer (see Fig. 6.25). Patterned monolayers produced from ω-mercaptohexadecanoic acid were modified by adsorbing an amino-functionalized disulfide onto the region not modified during microcontact printing. In a second step, enzyme horse radish peroxidase (HRP) was coupled to the amino groups. The UME could be positioned within this periodic pattern and was then used for electrochemical desorption in the direct mode [198]. The regenerated bare gold surface at the region of local desorption was modified by chemisorption of glucose oxidase (GOx) which was modified by a thiol group. The reactivity of the resulting multi-enzyme structure could be investigated in the generation-collection mode with respect to different surface reactions taking place at the sample (Fig. 6.26). The coupled enzymatic reactions of both enzymes could be analyzed as well as the activity of each enzyme. Recent improvements of the method are concentrating on hyphenating SECM with other scanning probe techniques. A number of groups tries to enable imaging

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Fig. 6.25. SECM image in the feedback mode of a self-assembled monolayer of ωmercaptohexadecanoic acid on gold. High currents (bright) reflect fast reactions at the bare metal surfaces, low currents correspond to slow reactions at passivated regions. The patterned monolayer was produced by microcontact printing. Reprinted with permission from [197]. Copyright (2002) American Chemical Society

Fig. 6.26. SECM image in the GC mode and line scan of a surface modified by a periodic pattern of HRP and a single spot with GOx. (a) Schematic of the layout, (b) possible reaction at the sample. GOx produces H2 O2 which can also be converted by HRP. The UME monitors the local conversion at the HRP by detecting a ferrocinium derivative (Fc+ ), (c) Image of the activity of HRP as a function of distance to the GOx spot, UME potential E T = 0.05 V, (d) possible reactions in the presence of glucose and O2 but without Fc; (e) line scan corresponding to (d). Only the activity of GOx is seen by directly recording the production of H2 O2 at E T = 0.7 V. (f) Possible reactions in the presence of H2 O2 and Fc in the solution bulk but in the absence of glucose; (g) line scan corresponding to (f). Only the activity of HRP is seen, E T = 0.05 V. Reprinted with permission from [199]. Copyright Wiley-VCH (2003)


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of rough or tilted surface by introducing a distance control loop. The measured Faradayic current can not be used for the distance regulation loop because depending on the nature of the material on the sample it may increase or decrease upon approach to the sample (Fig. 6.23). As a signal that is independent on the electrochemical processes, shear forces between a vibrating UME and the sample are used similar to the common practices in scanning near field optical microscopy [200–202]. However, since the probe is immersed in an electrolyte solution, the task is much more challenging for SECM. Nevertheless, such measurements have become increasingly common, although they have not reached the state of maturity. Another development is the coupling of confocal scanning microscopy with SECM [203–205]. This provides an ideal way of probing concentration distribution in the probe–sample gap, and it has numerous interesting perspectives for surface modification and fundamental electrochemistry. Following a general trend in scanning probe microscopy, there have been intensive attempts to integrate SECM probes into scanning force microscopy (SFM) probes [206–208]. This provides the ability to measure surface morphology and to correlate this with local surface reactivity. Figure 6.27 shows an advanced example from Kranz et al., where a tapping mode SFM image of soft gel pads was recorded at the same time as a reactivity image showing the activity of the enzyme GOx trapped in the gel pads. The unique features of SECM have significantly changed the way fundamental and applied electrochemical research is performed. In the future this kind of functional analysis of surface layers and interfacial reactions will be integrated into scanning probes that can also image surface morphology, allowing the direct correlation of surface morphology to surface reactivity for many important classes of technological and biological samples. We should expect that this will lead to

Fig. 6.27. Microphotograph of an SECM cantilever (a) and sketch of the set-up and the chemical reactions (b), together with an tapping mode SFM image of GOx-loaded gels (c). (d) A SECM reactivity image of GOx activity in the absence (top, no reaction) and presence of glucose (catalytic conversion at the sample). This experiment allows sample morphology to be related to local reactivity. Printed with permission from [206]. Copyright (2003) Wiley-VCH

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breakthroughs in our understanding of complex interfacial phenomena relevant to corrosion and catalysis.

6.7 Tips for Magnetic Force Microscopy In this section we elaborate on the tips used in magnetic force microscopy, starting with a discussion of the ideal tip shape for MFM imaging. The first MFM tips were realized individually by hand, and some of the best tips still are, even today. However, batch-fabricated tips are cheaper and more reproducible. We will discuss various magnetic coatings that can be used on standard AFM tips, and new batch fabrication methods that produce tips that approach the ideal tip shape. 6.7.1 Ideal Tip Shape Most scanning probe microscope tips are pointed and very sharp: STM and AFM tips are atomically sharp, while most SNOM tips are tapered optic fibers. One might therefore assume that an MFM tip should also have the shape of a sharp needle. Techniques such as STM, AFM and SNOM, however, measure very short-range effects, whereas MFM measures long-range magnetostatic forces (the same idea holds for electric force microscopy (EFM) and electrostatic forces). In STM and AFM, only the final atom determines the signal, but in MFM a much larger part of the tip takes part in image formation. Therefore, the optimum tip shape is not a sharp needle but a bar or cylinder with a flat front end [209], analogous to a hard disk head for perpendicular magnetic recording. The fact that the bar shape is the optimum shape for MFM can perhaps be most easily understood by considering the magnetic charge distribution in a tip with ideal uniform magnetization (see Fig. 6.28). In a bar-type tip, all magnetic charges are located at the front surface of the bar, and as close as possible to the sample surface. Therefore, all magnetic charges contribute equally well to the signal. In a pointed

Fig. 6.28. Point-sharp versus bar-type MFM tips, graph from [209]. The width of the base of the triangular tip is adjusted in such a way that the maximum signals of both tips are equal


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MFM tip, the charge is distributed over the complete tip. The charges located further away from the sample still contribute to the signal, but the effect of them is stronger at longer wavelengths. If we set the maximum signal of a pointed tip and a bar-type tip to be equal, the bar-type tip will perform better at high spatial frequencies, and therefore show a better resolution. Of course, when the wavelengths approach the gap zero of the bar tip, the pointed tip performs better. But we can always tailor the bartype tip in such a way that this happens at wavelengths above the critical wavelength, as indicated in Fig. 6.28 on the right. One can argue that we can allow the pointed tip to have higher signal at low spatial frequencies by increasing the tip length. This does not, however, significantly increase the signal at high spatial frequencies. In all cases the signal close to the critical wavelength will be highest for the bar-type tip. Of course, an ideal rectangular bar cannot be reproduced exactly. It turns out that a mild rounding off (elipsoidal shape) is actually beneficial in most cases [210]. The optimum tip shape is one thing; making such a tip is quite another. When realizing MFM tips, we need such small feature sizes that one cannot rely on conventional lithography (as used in the semiconductor industry). Instead we have to use tricks to obtain such small dimensions, and we enter the area of nanotechnology. The first scanning probe microscope tips were hand-made in a one-by-one fashion. The best tips available today are still made by hand. Batch-fabricated Si and SiN tips, however, have strongly improved the ease of use of MFM. In the following we briefly discuss a number of MFM tips, starting with hand-made tips, and continuing with batch-fabricated tips. 6.7.2 Hand-Made Tips The first MFM cantilevers were very thin Co or Ni wires that were etched down to a sharp point. The wires were bent around a razor blade edge, and the laser spot was deflected from the resulting “knee” – a far from stable operation. The tips obtained this way were, however, not very sharp, and contained a lot of magnetic material [211]. Much better resolution was obtained using tungsten tips coated with a magnetic layer from the side [212]. However, the resulting magnetic thin film was not always smooth, due to the surface roughness of the tungsten point caused by the etching process. A much smoother tip was obtained using contamination needles, which were already being used for AFM. These contamination needles were grown in a SEM that was accidentally or deliberately contaminated with organic gases. A thin magnetic layer was deposited on the sides of these carbon tips. Some call these tips “electron beam deposited” (EBD) tips [213], but to avoid confusion with electron beam evaporation (a layer deposition technique), others call them “electron beam induced deposited” (EBID) tips [214, 215]. The tips made by Skidmore et al. are particularly impressive, with a tip radius of less than 7 nm. Strangely enough, the resolution of the images is no better than that obtained with conventional tips [216]. Instead of depositing a layer on the side of the EB(I)D tip, one can also use the contamination as an etch protection mask [217]. In this way, one can leave a tiny disk of magnetic material at the top of an AFM tip. This MFM tip does not have the ideal shape for

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high resolution, but can for instance be used in situations where a low tip switching field is necessary. The arrival of the focused ion beam (FIB) instrument opened up new possibilities for MFM tip preparation. The two techniques known to the authors start with a magnetic coating on a commercial AFM cantilever. Folks et al. [218] start with a commercial MFM tip, then FIB etch a small hole at the apex of the tip. The tip is than magnetized in-plane (parallel to the cantilever) so that magnetic charges occur at the edges of the tip. In this way one obtains an in-plane tip that is only sensitive to stray fields parallel to the sample in the direction of the tip magnetization (x- or y-direction). In general one would like to measure the z-component, from which the x- and y-components of the fields can easily be derived. Such a tip has been prepared by Phillips [219], who starts with a thin Co film on a commercial AFM cantilever, and then etches away the unnecessary material. A 8 µm-long 50 nm-wide needle can be prepared in this way. 6.7.3 Coating AFM Tips The workhorse MFM cantilever is a Si batch-fabricated cantilever with a tip optimized for AFM, which is coated with a magnetic material. There are a wide range of magnetic coatings in use, and if we tried to give an exhaustive list here, we would certainly fail. We can, however, indicate the main coatings used. The most common magnetic coating is an alloy based on Co with additives such as Cr and Pt, based on the pioneering work of Grütter et al. [221, 222]. These alloys are often derived from hard disk recording layer materials and have a high coercivity. This makes the tip resistant to reversal by the medium stray field. To achieve a high coercivity, the material is prepared such that magnetically seperated grains are achieved. The exact locations of these grains on the tip apex vary from tip to tip, which explains why some tips have higher resolution than others. Companies sell tips with different coercivities and film thicknesses (and give them meaningful names such as the high coercivity tip or the low moment tip).

Fig. 6.29. MFM tips prepared by FIB. (a): Perforated tip [218], (b) Co needle [220]


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For some applications, the coercivity of the layers on the tips is not high enough; for instance when measuring stray fields from recording heads or permanent magnets. In this case, an alternative can be found in very soft magnetic coatings that switch in very small external fields. However, one result of this is that the tip is always attracted and so we lose information on the polarity of the field. Materials used include Co [222], NiFe [222], Fe [223] and granular Fe(SiO2 ) films [224]. The perfectly soft MFM tip is a diamagnetic tip (using for instance a Pt-coating). Of course, the magnetic moment for this tip is much lower than for ferro- or superparamagnetic tips, and the signal is very low [225]. Although magnetically coated AFM tips are widely used in MFM, they are a bad approximation to the ideal tip shape. In the following we discuss a new method of realizing batch-fabricated tips, in a method optimized for MFM. 6.7.4 Tip Planes: The CantiClever Concept A magnetic tip that is suitable for high resolution MFM should have lateral dimensions on the nanometer scale. One would like these dimensions to be controllable and variable in order to be able to customize the MFM tip for different types of measurements or samples. This can be accomplished by defining both lateral dimensions by thin film deposition techniques. The magnetic tip can than be made by depositing magnetic material onto the side of a free-hanging and very thin layer called the tip plane. The width and thickness of the magnetic tip are defined by the thickness of the tip plane and the magnetic layer respectively. The length of the tip is defined using photolithography. A schematic of the structure is shown in Fig. 6.30. Such a structure is very difficult to make, however, when the conventional approach is used to fabricate the cantilevers. During fabrication, conventional cantilevers are situated such that the oscillation direction is perpendicular to the surface of the substrate. Using this approach would require the very thin tip plane to be fabricated as a freestanding layer perpendicular to the substrate surface. Instead, the free-hanging tip plane is made in a completely new approach. During fabrication, the cantilevers are rotated at right angles compared to conventional cantilevers, creating a lateral oscillating cantilever with its oscillation direction parallel to the substrate surface, as shown in Fig. 6.31. These cantilevers are nicknamed CantiClevers. This approach makes the fabrication of the cantilever more difficult compared to conventional cantilevers, but also enables precise control over the cantilever resonance frequency. The resonance frequency is given by (6.1). In contrast with Fig. 6.30. The cantilever with the tip plane

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Fig. 6.31. Tip plane on the cantilever for both cantilever orientations. Left: conventional approach with horizontally vibrating cantilever. Right: new approach with laterally vibrating CantiClever

Fig. 6.32. SEM image of the CantiClever tip plane. The image is false-colored to highlight the SiN tip plane. The inset shows a zoom shot of the end-point

Fig. 6.33. SEM image of the Co tip (false colored

the fabrication method used for conventional cantilevers, the CantiClever method allows the dimensions of the cantilever (which determine the resonance frequency, the cantilever thickness t and the length l) to be controlled and varied, as both are defined parallel to the substrate surface. A single substrate can carry a large number of cantilevers with different resonance freqencies, suitable for different applications. Furthermore, standard deposition and etching techniques can be used to define the tip plane, as it is also oriented parallel to the substrate surface. The result is a reproducible manufacturing process that incorporates both the cantilever and the magnetic tip and allows for batch fabrication of the probes [226, 227]. A SEM photograph of the tip plane is shown in Fig. 6.32, while the Co layer deposited on the side of the plane can be seen in Fig. 6.33. Acknowledgements. The authors are deeply grateful to the following colleagues: L. Abelmann and A. van den Bos (MFM), G. Wittstock (SECM), Ch. Ziegler and N. Schwender (functionalized tips), who kindly supported us with up-to-date overviews in their specific active research fields.


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7 Quantitative Nanomechanical Measurements in Biology Małgorzata Lekka · Andrzej J. Kulik

Abbreviations PBS AFM FFM TMAFM FS MT µF IF ECM YM PSD

phosphate buffered saline atomic force microscopy friction force microscopy tapping mode atomic force microscopy force spectroscopy microtubules microfilaments or actin filaments intermediate filaments extracellular matrix Young’s modulus position sensitive detector

7.1 Stiffness of Biological Samples 7.1.1 Cell Structure 7.1.1.1 Cell Cells are the basic units of all living organisms. Although we know the structure of cells, our knowledge of how cells work, their properties, and how these properties change due to the actions of various factors is still incomplete. Several structural elements are similar for all animal cells (see Fig. 7.1). All cells contain a plasma membrane separating the cell interior from the environment. The membrane regulates the movement of water, nutrients and wastes to and from the cell, and is composed of a lipid bilayer in which numerous proteins that participate in cell functions are embedded. At the center of the cell there is a cell nucleus that contains the cell’s DNA, the genetic code that governs protein synthesis. In addition to the nucleus, inside the cell there are many other organelles – small structures that help carry out routine cellular operations. These include, for example, ribosomes (which participate in protein synthesis), lysosomes (which contain enzymes that help to digest nutrient molecules and other materials), and mitochondria (which are often referred to as the power-plants of the cell, as many of the reactions that produce energy take place in mitochondria) [1].

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M. Lekka · A.J. Kulik

Fig. 7.1. Structure of a cell

7.1.1.2 Cytoskeleton The shape of a cell is maintained by a three-dimensional protein structure called a cytoskeleton. The cytoskeleton is composed of three types of proteins forming different filaments (microfilaments, intermediate filaments and microtubules; see Fig. 7.2), organized into a network and bundled within the cytoplasm of eukaryotic cells. Although the term “cytoskeleton” gives an impression of a rather static structure, all of these three constituents are dynamic, constantly changing their shape through cycles of polymerization and depolymerization and through interactions with other proteins. As well as serving as the structural cell element, the cytoskeleton can participate in cell motility and it can regulate the activity of protein transport. In reaction to an extra- or intracellular stimulus, the cytoskeleton can self-assemble and adapt to a new situation within just a few seconds. Microfilaments (µF, [2]) are linear filaments constructed from the globular protein actin (G-actin, molecular weight 43 kDa). These actin molecules polymerize

Fig. 7.2. The structural elements of the cell cytoskeleton: (a) actin filaments, (b) intermediate filaments, (c) microtubules

7 Quantitative Nanomechanical Measurements in Biology

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to form a long fiber. Two such fibers are twisted around each other in a helix to form a single actin filament with a diameter of about 7 nm, the thinnest cytoskeletal filament. Microfilaments, together with other types of filaments, are responsible for the mechanical strength of the cell and play a role in cell movement. They participate in cell division into two daughter cells, where their role is to anchor centrosomes at opposite poles of the cell during mitosis. The actin filaments also participate in the linkage of transmembrane (such as cell surface receptors) to cytoplasmic proteins. Microtubules (MT, [2]) are small tubes constructed from tubulin dimer, composed of two subunits: α- and β-tubulin, both with a molecular weight of 55 kDa. They form so-called protofilaments. Each microtubule is typically composed of 13 protofilaments arranged around the circumference with diameters of about 24 nm and lengths in the range of 200 nm to 25 µm. A microtubule extends by the adding tubulin proteins to one of its ends. Microtubules are found in the cytoplasm of all eukaryotic cells, where they are often observed to spread out radially from a centrosome located near the nucleus. From this center, they provide a strong frame that supports the cell and determines its shape. They also serve as tracks along which cellular organelles can migrate. Microtubules can also form specialized structures such as centrioles, cilia, and flagella. Both cilia and flagella are cellular appendages consisting of a core of microtubules enclosed in an extension of the plasma membrane, and they play an important role in cellular locomotion. Intermediate filaments (IFs, [2]) are the third type of fibrous cytoskeletal components. They are about 10 nm in diameter, so they are typically intermediate in size between microfilaments and microtubules. Unlike the microfilaments and microtubules, intermediate filaments are made of several different proteins. Therefore, intermediate filaments can be divided into five major types. Type I and II are composed of acidic and basic keratin, respectively. They are produced by different types of epithelial cells (bladder, skin). Type III encompasses intermediate filaments distributed in a number of cell types, including: vimentin in fibroblasts, endothelial cells and leukocytes; desmin in muscle; glial fibrillary acidic factor in astrocytes and other types of glia, and peripherin in peripheral nerve fibers. Type IV are neurofilaments and type V are made of laminin. All three cytoskeletal components have distinct subcellular localizations. Actin filaments are usually present in a layer known as the “cell cortex”, immediately beneath the plasma membrane, and in cell projections such as microvilli. In certain situations they are associated with a number of proteins and other structures, such as the contractile ring in dividing cells, or focal contacts linking the cytoskeleton with surface receptors. Microtubules extend from the nucleus region towards the cell periphery. Intermediate filaments are distributed in a similar pattern to microtubules. IFs and MTs are excluded from the actively expanding leading edge of the moving or “ruffling” cell [3]. 7.1.1.3 Mechanical Properties of Cells Over the past few decades, an increasing amount of evidence has suggested that many cellular processes depend on (or are regulated by) mechanical forces arising from either the cell alone or from its environment – an extracellular matrix (ECM)

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and/or neighboring cells [4]. It has been reported that these processes influence the ability of cells to either maintain or change their shapes under different types of external stresses. One property that decides whether the cell maintains or changes its shape is the cell deformability. Cell deformation depends on the overall viscoelastic mechanical properties of the cell, which are not static, as they change depending on the cell’s state. The mechanical stability of the cell is maintained by its internal structure – the cytoskeleton [5–7]. Therefore, in order to resolve the cell mechanics at a molecular level, studies of the mechanical properties of the cell and its structural components are needed. Alterations in cytoskeleton organization, which manifest themselves by changes in mechanical properties, can be induced by different factors. Depending upon the source of cytoskeleton rearrangement, the resulting changes can be beneficial or detrimental for the cell. They can reveal the adaptation of the cell to new circumstances but also they can indicate internal changes in cytoskeleton structure [1, 2]. Therefore, measurements of cell mechanical properties may help us to understand the causes of cytoskeleton reorganization. There are different reasons for cytoskeleton reorganization. There may be a lack of a particular protein that is connected to the cytoskeleton and crucial to cell functioning (for example dystrophin, the absence of which is responsible for Duchenne muscular dystrophy, a serious muscle degenerative disease, [8]). As the cytoskeleton structure is not stable but changes according to cellular processes like cell division, cell locomotion, or transport of intracellular particles; many of these processes manifest themselves in altered cell deformability. It is known that the cancerous transformation introduces significant changes in cell structure and behavior. Differences in cell growth, morphology, cell-to-cell interaction and also in the cell membrane and organization of the cytoskeleton have been reported [9, 10]. These differences can also cause changes in the mechanical properties of the cell, usually leading to higher cell deformability (lower Young’s modulus). The changes are generally attributed to the altered cytoskeletal organization [5, 11]. The low stiffness of cancer cells may be caused by a partial loss of actin filaments and/or microtubules, and therefore by a lower density of the cellular scaffold. Poor differentiation of the cell and the reduced adhesive interactions characterize the vast majority of cancer cells [12]. A measurement of the Young’s modulus of cells can probably help to determine the range of cytoskeleton changes and allow us to quantify them. 7.1.2 Determination of Young’s Modulus Many techniques have been applied to study the mechanical properties of biological specimens, at both the macro- and the microscale. However, considering the increased role of mechanical properties at the subcellular level, developing techniques that can deliver very local information from the cell surface is still a challenge. The most popular techniques used for the determining local mechanical properties are magnetic twisting cytometry [13], micropipette aspiration [14], microplate manipulation [5], and, recently, optical tweezers [15] and atomic force microscopy [16].


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All of the above methods share the same idea of measurement: when a certain force is applied, the cell is deformed in some way and the deformation produced is recorded using various detection methods. Magnetic twisting cytometry applies a controlled mechanical stress to specific cell surface receptors using ligand-coated ferromagnetic beads, micropipette aspiration aspirates a single cell into a glass micropipette with a given suction pressure, microplate manipulation deforms a single cell situated between two microplates, optical tweezers use a strongly focused laser beam to manipulate cells, and atomic force microscopy produces deformation by sample indentation. The latter method, AFM, seems to be convenient method for studying the local mechanical properties of living cells. AFM can measure the mechanical properties of many types of biological samples, ranging from biomolecules through to small cellular structures (such as microtubules) to very large components, like living cells [16]. The great advantage of AFM is the possibility of performing measurements in a nondestructive manner, with very good force and lateral resolution, and in a liquid environment, which is the natural milieu for most biological samples. Apart from surface topography, AFM can be used to measure the binding force between a molecule and its ligand [17], to determine the elastic properties of cells [18], to evaluate the friction coefficients of materials [19] and to perform high-resolution imaging of isolated proteins and structures [20]. Another advantage comes from its real-time or quasi-real-time measurements, which provide not only static but also dynamic information about the investigated specimens and allows the investigation of chemical modifications to surfaces and their influence on biological systems. In particular, control over biomaterial surface chemistry can result in a regulated cell response [21, 22]. Atomic force microscopy provides three-dimensional images of the topography of the investigated sample and information about its physical and mechanical properties. In AFM, a sharp small tip, which is attached to a compliant cantilever, is placed almost parallel to the surface and then moved over it, performing a raster scan. The force acting between the tip and the sample causes the cantilever to be deflected, and this deflection may be detected using, for example, the optical system shown in Fig. 7.3. The laser beam is focused at the end of cantilever and then, after reflection from the cantilever, it is detected by a position-sensitive detector (PSD). The position of the beam spot on a PSD surface provides information on cantilever displacement. Usually, commercially available instruments use a PSD photodiode that has an active area that is divided into four quadrants, which enables the recording of two orthogonal cantilever deflections: a normal one, (perpendicular to the investigated surface, corresponding to surface topography), and a lateral one (torsion, related to frictional forces). The great advantage of the method is that immersion of the sample in (transparent) liquid does not change this principle of operation [23]. During measurements of the sample stiffness, the AFM tip mounted at the end of the cantilever approaches the surface and indents it. The loading force is controlled by the amount of cantilever deflection and the spring constant. The measurement may be carried out in several different ways. The most popular method involves force spectroscopy, where so-called force curves are recorded. A force curve gives the relationship between the recorded cantilever deflection and the relative sample position. This dependence can be then converted into a force (load)– indentation curve, which is the basis for the Young’s modulus determination [24].

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Fig. 7.3. Schematic view of an atomic force microscope

The other measuring technique uses AFM in a so-called nano Swiss cheese technique [25] that permits the axial and shear moduli to be determined. Although the latter technique was developed primarily for investigations of carbon nanotubes, it can be also be successfully applied to biological samples with similar fibrous structures (of course, providing that the experiment is performed in a liquid environment). The force modulation mode can also be applied to determine the mechanical properties of living cells, although this method gives only qualitative results [19, 26]. 7.1.2.1 Force Spectroscopy 7.1.2.1.1 Force Curves The most popular way of determining the stiffness of living cells uses so-called force curves to derive the Young’s modulus of the sample [23]. In this approach, the AFM probe, mounted on a cantilever, is placed over the sample surface and then pushed against the sample; data acquisition is usually performed sequentially. The force–distance curve shows the relationship between the relative sample position (realized by the movement of the piezoelectric scanner) and the cantilever deflection (measured using a position-sensitive detector). Typical force curves taken for a compliant sample are presented in Fig. 7.4. These curves are then used to determine the local Young’s modulus values or interaction forces. However, real experiments usually consist of measuring the voltage applied to the piezoelectric scanner (which is nonlinear and hysteretic) versus the relative cantilever deflection (which is uncalibrated). Therefore, a careful calibra-


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Fig. 7.4. Typical dependence between cantilever deflection and relative sample position (so-called “raw data”) recorded for a bladder cell in culture conditions. The approach (gray line) and retraction (black line) traces do not overlap, indicating the hysteresis and the nonlinearity of the piezoelectric scanner

tion and linearization procedure should be applied in order to obtain quantitative results [27]. 7.1.2.1.2 Scanner Linearization Correcting for the nonlinearity and hysteresis of the piezoelectric scanner is crucial to gaining an accurate reading of a displacement. In commercially available AFMs, scanner linearization is frequently an integral hardware component, and thus scanner calibration is not needed. Real displacements of the sample in all three axes are measured using optical, capacitive or other methods. One simple way to verify the precision of the scanner is to measure a force–relative sample position curve on a stiff, nondeformable surface, and then to check whether the approach curve is identical (overlaps with) to the retraction curve and whether it is a straight line. If this is the case, the piezoelectric scanner is hardware-linearized. If not, the piezoelectric scanner should be linearized off-line. There are several methods of scanner linearization; out of these, the method of z-calibration proposed by Jaschke and Butt [28] is very straightforward and does not require any additional equipment. However, it can only be applied to AFMs with an optical detection system. In this method, a mirror is placed at the sample position in such a way that the inclination of the mirror is the same as the inclination of the cantilever. The slightly unfocused laser beam produces two reflected beams: one from the back of the cantilever and one from the surface of the mirror (Fig. 7.5a). When the mirror moves up and down, the optical path length between these beams changes and the resulting interference signal varies (Fig. 7.5b). The distance between subsequent maxima reflects the nonlinearity of the scanner, and the Bragg law can be used to calculate the true scanner extension. One can obtain parameters that describe the scanner nonlinearity from the above data by fitting a polynomial function (usually a parabolic function is sufficient, see Fig. 7.5c). The method is exact and does not require additional equipment, but it does require knowledge of the geometry of the system.

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Fig. 7.5. (a) Schematic of the set-up used to linearize the scanner off-line. (b) Normalized photodiode signal of the extension and retraction of the scanner as a function of the applied voltage. (c) Scanner nonlinearity, fitted with a parabolic function (Reprinted with permission from [27], Copyright (1999) from Elsevier)

Fig. 7.6. Effect of scanner nonlinearity on force curves recorded on (a) a stiff (glass coverslip) and (b) compliant (cell) surfaces

The effect of scanner hysteresis and nonlinearity on force curves for stiff and compliant samples is shown in Fig. 7.6a,b. When a nonlinearized scanner is used on a stiff surface, the approaching trace does not overlap the retracting one. Both curves should overlap after being corrected for nonlinearity (Fig. 7.6a). Another effect is illustrated in Fig. 7.6b, where the scanner nonlinearity leads to an interpretation that


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is completely false for compliant samples. Both curves overlap in the upper plot of Fig. 7.6b, suggesting a purely elastic interaction. However, after correction, the true data exhibit an approach–retract hysteresis due to the contact having viscoelastic character. 7.1.2.1.3 PSD Signal Calibration AFM measures the cantilever deflection by monitoring the position of the laser beam reflected from the cantilever. The active area of the measuring PSD photodiode is usually divided into four quadrants. The cantilever deflection that is perpendicular to the investigated surface is found by measuring the difference between the signals from the two upper and the two lower quadrants. In order to calibrate this cantilever deflection, the force curve should be recorded on a stiff, nondeformable surface (such as glass or mica), where the deflection directly reflects the position of the sample. This dependence is linear and is usually also employed as a reference. The slope determines the calibration coefficient between the PSD signal (in volts) and the real displacement (in nanometers). 7.1.2.1.4 Spring Constant Determination Force calibration requires knowledge of the cantilever spring constant. The resonant frequency of a thermally excited cantilever is measured in the most widely applied method for spring constant calibration. The detailed procedure for a spring constant determination carried out in this way has been described in a publication from Sader [29]. Figure 7.7 presents noise spectra measured for two types of cantilevers (k = 0.03 N/m and 0.1 N/m). The spectra were obtained by applying a fast Fourier transform to an average of six noise spectra, each consisting of 4096 points [27]. The measured resonant frequencies are lower than the nominal values given by manufacturer: 11.8 kHz instead of 15 kHz and 27.7 kHz instead of 35 kHz. These

Fig. 7.7. Noise spectra for two types of cantilevers (k = 0.03 N/m and k = 0.1 N/m). The calculated resonant frequencies are lower than the nominal values: 11.8 kHz instead of 15 kHz and 27.7 kHz instead of 35 kHz (Reprinted with permission from [27], Copyright (1999) from Elsevier)


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changes in resonant frequency are used to calculate the normal spring constant. It is worth noting that these properties differ even between tips carefully chosen from the same series of production. 7.1.2.1.5 Force–Indentation Curves Force–indentation curves are obtained by measuring the deflections on stiff and compliant surfaces at a given relative sample position, and then noting the difference between these deflections (Fig 7.8). When a stiff material (not easily deformable like silicon or glass) is investigated, the deflection reflects the position of the sample. This is represented by a straight line and is usually employed as a reference line for the force calibration. For compliant samples, like cells, cantilever deflections are much smaller and the resulting force curve has nonlinear character. The difference between these curves determines the deformation of the sample surface [24].

Fig. 7.8. Indentation determination from AFM force curve data

7.1.2.1.6 Young’s Modulus Determination The force versus indentation curve describes the mechanical response to an applied load, which is characteristic of a particular material. The Young’s modulus, which characterizes the cell stiffness, can be evaluated using Hertz contact mechanics, which describes the deformation of two elastic spheres under an external load [30]. The Hertz model can also be extended [31] to account for an infinitely stiff indenter with a variety of AFM tip geometries (spherical, parabolic, conical or flat-ended) and a flat, deformable substrate. Usually, the AFM probe tip is a four-sided pyramid that can be modeled using either a cone or a paraboloid. For a conical tip, the force–indentation curve is given by: F(δ) =

2 · E · (δ)2 , π · tan(α)

(7.1)


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and for a tip that is a paraboloid: F(δ) =

4 √ 3 · R · E · (δ) 2 , 3

(7.2)

where F is the loading force, α is the open angle, R is the radius of curvature of the AFM tip, E is the reduced Young’s modulus, while δ is the indentation depth. The reduced Young’s modulus is given by: $ % 1 − µ2sample 1 − µ2tip 1 = + (7.3) E E tip E sample When E sample E tip (which is the case for living cells), then 1 − µ2sample 1 = E E sample

(7.4)

where µ is the Poisson ratio, which is related to the compressibility of the sample material. The Poisson ratio for cells is difficult to determine. Therefore this value is usually set to be 0.5, since cells can be assumed to be incompressible material. The final Young’s modulus distribution is formed from values from single cell force curves (see Fig. 7.11). 7.1.2.2 The Nano Swiss-Cheese Method Atomic force microscopy can be used to study the mechanical properties of not only cells but also their structural elements, like cytoskeletal components such as microtubules. The first application of AFM to investigating the mechanical properties of cell structural elements, by determining the Young’s modulus, was performed by Vinckier et al. [32]. They studied glutaraldehyde-stabilized microtubules. The force curves were recorded for a mid section of a single microtubule lying on the stiff substrate. However, the main problem with this method was that it produced only a very small deformation of the microtubule, resulting in very small vertical displacements of the tip, which was difficult to measure reliably. Furthermore, an analytical description of the contact between a spherical tip and an anisotropic tube is difficult to establish. A more suitable method of studying the mechanical properties of fibrous structures was proposed by Salvetat et al. [25]. The method was initially developed for determining the Young’s modulus of carbon nanotubes, but it was later shown to be useful for investigating the mechanical properties of microtubules. In this measurement method, single fibrils are deposited on the surface of a polished ultrafiltration membrane. The fibrils adhere to the surface, and some of them are deposited over holes in the surface spanning their edges (see Fig. 7.9, [33]). When a tube deposited over a hole in this way is found, a series of contact–mode AFM images are taken under increasing loads, and each image corresponds to the surface under a given load.


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Fig. 7.9. Schematic drawing of a long tube with a relatively small diameter deposited on a porous substrate (a). During AFM imaging in contact mode, the tube deforms under the applied force (b), and this deformation is recorded as change in the local height in the corresponding AFM image (Courtesy of A.Kis, from [34])

For a range of applied normal loads, the midpoint deflection of a thin long tube under a nominal point load F can be found using the clamped beam formula: δ=

FL 3 192 · E · I

(7.5)

where L is the suspended length, I is the second moment of the cross-section, which is d 4 /64 for a solid cylinder of diameter D. This variable load imaging technique is used to obtain quantitative information, as we can be sure that the AFM tip is in the desired location when the tube is deformed. The above equation is valid for small deformations if the tubes adhere well to the substrate, which is confirmed by images that reveal no displacement of the sections of the tube in contact with the membrane. Successive tube deflection measurements are usually performed for increasing and decreasing loads. Agreement between those measurements confirms the linearity and further establishes that the tube is firmly bound to the substrate. 7.1.2.3 Other AFM-based Methods Biological samples are very delicate and are sometimes destroyed during scanning in contact mode. This lead to the introduction of dynamic force microscopy [35], in which the amount of force applied during measurements is minimized since it does not require continuous tip–sample contact. Among various types of dynamic force microscopy, tapping mode AFM (TMAFM) has become the most widely used [36]. In TMAFM, the cantilever is oscillated near its resonant frequency. During scanning, the oscillation amplitude is held constant using a feedback loop by adjusting the vertical position of the sample. Thus, the feedback signal does not reflect - as frequently stated - the surface topography [37], rather the resulting image shows the surface representing the constant amplitude of an oscillating cantilever.


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Using this technique, the amplitude of oscillation and the phase can be recorded at the same time. The interaction between the tip and the surface depends not only on the topography of the sample but also on other material properties, such as stiffness, elasticity or adhesion. The phase shift recorded for a compliant sample can be used to detect, for example, differences in sample stiffness. However, in general it is difficult to measure the sample’s physical properties using tapping mode images. Five physical properties of the sample are used in the model describing the vibrating cantilever tapping sample surface [38], and the same oscillating cantilever behavior versus average tip–sample distance can be obtained for many different combinations of all five physical properties. This makes it difficult to use dynamic force microscopy quantitatively, although the phase lag between cantilever detection and excitation can be used to measure energy losses during tapping [39]. The main advantage of dynamic force microscopy is that only small lateral forces are produced during scanning, which prevents sample displacement, modification or damage. 7.1.3 Brief Overview of the Application of AFM to Studies of Living Cells Great efforts have been made to measure the stiffnesses of delicate biological samples, since this information can improve our understanding of cell mechanics. The examples described below represent only a small proportion of possible applications of the AFM technique to studying changes in the mechanical properties of living cells. 7.1.3.1 Living Cells Cells are very compliant and complex objects, and so cell indentation generates responses from a variety of cell structural components, including glycocalix, the cell membrane and the interior of cell (the cytoskeleton and organelles immersed in cytosol). The responses from the glycocalix and the cell membrane are the most significant for small indentations (100 nm and less). However, these components are usually not visible in AFM measurements due to their very low compliance and the low accuracy of the contact point determination. The large deformation range is dominated by the responses from the internal cell structures such as cytoskeleton and cellular organelles. Comparing the dimensions of all interacting objects with the indentation depth suggests that the response of the cellular scaffold is the most important, and therefore the elastic properties measured reflect mainly the properties of the cytoskeleton [2, 11]. In order to prove this hypothesis, a lot of effort has been made to investigate the role of cytoskeletal elements in the mechanical properties of the cell. The contribution of the cell cytoskeleton to the mechanical properties determined by AFM has been discussed in several studies, where the importance of both the structural proteins of the cytoskeleton and other proteins associated with the cytoskeleton was demonstrated [40, 41]. Since the cell cytoskeleton is composed of three main elements (actin filaments, intermediate filaments and microtubules), their individual roles have been investi-

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gated in order to find the main contributor to cell mechanics. The common method used for this is to apply so-called “cytoskeletal drugs” that promote or inhibit the formation of a given type of filament. Such measurements have been performed for various cell types [42–44], and complete and comparative studies on cytoskeleton integrity have been presented by Wu et al. [45] and Rotsch et al. [46], where the elastic properties of cells were measured after treatment with cytochalasin B, a protein that causes depolymerization of actin filaments (see Fig. 7.10), or nocodazole, which disassembles cell microtubules. The results have shown that the actin filaments are mostly responsible for the mechanical properties observed in AFM measurements, disrupting the microtubules produced no effect on the mechanical properties of cells as measured by AFM. The role of cytoskeleton-associated proteins in measured cell elasticity has been demonstrated for cells possessing vinculin, a protein that is required for actin stress fiber formation. The loss of vinculin produces a noticeable reduction in cell adhesion and spreading and the presence of stress fibers. The comparison performed by Goldman et al. [39] in 1998 showed that vinculin-deficient F9 mouse embryonic carcinoma cells had a lower Young’s modulus than wild-type cells. The authors attributed these changes to altered actin cytoskeletal organization [47], indicating that vinculin is an integral part of the cytoskeletal network.

Fig. 7.10. Time series of elasticity maps of a 3T3 fibroblast showing disaggregation of the actin network by 10 µM cytochalasin B. (a) Reference map with marked stress fibers (arrow). (b-i) Successive degradation of the elasticity maps, indicating a decrease in the cell’s Young’s modulus. The stress fibers clearly visible in images (a-f) disappear completely after 120 min (g-i) (Reprinted with permission from [46], Biophysical Society 2000)


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The cell elasticity determined via AFM is a very local feature; large discrepancies are seen between the Young’s modulus measured for a population of cells and that for a single cell. It has been reported that cells in vitro have Young’s modulus values in the range of 1–100 kPa [23, 48], which encompasses various types of investigated cells, including vascular smooth muscle cells, fibroblasts, bladder cells, red blood cells, platelets and other epithelial cells. Since different cell types are being measured, the large variation in Young’s modulus is easy to understood. Despite large errors (defined as the half-width of the YM distribution peak at half maximum, the HWHM), a comparison of the cell elasticity clearly shows a much smaller Young’s modulus (a larger deformability) for cancerous bladder cells than for reference cells [11]. The difference between the YM of these cells was almost one order of magnitude (Fig. 7.11). The significant differences in the YM values found enabled alterations within cell populations arising from the actions of drugs to be traced. This was demonstrated in studies that showed a correlation between cell stiffness and glycolytic activity as a function of various degrees of chitosan deacetylation (defined as the ratio of the number of amino groups to the sum of amino and acetyl groups). An increase in Young’s modulus was correlated with decreasing glycolytic activity (determined as the levels of ATP and lactate) in cancerous cells (Fig. 7.12). The YM values of the reference cells were unaffected, which highlights the cancer-specific action of chitosan [49]. It is known that the mechanical properties of cells depend strongly on the surface properties of the substrates used in cell cultures. There is also an increasing of evidence showing that substrates have a regulatory effect on cell growth, differentiation or proliferation. The surface properties of substrates are important, for example,

Fig. 7.11. Histograms showing the mean values and the distributions in Young’s moduli measured for two reference cell lines: normal (Hu609 and HCV29) and cancerous (T24 and BC). In this evaluation it was assumed that the Poisson ratio is µcell = 0 (Reprinted with permission from [11], Springer-Verlag, 1999)

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Fig. 7.12. Correlation between glycolytic activity and Young’s modulus for bladder cells treated with various types of chitosan preparations (three degrees of deacetylation: 56.6%, 88.7%, 97.7%). CC indicates control cells. Chitosan with a degree of deacetylation of 97.7% showed both the largest increase in Young’s modulus and the highest glycolytic inhibition

in implants, which need to be compatible with the living organism. The work performed by Domke et al. [22] investigated a whole range of parameters accessible by AFM, such as the cell morphology, the adhesion, and the cell elasticity of living osteoblasts. The properties of cells cultured on metallic surfaces were compared with those of cells growing on a standard glass surface and on a polystyrene surface in the tissue culture dish (Fig. 7.13). Figure 7.14 presents the effect of the substrate on cell elasticity. The Young’s modulus changes significantly depending on substrate type. The excellent lateral sensitivity and precision of AFM measurements allows the cell Young’s modulus to be mapped, highlighting the heterogeneity of single cells. It has been reported that the Young’s modulus of plasma membrane is 1–10 kPa, while the corresponding value for a membrane that is more strongly supported by the cytoskeleton is higher (15–50 kPa, [42]). This makes it possible to perform elasticity measurements within a single cell. The work described in [50] traces the alterations in elasticity during cell movement; the distribution of local fibroblast stiffness was found to be related to cell migration. When the cells were stationary, the

Fig. 7.13. The mean value of the Young’s modulus for osteoblasts cultured on different types of substrates (CoCr, Ti, TiV, glass and PS tissue culture polystyrene surfaces) (Reprinted with permission from [22], Copyright (2000) from Elsevier)


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distribution of stiffness on the cellular surface was more or less stable. Significant changes were observed during migration, where the stiffness in the fibroblasts’ nuclear regions decreased. This observation was explained in terms of contractile force variations. The contractile force is generated by stress fibers made of actin filaments. When the cell starts to move, this force drags the cell body forward, causing nuclear displacement. Another example of the application of AFM to investigations of cellular processes is given by by Mantzke et al. [51], where cell division was studied via direct measurements of furrow stiffening. 7.1.3.2 Bacterial Cells Another class of biological samples used in AFM studies of mechanical properties is bacterial cells [52]. However, while animal cells have been investigated extensively, bacterial cells have been investigated in only a limited number of AFM studies. One of the first studies investigated magnetotactic bacteria by means of force modulation microscopy [53]. In this study, the strong contrast observed in the force modulation images was attributed to the presence of stiff magnetosomes inside the bacterial cells. Later on, the elastic properties of the proteinaceous sheath of the archeon Methanospirillum hungatei were probed via indentation experiments [54], giving Young’s moduli of 20–40 GPa, indicating that the strength of the sheath could withstand an internal pressure of 400 atm. Another type of cell, gram-negative magnetic bacteria of the species Magnetospirillum gryphiswaldense, was investigated using AFM in buffer solution [55]. Multiple force–distance curves were obtained for the substratum and for the bacteria, which made it possible to determine the effective compressibility of the cell wall, which was about 42 mN/m. AFM studies of bacteria have not only been used to determine the Young’s modulus value of the bacterial wall. Changes during cellular processes, such as the germination of Phanerochaete chrysosporium spores [56], have also been studied. For dormant spores, no significant deviation from linearity was seen in the contact region, whereas a nonlinear curvature was observed for germinating spores, which was attributed primarily to sample compliance. This difference was consistent with the presence of a proteinaceous rodlet layer at the surface of the dormant spore and the absence of such layer on a germinating spore. Another example has highlighted the significant differences in mechanical properties between fibrillated S. salivarius HB and nonfibrillated S. salivarius HBC12 strains [57]. The cell surface appeared much more rigid due to the collapse of the fibrillar mass at various ionic strengths in 0.1 M KCl solution. The force–distance curves revealed that the nonfibrillated strain was much stiffer than the fibrillated one. Since elasticity maps can give information about the homogeneity of the investigated samples, a comparison of the maps for two types of bacteria (Lactobacillus family, [58]) has shown that bacteria L. crispatus expressing an S-layer have surfaces that are more homogenous than the surfaces of strains without such a layer. For another bacteria type, L. johnsonii, stiff and compliant regions have been found that have been explained by the composition of the surface.

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7.1.3.3 Cellular Structures Not only can the mechanical properties of whole cells be determined through AFM, but the high lateral resolution and precise control of the loading force afforded by AFM permit studies of the mechanics of cell organelles (like microtubules or stress fibers). Microtubules are the most commonly studied structural components. Their elastic properties were estimated by Vinckier et al. [32] in 1996 in buffer solution as a function of the concentration of glutaraldehyde (a cross-linking agent). The microtubules were more compliant when less cross-linking agent was present. Upon extrapolating to 0% glutaraldehyde concentration, the Young’s modulus of MTs in buffer was found to be 3.1 ± 0.9 MPa. Using the “nano Swiss-cheese” method, the bending modulus of the MTs was determined as a function of temperature (Fig. 7.14). The curve obtained showed a significant decrease in bending modulus upon a decrease in temperature from 37 ◦ C to 5 ◦ C. Results of bending experiments, using different suspended lengths of the microtubules, demonstrate that microtubules are highly anisotropic from a mechanical point of view. The tubulin subunits that form protofilaments are strongly bound along each single protofilament and weakly bound between neighboring protofilaments [33, 34]. 7.1.4 Summary AFM has established itself as a useful technique for determining the Young’s moduli of a broad range of biological samples, in particular living cells. Determining the

Fig. 7.14. Temperature dependence of the bending modulus for microtubules, showing the transition at 20–30 ◦ C (Courtesy of A.Kis, from [34])


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absolute value of Young’s modulus value using AFM. Even if the calibration procedure is performed very carefully and with high accuracy, the calculated Young’s modulus reflects only the elastic properties of the particular layer defined by the indentation depth (or layers that are deeper than the indentation depth). If we take into account the large heterogeneity of biological material too, the Young’s modulus determined in this way cannot fully describe the overall mechanical properties of the cell(s) investigated. Since one end of the cantilever is fixed, the tip does not move along a vertical line during the vertical movement of the sample but instead follows a circular trajectory. As a result, the tip does not remain at the same point on the sample surface, but rather slides laterally along the axis of the cantilever. This lateral movement can be estimated as being 10% of the vertical displacement. Some microscopes partially compensate for this tip displacement in the sample plane by coupling the movements along the z and y axes. Also noticing that that tip is usually not fully perpendicular to the sample plane, it is important to be very cautious during quantitative analyses of force-distance curves. The most popular model, the Hertz model, is used to determine the mechanical properties of living cells under the assumption that the cells are purely elastic and isotropic. Actually, the measured materials are either purely elastic or they have a strong plastic component and so they usually behave like a combination of both. This is reflected in the shape of the force curve. The pure elastic character manifests itself in the exact overlapping of the approach and retract traces (assuming that the calibration procedure has removed any piezoelectric scanner nonlinearity). When the approach trace does not overlap the retract one, the plastic component starts to appear in the response of the investigated sample to the indenting force. Another important source of errors originates from the physical models applied. Some approximation of the AFM tip shape must be assumed. Usually, the four-sided pyramid of the AFM tip is approximated by a cone or paraboloid. The cone fits the experimental data well for large deformations, but if the measured indentations are small, the paraboloidal approximation is used instead. For small indentations, the alternative “nano Swiss-cheese” method can be also applied. This method can be extended to measurements of other cytoskeletal filaments, like actin and intermediate filaments. Together with the tip shape assumption, there are two other assumptions in the physical model. The first one states that the deformation of the AFM tip is negligible. The Hertz model delivers the reduced Young’s modulus value, which is, in fact, a superposition of the Young’s moduli of the AFM tip and that of the investigated sample. However, the living cells are much more compliant than the AFM tip, with stiffness values of 1–100 kPa. Thus the stiff AFM tip (modules: 150 GPa) deformation can be neglected and the Young’s modulus determined reflects the Young’s modulus of the investigated cell. The second assumption is related to the Poisson ratio of the cell. The Poisson ratio, a characteristic material property that depends on the compressibility, ranges from 0 to 0.5, and is difficult to estimate for cells. Therefore, this factor is assumed to be constant for the same types of biological materials. It is usually set to 0.5, the value for materials that are incompressible.

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AFM may also encounter problems from stiff substrate below the investigated cell. The influence of the stiff substrate on the indentation data leads to a dramatic increase in the estimated Young’s modulus value. In the case of thin biological samples where the influence of underlying stiff substrate is not negligible, it is essential to take this effect into consideration. A strong correlation between the sample thickness and the Young’s modulus value was demonstrated by Domke et al. [59] for gelatin films of different thickness. The limitations of AFM when determining the Young’s moduli of compliant, biological samples, especially at the microscopic scale, are being investigated and considered. These limitations should be addressed through both theoretical developments of the applied model and some practical solutions. For example, sharp cantilevers cause high local strains that are far from the linear material regime. This problem can be avoided by using microspheres as probes. Recently, corrections to the Hertz model that take into account the sample thickness were introduced by Dimitriadis et al. [60]. They present a simple theory that can be applied to determinations of the elastic moduli of compliant samples independent of the sample thickness. These possible sources of error in quantitative estimations of Young’s modulus using force curve spectroscopy may raise doubts about the usefulness of this method for measurements and analysis of the mechanical properties of living cells. However, we should note that exact knowledge of the absolute Young’s modulus is not always needed. Very often relative changes or trends may provide valuable information about the state of a biological material. Also, living cells are highly heterogeneous and non-isotropic due to their internal structures, and this is reflected in different Young’s moduli values when it is measured at different locations on a cell’s surface. However, this diversity should not be considered to be a drawback; instead it should be seen as a spur to improve AFM in order to make it more suitable for measurements and analyses of the mechanical properties of living cells.

7.2 Friction Force Microscopy Friction can be described as a resistance to motion that occurs when two solid objects that are in contact are moved with respect to each other. At the macroscopic scale, the main mechanisms of friction are adhesion and deformation. Adhesion is responsible for forming junctions between contacting bodies, the shearing of which requires work. Deformation dissipates energy through either plastic deformation or elastic hysteresis [61]. However, the above explanations do not describe friction on the microscopic scale, leaving the underlying physical mechanism poorly understood. Contacts between sliding surfaces are of special interest to engineers designing mechanical devices. The lubrication of sliding contacts has always been a very important factor in machine efficiency. The studies of lubrication performed so far have targeted the minimization of friction forces, which has involved (among other things) investigating the origins of friction and searching for increasingly sophisticated lubricants based on organic molecules [62].


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One way to study friction is through atomic force microscopy (AFM), which offers the unique ability to measure the normal (surface morphology) and lateral (friction) forces that act between the probe tip and the sample at the same time [63]. The development of the scanning force microscope has resulted in an increasing number of studies directly related to friction, due the ability of the technique to measure sliding contacts at the nanometer scale. This very high spatial resolution, when combined with quantitative measurements of interfacial properties, should lead to a better understanding of microscopic friction and other related phenomena at the molecular level. AFM-based techniques can provide an insight into chemical inhomogeneities at small length scales. Variations in surface frictional properties may correspond to variations in surface chemical composition and/or molecular organization. The simultaneous information about surface morphology and friction afforded by AFM may clarify effects that are difficult to investigate using other techniques. The well defined “single asperity” contact between an AFM tip and a sample surface provides the possibility of an analytical description of these forces [64]. 7.2.1 Friction and Chemical Force Microscopy 7.2.1.1 Friction Force Microscopy A friction force microscope (FFM) is an extension of a standard atomic force microscope working in the constant force mode (a constant force is maintained between the probe tip and the investigated surface) [62]. As in the standard AFM technique, the probe tip is mounted at the end of an elastic cantilever which is moved over a surface. Interaction forces cause a deflection of the cantilever which is recorded by an optical deflection system comprising a laser and a position-sensitive detector. The deflection and torsion of the cantilever are measured (the deflection in the normal direction corresponds to the surface morphology, while the torsion of the cantilever gives indirect information about the friction forces). The normal and lateral signals are separated as follows. The active area of the position-sensitive detector is divided into four quadrants. The difference in signal between the two upper quadrants and the two lower ones is a measure of the normal deflection, while the difference in signal between the two left quadrants and the two right ones gives the torsion of the cantilever, related to the friction forces. However, it is important to note that the friction force is not measured directly. It can be only estimated or calculated from the cantilever torsion due to the friction force, and/or from the slope when the cantilever reaches the steep part of the slope (see Fig 7.15). To separate these two effects, the surface topography should be measured at the same time. Since the friction force is the lateral component of the total force acting on the moving tip, its value can be determined by subtracting the mean values (the average force values from a given region of the investigated surface) obtained from scans performed in two opposite directions (forward and backward scans) along the same path. The dependency observed upon performing such scans is called a friction loop, and the difference between two scan lines divided by two gives the value of the friction force (Fig 7.15, [65]).

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7.2.1.2 Calibrating Friction Force Quantitatively determining the friction force by AFM requires a lateral calibration that is similar to the calibration of the normal force. Two quantities are needed to calibrate the results: (1) the lateral photodiode signal (in order to convert the measured volts into the lateral displacement of the cantilever) and (2) the lateral (torsion) spring constant of the cantilever. There are many ways to calibrate the photodiode’s lateral sensitivity. A number of methods have been described for calibrating the photodiode’s response to cantilever torsion. Perhaps the most direct one is the method of Meurk et al. [66], where the stepper motor is used to tilt a mirrored substrate, and the output voltage is measured as a function of angle. Alternatively, Liu et al. [67] proposed a calibration procedure based on the relationship between the torsional signal (arising due to friction forces) and the normal bending signal (originating from forces acting perpendicular to the investigated surface). The lateral sensitivity of the photodiode signal is proportional to the normal sensitivity (which is easy to measure from the slope of the force curve). This ratio is influenced by the shape and size of the laser spot and may be used for calibration only when the total photodiode signal is constant. Some workers [68] have proposed deriving the calibration factor from the initial slope of a friction loop. Assuming that the tip is pinned to the substrate, and that the lateral movement has been accurately calibrated, then the angle of twist can be obtained. In contrast to the vertical spring constant measurement, the lateral spring constant is difficult to measure. Therefore, the lateral spring constant is usually calculated and not measured. For the most popular cantilevers, the lateral spring constant is calculated using finite element methods or by means of analytical formulae [69]. However, such calculations are only approximations, since they use a relatively

Fig. 7.15. The lateral force signal (LFM) in atomic force microscopy and the concept of the friction loop


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simple cantilever model; in other words they assume an ideal geometry, a cantilever Young’s modulus value for the bulk material neglecting coatings, a fixed thickness of cantilever, and so on. All of these factors may lead to a poor estimate of the lateral cantilever spring constant. In addition to the two-step calibration procedures mentioned above, there are a few alternative methods of either calibration or direct measurement of the friction force. Ruan and Bhushan [70] suggested that the friction could be measured by moving the cantilever in the direction parallel to the long axis of the cantilever (the usual approach is to perform an AFM scan in a direction perpendicular to the long axis). In their method, the vertical spring constant multiplied by the vertical piezo movement necessary to hold the cantilever deflection constant provides an estimate for the friction force. The friction measured in this way was later used to calibrate conventional (carried out in the perpendicular mode) lateral friction measurements. Another method was proposed by Ogletree et al. [71], where friction loops were measured as a function of applied load on substrates with two well-defined slopes (Fig. 7.16).

Fig. 7.16. The idea behind the wedge calibration method. Schematic of friction loops measured on flat, positively and negatively sloped surfaces for the same loading force. The friction loop half-width W is slightly different for the three cases, while the loop offset ∆ is substantially different and is indicative of the overall tilt of each surface. The values of W and ∆ are measured over a range of applied loads for known slopes and used to calibrate the lateral force response of the cantilever (Reprinted with permission from [71], Copyright (1996) American Institute of Physics)

7.2.1.3 Chemical Force Microscopy Another variation of AFM, called chemical force microscopy (CFM, [72]), involves a probe tip that has been chemically modified, and it is used to investigate surfaces that

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Fig. 7.17. The concept behind chemical force microscopy. The modified AFM probe is moved over a sample surface that is characterized by different chemical properties (such as different functional chemical groups). The different interaction forces between the probe and the surface produce different lateral signals

have particular chemical functional groups. In CFM the method of signal detection remains the same as in classical AFM, but the origin of the signal is different (Fig. 7.17). The chemical modification made to the probe tip makes it possible to image surfaces that are flat at the nanoscale but chemically different. This in turn makes it possible to map the chemical nature of surface in a variety of conditions, including different solvents, pH values, ionic strengths, temperatures, and different molecular structures on the surface. The experiments performed so far have largely focused on identifying specific functional groups on a surface constituting of organic layers, usually self-assembled monolayers (SAMs), as they form highly chemically defined surfaces [73]. To probe interactions between functional groups, the tip must be modified using well defined molecular layers. This modification can be accomplished by using monolayers of amphiphilic molecules adsorbed onto the surface of the AFM tip. Different types of interactions can be studied by varying the head groups of these molecules. This approach is somewhat limited because physisorbed layers are not very robust. Alternatively, stable molecular layers can be produced by covalently linking the molecular components to the tip and sample surfaces. Stable and robust monolayers of alkyl thiols or disulfides containing a variety of terminal groups can be readily prepared this way, enabling systematic studies of the interactions between basic chemical groups on the probe tip and similarly modified Au substrates [74].


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7.2.2 Applications of FFM/CFM 7.2.2.1 Solid Surfaces AFM/FFM has usually been used to study friction-related problems related to solid surfaces. AFM/FFM techniques are increasingly being used for tribological studies of engineering surfaces at scales ranging from atomic and molecular to the micron scale. The parameters studied include surface roughness, adhesion, friction, scratching, wear, indentation, detection of material transfer, and boundary lubrication. The most commonly investigated phenomena are the nature of single asperity contact and frictional slip-stick phenomena on the atomic scale. Other studies of friction under ultrahigh vacuum conditions (that provide well-defined surface chemical conditions) are performed in order to test point contact theories relating to load, friction, adhesion and contact area. At most solid–solid interfaces of technological relevance, contact occurs at numerous asperities of different shapes and sizes. The effect of the single asperity radius on the friction can be studied using AFM tips of different radii. A very sharp AFM/FFM tip provides a single asperity contact when sliding over a surface, which enables studies of friction force to be made at an atomic scale. Using the AFM/FFM technique, it has been proven that the friction laws for a single asperity are quite different from those known observed for macroscopic scale friction. The main result is that friction on the nanometer scale exhibits a sawtooth behavior: an atomic stickslip [75]. An example of this kind of experiment is presented in Fig. 7.18. The surface of NaCl (100) was scanned with a silicon cantilever at an applied force of 0.65 nN and scan velocity of 25 nm/s. The friction loop was obtained by subtracting the two

Fig. 7.18. (a) Lateral force image obtained for NaCl(100) for a loading force of 0.65 nN and a sliding velocity of 25 nm/s. (b) Friction loop obtained along the same scan lines recorded in forwards and backwards directions, showing the characteristic shape indicating stick-slip phenomena (Reprinted with permission from [75], Copyright (2000) by the American Physical Society)

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Fig. 7.19. Dependence of friction force on sliding velocity, which can be attributed to lateral loading rate. A gradual increase in friction force is visible as the velocity increases (Reprinted with permission from [75], Copyright (2000) by the American Physical Society)

lateral scan lines recorded in the forward and backward directions. The sawtooth character of the lateral signal highlights the slip-stick phenomenon (Fig. 7.18b). In this experiment, it was reported that the friction depends on the sliding velocity. Further studies [76] have shown that the friction that occurs during periodic hopping may be strongly influenced by thermal activation. When the AFM tip moves on a surface, the atoms in contact must overcome an energy barrier. Thermal activation leads to a logarithmic increase in the friction force as the sliding velocity increases (Fig. 7.19). Later on, it was shown that this increase in friction force stops at a certain critical velocity beyond which the friction remains constant. However, this effect was observed in measurements performed in vacuum or in air. In a humid environment, it has been reported that the friction force shows an exponential dependence on the sliding velocity when the surface is rough and hydrophilic (it is still logarithmic on a hydrophobic surface). The explanation for this involves spontaneous water adsorption [76, 77]. 7.2.2.2 Organic Surfaces Many papers [78–80] show good quality surface images that were obtained using chemical force microscopy, by recording lateral or friction forces. All of these studies used FFM/CFM to study interaction forces between organic surfaces and an AFM tip that had been previously modified with particular functional groups. The functionalization of the AFM tip provides a way to probe the chemical diversity of a surface with a spatial resolution of nanometers. The use of self-assembled monolayers for this purpose enables frictional studies in conditions of well-controlled surface chemistry, which enables the characterization of complex heterogeneous materials. One of the deposition methods involves adsorption of a monolayer of a specific organic molecule on a particular surface. Nakagawa et al. [81] demonstrated that the alkylsilane monolayers can be deposited onto the surface of a silicon nitride cantilever. Cantilevers prepared in this way were used to determine the interaction forces with methyl-terminated and perfluorinated alkylsilanes deposited on the silicon surface. In the other method of adsorbing organic molecules, where organic molecules are extensively used, gold is first deposited on the AFM probe, and then it is immersed in a solution of organic molecules [82]. The efficiency of this method


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of functionalization was demonstrated for modified probes with either carboxylic acid (COOH) or methyl groups (CH3 ) on their surfaces. These probes were used to image the surfaces with patterns composed of carboxylic- and methyl-terminated regions. The same pattern was observed in friction images for either tip functionality (COOH or CH3 ), but the contrast was different for different functionalities, which corresponded to a particular chemical interaction. For example, when the surface was scanned with the carboxylic-terminated tip, the acid-terminated regions exhibited a brighter contrast (related to the higher friction force) than the methyl-terminated regions [82]. The sensitivity of friction force microscopy to changes in surface chemical properties was demonstrated by Wong et al. [83], who compared the frictional behavior of SAMs of alkyl chains of different (odd and even) lengths, which differ in the angle of orientation of the dipole moment of the terminal C–C bond. The results presented show that monolayers formed from odd-length alkyl chains have significantly smaller friction coefficients than those with even-numbered chains of the same length. The difference was large enough to enable the imaging of patterned monolayers formed by microcontact printing thiols with only one carbon atom difference in length. The friction force is sensitive enough to image the molecular organization of organic molecules, which was demonstrated for thiolipid monolayers on mica [84]. These thiolipids form domains with different orientations of lipid molecules. The friction force measured on these domains showed unexpectedly strong friction depending on the angle of the tilt direction of the alkyl chains of the monolayer. The molecular tilt causing these frictional effects was less than 15◦ , demonstrating that even small molecular tilts can make a major contribution to the friction. The FFM/CFM combination has been used not only for surface imaging with high chemical resolution, but also for investigatiing the molecular origin of friction forces. There are many papers that explore the relationships between the friction force (determined for a variety of organic molecules) and parameters that are essential to the frictional behavior of the studied surfaces, such as the load force, the velocity of the scanned AFM tip, the solution pH, the humidity, or the temperature [85]. In particular, alkylthiols and alkylsilanes that form self-assembled monolayers (SAMs) are ideal model systems that can be used for investigating the molecular origin of friction. Many authors have presented results on the relationship between SAM structure and frictional behavior. The friction in these systems has been studied as a function of structural properties such as alkyl chain length or terminal functionality, and also as a function of load and sliding velocity. The results presented in the work of Brewer et al. [86] showed different friction–load and friction–velocity correlations as a function of the type of terminal end functionality and the length of the alkyl chain, indicating the importance of the latter. Friction forces are load-dependent, which means that the friction force increases with the load applied perpendicularly to the sample surface. For macroscopic friction, this dependence is linear and the coefficient of proportionality is called the coefficient of friction. However, in contrast to the macroscopic situation, the relationship between friction and loading forces is not expected to be linear for a single asperity contact. This conclusion was supported by experimental data obtained for some organic materials. Actually, a number of examples have shown a linear friction–

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load relationship for organic monolayers, including for alkanethiol on gold [87] or phenyl-terminated SAMs [88], octadecylchlorodimethylsilane deposited on the oxide of silicon [89] or alkylsilanes on silica and Langmuir–Blodgett films of stearic acid [90]. However, there are also studies where nonlinear behavior was observed; for example complexes of naphthoylene benzimidazole and stearic acid [91]. More recently, Wei et al studied the frictional properties of monolayers of alkylsilanes on mica surfaces [92]. Their data showed a nonlinear relationship with the external load. This behavior can be explained by considering multiple asperity contacts and/or the plastic deformation of the samples. While the work of Xiao et al. [91] supports the hypothesis that plastic deformation may play a role, the other work by Kiely and Houston [93] suggests that monolayers may deform viscoelastically during contact with an AFM tip. This is an important suggestion, because many of the fundamental mechanistic studies of nanoscale friction behavior have focused on inorganic materials, characterized by high elastic moduli and similar bulk and surface chemical properties. The deformation of monolayers, consisting of relatively compliant organic compounds, on substrates that are often much stiffer, is significantly more complex from a theoretical perspective. It is known that friction is independent of velocity at the macroscopic scale. At the microscopic scale, however, this is not true, and a dependence on velocity was observed for both solid [75, 76] and organic surfaces. Similar to the friction–load relationship, the character of the friction–velocity function depends on the type of organic molecules involved. As an example, the work performed by Brewer et al. [86] for a series of six self-assembled monolayers of alkanethiols with varying chain lengths and terminal group functionalities (−COOH, −OH, −CH3 ) is presented. The friction–velocity dependencies were similar for all SAM monolayers when the methyl-terminated probe was in contact – the friction force gradually increased as the velocity increased. However, a different behavior was observed for carboxylterminated probes. Depending on the type of SAM monolayer deposited on the surface (polar or apolar), the friction force reached a plateau (Fig. 7.20). The effect was characteristic of polar-terminated SAMs. The authors explained the results

Fig. 7.20. Variation in friction force with scan speed for a carboxylic acid (C10 COOH)functionalized tip in contact with six different SAMs (Reprinted with permission from [86], Copyright (2001) by the American Chemical Society)


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by invoking cohesive forces between polar-terminated groups. According to them, cohesive forces are responsible for the breaking and forming of the hydrogen bonds that are important to SAM deformation when acid-terminated tips are used. 7.2.2.3 Biomolecular Surfaces Proteins are molecules that participate in almost all biochemical reactions in cells. They are designed to bind to other molecules, from ions to large complex molecules such as other proteins or nucleic acid. The formation of enzyme–substrate complexes, recognition between antibodies and antigens, forces between complementary DNA strands, or the selectivity of lectins towards carbohydrates are just a few examples of this binding, which is involved in many biochemical phenomena, including molecular transport, signaling, and cellular adhesion. The basic working principle of a molecular recognition event is the specific binding of a ligand molecule to a receptor molecule that is usually embedded in cell membrane. The binding interaction can be characterized by two features, the specificity, describing the preferential binding of one or a few closely related molecules, and the affinity, defining the strength of binding. For highly specific interactions, the structure of the binding site of the molecule should exactly fit its complementary protein. Protein binding (the recognition between biological molecules) is responsible for the majority of the interactions of cells with other cells and/or with various proteins in the external environment [94]. The loss or alterations of these interactions may affect cell functioning and lead to many pathological states. Numerous diseases are related to the malfunctioning of these molecular recognition processes. Investigations of the interaction mechanism are key to understanding a wide spectrum of biological processes, such as inflammation or cancer metastasis, which should lead to the development of highly specific drugs [95]. Many methods have been used to investigate unbinding processes between two molecules, especially those induced by the action of an external force that causes bond breaking. These methods encompass, for example, a biomembrane force probe with pipette suction [96], a hydrodynamic flow based method [97], optical tweezers [98], and also atomic force microscopy (AFM [99]). Over the last few years, the latter method has become an indispensable tool to biological studies of molecular interaction forces. Functionalization of the AFM probe significantly enhances the microscope’s applicability, since molecular interactions can be studied with high chemical sensitivity. AFM offers a convenient way to measure forces involved in specific interactions by detecting unbinding events between the AFM tip, which is functionalized with one type of molecule, and the substrate, upon which its complementary partner is immobilized. The unbinding force, determined from the force required to separate the tip from the substrate, has been measured for single molecular complexes such as avidin–biotin [100], human serum albumin (HSA) and a polyclonal anti–HSA antibody [101], and even between complementary strands of DNA oligomers [102]. The chemical sensitivity of FFM suggests its possible applicability to measurements of interaction forces between biological molecules during recognition processes. Similarities in the molecular bond detection by means of friction and

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Fig. 7.21. Scheme of the molecular interaction between measurements of the pull-off (left, AFM) and friction force (right, FFM). The main difference is the direction of force separation(Reprinted with permission from [104], Copyright (2000) by the American Physical Society)

classical force spectroscopy in AFM are shown schematically in Fig. 7.21. Bonds are formed during contact between the tip and the surface. Tip–surface separation can be induced by the force applied to the cantilever, either by withdrawing it vertically (AFM) or by moving it over the surface (FFM). There are few examples of the application of chemical force microscopy to detecting (bio)molecular interactions. The data presented in 1999 by Mazzola et al. [103] show the potential use of CFM to characterize the interactions between oligonucleotides, short segments of DNA. A patterned array of oligonucleotides was scanned with a DNA-modified AFM tip. The elements of the substrate arrays were relatively easily differentiated by means of friction force. The complementary nucleotides exhibited stronger friction, which was correlated with the strength of the interaction force between the tip and the functionalized surface. The noncomplementary interaction was shown to be weaker in terms of both friction and adhesion forces. The observed friction force was strongly dependent on the loading force. In work by Lekka at al [104], both chemical (both the AFM tip and the surface were modified with a pair of proteins that recognize each other) and friction (the lateral signal was recorded in order to study molecular interactions) force microscopies were applied to study a typical carbohydrate–protein interaction. Measurements were carried out for two lectins: concanavalin A (from Canavalia ensiformis) and lentil lectin (from Lens culinaris agglutinin), which have the same type of specificity to mannose and glucose sugars. Carboxypeptidase Y was chosen as a complementary glycoprotein (a protein with a covalently attached carbohydrate chain), since its carbohydrate moiety is mainly composed of mannose. Proteins were deposited on the sample surface using by microprinting with PDMS stamps that had holes arranged periodically. The sample surfaces with printed proteins were imaged with a ConA-functionalized AFM tip. The surface topography of the immobilized proteins (CaY or BSA) reflected the patterns of the PMDS stamp used. Figure 7.22 presents the surface topography of the CaY deposited on the modified glass surface. The friction force map obtained for the Con–CaY complex showed the pattern corresponding to the chemical diversity of the sample. The areas outside the circles were covered with CaY, and the interiors of the circles corresponded to the silanized and glutaraldehyde-


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Fig. 7.22. Surface topography and friction force images recorded at the same lateral loading rate of 1.4 × 109 pN/s, measured for (a) ConA–CaY and (b) ConA–BSA molecular complexes in a buffer solution

treated glass surface. In order to verify the observed recognition interaction for the ConA–CaY complex, measurements were performed for the ConA-functionalized AFM tip and a BSA-printed surface. As was expected, the friction force did not show any pattern, while the surface topography reflected a similar pattern, just as for the patterned CaY surface. When molecular forces are involved in interactions between (bio)molecules, it has been reported that the resulting unbinding force depends strongly on the applied loading rate (defining how rapidly the applied force breaks bonds [96]), which provides an insight into the energy landscape of the interacting molecules. The dependencies obtained so far show mostly linear region(s), indicating the presence of one or more energy barrier [96, 105, 106]. Therefore, assuming that the observed pattern for the ConA–CaY interaction is due to discrete dominating adhesion bonds, the dependence between the friction and the lateral loading rate was expected to be similar to that measured using classical force spectroscopy. Results have shown that such a dependence occurs only for interactions between ConA and CaY, where recognition events were expected. ConA–BSA does not show this complementariness, and their binding strength stems from forces acting outside the binding site, such as electrostatic and hydrophobic forces. For this type of interaction there was no dependence on scanning velocity (Fig. 7.23). It has already been demonstrated by classical force spectroscopy that the molecular unbinding process manifests itself

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Fig. 7.23. Lateral loading rate dependence (measured in TBS buffer) for a lectin-functionalized (ConA, gray squares; LcH, black circles) tip and a CaY-modified surface. The specific interaction was inhibited by adding lectins (1 mg/ml) to the solution (gray squares and black circles, for ConA and LcH, respectively). After 30 minutes of incubation, the unbound or loosely bound lectins were removed by washing in buffer (Reprinted with permission from [104], Copyright (2005) American Institute of Physics)

in a characteristic dependence on the loading rate. The loading rate is usually determined as the product of the retraction velocity and the system spring constant (in the normal direction). For friction force, the loading rate can be defined as the product of the lateral spring constant and the scan velocity. On the other hand, the dominance of the adhesion component in the friction force was theoretically predicted by Filipov et al. [107]. The authors considered two plates connected by adhesive bonds, sliding against each other by the action of an external force. Their model describes the relationship between macroscopic frictional phenomena and the dynamics of formation and the rupture of microscopic bonds. The dynamic response of the system can be characterized by two constants that describe a competition between the rates of bond rupture and formation and the loading rate. 7.2.3 Summary The combination of FFM and CFM offers high spatial chemical resolution that can be applied in order to study the properties of self-assembled monolayers and other wellordered organic monolayers. It has yielded insights into the relationship between nanoscale friction, molecular structure and organization. FFM/CFM has proved to be exquisitely sensitive to changes in molecular organization and in chemical composition, and has several additional advantages: (i) the measurement is fast (information from one friction image is comparable to 103 friction–distance curves), (ii) it gives better control of normal force (as the measurement is carried out using feedback conditions), and (iii) the technique allows easy localization of binding sites (for example on cellular membrane). Its application to studies of interactions between biomolecules shows much promise, as the dependence between the friction


237

force and the sliding velocity can provide information on the molecular energy landscape.

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8 Scanning Microdeformation Microscopy: Subsurface Imaging and Measurement of Elastic Constants at Mesoscopic Scale Pascal Vairac · Bernard Cretin

8.1 Introduction Early acoustic microscopes were based on similar principles to the optical microscope: a lens focuses the acoustic beam at the surface of or inside the investigated object [1, 2]. After interaction with the sample, the transmitted or reflected acoustic field shows the presence of inhomogeneities or defects. Because the acoustic impedance of air is very small compared to that of solid material, this scanning technique needs a coupling liquid to be inserted between the lens and the sample. As with optical imaging, the image contrast is directly related to local discontinuities and refractive index. Imaging is usually qualitative, and the first acoustic microscopes were applied to NDE and medicine. Other acoustical methods have since been developed (for example, an interferometric technique called V(z) [3]) in order to quantify the local elastic constants of the investigated material. As with other imaging techniques based on wave propagation, the main limitation of the conventional acoustic microscope is the lateral resolution, which is given by Rayleigh’s criterion [4]: the ultimate resolution is about half the wavelength of the acoustic wave. In solid materials, a resolution of 1 micron requires a frequency exceeding 1 GHz. Such high frequencies lead to high acoustic wave damping. The emergence of near-field microscopy has enabled this physical limit to be surpassed. In modern near-field microscopes, the tip can be used as the source and/or the detector of vibrations, enabling high-resolution imaging. Because the excitation frequency is usually low (for example, the resonance frequency of the cantilever), the acoustic wavelength is greater than the radius of the tip and, to a first approximation, the spatial resolution is mainly related to the interaction area and not to the acoustic frequency. A resolution of 100 nm can be easily obtained with micron-sized tips and low contact forces. At the mesoscopic scale (lateral resolution of typically 0.1–1 microns) imaging is typically obtained at frequencies in the range of 10 kHz–1 MHz. Local elastic properties can be accurately quantified at this scale because the tip–sample contact can be modeled using the well-known theory of Hertz. After a short historical overview of near-field microscopy, this chapter presents some mesoscopic-scale microscopes, and then reviews the basic physical concepts behind near-field acoustics and the tip–sample interaction. The new concept of the scale parameter is introduced (the behavior during near-field acoustics depends upon the size of the acoustic source). Mechanical and acoustical approaches are discussed.

242

P. Vairac · B. Cretin

The main features of the microscope are analyzed: investigation depth, spatial resolution and the effect of topography. The configuration of a specific near-field acoustic microscope is then detailed: the scanning microdeformation microscope (SMM) which is a kind of ac contact force microscopy. SMM can image the close subsurface of the investigated sample or it can be used to characterize the local elastic constants. Some applications to the imaging and characterization of test samples are presented (measurements of local elastic constants of bulk materials and of thin films using a specific probe enables the local viscoelasticity and density to be quantified). Towards the end of the chapter, an example of ultimate metrology with SPM is presented: local measurements at the noise level.

8.2 Review and Physical Background of Near-Field Acoustic Microscopes 8.2.1 Review of Near-Field Microscopes Near-field microscopy emerged in the early 1980s, with the scanning tunneling microscope (STM) [5] for high resolution, but also with instruments capable of mesoscopic resolution in other domains (such as optics and acoustics). The invention of the first near-field probe is attributed to W. Dürr, D.A. Sinclair and E.A. Ash [6]; the spatial resolution of this instrument was about λa /4 at 4.5 MHz (close to Rayleigh’s limit). A few years later, Zieniuk and Latuszeck [7] replaced the conventional lens with a subwavelength source radiating in a fluid close to the sample surface. For a wavelength λ, this noncontact technique gave a resolution of λ/10 at 30 MHz (about 5 µm). A schematic of the probe is given in Fig. 8.1a. The main drawback of this technique was its limitation to surface investigations (note that the conventional acoustic microscope is capable of subsurface observation through appropriate positioning of the lens focusing plane). In 1989 Güthner et al. [8] developed a vibrating probe in order to map the topography of rough surfaces. Another method used to obtain a small acoustic source is base the design on an aperture. Such a probe has been studied by B.T. Khuri-Yakub et al. [9]. A small aperture (100 µm in diameter; very small apertures are not easily machined) is placed in the focal plane of an acoustic lens. Obviously, the interaction with the sample can be detected when the sample surface is close to the aperture. The demonstrated spatial resolution was about the size of the aperture (100 µm) at 3 MHz operating frequency. Following these first developments in the field, the resolution of the near-field microscope was clearly related to the size of the probe and the challenge was to investigate smaller scales. In terms of resolution, the best microscope was the STM. Designs based on STM operating simultaneously with near-field acoustics have been realized. Two systems that use a similar principle (K. Takata et al [10] and K. Uozumi et al. [11]) have been reported. Figure 8.2 shows the structure of the design from Takata et al. for the acoustic microscope that uses the tunneling effect. The microscope operates in transmission mode. The tip–sample distance is controlled with the tunneling current. The tip vibrates at 70 kHz frequency. Transmitted acoustic stress is detected on the opposite face of the sample with a piezoelectric transducer.

8 Scanning Microdeformation Microscopy at mesoscopic a)

b)

Transducer

Coupling liquid

243 Quartz tuning fork

Sample

Sample

c)

d)

STM tip Tip voltage Lens Coupling liquid Aperture Sample

Transducer

Fig. 8.1. (a) Sapphire tip for near-field acoustic microscopy, as used by Zieniuk et al. [8]. (b) Tuning fork of Güthner et al [9]. (c) Probe using an aperture, from Khuri-Yakub et al. [10]. (d) Principle of the microscope of Takata et al. that combines tunneling current and acoustics [11]

Fig. 8.2. Model of the baffled piston radiating in a liquid

The resolution obtained was about 100 nm. The microscope combines standard STM information with acoustic coupling. Uozimi et al. have used another operating mode, the excitation transducer, which has been used to detect the echoes from acoustic pulses at 1.4 MHz (this system operates in the reflection mode). In the designs of the first near-field microscopes, the probe has a conical geometry. However, it is well known that pressure transformer in conventional acoustics uses a Mason’s horn. A near-field acoustic topography system based on this principle was developed in 1992 by A. Kulik et al. [12]. In this microscope, the horn was excited at a frequency of 1 MHz. A resolution of 100 µm was obtained (λ/15


244

resolution). The Mason’s horn can also be used to characterize the viscosities of fluids (a sonotrode [13]) at the mm scale and to study transformations of materials. In recent years, further developments in acoustic microscopy have led to increased lateral resolution and the ability to measure local properties of the investigated material at the nm scale. Most of these systems have been derived from atomic force microscopes (AFM) operating in dynamic mode. The heads used facilitate the detection of the local elastic stiffness, defects located close to the surface, or surface acoustic waves. Several combined AFM and acoustics systems, known by different acronyms (UFM, SLAM, AFAM, . . . ), have been proposed since 1992 [14–20]. The research in this field is intense; many features are being discovered and investigated in both theoretical and experimental domains. All of the techniques presented here surpass the physical barrier given by Rayleigh’s criterion. The surface topography is easy to map with near-field acoustics because the local probe is very sensitive to the sample interaction distance. Surface and subsurface inhomogeneities and defects can be detected with a better resolution than possible with conventional methods (such as scanning acoustic microscopy) but the depth that can be investigated is obviously limited because the acoustic field is strongly divergent. 8.2.2 Physical Basis for Near-Field Acoustics and the Scale Effect In this heuristic approach, we will consider a simple model for the generation of longitudinal acoustic waves by a plane baffled piston. We assume a circular baffled piston that conforms to the model given in Fig. 8.2. In this model, the wall is assumed to be fixed (no displacement) and the emission occurs in a fluid. The calculation of the radiated field takes into account the cylindrical symmetry; an element of the source is a ring of internal and external radii r and r + dr . The movement of the piston is assumed to be periodic: u t (0, t) = U0 e jωt . After separating the variables: u T (z, t) = u (z) e jωt , the potential can be calcu− jkR lated by integrating the Green function: G = e4πR . The displacement is obtained by differentiating the obtained potential. The expression for the displacement on the z-axis normal to the piston is [21]: √ z − jkz − jk a2 +z 2 (8.1) −√ e u (z) = U0 e a2 + z 2 This expression is complicated. In order to express the approximate solution for conventional acoustics, we assume that the phase term prevails. Approximating the amplitude to the 0th order, and expressing u(z): 2 − jk 1+ a 2 z 2z

u (z) ≈ U0 e− jkz − e

≈ 2 jU0 e− jkz e− j

ka2 2z

The modulus of the displacement along the z-axis can be written: 2 |u(z)| ka ≈ 2 sin U0 4z

(8.2)

(8.3)

8 Scanning Microdeformation Microscopy at mesoscopic

245

Fig. 8.3. Modulus of the normalized displacement in front of the piston versus the distance for k = 100 m−1 and a = 0.1 m

Depending on the z-value, two regimes are observed, as shown in Fig. 8.3: (i) Fres2 nel’s zone when z < aλ (oscillations of the modulus are seen here); (ii) Fraunhofer’s zone: z

a2 λ

⇒

|u(z)| U0

≈

πa2 zλ

(the classical 1/z law is seen here). 2

The limit between the two zones is d = aλ , which is called the Fresnel distance. This approximation can be found in all books and courses on the fundamentals of acoustics. This model is valid for a λ. However, in near-field microscopy, this approximation is no longer valid and we use the complete expression of u(z) to express its adimensional modulus: + & ', |u(z)|2 1 2 2 /z 2 − 1 = 1 + − cos kz 1 + a 1 + a2 /z 2 U02 1 + a2 /z 2

(8.4)

A plot of the adimensional modulus of u(z) is given in Fig. 8.4. Obviously, if the piston has a diameter that is very small compared to the wavelength, the oscillations related to the interference disappear. Figure 8.4 shows that below z = a = 1 µm, the amplitude is close to that of the piston. Above z = 1 µm we observe a 1/z 2 law. This dependence can also be calculated by limiting the development of the expression of u(z) to the second term (in near-field microscopy, the amplitude is predominant and not the phase): |u(z)| ∼ a2 = 2 U0 2z

(8.5)

Of course, when z 1/k, we observe the 1/z law which is the conventional behavior for the amplitude of the displacement field (this part of the curve is not represented in Fig. 8.4). Thus, the scale effect yields two different laws, 1/z 2 and 1/z. To summarize, there are three characteristic limits defining the transitions: z = a2 /λ, the Fresnel distance for a source with a radius greater than the wavelength of the acoustic wave – z = a and z = 1/k for a source with a radius smaller than the wavelength

–


246

Fig. 8.4. Adimensional amplitude of the displacement versus the distance from the front of the piston for a = 1 µm and k = 100 m−1

Thus we can deduce the value a0 of the radius of the piston corresponding to the transition between the two physical behaviors: λ a0 = √ 2π

(8.6)

Figure 8.5 shows the clear difference between the two regimes. The value k = 100 m−1 has been included to allow comparisons with the above results. Oscillations appear when a > a0 , as expected. Note: The angular power distribution is calculated in the same way as in normal optics. When a is greater than λ, the Airy distribution is observed. Conversely, when

Fig. 8.5. Scale effect in acoustics; evolution of the adimensional field u(z)/U0 versus z (k = 100 m−1 ) where the radius of the piston is: (a) radius a = 4 × a0 ; (b) radius a = a0 /4


247

a is smaller than λ, the energy is distributed in the space in front of the piston and both longitudinal and transverse components of the field exist (transverse mode energy loss explains the huge decrease in displacement for the small piston). In most near-field acoustic microscopes, the source is a vibrating point in contact with the object. Experimentally, a subsurface defect at a depth greater than a can be detected if the “reflector” (defect or interface, for example) has a side-size greater than the radius of contact (the contact radius is defined in the chapter on sphereplane interactions). It should be noted that this evolution in the field (the 1/z 2 law) is a characteristic of all near-field acoustic microscopes. As mentioned previously, the lateral resolution is approximately the radius of the source. The major disadvantage of this technique is the inability (with only one source) to carry out observations deep inside the subsurface (we are limited to a few a). 8.2.3 Mechanical Approach Two different approaches can be considered in order to explain the dynamical response of a sample excited with a local source. The first one is based on acoustics, and the second one on conventional solid mechanics. This second approach is more pertinent when the size of the excitation source is negligible compared to the transition parameter a0 . In this part, we first review contact theories applicable to mesoscopicscale microscopy. Then we interpret the origin of the contrast in the images by using the theory of static beam flexion. 8.2.3.1 Models of Tip–Sample Interactions 8.2.3.1.1 Hertzian Model of Sample On the mesoscopic scale, van der Waals, adhesion and capillary forces can usually be neglected, and the basic theory of contact may be used in a first approach. Because the tip shape is often smooth, the standard model is a sphere contacting a plane (of course, this geometry is a rough approximation, but it can be used on the mesoscopic scale because geometries are easily checked). Let us assume a spherical shape for the tip extremity (radius R) and a planar surface for the object. According to the historic Hertzian theory of contact, both tip and object are strained due to the contact force F (Fig. 8.6); the contact radius a is expressed as [22]: a=

3RF 4E ∗

1/3 (8.7)

where E ∗ is expressed as: 1 1 − νt2 1 − νs2 = + E∗ Et Es

(8.8)


248 Fig. 8.6. Hertzian model of interaction between a sphere (tip) and a plane (object surface)

E t , νt and E s , νs are the Young’s moduli and the Poisson ratios of the tip and sample respectively. E ∗ ≈ 1011 N/m2 is typically for the interface between a sapphire or diamond tip and a silicon specimen in standard operating conditions. This old and simple model is valid as long as the interaction is kept at the mesoscopic scale. In dynamic AFM (in the case of nanometer-sized tips), JKR, DMT or Maugis [23] models have to be used because local forces (atomic, capillarity, adhesion) are predominant. Following the Hertzian theory, a 15 µm radius tip and a 500 µN force typically yield a value for the contact radius a of 1 µm. Physically, a is the radius of the static and dynamic source of stress and displacement at the sample surface. The JKR model (developed by Johnson, Kendall and Roberts [24, 25]) is valid when adhesion forces are not negligible, and for soft materials (such as polymers). This model is not reviewed here because the main applications of mesoscopic microscopy concern rigid materials. 8.2.3.1.2 Sneddon’s Model for High Loads Sneddon has proposed a model applicable to axisymmetrical tip shapes and increased strain compared to the Hertzian model. It also neglects adhesive surface forces. The following relations enable us to estimate the values of mechanical parameters: – Normal force between the tip and the sample: 1 ∗

F = 2E a 0

x 2 f(x) dx √ 1 − x2

(8.9)

where f(x) describes the shape of the tip (x = ar , r is the distance from the center of contact) – The indentation depth δ: 1

δ= 0

f(x) dx √ 1 − x2

(8.10)


In the case of a spherical tip, the following relations can be used: E∗ 2 R+a a + R2 Ln − aR F= 2 R−a and δ=

1 aLn 2

R+a R−a

249

(8.11)

(8.12)

Experimentally, we have shown that this model can be replaced with the Hertzian model so long as the surface strain is reversible (this is the expected case, and it is checked for after surface scanning) and the tip is not worn by friction (the tip damage is clearly visible with SEM after a few scans). 8.2.3.1.3 DMT Model for Adhesive Contact and Rigid Materials The theory developed by Derjaguin–Muller–Toporov (DMT), completing the Hertzian theory, takes into account adhesive forces by assuming that attractive forces act only outside the contact area. Thus, there is a repulsive contact area and an attractive action outside the contact. The geometry of the contact is supposed to remain Hertzian, so attractive forces are unable to deform surfaces. The DMT theory neglects the influence of attractive forces in the contact area and the computed contact radius is generally underestimated. The DMT theory applies to the case of small radii of curvature and weak adhesive forces. Nevertheless, we will note that the most realistic theory for AFM in contact mode gets close to the DMT model. With this model, the adhesive force (Fad ) is expressed by: Fad = 2πRW

(8.13)

The contact radius a can be written: 3 (F + Fad ) R 1/3 a= 4E ∗

(8.14)

and the indentation depth δ is expressed by: a2 δ= = R

$

9 (F + Fad )2 16 RE ∗2

%1/3 (8.15)

The Maugis model is more general and enables the behavior of both soft and stiff materials to be modeled. On the mesoscopic scale, this last model gives no experimental advantages, because most local forces are negligible.


250

8.2.3.1.4 Complete Model of the Cantilever and Mechanical Contact Basically, in all microscopes operated in contact mode, the cantilever is assumed to be clamped at one end while the other end is coupled to springs and dashpots modeling the tip–sample interaction. In a first approach on a mesoscopic scale, the effects of the dashpots can be neglected. Because a near-field acoustic microscope needs an excitation, a piezoelectric transducer is used to excite the cantilever base. Therefore a clamped end cannot be a realistic boundary condition. In order to take into account the effect of this piezoelectric transducer, the conventional model has been completed by replacing the clamped end by a spring k p with a high stiffness and a mass m p (this model is valid below the first resonance frequency of the transducer). Moreover, at the other end of the cantilever, the tangential stiffness of the tip–sample contact has been modeled using a lateral spring. The complete model is described in Fig. 8.7, where the spring k P represents the relative stiffness of the piezoelectric transducer and m p the relative mass. An additional mechanical guide is used to model the real clamped end (in the field of mechanical vibrations, a perfect clamped support is purely theoretical). The longitudinal interaction stiffness kiL and the lateral interaction stiffness kil have to be known in order to evaluate the stiffness of the sample. On the mesoscopic scale, and in the ideal case of a flat sample, kiL can be estimated using the classical contact theory of Hertz mentioned above. When the tip (assumed to be a spherical indenter of radius R) contacts the sample surface, the vertical spring is expressed by [22]: kiL = 2E ∗ a

(8.16)

Lateral effects are simulated using a second spring of stiffness kil . The calculated value for the lateral stiffness [26] of the contact between a sphere and a plane was given by Mindlin [22]: kil = 4kiL

G∗ E∗

with

G∗ =

2(1 + νt )(2 − νt ) 2(1 + νs )(2 − νs ) + Et Es

−1 (8.17)

Fig. 8.7. Complete model of contact resonance modes used to quantify the local mechanical stiffness of a material


251

In the case of very stiff tips (such as diamond tips), E s E t , and this relation can be simplified as follows: kil ≈

2(1 − νs ) kiL (2 − νs )

(8.18)

Consequently, the lateral stiffness is usually slightly smaller than the longitudinal stiffness. It should be noted that this model is valid for homogeneous and isotropic solids (the Young’s modulus represents the longitudinal stiffness of an isotropic material). For anisotropic solids, a projection of the elasticity tensor can be used. Analytically, estimating the contact stiffness requires that the following fourth order differential equation for flexural vibrations of the cantilever [27–29] be solved subject to appropriate boundary conditions: ∂4 y ∂2 y + ρ A =0 (8.19) C ∂x 4 ∂t 2 where E C is the Young’s modulus of the cantilever, ρC is the mass density, A is the cross-section, and I is the area moment of inertia with respect to the neutral axis (E C I = kc L 3 /3). The previous equation describes the propagation of disperse flexural waves; the modes are given by the following equation: EC I

E C Iµ4 − ρC Aω2 = 0

(8.20)

where ω is the angular vibration frequency. Assuming a general solution, then by separating variables in (8.19): y(x, t) = y(x) e jωt

(8.21)

with y(x) =C1 (cos(µx) + cosh(µx)) + C2 (cos(µx) − cosh(µx)) + C3 (sin(µx) + sinh(µx)) + C4 (sin(µx) − sinh(µx))

(8.22)

The parameter µ is defined by (8.19) and the constants C1−4 are determined by the boundary conditions applied to the cantilever at the excited end (x = 0): ⎧ ∂y(x) ⎪ ⎪ ⎨ ∂x = 0 (8.23) ⎪ kp m p ∂ 2 y(x) ∂ 3 y(x) ⎪ ⎩ = y(x) + ∂x 3 EC I E C I ∂t 2 while at the other end (the interaction end, x = L): ⎧ 2 kil mlc2 ∂ 2 y(x) ∂ y(x) ⎪ ⎪ = −l y(x) + ⎨ ∂x 2 EC I E C I ∂t 2 3 2 ⎪ ⎪ ⎩ ∂ y(x) = kiL y(x) + m ∂ y(x) 3 ∂x EC I E C I ∂t 2

(8.24)


252

In the case of small strain, and assuming that the displacement of the center of mass m in the x-direction is smaller than that of the tip extremity by a factor lc/l (where lc is the distance between the center of mass of the tip and the cantilever), the displacement of the tip end in the x direction is given by x = lδy/δx. Under usual measurement conditions, the tip contacts the sample, and we obtain the following relations between the constants C1−4 : C3 = 0

(8.25)

C4 = θ C1 ,

θ=−

kp 3 µ EC I

+

m p ω2 µ3 E C I

β(cos µL + cosh µL) − (sin µL + sinh µL) C1 = C2 β[(− cos µL + cosh µL) − θ(sin µL + sinh µL)] −[(− sin µL + sinh µL) + θ(cos µL − cosh µL)] (cos µL − cosh µL) − α(sin µL − sinh µL) C1 = C2 α[−θ(cos µL + cosh µL) + (sin µL + sinh µL)] −[θ(sin µL − sinh µL) + (cos µL + cosh µL)]

(8.26)

(8.27)

(8.28)

where the coefficients α and β are defined as: α=

mω2 µ3 − , kiL Ec I Ec I

β=

mlc2 ω2 µ − Ec I kil l 2 − Ec I

The eigenvalues µn can be calculated by using (8.19): 2 1/4 ωn ρA µn = Ec I

(8.29)

(8.30)

Finally, the characteristic equation of the system is obtained by equating (8.27) and (8.28). The computed solutions µn of (8.30) enable us to compute the resonance frequencies f n . The solution y(x, t) can be expressed as: (8.31) y(x, t) = y0 (cos µx + cosh µx) +

C2 (cos µx − cosh µx) + θ(sin µx − sinh µx) e jωt C1

Experimentally, this model has been validated for a silicon sample in the standard force range (0–800 µN) for mesoscopic tips. 8.2.4 Models of Subsurface Sensing Using Acoustic Waves and Surface Bending We have previously shown how to quantify local elastic properties of materials on the mesoscopic scale. But the microscope can also be used to investigate the subsurface


253

when the geometry is appropriate. The contrast observed can be explained with dynamic or quasi-static theories that are applicable on the mesoscopic scale. In all dynamic AFM, or the mesoscopically equivalent system, the tip acts both as a generator and a sensor. In a phenomenological approach, let’s consider a perfect reflecting plane (ideal subsurface defect) and a constant dynamic pressure over the surface of the piston modeled in Sect. 8.2.2. The pressure variation on the excitation piston is simply expressed using the image theory: a symmetrical piston on the back side of the reflecting plane and (8.5): κU0 a ∆p ∼ =− (2h)3

2

(8.32)

where h is the distance between the surface and the reflecting plane. If we assume that the cantilever can be approximated by a spring–mass system at a frequency close to a resonance frequency, the resonance frequency of the whole system depends on the acoustic (or vibrational) coupling. In our application, for low frequency shifts ∆ f of the cantilever resonance frequency f 0 , we can linearize the response of the cantilever and express the frequency shift: ∆p 3∆h ∆f =− ≈ f0 p0 h

(8.33)

This heuristic approach shows that subsurface planes normal to the motion of the piston and close to the surface are detected with an image contrast that depends on h −3 . Moreover, because the stability of the oscillator is about 10−5 in basic oscillating reflection mode, low variations in the surface stiffness can be detected (the ultimate experimental sensitivity is better than 1 N/m, or about 10−4 kiL ). Another approach is based on the deformation of the sample surface when the tip is applied with a given force. In this second analysis, the frequency variation of the resonating cantilever is explained using plate mechanics: the beam between the surface and the surface plane can be considered to be a clamped plate with a quasistatic motion. By using the formula for the deflection of bent clamped beams, we can approximate the variation of the detected frequency using: 16EL i h 3 (8.34) w3 where km is the stiffness of the membrane considered to be a beam, w is the apparent width, E is the Young’s modulus of the sample, and L i is the interaction distance. Typical values of km are in the 107 –108 N/m range. Because the usual value of kiL is about 104 –105 N/m, the applied force decreases when the tip is scanned over the grooves (the clamped end of the cantilever is moved to apply the force and no force control is used). The force variation is proportional to the relative variation in the global stiffness k: km ≈

kC km ∆ f ∼ ∆k ≈ 10−3 where k = = f k kC + km

(8.35)

Experimentally, a good agreement is obtained between this interpretation and the image contrast: in reflection mode, a subsurface defect (a chemically etched groove) is clearly detected [30].

254


8.3 Imaging and Measurement with Scanning Microdeformation Microscopy 8.3.1 Configuration The set-up for scanning microdeformation microscopy is shown in Fig. 8.8. As in some AFM set-ups, the sample is supported by a three-axis translation unit (the other way to obtain relative motion makes use of actuators fixed to the cantilever support). The vertical axis enables the value of the static contact force to be adjusted. The microscope can operate in either transmission or in reflection mode. In the transmission mode, the tip is vibrated by the low frequency (LF) generator and the transmitted stress is detected with a second piezoelectric transducer on the back side of the sample. The weak detected signal is processed to extract its amplitude and phase (signal processing is carried out with a double-phase lock-in amplifier). In reflection mode, a high sensitivity heterodyne interferometer is used as a noninvasive sensor to detect the amplitude and phase of the vibrating cantilever. The head of the microscope is composed of a piezoelectric transducer, the cantilever and the tip (made of diamond or sapphire and not of silicon, unlike standard AFM where the tip is obtained by chemical etching of the cantilever). As mentioned previously, this hybrid sensor, shown in Fig. 8.8b, can be considered to be an electromechanical resonator whose frequency is related to the tip–sample interaction [31]. The tips can be standard pickup needles (radius 15 to 45 µm), or specific diamond tips with radii down to 0.6 µm. Depending on the tip radius and the applied force, the contact radius is 50 nm–2 µm. In another configuration, not represented in Fig. 8.8a, the resonant frequency of the microscope head can be detected by replacing the generator with a specific amplifier; in this case, the electrical signal of the oscillator is simply counted (this is the basic reflection mode where the resonance frequency is extracted). This microscope, called a scanning microdeformation microscope (SMM), can be used to image surface and subsurface defects or to characterize the local elastic constants of the investigated sample. Of course, qualitative and quantitative observations are physically coupled and the sample must be well known a pri-

Fig. 8.8. (a) Principle of the SMM (b) Hybrid cantilever developed for SMM


255

ori. Experimentally, in the imaging mode, the generator frequency is kept constant and the sample is scanned. The amplitude and phase images are mainly related to the mechanical properties of the surface and subsurface, but surface roughness may play a predominant role because the force is not controlled (even in the case of force control, the contact radius depends on the geometries of the surface and the tip). In the first characteristic mode described, the excitation frequency is scanned and the tip pressure is increased so that the shift in the resonance frequency can be measured. For ideal operating conditions, the measured frequency shift depends on the contact force and on the sample stiffness. The sample stiffness can be estimated for a known contact force, which is easily estimated if the contact position and the vertical motion of the cantilever are known. The vibration of the cantilever is usually measured using the standard deflection method, but many optical methods have been developed to sense vibrations [32–41]. The deflection technique is sensitive to the surface gradient and needs a preliminary calibration in order to obtain quantitative results. Conversely, a heterodyne interferometer can be easily autocalibrated and we have developed a specific probe to sense the small vibrations of the cantilever (in contact mode, the typical magnitude of the vibration is in the 10–100 pm range). The principles behind the interferometer developed for out-of-plane vibration measurement are shown in Fig. 8.9 [42, 43]. The reference beam is reflected by the PBS (polarizing beam splitter) and directed towards the photodetector. The signal beam goes through the PBS, is upshifted by a Bragg cell (acousto-optic modulator in Bragg incidence) driven at a frequency f B , and reflected by the cantilever; it is phase modulated on the cantilever surface. During the return path, the laser beam is upshifted again so that the frequency difference of the two beams is 2 f B . After a second pass through the PBS and the two plates, this signal beam is reflected by the front mirror of the HeNe laser. The two plates are used to rotate the polarization so that the signal beam is completely reflected by the PBS after reflection onto the laser mirror. This probe is interesting because no polarizer is needed in front of the photodetector. Thus, ideally, all of the laser power can be photodetected. For SMM applications, an optical fiber is added so that the beam is focused with a “selfoc” lens onto the end of the cantilever, above the tip. As with all heterodyne interferometers, the phase and frequency of the output signal enables the displacement or the velocity of the surface motion perpendicular

Fig. 8.9. Heterodyne interferometer used to measure cantilever vibrations


256

to the probe beam to be evaluated. The phase shift is given by the following expression: ∆φ =

4πδ λ0

(8.36)

where δ is the displacement normal to the surface and λ0 is the optical wavelength. Sensing with heterodyne interferometry is suited to noncontact measurements of displacements with magnitudes below 20 nm, in other words when the modulation index is low enough (this magnitude limitation originates in the nonlinear behavior of the phase modulation: at high modulation several sidebands appear and conventional phase detectors are inaccurate). In ideal conditions (perfect laser source and detector, the two arms are balanced, noiseless preamplifier and a fully matched photodetector and preamplifier) the ultimate sensitivity of the two-wave interferometer is expressed by [43]: " √ λ0 eB ∼ (8.37) u ult = = 1.3 × 10−15 m/ Hz 2π I0 where B is the noise bandwidth, I0 is the optical intensity (static or dynamic component) and e is the electron charge. The numerical value applies to I0 = 1 mW and λ0 = 633 nm. Experimentally, we encounter some interference, the preamplifier is noisy (less than 2 dB NF), the photodetector has a quantum efficiency of 0.5, some of the power is absorbed by the cantilever, and the two beam powers are unbalanced. Moreover, as mentioned above, an optical fiber is connected, √ and the experimental sensitivity of the probe is, in this case, about 3.10−14 m/ Hz because of the losses in the connectors. In SMM or other forms of ac force microscopy, the laser probe enables the cantilever motion or the vertical component of the vibration/acoustic field at the sample surface to be measured quantitatively. The magnitude and phase of the cantilever vibration are easily estimated and they yield quantitative values for the spring and dashpot interactions between tip and sample. 8.3.2 Application to Subsurface Imaging As with all the methods of scanning microscopy, the surface to be inspected is scanned with a XY translation unit controlled by a microcomputer. The data are acquired point by point (in reflection mode, the probe signal is processed and the magnitude and phase are extracted) and are transmitted to the microcomputer. Finally, each data is 8-bit encoded and the byte is associated with a pixel on the screen. The image obtained enables us to relate the magnitude and phase, or the resonance frequency, to the local elasticity of the investigated sample. The acquisition of an image primarily depends on the size and the number of acquisition points on the scanned surface. Translation speed is adjustable, and an image of 100 × 100 points


257

can be obtained in about five minutes, depending on the size (the larger the scanned area, the longer the acquisition time). Additional time is required for large scanned areas, while a tilt correction is often necessary to avoid force gradients (in the system presented here, the force is not regulated). Applications of SMM imaging are related to surface elasticity variations and subsurface defects. Different kinds of subsurface defects have been simulated by chemically etching silicon (silicon is a well known material; silicon wafers are highly polished and micrometer accuracy is easily obtained with chemical etching). In the example presented here, the chemical etching of one side of the wafer (360 µm thickness) allowed the various grooves to be engraved in silicon. The other face of the wafer was protected, preserving its polished aspect for the scan. A cross-section of a realized sample (a grating) is represented in Fig. 8.10a. After machining, the sample is stuck onto the support. The polished surface on which no defect is apparent is scanned by the tip. The image in Fig. 8.10b was obtained with a sapphire tip of radius 45 µm and with a resonance frequency of around 18 kHz. The resonance frequency was measured at each point. After performing a surface scan of 1000 × 1000 µm2 , the grating was shown to take the form of parallel strips. We should note that the contrast is poor because the grating is deep (the depth is about twenty times the contact radius), but the result is significant. As mentioned above, the fact that the simulated subsurface defect can be observed is explained by the bending of the surface above the subsurface grooves. This is explained further in Fig. 8.11. When the tip force is applied above a subsurface groove, the surface is bent and the static force decreases because the force is not controlled in this system. Consequently, the frequency decreases because the stiffness of the interaction is expressed, following (8.7) and (8.16), as: kiL = (6RF)1/3 E ∗2/3

(8.38)

Fig. 8.10. (a) Cross-section of the etched silicon sample simulating subsurface defects. (b) Image of the resonance frequency of the subsurface grating (45 µm tip and 1 mN load). Scan size: 1000 × 1000 µm2

258


Fig. 8.11. Mechanical explanation for the contrast observed in the image: the contact force depends upon the local stiffness of the sample

Experimentally, a good agreement is obtained between this interpretation and the image contrast: in reflection mode, the subsurface grooves show up as darker zones in the images, as seen in Fig. 8.10b. The variation in frequency observed in reflection oscillating mode when scanning a sample with subsurface grooves is typically ∼ 20−50 Hz, as expected [30]. Note that the contact radius also decreases. For small variations ∆F of the applied force F, the contact radius a is approximately given by: 1 ∆F ∆F < 0 ⇒ a < a (8.39) a ≈a 1+ 3 F 8.3.2.1 Artefacts in the Images As in all other methods used to investigate materials, SMM imaging requires a fair amount of knowledge of the sample. The surface roughness strongly influences the observed contrast because the resonance frequency of the microscope head is essentially related to the contact stiffness. For rough surfaces, the local elasticity depends upon the local curvature of the surface, and artifacts are observable. Figure 8.12 shows the results from scanning a rough surface. The sample is a silicon wafer with a porous surface. The image shows dark and light gray areas corresponding to the value of the local curvature, giving different values for the stiffness: as the gray becomes darker the lower force drops and the equivalent spring becomes softer.

Fig. 8.12. Image of porous silicon; the contrast is essentially related to the topography


259

8.3.3 Characterization of Local Mechanical Constants Local probes can be used to characterize elasticity, viscoelasticity, adhesion and plasticity. The main issue with obtaining quantitative results on the investigated material is knowing how to accurately describe the behavior of the system with a realistic model. Because estimating local elastic properties is a great challenge, one important application of SMM is performing sample stiffness measurements with mesoscopic resolution. As mentioned in Sect. 8.2.3.1, in order to quantitatively measure the sample stiffness, quantitative calculations using the inverse model are required. A knowledge of the surface and tip geometries and of the properties of the material allows us to choose the appropriate model. For quantitative evaluation, the resonance frequency of the resonator can be the first vibration mode of the cantilever, which is selected by filtering the electric signal of the bimorph piezoelectric transducer. Typical oscillation magnitudes are in the range 1–10 nm without contact, and in the range 10–100 pm when the tip contacts the sample surface. Experimental results have been obtained using a silicon cantilever (the cantilever axis corresponds to the (100) plane of the monocrystalline silicon) with length L = 6.75 mm, width w = 456 µm and thickness t = 148 µm. The local Young’s modulus can be estimated by measuring the output signal from the oscillator (SMM head and amplifier) and by numerically solving the inverse problem. A comparison between experimental and known data from reference materials is given in Table 8.1 for a 45 µm radius tip and for a range of applied forces (the value of E depends on the crystalline orientation for silicon). The accuracy is observed to be better than 5% at the mesoscopic scale (typical contact radius is about 1 µm under these conditions). The measurements require a good knowledge of the geometries of the tip and the sample; the tip is new and the sample surface has been polished. The model presented here is used to quantify the elastic properties of materials based on the results from a typical near-field microscope which uses a contacting tip and has a geometry that is known (approximately): a sphere contacting a plane surface. Table 8.1. Comparison between measured and supposed values of the Young’s modulus Sample

E/1 − ν2 (Pa) (Theory)

Frequency (Hz) (Experimental)

K iL (N/m)

E/1 − ν2 (Pa) (Experimental)

Accuracy (%)

Silicon Silica

1.69×1011 7.8×1010

28,075 27,265

128,580 95,770

1.63×1011 8.05×1010

3.5 3.4


260

8.4 Specific Application 8.4.1 Thin Film Measurements One attractive potential application of this type of ultrasonic method is to determine the local properties of layered materials [45, 46]. In this section, we present some results from layer thickness and layer elasticity determinations achieved using SMM stiffness measurements. The model we have used takes into account the effect of the thin film attached to a substrate. The application of this model to the experimental results shows that the SMM allows us to characterize the elasticity of layer-substrate pairs with relatively good accuracy. 8.4.1.1 Theoretical Elements In the particular case of a sample made of a thin film deposited on a substrate, if the Young’s modulus E f of the film differs from that of the substrate E s , then the measured modulus E c for this composite (film + substrate) changes with the depth of penetration h. The value of E c equals E f for h = 0 and E c equals E s for h t, where t is the film thickness (Fig. 8.13). Except for relatively thick coatings, the influence of the substrate cannot be neglected. The composite modulus can generally be expressed as: E c = E s + (E f − E s ) φ(x)

(8.40)

Here φ is a function of the relative penetration x where φ = 1 for zero penetration and φ → 0 for very large depths. The relative penetration x is characterized by the ratio a/t. Equation (8.40) enables one to determine the film modulus E f from E c values measured for one or more indentation loads and depths of penetration, provided the substrate modulus E s and the function φ(x) are known. Gao et al. [47]

Fig. 8.13. Schematic of the contact of a sphere with a film bonded onto an elastic half-space


261

Fig. 8.14. Dependence of I0 in 8.41 on the ratio of film thickness to contact radius

have developed an asymptotic expression for the influence of a film on the elastic response as a function of the ratio of the film thickness to the contact radius (t/a). They show that the effective modulus for a film on a substrate is given by (8.40) with φ(x) = I0 (t/a): ξ 2 1 1 + ξ2 −1 − (8.41) I0 = tan (ξ) + (1 − 2ν)ξ ln π 2π(1 − ν) ξ2 1 + ξ2 with ξ = t/a. For the materials of interest here (silicon and copper), the Poisson ratios are very similar and are assumed to be ν. The functional dependence of I0 (t/a) in (8.41) is shown in Fig. 8.14. After introducing the thin film effect, the classical way to approach this mechanical model is to solve the fourth-order differential equation for flexural vibrations of the cantilever (8.16) with the different applied boundary conditions (8.23 and 8.24). 8.4.1.2 Experimental Results and Discussion The experimental results were obtained by operating the SSM in reflection mode (see Sect. 8.2.1). The electromechanical resonator used in this case was made from a silicon cantilever with a rectangular cross-section, length L = 6.67 mm, width w = 450 µm, and thickness h = 148 µm. The sapphire tip (radius 19 µm) was a rounded cone connected to a cylinder of diameter d = 410 µm, and the total length of the tip was l = 1195 µm with lc = 459 µm (distance between the center of mass of the tip and the cantilever) and m = 4.678 × 107 kg.

262


Several specimens layered with thin copper films, deposited by a magnetron sputtering onto silicon, were investigated. Figures 8.15 and 8.16 show the first results from these samples for different film thicknesses: the evolution of the first cantilever flexural mode and the contact stiffness respectively. From these figures, it is clear that the measured cantilever resonance

Fig. 8.15. Evolution of the resonant frequency detected for samples made of a substrate of silicon (E = 169 GPa, ν = 0.278) and a thin layer of copper (E = 112 GPa, ν = 0.370) for different loads

Fig. 8.16. Evolution of contact elasticity for samples made of a substrate of silicon (E = 169 GPa, ν = 0.278) and a thin layer of copper (E = 112 GPa, ν = 0.370) for different loads


263

frequency follows the trend expected. As the copper is softer than the silicon, the contact stiffness will decrease as the thin copper film thickness increases. Figure 8.17 shows the plotted for the solution to the inverse problem, in the case of a 200 µN load on the composite sample, which gives the value of the Young’s modulus. The elastic modulus of the copper thin films was estimated from (8.40). The average value obtained from calculations of I0 (t/a) for several thicknesses is 109 GPa for the thin copper films. This value is very close to the value of the bulk copper sample. This approach to measuring the Young’s modulus of a thin layer on a substrate using the analysis of Gao et al. seems to be reliable. The best results are obtained for loads as low as possible; but for various reasons, such as irregular tip shape and surface roughness (even for very smooth specimens), more reliable data are obtained for contact depths of several tens of nanometers or more. In summary, we have reviewed a method of characterizing the local elastic properties of thin layers of metal deposited onto silicon substrate using near-field acoustics. The elastic modulus of a thin homogeneous film can be determined by dynamically indenting (elastic regime) the specimen to several depths and extrapolating to the required E c value. We have used the analytical function developed by Gao et al. to describe the effect of the thin film on the substrate. The behavior of the resonant frequency shift is in good agreement with the theoretical calculations, which allows us to estimate the local elastic properties of the thin layer.

Fig. 8.17. Young’s modulus evolution for a sample made of silicon substrate (169 GPa) and a copper layer (112 GPa). Interaction force 200 µN

264


8.4.2 Shape Memory Alloy In this section, we present an application of the SMM to the study of samples that have mechanical properties that evolve as a function of temperature and/or stress: shape memory alloys. 8.4.2.1 Properties of Shape Memory Alloys Shape memory alloys exhibit a martensitic transformation between a low-temperature monoclinic phase called Martensite and a high-temperature cubic phase called Austenite. This transformation is responsible for the shape memory and pseudoelastic effects exhibited by deformed CuZnAl alloys. These transformations occur at transition points that are typically linked to temperature and/or stress. In the absence of an applied stress, the transformation begins upon cooling to Ms and finishes at M f (see Fig. 8.18). On heating, the transformation starts at As and is completed at A f , and there is a hysteresis between both transitions. On the other hand, the transformation can be induced by applying stress at a fixed temperature above M. In this case, the transition points are the critical stresses for the appearance of the first trace of Martensite [48–50]. Martensitic transformations of such alloys have already been observed in bulk samples by mechanical spectroscopy. Such measurements, also called “internal friction” allow us to probe mechanical energy dissipation during the phase transition. However, they only give the average behavior for all parts of the sample. This study, and the one published by Oulevey et al. [51] provide the first local measurements of the martensitic transformations in thermally cycled shape memory alloys. 8.4.2.2 Experimental Configuration In order to perform these measurements, the SMM was fitted with a low-cost single heating stage. This stage is illustrated in Fig. 8.19.

Fig. 8.18. Schematic presentation of domains associated with the transformations A to M and M to A


265

Fig. 8.19. Principle of the heating stage

The heater is a Peltier module, which allows us to modify the temperature with an electric current. It is possible to heat or cool the sample with this classical thermal system, and it is necessary to use a radiator to remove the heat. Moreover, we use a thermistor (TCS 605) to measure the instantaneous temperature. The aim of the design is to minimize thermal drifts in the support sample. Problems arise when a temperature ramp is desired to study the shape memory alloy as a function of temperature. In the scanning system, the Z-scan has to compensate for the vertical component of the thermal expansion of the sample support. Therefore, the accessible temperature range is limited by the maximum extension of the scanner (10 µm in our case). To minimize thermal drift, the heater system was realized in aluminum and Invar (the thermal expansion coefficient of Invar is very low). The Invar part of the system holds the support sample in place via nylon screws and springs. In this way, thermal fluctuations are directed towards the radiator, and the drift induced in the sample is minimized. This stage enables us to obtain a thermal drift on the order of 50 nm/◦ C and temperatures above 100 ◦ C with our 10 µm Z-range scan. 8.4.2.3 Experimental Results and Discussion The characteristics of the CuZnAl and CuAlNi(SiZn) samples used for this study are given in Table 8.2. We have performed two kinds of measurements on the shape memory alloys. We have measured the local elasticity as a function of the temperature, and the dilatometric evolution of the sample using the optical deflection system of the SMM. Figures 8.20a and b show the local elastic measurements.


266

Fig. 8.20. (a) Local Young’s modulus evolution versus temperature for the CuZnAl alloy. (b) Local Young’s modulus evolution versus temperature for the CuAlNi(SiZn) alloy Table 8.2. CuZnAl

CuAlNi(SiZn)

Cu Zn Al MS AS

Cu Al Zn MS AS

70.17 25.63 4.2 20 ◦ C 16 ◦ C

62.22% 27.94% 4.65% 64 ◦ C 91 ◦ C

The results show the behavior of the shape memory alloys for temperature variations at constant stress. We can see significant drops in elasticity , by about 20 GPa, that are correlated with temperature phase transitions of the samples. Macroscopic measurements of internal friction have shown that phase martensitic transitions produce peaks in the internal friction in the samples. This phenomenon explains the drops in local elasticity measured with the SMM. The volume change during the transformation was also measured. As can be seen from the temperature–elongation curves (Fig. 8.21a and b), the volume change is relatively small (∆l/l ∼ 1% in the two cases), and in these cases the variations in slope are correlated with martensitic transition phase temperatures. It is worth noting

Fig. 8.21. (a) Results of the dilatometric measurements for the CuZnAl alloy. (b) Results of the dilatometric measurements for the CuAlNi(SiZn) alloy


267

Fig. 8.22. Observation of the modification of the orientation of the martensite plates by scanning of the tip too. The plates are aligned in the direction of scanning (here vertical). Sample: CuZnAl. Surface: 25 × 25 µm2

that a relative volume change of 0.08% has been measured in CuZnAl alloy [52]; this value has a similar magnitude with the one observed here. The correlation between the transition temperature measured at the macroscopic scale and the local mechanical spectroscopy measured with the SMM suggests that the peaks observed in the elasticity and the change in dilatometric evolution originate from the martensitic phase transition. Nevertheless, it seems that the change in Young’s modulus (50 GPa) between temperatures above A F and temperatures below M F is not measured directly. A comparison of the local transformation temperature with the extent of transformation confirms that the forward and reverse transformations initially occur at the sample surface. Finally, we tested these samples (the shape memory alloys) by performing the same experiment at constant temperature but variable stress. Scanning the CuZnAl sample allows us to demonstrate that it is possible to modify the orientation of the martensitic plates with the tip of the SMM. Figure 8.22 presents the martensitic plates oriented along the scan direction. Scanning the tip in contact with the sample under a static force modifies the martensite plates, and the image shown reveals that the martensite plates have been oriented along the scanning direction (vertical in the image). This result suggests that SMM could be used to microstructure the surfaces of shape memory alloys. 8.4.3 Viscosimetry It is well known that a mechanical resonator can be used to determine the characteristics of a liquid. The liquid generally interacts with the resonator and acts as a complex acoustical load, which is converted into a frequency resonance shift and a resonance damping. Among such acoustic sensors, surface wave sensors [53] and bulk wave sensors [54, 55] have been widely investigated. Furthermore, the local vibrating probe sensor seems to be an interesting way to achieve density and viscosity measurements [56–59] which benefits from nearfield acoustics. The proposed microsensor is based on the local probe microscopy technique, which is used for the local characterization of materials. In our set-up,

268


a small vibrating sphere immersed in the sample replaces the tip classically used in ac force microscopy. The sphere acts as a subwavelength acoustic source interacting with the investigated medium, and it has been shown that the investigation depth is related to the source radius. Coupling the probe with an electromechanical resonator enables the interacting forces to be measured. The chosen resonator is a soft cantilever, whose behavior is sensitive to both the magnitude and phase of the interacting forces. Varying these forces yields a cantilever resonance frequency shift and a variation in damping. Physically, the interacting forces have two origins: the inertial effect from the movement of the fluid mass by the vibrating sphere and the dissipative effect due to the viscous drag force. From another point of view, a solid body vibrating in a fluid generates two types of waves: compression waves, characterized in our case by spatial dispersion, and shear (transverse) waves, which are evanescent. In this way, the acoustic energy is confined to the vicinity of the sphere, and the investigation depth does not typically exceed the size of the sphere. 8.4.3.1 Experimental Arrangement The immersed part of the sensor is a silica sphere connected to the cantilever via a thin wire (a 1-mm-long cleaved silica optical fiber). Their diameters are 500 µm (sphere) and 50 µm (wire). This configuration ensures a good mechanical coupling. The cantilever is a silicon chemically etched beam with a stiffness of 100 N m−1 (size: 8000 × 350 × 150 µm). The driven end of the cantilever is embedded into a piezoelectric (PZT) plate bimorph generating flexural vibration as previously described (see Sect. 8.2.1). Thus, the sensor head can be considered to be an electromechanical resonator. Experimentally, the cantilever flexural resonance mode 2 is preferred to mode 1, which produces excessive damping in viscous fluids. The use of a high flexural mode allows the cantilever damping to be reduced, but globally the resonator is less sensitive to boundary conditions, and thus the sensor sensitivity decreases. The choice of the second vibrational mode seems to be a good compromise. The geometrical parameters have been chosen in order to optimize the sensor sensitivity. The most important parameters are the cantilever stiffness, which must be compatible with the interaction force’s range, and the sphere radius, which determines the interaction surface. High sensitivity requires a large interaction surface between the sphere and the fluid sample, and a small mass (an excessive mass induces a decrease in the relative contribution of the fluid). Thus, an optimum value can be found for a given measurement range, using criteria that include the four definable relative sensitivities corresponding to the measured values (resonance frequency f and Q-factor) versus fluid characteristics (dynamic viscosity η and density ρ). The excitation is performed with a low frequency generator (amplitude of a few mV). Three types of detection have been investigated experimentally. (i) Optical measurement of the magnitude and the phase of motion of the cantilever using the heterodyne interferometer [11]. This method yields the mechanical parameters of the resonance directly. (ii) Electrical detection can be performed with a network analyzer connected to the PZT bimorph. (iii) The last envisaged set-up involves achieving an oscillating loop, including an automatic


269

gain control (AGC). Thus, the cantilever resonance frequency and damping can be extracted by measuring the oscillating frequency and the error voltage on the AGC [54, 55]. 8.4.3.2 Modeling Figure 8.23 represents the mechanical model of the electromechanical resonator, taking into account the behavior of the cantilever and the sphere–fluid interaction. The fluid motion is governed by the continuity equation: ∇υ = 0

(8.42)

and the Navier–Stokes equation: 1 η ∂υ + (υ∇) υ = − ∇ p + ∆υ ∂t ρ ρ

(8.43)

where υ is the sphere linear velocity, and p is the pressure. For a small Reynolds number, the quadratic term can be neglected. This assumption corresponds to creeping motion in the fluid. Here, for a vibrational particle, the Reynolds number is expressed by Re = ω

Arρ , η

(8.44)

where A is the vibrational magnitude, ω is the cyclic frequency, and r is the sphere radius. With standard parameter values (A ∼ 10 nm, ω ∼ 105 rad s−1 , r = 0.25 mm, ρ ∼ 1000 kg m−3 , η > 1 cP), Re ∼ 0.2, justifying the above assumption. The second assumption introduced by the use of (8.43) is the incompressibility of the fluid. The resistance force acting on a vibrational sphere immersed in a fluid could be deduced by solving (8.42) and (8.43) with these assumptions. This force is given by the Stokes formula [60] F = R (η, ρ) υ + M (η, ρ)

dυ dt

(8.45)

R and M represent, respectively, the viscous dissipative effect and the added mass. They are given by

Fig. 8.23. Scheme of the physical model describing the behavior of the microsensor


270

& r' 9δ 2 3 r , (8.46) R = 6πηr 1 + , M = π ρ 1+ δ 3 2r √ where δ = 2η/ρω is the penetrative length of the shear waves. The equation of motion for flexural vibrations of the cantilever is a fourth order differential equation [13]. ∂2Y ∂2Y ∂Y ∂2 E I + σa +b = 0, (8.47) ∂x 2 ∂x 2 ∂t 2 ∂t where E is the Young’s modulus, σ is the mass density, a is the cross-section, and I is the area moment of inertia of the cantilever. The last term expresses the damping effect that may be attributed to the air along the cantilever. The resistive force is proportional to the velocity and b is the damping factor per unit of length. The solution of (8.47), Y(x, t), is the deflection at the coordinate x. Y(x, t) can be expressed using (8.22). In this case the constants C1−4 are determined by the following boundary conditions. At the excitation end (x = 0): y(0) = y0 e jωt (excitation) , ∂y(0) = 0 (zero slope) , ∂x x=0

(8.48) (8.49)

At the interaction end (x = L): ∂ 2 y(x) = 0 (zero bending) , ∂x 2 x=L EI

∂ 3 y(x) ∂x 3

= M x=L

∂ 2 y(x) ∂t 2

(8.50)

+R

∂y(x) ∂t

(shear expression) (8.51)

The parameter R is defined by (8.46); the inertial term is added with the probe mass, m s : M = M +m s . The solution to the problem with these boundary conditions yields a nonhomogeneous system. Finally, we obtain the following parameters: y0 , 2 y0 α sin(µ) sinh(µ) + β(sin(µ) cosh(µ) + cos(µ) sinh(µ)) , C2 = − 2 α(1 + cos(µ) cosh(µ)) + β(cos(µ) sinh(µ) − sin(µ) cosh(µ)) C1 =

C1 = 0 , y0 α sin(µ) sinh(µ) + β(sin(µ) cosh(µ) + cos(µ) sinh(µ)) , C4 = − 2 α(1 + cos(µ) cosh(µ)) + β(cos(µ) sinh(µ) − sin(µ) cosh(µ)) (8.52)


271

where α and β are defined as: α = EI

µ3 , L3

β = M ω2 − jRω

(8.53)

(8.54)

This model allows us to calculate the mechanical transfer function between the clamped end and the interaction end of the cantilever. A frequency analysis provides the resonance frequency and quality factor for each mode versus the fluid characteristics. For undamped systems, the deflection y(x) is real and the phase is a discontinuous function of x, with a π step at each vibration node. Here, the deflection is complex, which means a vibration phase rotation along the cantilever. Thus, the deflection can be written: y(x) e jωt = |y(x)| e j(ωt+θ(x))

(8.55)

Figure 8.24 shows the deflection mode 2 calculated for silicone oil at resonance. By solving the inverse problem, it becomes possible to compute the fluid characteristics from the measured data (resonance frequency f and Q-factor). An algorithm based on the model returns the terms η and ρ, which corresponds to the measured f and Q. Calibrations of unknown model parameters are performed beforehand with reference liquids. Fig. 8.24. Theoretical deflection of the cantilever for silicone oil

8.4.3.3 Results Figure 8.25 presents the schematic design of the set-up used. The optical beam is guided by an optical fiber and is focused on the top of the cantilever by a selfoc lens. The interferometer, used with demodulation electronics, is used as a noninvasive sensor to detect the magnitude and the phase of motion of the vibrating cantilever. The signal at the probe output is averaged out using a double-phase lock-in amplifier.


272 Fig. 8.25. Experimental set-up used to perform the measurements

Fig. 8.26. Amplitude response for three different fluids

Table 8.3. Comparison between literature values and experimentally deduced values for viscosity and density for different fluids Sample

Viscosity (kg m−1 s−1 )×103

Density (kg m−3 )

Experimental Viscosity×103 Density

Relative errors δη (%) δρ (%)

1-Propanol Water NaOH 30% Silicon oil

2.23 1.00 13.5 100

803 998 1328 980

2.17 1.02 13.3 103

2.7 −2 1.5 −3

793 1038 1304 943

0.9 −3.8 1.8 3.8

A low frequency generator with a weak signal excites the electromechanical resonator to about 1 mV amplitude. The magnitude and phase of the displacement at the cantilever end are probed, digitized and stored in a microcomputer. Figure 8.26 represents the experimental resonance shapes recorded with the optical probe for three different fluids. The input data to the algorithm ( f and Q) were obtained by fitting the resonance shapes with a second-order transfer function. Table 8.3 summarizes the experimental measurements for various fluid samples and compares them with known data [61]. These results show that the experimental and the theoretical values are in good agreement, demonstrating that the developed model describes the behavior of our microsensor well. The sensor global accuracy is estimated to be about 5% in the following measurement range: viscosity of 1–100 cP


273

and density of 500–1500 kg m−3 . Typical sensor resolution (in water) is about 3% of the range for viscosity and 0.6% for the density. 8.4.3.4 Wall Effects The last part of the study concerns measurements performed when the sphere oscillates in the vicinity of a rigid obstacle immersed in a fluid, in order to quantify the investigation depth of the sensor. Figure 8.27 shows the real and the imaginary part of the drag force when the sphere approaches, respectively, a laterally immersed plane wall (parallel to the sphere oscillating direction) and a horizontally immersed plane wall (perpendicular to the sphere oscillating direction). The forces exerted on the sphere were computed by solving the inverse problem. As previously stated, experimental data were acquired with an interferometer. This study shows that the presence of a bounding wall introduces a measuring error of less than the sensor resolution if the sphere is at least 1 mm away from a horizontal wall and 0.3 mm from a lateral wall in silicone oil. The volume corresponding to the smallest cylindrical tank is about 3 µl in this case. In the case of quasiuniform motion, the correction of Stoke’s law necessitated by the presence of a plane boundary depends only on the geometrical parameters [62]. When the sphere vibrates, the correction is also related to the penetrative length, and thus to the characteristics of the investigated fluid. For water, for example, the tank could be made smaller with no ill effects. In conclusion, we have developed a sensor that allows us to obtain the fluid density and viscosity simultaneously by measuring the frequency shift and the Q-factor of an electromechanical resonator. The use of near-field acoustics allows us to benefit from a small investigation depth at low frequency, unlike microbalance-based techniques where the sensitive area is a surface and the acoustic energy can be radiated throughout the volume of the liquid if the surface is not smooth. In our set-up, the volume of fluid required to determine mechanical characteristics can be as small as 3 µl. Thus, biological and medical applications where only small quantities of fluids are available can be considered. We have developed a model describing the behavior of the microsensor. This model uses a linear approximation of the Navier–Stoke’s equation. The cantilever was studied as a continuous medium in forced motion with

Fig. 8.27. Interaction forces in silicone oil when the sphere approaches a wall

274


an external disturbing force. This model enables us to estimate the deflection of the cantilever versus the frequency, and to extract the viscosity and the density from measurements of the frequency shift and the Q factor. A good agreement has been obtained between theoretical and experimental values. Moreover, complete modeling of the system allows the geometrical parameters to be optimized for a given measurement range. After the complete system operating in a closed loop (oscillator) was calibrated, it could be used as a microsensor to characterize the densities and viscosities of microvolumes with relatively good accuracy, estimated to be better than 5%. This microsensor could become an interesting tool in the biological and medical domains.

8.5 Ultimate Metrology: Measurements at the Mechanical Noise Level It is known that the thermal vibrations of the cantilever are a fundamental source of noise in atomic force microscopy [63–65]. At room temperature, the thermal noise induces thermomechanical oscillations in microcantilevers, which are of a Brownian motion type. However, these thermomechanical oscillations can be analyzed in order to obtain information about the tip–sample interaction. Several applications have been presented in the last few years that demonstrate the potential utilization of thermomechanical noise. A common application of thermomechanical noise analysis is the fast nondestructive calibration of the cantilever spring constant [66–69]. Recently it has been shown that thermomechanical noise can be used to determine oscillatory hydration potentials [70] as well as to measure viscoelastic properties of polymers [71–73] and noncontact friction between closely spaced bodies [74]. In all of these publications, the authors describe the evolution of the resonant frequency of the microcantilever through the power spectrum density in the noncontact case and for gas/liquid interactions. The standard optical level technique was used for sensing. They observed that it is impossible to use this technique to detect the resonant frequency in the case of contact with hard samples because the ultimate sensitivity of standard optical probes is too weak. The first study presenting the thermomechanical noise of a surface-coupled cantilever at room temperature was published by Stark et al. [75]. They investigated mechanical sample properties by analyzing the thermomechanical noise of the first antisymmetric eigenmode of the cantilever. In this section, we present measurements of the thermomechanical noise of a cantilever while it is in contact with hard samples. The spectral power density measured with a specific probe clearly shows the resonant frequency of the first flexural mode of vibration of the cantilever. We demonstrate that it is possible to investigate the local elastic properties of hard samples without excitation. The microcantilever can be considered to be a rectangular beam defined by its geometric parameters (L: length, w: width, t: thickness), the Young’s modulus and the quality factor. At temperature T , the cantilever is subject to a random timedependent Brownian motion around its equilibrium position, which is usually called mechanical noise. The noise spectrum S(ω) can be measured using a spectrum


275

analyzer: 1 S(ω) = 2π

+∞

s(t) e− jωt dt

(8.56)

−∞

It is possible to rewrite the noise spectrum of the n th eigenmode of the cantilever by using the dissipation-fluctuation theorem [76] and expressing the rms noise of the measured in bandwidth B [28, 76]: ⎡ ⎤1/2 2 2k B TBω + ,⎦ , (8.57)

Sn (ω)1/2 = ⎣ 2 2 Qkn ω2 − ω2n + ωQn2ω where kB is the Boltzmann constant, T is the temperature, Q is the quality factor, kn is the mode-dependent spring constant, and ωn is the resonant frequency of the n th mode. The spectral density of the thermomechanical noise of the microcantilever was first measured with the tip free and in contact with a silicon sample. Figure 8.28 shows the experimental system used. All of the experiments were performed with a single micromachined silicon cantilever with a rectangular cross-section and a diamond tip of 2 µm radius, bonded to the free end with cyanoacrylate adhesive. The cantilever dimensions were: length L = 8 mm, width w = 456 µm and thickness t = 148 µm. The displacement amplitude of the tip was measured with a high-sensitivity heterodyne laser probe [42, 43], as previously described. After demodulation of the photodetected signal, the frequency spectra of the thermal noise were measured with a spectrum analyzer (HP 35670a).

Fig. 8.28. Principal of the setup


276 Fig. 8.29. Noise spectra for the first flexural vibration mode in the case of interaction with a silicon (100) sample. The three curves indicated the different cases of loading force P (diamonds 8.6 mN, triangles 750 µN, circles 170 µN)

In a first set of experiments, we detected the vibration modes without any tip– sample interaction. The resonant frequencies measured are in good agreement with the calculated ones. Moreover, the difference between these values and the results obtained using active excitation (piezoelectric transducer, PZT) are less than 3%. (see Table 8.4). Table 8.4. Comparison of calculated resonant frequencies with those measured using the noise spectrum and those measured using PZT excitation Mode 1 (Hz) Mode 2 (Hz)

Calculated 1954 15,441

Noise 1912 14,856

PZT 1896.5 14,577

In the second set of experiments, we took measurements with the tip in contact with a silicon sample. The amplitudes of the thermomechanical noise of the cantilever for several loading forces on the silicon sample are shown in Figure 8.29. As predicted, the resonant frequency increases as the force increases. Note that we have considered the cantilever to be a beam interacting with the sample though two variable springs. The other end of the beam was modeled with a clamped end (Fig. 8.30). We performed the measurements for two different samples of silica and silicon

100. Figure 8.31 shows the responses for the two samples versus the interaction force. As expected, the resonant frequency is higher on the silicon sample, which has a higher Young’s modulus (169 GPa) than the silica sample (71 GPa). Moreover, the trend in the average frequency shift obtained here is consistent with the trend measured using the active method [77, 78]. These results demonstrate the relationship between the local elastic properties of the samples and the resonant frequency shift. Figures 8.32 and 8.33 present comparisons between the detected frequencies and the theoretically evaluated frequencies for two samples made of silicon and silica.


277

Fig. 8.30. Model used in thermomechanical noise experiments

Fig. 8.31. Comparison between the silicon and the silica response versus the loading force on the sample (diamonds silica, squares silicon)

Fig. 8.32. Comparison between the fitted model and experimental results in case of interactions with a silicon sample

These first results demonstrate that the detected frequencies are of the correct order of magnitude, and that the frequency shift follows increasing stiffness as in the Hertzian model of elastic contact. The results confirm that, by solving the inverse

278


Fig. 8.33. Comparison between the fitted model and experimental results in case of interactions with a silica sample

problem with the proposed model, this passive method can be used to quantify the local elastic properties of the samples. We have shown that it is possible to detect the thermomechanical noise of a microcantilever in contact with a stiff surface, even for high loading forces. This method provides an interesting alternative approach to determining elastic sample properties by scanning force microscopy. The use of this passive thermal excitation allows us to minimize the tip amplitude and thus avoid nonlinearities and friction effects in the tip–sample contact. Moreover, classical experiments have shown that passive flexural overtone microscopy can provide complementary information on the mechanical contact properties. For a complete analysis of the contact properties, other resonances will also need to be analyzed. It is also possible to make use of the method to obtain information on viscoelastic properties.

8.6 Conclusion The use of ac force in near-field acoustic/mechanical microscopy provides an interesting tool for the imaging or measurement of local elastic properties on micron or submicron scales. On the mesoscopic scale, SMM has given a spatial resolution of between 100 nm and 1 mm in imaging mode, but the quantification of local elastic properties is more difficult below 100 nm because of the unknown geometry and the different physical interactions related to adhesion, atomic forces and the presence of adsorbed layers in ambient air (adsorbed liquid layers can be detected on the nanoscale on any surface after a few minutes in ambient air). On micron and mesoscopic scales, accurate measurements of elastic constants can be achieved without damaging the investigated surface, depending on the relative adhesion between the tip and the specimen (nevertheless, the investigated surface must be polished and the tip curvature must be known). The Hertzian theory has been successfully validated on this scale and used to model the stiffness from the tip–sample interaction. A specific heterodyne optical probe has been added to the set-up. This combination has enabled us to completely quantify the motion of the cantilever, tip and sample (the sensitivity of the optical probe is typically below the mechanical noise level). Mapping the cantilever has highlighted the large displacement of the


279

piezoelectric transducer end, corresponding to bad cantilever clamping. The results obtained yield a more complete mechanical model of the cantilever that is more representative of the physical shapes of the resonance modes. The experiments have shown that the experimental spatial resolution is approximately equal to the radius a of the contact area between the tip and the sample. Because typical frequencies are in the 10–100 kHz range, SMM permits resolution better than l/10,000 which corresponds to an unusually low value for near-field techniques (this value justifies the quasistatic approach of the model). Moreover, the investigation depth has a surprising large value due to the good sensitivity of the microscope head (the frequency can be detected very accurately): subsurface reflectors at distances of up to 50 µm have been detected (the detected depth can reach a few tens of the contact radius; conversely, small subsurface defects are not detected because the surface stiffness does not vary enough). Both the heuristic model and the mechanical approaches show that subsurface planes normal to the displacement of the piston are detected with an image contrast that depends on h −3 , where h is the depth of the “reflecting plane”. If the thickness h is large enough (typically above 50 µm, depending on the tip radius and the applied force), the simulated subsurface defect is not detected because the resonant frequency shift is smaller than the frequency noise. The SMM is sensitive to both local stiffness and surface/subsurface inhomogeneities; this microscope can therefore be used to measure the stiffness of a thin layers (or its thickness if the Young’s modulus is known). Because the cantilever is very sensitive to the interaction force, the microscope head can also be used to measure viscous properties and densities of fluids. We have demonstrated a set-up that was used to prove the accuracy of the method. In our first approach, micron-sized probes were used, but these techniques could be transferred to the mesoscopic scale. The properties of both solids and liquids can be quantified with the resonant cantilever, depending on the probe used. In our discourse on ultimate metrology, we showed how it was possible to overcome a great challenge: the passive measurement of elastic constants using the mechanical noise of the canitlever. The mechanical noise is detected with the optical probe and the noise response of the microscope head (with the tip in contact with the sample) enables us to estimate the local stiffness by solving the inverse problem. This new approach to passive local characterization should now be developed in other domains of near-field microscopy.

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9 Electrostatic Force and Force Gradient Microscopy: Principles, Points of Interest and Application to Characterisation of Semiconductor Materials and Devices Paul Girard · Alexander Nikolaevitch Titkov

In this paper, the physical principles of local electrical observations and measurements and related analytical formulas, as well as the optimal experimental conditions for electrical measurements, are first reminded. Secondly, it is shown, based on experiments and analytical expressions, how the observations obtained with the methods of force and force-gradient microscopy can be analysed. The advantage of the forcegradient microscopy over the force microscopy in better resolution and precision of measurements is demonstrated. The related analytical formulas are given, which explain the advanced capabilities of the force-gradient microscopy for precise electrical studies on local objects on the subnanometer scale. Finally, based on recent published data, the main applications of the electrostatic-force and force-gradient microscopy for analysis of materials and devices are briefly considered.

Abbreviations a AM AΩ A2Ω b C C = ∂C/∂z, C = ∂ 2 C/∂z 2 , Ccone , Ccanti ; Capex , Ccone , Ccanti Capex

C1 , C1 C2 , C2

damping maximum mechanical amplitude maximum electrical amplitude at pulsation Ω maximum electrical amplitude at pulsation 2Ω coefficient tip to sample capacitance which is composed of two capacitors in series: the tip to surface plus the surface to bulk capacitors. first and second derivatives of the tip to sample capacitance in the direction perpendicular to the surface, or capacitance coupling capacitance couplings with an infinite conducting surface and the apex, cone and cantilever parts of the sensor capacitance coupling with the area of interest (centre of a track or disk) or, in abbrevij ation, C1 , j = 1 or 2, respectively capacitance coupling outside the area of interest (centre of a track or disk) or, in abbrej viation, C2 , j = 1 or 2, respectively

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284

dC , dC dFdc , dFΩ , dF2Ω ; Fdc , dFΩ , dF2Ω dV0 d Dc , Darea Fapex , Fcone , Fcanti FM Fz Fdc FΩ F2Ω Grad Fz =

∂Fz ∂z

Grad Fdc = ∂F∂zzdc Grad FΩ = Grad F2Ω = ∂F∂zz2Ω H k K K1 m P(x, y) Q Qs R RaF , RaFG V0 Vdc Vdc eff = V0 + Vdc Vac z ∆z

∂FzΩ ∂z ,

small capacitance coupling variations small variations of force or force gradient dc or at pulsations Ω or 2Ω, respectively small surface voltage variations local sphere to sample distance diameter of a disk on sample surface totally or partially coupled electrically with the apex surface Parts of the electrical force applied on the apex, cone and cantilever, respectively mechanical force electrical force normal to the sample surface dc electrical force force at electrical pulsation Ω, force at the double of the electrical pulsation Ω first derivative of the electrical force in the direction of vibration (perpendicularly to the sample surface) first derivative of the electrical forces (dc, Ω, 2Ω respectively), in the direction of vibration (⊥ to the sample surface) tip cone height sensor stiffness ratio between apex and cone capacitance couplings = ratio between measured and real voltage differences sensor effective mass local electrostatic pressure quality coefficient of the resonance local surface charge tip apex radius lateral resolution obtained in force or force gradient tip to sample voltage due to work function differences (and sample polarisation) dc tip to sample voltage externally applied efficient dc voltage ac tip to sample voltage externally applied (pulsation Ω) mean tip to sample distance sum of the different amplitudes of vibration (mechanical and electric)

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measured and real voltage difference between a small track or disc and outside (infinite plane) real voltage difference between a small track or disc and outside (infinite plane) electrical pulsation of the ac voltage

9.1 Introduction Many local methods of Scanning Force Microscopy originated from the original Atomic Force Microscopy (AFM) [1]. Among them, we can mention the detection of long-range forces, such as magnetic [2] or electrostatic forces. The detection of local electrostatic effects involves charges [3–5] or dc voltages [6]. Such observations and measurements are of particular fundamental and technological interest for the physical and technological analysis of a variety of new solid materials and structures, especially modern or even futuristic semiconductor device structures. What is very important is that it is possible to characterise devices under working conditions at low [7] and high frequencies [8–11] operation, which opens up the way to optimisation of devices, as well as to analysis of their failure. Since the realisation of new structures of reduced size, i.e., in the nanometer range, is well underway, the electrical microscopy is forced to be able to keep pace with the development of these new and interesting devices, on which, fundamental studies using electrostatic microscopy can also be foreseen. A review of the domain appears necessary. In this paper, we first intend to remind the principles of operation of the electrostatic-force (EFM) and force-gradient (EFGM) microscopy methods in order to obtain local electrical data, i.e., local voltages and capacitances. Secondly, we shall illustrate by experimental data their main possibilities and compare them. Guidelines to interpret the EFM and EFGM image contrasts shall be also proposed. Thirdly, based on the recent experimental results demonstrating the advantage of EFGM over EFM, the advanced capabilities of EFGM will be discussed in terms of sensitivity and spatial resolution. Finally, before the conclusions, different applications of these methods will be shown and discussed. Since the local surface voltages are sensitive to the surface work functions, the possible applications of EFM and EFGM are seen in classification of materials and in analysis of elementary structures of device architecture. In addition, the fact that voltages resulting from the action of external factors, such as biases, can be explored, makes it possible to study working devices as well.

9.2 Principles Since the action of electrostatic forces extends over long distances, all the metallized parts of the AFM sensor have a priori to be taken into account. These parts are the cantilever plane, the surfaces of the tip cone and the tip apex. The sensor-to-sample


286

structure behaves like a capacitor, with the sensor being the mobile plate of this capacitor. Under a tip-to-sample voltage, the resulting attractive electrostatic force bends the cantilever, this bending being detectable with an Atomic Force Microscope. In addition, action of the electrostatic force on the sensor mechanically oscillating near the resonance frequency decreases the sensor stiffness relative to its nominal value for free oscillations. In the linear regime, i.e., when the oscillation amplitude remains much lower than the mean tip-to-sample distance, this change in stiffness is equal to the force gradient in the Z direction of oscillations [12]. This results in a modification of the resonance frequency which gives a second way to sense tip-to-sample voltages. The electrical methods based on the detection of the force or the force gradient are called the electrostatic force and force gradient microscopies, i.e., EFM and EFGM, respectively. In the following, it is demonstrated how and under what conditions electrical measurements with AFM can be undertaken with a success. 9.2.1 Basic Relations The primary parameter in electrical microscopy is the tip-to-sample voltage V which can be written as V = (V0 + Vdc ) + Vac sin Ωt. Here, V0 corresponds to the existing tip-to-sample work function difference, to which externally induced voltages (by polarisation, illumination, mechanical stress, etc.) can be added, whereas Vdc and Vac are external biases applied to the tip. Under the assumption that the tip-to-sample capacitance C and its derivatives C = ∂C/∂z and C = ∂ 2 C/∂z 2 remain unchanged upon addition of external voltages, which is generally the case even on semiconductors at low tip-to-sample electric fields, the following relationships for electrostatic forces and their gradients can be derived. They demonstrate the dc and ac contributions, the latter at frequencies Ω/2Π and 2(Ω/2Π): F = Fdc + FΩ + F2Ω GradFz = GradFdc + GradFΩ + GradF2Ω

(9.1)

where Fdc = 1/2C ((V0 + Vdc )2 + Vac2 /2)

GradFdc = 1/2C (V0 + Vdc )2 + Vac2 /2

(9.2)

In 9.2, the first term is related to the effective value of Vdc,eff = (V0 + Vdc ) and, consequently, depends on the surface voltage. At the same time, the second part only depends on the effective value of the ac voltage applied. FΩ = C (V0 + Vdc )Vac sin Ωt GradFΩ = C (V0 + Vdc )Vac sin Ωt

(9.3)

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If a counter voltage is applied, such as FΩ = 0 or GradFΩ = 0, then V0 = −Vdc and the local voltages can be measured. This procedure corresponds to the Kelvin force (KFM) or force gradient (KFGM) measurements, respectively. F2Ω = −1/4C Vac2 cos 2Ωt GradF2Ω = −1/4C Vac2 cos 2Ωt

(9.4)

From 9.4, the possibility of imaging of the tip-to-sample capacitance can be foreseen. That capacitance is the result of two in-series connected elements, i.e., is composed of the air-gap tip-to-surface and the surface-to-bulk capacitors. So, imaging of the subsurface capacitors is also possible. There is a certain parallelism between the EFM and EFGM methods. However, there are also important differences, particularly those concerning the measurement procedure, sensitivity and lateral resolution, especially for low dimensional structures and devices. We shall examine this point later. 9.2.2 Principles of Surface-voltage Measurements Equations (9.1) to (9.4) show that electric effects can be detected using either EFM or EFGM. However, quite frequently the main interest is in the detection of surface voltages by the Kelvin method, i.e., by KFM or KFGM. So, it is useful to consider the optimal conditions under which voltage data can be obtained. 9.2.2.1 Optimal Conditions of EFM and EFGM To obtain electrical data, the AFM sensor has to be excited by electrical or mechanical forces. Since such a sensor can be approximated by a mechanical mass-spring oscillator (see Fig. 9.1), it behaves like the well-known harmonic oscillator when the excitation force is independent of the oscillation amplitude. This condition is generally fulfilled for a mechanical force. However, an electric force, by contrast, increases as the tip-to-sample distance z becomes shorter. Then, depending on the amplitude of mechanical oscillation, A M , relative to the mean tip-to-sample distance z , two oscillation modes can be defined. When the amplitude of mechanical oscillations, as well as those of electrical oscillations, i.e. A M , AΩ , A2Ω , are much smaller than z,

Fig. 9.1. Mechanical spring-mass oscillator modelling the AFM sensor


288

an electrostatic force acting on the sensor can be considered nearly constant, and the harmonic-oscillator formalism is well satisfied. This so-called linear regime allows a precise description of the sensor behaviour in an electric field. As well, the sensor can also operate in the non linear regime, in which the mechanical amplitude is nearly equal to the tip-to-sample distance. In the non linear regime, the equations of sensor motion are not straightforward and sensor behaviour should be considered additionally. In the linear regime, the electric forces can be expanded in a Taylor series around the mean tip-to-sample distance z and the equation for the sensor oscillation amplitude ∆z can be written as ∂∆z ∂ 2 ∆z +a + [k − (GradFdc + GradF2Ω + GradF2Ω )]∆z ∂t 2 ∂t = FM sin ω M t + FΩ + F2Ω

m

(9.5)

where ∆z, m, k, a, ω M , FM are the sum of the amplitudes of mechanical and electrical oscillations, the sensor mass and stiffness, the damping coefficient, the mechanical frequency and the force, respectively. Then, two extreme and optimal conditions under which electrical measurements are possible can be derived. First, if the gradients on the left-hand side of (9.5) are negligible, compared with the sensor stiffness k, the sensor is subject to mechanical and electrical excitations and the resulting movement is simply a superposition of three independent movements at the frequencies ω M , Ω and 2Ω. The detection of sensor oscillations induced at the electrical frequencies enables EFM measurements. Secondly, if the electric forces induce only small movement amplitudes of the sensor, i.e., well below the mechanical amplitude, they can be neglected on the right-hand side of (9.5). Then, if the electrical gradients induce detectable changes in the mechanical resonance frequency by causing a phase shift of the mechanical oscillation, this makes possible EFGM measurements [13]. We have defined the conditions in which the sensor movements are simple and a linear relationship with electrical data can be established. First, when detecting electric forces (EFM), the amplitudes of electrical oscillations, AΩ , and A2Ω , must be kept smaller than the mean tip-to-sample distance z. Secondly, when detecting gradients (EFGM), the fact that a mechanical oscillation is required imposes additional conditions: i) the amplitude of the mechanical oscillation, A M , must be smaller than the mean tip-to-sample distance z and ii) the amplitudes of electrical oscillations must be smaller than the amplitude of the mechanical oscillation, i.e., AΩ , A2Ω A M . These optimal conditions have to be kept in mind when performing electrical measurements. In electrical measurements, the tip-to-sample distance should be kept constant during surface scanning if an analytical description of the results is desirable. Satisfying this condition also leads to several additional improvements. First, the Kelvin probe measurements presuppose that the first and second derivatives of tip-to-sample capacitance C or C are kept constant during scanning and do not change appreciably when nullifying the dc voltage (V0 + Vdc ). The reason is the following. The Kelvin feedback loop yields a counter voltage Vdc to make the product C (or C ) (V0 + Vdc ) ∼ = 0. If this zero is technically not perfect over the entire surface,

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changes in the capacitance coupling can give rise to surface voltage artefacts. Secondly, in electrical measurements the distance z is a key factor determining the minimum detectable voltage and spatial resolution, which are proportional to z 2 √ and z, respectively. These points shall be explained in Part IV, Sect. 4.2 and Sect. 4.3, respectively. Finally, since the air-gap capacitor remains connected in series with the sub-surface capacitor, a constant tip-to-sample distance is mandatory when the sub-surface capacitance imaging is required. In conclusion, a constant tip-to-sample distance z appears highly desirable for good-quality electrical measurements. Fortunately, this condition and the earlier mentioned relationships between the amplitudes of mechanical and electrical oscillations can be quite well satisfied in the existing double and single pass modes of AFM electrical measurements. In the double pass mode, a double line scan makes it possible to obtain: i) in the first scan, the surface topography in the intermittent-contact mode (so-called TM mode) and ii) in the second scan, electrical data by keeping the mean tapping tip-to-sample distance constant at some chosen value z 0 . The distance z 0 is computer controlled by moving the tip up and down relative to the sample surface, following the topography profile determined in the first scan. As the mechanical and electrical amplitudes and dc voltages can be independently controlled during the two scans, the linear conditions explained above can be obtained, especially in EFGM [13]. In the single pass mode, the surface topography and electric forces are simultaneously measured and the control of z is performed automatically. 9.2.2.2 The AFM Topography Background It should also be explained why the AFM topography data can define the mean tipto-surface distance when the sensor is mechanically oscillating at the sample surface in the presence of an electric field. The AFM topography is generally obtained in the non linear regime. However, in order to describe the fundamental aspects of the AFM experiment, with an emphasis on the behaviour of the sensor, we start from the linear regime, which allows us to foresee also the main features of non linear regime of sensor oscillation in close proximity to the surface. In accordance with (9.5), the main effect of a gradient, assuming that ω0 only to the dc component is present, is to shift the resonance frequency from 2Π !& ' ω0 dc . Since an attractive or a repulsive force induces a positive or 1 − GradF 2Π k negative gradient, respectively, an attractive force leads to a decrease in the resonance frequency, and conversely. As a result, depending on the presence of attractive or repulsive forces, the phase shift Φ decreases or increases, respectively. In this case, the sign of the phase shift is a signature of the nature of the main tip-to-sample interaction involved. The consequences for the mechanical amplitude and phase are shown in Fig. 9.2a and Fig. 9.2b, respectively. It is important to note that an attractive gradient has an opposite effect on the mechanical amplitude when the mechanical


290

Fig. 9.2. Behaviour of (a) the mechanical amplitude A M and (b) the phase Φ with the frequency of mechanical oscillation, ω M sg n the presence of force gradients. The centred curves correspond to the free resonance amplitude and phase, whereas the left and right curves are related to attractive or repulsive gradients for which the resonance is shifted to lower or higher frequencies. As shown by the arrows, an attractive force always decreases the phase, while the amplitude decreases or increases when ω M > ω0 or ω M < ω0 , respectively

frequency is set lower or higher than the free resonance value ω0 . It results in an increase or a decrease in the mechanical amplitude, respectively. Following (9.5), the mechanical amplitude A M and phase Φ (with the origin at ω0 ) are written [12]: resonance frequency 2Π A M = !& & ω20 1 − ⎛ Φ = arctan ⎝ Q

FM m

GradFdc k

& ω20 1 −

'

−

ω2M

GradFdc k

ω0 ω M

'

'2

+

− ω2M

&

ω0 ω M Q

'2

(9.6)

⎞ ⎠

(9.7)

0 where Q = mω a is the quality coefficient. The amplitude A M corresponds to a stabilised value. It must be remembered that, when such an oscillator experiences a local perturbation, for example, a sharp change in the force gradient, a transient time τ ∼ = 2Q/ω0 [14] is required to reach a new amplitude A M1 . The physical reason for such a behaviour is the following: the energy stored by the oscillator should be changed and it requires some time to reach another state of equilibrium. aconsequence, if the time interval between successive scan pixels, or the period

As 2Π Ω of an electric signal, is shorter than this transient time, the energy exchange has no chance to occur. So, a sharp lateral change in the dc voltage on the surface or voltage variation at a high electrical frequency have no appreciable effect on the amplitude of mechanical oscillations. Fortunately, the frequency or phase changes that do not require any energy transfer can be observed. Thus, since when changing from atmospheric pressure down to vacuum Q increases by some orders of magnitude, the use of the oscillator frequency is mandatory for gradient sensing and AFM operation under vacuum.

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However, it is necessary to mention that highly localised strong dc force gradients can influence mechanical oscillations if the scanning performed is sufficiently slow. The topography data can give the “signature” of such local electrical effects frequently related to charges [15], as it will be shown in Sect. 2.3. This gives an additional EFGM method for electrical imaging. In electrical measurements, it is also important to understand how the mechanical amplitude depends on the mean tip-to-surface distance in the presence of an electric field. It appears that this dependence is different for mechanical frequencies chosen higher or lower than the resonance value. With a decreasing of distances z, the dc gradient steadily grows, following the increase in electrostatic force composed of the 2 and Fcone ∝ apex-to-plane and cone-to-plane contributions [16, 17] Fapex ∝ Rz Vdc

A 2 Ln z Vdc , respectively. As a consequence, the effective resonance frequency ω0 ∼ = (

dc ω0 1 − GradF steadily decreases. If, initially, the mechanical frequency ω M is k slightly higher than the resonance value ω0 , their difference becomes larger, and, as it follows from (9.6), the amplitude A M steadily decreases to zero. In addition, the higher the tip-to-sample voltage is, the stronger the gradient becomes, and, therefore, the amplitude A M decreases further. This behaviour of the amplitude is the same when ω M = ω0 with an obvious exception that, at large distances z, the amplitude will start from a higher value. However, when ω M < ω0 , the behaviour of the amplitude is different. For example, at a frequency symmetrical to the first case, the amplitude starts from the same value at great distances and then increases up to Q FM , and, only after that, decreases to zero. This behaviour of mechanical A M = mω 0 ωM amplitude is illustrated in Fig. 9.3 where a good qualitative agreement is obtained between the analytical and numerical simulations for the linear regime of mechanical oscillations. Naturally, in accordance with the physical meaning of the experiment, the maximum amplitude A M must be smaller than the mean tip-to-sample distance z, which is really obtained when a numerical simulation is performed with the standard 4th order Runge–Kutta algorithm (RK4) [18]. At the same time, the simplified analytical model presented here can give higher values. A more realistic description of the sensor behaviour in AFM topography mode corresponds to the non-linear regime, in which the amplitude of oscillation approaches the tip-to-sample distance. In that case, physically, the mean attractive gradient is steadily increasing with the amplitude of oscillation. Anyway, at the same mean distance z, the gradient exceeds that in the linear case, so the frequency shift is more important and increases with the amplitude of oscillation. When ω M ≥ ω0 , the behaviour of the amplitude remains qualitatively the same as in the above considered linear case, with the following exceptions: i) the decrease in the mechanical amplitude is more pronounced than that in the linear regime and ii) the tip can vibrate without coming in contact with the surface, approaching it until distance of only several nanometers. The mean distance z corresponds to the amplitude of oscillations plus this “non-contact distance”. When ω M < ω0 , the amplitude is increasing under saving a non contact regime of sensor oscillations, and the tip is coming down to a very close distance from the surface, similarly to the case of linear operation. The only exception is that, here, non-linearities generate instabilities and hysteresis behaviours, i.e., the amplitude is not symmetrical when the tip approaches the surface or retracts. This case has been examined analytically taking into account attractive

292


Fig. 9.3. Mechanical amplitude A M versus tip-to-sample distance z for different mechanical frequencies ω M . The results obtained using the analytical formula and numerical simulations based on the 4th order Runge–Kutta algorithm (RK4) are reported. The calculations are performed using the following parameters: attractive electric force is calculated at Vdc = 2.5 V at sensor tip height H = 15 µm, half-cone angle θ = 0◦ , apex radius R = 20 nm, sensor spring stiffness k = 3 N/m, quality coefficient Q = 100, resonance frequency ω0 = 100Π103 rad/s and mechanical amplitude A M0 = 20 nm [From 18]. To make a comparison between the calculated value of mechanical amplitude and mean tip-to-surface distance z, the position of the sample surface is indicated by a dotted line

forces acting during the entire period of oscillation. They are the Van der Waals [19] and electric forces [18], which are generally encountered first when the tip-to-sample distance starts to decrease. At the highest amplitudes, repulsive forces are added. They act only during a part of the period of oscillation, when the tip comes in an intermittent physical contact with the surface and somewhat penetrates into the sample. This corresponds to the well-known “tapping mode”, which has proved to be of interest in imaging of the morphology or mechanical properties of a sample [20,21]. In the case in question, the mean tip-to-sample distance is virtually equal to the amplitude of oscillation, if the penetration is assumed negligible, which is the case for materials with a high Young modulus, like semiconductors. Anyway, in the non linear regime, changes in the phase shift appear to be a very discriminative test of the attractive or repulsive nature of the main force involved, as in the linear regime [21]. In ambient AFM, during scanning, the feedback loop maintains constant the mechanical amplitude by attracting or retracting the tip from the surface. As the loop works under the assumption that the amplitude increases with the tip-to-sample distance, there are three possible working conditions. In vacuum AFM, since the feedback loop maintains constant the shift of the resonance frequency, the first working condition is mainly used. If the mechanical excitation is chosen equal or above the resonance frequency under an attractive force, a true non-contact mode occurs, with the mean tip-to-surface distance z stabilised at values exceeding the mechanical amplitude by several

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nanometers. This corresponds to the standard condition used in single-pass EFM measurements. If the mechanical excitation is chosen below the resonance frequency and strong attractive forces are present, a true non-contact mode also can take place, but the minimum tip-to-sample distances are much shorter than those in the above case. However, when attractive forces are weak, the tip comes in intermittent contact with the surface and experiences repulsive forces. In this case, the mean tip-to-sample distance is determined by the amplitude of oscillation. This last working condition corresponds to the first scan of the double-pass EFGM and the mean distance of the second pass is controlled via an additional distance change, referring to the tapping amplitude in the first scan. 9.2.2.3 EFM and KFM The EFM in non contact single pass mode is usually obtained keeping the mean tip-to-sample distance constant under mechanical amplitudes set between 10 and 50 nm, and the minimum tip-to-sample distances, around several nanometers. Then, the main electric force is related to the cone-to-plane capacitance coupling, which has already been evidenced by the experimental behaviour of the force as a function of distance and by modelling of this behaviour [16, 17, 22, 23]. At the same time, the main gradient is related to the apex-to-plane capacitance coupling and depends on the ac and dc voltages. Concerning the ac gradients, the transient time of the stabilisation of the mechanical amplitude, τ = 2Q/ω0 ≈ 1 ms (ω0 ≈ 50 kHz and Q ≈ 150), is fortunately considerably longer than the electrical period chosen to be around 0.1 ms, i.e., 2(2Π/ω0 ) < 2Π/Ω < 20(2Π/ω0 ). Then, generally, the modulations of the mechanical amplitude remain unlikely, except under the particular conditions mentioned later. This conclusion is strengthened if the mechanical excitation of the sensor occurs slightly above the resonance, where the mechanical amplitude is only slightly dependent on the shift of the resonance frequency. Concerning the dc gradient (see (9.2)), changes in the dc voltage (ca. 1 V) on the sample surface can be kept below the effective ac value, so that the gradient remains nearly constant when scanning, or it leads only to small variations of the mean distance z (several nanometers). As another consequence, the perturbation of the topography data remains low when the Kelvin loop is working, but it is not improbable that changes in the local capacitance could influence the topographic data. In general, the experimental conditions should be always adjusted in order to keep the topographic data independent of the local voltages and, consequently, to maintain the tip-to-sample distance (mean and minimum) constant in the course of scanning. In the EFM measurements, the surface-to-tip cone interaction gives the main contribution to the electrostatic force.

In this case, the tip-to-surface capacitance is dominated by the term C ∼ = log Az and can be considered constant and virtually independent of the small local changes in z due to local gradients. As only the dc gradient affects the mechanical amplitude (see (9.5)), the sensor behaves as if the

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mechanical and electrical oscillations, i.e. A M , AΩ , A2Ω , were independent and superimposed. However, for the extreme conditions, i.e., in the case of high voltages, appreciable local charges and large apex radius or flattened tips, the influence exerted by the electric gradient on the mechanical amplitude is stronger than that of the other forces (Van der Waals, etc.). Such giant dc gradients can hide the local topography. In addition, some amplitude instabilities become possible because of a decrease in the time of mechanical stabilisation. This decrease may be due to a reduction in the quality coefficient by an extra damping induced by the air sheet localised between the flattened tip apex and the sample [24]. Such exceptional cases are generally avoided, which is confirmed by published results of electrical observations and measurements concerning capacitors or surface voltages, and also by the possibility of obtaining electrical data simultaneously with the topography on the same scan line [25–28] and [29–35], i.e., in the single pass mode. Since, the tip cone-to-surface capacitance coupling, which has a moderate level of localization of the interaction, is involved in force detection, the use of EFM and KFM remains essentially valid for observation and measurements of voltages on areas only in the submicrometer range. When a higher spatial resolution is necessary, i.e., if the nanometer range is considered, we have to find conditions in which the tip cone-to-sample capacitance would be reduced and the interaction would be concentrated in the region of the sensor situated at a closer proximity to the surface, i.e., at the tip apex. The way to obtain these conditions is to reduce the tip to sample distance and to work with the force derivative, i.e., with the force gradient. This leads to EFGM and KFGM. 9.2.2.4 EFGM and KFGM For the double pass mode, if the optimal linear conditions are satisfied and using low electrical frequencies, consideration of (9.5) gives the expression for the total variation of phase : & ' ⎞ ⎛ Ω +GradF2Ω − ω2M ω20 1 − GradFdc +GradF k ⎠ (9.8) Φ = arctan ⎝ Q ω0 ω M If the mechanical excitation occurs at the resonance frequency, i.e. ω M = ω0 , then GradFdc + GradFΩ + GradF2Ω (9.9) Φ = −Q k A suitable selection of the signal frequency allows us to detect any of the electrical gradients mentioned in (9.2) to (9.4). This is illustrated in Part III. 9.2.3 Detection of Strong Local Electrical Effect via the “Topographic” Data Since the topography is obtained with the mechanical amplitude maintained constant by changing the tip-to-sample distance by means of the feedback loop, the

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Fig. 9.4. Schematic of the electrostatic interactions for the cases when only a static charge is present on the surface (a) and that when an additional voltage V is applied (b)

topographic data can be changed by the presence of strong local voltages or charged areas. More precisely, the closed loop raises the mean tip-to-sample distance over a charged area in order to cancel the decrease in the local amplitude of mechanical oscillation, caused by electric gradients. Such an evidence has been reported recently [15]: the detected height of charged nanocrystals exceeded the real value by several nanometers. It can be mentioned that, in the presence of a local charge on an insulating surface, the dc force is composed of a capacitance component, as written in (9.2), to which Coulomb forces are to be added. In the presentation of Fig. 9.4, Coulomb forces are the forces between the surface charge Q s and i) its image charge −Q s on the tip, with the image charge on the conducting substrate plate underneath the insulator neglected, and ii) the tip charge resulting from the dc voltage (V0 + Vdc ) between the tip and the substrate plate. Then, the dc gradient Grad Fdc can be written as: GradFdc = (Vo + Vdc )[Adc ] + 1/4∂ 2 C/∂z 2 Vac2 + 2Qs2 /(4Πεoz 3 ) with:

Adc = 1/2∂ 2 C/∂z 2 (Vo + Vdc ) + [Q s /(4Πε0 z 2 )][2C/z − ∂C/∂z]

(9.10)

where Q s is the local surface charge, C is the tip-to-back substrate capacitance, z is the mean tip-to-surface distance and ε0 is the permittivity of free space. The influence exerted by a local charge on AFM topographic images of a flat surface is illustrated in Fig. 9.5, in which contact and non-contact topographic AFM and EFM force images are presented for the surface of a SiO2 layer with an artificially created network of local charges. The charges were deposited by bringing the AFM tip in contact with the oxide layer surface and by applying (−4 V) to the tip. The topographic image taken in the contact mode (a) shows the real morphology of the oxide surface, which is rather flat with a roughness of 1–2 nm.The so-called “topography” obtained in the non-contact mode (b) reveals the appearance on the surface of a network of rather high (10 nm) regularly distributed artificial hills. The locations of these hills well coincide with those places on the surface where an intentional local charging was initially performed, and this charging is confirmed by the observation in the EFM force mode (c). These striking observations show how the amplitude of mechanical oscillations can be sensitive to local electric effects due to the electric gradients.

296


Fig. 9.5. AFM contact (a) and non-contact (b) and EFM (c) images and profiles (white lines) of the surface of nano-thin (20 nm) SiO2 layer on the Si surface with a charged network on it. Experiments are performed in a vacuum (residual pressure 10−5 Torr)

9.2.4 Conclusions The principles of electrical measurements have been reminded and explained. First, the basic relations have been presented, which show the relationships existing between the local electrical parameters, i.e., the tip-to-sample voltages and capacitances, and the experimental data that can be obtained on an AFM set-up, i.e., the cantilever bendings and the resonance frequency shifts. Secondly, we have exposed under which optimal conditions the AFM experimental data have the maximum chance to satisfy these relations. These conditions define the small signal approximation, which is to be implemented experimentally: the amplitudes of mechanical and electrical oscillations must be lower than the mean tip-to-sample distance, i.e., the oscillations occur in the linear mode. In addition, they expect the stability of the air-gap capacitance couplings during scanning, which is achieved under a stable tipto-sample distance. This stability of the local distance can be obtained with an AFM in the two main modes of tip oscillation. The tip is oscillating at a close proximity to the surface either in the purely non-contact conditions (under domination of an attractive force) or in the intermittent-contact conditions (under domination of a repulsive force). Then, the AFM loop keeps constant the mechanical amplitude of oscillations in ambient conditions – or the resonance frequency shift under vacuum – by modifying the mean tip-to-sample distance. In the explanation, a description of the linear regime of sensor oscillations has been given, which also presents an intuitive idea of the non linear regime commonly used to explore the topography with an AFM. Some extreme cases in which these conditions are not satisfied have been exposed, and, for example, that strong localised electrical gradients can appreciably

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change the topographic observations. Topographic data can be used as a third way to explore electric effects, which has been illustrated. The standard electrical methods involve the detection of forces (EFM), which has been mainly used up to now, or force gradients (EFGM), an emerging method, in which a supplementary loop allows one to measure the surface local voltages. For implementation of the electrical methods, the AFM topographic images should also be obtained in the oscillating mode, either simultaneously with force measurements or preliminary to force-gradient measurements. These procedures constitute the EFM/KFM and the EFGM/KFGM methods, which are compared in the following paragraphs.

9.3 Observation and Interpretation Now, we consider typical observations made using the EFM and EFGM methods in single-pass and double pass modes, respectively, and offer some guidelines for the interpretation of the images obtained. In order to illustrate the capabilities of EFM and EFGM, we report observations on two different samples whose AFM morphologies obtained in vibration modes are shown on Fig. 9.6. Sample 1 with InAs nanoislands on GaAs (001) substrate (Fig. 9.6a) was used for single pass measurements. InAs nanoislands determining the surface morphology have heights and apparent diameters ranging from 1 to 8 nm and 15 to 90 nm, respectively. Measurements in the double pass mode were performed on laser-sintered stripes on 200-nm-thick film of Ge22 Sb22 Te55 deposited on a Pt coated Si substrate (sample 2, Fig. 9.6b). Two vertical stripes having the shape of a groove can be seen, whose maximum depth and width are 3 nm and 3.5 µm, respectively. In addition, there are randomly distributed hills (white spots in the image) about 40 nm in height on the surface.

Fig. 9.6. AFM surface topography images of (a) InAs nanoislands on GaAs (001) substrate (sample 1) from [35] and (b) laser-sintered stripes on Ge22 Sb22 Te55 film deposited on a Pt film (sample 2) from [13]. The images in (a) and (b) are taken in non contact and tapping modes, respectively

298


It is worth mentioning that topographic data may help in the interpretation of the EFM and EFGM electrical images. If the specific features of topographic images reveal different materials on the sample surface, variations of work function or surface voltages in the case of devices are to be expected on the surface. The local variations of the surface topography can change locally the tip-to-surface capacitance and derivatives of the capacitance. This capacitance is constituted by the air-gap tip-to-surface capacitance in series with a surface-to-bulk capacitance. Assuming a constant mean tip-to-surface distance, the air-gap capacitance decreases when the tip position over the surface shifts from a plane to a hill, and, conversely, increases when the tip encounters a crater on the surface. Since the electrical data can be obtained at different frequencies, we examine successively the data corresponding to dc signals, signals at an electrical frequency (Ω/2Π) and, finally, those at a doubled frequency 2(Ω/2Π ). The data examined were obtained on i) a Thermo Microscope Autoprobe atomic-force microscope for EFM observations in a single pass mode and ii) a Digital VEECO Dimension 3100 for EFGM observations in a double pass mode. The laser beam reflected from the cantilever falls on a double-section photodetector and gives rise to an A–B signal (difference of photoresponces of sections A and B). This signal is further analyzed, using an amplifier tuned to electrical frequencies (EFM detection) or first to the mechanical frequency and then to the electrical frequencies (phase, i.e., EFGM detection), respectively. The experimental set-up is shown schematically in Fig. 9.7. These signals result from a tip-to-sample polarisation V = (V0 + Vdc ) + Vac sin Ωt. Usually V0 merges in the 1 volt range (ca 0.5 V) and it becomes easy to change the sign of the effective voltage Vdc eff = Vdc + V0 and, consequently, the sign of the ac force or phase (9.3).

Fig. 9.7. Sketch of the experimental configuration used to detect either forces (detection of cantilever bending) or force gradients (detection of phase modulation) related to electrical forces from [13]

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9.3.1 DC Observations The dc force is generally undetectable directly because of the very small static bending of the sensor cantilever. However, the dc force gradient can be detected via a change in the phase of mechanical oscillations of the tip in an electric field. Figure 9.8a demonstrates such a phase contrast between the laser sintered areas and the untreated surrounding surface of sample 2, detected in the double pass mode. The contrast arises due to a difference of 90 mV in the surface potentials V0 between these areas, revealed in the KFM image of the surface shown in Fig. 9.11. Figure 9.8a also demonstrates an inversion of the relative contrast between the sintered areas and the untreated surrounding surface for the top and bottom parts of the phase image. In the top and bottom parts of the phase image, the externally applied dc voltage Vdc changes from +2V to −2V , which causes an inversion of the sign of the effective voltage Vdc eff . The effect of the sign of the effective voltage on the relative phase contrast of the areas with different surface potentials is illustrated in Fig. 9.8b. It is related to the different shifts of the parabolas (V0 + Vdc )2 in (9.2) along the Vdc voltage axis for the areas with different values of V0 . On the basis of (9.2), the local lateral variations of the dc force gradient dFdc can be related to local variations of dV0 and dC by dFdc = C (V0 + Vdc )dV0 + 1/2 (V0 + Vdc )2 + 1/2Vac2 dC (9.11) Here and further in this section, dF has a meaning of a local lateral variation of the parameter F , etc . . . . Equation (9.11) explains why a change of the sign of Vdc eff inverts the contrast of phase (force-gradient) image at the surface areas with different values of V0 .

Fig. 9.8. (a) DC phase image of laser sintered stripes on surface of sample 2, the image top and bottom parts are obtained at Vdc voltages +2V and −2V , respectively from [13]. (b) Sketch of two parabolic phase dependences on Vdc for two areas on the surface having different tip to surface work functions which explains the phase contrast inversion with a change of sign of voltage Vdc eff


300

It is interesting to note that the relative contrast of the hills on the surface in Fig. 9.8a is not changed in the top and bottom parts of the phase image. This difference in the behaviour of the contrast is related to the other origin of the phase contrast in this case. Having the same surface potential, the hills on the surface induce a considerably lower tip-to-surface capacitance, which leads to a local decrease in the force and force gradient for any polarity of Vdc eff . The absence of a contrast inversion in this case is also consistent with (9.11). Similar phase contrast inversion upon a change of the sign of the effective voltage Vdc eff has been observed when examining local charges on insulators [36– 38]. In that case, the electrostatic force existing between the tip and the sample is constituted by the Coulomb force between the charge Q s deposited on an insulator and its image −Q s on the tip plus the capacitance force (9.2) between the tip and the conducting plane supporting the insulator (9.10). The lateral variation of the force gradient dFdc , contains a term with (V0 + Vdc )(2 Cz − C ) dQs, which is sensitive to the sign of the effective voltage Vdc eff at the place where the charge resides. 9.3.2 Ω Observations The results of Ω-EFM and EFGM observations are shown in Figs. 9.9a,b (for sample 1) and 4c,d (for sample 2), respectively. A contrast inversion is observed on steep topographic irregularities, when the effective tip-to-sample voltage changes from positive to negative. These steep topographic changes are InAs nanoislands on the surface of sample 1 and hills on the surface of sample 2. Such a similarity of the results of force and force-gradient observations finds explanation after performing a differentiation of (9.3): dFΩ = Vac [(V0 + Vdc )dC + C dV0 ] dFΩ = Vac [(V0 + Vdc )dC + C dV0 ]

(9.12a) (9.12b)

The obtained relations show that topography-related differences in the local capacitance and the derivatives dC and dC induce changes in contrast in the FΩ and FΩ images, respectively, when sign of the effective voltage Vdc eff is reversed. At the same time, ( 9.12a,b) contain no terms in which Vdc eff and dV0 are present together. Therefore, a lateral variation of the surface potential dV0 in Ω-force and force-gradient observations should not cause image contrast inversion upon a change of the sign of Vdc eff . Indeed, no contrast inversion is observed in Fig. 9.9c,d between the sintered and untreated areas on the surface of sample 2. 9.3.3 2Ω Observations In Figs. 9.10a and b, local drops in capacitance and the derivatives dC and dC are observed on the InAs nanoislands and on the surface hills, respectively. These

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Fig. 9.9. The EFM images of InAs nanoislands on GaAs surface of sample 1 at Vdc = −1 V (a) and Vdc = +1 V (b) from [35] and the EFGM images of laser sintered stripes on surface of sample 2 at Vdc = +2 V (c) and Vdc = −2 V (d) from [13] taken at the electrical frequency Ω/2Π. The EFM image contrast inversion for InAs nanoislands in (a) and (b) corresponds to a change in local capacitance coupling due to the topography shape of InAs nanoislands and occurs with a change of the sign of effective voltage Vdc eff . In the same situation, there is no any contrast inversion on the laser sintered stripes with a smooth shape but on the left hand side topographic accidents generate a contrast inversion

observations are accounted for in a natural way by the topographic shape of these objects and by the formulas derived by differentiation of (9.4): dF2Ω = −1/4Vac2 dC

(9.13a)

dF2Ω

(9.13b)

=

−1/4Vac2 dC

302


Fig. 9.10. The EFM (a) from [35] and EFGM (c) from [13] images, respectively of surfaces of sample 1 (.4 × .4µm2 ) and sample 2 (8.5 × 2 µm2 ), at the electrical frequency 2(Ω/2Π). Between EFM and EFGM images the color code is reversed: the minimum is black or white, respectively. Capacitance coupling reductions are seen both at InAs nanoislands for sample 1 and at surface hills for sample 2. The image (b) presents the surface topography of sample 2. Comparison images (b) and (c) shows that sintered stripes with a smooth topography do not initiate reduction in a capacitance coupling

9.3.4 Surface Voltage Observations When making the EFM signals at a frequency Ω/2Π equal to zero by injecting a counter voltage Vdc during scanning, variations of the surface voltage can be observed. The corresponding surface voltage images for samples 1 and 2 are shown in Figs. 9.11a and b, respectively. 9.3.5 Guidelines for Interpretation We presented several typical examples of how the data obtained in EFM and EFGM measurements can be related to the electrical properties and topography of the sample surface. Here, the voltage differences on the surfaces are related to a difference in the work functions, but voltages of any other origin can also be involved (piezoelectric, external polarisation, photovoltages, etc.). Here, changes in the capacitance coupling are associated with topographic features, but subsurface effects (depletion regions in semiconductors, fluctuations of the thickness of the surface dielectric layer or of the dielectric constant, etc.) may also be significant.

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Fig. 9.11. Surface voltage variations obtained under detection in KFM for sample 1 (a) and sample 2 (b) from [35] and [13]

It is important to remember that variation of the local voltage and capacitance on the surface may produce a similar variation of the contrast in electrical images obtained in different measurement modes. However, we need to determine the real origin of the effect in each case. This can be done by addressing to the Table 9.1 in which the results of observations in different modes are systematised and certain conclusions are made for characteristic combinations of data obtained in these modes. It can be noticed that these observations can be easily made with any AFM device: a tip-to-sample electrical polarisation plus detection of electric signals by means of Table 9.1. Manifestations of the local surface voltage and tip to sample capacitance changes in EFM (F) and EFGM (F ) observations There are local capacitor variations When dF2Ω = 0 or

dF2Ω

= 0

Consequences: i) If dFΩ = 0 = 0 or dFΩ ⇒ possible changes in local voltage dV0 and capacitance coupling dC or dC .

There are no local capacitor variations =0 When dF2Ω = 0 or dF2Ω = 0 Consequences: i) If dFΩ = 0 or dFΩ

⇒ there are local voltage changes dV0 .

If contrast is reversal with Vdc ⇒ localisation of capacitance coupling dC or dC . = 0 ii) If dFdc = 0, or dFdc ⇒ possible changes in local voltage and capacitance coupling.

= 0 ii) If dFdc = 0 or dFdc ⇒ there are local voltage changes dV0 .

If contrast is reversal with Vdc ⇒ localisation of voltage change dV0 .

If contrast is reversal with Vdc ⇒ localisation of voltage change dV0 .

If there is no contrast reversal with Vdc ⇒ localisation of capacitance coupling dC or dC .

304


popular lock-ins do not require any specialised knowledge. Of course, the applied voltage Vdc must change the sign of Vdc eff to achieve a contrast reversal, whereas a low-noise Kelvin set-up is a little bit more complex to implement. However, Kelvin measurements can be useful in analysis of electrical data.

9.4 Future Opportunities In the following, we, first, show that the KFGM method is of interest for measurements of local surface potentials, then extend the conclusions to the case of characterization of small devices and, finally, examine the spatial resolution limits of the method. 9.4.1 Interest in the KFGM Method Figure 9.12 presents the results obtained in measurements of the Kelvin voltages by the KFM and KFGM methods on the mirror of an n-InAs/n-AlGaSbAs/p-InAs laser structure. These voltages correspond to the difference of the surface voltages on the mirror between the semi-infinite area of n-InAs substrate and the centre of a 2-µm-wide track. This track is the opening of the n-AlGaSbAs layer on the mirror. The dependence of this surface voltage difference on the mean tip-to-surface distance z was studied. To make the results comparable, both the measurements were performed using the double pass mode. In the second scan, the minimum mean tipto-sample distance z was 40 nm and the mechanical and electrical amplitudes were smaller than 5 nm. So, the KFM and KFGM measurements have been performed in the linear mode. The dependences of the surface voltage difference on z, presented in Fig. 9.12, exhibit different behaviours in the cases of measurements by the KFGM and KFM methods. In the first case, the measured surface voltage difference of 0.4 V is quite insensitive to the distance z until it becomes as large as 300 nm. In the second case, the measured surface voltage difference is always smaller than that in the first case and monotonously decreases with increasing z without demonstrating any stable level at small z. The decrease in the measured voltages with increasing distance z and the noticeable difference between the KFGM and KFM values can be explained by the mixing of several contributions in the electrostatic interaction of the sensor with the surface. Each element of the sensor, i.e. the tip apex, tip cone and cantilever plane, is in a capacitance coupling with different areas on the surface, which have different surface voltages. Furthermore, when the mean distance z is changing, the relative importance of these different contributions is also changing. Generally, the sensor-to-surface interaction is mainly the apex-to-surface interaction at small distances of several nanometres, the cone-to-surface interaction begins to dominate at larger distances of several tens of nanometres and, finally, the cantilever-to-surface interaction becomes the strongest at distances in the micrometer range [16]. In fact, because of the mixing of these different interactions, the potentials measured always

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Fig. 9.12. Comparison of the experimentally obtained values ∆Vmeas of a potential step on a surface and dependences of ∆Vmeas on the mean tip-to-surface distance z for measurements by the KFGM and KFM methods. ∆Vmeas is the difference of two measurements achieved: i) at the centre of a track of 2 µm wide and ii) above the outside substrate. The measurements were performed in a double pass mode: a topography line is acquired first and secondly the voltage measurements are obtained after retracting the tip away from the surface for a given “lift” value. The dependences start with mean tip to sample distance z of 40 nm. A typical PtIr5 tip [40] with height of 10 µm and apex radius of ca. 40 nm is used

have averaged values taking into account all the contributions. For a realistic surface with a nonuniform distribution of the surface potential and a nonuniform topography, the effective surface potential determined by KFGM or KFM is a weighted value j

averaged over the surface, with the role of coefficients played by n

Vdc = −

∂Cis ∂z j

[39]

j

is Vi ∂C ∂z j

i=1 n

i=1

j

∂Cis ∂z j

(9.14)

where Cis is the partial capacitance between the sensor (subscript s) and ith region on the surface at voltage Vi and j is the order of the derivative, equal to 1 or 2 for KFM and KFGM, respectively. The main problem in making generalisations concerning the capabilities of the measurement technique is to derive analytical formulas for the sensor-to-sample capacitances. This problem has been solved for the case when the sensor is over an infinite isovoltage surface, in which the axial symmetry of the experimental geometry makes the analysis easier [16, 17, 41, 42]. In the following, we explain semi-quantitatively the experimental observations in Fig. 9.12 on the basis of these analytical formulas. It has been shown [39] that the capacitance couplings are equivalent when the object is either a track or a disc with a diameter equal to the track width. So, a track is also substituted with a disc in our consideration, which permits us to analyse a simplified Z-axis-symmetric configuration of the electric field.


306

Fig. 9.13. Sketch of (a) the field lines distribution between tip and surface and (b) the corresponding capacitors scheme. It is assumed that V3 > V1 > V2

In our case, the distribution of the electric field depends on the three different voltages V1 , V2 and V3 in the experimental configuration shown in Fig. 9.13a. In fact, the edge effects between the disc and the surrounding surface can make the field distribution more complex [43]. Fortunately, these edge effects are weakened when the field component normal to the surface dominates over the lateral one, i.e. V3 − V1 > V2 − V1 , which is usually the case in the experiment. Then the measured surface voltage can be written as j

Vdc = −

j

C1 V1 + C2 V2 j

(9.15)

j

C1 + C2

where C1 is the capacitance between the centred sensor and a disc of diameter equal to the track width, and C2 is the capacitance between the sensor and the surface surrounding the disc, as shown in Fig. 9.13b. It is important, that, for a voltage distribution at any instant of time, there is a possibility to define areas on the sensor and on the surface that are in an electrostatic interaction. These areas are defined by a bunch of field lines that connect them. So, the areas on sensor and surface which are in an electrostatic interaction can be defined and the related capacitances C1 and C2 can be found. When the sensor shifted over the surface far outside the disc, the voltage is measured under the conditions of a sensor over an infinite surface having homogeneous and constant potential, which gives a correct surface voltage Vdc outside = −V2 . Then the absolute value of the measured voltage difference between the centre of the disc and an area far outside the disc can be written as: ∆Vmeas =

1 1+

j

C2

[V1 − V2 ]

(9.16)

j

C1 j

j

In order to find C1 and C2 , the surface areas involved in coupling with different parts of the sensor have to be determined. First, the diameter of a round area on the surface electrostatically coupled to the apex can be obtained. In the case of small tip-to-sample distances, numerical simulations of field lines make it possible to determine the radius of this circle on the surface [44]. The geometrical illustration in Fig. 9.13 helps us to understand the procedure by which the radius is determined. It can be found as a position on the x x axis to

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which the field line originating from the boundary between the tip apex and the cone arrives. This field line is determined by the arc of a circle with a centre on the x x axis at the point of intersection of the x x axis with the tangent to the tip cone surface. Then, from simple geometrical considerations, the diameter Dc of the disc on the sample surface, which is electrostatically coupled with the apex, can be written as z 1 Dc = 2R cos θ + 1 − sin θ + − tan θ (9.17) R cos θ where θ is the half-cone angle; z, the tip-to-sample distance; and R, the apex radius. In the experiments corresponding to Fig. 9.12, θ = 10◦ , R = 40 nm and the diameter of the interacting area, Dc , is in the range from 200 to 640 nm for z varying from 40 to 300 nm (i.e., until the maximum signal in KFGM measurements decreases by 10%). Thus, in the experiment described, the apex always remains totally coupled with a track of width 2 µm. Let us now consider the relationship between the apex-to-surface interaction and all the other contributions important in KFGM measurements. It can be mentioned that there are analytical expressions for the capacitance coupling C between the apex, cone and cantilever and an infinite surface [16,17]. So, C can also be deduced C for all these parts. In order to deduce the C2 ratio in (9.16), we have first to compare 1

the second derivatives of the apex, cone and cantilever capacitances, i.e. Capex , Ccone and Ccanti related to the infinite plane, and those parts of the capacitances, which are involved in the coupling between the sensor and the disc and between the sensor and the outside area. = KCcone As regards the sensor-to-surface force gradient, the relation Capex ◦ can be written for small cone angles (θ ≤ 10 ) as a function of the tip-to-sample distance z and of the tip radius R: 2 2 1 R cos θ 1 ∼ (9.18) [2z/R + .826] = K 2 z sin θ (1 − sin θ) (ln cot g(θ/2))

From (9.18), it follows that the normalised distance z/R is to be equal to, or smaller than 0.3 for the apex-to-surface interaction to be dominant (K ≥ 10), and z/R = 2.1 for the apex-to-surface and cone-to-surface interactions to be equivalent (K = 1). Then, for a reasonable apex radius of a tip, R = 40 nm, z values satisfying the above conditions can be found to be 12 and 85 nm, respectively. Since the cone and cantilever are, a priori, partially coupled with the inner disc, we have to introduce the corresponding contributions into C1 , i.e., C1 = Capex +ε1 Ccone +ε2 Ccanti where 0 ≤ ε1 and ε2 ≤ 1. Then it follows for the capacitance coupling C2 between the sensor and the rest of the surface that C2 = (1 − ε1 ) + (1 − ε2 )Ccanti . So, the ratio between the capacitance coupling between the Ccone sensor and the sample surface outside the disc and that between the sensor and the centred disc can be written as C

(1 − ε1 ) + (1 − ε2 ) Ccanti C2 cone = Capex C C1 + ε + ε canti Ccone

1

2 C

cone

(9.19)


308

In the experiments corresponding to Fig. 9.12, at the shortest distances z/R ∼ = 1, the Ccanti ∼ coupling of the cantilever is negligible versus that of the cone [45], i.e., C = 0 and, cone ∼ /Ccone as follows from (9.18), K = Capex = 2.5. In addition, it can be considered that the cone is mainly coupled with the track, i.e., ε1 ∼ = 1. Then C2 /C1 ∼ = 0 and it follows ∆Vmeas ∼ from (9.16) that ∆Vreal = 1. So, the potential difference experimentally found in KFGM measurements near the surface should be considered true. This conclusion finds an additional confirmation in the observation that the voltage tends to a constant value at small z. Physically, this means that, in the KFGM measurements, the inner disc is seen as an infinite plane by the sensor. decreases notably while Ccone remains nearly As the distance z increases, Capex constant and ε1 starts to decrease slowly from a value close to unity. Then C2 /C1 increases slowly (9.19) and the measured voltage difference steadily decreases relative to the true value. Finally, at the largest distances, at which the cantilever coupling 1−ε becomes predominant over that involving the cone and apex, C2 /C1 ∼ Ccanti = ε2 2 meas ∼ and ∆V ∆Vreal = ε2 . Because ε2 is related to the ratio between the surface areas of the cantilever coupled with the disc and with the outside of the disc, the ratio of the meas ∼ voltage differences is decreasing towards ∆V ∆Vreal = ε2 = .1 [44]. Let us consider now the results of KFM measurements shown in Fig. 9.12. For = KCcone similarly to (9.18) gives: z/R 1, writing Capex z + R(1 − sin θ) R ∼ 1 − ln −1 =K 2 z H (ln cot g(θ/2)) cos2 θ z + R(1 − sin θ) +R sin θ

(9.20)

At typical tip heights H = 10 µm and θ = 10◦ , the estimations of z/R for K = 10 and 1 are 0.05 and 0.5, which means that z = 2 and 20 nm, respectively. Then, the apex coupling is unlikely to dominate in KFM measurements and measurement at the shortest distances of 40 nm involves mainly the cone-to-track interaction, which is progressively replaced by the cantilever coupling as the distance z increases. So, it seems impossible to obtain the true voltage difference in the KFM measurements even at the minimum tip-to-surface distance z achieved and, as the distance increases, a tendency towards a low saturation limit, explained for the case of the force gradient, is clearly observed. In summary, i) it is difficult to determine the true potentials by KFM, but this can be done with KFGM and ii) the potentials measured by KFM steeply decreases as the tip is retracted from the surface. Similar KFM observations have been reported by other authors [39, 46]. The interest in the gradient KFGM method for correct voltage measurements has been expressed. The easiness with which suitable working conditions are established is due to the fact that, for typical EFM tips, the requirement that the apex-to-surface interaction is predominant is satisfied at reasonable tip-to-sample distances z of R/3, instead of R/20 in the KFM method. Consequently, this method is the most suitable under ambient conditions, in which it remains difficult to make the tip approach a surface to distances less than several nanometres, because the water film on the

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sample often sticks to the tip. It can be noted that the advantage of the gradient KFGM method over the KFM has already been accepted by many authors [23, 47–50]. 9.4.2 Spatially Resolved Observations The question of whether KFGM can be used to observe low-dimensional structures, even with some error in voltage measurements permissible, has to be examined. It leads to a new definition of the spatial resolution, related to the quality, or the precision, of a measurement, a point of importance for the future of electrical observations. If, first, the apex-surface interaction dominates over that between the cone and the surface interaction and, second, the voltage differences on the surface are small in comparison with the tip-to-sample voltage and the measurement adjustment is perfect, then the lateral resolution is determined by the surface area that is in a full interaction with the apex, i.e., by the diameter Dc of a disc lying immediately below the apex. From (9.17), when the half-cone angle of the tip approaches zero, Dc decreases to Dc = 2[2R + z]. As z can be reasonably reduced down to circa 2 nm, the main condition for observation of nanostructures is that the apex diameter R is to be diminished. The standard tip diameter is in the 20 nm range, but fortunately, carbon whisker tips can go down to several nanometres [51–53]. Another important property of carbon whisker tips is that they behave like quasi-cylinders and the R/z condition becomes obviously less critical (see (9.17)). As the whisker height is about 400 nm, the coupling of the nanotube apex remains predominant even at R = 3 nm. Let us consider now the precision of measurement when the linear dimensions of the surface area of interest, i.e., of a disc of diameter Darea , become smaller than the apex-to-surface coupling diameter Dc . A predominant apex-to-surface coupling is again assumed, but, the field lines originating from within the area of interest would reach only a part of the apex, i.e., the coupling C1 is related to much less than half with K 1 < 1. Then C2 = (1 − K 1 ) Capex , and (9.16) a sphere, i.e. C1 = K 1 Capex ∆Vmeas gives ∆Vreal = K 1 . As a result, the measured voltage difference becomes K 1 times smaller than the true value. Geometrically, the field line originating from the area of interest impinges the apex below the boundary between the apex and the cone (see Fig. 9.14) and the interaction C1 can be described using a tip with a larger half-cone gives a relationship between angle θ1 > θ. Then, writing the equality C1 = K 1 Capex ◦ the partial apex coupling and the full one for θ = 0 : 1 − sin θ1 (1 − sin θ1 ) +

1+ z R

1+

2z R(1−sin θ1 ) z R(1−sin θ1 )

1 + 2z R = K1 2 1 + Rz

(9.21)

If K 1 is known, the values of θ1 can be deduced for different z/R, and then (9.17) can be used to determine the diameters Darea . Figure 9.15 shows how Darea varies with K 1 for the experimentally accessible z/R range. Consequently, as the involved quantities, i.e., z, R and the diameter of the object Darea , are experimentally meas known (Darea can be found from the AFM topography image) K 1 = ∆V ∆Vreal can be

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Fig. 9.14. Geometrical sketch used for the analytical formulation the diameter Dc of the disk on surface electrostatically coupled with the spherical apex, the apex of radius R being tangent to the cone of half angle θ

Fig. 9.15. Relationship between the diameter Darea of the object examined, the tip radius R and the necessary tip to sample distance z permitting to achieve experimental accuracy K 1 = ∆Vmeas /∆Vreal in the range from 0.1 down to 1 for electrical measurements on objects of nanosize dimensions. For K 1 √ = 1, the relation Darea /2R = 2 + z/R is obeyed. The dotted line slope corresponds to Darea ∝ Rz

determined. Such a knowledge is a first step towards solving the problem of interest, i.e., determining the true voltage difference. Let us estimate the possible object sizes for the best experimental conditions of measurements in a vacuum. Assuming reasonable conditions under vacuum: z/R = 0.1, i.e., z = 0.3 nm and the carbon tip has a diameter R = 3 nm, and

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K 1 = 0.5, we obtain Darea /2R = 0.3, which means that a nano-object with a diameter of 1.8 nm is very likely to be electrically observed. If we refer to the case in which no error is permitted, the diameter increases to 12.6 nm, i.e., Darea = Dc = 2(2R + z), which shows a noticeable improvement of the capacity for voltage observations of nano-objects and measurements for the KFGM technique. The optimum voltage sensitivity can be found by substituting the estimate F = 3 × 10−6 N/m corresponding to the limitation imposed by the thermal noise in a 1 Hz bandwidth at 300 K [12] into the expression for the apex force gradient FΩ = 2Πε0 R/z 2 Vac Vdc (9.3). Assuming that Vdc = 1 V, for R = 3 nm and z = 1 nm the estimation of minimum detectable voltage of 0.5 mV in a 10 Hz bandwidth can be obtained, which appears to be quite acceptable. 9.4.3 Another Way to Estimate the Maximum Possible Spatial Resolution For very small tip-to-sample distances, it can be assumed that the force (or gradient) interaction comes mainly from a small area on which the field lines are perpendicular to the apex surface, then analytical formulas can be derived. So, the results obtained with force and force gradient may be compared in measurements on very small objects. As regards a comparison of the force and force gradient, a first approach can be made by assuming the equivalence of the sphere-to-plane interaction, with F = −Πε0 (R/z)V 2 , and of the plane capacitor constituted by two discs of radius Ra 2 separated by a distance z, where F = 1/2 ∂C ∂z V . Taking into account the equality 2 ∼ 2 2 of the two force expressions, i.e., −Πε 0 (R/z)V = −1/2Πε0 (Ra /z ), the disc √ √ ∼ radius for the force detection RaF = 2 Rz can be estimated. A similar result is √ obtained for the force gradient detection , i.e., RaFG ∼ = Rz. This remains a rough approximation, but √ it suggests (and this is confirmed by using more refined models) ∼ that RaFG,F = b Rz, with b being larger in force measurements, compared with the gradient technique. It suggests also that, in the case of continuous potential profiles, when the distance z approaches zero, RaFG,F goes towards zero too, i.e., a very small size object could be observed. More precise data can be obtained using the analytical formulas. The electric force F can be written as the sum of the electrostatic pressure on the surface S of the apex, i.e., as F = S P(x, y)dxdy, where P(x, y) = ε0 E 2 /2 is the local electrostatic pressure, and E = V/d is the electric field at the tip under a voltage difference V . The local tip-to-sample distance d can be expressed as d = z + x 2 +y2 2R , assuming that the field line is straight and perpendicular both to the plane and to the tip apex. However, at a constant distance z, there are different ways to determine the spatial resolution, depending on how the voltage varies over the surface. First, in the case of a step of voltage, the resolution RaF,FG can be defined as the spatial response of the electric signal to such a step, for example, as the distance over which the force varies from 25% to 75% of its maximum value. For the resolution


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√ defined in this way, it has been shown that RaF ∼ = 1.63 Rz for force √ [54]. In ∼ a similar way, it has been found for the force gradient that RaFG = 1.05 Rz. Second, in the case of an infinitely thin and infinitely long line under a potential of 1 V over the surface under a zero potential, the resolution can be defined as the lateral width of the electric signal at mid height. The relative force Frel at a distance x (2Rz)3/2 from the centre of the line can be written as Frel = (2Rz+x 2 )3/2 , whereas the relative √ (2Rz)5/2 gradient becomes G rel = (2Rz+x 2 )5/2 . Then, the resolutions are RaF ∼ = 2.17 Rz and √ RaFG ∼ = 1.6 Rz, respectively. From the considerations presented, it follows that the lateral resolution in KFGM is around 0.7 of that in KFM (see Table 9.2). If the resolution is defined as the ability to obtain quantitative data on voltages in terms of the precision of measurement, as discussed previously, the same calculations can be made. For a track of width wtrack = X +X/2 +∞ and of infinite length, the force F can be written as F = 1/2ε0 dy E 2 dx. This −∞ −X/2 & ' X(12Rz+X 2 ) X and F = F∞ (8Rz+X 2 )3/2 . Here, F∞ = −Πε0 R/zV 2 gives F = F∞ √ 2 8Rz+X

and F∞ = Πε0 R/z 2 V 2 are the sphere-to-infinite plane force and force gradient, respectively. = 0.9, which is a high-level If a 10% error is assumed,√i.e., F/F∞ = F /F√∞ criterion, we have w F = 5.8 Rz and w FG = 3 Rz for the force and gradient measurements, respectively. For electrical observations, if the visibility √ threshold is defined as detection of 10% of the ideal signal, this gives w FG = .2 Rz, which is consistent with Fig. 9.15, under the assumption that w FG ∼ = Darea , as mentioned in Paragraph 4.1. It should be noted that on three-dimensional “mesa-like” structures, i.e., those from quantum boxes down to groups of several atoms, the involved electrostatic coupling with the surrounding area is likely to be reduced versus that for the object of interest immediately below the tip. Then the above expectations on the resolution and sensitivity of measurements do not seem to be unrealistic. This could meet the demand of some recent studies for measurement of highly localised surface potential differences on 7 × 7 silicon surfaces [55]. In summary, it has been shown in this section that the electrostatic interaction between the surface and the probe can be measured with a considerably higher spatial resolution if the force gradient, rather than the force itself, is used as the signal source. The important improvement in resolution is related to the

Table 9.2. Resolutions of the EFGM and EFM methods, obtained with different definitions. R and z are the spherical apex radius and the minimum apex-to-plane distance Spatial Resolution in: Force Method √ √ First-order model 2 √ Rz Voltage step 1.63 √ Rz Infinitely thin line 2.17 Rz

Gradient √

Resolution Gradient/ Resolution Force

Rz √ .707 1.05√ Rz .644 1.6 Rz .737

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fact that the apex-to-plane interaction is involved in gradient observations, instead of the cone-plus-cantilever interaction when using the force detection. More precisely, the fact that the full apex-to-sample coupling is obtained up to relatively large distances (typically z/R ≤ 0.3) allows simultaneous use of small distances z and tip radius R, which leads to a decrease in the diameter Dc of the coupled area on the surface. In addition, if the tip-to-sample distance z is strongly diminished, then, first, the resolution becomes better since it is proportional to √ Rz, and, second, there is a gain of more than 30◦ when gradient is used instead of force. The gradient-force detection takes the benefit of the double pass mode in which the distance z is well defined thanks to the tapping mode. We believe that the experimental results and theoretical considerations presented here clearly demonstrate the enhanced resolution of KFGM and the possibility of a quantitative determination of electrical properties of materials and structures on a nanometre scale.

9.5 Some Applications As it has been shown, EFGM and EFM are the methods that can be used to obtain simultaneously three kinds of data: the surface morphology, which allows the area of interest to be located, the tip-to-sample capacitance, which is related to the subsurface capacitors connected in series with the tip to surface air gap and changes in the surface voltage. Electrical data can furnish information about i) (the first and quite striking point) the structure and lateral dimensions of the areas having specific electrical properties and ii) the surface voltages or capacitances. For semiconductors, these two types of quantitative data can be related to the nature of the material and its doping levels (or local charges in insulators) but, of course, surface band bendings can alter the information [56]. However, capacitance measurements can clarify this point because, even if the Fermi level is pinned at the surface by surface traps, the width of the space charge region within the semiconductor is material- and dopingdependent. In addition, there is a direct possibility of revealing the band bendings by studying changes in the potential under illumination of the surface with light absorbed by the semiconductor under consideration, i.e., photovoltages [57]. Moreover, EFGM and EFM can be used to study the built-in and externally applied potentials in semiconductor device structures. For this purpose, the differential voltage or capacitance can be measured, i.e. the difference in the values for the external bias switched on and off. The differential methods obviate the need to consider the tip-to-sample work functions, which are generally complex quantities, especially under ambient conditions. The tip work function, which serves as a reference (cf. (9.2)), may vary between tips and, in addition, the sample work function can be modified by surface ageing or contamination. In the following, we outline several, though certainly not all, important domains in which EFGM and EFM have been shown to be quite useful. Starting from ambient measurements and proceeding to a vacuum and Ultra High Vacuum (UHV), we show that these domains are material characterisation, studies of semiconductor devices, injection and storage of local charge in oxides and semiconductor quantum dots and

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high-sensitivity non-contact potentiometry in UHV with a nearly atomic resolution. We should also mention that EFGM and EFM have a rather short history, i.e., less than two decades, so we foresee a further development of these methods with new applications. 9.5.1 Applications Under Ambient Conditions When considering the EF(G)M applications for material analysis, we can start from such a semiconductor as GaN, which has been an object of a strong interest in these recent years owing to its optical properties. On thin GaN films, a clear correlation has been shown between the uniformity of the surface voltages and the Hall mobility in the bulk, which reveals a strong interaction of impurities with the interfaces between bulk structural domains [58]. Further, the following more direct measurements of the density of surface states, surface band bending, inversion domains, minority carrier diffusion length, and electric fields piezoelectrically induced at structural defects have been undertaken [59–62]. It is worth mentioning that, owing to the wide variety of its electrical properties, GaN has been an excellent model material for testing and validation of the EFM and EFGM methods [49]. To illustrate the diversity of applications of EFGM and EFM methods in studies of the material quality, we can also mention the following: two-dimensional dopant profiling of Si wafers patterned by photolithography and a subsequent ion implantation [63], revealing and characterization of defects in GaAs films grown on Ge [32], identification of defects in Bridgman-grown single crystals Cd0.9 Zn0.1 Te [64] and electrostatic characterisation of InAs nanoislands MBE-grown on n-type (001) GaAs surfaces [65]. For semiconductor devices, it is crucial to monitor the doping and the potential profiles in the actual device in order to confirm the technological achievement of the expected design. This can be easily done by the EFGM and EFM methods via measurements of changes in the surface potential and capacitance. For planar device structures, these measurements are performed just on the top surface. For devices with a vertical geometry, the measurements are performed on cleaved cross-sections of the device area of interest. In the case of semiconductor light-emitting devices, the measurements are performed simply on the optical mirrors. In the last decade, EFGM and EFM methods have been used to study p–n Si diodes [26, 28, 30, 66, 67], III–V heterostructures [68–70], n–i–p–i structures [71, 72], solar cells [34, 73], and light-emitting diodes and lasers [7, 33, 74–77]. Moreover, it must be mentioned that different contact electrical methods derived from the scanning probe microscopy are now available. We can mention scanning capacitance microscopy , i.e., nano C(V) [78] or Schottky C(V) [79], in which observations on the 20-nm scale have been made; nanopotentiometry, in which a conducting tip is used as a probe measuring the voltage relative to contacts of a working structure (even a working MOS transistor has been examined) [80]; and a spreading resistance (SSRM), which consists in measuring the tip-to-semiconductor contact resistance, which, in turn, depends on local doping levels [81]. As the silicon technology is reducing its dimensions, there is strong necessity for doping characterisation on the nanometer scale. Use of spatially resolved methods is mandatory, among which the electron holography [82] must be also mentioned.

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EFGM and EFM are now becoming powerful techniques for electrical characterisation of electronic and optoelectronic devices in equilibrium and under applied external electric biases. Explicit information on the position of hetero- and electrical junctions can be obtained, the areas of parasitic potential drops can be found and the current leakage through dielectric protection layers and nonequilibrium carriers that escape from the active regions can be monitored. The characteristic parameters obtained can be directly compared with the results of a computer simulation, as reported for p–n junctions on Si [83, 84], InP diodes [85], and GaAs/AlGaAs lasers [76]. The failure analysis of laser diodes has also been performed [76,86]. These recent observations have helped to improve the device quality and optimise their design. It must be noted that, at the very beginning of EFM studies, observations on gigahertz devices were reported in the literature [8,9] and showed a practical interest in the proper adjustment of these devices. By mixing the sinusoidal signals from the device and the tip, a down conversion of frequency occurs and a gigahertz voltage distribution over a device becomes achievable [87, 88], even for non-sinusoidal signals [89]. Among the externally induced constraints, photovoltages are to be considered first, in addition to the external field mentioned above. They include effects that are the subject of the surface science [57] and, as well, have given rise to applications in semiconductor characterisation, even at the millimeter scale when using the Kelvin capacitor method [90]. However, the appearance of EFM techniques brought these characterisations to the submicrometer scale [91], and the gradient operation will improve it. For example, very small semiconductor objects, such as quantum boxes, are certainly concerned and could be characterised individually. Secondly, since the tip and sample behave as an MIS structure [56], band bendings induced by local voltages are likely to occur [92]. They have been recently observed in non-contact scanning by EFGM of III–V structures [93], which clearly reveals the subsurface energy structure. In any case, in the future, additional theoretical concepts have to be developed, especially those concerning linear and non linear tip-to-sample interactions near a semiconducting surface, as well as their influence on the local observations and measurements. It is important to mention the ability of EFM to deposit and image localised charges in insulators [94]. Charge deposition occurs by bringing the tip in contact with the surface and applying a bias of any polarity to the tip. The charged surface area may be imaged later by the EFM in force- or voltage-detection modes. Over the past decade, this capability of EFM has attracted a considerable interest for many scientific and technological reasons, one of which being the intention to create high-density electrical memories and, as a first implementation, non-volatile floating-gate transistor memories [95, 96]. In this context, numerous EFM studies of the charge injection efficiency, localisation and retention duration in and on thin insulator layers on semiconductor surface have been performed [97–100]. In parallel, the use of an array of silicon nanocrystals as discrete storage nodes for electric charges was proposed as a means for making the writing time shorter and the retention time longer [101, 102]. It was predicted that the additional charge confinement in nanocrystals would lead to these improvements. This prediction initiated EFM experiments on charging and charge mapping of isolated semiconductor nanocrystals in and on nano-thin oxide layers [15, 31, 36, 103–106].

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9.5.2 Vacuum or UHV Applications It should be noted that most part of the published EF(G)M experiments have been performed in the ambient conditions. However, the surface contamination possible in this case and the presence of an adsorbed liquid layer on the surface does not allow the tip to approach the surface very closely, which impairs the sensitivity and lateral resolution of the electrostatic measurements. Some measurements have been performed in the atmosphere of an inert gas. However, purity of the surface is still not guaranteed even in this case and the cantilever quality factor is nearly the same as that in the ambient atmosphere. A considerable improvement of the sensitivity and lateral resolution would be achieved if EFGM experiments were performed in ultrahigh vacuum conditions, which would ensure both better physical conditions and an opening towards fundamental physics, though at the cost of some experimental constraints. It must also be noted that a vacuum drastically diminishes the damping of cantilever oscillations, so the resulting increase in the quality coefficient improves the sensitivity in detection of the electric gradient. Several EFGM and EFM experiments in UHV conditions have demonstrated a drastic improvement in the quality of the results obtained. In these conditions, both the topography and EFGM EFM data can be obtained with an atomic resolution. On a clean Si(111) surface on which non contact AFM has shown atomic resolution [107], a pronounced difference in surface potential was found for nanosize areas with 7 × 7 and 5 × 2 reconstructions [47]. Furthermore, an increase in the surface potential at the points of adsorption of Ag atoms was observed. In the studies of GaAs(100) surface [108], the charged atomic-scale point defects were clearly resolved with the true atomic resolution. Recently, a very local, on a scale of 1 nm, increase in the surface potential at the step edges, compared with that in the terrace area, has been observed on a clean Si(111) surface [109]. This fundamental result is rather interesting because it is the opposite to the situation for many metal (Au, Cu and C) surfaces, for which the work function measurements reveal a reduced work function at the step edges. The above numerous examples of the EFM and EFGM applications show that, owing, among other things, to their relative easiness of operation and non-contact and non-destructive nature, these techniques are now becoming a well-accepted method for electrical characterisation of semiconducting materials and devices with a high spatial resolution. Very interesting results have been obtained even on the submicrometer scale, especially regarding the technological analysis of structures, as well as on working devices. The next step is certainly the control over the operation and fabrication of new nanometer-scale structures and devices, including biological structures, which adds a powerful interest to the research domain of highly localised electrical characterisation.

9.6 Conclusion The electrostatic-force microscopy and electrostatic-force-gradient microscopy provide a simultaneous probing of the surface topography and of the local electrostatic

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interactions on it, with a nanometer-scale resolution. This enables a straightforward correlation of the electrical properties with the architecture of both the surface and subsurface layers. Measurements of local electrostatic interactions on the surface are particularly important for determining such electrical properties as the lateral variation of the surface potential, dopant concentration and surface charge distributions over the entire surface. As the dimensions of electronic devices are continuously decreasing (to a few tens of nanometers at present), the ability to characterise electrical properties on a deeply submicrometer scale gains a great significance. In this context, the EFGM and EFM methods can play an important role in the electrical characterisation of materials, nanosize inclusions and defects in them, integrated circuits, and nanometer-scale electronic and optoelectronic devices. In this presentation, the capabilities of the EFM and EFGM methods have been shown and compared, and guidelines have been given for the interpretation of the observations made. Although the EFM directly measures the electrostatic force, its spatial resolution is relatively poor due to the long-range nature of the Coulomb forces: all parts of the AFM sensor, including the tip apex, tip cone and cantilever, contribute to the measurement, which results in smearing of the actual potential distributions. The EFGM method, on the other hand, measures the force gradient instead of the force itself and, because of that, it can measure the surface potential with a higher resolution. It appears that the EFGM method is becoming a pertinent way to obtain quantitative data about changes in local voltages. New numerical criteria for evaluation of the spatial resolution of the electrical methods have been proposed and the EFGM capabilities for quantitative study of nanosize objects have been shown. An overview of the numerous applications of the EFM and EFGM methods to characterisation of semiconductor materials and devices has been presented, showing that they are now becoming the established methods for exploring the local electrical properties of semiconductor materials, structures and devices. This field can be further widened by new developments. It is worth noting here that the EFGM method still has a potential for further improvement in sensitivity and spatial resolution. This can be achieved by working in vacuum conditions on clean surfaces at much shorter tip-to-surface distances of few nanometers and with an increased value of the cantilever quality factor and, accordingly, with enhanced sensitivity. Also, further modelling of electrostatic tip-to-semiconducting surface interaction in ambient and vacuum conditions is also desirable. These efforts would go far in meeting the growing demand for studies of the fundamental and practical properties of really nanosize objects, products of the modern technological shift into nanofabrication. Acknowledgements. The authors gratefully acknowledge Dr M. Ramonda and Dr E. Ph. Cadet from Montpellier University for their contributions to the experimental and simulation results mentioned here. A.N. T. is also grateful to Dr A. Ankudinov from the Ioffe Institute at St Petersburg for creative discussions and to Dr V. A. Bykov, President of NT-MDT Co., Moscow, for providing the technical opportunity to become involved in the exciting field of atomic force microscopy and for the stimulating interest in the results obtained, as well.


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10 Polarization-Modulation Techniques in Near-Field Optical Microscopy for Imaging of Polarization Anisotropy in Photonic Nanostructures Pietro Giuseppe Gucciardi · Ruggero Micheletto · Yoichi Kawakami · Maria Allegrini

Abbreviations AFM BC CB CD EOM HRC IC LB LD LED NPB PEM PL PM PS-b-PI QD SEM SNOM

Atomic force microscopy Block copolymers Circular birefringence Circular dichroism Electro-optical modulator Radiative high recombination center Illumination/collection Linear birefringence Linear dichroism Light emitting diode naphthaphenylene benzidine Photoelastic modulator Photoluminescence Polarization modulation Polystyrene-b-polyisoprene Quantum dot Scanning electron microscopy Scanning near-field optical microscopy

10.1 Introduction Photonic structures are arrangements of shapes repeated uniformly in an optical conducting material. Such systems create an “optical crystal”: a discrete structure that limits the optical modes allowed. This fact gives rise to extraordinary optical properties [1, 2]. Usually these structures must be small and are difficult to produce, so the fabrication process has to be verified through observation. The most common techniques used are SEM (scanning electron microscopy) and AFM (atomic force microscopy), but these techniques do not provide direct information on the actual optical behavior of the system. In particular, photonic materials are generally characterized by the presence of shapes set along a regular periodic structure. This fact induces polarization effects that, with standard far-field measurements, can be lost. In fact, when sizes are smaller than optical wavelengths, conventional optical microscopy becomes insufficient as a tool of investigation. Near-field optical techniques have proved

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more successful in this case. Polarization modulation near-field technique represents a promising approach that can be used to investigate the polarization properties of photonic systems, while scanning near-field optical microscopy (SNOM) [3–5] provides a spatial resolution that can overcome the diffraction limit. Therefore it has a broad spectrum of possible applications in nano- and biotechnology (for a review see [6,7]). The tremendous improvement in the resolution compared to conventional far-field techniques, such as confocal microscopy, is obtained by exploiting a nanosized light source or antenna, and imaging the sample within the near-field of the probe (at a distance much smaller than the wavelength of light). The sample is raster scanned at a constant distance beneath the SNOM probe [8–10] and images are reconstructed by monitoring, point-by point, the optical probe–sample interaction. The most common SNOM probes consist of tapered, metal-coated, single-mode optical fibers ending with a circular aperture that has a diameter in the 20–200 nm range [5, 11–13]. Since visible light is employed in SNOM, a number of different contrast mechanisms are available [6]. Incorporation of polarization-dependent imaging provides a new tool for polarimetric investigations at the nanometer scale. Polarimetry is a branch of optics that involves the assessment of four phenomena, linear birefringence (LB), circular birefringence (CB), and the correspondent linear dichroism (LD), and circular dichroism (CD) [14]. These are respectively the anisotropy in refraction and in absorption, the difference between the refractive indices of left and right circularly polarized light, and the difference between the absorption of left and right circularly polarized light. Access to the 50–100 nm scale has permitted new exciting studies on samples with mesoscopic order such as metal nanostructures [15], magnetic films [16], lipid films [17], J-aggregates [18], conjugated polymers [19–21], liquid droplets [22, 23], and it has allowed single magnetic bits to be read and written [24, 25]. In the aforementioned experiments, SNOM imaging with light at a fixed polarization has demonstrated its unique capabilities, giving valuable structural information. In 1996, Ado et al [26] introduced a modulating analyzer that could be used to obtain both parallel- and cross-polarized images simultaneously. Polarization-modulation SNOM (PM-SNOM) [27] provides more quantitative information in less time by rotating the polarization of the light which illuminates the sample at high frequency [28]. The true birefringence, dichroic ratio properties, and the size and relative orientations of single mesoscale domains can be mapped with unprecedented spatial resolution in PM-SNOM. In this chapter we will review the different experimental approaches to PMSNOM. For each configuration, a theoretical framework is put forward that allows quantitative information to be extracted from the SNOM signals. New methods used to deconvolute the intrinsic anisotropy of the fiber probe are described which can dramatically increase the accuracy of the measurements. Finally, we review the most notable applications of PM-SNOM to photonic structures such as thin solid films, polymers, light waveguides, nanocrystals, and quantum confined heterostructures.

10.2 Polarimetric Imaging The analysis of crystals using visible light was an art developed prior to the discovery of X-ray diffraction. Developments in polarimetry, polarized light microscopy,

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and the advent of high sensitivity detectors coupled to microcomputers have signaled a renaissance in the science of crystal optics. Indeed, the vivid colors of polarized light maps frequently grace the covers of company catalogs. In a typical polarimeter, the sample is placed between a polarization state generator and a polarization state analyzer. This allows the changes in the polarization state of light passing through a sample to be studied, giving information on its anisotropy. It is only within the past 20 years that tools have been developed that can separate the various optical effects from one another, thereby maximizing the information about crystal structure and growth contained in visible light micrographs. Several review articles have been devoted to macro- and micropolarimetry [29, 30]. This body of research was begun in 1985 when Mickols et al introduced the imaging differential polarization microscope [31]. In this device various voltages were applied to a Pockels cell (electro-optical modulator, EOM) in order to modulate between orthogonal polarization states or between left and right circular polarization states at a frequency of ∼ 1 kHz. The input polarization was passed through the sample on a microscope stage, and the transmission was analyzed. The difference between the orthogonal polarization states provides, in principle, the two LD and CD maps. First applications of this device concerned the analysis of polycrystalline biological structures, such as intracellular polymeric hemoglobin in subjects affected by sickle cell anemia [32], or spermatocyte nuclei [33]. The development of CD imaging microscopy, based on coupling a Carey spectropolarimeter to a microscope [34], faced severe instrumental artifacts [35]. In fact, the residual LD of the sample generates fictitious CD signals. The strain in the photoelastic modulators (PEM) used to modulate the polarization state of light compounds these artifacts [36]. CD images with submicron spatial resolution have only recently been obtained [37]. Linear anisotropies can be quantified by the rotating polarizer technique embodied in the Metripol microscope [38]. The optical assembly (Fig. 10.1) consists of

Fig. 10.1. Schematic of the Metripol optical chain

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a spectral filter that monochromates the unpolarized light source, a mechanically rotating linear polarizer, a sample which induces a phase shift δ, a quarter wave retarder and a polarizer aligned at 45◦ with respect to the quarter wave plate’s principal axes. The polarization is rotated at discrete steps over full periods generating images which, pixel-by-pixel, are Fourier-analyzed, permitting the extraction of the LB parameters. The anisotropic absorption (LD, or CD in special cases) can be accessed by removing the quarter-wave plate and the analyzer. Recently the Metripol has been applied to crystallographic investigations of ferroelectric Na1/2 Bi1/2 TiO3 [39], and phase transitions in Pb(Mg1/3 Nb2/3 )(1−x) Tix O3 [40] and K2 Mn2 (SO4 )3 [41]. An application of Metripol to the screening of birefringent crystals is illustrated in Fig. 10.2. The two micrographs (a, b) refer to K2 SO4 crystals grown from aqueous solution on a glass cover slip [30]. Figure 10.2a displays the phase difference δ of the two eigenmodes propagating through the birefringent, orthorhombic crystals, plotted as | sin δ| in false colors. In Fig. 10.2b the orientations of the cross-sections of the optical indicatrices are given as the angle (φ) of the slowest vibration direction (the largest refractive index) measured counterclockwise from the horizontal axis. Crystallographic information can be extracted from these images: we can see for example that some crystals are aligned with (010), whereas others are aligned with (021). Another application of the Metripol apparatus to determining the anisotropic sample properties relates to the optically anomalous crystal 1,8-dihydroxyanthraquinone. The crystals form square plates precipitating simultaneously with an orthorhombic polymorph with needle-like shapes. The needles are linearly dichroic. The LD and the orientation of the most strongly absorbing direction are clearly displayed by the Metripol pictures in Figs. 10.2c and 10.2d respectively.

Fig. 10.2. Metripol LB micrographs (a, b) of K2 SO4 crystallites, and LD micrographs (c, d) of 1,8-dihydroxyanthraquinone [30]; reproduced with the permission of The Royal Society of Chemistry


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10.2.1 The Jones Formalism As we have seen above, polarimetry measures the polarization state of light and the polarization-altering properties of materials and optical systems. A variety of methods have been developed to track the polarization state as it propagates through successive optical elements and complex samples. These are the Poincaré sphere [42], the Müller calculus [43], and the Jones formalism [44]. For a detailed treatment of these tools, the reader is referred to [45]. Here we give an overview of the Jones formalism which will be used later. The x and y polarization components associated with a monochromatic plane wave propagating in a non-depolarizing medium can be represented by a two-element Jones column vector. Assuming that the amplitudes of the two components are given by A x exp(iφx ) and A y exp(iφ y ), the Jones vectors associated with the most general polarization state are easily defined, as reported in Table 10.1. The polarization properties of a homogenous, stationary material is described by a 2 × 2 Jones matrix with four complex elements (eight degrees of freedom). Two degrees of freedom represent polarization-averaged quantities: the global phase accumulation (related to the average refractive index of the material) and the global absorption or transmission. The remaining six parameters describe the two eigenpolarization properties of the material (polarizations for which light propagates through without any polarization state change). Three parameters describe the optical retardance, and three the diattenuation of the sample. Retardance is the dependence of the optical path length on the incident polarization state. It occurs on transmission if the optical medium is birefringent (the refractive index is polarization-dependent). It produces a more rapid optical path-length accumulation for one polarization state with respect to the other. The degrees of freedom associated with the retardance can be expressed as the faster axis orientation, the ellipticity of the faster polarization eigenstate, and the magnitude. Similarly, diattenuation is the dependence of the transmission on the incident polarization. Diattenuation occurs if dichroism is present; in other words if one polarization state is preferentially absorbed. Note that the term diattenuation refers to the difference in transmittance of the two polarization modes, while dichroism more traditionally refers to the difference in absorption coefficients [46]. Therefore the use of the term diattenuation is more accurate in transmission experiments [47], while we can refer to dichroism as the measured quantity if fluorescence is detected (as a result of light absorption and re-emission from the sample) [48]. Given an input polarization Jin the output polarization state Jout of the light propagating through an homogenous medium characterized by a Jones matrix M Table 10.1. Jones vectors Linear Linear Right circular Left circular Elliptical polarization polarization polarization polarization polarization along x along y 1 1 1 1 1 0 1 A x eiφx Jxp = Jyp = Jlcp = √ Jlcp = √ Jlcp = ( iφ y 0 1 2 −i 2 i A2x + A2y A y e

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is given by the linear relation Jout = M · Jin . Moreover, an arbitrary polarization element M can be decomposed into two elements M = Mr · Md where Mr , describing the phase retardation, is represented by a unitary matrix (energy is conserved). Md , describing the diattenuation, is represented by a non-negative definite Hermite matrix. We list in Table 10.2 the Jones matrices of the most common elements used in the optical chains of PM microscopes. The transmittance will thus be given by 2 2 T(Jin ) = M · Jin / Jin . In particular the diattenuation can be calculated as [49]

T Jinmax − T Jinmin

Q(J) = max (10.1) T Jin + T Jinmin where Jinmax and Jinmin are the incident polarization states with maximum and minimum transmittance respectively. For the case of linear diattenuation only, given the Jones matrix in Table 10.2, from (10.1) we obtain Q = (u − v)/(u + v) [50]. Here u and v are defined as the transmittances for light polarized parallel and perpendicular to the diattenuating axis; since u is assumed to be the larger transmittance, the diattenuating axis represents the polarization direction of higher transmission. The calculation of the retardance is more subtle since it involves the relative phase difference accumulated by the two eigenpolarizations. Since Mr is unitary, we can define the phase retardance as [49] tr Mr (10.2) ϑ(M) = 2 arccos 2 For a linear retarder only (see Table 10.2) we get the expected result ϑ[B] = δ =

2πL∆n λ

(10.3)

where L is the sample thickness, ∆n is the difference in refractive index for the two eigenpolarizations due to the birefringence, and λ is the light wavelength. It is important to observe that, while the eigenpolarization axes of Mr and Md must be orthogonal, in the most general case the matrix M need not have orthogonal eigenpolarization axes. The eigenpolarization axes of the system M are orthogonal only if Table 10.2. Jones matrices x-Linear polarizer 10 L= 00 EOM

Rotation of an angle θ from x axis cos θ − sin θ R(θ) = sin θ cos θ

Linear retarder, fast x axis, retardance δ iδ/2 e 0 B(δ) = −iδ/2 0 e

Quarter-wave retarder −i 0 QW = 0 1

Linear diattenuator, Rotating linear polarizer PEM fast axis x angle θ from x axis −id sin ωt iωt √ e 0 cos2 θ sin θ cos θ u √0 e 0 PE = EO = RL(θ) = D(u, v) = v 0 1 0 0 1 sin θ cos θ sin2 θ


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the fast and the diattenuation axes are aligned. However, this is the exception rather than the rule, so no assumptions about the relative alignment of the eigenvectors of Mr and Md should be made a-priori. In the following paragraphs we will see how to apply the Jones formalism in more detail to the different implementations of PM-SNOM, in order to extract quantitative information from the measured SNOM signals.

10.3 Electromagnetic Field Diffracted by a SNOM Aperture The control of the polarization state at the output of the SNOM probe is an issue of crucial importance for polarimetric applications. In fact, for each kind of SNOM probe, we need to know what polarization state to expect in the near-field of the SNOM aperture as a function of the polarization properties of the excitation light. In this paragraph we will first provide an overview of the fabrication process of the most common SNOM probes, and then focus on a theoretical description of the field emitted by small apertures, paying particular attention to the polarization issues. The most widely used probes for aperture SNOM consist of optical fibers, tapered and metal-coated on the apical part in order to create an nanometer-scale aperture at the edge. Several methods have been proposed for producing a smoothly tapered surface allowing for a homogenous metal coating. Mechanical pulling assisted by a high-power CO2 laser was been first introduced by Betzig et al [5]. However, one serious inconvenience associated with this kind of probe is the extremely low throughput (∼ 10−6 for 50 nm diameter probes), due to the small cone angle (a few degrees). This leads in fact to the presence of a long tract in which the optical fiber acts as an over-damped waveguide, yielding an exponential power decrease. Chemical etching techniques have proved capable of producing larger cone angles, thus increasing the overall throughput by several orders of magnitude. Chemically etched fibers have been sharpened by making use of different etching rates between core and cladding in a selective solution of HF acid [51]. High quality tapers are also produced by the “tube etching” method introduced by Stöckle et al [52] and Lambelet et al [53]. A further increase in the throughput (by an order of magnitude) is obtained by employing the double taper method [54], in which the cone angle is controlled during the etching process by buffering the etching solution, adjusting the volume ratio of NH4 F, and maintaining that of HF to H2 0(OH) at 1 : 1. A triple taper procedure [55] has also been introduced in order to overcome the deterioration of resolution due to the large apical aperture produced by the increased cone angle. The fabrication process consists of several etching steps staggered by focused ion beam irradiation to form a very sharp tapered core on a flat floor, used as the third taper. The coating of the tapered fibers is usually accomplished by thermal evaporation or sputtering of aluminum or gold. A low-cost fabrication method, exploiting acrylic paint as a coating element, has recently been suggested by Micheletto et al [56]. Different approaches to producing innovative SNOM probes have been based on the modification of atomic force microscopy (AFM) cantilevers in a single manufacturing process [57]. SNOM probes have been fabricated by producing a miniaturized aperture (50–100 nm) at the apex of a hollow metal pyramid integrated at the end of

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a silicon cantilever. The main advantages related to such probes relate to their low degradation properties, as well as on the extremely high spatial resolution achievable in contact mode topography and friction force mapping simultaneously with optical transmission (or collection) measurement. The calculation of the electromagnetic field arising from a subwavelength aperture is a pure diffraction problem first tackled by Bethe [58] and Bouwkamp [59]. In their calculations, the SNOM aperture is approximated as a circular hole in a infinite, perfectly conducting screen. Afterwards, more realistic numerical models have been proposed using the modal expansion of the fields, the localized Green’s dyadic method, and the multiple multipole approach [60, 61]. The Bethe–Bouwkamp approach is however the most extensively used in near-field calculations because it leads to analytical formulae. We assume that the xy plane contains the aperture (diameter d = 2a), which is irradiated by a plane wave propagating along the z direction, polarized along the x-direction. Let us call E 0 = |E0 | and H0 = |H0 | the amplitudes of the fields and K = (k, k z ) = (0, k z ) the wave vector, with K = |K | = (ω/c) = 2π/λ. The electromagnetic field at the aperture (z = 0) can be written in cylindrical coordinates (r, θ, z) as [59]: 4iω 2a2 − r 2 Er (r, θ) = − cos θ E 0 √ 3πc a2 − r 2 8iω 2 (10.4) E θ (r, θ) = a − r 2 sin θ E 0 3πc 4r Hz (r, θ) = − √ sin θ H0 π a2 − r 2 where the tangential magnetic and the normal electric fields in the aperture are identical to the corresponding quantities of the undisturbed incident field. We can rewrite the near-field components in Cartesian components as: Ex = −

4iωE 0 2a2 − x 2 − 2y2 3πc a2 − x 2 − y 2

Ey = −

xy 4iωE 0 3πc a2 − x 2 − y 2

Hz = −

4H0 y 2 π a − x 2 − y2

(10.5)

Figure 10.3 shows density plots of the fields at the aperture, assuming that the incident radiation is polarized along the horizontal x-direction. The fields show divergences at the aperture’s edge (x 2 + y2 = a2 ). At the aperture’s center the emission is dominated by the x-component of the electric field. The calculation of the field propagated from the aperture is a typical propagation problem which is best approached in the reciprocal Fourier space. We define the angular spectrum of the field at the aperture as ˆ 0 (k) = E(R) exp(−i k · r) dr (10.6) E z=0


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Fig. 10.3a–c. Density plots of the electromagnetic field (E x , E y , Hz ) intensity at the aperture plane

where R = (r, z). Assuming K = (k, k z ), the spectrum of the propagated field will be given by ˆ ˆ 0 (k) exp(ik z z) E(k, z) = E (10.7) where k z (k) = (ω/c)2 − k2 , giving rise to both propagative (|k| < ω/c) and evanescent (|k| > ω/c) waves. The propagated field in the real space can thus be obtained by summing the spectral Fourier components: E(R) =

1 4π

ˆ 0 (k) exp [ik z (k) · z] dk E

(10.8)

Fortunately, an analytical expression holds also for the Fourier transforms of the fields in the aperture plane (10.5) [62], giving: 3k2y a2 k4x + 3k2y + a2 k2x k2y 8ia3 ωE 0 cos (ak) 2 4 − sin (ak) Eˆ x (k x , k y ) = 3c a k a3 k 5

k x β −3 + a2 k2x + a2 k2y 3k x k y 8ia3 ωE 0 Eˆ y (α, β) = − cos (ak) 2 4 + sin (ak) 3c a k a3 k 5 8ia3 ωE 0 kx sin (ak) Eˆ z (α, β) = 3c ak (ω/c)2 − k2 (10.9) ( where we have used k = (k x , k y ) with k = |k| = k2x + k2y . The calculation of the field spectrum Eˆ z is done via the Maxwell equation div E = 0 which, in the Fourier space, reads as [62]:

(10.10) k x Eˆ x k x , k y + k y Eˆ y k x , k y + k z k x , k y Eˆ z k x , k y = 0 In Fig. 10.4 we show density plots of the modulus of the field’s spectral components | Eˆ j | ( j = x, y, z), at different distances from the aperture z = 0.02a, 0.2a, 2a.

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Fig. 10.4. Density plot of the Fourier-transformed electric field intensity components as a function of the distance from the aperture. The attenuation of the high spatial frequency components as a function of the distance is clearly visible

In our simulations we have assumed a = λ/10. Such an assumption reflects a typical experimental situation of green light illumination (λ = 515 nm), and a probe diameter of 2a = 100 nm. It is interesting to observe the effect of propagation on the three spectral components of the electric field. From the value of the full width at half maximum (FWHM) ∆k x,y of | Eˆ j |2 , using the Heisenberg principle ∆r · ∆k ∼ 2π we can in fact calculate the spatial broadening of the field as a function of the distance from the aperture. At close distances from the aperture (z = 0.02a = 1 nm) the spectral distribution of | Eˆ x |2 looks like a sinc function asymmetric in the two variables. This asymmetry, also observable in the field equations (10.5), is due to the polarization direction of the incident field, which breaks the cylindrical symmetry of the system. The spectral broadening (∆k x ∼ 0.5(2π/a), ∆k y ∼ 0.6(2π/a)) indicates that the field E x is slightly more confined in the y-direction (∆y ∼ 1.7a = 0.85d) than in the x-direction where ∆x ∼ 2a = d. At a distance z = 0.2a = 10 nm, E x broadens and is still asymmetric with ∆k x ∼ 0.35(2π/a), ∆k y ∼ 0.4(2π/a), yielding ∆x ∼ 1.4d, ∆y ∼ 1.25d. Moreover, we see how the evanescent high-frequency


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spectral components (k x,y > (2π/λ)) begin being damped by the exponential decay propagator exp(−k z z). Finally, at a distance equal to the aperture diameter (z = 2a), the only non-vanishing spectral components are those with |k x,y | < 0.1(2π/a), in other words |k x,y | < (2π/λ). Such spectral content (∆k ∼ 2(2π/λ)) is typical of a diffraction-limited spot (∆r ∼ λ/2), as expected, meaning that at a distance from the probe equal to the aperture diameter, we are already in the far-field propagation regime. As can also be observed from (10.5), a non-vanishing electric field along the y-direction is expected at the aperture, although its maximum spectral intensity | Eˆ y | is one order of magnitude smaller than | Eˆ x |. Together with a decrease in the spectral content as we propagate in the z-direction, we observe that | Eˆ y | vanishes in the far-field region. In conclusion, according to this model, the in-plane electric field in the near-field region is dominated by the x-component, so we can assume that the incident polarization is conserved. The same model predicts that the far-field emission is represented by the superposition of an electric dipole field and a magnetic dipole field from an electric dipole (Peff ) and a magnetic dipole (Meff ) Peff = Meff

a3 E 0⊥ 3π

2a3 =− H0// 3π

(10.11)

located at the aperture’s center, and directed along the normal and along the plane aperture, respectively, as indicated in Fig. 10.5. In particular, for the case of a plane wave at normal incidence E 0⊥ = 0, the radiation will thus be described by the single magnetic dipole emission. Disagreements between the Bethe–Bouwkamp theory and experiment have been seen when monitoring the angular distribution of the far-field radiation emitted by a 60 nm diameter aperture SNOM fiber-probe [63]. The model in fact predicts an angular distribution I(θ) = cos2 θ when the detector is scanned in the plane containing Meff , and I = 1 when scanning in the plane perpendicular to Meff .

Fig. 10.5. In the Bethe–Bouwkamp model the far-field emission of a subwavelength aperture is described by two dipole fields, one electric and one magnetic, aligned orthogonal and parallel, respectively, to the screen plane

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Interestingly, the experimental data are well fitted by a linear combination of two orthogonal dipoles, one magnetic and one electric, both in the aperture plane, with the electric dipole aligned parallel to the incident electric field. A more exhaustive explanation of the experimental results is given later [64] by using the quasi-multipole method applied to a conical geometry, which is closer to the actual shape of the SNOM probe than the infinite screen assumed by Bethe. The two effective dipoles associated with the far-field emission of an aperture located at the apex of a cone with half angle β turns out to be: a3 1 E0 Peff = √ 4 2 sin3 β

1 a3 Meff = 2ˆz × Peff = √ H0 2 2 sin3 β

(10.12)

Figure 10.6 shows the spatial dependence of the radiated power in the plane containing the electric dipole Peff (the P plane). In particular, the far-field emission in the z-direction is polarized along the incident polarization direction: 3k2 a3 exp(ikr) EFF = √ E0 r 4 2 sin3 β

(10.13)

with k = 2π/λ. This means that the polarization of the incident field is preserved in the far-field emission of the SNOM when we observe along the probe axis. Problems with the Bethe–Bouwkamp representation also arise in the near-field. Discrepancies have been found, even from a qualitative point of view, between theory and experiment [65, 66]. Drezet et al, in particular, have proposed a simple static model capable of explaining the spatial patterns of fluorescence measured on small fluorescent beads, in disagreement with the Bethe–Bouwkamp representation of the fields. The field is here characterized by a charge distribution located along the metal coating (see Fig. 10.7a) whose density is given by σ = σ0 /Rn cos φ

(10.14)

for any point (ρ, φ, z) on the metal coating. Here φ is the angle with respect to the polarization plane, R the distance from the rim, and n = 1−π/(2π −β) is a function

Fig. 10.6. Orbital representation of the far-field angular power radiated in the P plane by the SNOM aperture. Figure redrawn from [64]


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Fig. 10.7. (a) Sketch of the tip. (b) xz-plane cross-section of the electric field lines (E x black lines, E z gray lines) in the proximity of the tip. The incident field propagates along the zdirection and is x-polarized. Reprinted with permission from [66]. Copyright (2004) EDP Sciences

of the corner angle β related to the cone angle α = π − 2β [67]. The electric field produced by such a charge distribution can therefore be numerically computed from E=

σ

r dS r3

(10.15)

where r is the distance vector from the surface element dS. The results (see Fig. 10.7b) show that, in contrast to the Bethe–Bouwkamp model, a strong E z component is expected in proximity of the aperture, as for the case of tightly focused beams [68–71]. The |E|2 distribution in the near-field is equally dominated by the x- and the zcomponents, and in particular (see Fig. 10.7b) E x is highest at the center, while E z is a maximum at the aperture rim. In summary, we have seen that, for the case of a linearly polarized incident beam, the aperture preserves the polarization state for both near- and the far-field emission. In addition a strong longitudinal component of the electric field appears in the near-field. This result is of high importance from an experimental point of view, since it suggests that the near-field polarization state (the one that actually excites the sample, but which is difficult to measure directly) can be controlled by measuring the polarization of far-field emission along the probe axis direction (which can be easily done).

10.4 Experimental Implementations Since the first demonstration of polarization contrast in SNOM [15], several experimental implementations have been proposed. In this section we will review some of most the relevant schemes, from the static polarization approaches to the polarization modulation designs in both the illumination and collection operation modes. 10.4.1 Static Polarization SNOM The first demonstration of polarization contrast was provided in 1992 by Betzig et al [15] using the apparatus sketched in Fig. 10.8a. A laser beam is coupled to

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Fig.10.8.(a) Schematic of the SNOM apparatus used for static control of both the illumination and of the transmitted polarization states. ND: neutral density filter; PMT: photomultiplier. Redrawn from [15]. (b) The reflection-mode apparatus uses a parabolic mirror for light collection. The incident polarization state is static, while a PEM is used to analyze the Kerr rotation of the scattered light. Reprinted from [73], Copyright (1998) with permission from Elsevier

an optical fiber SNOM probe ending with an aperture with a diameter in the 50– 250 nm range. The aperture is thus scanned by means of a piezoelectric actuator on top of the sample within the near-field region, at a constant distance or at a constant height. Alternatively, the fiber probe can be attached to a static holder and the sample is then scanned through the piezo actuator. Tip–sample distance control is accomplished by shear-force detection and stabilization based either on optical [8, 9] or non-optical means [10]. In the transmission-mode set-up of Fig. 10.8a, light scattered by the sample is collected underneath by a microscope objective and sent to the detector. Due to the low light levels, usually in the 10–100 nW range, a photomultiplier tube (PMT) is employed. The use of tapered fiber probes has made a straightforward application of the polarization microscopy concept to SNOM possible. The polarization contrast is in fact implemented by carefully controlling the polarization of the light transmitted by the aperture, and by using a polarization analyzer in front of the detector. Controlling the polarization state of the radiation emitted by the aperture is quite a delicate task. Although the laser light injected into the fiber can be polarized with an extinction ratio of several orders of magnitude, this ratio strongly decreases at the output of the aperture due to the birefringence of the fiber and dichroism from the aperture’s asymmetry. This phenomenon is shown in Fig. 10.9. The three polar plots refer respectively to the polarization states of light emitted by the laser (a), laser light propagating through 5 m of a single-mode optical fiber (b), and laser light coming out of a SNOM aperture located at the apex of 1.5 m of a single-mode optical fiber (c). The laser beam extinction ratio


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Fig. 10.9. Polar plots showing the modification of the laser beam polarization state (a) when passing through 5 m of single-mode optical fiber (b), and through a SNOM fiber probe with a nominal apical aperture of 100 nm (c). The extinction ratio decreases to 1 : 4 in the last case

(> 1 : 2000) decreases to 1 : 6 when passing through the optical fiber, and to 1 : 4 at the output of the SNOM aperture. This phenomenon has driven the development of techniques designed to compensate, partially, for such undesired polarization changes. As shown in Fig. 10.8a, a set of half- and quarter-wave plates can be used to change the polarization state of the injected light, obtaining a linear polarization state with a high extinction ratio at the SNOM aperture. Reflection mode operation, suitable for both transparent and opaque samples, was first implemented with the help of a parabolic mirror [72, 73], allowing for a maximum numerical aperture. The set-up shown in Fig. 10.8b, quite similar to that of Fig. 10.8a, employs a different technique both to control the incident polarization, and to analyze the polarization of the reflected light. The first task is accomplished by means of a fiber-loop polarization controller. This commercially available device contains three fiber loops: the first is a single loop, approximately representing a quarter-wave plate; a second double loop corresponds to a half-wave plate; the third one is a further quarter-wave plate. All three wave plates can be adjusted independently by mechanical rotation of the loops, altering the stress and thus the birefringent properties of the quartz fiber core. The polarization state of the scattered light is analyzed by modulation techniques that employ a PEM and a linear analyzer prism, combined to give lock-in detection. Defining φ as the rotation angle of the PEM, θF as the Faraday polarization rotation, and η as the ellipticity of the sample, the first and second harmonic signals (ω, 2ω) provided by the lock-in will give: Iω ∝ −J1 (∆0 ) sin(2η) I2ω ∝ J2 (∆0 ) cos(2η) sin [2 (φ − θF )]

(10.16)

where ∆0 is the retardation amplitude of the PEM, and Jn are the nth-order Bessel functions. Therefore Iω represents changes in the Kerr or Faraday ellipticity, and I2ω can be considered to be a measure of the Kerr or Faraday rotation. This detection scheme should, however, be used with extreme care when the sample reflectivity shows local changes, since the detector is still sensitive to overall changes in the total intensity of the light reflected by the sample, producing fictitious optical contrast. In order to remove this problem, the scattered light must be split

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into two orthogonal polarization beams, whose intensities (A, and B) are measured by two PMTs. The two signals are thus sent to an analog converter that performs the operation (A − B)/(A + B), which turns out to be proportional to 2θF [73]. Depolarization measurements are a very effective way to “view” single nanostructures, due their anisotropic light scattering properties. Such a task can also be accomplished in the so-called illumination-collection SNOM configuration [74,75]. Here the aperture acts as both a nanosource and a nanoantenna, illuminating the sample and collecting the scattered light. A typical apparatus is shown in Fig. 10.10 [75]. A Glan–Thompson polarizer (P1) is used to define the polarization of a laser beam with a high extinction ratio. The polarization state of the laser beam is thus adjusted by a half- and a quarter-wave plate, in order to cancel out the retardation effects occurring during propagation into the fiber. The polarization state at the aperture is controlled by monitoring the polarization of the far-field light emitted by the aperture. The use of 1 µm core fibers for double-tapered fiber probes, as well as short fiber lengths, usually contributes to the preservation and stabilization of the light polarization state. The excitation beam is reflected by a beam splitter (BS) and coupled to the fiber. The backscattered light, collected by the aperture itself, is compensated for the retardation induced by the fiber probe through a set of quarter-wave half-wave plates, and then detected. An interesting detection scheme employs a balanced polarization bridge for high sensitivity detection of the polarization changes. A half-wave plate is used to rotate the polarization by 45◦ and a polarizer prism (P2) splits the beams with orthogonal polarizations toward two detectors. Looking at the difference between the two signals, the depolarization properties of the sample provide high

Fig. 10.10. Experimental set-up of the IC depolarization SNOM. BS: beam splitter; P1 and P2: Glan– Thompson polarizers; PD: detectors. The inset shows a double-tapered SNOM fiber probe with an aperture in the 100 nm range. Figure reprinted from [75] with permission of the author. Copyright (2004) Institute of Physics


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optical contrast that can be used to locate single structures with nanometer-scale dimensions. 10.4.2 Polarization-Modulation SNOM: Illumination Mode The main advantage of PM-SNOM over conventional polarization microscopy is clearly related to the enhanced spatial resolution. Moreover, acquisition is performed point-by-point, as opposed to imaging microscopy based on CCD detectors, which allows lock-in demodulation techniques to be used, enhancing the signal-to-noise ratio. In PM-SNOM the direction of the linearly polarized near-field radiation is rotated at high frequency for all the sample locations. Therefore, information about the magnitude and orientation of the optical anisotropy can be recovered simultaneously. Two experimental approaches have been used to rotate the polarization of the probing light, using PEMs and EOMs. The PEM [50,76] provides a sinusoidal phase change δ ∝ sin(ωt), while the EOM [28,48,77,78] is characterized by a linear phase change δ ∝ ωt. The respective Jones matrices are reported in Table 10.2. A less expensive scheme, employing a rotating linear polarizer [79], will be described in Sect. 10.4.3. From a physical point of view it is equivalent to an EOM since the retardation phase varies linearly with time. Our EOM-based PM-SNOM is depicted in Fig. 10.11. Polarization control is accomplished by applying a linear polarizer followed by the EOM (Quantum Technology, Lake Mary, FL, USA) and a quarter-wave plate. The principal axes of the linear polarizer and the quarter-wave plate are aligned in the same direction, and rotated by π/4 with respect to the EOM ones. The resulting polarization state will thus be Jin = QW · R(−π/4) · EO · R(π/4) · L · Jlas

(10.17)

In other words Jin = [cos(ωt/2), − sin(ωt/2)], which corresponds to a linear polarization state whose direction rotates with time to an angle ωt/2. The 0 − 2π phase delay is achieved by varying the voltage applied to the EOM with a periodic saw-tooth with a typical frequency of a few kHz. The light is thus coupled into the SNOM fiber probe. In order to attenuate the intensity modulations related to the spurious effects inherent to the polarization-dependent coupling efficiency, an electronic feedback loop is used. A photodiode detects part of the light coupled into the fiber [79], and the feedback loop acts on an acousto-optic modulator in order to keep the coupled power constant. A fiber paddle (not shown in the figure) acts on the SNOM fiber to compensate for its birefringence. The light transmitted by the sample is detected through a PMT whose signal is split into two channels. A low-pass filter is used to extract the DC component. Lock-in detection, synchronous to the EOM modulation, is used to extract the AC modulated component, providing both amplitude and phase. Assuming the birefringence and dichroism principal axes of the sample to be coincident, using the notation of Table 10.2, we can calculate the PMT signal as T(Jin ) = |M · Jin |2 /|Jin |2 where the sample’s Jones matrix is Msample = R(−θ) · B(δ) · D(u, v) · R(θ)

(10.18)

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where θ is the angle between the major axis and the reference direction. The measured signal will thus be T = DC + AC cos (2θ + ωt) u+v 2 u−v AC = 2

DC =

(10.19)

where the DC and AC (amplitude and phase) are the true measured signals. We note that, since no analyzer has been placed in front of the PMT, the sample birefringence has no effect on the measured signal. The sample’s dichroic ratio is thus given by: u − v = AC (10.20) γ = u + v DC Equation (10.20) does not take into account the residual dichroism due to the aperture’s asymmetry. To get a quantitatively significant measurement of the sample’s dichroism it is necessary to deconvolute the tip’s dichroism, especially if the latter is

Fig. 10.11. Experimental set-up of a PM-SNOM using an electro-optic modulator (EOM) to modulate the polarization state of the incident light. The acousto-optic modulator (AOM) is used to modulate the laser power, in a feedback loop stabilizing the actual power injected into the SNOM probe. Figure reprinted from [48] with permission of the author. Copyright (2004) Institute of Physics


339

greater than the sample’s dichroism. To accomplish such a task [80] we first describe the tip’s dichroism with the Jones matrix √ ut 0 √ (10.21) Mtip = 0 vt Assuming that the major absorption axis of the tip correlates with the reference x-axis, the Jones matrix of the overall system will be given by Msystem = Msample · Mtip

(10.22)

with Msample given by (10.18). If we define a = (u + v)/2, ∆a = (u − v)/2, b = (u t + vt )/2, ∆b = (u t − vt )/2, a will describe the polarization-averaged transmission coefficient of the sample, and ∆a/a its dichroic ratio. b and ∆b/b will be the corresponding parameters for the tip. Assuming the tip’s dichroic ratio to be small, so (∆b/b)2 1, we will have ∆a∆b ab and thus T = a · b + ∆a · ∆b · cos 2θ + a · ∆b cos ωt + b · ∆a cos (2θ + ωt)

(10.23)

≈ a · b + a · ∆b cos ωt + b · ∆a cos (2θ + ωt) = DC + AC cos(2φ + ωt) (10.24) In order to measure the tip’s dichroic ratio, the sample must be removed. In the Jones formalism this corresponds to assuming Msample for the identity matrix, so a = 1, ∆a = 0. Equation (10.23) reduces to (10.20), so γtip = ∆b/b = AC/DC. To deconvolute γtip from the measured signal, and obtain sample properties, we can follow the method suggested by Wei et al [80]. We acquire both the amplitude AC/DC and the phase φ of the optical signal by means of a lock-in amplifier referenced to the modulation frequency ω. Solving (10.23), we can calculate and reconstruct (via software) the dichroic ratio and the orientation angle maps " 2 2 ∆a AC AC γsample = = cos (2φ) − γtip + sin (2φ) a DC DC (10.25) 2θ = arctan

AC DC AC DC

sin (2φ)

cos (2φ) − γtip

Similarly, the sample’s dichroic properties can be reconstructed by acquiring both the AC 0 in-phase and the AC π/2 quadrature component from the lock-in and dividing by the DC signal coming from the low-pass filter. In fact, rewriting (10.23) as T = ab + (a∆b + b∆a cos 2θ) cos ωt − b∆a sin 2θ sin ωt = DC + AC 0 cos ωt + AC π/2 sin ωt we obtain

(10.26) (10.27)

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γsample

∆a = = a

"

AC 0 − γtip DC

2

AC π/2 + DC

2 (10.28)

2θ = − arctan

π/2

AC /DC AC 0 /DC − γtip

Finally, we observe that for γtip ∼ 0.1 (typically γtip ∼ 0.05 [77, 80]), the error on γsample due to our approximations is less than 1%. Moreover, we note that if γtip 1 (10.25)–(10.28) are still valid even if the sample’s dichroic ratio is smaller than γtip . The second approach to PM-SNOM employs a PEM to modulate the polarization state of the incident beam. The experimental apparatus, very similar to the one using the EOM, is depicted in Fig. 10.12. A monochromatic laser source (in our case a laser beam) is passed through a linear polarizer oriented at 90◦ . The beam thus encounters the PEM (Hinds Instruments, Hillsboro, OH, USA) whose modulated axis is oriented at 45◦ . The PEM introduces a phase retardance varying sinusoidally with time into the +45◦ polarization component, leaving the −45◦ one unaffected (see the associated Jones matrix in Table 10.2). Therefore, the PEM acts as a linear retarder with sinusoidal retardance d sin(ωt), where d is the modulation amplitude and ω = 2π f the angular frequency. A quarter-wave plate oriented at 0◦ transforms

Fig. 10.12. Set-up of a PMSNOM working in transmission mode. A photo-elastic modulator (PEM) generates a linear polarization state whose direction is modulated with time. Reprinted with permission from [93]. Copyright (2002) American Institute of Physics


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the elliptically polarized light into a linearly polarized beam which is subsequently coupled to the SNOM fiber probe. The polarization state of the injected beam is thus Jin = QW · R(−π/4) · PE · R(π/4) · R(−π/2) · L · R(π/2) · Jlas

(10.29)

yielding Jin = [− sin(d/2 sin ωt), cos(d/2 sin ωt)]. Fiber paddles allow us to compensate for any retardance induced by linear birefringence in the SNOM fiber. The light transmitted by the sample is collected by the microscope’s objective. Sample birefringence variations are measured by placing a circular polarization analyzer (or a quarter-wave plate at 0◦ followed by a linear polarizer at −45◦ ) before the detector, while dichroism is studied without such an analyzer. With the circular polarization analyzer in place, changes in the magnitude and orientation of the sample LB provoke a change in the intensity of the light detected by the PMT. In the absence of LD the intensity measured by the PMT can be calculated as T(Jin ) = |M · Jin |2 /|Jin |2 where M = B(δ) (see Table 10.2). Thus T=

1 1 1 + (sin δ sin 2θ) cos (d sin ωt) − (sin δ cos 2θ) sin (d sin ωt) (10.30) 2 2 2

where δ is the magnitude of the phase retardance induced by the sample, and 2θ is the orientation of the faster optical axis. The Fourier expansion of (10.30) leads to 1 T = (1 + J0 (d ) sin δ sin 2θ) + (sin δ sin 2θ) J2n (d ) cos (2nωt) + (10.31) 2 n − (sin δ cos 2θ)

J2n+1 (d ) sin [(2n + 1) ωt]

(10.32)

n

containing a DC term, an AC term proportional to (sin δ sin 2θ) that repeats at even harmonics, and an AC term proportional to (sin δ cos 2θ) that repeats at odd harmonics. Here Jn are the nth-order Bessel functions of the first kind. By choosing a modulation amplitude d corresponding to a zero of J0 (such as d0 = 2.405) the DC term becomes independent of the sample retardance. Therefore, it can be used to normalize the first and the second harmonic AC amplitude signals provided by the lock-in, corresponding to √ (AC1 /DC)rms = − 2 · J1 (d0 ) sin δ cos 2θ (10.33) √ (AC2 /DC)rms = 2 · J2 (d0 ) sin δ sin 2θ providing (AC2 /DC)rms 2θ ≈ − arctan 1.202 · (AC1 /DC)rms " (AC2 /DC)rms 2 (AC1 /DC)rms 2 δ ≈ arcsin + √ √ 2 · J2 (d0 ) 2 · J1 (d0 )

(10.34)

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Such expressions are no more valid if the sample is significantly dichroic. In that case the sample’s dichroism can be measured by removing the circular polarization analyzer in front of the detector. Repeating the same calculations we find that √ u−v sin 2θ (AC1 /DC)rms = − 2 · J1 (d0 ) u+v √ u−v cos 2θ (AC2 /DC)rms = − 2 · J2 (d0 ) u+v

(10.35)

which permit the sample diattenuation (u − v)/(u + v) and the orientation of the dichroic axes 2θ to be calculated in a straightforward manner. Finally we note that (10.35) is valid even if the sample has birefringence, as long as the birefringence principal axes are coincident to the dichroism ones. Again, the fiber-induced retardance can be taken into account by removing the sample and acting on the paddles in order to zero the harmonic signals [76]. A procedure similar to the one described for the EOM case can be followed to deconvolute the residual tip dichroism [76], and determine the true retardance or diattenuation properties of the samples. 10.4.3 Polarization-Modulation SNOM: Collection Mode Quantitative measurements of the optical anisotropy using illumination-mode PMSNOM suffer several limitations. The phase retardation induced by the SNOM fiber must be compensated for by manipulating the polarization state of the light injected into the fiber. Moreover, the aperture’s dichroism must be measured and deconvoluted from the optical signal for each SNOM probe in order to obtain true values for sample properties. A further problem, common to confocal micropolarimetry, arises from the use of the Jones formalism itself for the calculations. The Jones formalism, as we have seen in Sect. 10.2.1, is based on a plane wave approximation, so it includes the assumption that light passing through the sample is collimated with a single well defined wave vector. The components of the Jones vectors can be defined only if such an approximation holds. In the vicinity of a SNOM aperture the field is far from being a plane wave. As we have analyzed in Sect. 10.3, the near-field emission can be considered to be a sum of plane waves, each component having its own wave vector and intensity. Moreover, a non-negligible longitudinal component of the electric field (E z ) is present. Confocal micropolarimetry has a similar drawback, related to the use of high numerical aperture objectives (NA = 1.4), which results in noncollimated radiation with strong longitudinal electric fields at the focus. Török et al [81] have proposed a generalization of the Jones formalism valid for use with confocal microscopy. A similar treatment for SNOM is yet to be developed due to the difficulty of modeling SNOM probe fields within the Jones formalism. The use of SNOM fibers to collect the light scattered by the sample bypasses such problems. Collection mode SNOM [82] is the Babinet complement of the illumination mode approach [83]. As shown in Fig. 10.13, light is focused by a lens onto the sample and then scattered light is collected point-by-point by the SNOM


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Fig. 10.13. (a) Set-up of a PM-SNOM working in collection mode in the transmission configuration, using a thermal xenon light source to excite the sample. Reprinted with permission from [84]. Copyright (2002) American Institute of Physics. (b) Collection mode PM-SNOM apparatus implemented for orthogonal light injection. See text for details

aperture. The incident polarization state is therefore prepared well. The use of lenses or long working distance objectives, characterized by a relatively small aperture, fulfill the plane wave approximation, justifying the use of the Jones formalism. From the SNOM point of view, this configuration cancels out the influence of the fiber birefringence, although the aperture’s intrinsic dichroism is a problem that is still present, requiring the application of the deconvolution procedures described above. The collection mode PM-SNOM scheme suggested by Wei et al [84] works in transmission and is depicted in Fig. 10.13a. Light coming from a xenon lamp is spectrally filtered by a monochromator and can be considered to be totally unpolarized (a glass slide can be used to correct the polarization effects of the monochromator mirrors). The light beam passes through a rotating linear polarizer (ω = 140 Hz). The polarization-modulated beam is then focused on the sample by a lens (again a glass slide is used to compensate for mirror-induced polarization effects). Light transmitted by the sample is collected by the SNOM fiber probe and fed directly into the PMT. The photocurrent is amplified and fed to both a photon counter, providing the DC component, and to a lock-in amplifier in order to get the AC amplitude and phase (2φ) components. The AC/DC ratio is calculated point-by-point and recorded together with the phase signal 2φ, providing the map of the true sample’s dichroic ratio. The use of a polychromatic light source is a significant advantage when measuring various crystalline materials, particularly when obtaining information on the resonant absorption properties. The use of a rotating polarizer makes the apparatus independent from the wavelength. We have recently introduced a different collection mode PM-SNOM design, more suitable for studying the light conduction properties of photonic structures [85]. The apparatus (Fig. 10.13b) has been expressly designed for the characterization of photonic waveguides in which the light must be injected and where it propagates parallel to the sample surface, orthogonally to the SNOM probe. In order to obtain a rotating

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linear polarization state, the linearly polarized beam (from a HeNe (632.8 nm) or a HeCd (442 nm) laser) is first passed through a quarter-wave plate rotated at 45◦ with respect to the laser polarization direction. The circularly polarized beam generated in this way is then fed through a rotating linear polarizer. The rotary motion of the polarizer is achieved using the motor of a modified chopper device, providing accurate control of both the rotation frequency ( f = 100−200 Hz) and phase. Within the Jones formalism, assuming a laser polarization state Jlas = (1, 0), the final polarization is given by Jin = R−1 (ωt) · L · R(ωt) · R−1 (π/4) · QW · R(π/4) · Jlas

(10.36)

yielding Jin = (cos ωt, sin ωt), corresponding to a linear polarization state whose direction rotates with time of to angle ωt. The laser beam is focused perpendicularly onto the sample by means of a 50 mm focal length lens, generating a spot size of about 30 µm. The scattered light is collected by home-made pencil-type probes [11, 12] or by CrAl- and gold-coated probes purchased from Nanonics Inc. and JASCO Inc., respectively. The light is detected by the PMT and fed to a photon counter whose analog output is both passed along to a lock-in amplifier (time constant ∼ 100 ms) for phase and AC amplitude analysis, and is used in order to derive the DC component. The sample is scanned by means of a 1 inch piezo-cylinder capable of 32 × 32 × 8 µm3 , or by means of a set of three smaller piezo-tubes capable of 15 × 15 × 6 µm3 . Our experimental approach is especially useful for obtaining qualitative and quantitative maps of the anisotropic scattering properties. The phase map in this case will provide information on the preferential polarization direction of the scattered light.

10.5 Applications of SNOM Polarimetry The application of polarization contrast has led to new ideas in the SNOM field. Recent discussions about whether it is possible to obtain true subdiffraction resolution with uncoated fiber tips [86] have in fact found a partial positive answer through the implementation of the depolarization configuration. Here [74] linearly polarized light is used to illuminate the sample and, with the IC-SNOM scheme, the orthogonal polarization is detected. Since the radiation back-reflected by the probe taper is still polarized along the direction of incidence, the depolarization configuration is in principle capable of completely suppressing the huge background, enhancing the scattering from small objects that partially depolarize the incident radiation. Moreover, the depolarization mode is capable of imaging best those edges perpendicular to the incident field polarization, especially in metallic samples. Figure 10.14a,b illustrates the polarization dependence of the near-field scattering from a rectangular grid sample produced by evaporation of chromium on glass followed by electron beam lithography and subsequent wet etching. We note how rotating the incident polarization by 90◦ leads to the observation of those structures orthogonal to the electric field. Such images provide an unambiguous demonstration of the possibility of carrying out near-field imaging with uncoated probes, and of the added value from the polarization contrast.


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Fig. 10.14a,b. Illumination/collection depolarization SNOM images of a chromium test pattern engraved on a glass substrate. The light polarization is slightly elliptical, as indicated in the figure. The two insets show the line profiles drawn in the same spatial location, outlining the different optical responses of the sample to the different incident polarization states. Reprinted with permission from [74]. Copyright (2002) American Institute of Physics

10.5.1 Polarization Responses of Photonic Waveguides Light sources like bulbs or fluorescent lights are quite inefficient despite being relatively modern. Semiconductor light sources, like LEDs or laser diodes, are much more efficient. These light sources have internal quantum efficiencies of about 90%. However, only 5–30% of the generated light can escape; the rest is reabsorbed or lost as heat. Photonic waveguides become very important in this context. Distributed Bragg reflectors (DBRs), for example, are gratings used in integrated laser technology to decrease standard mirror losses, resulting in a reduced lasing threshold and in a further reduction in the size of the device [87, 88]. It has been shown in fact that high reflectivity mirrors can be formed by deeply etching semiconductor/air DBRs, enhancing the optical feedback into the laser cavity [89]. The complete optical characterization of such photonic structures is thus very important. The possibility of characterizing the reflectivity response of DBRs, and in particular the dependence on the polarization state, prior to their implementation in actual laser systems is even more appealing [90, 91]. In the experiment presented here we have applied PM-SNOM to get an insight into the polarization properties of DBR structures. The DBR sample was bprepared by focused ion beam lithography on a 4 µm GaN layer deposited on a sapphire substrate. The DBR slabs are about 20 µm long and are separated by about 1 µm with an air gap between them of 244 nm; the thickness of each slab is about 680 nm and they are 1.5 µm deep. The peculiar structure of DBRs – the presence of alternated air/semiconductor (n GaN = 2.5) structures with a thickness comparable to the wavelength of light and a length on the 10-µm scale ( λ) – makes them suitable for theoretical 2-D descriptions, and for test experiments to prove the different polarization-dependent light scattering properties. First experiments have been performed with an experimental set-up similar to the one shown in Fig. 10.8a [92]. Excitation is accomplished by means of a CrAl-coated SNOM fiber probe (100 nm nominal aperture) at λ = 670 nm. The light injected into the fiber is p-polarized. No polarization analyzer has been placed before the detector. To show the strong dependence of the optical response on the incident polarization we have carried out line-scan mapping where we periodically change the polarization of the near-field excitation, keeping it fixed while always scanning

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Fig. 10.15. 3-D rendering of the optical transmission profiles (a) acquired scanning the same line (10 µm), while varying the near-field incident polarization. The corresponding topography profiles (b) show only slight changes due to mechanical and thermal drifts

the same spatial line. Every ten line-scans, the near-field polarization is changed by bending and twisting the SNOM fiber (by exploiting the stress-dependent birefringence properties of the SNOM fiber). Figure 10.15a shows a 3-D rendering of the spatial patterns of the optical transmission signal, showing different modulation patterns and shifts as a function of the incident polarization. The topography pattern (Fig. 10.15b) is reported to show that only a slight continuous shift takes place during the repeated scans of the same line, with no relevant implications for the optical signal. Further investigations have been performed with the collection mode SNOM apparatus (Fig. 10.13b). This is definitely the most suitable configuration for characterizing the optical properties of DBRs, since the illumination geometry exactly reproduces the operational conditions in real laser devices. First measurements have been aimed at highlighting the variation in behavior of the DBR as a function of incident fixed orthogonal polarization states. Since the laser beam (632.8 nm) has been focused into the GaN layer, light gets confined therein, emerging from the waveguides only when scattered from the DBR planes. As a result the DBR appears as a bright structure on a dark background. Figure 10.16 shows the topography (A) and the SNOM light collection maps, observed upon exciting with p- (B) and s-polarized light (C). The polarization plane is defined here by the laser beam and the orthogonal SNOM probe axis directions, as usual. With the aid of the line profiles (Fig. 10.16G), we first observe that light emission arises in the vicinity of the air gaps; moreover, we note that the influence (interference) of the plane reflections shows up for the fourth air gap (counting from the left). Differences between the two different illumination polarizations are apparent in the zoom shown in Fig. 10.16E,F and in the respective line profiles (Fig. 10.16H, squared zones), together with more detailed evidence of the collective effects of the DBR planes. To get more of an insight into the anisotropic scattering properties, we have applied the modulation techniques. The polarization sensitivity of the sample to the incident polarization (λ = 442 nm) is better evidenced in the AC phase map (Fig. 10.17B), acquired simultaneously with the DC optical map (Fig. 10.17A). The modulations observed in the DC map show the different scattering properties of the DBR slabs as a function of the position. The AC phase map qualitatively shows that the scattering properties are also polarization-dependent. Using the line profiles in Fig. 10.17C, we can see how the phase map accurately reproduces the features of the DBR, and these are slightly shifted with respect to the DC map (∼ 150 nm). The phase fluctuations evident on the right hand side of Fig. 10.17B are due to the very low light levels detected outside the DBR planes. The sudden phase changes, corresponding to the dark areas evident in the zoom of Fig. 10.17D, are simply due to 2π phase wraps.


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Fig. 10.16. Topography (a, d), and collection mode optical maps related to (b, e) p-polarization and (c, f) s-polarization excitation at 632.8 nm. The white arrows indicate the direction of propagation of the excitation light. (g) Line profiles extracted from the 15 × 12 µm2 scans (a, b, c) and (h) from the 5 × 3.5 µm2 scans (d, e, f)

Fig. 10.17. DC optical (a) and AC phase maps (b, d) acquired with the PM-SNOM in collection mode, exciting at 442 nm. Profiles (c) extracted in correspondence with the lines in (a), and (b). Scan size: (a, b) 8.6 × 8.6 µm2 , (d) 4.0 × 4.0 µm2

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10.5.2 Measuring Stress-Induced Birefringence Silicon nitride (SiN) is a material commonly used in microelectromechanical systems and micromechanical devices. Mechanical antireflection switches are optomechanical devices in which the reflection of light off SiN membranes is altered by deflecting the membrane, thus changing the air gap between the membrane and the substrate. Amorphous SiN films grown on silicon by chemical vapor deposition have high intrinsic stress, the nature and magnitude of which depend on the growth conditions. Stress can cause cracking and peeling in microelectronic devices, and causes warping in devices that contain unsupported SiN membranes. The presence of anisotropic stress causes anisotropic optical properties – local variations in the refractive index – that usually occur at subwavelength scales. Campillo et al [93] have used PM-SNOM to probe the anisotropic stress distributions in SiN membranes. The experimental setup is depicted in Fig. 10.12, exploiting a PEM oscillating at 50 kHz to modulate the incident polarization state (λ = 670 nm). The transmitted light is collected through a 100× microscope objective. Figure 10.18 shows a 1.5×1.5 µm2 scan of (a) the DC transmission, (b) the phase retardance magnitude, and (c) the diattenuation acquired on a SiN film 50 nm thick, in which an array of circular holes have been produced by electron beam lithography. The phase retardance and the diattenuation maps have been reconstructed from the ratios AC1,2 /DC of the measured quantities AC1,2 and DC from (10.33) and (10.35). The images show a regular pattern surrounding the holes. The retardance image shows spiral features of higher retardance, centered around the holes. The regions among the sets of holes are zones of low retardance. The retardance magnitude can be quantitatively estimated by PM-SNOM, and is of the order of 2 × 10−2 rad. The diattenuation image shows connected four-lobe structures, centered about the holes, with high diattenuation directly between two adjacent holes. The diattenuation variation for these lobed patterns is of the order of 1.6 × 10−2 .

Fig. 10.18. PM-SNOM images (1.5 × 1.5 µm2 ) of (a) the DC transmission, (b) the retardance magnitude, and (c) the diattenuation of a SiN film patterned with an array of circular holes with 70–10 nm diameter. Reprinted with permission from [93]. Copyright (2002) American Institute of Physics


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10.5.3 Polarization Anisotropy in Mesoscale-Structured Materials PM-SNOM imaging experiments of samples absorbing the excitation wavelength are usually referred to as “resonant PM-SNOM”. In such experiments the AC amplitude signal measures the local anisotropy of the extinction coefficient. The phase image can instead give an insight into the spatial orientation of the absorbing domains. Higgins and coworkers [28] have used resonant PM-SNOM to image single crystals of rhodamine 110. Figure 10.19a shows a topography image of crystals evidencing lateral dimensions on the micron scale and a thickness varying between 10 and 140 nm. Optical excitation at 514 nm is within the rhodamine absorption band. PM-SNOM measurements are carried out in illumination mode, collecting the transmitted light. An EOM rotates the incident linear polarization direction. The AC modulation amplitude (Fig. 10.19b) shows interesting structures. However, it is difficult to interpret, especially due to the lack of normalization with respect to the DC average transmission map. Modulations could in fact be due to cracks, or represent density concentration variations, leading to absorption modulations. On the other hand they could be simply due to changes in the light intensity scattered by the sample in the near-field, related to spatial differences in the average refractive index of the crystals, independent from the polarization anisotropy properties.

Fig. 10.19. (a) Topography, (b) AC amplitude and (c, d) phase maps of single crystals of rhodamine 110. The orientations of the dipoles shown by the arrows in (d) are calculated from the phase image (c). Reprinted with permission from [28]. Copyright (1996) American Chemical Society

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The phase map (Fig. 10.19c), being independent from spatial modulations of the polarization-averaged refractive index, gives a reliable measure of the local orientation of the absorbing domains. After calibrating the reference direction, achieved by substituting the sample with a linear polarizer sheet, the domain orientation can be reconstructed, as shown in Fig. 10.19d. Valuable information about the preferential alignment of the domains along the short edge of the crystals can be extracted from the phase map. As we have seen in the previous sections, the crystallization properties of organic/inorganic thin films play an important role in the performance of optoelectronic and microelectromechanical devices. Owing to the different absorptions of the two major axes of the crystalline domains, PM-SNOM is a powerful tool for investigating the crystalline structure of mesoscale domains formed during the growth process of thin films. Wei et al [84] have applied collection-mode PM-SNOM to investigate the crystallization domains of thermally annealed naphthaphenylene benzidine (NPB) thin films. These are commonly used as hole transport layers in organic light emitting diodes. It is thus very important to understand the properties of the crystal domains, since they strongly affect the hole mobility and efficiency of the final devices. NPB films of 80 nm thickness annealed at 120 ◦ C have been studied with the SNOM apparatus shown in Fig. 10.13a, using a rotating linear polarizer as a modulator. The analyzed sample is characterized by mesoscale aggregated domains of NPB molecules with an average roughness of 10 nm. Figure 10.20a,b provides precious information, since it clearly indicates that the domains have a crystalline structure. The dichroic image (Fig. 10.20a) has been corrected for the actual tip’s dichroism (∼ 8%) and provides information on the degree of crystallization of the domains. It worth noting that, since it has been constructed from the AC/DC ratio of the AC magnitude and the DC optical signal, it is insensitive to the polarizationaveraged changes in the refractive index. The phase change (Fig. 10.20b), also corrected for the tip intrinsic dichroism, provides quantitative information on the domain orientation. The apparent discontinuities are due to phase wraps of the lock-in amplifier.

Fig. 10.20. True dichroic ratio (a) and phase (b) images of a 80 nm NPB film, after deconvolution of the tip’s intrinsic dichroism. Reprinted with permission from [84]. Copyright (2002) American Institute of Physics


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10.5.4 Polarization Anisotropy in Polymers Highly conjugated terthiophene-based chromophores, which have been functionalized with alkyl linear lateral chains, can be dispersed in both the amorphous phase, and at the interface between the amorphous and the crystalline phase of polyethylene (PE). They can also be oriented by elongation stretching along the macromolecular direction, according to the pseudo-affine deformation scheme [94]. Stretched films have excellent dichroic properties in the absorption and fluorescence region of the chromophore molecule, suggesting interesting applications of these systems as linear polarizers. Several issues concerning the application of such materials are still the subject of debate, however. The homogeneity of the chromophore dispersion is crucial where device miniaturization is concerned, as in liquid crystal displays. Therefore, the molecular alignment efficiency must be carefully investigated and correlated with the growing parameters. Conventional micropolarimetry, being limited by diffraction, cannot be used to gain an insight into the features of the sample at the sub-200 nm scale. SEM, while offering suitable spatial resolution, does not provide information on the optical characteristics of the samples. Our group has recently applied PM-SNOM to stretched polymer films in order to provide a quantitative measurement of the local dichroic ratio in PE samples [48]. Samples are produced starting from a p-xylene solution of a terthiophene-derived chromophore and UHMW-PE. A casting process with solvent evaporation is used to obtain films of ∼ 100 µm thickness, followed by mechanical drawing carried out by hand-stretching over a hot plate. PM-SNOM in the transmission configuration (Fig. 10.11) was carried out to investigate the anisotropic absorption at 488 nm. We recorded the modulation amplitude AC and the phase signals from the lock-in synchronous with the EOM modulator. The DC signal, representative of the polarization-averaged transmission, was provided by a low pass filter. Quantitative information on the optical anisotropy, namely the dichroic ratio γ and the domain orientation θ, is thus recovered using (10.25). In our set-up, the fiber birefringence is compensated for by mechanically stretching the tightly bent fiber around a fixed point. The tip’s dichroism has been found to be much smaller than the expected dichroic ratio values for our samples (10–20%). In Fig. 10.21 we compare the maps acquired for as-cast samples (top pictures) and for films obtained after drawing (bottom pictures, mechanical drawing ratio of 30×). Elongated structures appear in the topography map (Fig. 10.21d) corresponding to the drawing axis, whereas a random morphology shows up in the as-cast sample (Fig. 10.21a). Such elongated structures appear in all of the samples, independently from the drawing ratio and the chromophore mixture, a result related to the particular growth mechanisms of the PE chains, which has been also demonstrated by other microscopy techniques. The maps of the optical quantities also show rather different features in the two samples. Once more the mechanical drawing leads to the appearance of elongated domains in both the γ and θ maps (Fig. 10.21e,f ) corresponding to regions with rather homogeneous linear dichroism values. In contrast, randomly shaped domains appear in the as-cast sample (Fig. 10.21b,c). In particular the phase angle spans over all the allowed range, and the dynamics of the dichroic ratio are larger (12% compared to 8% in the drawn sample).

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Fig. 10.21. Top pictures: (a) topography, (b) dichroic ratio, and (c) phase angle maps acquired for an as-cast sample. Bottom pictures: corresponding maps for stretched film. Reprinted with permission of the authors [48]. Copyright (2004) Institute of Physics Publishing

Both findings suggest a random distribution and alignment of the dye molecules. On the other hand, formation of elongated optically active domains in stretched films confirms the effect of mechanical drawing on the mutual organization of the molecules, possibly induced by a quasi-epitaxial growth of terthiophene molecules over the oriented polymer chain, enhanced by sample heating during the drawing process [95]. Block copolymers (BC) offer a particularly versatile way of creating the patterned arrays of refractive index required for photonic properties [96]. BC microphase separation, driven by the immiscibility of the end-connected constituent polymer chains (the blocks), produces a variety of domain motifs with a related set of 1-D, 2-D, and 3-D band structures [97], tunable through the BC composition [98]. Since the optical activity of single microphase domains and defect structures may dictate the device’s function, the advancement of photonic BC systems requires characterization tools specially targeted at analyzing the organization properties of materials, with deep submicron spatial resolution. By exploiting PM-SNOM techniques Fasolka et al [47] have provided optical visible light images with subdiffraction resolution of the BC morphology, simultaneously mapping the polarimetric parameters of major interest. The PM-SNOM measurements have been carried out with an apparatus similar to the one depicted in Fig. 10.12. A PEM is used to modulate the light polarization state, and Fourier decomposition [50] of the transmitted intensity is used to sample the diattenuation or, if a post-sample circular analyzer is in place, the birefringence of the sample. Figure 10.22 shows diattenuation measurements performed on a polystyrene-b-polyisoprene (PS-b-PI) BC sample, exhibiting a lamellar structure with an equilibrium periodicity of 240 nm. Thin (100 nm) sections have been sliced and deposited on a glass cover slip support. Images are centered on part of a single


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Fig.10.22.(a) Diattenuation and (b) diattenuation axis alignment of a single grain in a block copolymer sample. (c) Retardance and (d) fast axis alignment of a single polystyrene spherulite. Images are of size 4 × 4 µm2 . Reprinted with permission from [50]. Copyright (2003) Optical Society of America

grain showing occasional edge dislocations and lamellar separations, and they display stripe structures. The diattenuation and phase maps (Fig. 10.22a,b respectively) are calculated based on the simultaneous acquisition of the DC and the AC1,2 harmonic signals, as described in Sect. 10.3 (10.35). The diattenuation of this sample can be understood with the help of the theoretical near-field models described in Sect. 10.3. When the near-field polarization is oriented along the stripe direction, maximum absorption occurs. On the other hand, if the polarization is orthogonal to the stripes, a larger part of the field leaks out over the nonabsorbing PS domain, and more light is transmitted. Such an anisotropic absorption results in an effective diattenuation with the diattenuating axis (defined as the maximum transmittance axis) oriented perpendicular to the stripe. In this model, the diattenuation amplitude at the center of the stripe depends on its width. For stripes of width equal to or smaller than the tip aperture diameter, the diattenuation is largest when the tip is centered at the stripe. As the width surpasses the aperture in size, the diattenuation is maximum at some fixed distance from the stripe edge, and decreases towards zero in the center of the stripe. Over the less absorbing PS domain the situation is reversed. Retardance measurements have been demonstrated with the same technique [50] on thin-film PS spherulites. These disk-like crystallites consist of radially arranged lamellae of folded chains with intermediate amorphous domains. In this case, PMSNOM can give an insight into the crystal structures of these 2-D crystallites with extremely high spatial resolution. The pictures in Fig. 10.22c,d show the retardance and the fast axis alignment maps collected from a semicrystalline PS sample respectively. The maps have been reconstructed based on the acquisition of the DC and the AC1,2 harmonic signals acquired simultaneously, as described in Sect. 10.3 (10.34). The background retardance is assumed to be due to the SNOM probe. A single spherulite is visible. In this specimen, crystal growth was arrested before crystallite impingement was complete, so the spherulites are surrounded by amorphous PS. The images reveal that this amorphous material is also intercalated into the spherulite, as expected for semicrystalline specimens. Moreover, the radial arrangement of the crystallite lamellae is clearly apparent in the fast axis image, where defect structures near the nucleus can also be seen.

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10.5.5 Polarization Anisotropy in Photoluminescence Emission Luminous blue light emitting diodes (LEDs) have been recently realized, due to breakthroughs in the technology of growing Inx Ga1−x N/GaN/Al y Ga1−y N heterostructures [99–101]. Although a high threading dislocation density (108 – 1010 cm−2 ) is observed in GaN epilayers grown on sapphire substrates, an external quantum efficiency (ηext ) of about 20% is currently commercially available. Typically these devices emit over a spectral range from violet (400 nm) to blue (465 nm) [102]. The incorporation of indium at GaN sites is an effective way to suppress nonradiative recombination centers related to defects in the crystal [103]. The fluctuation in indium concentration along the surface also contributes to the capture of excitons in localized areas of potential minima where nonradiative pathways are blocked [104, 105]. Understanding the fundamental phenomena involved in the light emission process is important if we wish to achieve further development. Various high resolution optical investigations have been performed on these samples using SNOM in several configurations and other techniques in order to obtain novel information and to try to elucidate the optical mechanism involved [106–108]. As a novel application of polarization modulation techniques, we show here how the PM-SNOM approach can be used to investigate these intriguing solid state samples. To investigate our samples we implemented the fluorescence contrast approach on a PM-SNOM. The apparatus can operate with definite excitation polarization (circular polarization) followed by a rotating polarization analyzer (Fig. 10.23a), or as

Fig. 10.23. Experimental apparatus for photoluminescence PM-SNOM. (a) Circularly polarized light excites the sample and the PL polarization properties are analyzed through a rotating polarizer. (b) Rotating linear polarizer is used to excite the sample fluorescence


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a more conventional PM-SNOM in which the incident light polarization is modulated (Fig. 10.23b) and the analyzer in front of the PMT is removed. While the first configuration permits us to obtain information about the polarization anisotropy of the photoluminescence (PL) emission, the latter is aimed at gaining an insight into the dichroic absorption properties of the sample. The configuration in Fig. 10.23a is the most straightforward. A laser beam λ = 442 nm, is passed through a quarterwave plate in order to obtain a circularly polarized beam. The light is thus injected into the SNOM fiber probe and induces PL on the sample as in standard illumination mode SNOM. The PL, centered at λPL = 540 nm, is first filtered through a colored 500 nm film (a low-pass filter), and subsequently analyzed by a rotating linear polarizer. A PMT detects the light. The PMT signal is split and fed both to a lock-in amplifier synchronous with the polarizer rotation (providing the AC and phase components), and to the acquisition (this is the DC component); in this way we have two simultaneous images, a standard PL SNOM image (the DC signal) and its corresponding PM-SNOM one (either the AC or phase signal). Comparing the two images gives us extremely interesting information about the inner optical characteristics of our crystals. Figure 10.24a shows a typical DC photoluminescence SNOM map of InGaN. We can distinguish indium concentration fluctuation inhomogeneity, with large nonradiative domains in the center. Also, a localized radiative high recombination center (HRC) in the bottom right of the region is noticeable. Interestingly, the AC intensity polarization modulation map – Fig. 10.24b – shows a uniform pattern and only the radiative HRC is visible. This indicates that the whole region has no polarization properties; however the presence of a strong peak corresponding to the HRC position points toward the fact that the HRC emission has strong orientation properties of some kind. Photoluminescence phenomena can

Fig. 10.24. (a) DC PL map. Besides typical emission fluctuations due to indium concentration inhomogeneity, note the high recombination center associated with a quantum dot (QD). (b) AC amplitude map. Only the QD emission is apparent, indicating asymmetric carrier confinement for the particular QD center. Cross-sections along the line are reported below each map. Images are 7.5 × 7.5 µm2 , 256 × 256 pixels

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Fig. 10.25. DC PL map (a) and AC amplitude (b) acquired with the PM-SNOM. Domains of different polarization properties are indicated by the arrows. Cross-sections along the line are reported below each map. Images are 7.5 × 7.5 µm2 , 256 × 256 pixels

involve polarization properties in association with quantum dots (QD). Models point out that anisotropic strain can induce asymmetry in the carrier confinement, resulting in polarized PL emission, compatible with what is observed here [109]. The PM-SNOM apparatus, implemented for fluorescence contrast, is shown in Fig. 10.23b. In this case the excitation wave is a linearly polarized field produced by a quarter-wave plate followed by a rotating linear polarizer. This configuration is complementary to the previously described one, so the physical mechanisms involved in the photoluminescence process are different. Also, the polarizationmodulated beam travels a longer optical path before actually interacting with the sample. This fact complicates the spurious interactions and the interpretation of the images is then more subtle in this case. We show in Fig. 10.25 raw experimental results as an example of the interesting and complex data obtained. Figure 10.25A shows the DC photoluminescence SNOM map. The typical indium concentration fluctuations, modulating the PL emission intensity, are visible, and a linear domain of depleted radiation crosses the region diagonally. The corresponding AC PM-SNOM map (Fig. 10.25B) shows a complex behavior in which radiative recombination areas are represented as dark areas in some cases and as bright areas in others, outlining the absence of cross-talk between the two maps. Such a peculiar fact means that recombination can occur with or without a preferential polarization direction, depending on the emitting centers. Two domains that have different behavior in the polarization map are indicated by arrows. Interestingly, thin structured paths are visible in the lower part of the AC map. These structures are not apparent in the corresponding optical map, suggesting that polarization properties are structured in the inner crystal. These very attractive facts are provided by the sole PM-SNOM approach, which is therefore a promising method for future investigations on solid state optical materials that may lead to the development of new and more efficient optical devices.


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10.6 Conclusions In the near future, it is expected that photonic devices such as integrated light sources and waveguides, which are much faster and consume less power than their electronic counterparts, will be used extensively to link current silicon-based technology devices for information transport and elaboration. Near-field microscopy, which is capable of accessing the nanometer scale in the optical domain, allows us to enhance our comprehension of the local optical properties of such devices. In this chapter we have seen how implementing the polarization contrast on a SNOM framework can yield a new, powerful tool that can give us insights into the anisotropic optical properties of micro- and nanophotonic structures. In fact, absorption and/or emission processes at the nanometer scale define the optoelectronic behavior of these materials, and their anisotropy represents one of the most important parameters to be studied and understood for final device engineering. PM-SNOM tailors micropolarimetry to the nanometer scale. Quantitative measurements of the birefringent and dichroic properties of single nanocrystals, thin films, organic/inorganic photonic crystals and waveguides demonstrate the wide application spectrum of PM-SNOM in materials science and engineering. With its several experimental designs, PM-SNOM is still a topic of strong scientific interest. The development of PM techniques on SNOM frameworks capable of higher resolution and new, better polarization-conserving probes are just a few examples of the current research in this field. Acknowledgements. Many colleagues in our laboratories have contributed their thoughts and work to this short review. The authors warmly thank M. Labardi, S. Patanè, A. Ambrosio, M. Alderighi, N. Maghelli, L. Ramoino, E. Cefalì, F. Fuso and L. Pardi. We are indebted to C. Spadaro for the polarization characterization of the optical fibers. We are grateful to L. Vaccaro, P. Royer, G. Lerondel for stimulating discussions and to S. Fujita and N. Micali for carefully reading the manuscript and for fruitful discussions. We are also indebted to T. Nakagawa of Unisoku Ltd Japan for fixing and improving the electronics of our SNOM system. We acknowledge financial support from the Italy-Japan bilateral Project 2A2 “Surface analysis and nano-characterization of innovative materials by scanning probe microscopy techniques”. PGG acknowledges the Venture Business Laboratory for financial support.

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11 Focused Ion Beam as a Scanning Probe: Methods and Applications Vittoria Raffa · Piero Castrataro · Arianna Menciassi · Paolo Dario

11.1 Introduction When energetic ions hit the surface of a solid sample they lose energy to the electrons of the solid, as well as to its atoms. The most important physical effects of incident ions on a substrate are the following: sputtering of neutral and ionized substrate atoms (this effect enables substrate milling), electron emission (this effect enables imaging, but may also produce charging of the sample), displacement of atoms in the solid (induced damage) and emission of phonons (heating). Chemical interactions include the breaking of chemical bonds and consequently the dissociation of molecules (this effect is exploited during deposition). Thanks to advances in optical components, the focusing accuracy of an ion beam has steadily improved, thus facilitating the development of instruments in which ions are used for imaging, analysis and modification of near-surface properties with high spatial resolution (down to 10 nm). The ion source most frequently used with such equipment is liquid gallium, with a typical energy between 4 keV and 50 keV, although other metal sources and higher energies are also encountered. This technology is commonly called scanning ion or focused ion beam (FIB) technology. FIB systems based on high-brightness gallium liquid metal ion sources became commercially available in the late 1980s, although even today such instruments are relatively rare apart from semiconductor manufacturing facilities. The use of FIB systems as precision sectioning tools that can be used down to submicron scale is well documented, thanks to their ability to deposit metals and insulators on a micron scale. Recently, FIB systems have achieved a spatial resolution rivalling that of the traditional scanning electron microscope (SEM), giving them a respectable role as an imaging tool, in addition to sectioning and deposition capabilities. The following contribution includes a description of the main aspects of ion beam technology. It starts with an overview of the system and the typical processes involved (imaging, milling, gas assisted deposition and ion beam lithography). Then a review of the main FIB process applications is given, illustrated by practical examples: its use in materials science as an analytical technique, applications in the semiconductor industry, and its use in micromachining as fabrication tool.

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11.2 Description of the System 11.2.1 System Overview The main components of a traditional FIB system are the ion column, the working chamber, the gas injection system, the vacuum system and the workstation for the user interface. The ion columnis the key component of a FIB system (a schematic diagram is shown in Fig. 11.1). Generally the ion column is similar to the one used with a SEM and comprises a liquid metal ion source, two electrostatic lenses, one set of beam blanking plates, a beam acceptance aperture, an automatic variable aperture and an octopole deflector. The ion beam currents vary between 1 pA and 10 nA while the ion beam energy is usually between 4 and 50 keV. A vacuum of about 5 × 10−5 Pa is maintained inside the column. Commercial FIB systems generally use a gallium ion (Ga+ ) beam generated from a liquid metal ion source (LMIS) (see Fig. 11.2). The ion beam first undergoes refinement through the spray aperture and then it is concentrated in the first electrostatic lens. The variable aperture mechanism can tune the beam current over four decades, thus allowing high resolution imaging with fine beams and rough and fast milling with larger beams. In order to protect the sample from constant milling, the blanking deflector and the blanking aperture deflect the beam by diverting it from the aperture to a Faraday cup. The beam is focused to a fine spot, less than 10 nm, by the second electrostatic lens. A multichannel plate is placed near the sample to collect secondary electrons for imaging.

Fig. 11.1. Schematic diagram of a FIB ion column

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Fig. 11.2. A gallium liquid metal ion source (LMIS) (not to scale); commercial Ga+ LMISs have typical lifetimes of 1500 hours

The vacuum inside the column and work chamber is maintained through a system of vacuum pumps: a fore pump is used in conjunction with a turbo pump to evacuate the work chamber, and one or two additional ion pumps are used to reach the optimal pressure value. FIBs are also provided with a gas injection system which is able to deliver different type of gases used for metal or insulator deposition and gas-assisted etching. The injection system controls the optimization of gas parameters and makes the gas travel from a crucible through a needle to the point where the beam intersects with the sample surface. 11.2.2 Liquid Metal Ion Source (LMIS) The Ga+ ions are produced on the tip of a conical metal substrate (the emitter, usually made from tungsten) that is wetted by a thin liquid film of gallium from a reservoir (Fig. 11.2). By applying a high negative voltage to the extraction electrode, the gallium is stressed into a Taylor cone due to the equilibrium between electrostatic forces and surface tension. A tiny cusp (incipient jet) with an end radius of approximately ˚ protrudes, forming the end of the cone. Ion emission occurs at this incipient jet 20 A due to field evaporation [1]. The potential barrier that holds atomic electrons near the nucleus is deformed by applying very high electric fields to the field emission ion sources (about 1010 V m−1 ). As a consequence some electrons tunnel through the barrier, leaving behind a positively charged ion. The extraction current is typically 2 × 10−6 A. The gallium LMIS provides a brightness of about 106 A cm−2 sr−1 and a high angular intensity of about 17 × 10−6 A sr−1 [2]. Because of the high current density at the surface of the LMIS, the ion beam has an energy spread of 5 eV or more [3]. This energy spread is unavoidable and it is the most important factor that

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limits the beam diameter and beam current density [4]. The mechanism for this limitation is chromatic aberration: ions of different energies are focused at different point in the column, thus increasing the beam diameter. On the other hand, spherical aberration is not a limiting factor in ion column performance. In general the energy spread increases with increasing ion current from the source [5]. Virtual source size is another LMIS-related fundamental property. It seems to be caused by Coulomb repulsion normal to the direction of travel. It will ultimately limit the beam diameter and make it necessary to extract only the very central part of the emission cone through the beam defining aperture (spray aperture in Fig. 11.1). The source is responsible for another beam characteristic, which is the tail in the beam current profile. The beam current profile is Gaussian in the center, but several orders of magnitude below the peak the current distribution drops more slowly, in an exponential manner [6]. 11.2.3 Ion Optics The theory and design of charged particle “optics” (focusing systems) is a longestablished field [7]. Here we will only examine basic relationships dealing with the most important parameters of focused ion beams, such as probe size, beam current, beam energy and in particular the aberration that places a threshold on the probe size. In order to use FIB systems to produce nanostructures, the ions in the beam must be focused into a probe that is as small as possible [8]. The current in this probe defines the rate at which the structures can be made. The probe size (d) and current (Ip ) are directly related to the brightness of the ion source (Br ) and the set-up of the ion optics. As already mentioned, the LMIS can guarantee 106 A cm−2 sr−1 thanks to its small emission area (about 50 nm). Ion beams have negligible diffraction effects. In fact, the De Broglie relation in the nonrelativistic region is λ = h/ p = h/(2m e Φ), where λ is the wavelength, h is the Planck’s constant, m is the particle mass, e is the particle charge and Φ is the applied potential. Using the above equation, protons in the energy range 100 eV–30 keV correspond to λ values of 2.8 pm to 0.16 pm. For comparison with electrons in the same energy range, λe varies from 0.12 nm to 7 pm. At this very short wavelength, diffraction effects can be neglected when considering the interaction with electrostatic or magnetic fields that define the particle trajectories. In fact, the contribution of the diffraction to the probe size (dd ) is related to the ion wavelength (λ) and the half-opening of the beam at the target (α) using the following relation: λ (11.1) α Using λ = 0.16 pm, α = 4 mrad, we obtain dd = 24 pm. LMIS has a drawback from its initial energy spread (5 eV). Therefore, the resolution of the optical system is determined primarily by the contribution of chromatic aberration to the probe size (dc ) of the objective lens, the final lens above the specimen, given by Barth [9]: dd = 0.6

dc = 0.34Cc

∆U α U

(11.2)


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Realistic figures for (11.2) are U = 30 keV (mean energy of ions), Cc = 50 mm (chromatic aberration coefficient of the objective lens), ∆U = 5 eV (energy spread of the ions). This would result in a dc = 11 nm. Some methods can be used to minimize chromatic aberration, such as smaller angular limiting aperture, larger operating energy, element with negative Cc and energy filtering. The main optical difference between these methods is their influences on the probe current, which can be derived as: dpar 2 (11.3) Ip = Br Uπα2 π 2 where dpar is the demagnified source size at the specimen level. The method in which the opening angle is limited by an aperture limits the probe current and therefore the production rate (milling and deposition rate) drastically. Increasing the beam voltage appears advantageous because it does not influence the probe current. With a beam energy of more than 300 keV the probe size decreases to 1 nm but, for a feature of fixed area, the working time increases. The total chromatic aberration coefficient can be minimized by adding an element with negative Cc , like a set of quadrupoles or an electrostatic mirror, as described by Zworykin [10]. However, this method has a large sensitivity to changes in the individual chromatic aberration coefficients. Energy filtering can be realized with a slit positioned at the image plane of a 90◦ -deflector. This combination can work as an energy band pass filter. Using this method, the chromatic aberration can be decreased until the next limitation comes in. Aside from the chromatic aberration, the spherical aberration (ds ) is dominant, as described by the equation given by Barth [9]: ds = 0.18Cs α3

(11.4)

The coefficient of spherical aberration (Cs ) is generally larger than Cc , for example 125 mm, which would result in ds = 0.72 nm. Assuming that a quadratic summation of the contributions is appropriate, the total spot size is given by: ( 2 + d2 + d2 (11.5) d = dpar c s For FIB commercial systems, the minimum spot size is approximately 10 nm [11]. 11.2.4 Dual Beam Systems One of the main disadvantages of the FIB system is the continuous milling of the sample surface even in the imaging mode, which uses low beam current. This drawback cannot be avoided, because it is strongly connected with the large masses of the beam ions: even using low mass ions, the interaction with the sample is much stronger than the interactions produced by electron beams.

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In the last decade, systems with an ion column and an electron column have begun to appear in industrial and academic research. These dual beam systems offer more than the features of a SEM and a FIB individually (Fig. 11.3). Complete sample preparation, imaging and analysis can be performed in a single instrument thanks to the use of precision ion beam milling, high resolution nondestructive electron beam imaging and energy dispersive systems (EDS, based on the detection and measurement of X-rays) for compositional analysis. Electron beam gas chemistry augments the capabilities of FIB in relation to charge reduction and sample protection. The SEM column and the tilted FIB column have a single coincident point on the sample (Fig. 11.4). This configuration enables SEM imaging during FIB cross-sectioning, and therefore real-time monitoring of the milling process. State of the art dual beam systems also have a multivacuum mode that offer researchers the ability to conduct in situ dynamic experiments in wet, cold or hightemperature conditions, and to image or analyze any sample without charging. Dual beam systems are quickly becoming required scientific instruments in the field of nanotechnology, since they allow researchers to carry out difficult and complex tasks in a relative easy and quickly way.

Fig. 11.3. DualBeam chamber with an inserted Gas Injection System (GIS)

Fig. 11.4. A SEM column and a tilted FIB column have a single coincident point on the sample


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11.3 FIB Processes 11.3.1 Imaging 11.3.1.1 FIB Imaging During FIB imaging the finely focused ion beam is raster scanned over a substrate, and secondary particles (neutral atoms, ions and electrons) are generated in the sample. As they leave the sample, the electrons or ions are collected on a biased detector. These secondary particles possess energies of only a few eV, and are readily detected on an ‘all or nothing’ basis using a collector such as a microchannel plate biased positively at a few hundred volts for secondary electron detection or negatively by a few thousand volts for positive secondary ion collection. The secondary ions that are emitted can be used for secondary ion mass spectroscopy (SIMS) of the target material in a mass spectrometer attached to the system [11]. When collecting secondary ions for imaging, to prevent positive surface charges from building up, the substrate can be flooded with electrons from a separate electron source. The system thus prevents damage due to electrostatic discharge and it also enables reliable imaging of nonconducting materials. Several contrast mechanisms interact simultaneously in typical FIB images and, as in most beam instruments, details in the image can be enhanced by suppressing or enhancing one particular mechanism. The contrast mechanisms reported in the literature [12, 13] fall largely into the categories of crystallographic orientation (channelling) contrast, material contrast and topographic contrast. Orientation contrast arises from channelling the incident ions between lattice planes of the specimen. The depth of penetration of the channelled incident ion varies with the relative angle between the ion beam and the lattice plane and the interplanar spacing of the lattice. Although channelling contrast is exhibited in both the secondary electron and secondary ion images, secondary electron yield is significantly larger than secondary ion yield for most materials, and is less sensitive to changes in chemistry. In crystalline materials channelling contrast can be used for grain structure characterization, to image crystal grains and thus reveal different crystal orientations. Figure 11.5 illustrates this phenomenon in a hematite sample. Grains of different brightnesses are evident in Fig. 11.5a. This comes from changes in grain orientation with respect to the beam. Material contrast arises from differences in the yields of secondary particles as a function of specimen chemistry. The main application is voltage contrast imaging, used to image semiconductor devices, as shown in Fig. 11.6. The low brightness part corresponds to the insulating dielectric layer over the metal conductor. The metal is the high intensity part. Although this effect can be significant in FIB secondary electron images, it is most readily observed in FIB secondary ion images, where it is frequently the dominant contrast effect. Topographic contrast in FIB has been explained in terms of differences in signal (secondary electron and ion production) as a function of primary beam local angle of

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Fig. 11.5. Channelling contrast imaging in secondary electron mode on a hematite sample (a), and its cross-section as realized by FIB (b)

Fig. 11.6. Voltage contrast imaging in secondary electron mode on a chip cross-sectioned by FIB

incidence relative to the specimen surface normal. This effect is then convolved with detector geometry to produce an image with many features familiar to the casual observer. Figure 11.7 illustrates topographic contrast imaging on an AFM tip: the shape of the tip is easily discernible due to the topographic contrast. It should be mentioned that imaging with FIB inevitably induces some damage to the sample. Most of the Ga+ ions that arrive at the sample surface enter the


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Fig. 11.7. An image of an AFM tip. The tip shape is readily discernible due to topographic contrast

sample; thus ion implantation occurs. The depth of this implanted region is related to the ion energy and the angle of incidence. Besides implantation, some milling always occurs when the ion beam is scanned across the sample surface. Of course this milling effect can be drastically reduced when using a fine ion beam (fine spot and low ion current). The best resolution achieved with FIB imaging equals the minimum ion beam spot size (typically 10 nm). Despite a lower resolution compared to other techniques (such as SEM imaging), FIB imaging offers the best voltage contrast and the best channelling contrast. However, the key factor which makes the FIB system a unique tool is the combination of imaging and microfabrication capabilities in a unique instrument, as will be discussed more in detail in the following. By way of illustration,

Fig. 11.8. Images at different working times of a thin gold film deposited on a glass substrate deposited by sputtering. The film is scanned by FIB using a medium energy beam (150 pA). This test allows us to follow the sublimation of the film from the glass substrate in real-time. Working times: (a) 30 s; (b) 1 min 30 s; (c) 1 min 52 s

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a simple example is reported in Fig. 11.8. A polycrystalline gold film, prepared on a glass substrate by sputtering deposition, with a transverse feature (1 µm width) in the middle to free the glass substrate (used as reference in the view), is scanned using a medium energy beam (150 pA). This test allows us to follow the sublimation of the film from the glass substrate in real-time. This real time imaging allows us to achieve close control and great reproducibility, which are not possible with traditional micromachining tools. The main drawback of FIB as imaging technique is that the process is destructive due to the continuous removal of material from the sample during the imaging. As the feature or defect being imaged becomes small (under 0.1 µm), the feature will be damaged or altered in the time needed to collect a high-resolution image. This problem can easily be overcome by using a dual beam system. 11.3.1.2 Dual Beam Imaging SEM is the preferred tool for defect review and it provides the image resolution needed for process monitoring. The high-resolution imaging and nondestructive nature of the electron beam makes it well suited to these tasks. However, SEM is limited to 2-D analysis and metrology. Owing to the limitations of single beam systems, new dual column systems, with both ion and electron columns, have become an important part of the tool set in laboratories and fab clean rooms. Figure 11.9 shows how the dual beam’s can reveal that a “surface defect” is located under the surface. As mentioned above, a well designed dual column system is able to integrate the ion beam’s milling and deposition capabilities with the electron beam’s high-resolution nondestructive imaging, without compromising either column’s performance. The cross-beam operation feature can be used, where both beams are scanned completely independently from each other and the secondary electron detection (SED) signal is synchronized with the SEM scan. The real-time ion milling process is known as the “slice and view” technique. The main purpose

Fig. 11.9. A dual beam image of a “surface defect” located below the surface


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of “slice and view” is to help the operator to decide when to end the milling process. Moreover, complementary information can be obtained by coupling SEM and FIB imaging. 11.3.1.3 Scanning TEM (STEM) Imaging A recent advance in FIB imaging concerns the merging of transmission electron microscopy (TEM) detection with FIB features in a single instrument. TEMs that incorporate liquid metal ion sources for direct imaging and specimen preparation [14] and dual beam instruments that incorporate a TEM detector are already available. The latter type have an electron gun and an ion gun, allowing the specimen to be observed with the electron source, and sputtering will occur only when the ion gun is on. These new FIB systems allow TEM specimen preparation and analysis in the same instrument without specimen transfer. Generally, the TEM detector is given as an optional component of a FIB system, which is placed in the specimen chamber. This configuration gives additional and special imaging capabilities to the system. An imaging resolution of below 1 nm can be achieved. The voltage contrast and topographic contrast are enhanced [15]. Figures 11.10a and b show a bright field image and a dark field image of a nanotube, respectively. The enhancement in channeling contrast and resolution means that the nickel particles present as a catalyst in the single wall nanotube can be clearly seen; these are not visible with a normal detector. It is easy to foresee the strategic benefits that could arise from these new FIB systems. As the use of FIBs expands from the microelectronics industry to “general” industry and university laboratories, the requirements of other disciplines such as nanotechnology, life science and material science must be addressed. Figure 11.11, showing a TEM sample of liver acquired by a dual beam FIB system, is an example of these new imaging capabilities [16].

Fig. 11.10. (a) Bright field image of a nanotube; (b) Dark field image of a nanotube [15]

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Fig. 11.11. TEM sample of liver acquired by a Gemini dual beam FIB system [16]

11.3.2 Milling 11.3.2.1 The Process The removal of sample material is achieved in an FIB system using a high ion current beam. The result is a physical sputtering of sample material (Fig. 11.12). An arbitrary shape can be etched by scanning the beam over the substrate. The resolution of the milling process is a few tens of nanometers. The maximum aspect ratio of milled holes is typically 10–20. The working current densities available range from 6 pA/cm−2 up to 3–15 A/cm−2 . Detailed studies that can be used to predict the sputtering yield can be found in the literature [17, 18]. Since these are mathematical dissertations beyond the level of this work, in the following we refer to other existing literature in relation to parameters or factors that influence the milling process. The milling process is regulated by sputtering phenomena – surface solid erosion. This occurs when a surface atom receives enough energy to overcome the bond barrier, leaving the solid. This energy could come from ions or from secondary atoms moved by ions. The kinetic regime that accurately describes the ion–solid interaction in the FIB process is the “linear cascade”, where each atom that is hit has enough energy to spark a cascade of collisions. When describing this regime we assume that two atoms


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Fig. 11.12. Illustration of the physical effects of incident ions on the substrate

in motion never meet and that collisions are always between an incoming bullet ion and a recoiling atom. The description of the ion–solid interaction begins by defining an energy loss function for incident ions through the solid. At the typical energies of the FIB process, the effect is well described by taking into account only the nuclear energy loss, defined as the energy transferred by an elastic collision between the incident ion and the nucleus (screened) of the target atom (omitting the electronic and charge exchange energy losses). It follows that: d d (11.6) E= E dx dx n where E is the energy of the bullet ion, x is the depth of the ion path into the solid and ( dE/ dx)n is the nuclear energy loss of the ion. The variables that enter into this function are the molecular weights of the bullet and the target atom (Mi , Ma respectively) and the angle of incidence between the normal surface and the ion beam direction (ϑ). The model of the ion–solid interaction is based on scattering theory: by considering elastic collisions and introducing the interaction potential between the ion and the target atom, it is possible to solve the particle motion completely. Knowing the paths, the angles and their probability functions, the collision cross-section (σ) and the energy transferred from the bullet to the target (T ) are easily calculated. The relation that relates these parameters to the nuclear energy loss is the following: d E=N dx

T dσ

(11.7)

where N is the atomic density of the target. It is useful to calculate the sputtering yield Υ , defined as the ratio between the number of atoms emitted and to the number of incident ions. On the basis of the above assumptions, the sputtering yield can be expressed as the product of two terms: Υ = Λ(U ) FD (E, ϑ, x = 0)

(11.8)

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where Λ is a constant that depends on the material, which is proportional to 1/U (U is the sublimation energy) and FD (x), the damage profile, given by: d FD (E, ϑ, 0) = (11.9) E α (Ma , Mi , ϑ) dx where α(Ma /Mi , ϑ) is a function that takes into account the parameters of the incident ions. The damage profile FD (x) is a Gaussian function that gives information about the energy transferred as a function of the depth x. It includes two factors: the contribution from the prime ion and the contribution from the solid atoms moved by it (the cascade). Having briefly summarized the physics of the process, we can discuss the parameters that influence the sputtering yield in FIB processes: 1. Energetic dependence Υ(E). This dependence is contained in dE/ dx, which is a combination of two substantially overlapping functions. The maximum of this function shifts to higher energy for increasing values of Mi and Ma . 2. Angle of incidence dependence Υ(ϑ). This dependence is contained in the function α: it increases roughly with 1/ cos(ϑ). The maximum is around ϑ = 70−80◦ , following a drop due to the reflection of the ions off the solid surface. 3. Dependence on ion source and target materials Υ(Mi , Ma ). This dependence arises from the functions α and ( dE/ dx)n . The sputtering yield increases with increasing values of the ratio Ma /Mi due to the increased bullet scattering at large angles which facilitates the energy transfer at the surface. Typical sputtering yields for various materials and normal incidence are given in the literature [18]. However, these numbers and correlations cannot be used to directly calculate the sputtering rate, because there are some factors that cause deviations from the ideal case. These include the shape of the ion beam [19] and the redeposition of sputtered material. A FIB system produces a beam with a profile that deviates somewhat from a Gaussian profile. Figure 11.13 shows two polydimethylsiloxane (PDMS) molds covered with thin gold film. These embossing masters were obtained from cavities produced by FIB milling a silicon wafer. These two cavities result from the patterning of two cylindric holes (respectively 5 and 2 µm in diameter for the left and right mold of the figure). The positive master, released from the cavity, presents a conical shape due to the combined effect of ion beam shape and sputtered material redeposition. The beam current distribution is Gaussian in the central portion, but exponential with a decay length of a few 100 nm at the tail region, where the current intensity is below < 1/100 of that at the Gaussian peak. Redeposition of sputtered materials may also be a problem because it is difficult to estimate. Sputtered atoms are redeposited at the walls of patterns and they distort the pattern profiles; this distortion depends on the beam scanning procedure [20]. The amount of redeposited material is dependent on the geometry of the structure being milled, and the process of redeposition is also nonlinear with time.


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Fig. 11.13. PDMS molds covered with thin gold film

11.3.2.2 FIB Milling as a Fabrication Tool FIB systems represent a unique tool for micromachining three-dimensional structures, offering a combination of real-time imaging and microfabrication capabilities in a single instrument. In addition, recent FIB systems can produce focused ion beams with diameters smaller than 10 nm, and they are becoming increasingly important nanofabrication tools. An example of a nanometric hole produced on a submicrometric sphere is shown in Fig. 11.14. Many other examples will be described in the following paragraphs. The small spot sizes (10–500 nm) and the very large range of available working current (from 1 pA up to 10 nA) could make FIB systems the ideal machining tools for both micro- and nanoengineering. Practically, however, there are still some obstacles to overcome before this goal is achieved. Due to the small spot size, FIB milling is a relatively time-consuming process if we wish to machine large volumes of material. For example, when using a beam current of 1 nA, it takes 60 min to mill out a volume of 50 µm−3 in silicon. This, therefore, makes FIB milling an unrealistic process for the mass production of large structures that require substantial milling. An alternative and realistic approach to producing large numbers of FIB-milled shapes would be to use the FIB system to mill either embossing tools or molds (Fig. 11.13). Although the initial FIB milling would be time-consuming, it could then be used in a mass production process by replication. The challenge lies with controlling the ion milling in three dimensions in order to produce cavities with predefined, arbitrary geometric cross-sections [21]. Nevertheless, when creating a FIB shape mill there are numerous factors to consider. In addition to the shape of the ion beam and the redeposition of sputtered material, the changes in sputter yield as the shape is milled must be taken into account [22]. Vasile et al [23] explain that the changes in the sputter yield are due to changes in the angle of incidence of the ion beam, and these can be accounted for by changing the dwell time (the time the ion beam remains fixed on one pixel position) at the pixels during

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Fig. 11.14. A hole (200 nm in diameter) produced by FIB milling of a submicrometric sphere

the milling. The authors propose an algorithm, based on a mathematical model of the FIB milling process, which yields a variable dwell time in a deflection pattern for a prespecified geometry. Given the material, ion beam dimension, energy and current as the input data, the model solves, pixel by pixel, a dwell time array for the ion beam and transfers this array to the control microprocessor. Solutions obtained by the model depend strongly on the size of the pixel in the deflection control scheme, on the ion beam dimensions and also on the absolute sputter yields for the material and incident ion. These sputter yields must also be known as a function of the ion angle of incidence. For sinusoidal curves and hemispheres, a good agreement between the model and the milled shape was found. However, upon decreasing the minimal feature size or increasing the complexity of the shape, deviations from predictions became apparent. The critical variable is the change in the ion angle of incidence during the milling process. Redeposition also compounds the deviation from ideality, and the magnitude of this effect is difficult to estimate. 11.3.3 Gas-Assisted Etching The problems related to FIB milling can be reduced by ion beam-assisted etching. The etching is performed by irradiating FIB in an ambient reactive gas as described above (in the “System Overview” section). This results in an increase in the etching rate (a few times up to several hundreds of times higher than for physical sputter etching) and this makes it possible to reduce the required dose. The selectivity towards different materials can be enhanced. Moreover, because substrate materials are removed by forming volatile products via etchant gas ion chemistry, redeposition is also significantly reduced. Thus very narrow and well defined grooves that cannot be obtained by normal FIB milling with the same accuracy, with aspect ratios of around 10, become possible [24].


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Fig. 11.15. Comparison of two identical trenches produced in a silicon wafer via FIB milling (right trench) and via FIB gasassisted etching (left trench)

The etching process follows a mathematical model proposed by Harriott [25]. According to this model the gas-assisted enhanced etching depends on the adsorption, evaporation and etch removal of the absorbed molecules. Lipp et al [26] investigated the focused ion beam enhanced etching of silicon using iodine as an enchant. They found that the material removal rate increases by a factor of up to 30 compared to convectional sputtering. To estimate the precision of the removal of material, they calculated the slope angles of milled craters. In FIB milling, the beam diameter of the Gaussian beam distribution increases with increasing beam current and the slope angle also worsens with increasing beam current, but for gas-assisted enhanced etching no dependence of the slope angle on the beam current was found. Moreover, the authors verified that using gas-assisted enhanced etching to form thin lamellas for TEM inspection ensured that no stress-induced defects were produced in the lamellas. The advantages of ion beam-assisted etching is clearly demonstrated in Fig. 11.15, which compares a feature produced with and without gas-assisted etching (left and right trench respectively). The cross-sectional profiles shown in the right trench are distorted by redeposition, which occurred during the cross-sectioning; this effect is not observed in the left trench, formed by cross-sectioning with gas assistance. 11.3.4 Gas-Assisted Deposition Commercial FIB systems enable maskless deposition of both metal and insulator materials. The basic principle of FIB deposition is localized chemical vapor deposition (CVD) [27]. The reactions involved in FIB deposition are very similar to those involved in laser-induced CVD [28, 29]. Platinum (Pt) and tungsten (W) are the usual metals deposited, while silicon dioxide (SiO2 ) is commonly used for insulator deposition in commercial systems. For W deposition the precursor gas is W(CO)6 ; in SiO2 deposition the most widespread precursor gases

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are 1,3,5,7-tetramethylcyclotetrasiloxane (TMCTS) and oxygen (O2 ) or water vapor (H2 O). In the deposition process (Fig. 11.16), the gas is sprayed into the work chamber through a fine needle; in this way the ion beam interacts with the gas at the same point on the sample surface. The gas is absorbed by the surface and the incident ion beam decomposes it, leaving the metallic or insulator part on the sample surface. The volatile reaction products desorb from the surface and are removed by the vacuum system. A drawback of this process is that the material obtained is not completely pure, due to organic contaminants and Ga+ ions that are included in the deposited film. Deposition is a constant competition between the decomposition of the adsorbed molecules and physical sputtering (milling). Adsorption and desorption occur continuously; the decomposition process is instead discontinuous because the ion beam is scanned on the surface with a particular pixel spacing, dwell time and refresh time (time between two subsequent complete pattern scans). These parameters are of crucial importance for limiting the milling during gas deposition and optimizing the deposition process; which is why the pixel spacing is five times wider for deposition than for milling (50 nm versus 10 nm) and the dwell time is typically ten time shorter (0.5 µs versus 5 µs). In order to enable the precursor gases to adsorb on the sites where the decomposition has occurred, the refresh time is much longer (typically 5000 µs versus 10 µs) than for the simple milling process. Other important parameters for deposition are the gas precursor temperature and the beam current [30]. The deposition process has a high resolution but a small deposition rate. The smallest features that can be deposited have lateral dimensions of the order of 100 nm and thicknesses well below 100 nm; typical deposition rates are 0.05 µm3 s−1 . Any type of two-dimensional shape can be realized by FIB gas-assisted deposition; threedimensional objects can be created by depositing multiple layers, as in microstereolithography, with the advantage of thinner layers and higher lateral resolution. The main disadvantage relates to the aspect ratios that can be obtained (between 5 and 10), and therefore the vertical dimensions that can be achieved. In addition the small deposition rate and the sequential nature of the process result in long and expensive deposition times for relatively big features (of the order of 1 mm), limiting the realistic dimensions of the objects that can be obtained to approximately 100 µm.

Fig. 11.16. Principles of FIB gas-assisted deposition


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11.3.5 Ion Beam Lithography The relentless commercial drive for ever smaller, faster and cheaper microelectronic components is pushing existing technologies to their limits. However, optical lithography is far from reaching the end of the road, and will continue to be used for some years with light sources of smaller wavelengths (such as ArF and F2 excimer lasers) and other optical techniques that can enhance the resolution. The microelectronics industry is still exploring next-generation lithographic technologies, such as X-rays, electron and ion beams or extreme ultraviolet light, but it is not clear which one of these technologies will be the most successful [31]. Among all of the types of particles (photons, electrons, ions) that are employed in lithography, ions have the shortest effective wavelength; this is the reason that, in principle, they can achieve the smallest features. In charged particle lithography, the ultimate resolution is influenced by various factors, including the scattering of the primary particles, the spread of the recoil ions and the secondary electrons, the beam profile, and resist properties such as the molecular size and the contrast. Primary beams are spread by forward and backward scattering. For electron beams, scattering is significant and causes a proximity effect which degrades resolution. Electrons tend to scatter their energy into a relatively large volume: this is because they are incident on a medium largely filled with particles of equal mass that give optimal energy transfer during scattering [4]. Ion beams undergo less scattering than electron beams and yield high resolution. With FIB lithography on polymethylmethacrylate (PMMA) organic resists, line widths of 12 nm are obtainable [32]. In recent years inorganic resists, such as AlF3 [33], have also been used for FIB lithography with a line resolution of 10 nm. Energetic ion beams have a much higher energy deposition rate for both electron and nuclear collisions compared to electronic beams, and this results in a high exposure sensitivity. Figure 11.17 shows a comparison of the PMMA resist exposure sensitivity for ion beam and electron beam lithography [34]. In this figure the

Fig. 11.17. PMMA thickness after development as a function of dose using various 100 keV focused ion beams and 20 keV electron beams. The development was performed using a 1:3 methylisobutyl ketone (MIBK) and isopropyl alcohol (IPA) mixed solution [34]

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normalized thickness remaining after development is plotted as a function of the exposure dose. Ion lithography has another important advantage in the absence of proximity effects. In electron beam lithography fine features are exposed by using the high energy density deposited near the point of entry. The widely scattered electrons produce a lateral substrate background which is below the threshold for exposure. For patterns different from isolated lines these backgrounds add up and the e-beam exposure dose has to be lowered due to the proximity to other parts of the pattern. Exposure times for insensitive resists like PMMA are of the same order of magnitude for ion and electron beams. PMMA is two orders of magnitude more sensitive to ions than electrons, but in commercial machines the current densities of ion beams (0.4–4 A cm−2 ) are two order of magnitude lower than electronic ones (10–100 A cm−2 ). E-beam lithography in the nanometer range is a well established technology thanks to the great effort that has been made to improve the e-beam column. Even if, theoretically, ion beam lithography can produce at least the same resolution as e-beam lithography, state of the art machines are not generally used for lithography because they are still a step behind in terms of technological development.

11.4 Main Applications FIB technology emerged in a unique way. Differently from many other tools and techniques, such as SAM (scanning atomic microscopy), SEM, EDS, SIMS and TEM, which have been applied in the semiconductor industry after maturing elsewhere, FIB was initially developed to satisfy the demands of the integrated circuit (IC) industry, and only later found diverse applications in materials science laboratories. The original application of X-ray and photomask mask repair [35] is a niche application nowadays, if compared to other new uses that have been found for this technology. These uses include electro-optic device fabrication [36,37], failure mode analysis in IC manufacturing processes [38], reverse engineering in the IC industry, step-by-step diagnostics of IC fabrication, IC test modifications, TEM sample preparation [39, 40], and magnetoresistive head (MRH) trimming. The application of FIB to the formation of the pole gap in MRH is the first example of a nondiagnostic use for FIB technology in industry. Concerning relatively new applications, there is significant interest in using FIB machines as machining tools at the micro- and nanoscale. FIB machining is well suited to rapid alteration and modification of existing microdevices. It can be used to perform one critical single step in a more complex production process (as in the MRH application) or as the dominant process in a relatively simple production process. M.J. Vasile et al [41] give an useful review of FIB applications in micromachining and explore the possibilities for the production of microtool bits for further machining applications and microsurgical tools in modified serial processes. Currently, molds fabricated by FIB machining are often applied to the production of microstructures in high volume batch processes, such


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as molds for micro-optical lenses or other small-scale optical components. Despite the significant potential for producing small objects in a serial process, FIB machining has not yet been exploited in mass production, which requires fast processes. Besides the possibility of modifying local structures, the FIB process has recently attracted great interest among materials scientists as an analytical technique. Bombardment with ions has side effects that can be explored by microscopy. The creation of damaged layers and ion beam-induced phase transformations belong to this category of analysis. SEM, SPM (scanning probe microscope) and SIM itself can be used to visualize the effects of the ion beams. The strong channelling contrast of ions makes imaging in the FIB ideal for accurate measurements of grain size distributions. In order to reduce damage during inspection, dual beam instruments have been employed. 11.4.1 FIB as an Analytical Technique A great deal of interest is now being shown in the application of FIB systems to materials science problems [42]. FIB was virtually unknown in the materials science community up to a decade ago. Now, conference presentations and journal articles that mention the use of FIB in material science applications are increasing in number at a dramatic rate. The reason for this lies in the intrinsic ‘physics’ of the FIB. The FIB microscope operates along the same principles as the SEM, where a beam of charged particles is scanned across a specimen, and the resultant signal(s) at each raster position are plotted to form an image. The larger masses of the ions (in comparison with electrons) results in greater energy transfer between the projectile particle and target atoms, and therefore a short deceleration range. Consequently, an analysis based on ion beams can be more surface sensitive than the corresponding technique that uses electrons of similar energy. Moreover, using an ion beam instead of an electron beam confers additional capabilities: bombarding a material with ions provides a controlled method of doping regions close to a surface (ion implantation), sectioning and deposition, while the ejected ions and the generated secondary electrons provide two destructive surface analytical techniques, SIMS and SEM, respectively. Nevertheless, a great deal of fundamental research is needed to achieve a more detailed understanding and harnessing of beam–specimen interactions, and the precision with which an ion beam can be focused should be further improved. The challenge is to construct instruments in which ions are used for imaging, analysis and modification of near-surface properties with a spatial resolution rivalling that of the traditional SEM. Present research into using gas field ion sources to replace liquid metal ion sources holds the promise of nanoscale beams of ions ranging from hydrogen to oxygen or xenon, which, compared to that presently available with gallium beams, should greatly enhance the SIMS capability. These sources also open up the possibility of Rutherford backscattering (RBS)-like analysis and chemical analysis via ion-induced X-ray emission.

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11.4.1.1 3-D Reconstruction For many applications it is important to measure the final shape of a microstructure, for example measuring a FIB-milled structure. For a symmetrical structure this can be done by simply cross-sectioning through its center using the FIB system, while for some nonsymmetrical structures (where taking a single FIB cross-section would not be representative) atomic force microscopy (AFM) can be used. However, for nonsymmetrical structures with high aspect ratios, AFM cannot be used because the geometry of the AFM tip affects the measurements. One solution is to use three-dimensional (3-D) reconstruction from a set of sequential two-dimensional (2-D) cross-sections milled through the structure in order to measure its shape [43]. Slicing and 2-D image collection is generally performed automatically by dedicated software. The images collected are processed in a software package that allows to create a 3-D reconstruction. This principle is illustrated in Fig. 11.18 [15]. An example of this application is given by Vasile et al [23]. The authors propose an algorithm, based on a mathematical model of the FIB sputtering process, for the patterning of a prespecified geometry by FIB milling. The software performs in real-time, during milling, a 3-D reconstruction from a sequential set of FIBmilled 2-D cross-sections in order to determine the shape of a FIB-milled structure. A comparison of the reconstructed and the intended shapes enables the FIB milling process to be controlled so that better correlation can be achieved between the two. Another example of this kind of application is fiber inclusion reconstruction. In this application nondestructive 3-D analysis is required. In this case a 3-D reconstruction can be obtained by processing RBS data using software [44]. When FIB is scanned on the sample, micro RBS spectra for random or channelling directions can be stored for each scanned position. RBS mapping data can be obtained at an arbitrary depth by adjusting the software energy window. Cross-sectional structure (RBS tomography) can be obtained by choosing an arbitrary data set of scanning lines. The RBS maps collected are processed in a software package which corrects the distortion arising from RBS kinetics, allowing to create a 3-D reconstruction.

Fig.11.18.(a) Collection of 2-D cross-sectional images; (b) 2-D cross-sectional image processing; (c) 3-D reconstruction [15]


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11.4.1.2 Structural Cross-section and Plan Section FIB offers several advantages over other techniques in the cross-sectioning of samples. While cleaving or dicing and polishing for SEM inspections can take several hours, FIB enables cross-sections to be realized within a few minutes. Furthermore, the precise location of the cross-section can be defined without constraints, and an arbitrary number of subsequent cross-section ‘slices’ can be taken on the same sample. The procedure for the cross-sectioning technique includes the following steps. In a first step, bulk material is removed with a rough and fast single pass mill adjacent to the location of interest. This results in a steep sidewall on the side near the feature of interest. This sidewall is made perfectly vertical and is smoothly polished in the subsequent milling step(s), using a finer ion beam and milling regime (polish mill) each time. Finally the sample is tilted and the polished sidewall is visualized (Fig. 11.6). For normal cross-sections (depth of a few micrometers and a width of 10–15 µm), the whole procedure takes no more than 15 minutes. This technique can be used for to analyze microstructures. It is also possible to use a similar technique on both sides of the region of interest. This procedure allows us to fabricate a thin lamella, suited to inspection by TEM. The use of FIB becomes fundamental when materials different from silicon, such as Al, Cu, and W are used in the fabrication process. While the manual cut can be easily performed for silicon along the crystallographic planes, polycrystalline materials break following their grain structure. Figures 11.19a and b show the same cross-section after a manual cut and FIB polishing: FIB helps to avoid image misinterpretation. The FIB technique also offers the ability to get the “plan section” of a sample; in other words to progressively expose the top layers of a device (reverse engineering). The plan section is a complement or substitute for conventional methods (chemical or plasma etches) of sequentially peeling off layers. FIB allows the position being investigated within the deposited layers to be tightly controlled. The notches produced by traditional delayering are avoided. On the other side, the process is quite time-consuming, so only a small area can be analyzed. FIB cross-sections and plan sections are also suitable for monitoring inspections. It is important to underline that the high spatial resolution and contrast mechanisms provided by FIB imaging, combined with the ability to perform ‘stress-free’ sectioning in situ, makes FIB a very attractive specimen preparation instrument for material characterization.

Fig. 11.19. (a) Cross-section image after manual cut; (b) Cross-section image after FIB polishing

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11.4.1.3 TEM–SEM Sample Preparation TEM is a powerful technique that can be used to investigate a wide range of materials (including semiconductors, metals, ceramics, polymers, biological materials and tissues), effects of different treatments, solid state reactions, and so on. The limiting factor when exploring sample microstructure using TEM is often the preparation of the sample, rather than the resolution of the electron microscope [45, 46]. To use TEM properly, samples must be thin enough (typically 100 nm or less) to transmit an electron beam. The traditional preparation procedure involves complex operations that include cleaving, mechanical polishing, dimpling, and broad ion beam thinning. These processes are time-consuming and tedious: once the defective area is located, it is cut from the wafer. This obviously destroys the entire wafer. The sample then undergoes several grinding and polishing steps until both sides approach the defect. Finally, broad beam ion milling thins the sample until it is transparent to electrons. The only way to monitor the thinning processes is to move the sample frequently to an optical microscope and inspect it. Many defects are almost invisible until revealed by the polishing process, and the slightest misjudgement during polishing or ion milling can destroy the sample. Furthermore, soft materials in the sample smear due to the polishing process. The milling capability of FIB offers a faster and more precise way to prepare TEM samples [47–49]. FIB preparation uses the ion beam to image the sample and to remove material from both sides of the desired section. It also allows the sample to be ‘polished’ to make it transparent to electrons. Moreover, the removal of material is monitored in real-time in FIB. A detailed description of FIB–TEM specimen preparation has been given by L.A. Giannuzzi et al. [50]. Two approaches to FIB sample preparation are currently used: “pre-thinning” and “lift out”. Pre-thinning uses traditional mechanical methods to cut a small sample from a wafer and then to thin it down to about 10–50 µm. After this, the sample is placed in the FIB. The ion beam removes material from both sides of the desired section using successively lower ion beam currents and thus leaving a thin membrane for TEM examination. The main advantage is very accurate positioning of the TEM slice; the sample can even be returned to the FIB for further thinning, if needed. Both cross-section and plan view TEM specimens can be prepared [51]. The lift-out technique requires little or no initial specimen preparation [52, 53]. For the lift-out approach, the intact wafer or device is placed in the FIB. Thus, lift-out specimen preparation can be used for serial sectioning of a bulk material. The main steps in the process are illustrated in Fig. 11.20. First, in order to prevent top surface damage to the samples, a layer of platinum or other metal is deposited (Fig. 11.20a). Trenches on both sides of the target are milled (Fig. 11.20b) by using successively lower ion beam currents. When the target reaches a thickness of about 0.5 µm, the wafer is tilted and the target is cut free along its bottom and along one side (Fig. 11.20c). After further thinning, the remaining side is also cut (Fig. 11.20d). The sample, 10–20 µm in length and width and less than 100 nm


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Fig. 11.20. (a) Protective coating; (b) coarse trench milling; (c) cutting the bottom and sides; (d) further thinning and extraction with a thin glass probe; (e) TEM specimen

thick, is lifted out using a fine glass probe and mounted on a TEM grid (Fig. 11.20e). The specimen is ready for TEM analysis. Lift-out TEM specimens may be prepared within 3 hours. It is advisable to point out that a dual column FIB system has even more potential for FIB-TEM specimen preparation than a single beam ion column system. In fact, nondestructive e-beam high resolution imaging during the milling process is provided in real-time. The location of the TEM sample can be established with nanometer accuracy. E-beam-induced Pt deposition could replace ion Pt deposition in order to avoid ion implantation damage. In addition, by monitoring the process with the electron beam, internal stresses in the sample, which cause the lamella to bend or be warped during the preparation process [54], can be identified and corrected. Although adequate chemical etching procedures are available for some materials when preparing TEM samples, ion milling has become the most common sample preparation method. More and more complex materials are being investigated, for which uniform chemical etching conditions are not known. On the other hand, ion milling can be applied to a wide range of materials. Studies performed on TEM sample preparation and the subsequent analysis of photoresist illustrate this point. Photoresist is a challenging material to study due to its polymer nature. Because it is soft compared to most other materials, preferential etching will occur and there is also the potential for deformation during mechanical thinning. The photoresist is also beam-sensitive, to both electron beams in the TEM and gallium beams in the FIB machine. A study by De Veirman et al [55] gives a comparison of the effectiveness of three different techniques (FIB, ion milling and tripod polishing) for the preparation of TEM and SEM semiconductor samples coated with photoresist. This work confirms that the FIB technique produces the best TEM samples in terms of homogeneity, thinness and speed. It also provides the least

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contaminated samples. No mechanical deformations of the thin polymer membrane are observed. The main drawback of the FIB method compared to traditional preparation procedures concerns the artefacts induced in TEM specimens by a Ga+ beam. Ga ions are implanted into (and mix with) the specimen as a result of the sputtering mechanism. Therefore, it is possible that Ga+ may influence local compositions within the specimen, although the extent of this effect is not known. The Ga+ implantation also amorphizes the outer layer of the specimen. There is still some uncertainty in the literature about how much amorphous damage is produced during FIB sputtering [56,57]. A review of the damage caused by ion milling was published by Barber [58]. Although a large number of articles have been published on ion damage to different materials, most of these discuss the damage caused by ion implantation of high-energy ions. Low-energy studies performed by Barna et al [59] prove how the thickness of the damaged layer can be reduced by decreasing the ion energy. This work demonstrates the improved quality of high-resolution electron microscopy (HREM) samples, prepared by reducing the ion energy (to 125–250 eV) during the final stage of ion milling. This improvement is due to reduced damaged layer thickness and surface roughening. An interesting advantage of TEM specimens prepared by FIB over those prepared traditionally is the ability to map out compositional dopant variations at the 10 ppm level. This has been observed for InP-based and similar materials (such as InGaAs, InGaAsP) prepared by FIB specimen preparation techniques [60–62]. In terms of advances in FIB instrumentation, new low-energy ion guns are available that could be applied to ion milling devices in order to minimize ion beaminduced artefacts or to remove damaged layers [11]. These new guns permit smooth, thin TEM samples with minimal surface amorphization, useful for further analysis (such as in demanding HREM applications) to be produced. The thickness of the damaged layer can be reduced to around 1 nm (a damaged layer around 20 nm thick is frequently reported for convectional guns). These new guns apply to low-energy Ar+ ions at a grazing angle of incidence. Special TEM/FIB instruments are available that allow conventional specimens to be prepared and analyzed in the same instrument without additional handling. Dual beam instruments that incorporate a TEM detector are already commercially available. In these instruments, the specimen can be observed with the electron source, and sputtering will occur only when the ion gun is on. It is possible to automate bulk trench cuts and most polish cuts in new FIB systems. Multiple specimen holders can be arranged in the FIB chamber: bulk cuts of several samples may be achieved quickly and without operator intervention. This reduces the time and cost of the process, and increases the sample throughput. It is conceivable that in the future routine TEM specimen preparation times may be reduced down to 30 min. 11.4.1.4 FIB Contrast Mechanisms for Material Characterization Specimen preparation by FIB milling is not limited to imaging applications. A huge number of examples of the application of FIB to material characterization can be


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found in the literature. A review of the main applications of FIB microscopy for material characterization, illustrated by practical examples, has been written by M.W. Phaneuf [42]. The primary benefit of combining in situ FIB sectioning with FIB imaging is straightforward: FIB cross-sectioned surfaces reveal grain morphology, submicrometer voids and other fragile specimen features, without the stress and potential damage involved in mechanical sectioning. FIB sectioning and imaging can reveal details about the undesired inclusion of particles through a combination of topographic and chemical contrast. For example, this can reveal the entrainment of reaction products during the production of a composite. Despite the lack of true chemical analysis, secondary ion chemical contrast can be used to make qualitative judgements on the chemical species present in investigated samples. Since secondary ion production is heavily influenced by matrix chemistry, a great deal of information is available in the FIB ion image. In particular, it is possible to establish the different phases present in such a material. This “optical metallography” analysis requires references (such as X-ray maps). In his work, Phaneuf reports an example of using FIB secondary ion chemical contrast as an aid to qualitatively determine the carburization of heat-resistant alloys. In the IC industry, a limited set of materials have traditionally been used that have well characterized properties. In microsystem applications, the range of materials available for designers is huge and there is a clear need to develop standard test methods to characterize the mechanical properties of microfabricated material. The main issue is that microfabricated materials have properties that are strongly dependent on the fabrication route and the dimensions of the structures that they constitute. The use of FIB micromachining to produce extremely sharp pre-cracks in impact test specimens [63] or to produce specimens for microtensile testing [64,65] has been described. Muroga and Saka [66] produced H-bar style specimens to observe the defect structures produced by rolling contact fatigue in steels, and Saka [67, 68] examined crack tips produced by Vickers indentation into dislocation-free single crystals of silicon. When studying material stiffness, it is a common practice to study deformation by creating nano-indentations with a suitable indenter and with different loads. The nano-indentation produced can be then scanned with both the electron beam and the ion beam, thus allowing the feature size of the microstructure at the surface near the indentation to be measured. This technique coupled with the delayering process allows us to see into a multilayered structure and to study the effects of deformation on each sublayer. FIB stress-free sectioning could be used to reveal failure mechanisms after fracture toughness testing. Phaneuf reports an example of this application. A fracture toughness sample of a composite was cut perpendicularly to the fracture surface using a low-speed diamond saw in order to obtain a specimen for examination in the FIB instrument. FIB polishing was used to create a box just below the fracture plane. About 40 µm of material from the cut surface was removed by FIB milling to eliminate the damage produced by the diamond saw. Secondary electron FIB images

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obtained from the polished cross-section allowed the extension of the secondary cracks resulting from the fracture to be localized and quantified. A failure mechanism was envisaged based upon these observations. FIB precision sectioning and orientation contrast also allows to implement a suitable procedure for performing corrosion tests. Secondary ion yields from gallium beams can increase by up to three orders of magnitude for metallic species in the presence of oxygen, making material contrast a valuable technique for studies involving corrosion or grain boundary oxidation of metals [42]. Long-term corrosion studies can be performed on cyclically loaded samples. Cracks developing on samples can be imaged in plan view, precision cross-sectioned at different locations along the length of a given crack and then tilted to permit the cross-sectioned surface to be viewed. Information regarding the grain structures around the crack tip and accurate measurements of the crack length and depth can be obtained without disrupting neighboring cracks. The granular crack path and other microstructural features associated with crack development can be reported. Iteration of this process provides crack aspect ratio and evolution data on the same specimen in an efficient manner [69]. Another study reported by M.W. Phaneuf concerns analysis performed on nickel hydroxide particles in a nickel hydroxide-based battery electrode after numerous charge–discharge cycles. In order to examine changes in the morphology of particles, a battery electrode that had reached the end of its useful lifetime (more than 6000 charge–discharge cycles) was sectioned in a stress-free manner without any additional mechanical preparation other than critical point drying. After tilting to view the cross-sectioned face, FIB topographic contrast revealed the ‘pulverized’ aspect of the nickel hydroxide particles, which was initially composed of radially oriented plates. 11.4.1.5 Defect Characterization and Structural Analysis Particles included in full processed wafers or mounted devices are often investigated in failure analysis. The location and characterization of defects using a particle beam has traditionally been conducted using a SEM and an X-ray detector for compositional analysis. However, there are limitations on the characterization capabilities of a system with just an electron column. Cross-sectioning a defect in order to analyze its internal composition is not possible. This restricts the analysis since important compositional information on the particle defect is contained inside the specimen, in the buried seed particle. In addition, subsurface defects cannot be observed directly using the SEM column. In order to overcome these limitations, other common methods of characterizing these particles are in use. One of the most popular methods is Auger depth profiling [70]. In Auger depth profiling the specimen is bombarded by ions, and the freshly created surface is analyzed by means of Auger electron spectroscopy (AES). The repetition of these steps results in measured values for concentration versus depth, called the “depth profile”. The biggest problems are related to sputtering-induced alteration of the


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specimen: ion sputtering, beside removing atoms from the surface in a random way, introduces various kinds of defects into the surface region of a studied specimen. Another method commonly used to characterize these particles is energy dispersive X-ray (EDX) in combination with sequentially peeling off the different layers. Reverse engineering can lead to contamination, which might affect the results. Moreover, FIB delayering allows precise plan sections to be obtained, which can then be used for further examination by EDX analysis. Nowadays, commercially multifunction instruments with integrated capabilities are available. ‘Dual beam’ FIB instruments, consisting of both a FIB column and a SEM column with an energy dispersive X-ray (EDX detector), exist, and these permit sectioning, imaging and chemical analysis all in the same chamber. FIB cross-sectioning and imaging reveal the grain structure, the chemistry of which was determined by energy dispersive X-ray spectroscopy during subsequent SEM analysis. Integration of a quadrupole SIMS into a FIB instrument is very common and provides the opportunity to make fast SIMS depth analysis, volume analysis, and to take ion-induced electron (IIE) images using the same instrument [71, 72]. All of these instruments have a dual column system in common, with an ion column to remove material and expose a subsurface defect for further imaging and compositional characterization. By combining defect-review software (which is able to read defect files generated by optical inspection tools and drive the system stage to the coordinates of the defect) with milling operations and composition identification, the dual column system has become an effective in-line defect characterization system. 11.4.2 FIB in the Semiconductor Industry As mentioned above, applications in the semiconductor industry have been the driving force behind the commercialization of FIB systems, chronologically, first as a tool for X-ray and photomask repair, followed quickly by their application to IC manufacturing processes for direct device fabrication, circuit diagnostics, failure mode analysis, IC test modifications, TEM sample preparation or MRH trimming. A detailed review of the applications of FIB in the semiconductor industry can be found in the work of S. Matsui et al [73]. Tables 11.1–11.4 summarize the applications of FIB lithography, etching, deposition, and doping in semiconductor industry. Briefly, FIB etching and deposition have become indispensable in fields such as failure analysis, mask repair, and TEM sample preparation. FIB lithography has several advantages over electron beam lithography, such as high sensitivity and negligible proximity effects, but damage is a serious problem and throughput for device fabrication is lower than that attained with electron beam lithography. FIB lithography is therefore rarely used in the fabrication of electron devices (but rather more in micromachining, because these microdevices are not damage-sensitive). The throughput of FIB doping is lower than that of conventional ion implantation, so it is difficult to put this technique to practical use when making Si and GaAs LSL devices. It can, however, be used to make quantum effect devices, at least in the research phase.

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Table 11.1. Applications of FIB lithography [73]

Resist lithography Organic resist

Inorganic resist Modification lithography IBEE

Selective epitaxy Vacuum lithography

Material (energy, ion)

Application

PMMA (50 keV Ga+ ) (260 keV Be++ ) SAL601 (260 keV Be++ )

Surface-structured quantum well Si, GaAs gate HEMT MMICs X-ray mask Wiring

MoO3 , WO3 (50 keV Ga+ ) SOG (100 keV Ga+ ) BiSrCaCuO (200 keV Si++ ) Diamond (25 keV Ga+ ) (260 keV Si++ ) InP (20 keV Ga+ ) GaAs (15 keV Ga+ )

H-Tc bridge GaAs/AlGaAs (200 keV Si++ ) Quantum structure Quantum wire Semiconductor laser

Table 11.2. Applications of FIB etching [73] Material (energy, ion)

Source gas

Physical sputter etching Si, GaAs, InP (25 keV Ga+ ) Si MOS device (30 keV Ga+ ) Resist (30 keV Ga+ ) MoSi2 , Cr (42 keV Ga+ ) COG (25 keV Ga+ ) Cr (0.5 keV Ga+ ) Au (100 keV Ga+ ) Chemical assisted etching Si (5, 20, 35 keV Ga+ ) GaAs (30 keV Ga+ )

Cl2

GaAs/AlGaAs (30 keV Ga+ ) SiO2 (20 keV, 30 keV Ga+ )

Cl2 XeF2

Application Device transportation Failure analysis TEM specimen preparation Photomask repair Phase-shift mask repair X-ray mask repair

Cl2 TEM specimen preparation Quantum wire Semiconductor laser mirror Phase-shift mask repair

In the following we report some practical examples of the commercial use of FIB technologies in the semiconductor industry. Moreover, the application of FIB-based fabrication as a powerful tool for the mass production of electronic devices (similar to optical and e-beam-based lithography) and for the rapid fabrication of nanoscale device prototypes for future electronic applications is considered.


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Table 11.3. Applications of FIB deposition [73] Material (energy, ion)

Source gas

Application

Chemical assisted deposition W (25, 30, 42 keV Ga+ )

W(CO)6

Device transportation Failure analysis X-ray mask repair X-ray mask repair

Au (15, 40, 100 keV Ga+ ) Pt (35 keV Ga+ ) Carbon (42 keV Ga+ ) SiO2 (60 keV, Ba++ , Si++ )

C7 H7 F6 O2 Au (Methylcyclopentadientyl) trimethyl platinum Hydrocarbon gas SiOCH3/4 + O2

Direct deposition Ga (50 eV Ga+ ) Au (54 eV Au+ ) Nb (108–428 eV Nb++ )

Photomask repair Rewiring

Superconductive thin film Josephson junction

Table 11.4. Applications of FIB doping [73] Material (energy, ion) Ga+ )

GaAs (100 keV AlGaAs/GaAs (100 keV Ga+ ) AlGaAs/InGaAs/GaAs (100 keV Ga+ ) AlGaAs/GaAs (30, 100 keV Ga+ ) InGaAs/InP (100 keV Ga+ ) GaAs/AlGaAs (150 keV Ga+ ) GaAs/GaAlAs (20, 100 keV Ga+ ) Si (5 keV Ga+ ) GaAs/AlGaAs (160 keV Si++ ) GaAs/AlGaAs (0.1–1 keV Si++ ) GaAs (140 keV Si++ ) Si (140 keV Si++ ) GaAs (100 keV Si++ , Be++ ) AlGaAs/GaAs (35 keV Si++ ) Si (220 keV As++ ) Si (75 keV Pd+ , Pd++ )

Application Isolation Resonant tunneling Ballistic transport Quantum wire Nanostructure processing Mixing of quantum well In-plane gated wire In-plane gated SET Si p–n junction 2D electron-gas formation Buried electron waveguide Gun diode Selective epitaxy In-plane gated FET Quantum wire CCD Silicide formation

11.4.2.1 IC Failure Mode Analysis FIB, when coupled with SEM and TEM, is a good method for localizing, crosssectioning and observing circuit failure. Electrical failure localization is performed through “Active potential contrast”. It is clear that the number of electrons ejected by a sample irradiated in FIB depends not only on the material, but also on its electrical state. For the same conductive material exposed to FIB, the number of

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ejected electrons will be low or high depending on whether the sample is floating or grounded. Putting one side of the structure at a fixed voltage and the other one at ground, the voltage loss will occur where the structure is resistive, allowing the failure point to be localized. An example of failure localization is shown in Fig. 11.21. Defect characterization is generally carried out using potential contrast and TEM techniques. The imaging and ion milling capabilities of the FIB technique can be used to cross-section the failure area with accurate control of the localization and to observe defects in the cross-section. To obtain a high resolution, the defect area is ion milled using the FIB technique to produce a thin lamella containing the defect, which can be then observed with subnanometer spatial resolution using TEM. Chemical analysis can also be carried out in the TEM using electron energy loss spectroscopy (EELS) and electron energy filtering (EEF) in order to identify the failure. High resolution compositional maps clearly show the materials in the defect area, which should lead to some hypotheses about the origin of the circuit failure. Identifying the failure mechanism leads to appropriate process adjustments and yields process enhancements. A typical example of this application is the failure analysis of memory arrays of integrated circuits. Many modern microelectronic chips contain embedded memory arrays. A typical memory bit cell contains several transistors. Failure of a single transistor or a contact within a bit cell makes the entire bit cell inoperable. During manufacturing, certain types of defects can be identified by in-line metrology, while others cannot. Integrated memory arrays can only be fully tested when the entire device is completed. This is why post-production characterization and failure analysis has become extremely important, particularly for memory arrays in

Fig. 11.21. Example of electrical failure localization by voltage contrast


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IC. A detailed description of the application of the slice and view FIB technique to failure analysis (FA) of memory arrays is reported by A.A. Volinsky et al. [74]. Using the milling and imaging capabilities of a FIB/SEM system, the authors identified several failure mechanisms, including missing patterns, killer particle defects, and shorted and open stack vias. The defects that account for most bit failures are shown in Fig. 11.22. Figure 11.22a illustrates an optical image of a cluster defect (a killer particle). Killer particles can be generated by a tool or a process and, since they are typically large in size, are normally recognized using in-line optical metrology. High particle counts result from a tool malfunction, and corrective action is required that normally includes a temporary process/tool shutdown. Figure 11.22b is a SEM image of via contacts from transistors from a second level metallization. The etch used to create the via on the left was incomplete, resulting in an electrical open. The incomplete etching was traced back to a photoresist process problem. Finally Figs. 11.22c and d show, respectively, a SEM image of two open bits due to a pattern defect and a SEM image of shorted bits (the middle metal lines are bridged together). This bridging was caused by a photoresist problem when patterning the trenches of the second metal layer, and ultimately the bridging led to the shorting of the bits. In multilayer structures, the physical defect can be buried under several layers of interconnections. In some cases, SEM is capable to image a defect but not to resolve it. For example, Fig. 11.23 shows a defect behind a layer of dielectric. While this defect is visible in the SEM due to a large interaction volume, it is not clear exactly where this defect is located in the stack, and what has caused it. In this case, FIB milling comes in handle in determining the failure root cause. Alternatively TEM

Fig. 11.22. (a) An optical image of a cluster defect (particle); (b) SEM images of via contacts: the etch used to create the via on the left was incomplete, resulting in an electrical open; (c) SEM image of two open bits due to a pattern defect; (d) SEM image of shorted bits: here the metal lines in the middle are bridged together [74]

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Fig. 11.23. SEM image showing a defect shadow before milling through it [74]

analysis, along with EDS and EELS, is able to provide additional information for which SEM resolution is not sufficient. In most cases the physical defect is located below the first metal layer and it is not visible in the optical microscope. In these cases, locating defect coordinates on a chip can be a challenge. The absolute memory address of a failed bit is typically available from electrical testing. However, this address needs to be translated into a physical address on the chip (x and y coordinates) for defect analysis and failure, cause identification. Memory translators capable of converting an electrical bit address into a physical location on the die can come in handy. 11.4.2.2 IC Diagnostics: Destructive and Nondestructive Analysis Modern silicon circuits mostly comprise interconnection structures. In 0.25 µm CMOS technology, for example, at least five metal levels are used. Moreover, the electrical resistance and capacitance of these interconnections are the main limiting factors of high speed circuits. It follows that great efforts have been made in the semiconductor industry to develop and characterize new metallization architectures. TEM of samples prepared using FIB has proven to be a powerful technique for circuit and interconnection analysis. The main advantages are the very precise localization and thinning of hard materials down to a thickness of 100 nm. The very low thickness obtained allows high magnification observation and chemical analysis of the diffusion barrier by means of electron energy spectroscopy. This technique has been accepted for the characterization of memory and processor IC chips because it makes it easy to localize cross-sectioning to the micron level. In such applications nondestructive analysis is required. FIB at medium and low energies can be used to investigate localized modification of microelectronic materials [75]. As mentioned above, three-dimensional analysis can be obtained by processing Rutherford backscattering (RBS) data using software. One of the most important applications of nondestructive analysis using FIB has been soft error and charge collection evaluation in DRAMs (dynamic random access memories) [76]. A soft error is produced by energetic particles from IC packaging or wiring metals


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that induce electron–hole pairs along their tracks in the silicon substrate, beneath the active memory cell. The charged carriers created upset the bit state (soft error) after reaching the active memory elements. The advantage of irradiation with FIB scanning is that the exact positions of soft errors can be determined. High-energy ion implantation has been used to form soft error immune structures in DRAMs, called retrograde well structures [77]. Soft error mapping proves that the soft errors are suppressed in a DRAM with a retrograde well structure. Therefore, measurement the ion induced current (the current generated by incident ions, that flows into the active n + region of the memory cell) and soft error mapping can be used to design retrograde well structures that are immune to soft errors in high density DRAMs. 11.4.2.3 IC Repairing Performing modifications on ICs using FIB is a well developed technology in the semiconductor industry [13] (Fig. 11.24). In particular, gas-assisted deposition is used to repair metal lines in integrated circuits. In the repair, a Ga+ focused beam is commonly used in the energy range 10 eV–10 keV. Numerous precursor gases can be used to deposit W, Al, Au or Pt with minimum line widths down to 0.1 micron. In all cases the metal films are impure and often contain 10–50% carbon; thus the resistivity of the film is 20–200 times larger than that of the pure metal. These films are still usable as conductors for circuit repair applications since they can be made thick to compensate for the poor conductivity. The repaired lines made from these metal–carbon films can be studied and characterized [78]. 11.4.2.4 Device Fabrication Traditional methods of device fabrication by lithography and multilevel processing usually depend on the specific chemical and physical properties of the etchants and

Fig. 11.24. Example of circuit modification and repair by FIB deposition

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materials used, and normally require a number of processing steps. In this regards, the FIB technique, which is not a material-specific process, is increasingly being used to fabricate devices [79]. It is worth mentioning that nanoscale junctions achieved using the threedimensional FIB etching technique can be fabricated from many different materials and heterostructures [80]. The field of magnetic recording has greatly benefited from the industrial exploitation of FIB technology. The application of FIB to the formation of the pole gap in MRH is the first nondiagnostic commercial use of the FIB technology. The production procedures, which use sophisticated pattern recognition programs, allow considerable flexibility and accuracy during gap (or pole shape) formation. MRH production requires highspeed machining with submicron accuracy and resolution, and it is a serial process. Figure 11.25 reports a FIB image and a schematic diagram of a ring-type MRH with nanometer dimensions, realized by FIB technology [81]. The probe has two narrow pole tips separated by a 60 nm gap length, and each tip has a cross-section of 140 × 60 nm2 . Tungsten was utilized as a nonmagnetic material to fill in the trenches surrounding the narrow pole tips. FIB etching was used to define the narrow pole tip, and this was followed by FIB deposition of nonmagnetic tungsten into the etched trenches. Direct observation of the ‘easy’ magnetization switching via magnetic force microscopy demonstrates that favorable magnetic properties are preserved during FIB processing. Another industrial use of FIB micromachining in the field of probe recording is magnetic head trimming [82]. FIB milling is used to reduce the pole width of an existing (giant) MRH to 100 nm, thereby increasing the magnetic resolution of

Fig. 11.25. A FIB image (a) and a schematic diagram of a probe with two narrow pole tips, each with a cross-section W × T of 140 × 60 nm2 , separated by a 60 nm long gap [81]


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the head (up to areal densities of 100 Gbit in−2 ). The main advantage is that the existing production infrastructure can be maintained while the critical dimension is drastically reduced. The process is fully automated, enabling magnetic head trimming on wafers and lapped rowbars through the use of pattern recognition software. Typical cycle times are between 2 and 5 s/head. An example of a FIB-trimmed thin film magnetic head is shown in Fig. 11.26 [83]. U. Dotsch et al [84] describe the fabrication of an in-plane gate (IPG) transistor by direct writing FIB. By directly writing a FIB line across a mesa that contains a charge carrier layer close to the surface, it is possible to overcompensate for these carriers with an appropriate ion species. This leads to lateral pn-junctions that divide the former homogeneous charge carrier layer into two electrostatically independent sheets (Fig. 11.27a). The current–voltage characteristic of this lateral interface is given at room temperature in the same figure. Up to positive and negative biases of several volts, the currents are negligible. Figure 11.27b shows the preparation of a FIB line with a gap in the middle. When the bias voltage is approximately zero, this constriction, typically 1 µm in width, shows clear ohmic behavior and runs into a saturation current for larger bias. By combining insets (1) and (2) of Fig. 11.27, inset (3) is obtained: the constriction is tunable by the adjacent in-plane gate (IPG) and shows current–voltage characteristics similar to those of a field-effect transistor. It is worth noting that this transistor concept is not associated with a particular semiconductor material and, up to now, has been realized in Alx Ga1−x As/GaAs heterostructures, Inx Ga1−x As/GaAs quantum wells, SiGe heterostructures, nonepitaxial silicon-on-insulator, and on standard bulk silicon. The fabrication of this type of transistor is therefore completely maskless and does not require any alignment procedures, since the gate, source and drain are all written in the same writing process by FIB. It is worth noting that, with typical beam currents of 30 pA, it is realistic to assume that millions of devices can be written per second by FIB. Therefore, the sequential character of the FIB implantation does not limit the fabrication speed.

Fig.11.26.(a) FIB-trimmed longitudinal magnetic head; (b) FIB-trimmed perpendicular magnetic head [83]

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Fig. 11.27. Creation of an IPG transistor by FIB: (1) line writing; (2) definition of a unipolar channel; (3) combination. The corresponding current–voltage characteristic is given at each stage [84]

11.4.2.5 Rapid Prototyping of Nanoscale Devices in the IC Industry In the submicrometer range, the physics governing principal device functions deviates from classical laws towards quantum-mechanical interactions with discrete energy states. Therefore, to establish fabrication technologies at the nanoscale, there is a strong need for other fundamental approaches besides the traditional scaling approach. Such a need for novel nanoscale technologies demands numerous experiments and prototype building. Hence, the ability to quickly fabricate a nanoscale prototype device for further characterization and optimization becomes a critical factor for achieving technological progress. For successful prototyping, it is vitally important to be able to fabricate an individual device in a relatively short period of time. Convectional e-beam-based


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fabrication is a fairly ‘slow’ process for making the prototype nanoscale devices needed to prove a concept. This is where FIB becomes helpful and/or complementary to other technologies. The multibillion dollar data storage industry is one example of a technology field that has strongly benefited from implementing FIB in the role described above. FIB has been put forward as a rapid way to fabricate a nanoscale magnetic device for further prototyping [82]. In particular, magnetic recording at areal densities beyond 1 Tbit in−2 provides an example of a technology in which the implementation of FIB prototyping has played a critical role in its successful transition to the nanoscale range [83]. It is believed that perpendicular recording can extend magnetic recording to as much as 1 Tbit in−2 areal density, which represents the superparamagnetic limit of conventional magnetic technology based on longitudinal recording [85]. Both perpendicular and conventional longitudinal recording systems rely on using some form of ‘soft’ magnetic head and ‘hard’ magnetic recording media with information written into the recording media through the magnetic field generated by the recording head [86]. The change from longitudinal recording media to perpendicular recording media demands a change in the recording head from the ring type shown in Fig. 11.28a to the perpendicular single pole type shown in Fig. 11.28b. While an optical or e-beam-based fabrication process for the production of a recording head includes approximately 30–40 steps, many different head designs can be explored by just trimming a thin film head from the air bearing surface (ABS), by exploiting the direct etching capability of FIB. A demonstration of how to use FIB for trimming longitudinal and perpendicular magnetic recording and playback devices to dimensions of less than 100 nm is given by D. Litvinov et al [86]. In that paper, regular shielded giant magnetoresistive heads (G-MRH) are FIB-trimmed from the air bearing surface (ABS) to define ultranarrow track width longitudinal (RH) and perpendicular (SPH) heads. The trimmed pole

Fig. 11.28. A schematic diagram of typical longitudinal (a) and perpendicular (b) recording media [89]

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track width is around 100 nm. The authors report that for a beam spot size of approximately 10 nm and an ion current of approximately 10 pA, FIB trimming one recording head to a track width of 100 nm takes about 5 hours. Moreover, other emerging and fast-growing technologies are considered to be examples of nanoscale technologies whose future could strongly depend on using FIB as a nanoscale fabrication tool for rapid prototyping, such as ‘ballistic’ magnetoresistance and magnetic nanotubes. ‘Ballistic’ magnetoresistance refers to the large magnetoresistance observed for nanoconstrictions (Fig. 11.29) at room temperature [87]. When an electron passes through a nanoconstriction with no magnetic domain wall, it experiences no scattering (or ‘ballistic’ passing) because the nanoconstriction is much shorter than the electron mean free path. On the other hand, in the presence of a magnetic field, electrons passing through a nanoconstriction begin to scatter, thus substantially changing the electrical resistance of the nanoconstriction. It should be noted that this type of sensor displays a relatively large magnetoresistance even at room temperature because of the nanoscale dimensions of the sensor [88]. If successful, ‘ballistic magnetoresistance’ promises not only to become the core physics mechanism used to sense information at areal densities beyond 1 Tbit in−2 , but also to have a major impact on the progress of the fast-growing area of magnetoresistive random access memory (MRAM) and the emerging area of spintronics (the electronics of the future). Magnetic nanotubes could be used in the process and control of magnetization switching in nanoscale magnetic probes for a number of emerging technologies, including MRAM, magnetic nanoelectromechanical systems (NEMS), and magnetic recording at densities beyond 1 Tbit in−2 . Magnetic nanoprobes could be fabricated using FIB trimming of regular MRH. The shape of the nanotubes can easily be changed by FIB. These FIB-machined nanotubes are perfect prototypal devices for studying the physics of magnetization switching in different types of MRAM and other magnetic devices. Figure 11.30 shows an example of a magnetic nanotube that has a 40 nm-deep cavity with a diameter of 40 nm. The nanotube was trimmed via FIB at the ABS of regular MRH with a rectangular cross-section of 60 × 60 nm2 and with a probe length of between 100 nm and 1000 nm [89].

Fig. 11.29. Nanoconstrictions realized by FIB trimming [89]


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Fig. 11.30. Magnetic nanoprobe fabricated by FIB trimming of a regular MRH [89]

11.4.3 Micromachining The miniaturization of mechanisms, actuators, sensors, and so on, and the integration of these micromechanical components with electronic devices (microelectromechanical systems and nanoelectromechanical systems – MEMS and NEMS) is considered to be a new high technology growth area of enormous potential [90, 91]. While many techniques such as laser beam and plasma beam machining and electrodischarge wire cutting have been very successful when applied to large structure micromachining, the need for micromachining techniques that are capable of producing submicron structures has not yet been completely satisfied. FIB has been recently included in a review of new and promising techniques for submicron fabrication [92]. A summary of the characteristics of some micromachining techniques is given in Table 11.5. The extensive subject of microfabrication, microlithography and micromachining is covered comprehensively in two excellent recent handbooks by Madou and Rai Choudhury [93, 94]. Reported data allow us to make some comparisons between FIB and other advanced micromachining techniques. FIB (milling and deposition) is a direct write micromaching technique, capable of sub-100 nm resolution (below 100 nm for deposition, and even lower for milling and implantation). FIB has the advantage that its penetration into material is limited to a few atomic layers and the resulting secondary electrons have low energies and therefore relatively minor lateral diffusion. The consequence of very shallow penetration is that the high spatial resolution of the beam at the surface does not deteriorate below the surface, in contrast to e-beam writing.

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Table 11.5. Summary of the features of main micromachining techniques

FIB deposition FIB milling LCVD X-ray lithography (LIGA) UV lithography Microstereolithography Micro EDM Laser ablation

Resolution/ maximum feature size

Direct Aspect Deposition rate/ write ratio etch rate

100 nm/20 µm 20 nm/50 µm 10 µm/1 mm 50 nm/5 mm

Yes Yes Yes No

5–10 10–20 11 50

0.05 µm3 s−1 1 µm3 s−1 100 µm3 s−1 /

Scarse Scarse Diffuse Diffuse

No No Yes Yes

50 nm 5 µm/5 mm

No Yes

1 n/a

/ 100 µm3 s−1

Scarse Diffuse

Yes Yes

10 5

100,000 µm3 s−1 Diffuse 500 µm3 s−1 Diffuse

Yes Yes

100 µm/1 mm Yes 5 µm/5 mm Yes

Rate of use High volume in micro- production machining capabilities

The main benefits of FIB micromachining are the high aspect ratio and the wide range of shapes and materials that can be machined. In general, any complex geometry is possible. The size of the structures that can be obtained is limited by the available processing time (low deposition rate and milling rate). The slow processing is the main drawback of FIB. Hence only relatively small structures (typically tens of micrometers) can be realized within a reasonable time (less than 10 hours). For this reason, the technique is best suited for small-scale post-processing or prototype fabrication. 11.4.3.1 FIB Milling Micromachining Nowadays, the versatility and usefulness of FIB milling in microfabrication is well known and has been extensively demonstrated [95]. FIB milling is largely used to fabricate cutting tools, because this technique affords precise control over feature sizes, permits a variety of tool geometries and defines sharp cutting edges. Cutting edge radii of curvature are less than or equal to 0.1 µm. Moreover, inside the FIB chamber, tools can be moved with sub-µm accuracy and can be rotated over a large range of angles [96]. Tool fabrication via FIB milling provides precise beam positioning and dose allocation. Ion sources, focusing optics and beam deflectors are extremely stable over hundreds of hours. The FIB technique allows observation of the tool during fabrication and, consequently, microtool fabrication is highly reproducible. Because the tools are shaped by an atom-by-atom removal process, and mechanical forces that could lead to microtool fracture during FIB fabrication are negligible, ion sputtering is a relatively ‘stress-free’ process. This technique is therefore also beneficial for shaping brittle tools. In summary, the main advantage of using FIB as a micromachining tool for micro-end mills is the ability to fabricate: – well-aligned, nonplanar, cutting and tool-end clearance facets; – trenches several millimeters long;


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– trenches with nearly vertical sidewalls and low bottom surface roughness; – tools with submicrometer minimum feature sizes. Many researchers have demonstrated micro-end mills and cutting tools shaped by FIBs [97, 98]. Traditional materials for microtool fabrication are commercially supplied, generally in the form of a cylindrical piece of hard material such as stainless steel or tungsten carbide or single crystal diamond, attached to a mandrel. The tool blank can have virtually any diameter, typically between 20 µm and 30 µm. In order to fabricate a micro-end mill, the FIB stage and tool are fixed and the gallium beam impinges normally to a plane containing the tool axis, but tangentially to the tool circumference. This ion beam–solid geometry is selected because it produces one extremely sharp edge per facet. The ion beam is rastered under computer control within a rectangular field near one edge of the blank. As the beam is scanned, blank material is removed, leaving a vertical plane. The ‘upstream’ edge of the plane is rounded because of the Gaussian beam distribution and it cannot therefore be used as a cutting edge. As the plane develops, it acts as a cut-off filter and when it reaches the ‘downstream’ edge of the plane, the effects of the Gaussian distribution are greatly minimized. This intersection results in a sharp intersection between the plane and the ‘downstream’ side of the blank. Figure 11.31 shows evidence of a sub-µm radius of curvature along a cutting edge. After the first cutting edge is generated, the tool is rotated at this angle and the second cutting edge is generated. Accurate placements of facets on tools and a proper stage rotation sequence between ion sputter steps are used to make micro-end mills of different geometries. By machining two facets that intersect at the ‘downstream’ edge, a microscalpel can be produced. If the blank is positioned with its axis parallel to the incident ion beam and the beam is programmed to machine an annulus, very sharp probes can be produced, that can be used as catheters for biomedical applications. Many special tips for SPM have also been produced. These examples demonstrate the versatility and usefulness of this tool in microfabrication applications. Although fabrication of curved surfaces remains a difficult task, made more complicated by the dependence of the sputter yield on incidence angle, recent progress [23] has been made toward ensuring that 3-D micromachining can be performed for any pre-specified geometry via FIB sputtering. The final part of this section illustrates some practical examples of microtools fabricated by FIB. D.P. Adams et al [98] describe micro-end mills with two, four and five cutting edges created from 25 µm-diameter tool blanks of cobalt M42 high speed steel

Fig. 11.31. Sharp cutting edge of a tungsten carbide micro-end mill tool, intersection of two FIB-sputtered facets [98]

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and C2 micrograin tungsten carbide (Fig. 11.32). The authors report that, using a 20 keV focused gallium ion beam, the micro-end mill tools were fabricated in 2–3 h depending on micro-end mill design and tool material. In general, a tool with a large number of facets is fabricated in a relatively short time, because less material is removed. These micro-end mill tools were successfully used to machine millimeter-long trenches in aluminum, brass, steel and PMMA. Machined trench widths of 200 nm were verified (approximately equal to the tool diameters and the surface roughness (Ra) at the bottom of the micromachined feature). These microtools showed that FIB milling is not limited to micro-end mills but can also be used to make more complex micro-cutting tools. For example, high-precision threading and grooving tools for lathe machining of nonplanar work pieces are feasible. Y.N. Picard et al [100] demonstrate tools with rectangular, triangular, and other complex-shaped face designs, made from tungsten carbide, high-speed tool steel, and single-crystal diamond. SEM reveals that these tools have cutting edge radii of curvatures equal to 40 nm or less and cutting edge roughnesses (Ra ) of 0.05 µm or less. The cutting edge radius of curvature is then a small fraction of the microtool width and length, making the cutting edge sharp compared to the depth per pass used during machining. Some examples extracted from this work are shown in Figs. 11.33–11.34. Figure 11.33a shows a microlathe tool that has been ion milled from a diamond tool blank. The starting material was a single crystal segment that was brazed into a tapered tungsten carbide mandrel, and ground to an approximately 40 µm × 40 µm square cross-section. The width of the tool after ion shaping was 23 µm. This tool was made to have a small side rake angle (2.0◦ ) and relief behind all cutting edges. Figure 11.33b demonstrates the ability to tailor tool geometry by FIB milling. This tool was also shaped from a single-crystal diamond blank, but the ion milling patterns and sequences were chosen so that two rectangular cutting surfaces with 10 µm-wide tips separated by 7.7 µm result. Each of the tips on the tool measures 10 µm (wide) × 12 µm (long). The right tip is 29 µm thick, while the left tip is 26.5 µm thick. The variation in thickness is a result of the original cross-section of the tool blank, and its orientation to the ion beam. Nevertheless, relief is established behind all cutting edges on this two tip tool. Figure 11.34 shows a high-speed steel two-tip tool (Fig. 11.34a) and a portion of the triangular grooves cut into a 3 mm diameter PEEK rod using this tool (Fig. 11.34b). Each groove has a triangular cross-section and an internal angle of 62.5◦ . A back rake angle of 10◦ is also created for both tips. A relief angle of ap-

Fig. 11.32. Micro-end mills with two, four and five cutting edges fabricated by FIB milling 25 µm-diameter tool blanks [98]


a

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b

Fig. 11.33. Single-tip (a) and two-tip (b) single-crystal diamond tools shaped by FIB milling [100]

a

b

Fig. 11.34. A high-speed steel two-tip tool (a) and a portion of the triangular grooves cut into a 3 mm-diameter PEEK rod using this tool (b) [100]

proximately 4◦ is established behind all cutting edges and behind the two end points of the tool. The two tips are 24 µm long. Qualitatively, the tool shape appears to be replicated very well in the crosssections of the grooves. Both grooves shown in Fig. 11.34b have an angle of approximately 62.5◦ . The depths of the two grooves are also similar and very close to the intended depth. The groove pitch of the pair is 150 µm. The applications immediately devised for this tool include extremely small solenoids and rotary elements of microscrew pumps. Microcoils and microsolenoids are good examples of structures that are characteristically difficult to produce by classical methods. For this reason, processes consisting of machining and metal deposition have been investigated because they have the potential to produce small diameter solenoids with a large number of turns per unit length. Y.N. Picard et al [100] describe some processes for fabricating microcoil structures to be used in prototyping or small lot production. These processes combine FIB machining techniques with additive deposition techniques (such as PVD and electroplating). Figure 11.35 shows a helical cut performed in a copper-coated PMMA rod (3.0 mm in diameter). A 20 µm-thick Cu conducting layer is deposited by electron beam evaporation and by electroplating. A FIB-shaped tool is then employed in a precision lathe process to cut the conductive layer at sequentially increasing depths until the (soft) insulating support is reached. The result is an electrically isolated

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Fig. 11.35. Helical cut in a PMMA cylinder (diameter 3 mm and Cu coating 20 µm thick) [100] performed by a FIB-shaped tool

helical winding in which the conductor width is determined by the pitch of the microturning operation. The thickness of the metal conductor is set by the previous electroplating procedure and etching (if removal of debris is deemed necessary). This helix has a pitch of 74 µm, while the conductor has a rectangular cross-section of 45 µm × 15 µm. In general, it is possible to assert that cross-sections machined by FIB milling closely match tool face shape. Metrology of machined workpieces shows nearly identical feature widths and includes angles comparable to the FIB-fabricated microtools. Measurements of groove depth around the circumferences of cylindrical workpieces show good matching independently of arc length, thus demonstrating the ability of FIB-milled precision lathe machines to fabricate microscale and miniature cylindrical components. In terms of the time required for milling/lathe tool fabrication, a FIB column producing 20 nA would allow the fabrication of a 25 µm-wide high speed steel and tungsten carbide tool in less than 30 min. Diamond cutting tools can be shaped by FIB milling with extremely precise dimensions and complex tool face shapes. In general, a diamond tool can be machined in order to have good rake, sharp relief, minimal cutting edge roughness, and an extremely small radius of curvature (< 40 nm), features which are not achievable by conventional methods, such as grinding. Based on the known effects of FIB H2 Oassisted sputtering [99] on diamond, it is reasonable to expect that a diamond tool can be shaped in a time comparable with that used for other materials. 11.4.3.2 FIB Deposition Micromachining As mentioned above, FIB deposition allows the deposition of complex 3-D shapes with overhanging features [27]. The key feature of this technique is the ability to deposit material structures that extend beyond the present or previously deposited structure underneath. An ‘overhang’ can be created in this way. As the deposition proceeds layer-by-layer, the new material extends a bit more over the previously deposited layer in each pass. Various 3-D structures have been fabricated by exploiting this technique. Two examples are shown in Fig. 11.36 [92]. For the overhanging deposition, a trade-off must be reached between accuracy and cleanliness on the one hand, and processing time on the other hand. If a very fine (low current) ion beam and a very small overlap between the deposited layers is used, a very accurate (no more than 100 nm) deposition is obtained with little or no debris underneath the


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overhanging parts. On the other hand, if high deposition rates are required, a higher current ion beam is preferred, which will result in a coarser definition of the deposition pattern and a larger amount of debris. Moreover, the deposition times required are different, although the difference is very small for small area depositions (below 1 mm2 ): the ion current ranges from 100 pA up to 1 nA while the deposition time changes by about an order of magnitude (from 1–2 hours down to a few minutes). FIB deposition also offers the possibility of producing hermetic encapsulation. Here, the ability to fabricate extending or overhanging features is used to deposit ‘roof’-like structures [92]. Such a roof structure can be used to encapsulate, for example, a resonating beam: an interesting application of this technique is the transformation of a trapped gas pressure sensor into an absolute pressure sensor. It is worth mentioning here that S. Reyntjens et al [27], while investigating the mechanical properties of FIB-deposited materials, and more specifically the Young’s modulus of deposited silicon oxide, found a value that was a little lower than the literature values for other types of SiO2 : 45 GPa for FIB-deposited SiO2 versus about 60 GPa for thermal dry or wet SiO2 and 92 GPa for sputtered SiO2 . Based on the above analysis, FIB milling and FIB-induced deposition are valuable and powerful techniques for the fabrication of 3-D microstructures. The main benefits are the wide range of shapes that can be realized, the high aspect ratio and the high resolution attainable. The size of the machined structure is limited by the available processing time. Hence only relatively small structures (typically tens of micrometers) can be realized within a reasonable time. The technique is therefore best suited to small-scale post-processing or prototype fabrication. With the growing importance of MEMS and NEMS, the question that arises is whether FIB can be successfully and economically applied to prototyping and production in these areas. A nice demonstration of how FIB can be used for a limited, but

Fig. 11.36. Example of three-dimensionally shaped depositions with overhanging features [92]

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highly accurate, post-processing step on a micromechanical device, is represented by cantilever-based microaccelerometers. Such devices were proposed, demonstrated and placed into production at least ten years ago [101, 102]. A microaccelerometer with a tunnel junction readout, consisting of a perforated square proof mass suspended on two arms, is shown in Fig. 11.37 [103, 104]. The device was realized from a proof mass of a silicon-on-insulator (SOI) material (3 µm thick), combining silicon surface micromachining (to pattern the proof mass) and FIB milling (to mill the readout gap of the accelerometer and release the proof mass). The gap is used to read out the acceleration. In this application, the main requirement is the ability of the FIB to mill narrow and deep trenches at any angle. This is an example of how FIB precision cuts to free cantilever arms permit the incorporation of a robust structure with a precision free arm motion measurement. The time required to make the oblique cut in the 10 µm × 7 µm beam by FIB is around three minutes with a 30 keV, 4 nA gallium ion beam assisted by iodine vapor (used in order to increase the reactive etch rate). The large application and mass production of this sensor, and of other general MEMS devices, could therefore justify the use of FIB in a serial industrial process.

Fig. 11.37. An image of a microaccelerometer postprocessed by FIB milling [103]

11.5 Future Directions The controlled and reproducible fabrication of nanostructured materials will constitute one of the main challenges to industry in the near future [91]. The industrial community has great expectations regarding the implementation of ion beam technology as a nananoscale fabrication tool for rapid prototyping. As one of the most illustrative examples, it is believed that the multibillion dollar data storage industry will strongly benefit from implementing FIB in the role described above. For direct-write 3-D micromachining at submicron resolution, the most promising approach appears to be high-energy ion beam micromachining (proton micromachining [105]). This technique should be the first to be used as a substitute for conventional mask processes in the high-volume batch production of microcomponents. High-energy ion beam micromachining may have a significant role to play in rapid prototyping and in the manufacture of molds, stamps and thick masks.


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At research level, manipulation capabilities at the nanometer scale are required for many electronic, chemical and biological applications, but current methods derived from the IC field do not achieve the dimensional control achieved via FIB. For this reason, the field of FIB-enabled nanomanipulation is growing rapidly [106]. At these dimensions, the use of low-energy ion beams has revealed surprising atomic transport phenomena that occur in a variety of materials and geometries [107]. This method (often called “ion beam sculpting”) could be applied to the problem of fabricating a molecular-scale hole, or nanopore, in a thin insulating solid-state membrane. As discussed by Jiali et al [107], such pores could be used to localize molecularscale electrical junctions or switches, and they could be used as masks to create other small-scale structures. Nanopores are also used as membrane channels in all living systems, where they serve as extremely sensitive electromechanical devices that regulate electric potential, ionic flow and molecular transport across cellular membranes. The future of FIB in material analysis is bright, and the present holds many opportunities for both research- and application-driven usage of this technology. In the near future, we can expect commercially available dual beam multifunction instruments that integrate the following subsystems: 1. different modules such as hot, cold and tensile stages for mechanical characterization, and wet and dry working chambers for organic and inorganic applications 2. structural analysis detectors (such as electron-induced chemical analysis or electron backscatter pattern (EBSP) crystallographic analysis) 3. higher current columns (Ar+ or proton guns) capable of faster bulk sectioning of dense materials in order to produce cross-sections with lengths and depths of as much as hundreds of micrometers Increasing numbers of today’s researchers believe that more efficient use of FIB technology will lead to major discoveries in the future.

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Subject Index

abalone nacre, II 110 activation energy, III 89 active near-field optical probes, II 178 actuation, IV 251 adhesion, III 272, III 282, III 300, III 303–305, III 309, III 313, III 316, III 323 adhesion corrected, III 99 adhesion length, III 283 adhesion meter (CAM), III 306 adhesion paradox, III 283 adhesives, III 31 adsorbed layers, III 284 Al2 O3 , III 42 alkane derivative, IV 170, IV 171 n-alkane derivatives, IV 167 amino acid, IV 174 Amontons’ law, III 272 amplitude modulation, II 1 analytical technique, II 381 anharmonic signals, II 23 antenna structures, II 177 aperture, III 236 aperture-based near-field optical probes, II 175 apertureless near-field microscope, III 235 apertureless SNOM, III 236, III 242 array of silicon cantilever, II 169 artifacts, III 242 artificial hip joints, III 286 asperity, III 346–348, III 352, III 354–357, III 367 asphalt, III 287 atom orbitals, III 45 atomic force microscopy (AFM), II 1, II 143–147, II 149, II 152, II 155, II 158, II 159, II 161, II 165, III 28, IV 251 AFM tips, III 288 AFM topographies, III 270 AFM based lithography, IV 105

blunt AFM tips, III 289 dynamic AFM, II 1 α atoms, III 41 β atoms, III 41 autocorrelation function (ACF), III 357 auxiliary electrode, II 187 average, III 353 barium titanate, III 240, III 250 barrier-hopping fluctuations, III 100 basic principles of STM and STS, III 185 bearing ratio, III 355 benthic, III 30 bias-induced nanofabrication, IV 107 bidirectional optical lever, III 225 binary mixture, IV 170 biochemical sensors, II 185 biocompatibility, III 286 biology, III 283 biomineralization, II 109 biomolecules, II 183 bipyridyl, IV 167 bitter pattern, III 359 blind friction calibration, III 93 boron carbide, III 286 boundary conditions, II 6 bow-tie antenna, II 177 BSE, III 34 buckling, III 227 calender, III 348 calendering, III 345, III 348 calibration of lateral forces, III 93 calix[8]arene, IV 163 calorimetry, IV 252 CAM, III 306 canticlever concept, II 195 cantilever bending, III 222 dynamics, II 33

414 probes, II 165 cantor set, III 280 capacitance, III 231 capacitive forces, III 233 capillary neck, III 103 carbon, II 170 clusters, III 270, III 272 nanotube, II 170 nitride, III 286 thin films, III 270 carrier lifetime, II 181 cavity model of elastic-plastic indentation, III 117 cells, III 287 in vivo, III 28 channelling contrast, II 367 chaperones, III 34 characterization of the magnetic tape with MFM, III 359 charge carrier transport, III 109 chemical force mapping, II 183 chemical force microscopy (CFM), II 183 chemical functionalization, II 183 chemical mapping, II 184 chromatic aberration, II 364 cluster-assembled, III 269 co-adsorbed, IV 167 collocated systems, II 6 colloidal probes, III 291 commensurate, III 288 compliance, II 167 confined polymer systems, III 85 confocal scanning optical microscopy (CSOM), III 238, III 245 time-resolved CSOM, III 241 contact area, III 299, III 302, III 308, III 309, III 311 junctions, III 274 mechanics, III 90 mode, III 220, III 221, III 225 potential, III 222 potential difference (CPD), III 223, III 231 pressure, III 117 problem, III 303 stiffness, II 8 contamination needles, II 193 continuum models, III 288 contrast, III 229 mechanisms, II 367 converse piezoelectric effect, III 220

Subject Index cooperative molecular motion, III 107 cooperatively rearranging regions (CRRs), III 103 cooperatively rearranging regions (CRRs), III 85 coronene, IV 162 correlation length, III 357 corrosion phenomena, II 189 Coulomb explosion, III 43 creep models, III 100 Creutzfeldt-Jakob disease, III 33 vCJD, III 34 critical wavelength, II 193 cryoelectron microscopy, III 35 current vs. distance (IZ) curve, II 97, II 101 cut-off effect, II 175 data storage devices, II 169 Deborah number, III 105 decacyclene, IV 162 defect characterization, II 388 deformation, III 300, III 306, III 309, III 316, III 323 dendrimers, IV 23 Derjaguin–Muller–Toporov (DMT), II 7 detachment stress, III 282 detection of higher eigenmodes, II 20 device fabrication, II 395 device under test (DUT), II 179 dewetting kinetics, III 110 diamond, II 167 probes, II 171 diamondlike, III 286 diatoms, III 28 dielectric permittivity, III 233 diffusional shielding, II 188 dimensional constraints, III 109 dip-pen nanolithography (DPN), IV 2, IV 105, IV 251 Dirac’s delta function, II 181 direct writing, IV 8 directional diffusion, IV 176 disentanglement barriers, III 111 dissipation, III 274 lengths, III 108 dissipative, III 231 distribution of heights, III 264 DLC coating, III 322, III 323 DMT model, III 303 DNA, IV 8 domains, III 229

Subject Index boundary, III 227, III 234 contrast, III 221–223, III 225, III 235–237, III 239, III 245, III 250 structure, III 218, III 235–237, III 242, III 244, III 245, III 247, III 251 walls, III 218, III 219, III 221, III 230, III 236, III 237, III 245, III 246, III 248, III 249, III 251 dual beam systems, II 365 dynamic force microscopy (DFM), II 143–147, II 158 dynamic-contact electrostatic force microscopy, III 222 eigenfrequency, II 167 eigenmode, II 3, II 5 eigenvector, II 3 elastic–plastic materials, III 120 elasto–plastic deformation, III 279 electrocatalysis, II 189 electrochemical DPN (E DPN), IV 22 electrochemical microscopy, II 166 electrode, II 96 electron beam deposited tips, II 170, II 193 electrooptic contrast, III 234 modulation, III 238 electrostatic force, III 222 interaction, II 8 electrostatic force microscopy (EFM), III 219, III 220 energy barrier, II 152–155 ergodic, III 285 errors, III 367 correction, III 344, III 346 rates, III 348 Escherichia coli, III 35 etched, III 269 Eunotia sudetica, III 31 evanescent, II 175 waves, II 174 Eyring model, III 99 far-field optics, II 174 ferroelectric, III 217, III 238, III 240, III 241, III 244, III 251 ferroelectric random access memory (FeRAM), III 218, III 252 films, III 246, III 249 finite element modeling, II 99

415 flat tips, III 291 flexural mode, III 225 focused ion beam (FIB), II 194, II 361 force spectroscopy, II 143, II 144, II 149, II 152, II 153, II 155, II 156, II 158, II 161 force–distance curve, II 185 force-induced nanofabrication, IV 108 Fourier coefficient, II 1 Fourier optics, II 174 fractal, III 266, III 357 dimension, III 266 morphology, III 291 surfaces, III 277 fractures, III 267 free volume, III 88 free cantilever, II 2 friction, III 221, III 227, III 248 coefficient, III 104, III 272 force, III 272 friction-velocity analyses, III 104 internal, III 103 monomeric, III 103 theories, III 274 friction force microscopy (FFM), III 85, III 98 frictional dissipation, III 103 fullerenes, III 286 functional head group, II 183 functionalized probes for biological applications, II 166 functionalized tips, II 182 gallium arsenide, II 167, II 173 gas-assisted deposition, II 377 GASH, III 221 Gaussian distribution, III 354–356 geofractals, III 266 glass transition, III 85 graphite, III 41 GroEL, III 35 GroEL GroES complex, III 35 GroES, III 35 guest-host interactions, IV 160 GW model, III 303, III 308, III 311 hard disk recording, II 194 hardness tester, III 292 harmonic, II 1, II 3 signal, II 18 head-tape interface, III 350, III 352, III 353, III 357

416 spacing, III 352, III 354, III 355, III 357 Heaviside function Θ(t − t0 ), II 181 height–height correlation function, III 267 Hertzian, III 275, III 288 hetero epitaxial nucleation, II 110 heterogeneous dynamics, III 85 glass formers, III 103 heterogeneous molecular arrays, IV 166 high coercivity tip, II 194 high spatial frequencies, II 193 high spatial frequency near-field components, II 175 high-speed imaging, IV 252 higher harmonic images, II 25 higher harmonics, II 16 highly oriented pyrolytic graphite (HOPG), III 38 hollow atoms, III 41 host network, IV 160 hydrocarbons, III 284 hydrogen bond, IV 160 IC diagnostics: destructive and nondestructive analysis, II 394 IC failure mode analysis, II 391 immobilization of peptides, IV 14 implants, III 286 in situ polymerization, IV 22 in-plane gate (IPG) transistor, II 397 inclusion effect, IV 162 indentation, III 291 hardness, III 274 indirect patterning, IV 8 inorganic materials, IV 25 inorganic overlayers, III 188 instability, II 16 insulating surfaces, III 38 interfacial boundaries, III 109 constraints, III 84 energy, III 282 glass transition profiles, III 113 plasticization, III 109 reactions, II 186 Tg profile, III 125 intermittent contact mode, II 1, III 234 intermolecular forces, II 149 intralayer array, IV 168 intramolecular forces, II 143, II 149, II 156, II 158, II 161 ion

Subject Index bombardment, III 38 column, II 362 current, II 97 optics, II 364 ion beam, II 408 ion beam lithography, II 379 ion-blasted, III 269 JKR model, III 303 Kelvin probe force microscopy, II 3 kinetic sputtering, III 38 kurtosis, III 265, III 352, III 354, III 355, III 357, III 367 kuru, III 34 lamella templates, IV 174 lamella-type structure, IV 167 Langevin equation, III 101 Laplace transformation, II 13 large organic molecules, III 33 Larmor frequency, III 47 laser scanning microscopy, III 236 lateral contact stiffness calibration, III 93 forces, III 228 twisting, III 225 lateral force microscopy (LFM), IV 18 layered compounds, III 288 LB films, III 288 leakage current, II 93 light lever detection, II 22 light lever readout, II 20 light-emitting diodes, III 109 linear creep model, III 100 linker molecule, II 183 lithium niobate, III 219, III 238, III 249 living cells, III 27 LMIS, II 362 lock-in, III 225 low moment tip, II 194 low temperature grown (LT) GaAs, II 179 luminescence spectroscopy, III 245 machined, III 269 magnetic charges, II 192 material, II 195, IV 27 microscopy, II 166 particles, III 343, III 345, III 348, III 349, III 359, III 361 recording tape, III 343

Subject Index resonance, III 48 resonance imaging, III 48 storage devices, III 286 tape, III 343 magnetic force microscopy (MFM), II 192, III 345, III 358–367, IV 27 magnetic resonance force microscopy, III 48 magnetization, II 192 magnetoresistive, III 344, III 350 magnetoresistive head (MRH), II 380 manufactured metal surfaces, III 267 mask defects, IV 28 material aspects, II 166 material characterization, II 386 material contrast, II 367 mediator, II 188 meniscus force nanografting (MFN), IV 9 mesoscopic contacts, III 293 metal, III 274, III 288 metal-in-gap (MIG), III 343 method of reduced variables, III 104 mica, III 288 micro-Brillouin, III 245 micro-Raman, III 245, III 248 microarrays, IV 7 microcontact Printing (µCP), IV 28 microelectromechanical systems (MEMS), II 167, III 265 micromachining, II 401, III 286 micromechanical properties, III 30 micromotor, III 286 MIG, III 364, III 365 head, III 365, III 366 milling, II 361, II 407 millions of nanometric oscillators, II 169 millipede project, II 169 mineral bridges, II 110 mixed monolayer, IV 19 mobile atoms, III 284 modal harmonic distortion, II 32 modulated contacts, III 92 modulus-matched interface, III 124 molecular assembled, III 269 dynamics (MD), III 284 electronic devices, III 109 friction, III 102 mobility, III 84, III 85 networks, IV 160 overlayers, III 201 recognition, II 149, II 156, II 161

417 relaxations, III 85 templates, IV 159 molecular recognition force spectroscopy, II 158 monolayer protected clusters, IV 177 monomers, IV 22 morphology, III 263, III 274 morphotropic phase boundary (MPB), III 219, III 250 mother of pearl, II 109 multiple degree of freedom (MDOF), II 9 multiply charged ions, III 38 multiscale, III 266 nacre, II 109 nanodefects, III 37 nanoelectromechanical systems (NEMS), III 85, III 103 nanofabrication, IV 103 nanografting, IV 105, IV 109 nanoimpact studies, III 123 nanoindentation, III 312, III 316, III 317 nanomanipulation, II 409 nanometric oscillators, II 169 nanoparticle inks, IV 25 NanoPen Reader and Writer (NPRW), IV 19, IV 109 nanopipette, II 95 nanoroughness, III 286 nanoscience technology, II 165 nanoscopic constraints, III 85 nanoscratching, III 316, III 317 nanoshaving, IV 109 nanostructuring, III 45 nanotube tips, III 272 nanotubes, III 286 nanowear, III 316–318 Navicula seminulum, III 31 near-field electrooptic microscopy, III 242 near-field optics, II 166, II 174 near-field scanning optical microscopy, III 219 noise, III 345, III 357, III 360–362, III 367 nominal contact area, III 272 non-Markovian behavior, III 102 noncollocated systems, II 6 noncontact, III 220, III 223 nonlinearity, II 16 nonminimum phase, II 6 nonspecific tip modification, II 182 object spatial frequencies, II 174

418 oligonucleotides, IV 8 optical aberrations, III 240 optical microscopy, III 235 order–disorder transitions, III 87 organic thin film transistors, III 109 output feedback, II 11 matrix, II 10 2D overlayers, III 208 paraelectric, III 238, III 240, III 241, III 251 parametric excitation, II 33 patch clamp technique, II 107 PEG, II 151, II 158 pentacene, IV 162 phase separation, IV 167 phase transition, III 219, III 236, III 238, III 240, III 244, III 249, III 251, III 252 photoconductive switch, II 179 phthalocyanine, IV 163, IV 168 physicochemical parameters, II 186 piezoelectric force microscopy, III 219 piezoelectric SPM, III 247 piezoresistive cantilevers, IV 252 piezoresistive Wheatstone bridge, II 169 piezoresponse, III 243 piezoresponse force microscopy (PFM), III 220 plastic, III 274 failure, III 280, III 281 flow, III 117 yield, III 116 plasticity index ψ, III 276 pointed tapered metal-coated waveguide, II 175 poisson equation, II 99 polarized light microscopy, III 235 pole zero, II 13 pole-tip recession (PTR), III 350, III 351 poly(ethylene glycol) (PEG), II 150 poly(styrol) (PS), II 28 polyelectrolytes, IV 23 polymer, IV 21 coatings, III 315 polymeric, III 288 polysilicon, III 286 polystyrene, III 103 porous sample, II 94 porphyrins, IV 163 potential barrier, III 102 potential sputtering, III 38

Subject Index power spectrum, III 269 preferential adsorption, IV 160 projection mask technique, II 172 prostheses, III 286 proteins, IV 10 arrays, IV 127 immobilization, IV 129 nanoarrays, IV 11 nanopatterns, IV 128 unfolding, II 157 pull-off force, III 282 pump/probe experiment, II 179 pyroelectric probe, III 236 quality factor, III 223 quantum computing, III 47 quartz microbalance, III 43 ramped creep model, III 100 randomness, III 264 Rayleigh’s criterion, II 175 reactive ion plasma etching, II 177 real area of contact, III 273, III 355, III 356, III 358 recognition image, II 143, II 159–161 reconstruction, II 33 3D, II 382 recording head, III 343, III 344, III 364 redox couple, II 187 relaxation, III 102, III 104 relaxor, III 219, III 251 resolving, II 175 power, II 174 resonance modes, III 224 resonant frequencies, II 5 resonant modes, III 232 rheological boundary layers, III 109 rheological gradients, III 119 rim formation during indentation, III 120 rms roughness, III 353, III 354, III 357 road pavements, III 288 rock surfaces, III 287 root locus map, II 13, II 15 roughness, III 264, III 299–301, III 303, III 304, III 309–311, III 316, III 324, III 344–347, III 351–353, III 355, III 357, III 358, III 360, III 367 rubber, III 288 saturation, III 364, III 365 scanning electrochemical microscopy, II 186 scanning electron microscope (SEM), II 366

Subject Index scanning electrooptic microscopy, III 237, III 248 scanning force spectroscopy (SFS), II 183, II 184 scanning ion conductance microscope (SICM), II 91 scanning near-field optical microscopy (SNOM), II 174, III 236 scanning probe lithography, IV 103 scanning thermal microscopy, IV 251 scanning tunneling microscopy, II 165, III 28 sealing effects, III 283 second harmonic, III 233, III 245, III 248 secondary ion mass spectroscopy (SIMS), II 367 self-affine surface, III 267 self-assembled monolayers (SAMs), III 288, III 306, III 313, III 315, IV 162 self-lubricating coatings, III 317 self-similar, III 266 semiconductor device characterization and nanofabrication, II 166 semiconductor industry, II 389 shear bands, III 120 shear force microscopy, II 111 shear modulation force microscopy (SM FM), III 85, III 97 shear strength, III 274 silanes, IV 19 silazanes, IV 19 silicon, II 167, II 168 silicon (111)-(7 × 7) surface, III 46 silicon-carbide, III 286 silver/silver chloride electrode, II 96 single degree of freedom (SDOF), II 3 single electron spin, III 48 detection, III 47 single molecular array, IV 169 single molecule level, III 34 single molecules, IV 163 single-crystalline diamond, II 171 site-selective adsorption, IV 172 skewness, III 264, III 354, III 355, III 357, III 367 slope-detection method, III 223 small cantilever, III 35 solid-state materials, IV 26 specific tip modification, II 183 spectrum, II 27 spherical aberration, II 365 sputter-deposited, III 269

419 squeezing effect, II 92, II 100 standalone cantilever probes, II 169 state space, II 4, II 9 static contacts, III 90 static friction, III 284 statistics, III 277 steady-state diffusion-limited reaction, II 187 stearic acid, IV 167, IV 173 stiffness, III 292 strain hardening, III 121 rate effects, III 123 shielding, III 124 softening, III 120 stress distribution, III 280 stroboscopic mode, III 241 structural analysis, II 388 structural anisotropy, III 125 structural heterogeneity, III 105 structure function, III 268 structures 1D, III 188, III 202 2D, III 196 subatomic features, III 45 subatomic range, III 27 subharmonics, II 19 substrate constraints, III 122 subwavelength aperture, II 175 superposition of friction-velocity isotherms, III 104 supramolecular architectures, IV 159 compounds, IV 24 surface charge density, III 231 surface energy, III 306 surface force apparatus (SFA), III 273 surface forces, III 309 switching, III 241, III 248, III 249, III 252 system matrix, II 9 tape drive, III 343, III 346, III 348, III 351, III 362 tape roughness analysis, III 351 tapping-mode, II 1, II 108, III 225, III 234 TEM -SEM sample preparation, II 384 template, IV 168 texture, III 299, III 300 TGS, III 223, III 225, III 242, III 245, III 250 the glass transition, III 85 thermogravimetry, IV 252

420 thermomechanical data storage (TDS), III 115 thin films, III 218, III 219, III 237, III 238, III 240, III 251, III 267 head, III 343, III 366 recording, III 362 recording media, III 361, III 362 thiol, IV 167 third-body, III 273, III 284 tip modification, II 182 tip–sample interaction, II 7, II 12 tips, II 193 tire friction, III 283 top-to-bottom nanostructure fabrication, II 176 topographic characterization of heads, III 349 topographic characterization of the magnetic tape, III 345 topographic contrast, II 367 topography, III 299, III 301, III 307, III 318, III 323 topothesy, III 268 total harmonic, II 32 total harmonic distortion (THD), II 17 transfer function, II 13, II 22 transmission electron microscopy (TEM), II 371 transmission minima, II 5, II 12 transmission zeros, II 13 transport phenomena, II 186 transport properties of DPN, IV 4 tribological models for FFM, III 99

Subject Index tribology, III 263 truncated, II 5 model, II 15 twisting, III 227 two-level model, III 303 two-level roughness, III 303, III 307 ultrafast electrical field sampling, II 166 ultrafast scanning probe microscopy, II 179 ultrahigh vacuum, III 42 ultramicroelectrode (UME), II 186 unbinding force, II 149, II 151–154 van der Waals interaction, II 7, IV 163 VCSEL, II 178 virus, IV 15 viscoelastic materials, III 121 vitrification, III 106 voltage modulation, III 220 voltage-modulated scanning force microscopy (VM SFM), III 222 water, III 284 waveguide properties, II 175 wet-chemical surface modification, II 186 Williams–Lendel–Ferry (WLF), III 89 WLF behavior, III 104 writing, III 219, III 248 X ray crystallography, III 35 yield stress, III 280 Young’s modulus, III 276

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