Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy

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Scanning Probe Microscopy of Functional Materials

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Sergei V. Kalinin • Alexei Gruverman Editors

Scanning Probe Microscopy of Functional Materials Nanoscale Imaging and Spectroscopy

Editors Sergei V. Kalinin The Center for Nanophase Materials Sciences and Technology Division Oak Ridge National Laboratory 1 Bethel Valley Road 37831 Oak Ridge Tennessee USA [email protected]

Alexei Gruverman University of Nebraska - Lincoln Department of Materials Science and Engineering Department of Physics & Astronomy 202 Ferguson Hall 7 920 68588 Lincoln Nebraska USA [email protected]

ISBN 978-1-4419-6567-7 e-ISBN 978-1-4419-7167-8 DOI 10.1007/978-1-4419-7167-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938721 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Scanning probe microscopy (SPM) has become a mainstream technique of nanoscience and nanotechnology by providing easy to use methodology for noninvasive imaging and manipulation on the nanometer and atomic scales. Beyond topographic imaging, SPM techniques have found an extremely broad range of applications in probing electrical, magnetic, optical and mechanical properties – often at the level of several tens of nanometers [1, 2], opening the way to an understanding material functionality and interactions at their fundamental length scales [3]. For more than a decade after the introduction of the first commercial microscopes in late 1980s, SPM evolved as a primarily qualitative imaging method. The surface topographic and functional (e.g., magnetic, electrostatic, or mechanical) images were acquired in parallel and were interpreted by an observer. A common feature for these measurements was that only a single or a small number of parameters describing the local properties were obtained; furthermore, information contained in complementary images was usually ignored (or interpreted solely within the limits of a cursory examination). These limitations stemmed primarily from the inherent limits of data processing electronics available at the time, the dearth of well-characterized probes, relative novelty of the field, and only a small number of available microscopic platforms. Nevertheless, even qualitative imaging capabilities have provided multiple opportunities in research for almost a decade. Ironically, this multitude of research opportunities has somewhat shifted the focus of research and development away from further technological advances in SPM. In contrast, the last several years have seen tremendous progress in force-based SPMs. The emergence of digital control and field-programmable gate array electronics have greatly increased the data acquisition and processing speed, allowing multiple information channels to be acquired without compromising image acquisition speed or quality. Similarly, recent advances in the theoretical understanding of contrast mechanisms in SPM and increasing market competition have lead to the rapid emergence of multimodal and spectroscopic SPM methods, including dual excitation frequency SPM (Asylum) [4–6], HarmoniX (Veeco) [7, 8], and configurable multiple frequency lock-ins by Agilent [9] and Nanonis [10]. This progress in fast data acquisition electronics and signal processing in SPM has allowed multiple information channels to be collected in the 1–10 ms range of a single pixel. This development in turn enabled several families of rapid multimodal and spectroscopic v

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imaging SPM techniques. Examples include band excitation [11] and digital lock-in [12] SPM, which allow rapid sampling of a response–frequency curve at each location on a surface, switching spectroscopy PFM [13] for mapping the ferroelectric behavior, rapid force–volume imaging [14] modes ushered in by small (high frequency) cantilever technology, torsional resonance imaging for mechanical property characterization [7], and many others. Advances in ultra-stable STM platforms resulted in a resurgence of STM-based spectroscopic methods, such as continuous imaging tunneling spectroscopy (CITS), dI/dV (density of states), dI/dz (work function), and d2I/dv2 (vibrational) imaging [15]. It is not an exaggeration to say that most of the recent advances in nanoscale science and condensed matter physics have been linked to the development of particular spectroscopic imaging modes – including imaging high-temperature superconductivity by Davies [16] and Yazdani [17], optically assisted SPM introduced by Ho [18], mechanical HarmoniX imaging introduced by Sahin [7, 8], and many others. In parallel with these instrumental developments, significant progress was achieved in development of SPM methods that combine novel experimental modalities, including thermal and mass-spectrometry assisted methods, novel electrical characterization modes, and combinations between SPM and beam techniques including focused X-ray and electron microscopy. Common to all of these methods is the acquisition of complex multidimensional data sets, typically comprised local spectroscopic responses of materials to external stimuli, or multiple parallel channels of information. At the same time, this allows not only the visualization of the structure of surfaces on the nanometer scale, but also insight into their functionalities. In this book, we aim to provide an overview of several notable recent developments in the field of functional SPM enabled by the advances in sample preparation and platform development, ultra-high resolution imaging, novel combined imaging modes, signal detection, data interpretation, and novel dynamic modes. In Chap. 1, Maksymovych delineates the applications of scanning tunneling microscopy and spectroscopy for probing chemical processes on a single-molecule level. While applications of STM for imaging surface structures on the molecular and atomic level has become common, he illustrates how STM can provide insight into chemical functionality of molecular systems. These range from tip-induced surface chemical reactions including long-range hot-electron induced phenomena to time spectroscopies of single molecule transformations to the minute details of the vibrational spectra probed by inelastic electron spectroscopy. High-resolution studies of biological functionality are addressed in the contribution by Malkin and Plomp (Chap. 2). Creatively combining the insights from the crystallization theory and high resolution atomic force microscopy imaging, the authors demonstrate that the molecular structure of the biological objects such as bacterial spores and viruses contains a wealth of information on their functionality and life cycle. Beyond providing a highly illuminating and often spectacular view of microscopic structure of these systems, these studies can be used to identify individual strains of bacterial systems, and establish their developmental pathways in response to changes in environment, chemical stimulants, and therapeutics.

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The recent advances in spectroscopic and multimodal SPMs enabled by novel data acquisition and analysis methods are summarized in Chaps. 3–6. Holscher et al. provide an in-depth description of dynamic force spectroscopy and microscopy in ambient conditions. Based on the precise measurements of the dynamic response of the cantilever, the complete force–distance curve and associated mechanical functionalities can be extracted. This topic is further developed in the contribution by Hurley (Chap. 4) who discusses probing mechanical functionality on the nanoscale, including mechanical properties and adhesive behavior, using Atomic Force Acoustic Microscopy-based methods. The signal formation mechanisms, detailed data interpretation, and multiple experimental examples are discussed. The new paradigm in dynamic SPMs – multiple frequency methods – is discussed in Chap. 5 by Proksch. While the classical SPMs utilize purely sinusoidal excitation signals corresponding to a single frequency in the Fourier domain, the use of multiple excitation and detection frequencies allows systematic mapping of frequency dispersion of the signal. Strategies for nanoscale mapping of dissipative interactions via multifrequency detection are discussed in detail. Finally, in Chap. 6 Sahin describes the tensional resonance method for probing dynamic mechanical properties. Utilizing decoupling between the flexural and torsional oscillation modes and difference in the corresponding resonant frequencies, the dynamic probing of the force–distance curve at each spatial pixel is possible. This approach is demonstrated for multiple applications, including phase transitions in polymers and high-resolution imaging of biological systems. The contribution by Ovchinnikova in Chap. 7 discusses in depth the rapidly emerging chemical imaging methods based on the combination of SPM and mass-spectrometry. While SPM is renowned for high spatial resolution, the amount of chemical information is typically limited. At the same time, modern massspectrometry methods provide ultimate information on the chemical structure of complex biological and pharmaceutical systems, often using minute amounts of material. The SPM-MS approach combines local thermal or optical excitation directed by an SPM tip, with subsequent pick-up of locally emitted products by the mass spectrometer, thus allowing local chemical identification. Critical for broad implementation of this approach is mass spectrometry at atmospheric pressures, and these methods are reviewed in detail. SPM methods for probing thermal phase transitions locally are summarized in the contribution by Nikiforov and Proksch (Chap. 8). Recent advances in SPM tip fabrication lead to the development of heated SPM probes with high heating– cooling rates. These probes enable a broad spectrum of thermal imaging methods. In one approach, the SPM tip concentrates the thermal field within the material, while the resulting surface deformation is detected by SPM electronics. The onset of melting transition below the probe results in probe penetration in the material, allowing the transition temperature to be identified. The combination of periodic heating and dynamic driving modes allows mapping of the glass transition temperatures as well. Beyond thermomechanical effects, these methods can be extended to probing local sample temperatures and heat conductivity, suggesting broad applicability for high-energy density material sand devices.

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The applications of SPM methods to probing electrical and electromechanical functionalities are discussed at length in Chaps. 9–13. In Chap. 9, Magonov et al. extend multiple frequency SPM to in-depth quantitative studies of electrical properties of semiconductors, ferroelectrics and self-assembled monolayers. Along with the overview of Kelvin Probe Force Microscopy and Electric Force Microscopy applications they discuss how frequency modulation realized in these modes can overcome uncertainties related to various mechanisms of response signal formation and improve spatial resolution in functional imaging. The contribution by Tian et al. (Chap. 10) describes the quantitative measurements of ferroelectric polarization distribution on the nanoscale by piezoresponse force microscopy (PFM). The force-based SPM signals scale linearly with tip-surface contact area resulting in a dearth of quantitative measurement capabilities in the range from molecular to mesoscopic (~100 nm) length scale. At the same time, the electromechanical signal in continuous approximation does not depend on the contact radius, enabling quantitative measurements of ferroelectric properties in the PFM mode. Using finite element simulation of the electric and elastic fields for various tip-sample interaction models, Tian et al. show that the real domain wall thickness can be extracted from experimental PFM line profiles across domain walls. This topic of PFM characterization of ferroelectrics is further developed by Huey and Nath in Chap. 11, who systematize a broad range of experimental studies of domain switching dynamics in ferroelectric thin films. By introducing high speed PFM, the rapid mapping of instantaneous domain patterns is possible at a rate of at least 100 times over standard PFM imaging. This is one of the most promising approaches in overcoming the PFM limitation in revealing the parameters of nucleation and fast domain wall motion as the basic mechanisms of polarization reversal in ferroelectric-based devices (notably ferroelectric memories) and understanding the role of structural defects in the thermodynamics of ferroelectric switching. The polar structure and polarization dynamics in relaxor ferroelectrics, one of the most mysterious classes of ferroic materials, are discussed in Chap. 12 by Shvartsman et al. The nanoscale ferroelectric ordering in relaxors presents a significant exploratory challenge but at the same time makes them the textbook example materials for demonstrating the superior capabilities of PFM in discerning the relationship between the polar structure and the unconventional dielectric properties. A detailed review of the PFM studies of several important groups of relaxors is presented highlighting experimental observation of temperature-induced transformation between ferroelectric and relaxor states. In Chap. 13, Ruediger reviews a complex problem of PFM image interpretation stemming from the tensorial nature of the electromechanical response, asymmetry in the local field distribution due to the sample defect structure or tip shape, surface modification and cantilever mechanics. Understanding these contributions allows one to avoid image misinterpretation and identify imaging artifacts while providing additional means for structural and electrical characterization of electronic materials.

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The novel functional SPM methods are discussed in Chaps. 14–17. In Chap. 14, Rose et al. discuss the perspectives of combined STM-focused X-ray measurements. The X-ray methods have evolved to provide in-depth information on the crystallographic structures and chemical identity of the surfaces, often with extremely high temporal resolution. However, spatial resolution is limited to several tens of nanometers. At the same time, STM-based methods routinely yield atomic spatial resolution, but are limited by 10–100 kHz bandwidth of amplifiers and are limited in chemical sensitivity. The potential for X-ray-STM combination and corresponding operational mechanism are discussed. The scanning ion conductance microscopy and its application for mapping surface structures and biological systems are discussed in detail by Rheinlander and Schäffer in Chap. 15. This method allows mapping ionic flows through the microcapillary and is ideally suited for studying the biological and electrochemical systems. This topic is further extended by Beyder and Sachs (Chap. 16), who describe the techniques that combine classical patch-clamp and AFM methods to probe electrophysiological properties on the cellular and subcellular levels. The contribution by Rodriguez et al. (Chap. 17) discusses in detail the novel problems that appear in the context of analysis of the multicomponent spectral data, and illustrates their applications for the voltage and time spectroscopies in PFM. The direct functional fits methods are discussed and compared with multivariate statistical methods including principal component analysis and correlation function analysis. Finally, the contribution by Gruverman (Chap. 18) reviews recent advances in probing and understanding polarization dynamics in ferroelectric capacitors. Although the PFM capability to detect the polarization state through the top electrode allows for direct studies of the dynamics of domain structure under the uniform field conditions, a major limitation was low time resolution. Discussion of the approach to extend the PFM studies into the 100 ns range is presented. The role of inhomogeneous domain nucleation and measurements of the switching parameters in conjunction with microstructural and scaling effects is discussed. Taken together, this book aims to give a prospective of new directions in functional SPM imaging. Many of the applications described in the book have appeared only in the last several years, and the future will undoubtedly see the emergence of a number of SPM modes for addressing materials functionality at the nanoscale. Sergei V. Kalinin Alexei Gruverman

References 1. E. Meyer, H.J. Hug, R. Bennewitz, Scanning Probe Microscopy: The Lab on a Tip (Springer, 2003) 2. S.V. Kalinin, A. Gruverman (eds.), Scanning Probe Microscopy: Electrical and Electromechanical Phenomena at the Nanoscale (Springer, 2006) 3. C. Gerber, H.P. Lang, Nat. Nanotechnol. 1, 3 (2006)

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4. R. Proksch, Appl. Phys. Lett. 89, 113121 (2006) 5. B.J. Rodriguez, C. Callahan, S.V. Kalinin, R. Proksch, Nanotechnol. 18, 475504 (2007) 6. http://www.asylumresearch.com 7. O. Sahin, S. Magonov, C. Su, C.F. Quate, O. Solgaard, Nat. Nanotechnol. 2, 507 (2007) 8. http://www.veeco.com 9. http://nano.tm.agilent.com/index.cgi?CONTENT_ID=253 10. http://www.nanonis.com 11. S. Jesse, S.V. Kalinin, R. Proksch, A.P. Baddorf, B.J. Rodriguez, Nanotechnol. 18, 435503 (2007) 12. A.B. Kos, D.C. Hurley, Meas. Sci. Technol. 19, 015504 (2008) 13. S. Jesse, B.J. Rodriguez, S. Choudhury, A.P. Baddorf, I. Vrejoiu, D. Hesse, M. Alexe, E.A. Eliseev, A.N. Morozovska, J. Zhang, L.Q. Chen, S.V. Kalinin, Nat. Mater. 7, 209 (2008) 14. D.A. Bonnell (ed.), Scanning Probe Microscopy and Spectroscopy: Theory, Techniques, and Applications (Wiley-VCH, 2008) 15. J.A. Stroscio, W.J. Kaiser (eds.), Scanning Tunneling Microscopy (Academic, Boston, 1993) 16. K. McElroy, R.W. Simmonds, J.E. Hoffman, D.H. Lee, J. Orenstein, H. Eisaki, S. Uchida, J.C. Davis, Nat. 422, 592 (2003) 17. K.K. Gomes, A.N. Pasupathy, A. Pushp, S. Ono, Y. Ando, A. Yazdani, Nature 447, 569 (2007) 18. S.W. Wu, N. Ogawa, W. Ho, Sci. 312, 1362 (2006)

Contents

Part I Spectroscopic SPM at the Resolution Limits 1 Excitation and Mechanisms of Single Molecule Reactions in Scanning Tunneling Microscopy.......................................................... Peter Maksymovych 2 High-Resolution Architecture and Structural Dynamics of Microbial and Cellular Systems: Insights from in Vitro Atomic Force Microscopy.......................................................................... Alexander J. Malkin and Marco Plomp

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Part II Dynamic Spectroscopic SPM 3 Dynamic Force Microscopy and Spectroscopy in Ambient Conditions: Theory and Applications.................................. Hendrik Hölscher, Jan-Erik Schmutz, and Udo D. Schwarz

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4 Measuring Mechanical Properties on the Nanoscale with Contact Resonance Force Microscopy Methods............................. D.C. Hurley

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5 Multi-Frequency Atomic Force Microscopy .......................................... 125 Roger Proksch 6 Dynamic Nanomechanical Characterization Using Multiple-Frequency Method.......................................................... 153 Ozgar Sahin Part III Thermal Characterization by SPM 7 Toward Nanoscale Chemical Imaging: The Intersection of Scanning Probe Microscopy and Mass Spectrometry........................ 181 Olga S. Ovchinnikova xi

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8 Dynamic SPM Methods for Local Analysis of Thermo-Mechanical Properties.......................................................... 199 M.P. Nikiforov and R. Proksch Part IV Electrical and Electromechanical SPM 9 Advancing Characterization of Materials with Atomic Force Microscopy-Based Electric Techniques.................................................. 233 Sergei Magonov, John Alexander, and Shijie Wu 10 Quantitative Piezoresponse Force Microscopy: Calibrated Experiments, Analytical Theory and Finite Element Modeling.......... 301 Lili Tian, Vasudeva Rao Aravind, and Venkatraman Gopalan 11 High-Speed Piezo Force Microscopy: Novel Observations of Ferroelectric Domain Poling, Nucleation, and Growth . ................. 329 Bryan D. Huey and Ramesh Nath 12 Polar Structures in Relaxors by Piezoresponse Force Microscopy..... 345 V.V. Shvartsman, W. Kleemann, D.A. Kiselev, I.K. Bdikin, and A.L. Kholkin 13 Symmetries in Piezoresponse Force Microscopy................................... 385 Andreas Ruediger Part V Novel SPM Concepts 14 New Capabilities at the Interface of X-Rays and Scanning Tunneling Microscopy..................................................... 405 Volker Rose, John W. Freeland, and Stephen K. Streiffer 15 Scanning Ion Conductance Microscopy................................................. 433 Johannes Rheinlaender and Tilman E. Schäffer 16 Combined Voltage-Clamp and Atomic Force Microscope for the Study of Membrane Electromechanics...................................... 461 Arthur Beyder and Frederick Sachs 17 Dynamic and Spectroscopic Modes and Multivariate Data Analysis in Piezoresponse Force Microscopy............................... 491 B.J. Rodriguez, S.Jesse, K. Seal, N. Balke, S.V. Kalinin, and R. Proksch

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18 Polarization Behavior in Thin Film Ferroelectric Capacitors at the Nanoscale.................................................................... 529 A. Gruverman Index.................................................................................................................. 541

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Contributors

John Alexander Agilent Technologies, 4330 W. Chandler Blvd., Chandler, AZ 85226, USA Vasudeva Rao Aravind Materials Science and Engineering, Pennsylvania State University, University Park, PA 16803, USA; Clarion University of Pennsylvania, Clarion, PA 16214, USA N. Balke Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA I.K. Bdikin Depto de Engenharia Cerâmica e do Vidro, CICECO, Universidade de Aveiro, 3810-193 Aveiro, Portugal Arthur Beyder Department of Medicine, Mayo Clinic, 200 First Street SW, Rochester, MN 55905, USA [email protected] John W. Freeland Advanced Photon Source Argonne National Laboratory, Argonne, IL 60439, USA Venkatraman Gopalan Materials Science and Engineering, Pennsylvania State University, University Park, PA 16803, USA [email protected] A. Gruverman Department of Physics and Astronomy, University of Nebraska, Lincoln, NE 68588-0111, USA [email protected] Hendrik Hölscher Institute for Microstructure Technology, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany [email protected]

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Bryan D. Huey Materials Science and Engineering Program and Institute of Materials Science, University of Connecticut, Storrs, CT 06269, USA [email protected] D.C. Hurley National Institute of Standards & Technology, 325 Broadway, Boulder, CO 80305, USA [email protected] S. Jesse Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA S.V. Kalinin The Center for Nanophase Materials Sciences and Technology Division, Oak Ridge National Laboratory, 1 Bethel Valley Road, 37831 Oak Ridge, Tennessee USA A.L. Kholkin Center for Research in Ceramic and Composite Materias (CICECO) & DECV, University of Aveiro, 3810-193, Aveiro, Portugal [email protected] D.A. Kiselev Depto de Engenharia Cerâmica e do Vidro, CICECO, Universidade de Aveiro, 3810-193 Aveiro, Portugal W. Kleemann Angewandte Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany Sergei Magonov Agilent Technologies, 4330 W. Chandler Blvd., Chandler, AZ 85226, USA [email protected] Peter Maksymovych Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA [email protected] Alexander J. Malkin Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] Ramesh Nath Materials Science and Engineering Program and Institute of Materials Science, University of Connecticut, Storrs, CT 06269, USA M.P. Nikiforov Oak Ridge National Laboratory (ORNL), Oak Ridge, TN 37831, USA [email protected]

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Olga S. Ovchinnikova Department of Physics and Astronomy, University of Tennessee, Knoxville 401 Nielsen Physics Building, 1408 Circle Drive, Knoxville, TN 37996-1200 [email protected] Marco Plomp Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] Roger Proksch Asylum Research, Santa Barbara, CA 93117, USA [email protected] Johannes Rheinlaender Institute of Applied Physics, University of Erlangen-Nuremberg, Staudtstr. 7, Bldg. A3, 91058 Erlangen, Germany B.J. Rodriguez University College Dublin, Belfield, Dublin 4, Ireland [email protected] Volker Rose Advanced Photon Source Argonne National Laboratory, Argonne, IL 60439, USA [email protected] Andreas Ruediger Laboratory of Ferroelectric Nanoelectronics, Institut National de la Recherche Scientifique, Université du Québec, 1650, Blvd. Lionel-Boulet, Varennes, Canada J3X 1S2 [email protected] Frederick Sachs Center for Single Molecule Biophysics, Physiology and Biophysical Sciences, 301 Cary Hall, University at Buffalo, State University of New York, Buffalo, NY 14214, USA Ozgur Sahin The Rowland Institute at Harvard, Cambridge, MA, USA [email protected] Tilman E. Schäffer Institute of Applied Physics, University of Erlangen-Nuremberg, Staudtstr. 7, Bldg. A3, 91058 Erlangen, Germany [email protected] Jan-Erik Schmutz Center for Nanotechnology (CeNTech) and Physikalisches Institut, University of Münster, Heisenbergstr. 11, 48149 Münster, Germany

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Udo D. Schwarz Department of Mechanical Engineering, Yale University, New Haven, CT, USA K. Seal Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA V.V. Shvartsman Angewandte Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany Stephen K. Streiffer Argonne National Laboratory, Center for Nanoscale Materials, Argonne, IL 60439, USA Lili Tian Materials Science and Engineering, Pennsylvania State University, University Park, PA, 16803, USA Shijie Wu Agilent Technologies, 4330 W. Chandler Blvd., Chandler, AZ 85226, USA

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Part I

Spectroscopic SPM at the Resolution Limits

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Chapter 1

Excitation and Mechanisms of Single Molecule Reactions in Scanning Tunneling Microscopy Peter Maksymovych

Introduction Scanning tunneling microscopy (STM) achieves atomic-scale resolution due to the exponential dependence of the tunneling current on the distance from the tip to the surface. The majority of tunneling electrons traverse the junction elastically via coherent quantum mechanical coupling between the electronic states of the tip and the conducting substrate. However, a small fraction of tunneling electrons undergoes inelastic scattering, losing parts of their energy to available dynamic modes in the junction with the energy that is less or equal to the electrochemical potential of one of the tunneling leads relative to the Fermi level of the other. Depending on the atomic electronic structure of the tunneling junction and the tunneling conditions, the excited processes may include localized plasmons with subsequent photon emission [1–4], frustrated [5] and free [6] adsorbate motion, formation of charged species [7], molecular fluorescence [8], rotation [9], vibration [10], bond breaking [11, 12], and isomerization [13, 14]. The STM can therefore glimpse far beyond the local electronic structure of the junction and it has been extensively used to explore the dynamic functionality of surfaces, nanoparticles, and single molecules. This chapter will address a subset of such studies, focused on the electroninduced excitation of one or several chemical bonds of molecules located in close proximity of the STM tunnel junction. Such localized chemical reactions have been used to understand the fundamental aspects of molecule surface and intermolecular chemical interaction, manipulate single molecules or molecular aggregates to construct artificial nanostructures with new chemical or electronic properties, and control molecule-surface electronic coupling. Representative examples of such studies are shown in Fig. 1.1. Despite the striking diversity of the observed phenomena, the elementary processes initiating the motion of the molecule along the

P. Maksymovych (*) Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_1, © Springer Science+Business Media, LLC 2010

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Fig. 1.1 (a) Linked-chevron arrangement of CO molecules assembled on Cu(111) surface using STM-tip-induced manipulation at T < 10 K (image size: 1.7 × 3.4 nm2). Blue (red ) dots mark the positions of the surface lattice atoms (CO molecules). Shifting of one molecule as shown by the green arrow triggers spontaneous cascade motion of CO molecules (blue arrows) along the assembled arrangement, until all the chevrons have decayed forming the structure in B. CO cascades were assembled into logic circuits, such as AND and OR gates (reproduced with permission from Heinrich et al. Science 298 (2002) 5597. Copyright (2002) by the American Association for the Advancement of Science. (b) Local manipulation of Kondo resonance by STM-tip-assisted attachment of single CO molecules to a Co atom on the Cu(100) surface. Models, STM topographies, and STS spectra in the center of the Co adatoms and complexes under investigation: (a) Co-adatom, (b) Co(CO)2, (c) Co(CO)3, (d) Co(CO)4. Models and topographies in (a)–(d) drawn to the same scale. The solid lines in the spectra are fits of a Fano function (redrawn with permission from Wahl et al. Phys. Rev. Lett. 95 (2005) 166601). Copyright (2005) by the American Physical Society. (c) Picometer-scale control of single-molecule dynamics. The image is STM topography of a biphenyl molecule on Si(100) surface (2 × 2 nm2, VS = –2 V, I = 0.56 nA). The tunnel current during the surface voltage pulse (VS = –2.5 V, duration 10 s) is shown for three different STM tip positions, P1, P2, and P3. Each step in the current trace corresponds to a single rotation of the molecules on the surface. Simply moving the tip between the three positions was found to change the rate of rotation by two orders of magnitude (reproduced with permission from Lastapis et al. Science 308 (2005) 1000) Copyright (2005) by the American Association for the Advancement of Science

1 Excitation and Mechanisms of Single Molecule Reactions

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reaction coordinate (RC) can be traced to a direct excitation of the RC mode by coherent [15] or incoherent [16] vibrational ladder climbing and an indirect excitation through anharmonic coupling to a mode excited by tunneling electrons [17]. Localized chemical reactions are therefore closely related to vibrational excitation of molecular bonds, which form the basis of inelastic electron tunneling spectroscopy (IETS). Inelastic spectroscopy has been successfully used for chemical identification of single adsorbed molecules and reaction products [18] and isotope purification on the nanoscale [19]. Conversely, the so-called action spectroscopy of single-molecule reactions [10] can provide additional information to what is directly accessible by IETS, as we will discuss later in the chapter. Several extensive reviews can be found in the literature that have highlighted the historic aspects of single-molecule chemistry and many examples of the kinds of reaction that can be induced and measured in the tunneling microscope [11, 12, 20]. Our goal here is to focus on the experimental approach to the measurement of the reaction rates and excitation thresholds, highlight the recent theoretical efforts to understand the mechanism of single-molecule reactions, and discuss emerging experiments on the collective reactivity in molecular complexes self-assembled on a surface.

Measurement of Single-Molecule Reactions The STM is usually operated in a constant current mode, where the resistance of the tunneling junction is maintained constant as the tip scans across the surface. To carry out a single-molecule chemical reaction, the tip is placed roughly on top of the feature that corresponds to an adsorbed molecule and the feedback loop is opened so as to maintain a roughly constant tip-molecule separation. The reaction is induced by increasing the tip bias which also leads to the increase of the tunneling current. After a certain time delay, the current abruptly drops or increases to a different constant value (Fig. 1.2). Once the voltage pulse ends, the feedback is reengaged. The rescan of the same area after the excitation pulse may reveal that the molecule has diffused, rotated, changed conformation, split into several other species, or desorbed. The procedure is thus very simple allowing thorough statistics to be derived from hundreds and even thousands of individual reaction events. Identifying the reaction products is not a straightforward problem, but it can often be solved from a careful inspection of the respective STM images (their shape, symmetry, orientation, and position relative to the underlying surface lattice), directly by applying conventional chemistry rules (law of mass conservation and knowledge about possible reaction scenarios) or through a direct measurement of the vibrational modes in the reaction products [10, 18]. For example, the author has done several experiments with electron-induced reactions of dimethyldisulfide (CH3SSCH3) molecule adsorbed on Au(111) [21, 22], shown in Fig. 1.2a, b. Electron excitation can in principle dissociate either a C–S, C–H, or an S–S bond in the molecule. However, the reaction products are always two identical species (Fig. 1.2b). This would be the case only upon rupturing of the S–S bond that produces two equivalent CH3S fragments. Curiously, the reaction itself is very likely initiated through the vibrational excitation of the C–H bond (which was confirmed

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Fig. 1.2 STM images (4.8 × 5.2 nm2, U = –40 mV, I = 20 pA) of CH3SSCH3 molecules adsorbed on Au(111) surface before (a) and after (b) electron-induced dissociation of the S–S bond by a 1–s current pulse at 1.0 V. The reaction products are two identical CH3S fragments. CO molecules co-adsorbed with CH3SSCH3 molecules have been used as markers of gold surface lattice atoms, allowing to identify the adsorption sites of the reactant molecules and product species on the surface (when combined with DFT calculations). Reproduced with permission from [P. Maksymovych, D. C. Sorescu and J. T. Yates, Jr. J. Phys. Chem. B 110 (2006) 21161]. Copyright (2006) by the American Chemical Society. [21]. (c) Seven consecutive traces of tunneling current acquired during electron-induced dissociation of CH3SSCH3 molecules on Au(100) surface. The abrupt drop of current marks the instant of the dissociation event, while Dt is the excitation time (Maksymovych, unpublished)

for a similar reaction on Cu(111) [10]), while the subsequent dissociation of the S–S bond occurs through the anharmonic coupling between two modes. Singlemolecule reactions are subject to detailed kinetic measurements, which can collectively reveal excitation mechanism with quantitative assignment of vibrational and electronic excitations leading to the reaction. One can measure the electron kinetics, rate constant, and energy threshold of the reaction.

Excitation Rate To overcome a potential barrier along the RC, the molecule needs to possess sufficient potential energy placed in the correct vibrational or electronic (or both) state. Once excited, a single molecule reacts on timescales that are comparable with the vibrational relaxation time of an adsorbate, on the order of 1–100 ps [23, 24]. This is at least nine orders of magnitude faster than the time resolution of even the best STM setups, which are typically bandwidth limited by the parasitic capacitance of conventional transresistance current preamplifiers to <10 kHz. Therefore, STM registers the reaction as an abrupt change of current, Fig. 1.2c, leaving the rate of actual atomic rearrangements inaccessible.

1 Excitation and Mechanisms of Single Molecule Reactions

7

One can, however, access the initial excitation rate, provided it is sufficiently slow. The excitation rate is certainly dependent on the molecule and the detailed experimental conditions, particularly the tip-surface bias and tunneling current. Practically, accessible reactions have a probability of being excited by inelastic electron scattering between 10 –12 and 10 –8 events/electron. The reaction events would therefore occur on an average once every 60 s to 6 ms at a tunneling current of 1 nA, which is well within the measurement bandwidth. Notably, it is the very high current density in the tunneling junction that allows STM to drive electroninduced reactions with such a low probability. By comparison, photochemical processes on surfaces induced by photoexcited hot electrons typically have yields in excess of 10-6 events/electron [25]. With this in mind, the initial plateau of constant current during the voltage pulse above a single molecule, Fig. 1.2c, corresponds to the time that it takes to excite the molecule via inelastic electron tunneling into some state that will trigger the followup reaction. Single-molecule events are stochastic [26–28] with an equal probability of occurrence at any instance of time described by the Poisson distribution: f (k , λ) = (λ k / k !)e −λ , where l is the average number of events in time t (or reaction rate) and k is the observed number. In the measured time interval, the observed number of events is k = 0 (the molecule has not reacted) while the expected average equals R·t , where R is the average reaction rate. The Poisson distribution is thus reduced to f (0, R·t ) = e− R ·t or the probability of observing longer excitation time is exponentially decaying with the observation time. Indeed, the experimental results confirm this hypothesis, Fig. 1.3a, and one can recover a reliable exponentially decaying function of number of events as a function of the delay time from about ~100 reaction events [11], and thus directly measure R.

Fig. 1.3 (a) Poisson statistics of single-molecule reaction events [CH3SSCH3 dissociation on Au(100)] manifested as an exponentially decaying probability of observing a longer excitation time. The curves correspond to a different excitation current, and are based on ~100 reaction measurements each. (b) Electron kinetics of CH3SSCH3 dissociation on Au(111) and Au(100) surfaces, which proceeds through a two-electron excitation at a tunneling bias <1 V and a singleelectron excitation at a bias exceeding 1.2 V (reproduced with permission from Maksymovych et al. [32]

8

P. Maksymovych

To satisfy the constancy of R during single-molecule reactions, one ideally needs to reproduce exactly the initial quantum state of the tip-surface–molecule junction, with the same degree of wavefunction overlap, tip configuration, tunneling current, etc. This is, of course, practically impossible. However, if the STM image of a single molecule has a traceable and reproducible shape, one can invoke the atomtracking technique [29] to assure the reproducibility of the tip’s position above the molecule with sub-angstrom accuracy. The absolute value of reaction rate, R, determined from the statistical analysis of data is sensitively dependent on experimental parameters, with a particularly strong dependence on the tip state [30]. However, the functional dependence of the R on the tunneling current and tunneling voltage are both much more reliable and provide key insight into the mechanism of the single-molecule reaction. In the simplest picture, R ~ k * IN [15], where I is a tunneling current, k is a rate constant, and N is an integer power exponent (Fig. 1.3b). When N = 1, the reaction is triggered by one electron, two electrons, etc. N can be extracted from a total of ~500 reaction events as a function of tunneling current. Although single-electron processes are dominant, multiple-electron excitation has been observed in many systems [31]. A transition from single- to multiple-electron processes can often be achieved by decreasing the excitation bias as demonstrated for oxygen dissociation on Pt(111) [15], CH3SSCH3 dissociation on Au(111) [32], and dissociation of chlorobenzene on Si(111) [33] among others.

Energy Threshold for a Vibrationally Mediated Reactions and Action Spectroscopy Most tip-induced reactions have a definite bias (energy) threshold, below which the reactions are not observable on typical experimental timescales. The threshold can be roughly estimated by measuring the average excitation time, and more rigorously by determining the excitation rate R, as a function of tunneling bias. Seminal examples of such measurements can be found in the work of Stipe et al. [34], who have detected that the onset of acetylene rotation on Cu(100) corresponded to the energy of the C–H stretch, Fig. 1.4, and demonstrated strong isotopic effect, manifested as a redshift of the corresponding threshold in the monodeuterated acetylene, HCCD. When comparing the measured reaction thresholds, one can invoke the known gas-phase vibrational modes for the same molecule [10, 34–36] or compare them to theoretical predictions for the adsorbed species. The latter case, however, can be complicated by the intricate charge dynamics responsible for inelastic excitation and relaxation of the adsorbed species, particularly when strong electron–phonon coupling or excited states are involved. Density functional methods based on the Hohenberg–Kohn–Sham formalism derive the physical observables from the ground-state density of the electronic system and, as such provide a mean-field single-electron description of the ground state. Predictive calculations of inelastic scattering have been demonstrated in several specific cases of conjugated molecules in a two-lead junction [37, 38], acetylene on

1 Excitation and Mechanisms of Single Molecule Reactions

9

Fig. 1.4 Thresholds for tip-induced rotation of acetylene molecules on Cu(100) surface measured from the yield of the reaction as a function of bias at a tunneling current of ~40 nA. (b) The slopes of the data in (a). (c) STM–IETS spectra for C2H2 and C2D2 showing vibrational peaks at 358 and 266 mV, respectively (reproduced with permission from Stipe et al. [42])

Cu(100) [39], one-level model systems, and very recently at the ab initio level using the lowest order approximation to calculate local photon emission rate based on a non-equilibrium Green’s functions treatment of the transport problem [40]. Much progress has also been made in the description of excited electronic states, although such calculations are not mainstream at the moment [41]. The cases of strong electron–phonon coupling and electron–electron correlations present their own major challenges. In this regard, it is highly desirable to produce an experimental reference for the comparison to reaction thresholds. IETS is an example of such a reference, where one can directly measure inelastic losses associated with molecular vibrations. Stipe et al. have shown that the energy for acetylene rotation on Cu(100) coincided with distinct peaks in the inelastic electron tunneling spectra of the respective molecules, Fig. 1.4c, including the isotopic shift [42]. Similar conclusions were reached for many other single-molecule reactions [43–45]. At the same time, IETS is a rather complicated experimental technique. Inelastic scattering occurs because of the vibronic modification of the

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electronic levels rather than field–dipole interactions in optical spectroscopies and the selection (or propensity) rules are quite different. On one hand, the fraction of the inelastic current is very small, imposing stringent requirements on the measurement equipment. On the other, many vibrational modes are not observed in the IETS spectra. The inelastic fraction of the tunneling current is intrinsically connected with the nature and alignment of molecular electronic states relatively to the Fermi level of the surface and the energy of the vibrationally excited molecular state. Using the Keldysh Green’s function method, Mii et al. [46, 47] have shown that the inelastic contribution to the tunneling current consists of two parts: enhancement of conductance due to vibronic modification of the electronic level (thus excitation of the vibration) and suppression of conductance due to elastic backscattering of tunneling electrons, where a virtual phonon is emitted and reabsorbed by the tunneling electron. Depending on the position of the adsorbateinduced resonance relative to the Fermi level and its intrinsic width, the competition between the two components may produce a dip (elastic larger), a peak (inelastic larger), or no detectable signature (compensation) in d 2 I / dV 2 spectrum [48–51]. Single-resonance models predict that the crossover between conductance enhancement and suppression occurs at the transmission probability of 0.5. This mechanism explains why the increase of the tunneling conductance of the C≡C stretch mode in acetylene on Cu(100) is very small compared to that of the C–H stretch mode [52], and why no other modes (C–H bend and C-metal stretch) are observed, where the elastic and inelastic contributions tend to cancel. An alternative approach to correlate reaction thresholds and vibrational modes of the molecule is a method of action spectroscopy, Fig. 1.5, defined as the number of molecular motions per injected electron as a function of tunneling bias. Action spectroscopy is particularly useful when the stability of the molecule on the surface is insufficient to acquire an IETS spectrum, for example, as is the case of CO hopping on Au(111) [53] and Ag(110) [54], biphenyl on Si(100) [55], CH3S on Cu(111) [56], CH3SH and benzene on Au(111) [21], and many others. Although such measurements were carried out since the earliest single-molecule studies [34], the true power of action spectroscopy was first demonstrated by Sainoo et al. [10]. The authors acquired action spectra for a relatively complex molecule, cis-2-butene on Pd(110) (Fig. 1.5a). Upon low-energy excitation the molecule exhibited two types of conformational isomerization relative to the crystallographic directions of the surface, which could be distinguished by the relative change of the tunneling current by the reaction (Fig. 1.5b). The action spectrum of the molecule revealed that the efficiency of the first reaction type significantly increased at ~37 and ~115 mV, while the other motion was enhanced at ~115 and ~360 mV. The IETS spectrum (which could be obtained in this particular case) revealed a significant feature only at 358 mV (Fig. 1.5c). From a subsequent comparison between action spectra, IETS and HREELS of the cis-2-butene and its fully deuterated analogue, it was determined that the 37-mV excitation corresponded to the metal–carbon stretch, 115 mV to the C–H bending mode and 360 mV to the C–H stretching mode. The stretching of the double bond C=C was not found in either IETS or action spectra. It was suggested that the p-orbital of the double bond couples

1 Excitation and Mechanisms of Single Molecule Reactions

11

Fig. 1.5 (a) Action spectra for the two types of motion (LB low-barrier; HB high barrier) of butadiene (C4H8) on Pd(110) and its deuterated analogue C4D8. Data were taken under fixed tunneling current of 3 nA for C4H8 and of 2 nA for C4D8. (b) Schematics of the molecular transformations corresponding to the LB (CUR ↔ CDL and CUL ↔ CDR) and HB tip-induced motions. (c, d) STM–IETS spectra taken over (c) C4H8 and (d) C4D8. Solid lines and dashed lines represent vibrational spectra of the positive and negative bias range, respectively. Note that the action spectra reveals the 115-mV mode corresponding to CH-bending and C–C stretching, which is completely missing from IETS (reproduced with permission from Sainoo et al. [10])

strongly to the metal surface, substantially increasing the rate of vibrational relaxation and suppressing the coupling of this bond to molecular motions. A theoretical analysis of the action spectroscopy for single-molecule motion was subsequently carried out by Ueba and Persson [57] using vibrational spectroscopy of acetylene on Cu(100), NH3 on Cu(100) [58], and Co hopping on Cu(111) [59] as model examples. In particular, these authors have analytically calculated the expression for a inelastically induced reaction rate for an adsorbate along the RC mode with energy ω as R(V ) = I in (V ) / τ v , where Iin is the inelastic fraction of the tunneling current that promotes the reaction mode from its lowest to the first vibrationally excited state and 1/tv is the rate of the anharmonic coupling to the RC. Expressing the inelastic tunneling current Iin(V ) in terms of tip-molecule and molecule-surface coupling constants (∆s and ∆ t for definition in terms of the tunneling Hamiltonian see [16], ∆ st = ∆ s + ∆ t ), density of states of the adsorbate-induced resonance (ρa (ε)) and the energy dissipation upon inelastic scattering, the approximate expression for the second derivative of the inelastic conductance (and by extension the experimentally observable d2R(V)/dV2) is d 2 I in (V ) 2 πe3 ∆ s ∆ 2t 2 ≈ χ ρa (µ s )ρa (µ s − eV )ρph (eV ).  ∆ 3st dV 2 Assuming a sufficiently broad adsorbate-induced resonance and low temperatures (thus a ground vibrational state of the excited molecule), the only quickly varying term in the above expression will correspond to the phonon density of states ρph (eV ).

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The central point of the analysis was that the action spectrum is quantitatively a better measure of the vibrational behavior of a single molecule than IETS. As mentioned before, the width and amplitude of the IETS peaks, Fig. 1.4c, originate from the competition between the elastic and inelastic contribution to the vibrationally induced modification of the tunneling current. The peaks in action spectra are proportional only to the inelastic contribution, because only inelastic losses can excite the respective vibrational modes and a chemical reaction upon subsequent anharmonic coupling to the RC. Thus, action spectra obey a different kind of “selection rules” and potentially can yield vibrational relaxation time from the width of the peaks in d2R/dV2.

Reactions Involving Excited Electronic and Anionic States Energy thresholds of many single-molecule reactions were found to be much larger than the single-vibrational quanta of the ground electronic state [9, 32, 33, 60, 61]. This may be evidence that molecular excitation proceeds through a two (or more) electron process in the ground electronic state, i.e., vibrational ladder climbing that finally leads to a successful anharmonic coupling to a RC. Alternatively, thresholds in excess of 1 eV can be a hallmark of a reaction that is initiated by the excited electronic or anionic state of the adsorbed molecule [20]. Kinetic measurements provide a very good way to distinguish between these possibilities, because the reaction rate for a multielectron process has a power-law dependence on the tunneling current. Electronic excitation of single molecules or surface atoms leading to chemical reactions has been observed in a vast number of systems, among them isomerization of azobenzene derivatives on Au(111) [62], hydrogen and deuterium desorption from Si(111), Si(100) (thresholds at 2–3 and 6 V) [63–65] and Ge(111) surfaces (>4 V) [66], CO desorption from Cu(111) (threshold of 2.4 V) [67], dissociation of iodobenzene at 1.5 V [68], dissociation of dimethyldisulfide (>1.3 V) [32] and methanethiol (>2.5 V) on Au(111) [69] and aromatic thiols on Cu(111) [70], localized reactions of aromatic halides on Si(111) [71], reversible displacement of a Sn ion through an adsorbed phthalocyanine molecule [72], isomerization of azobenzene on Au(111) [73], and others. The processes underlying these higher energy reactions, schematically shown in Fig. 1.6, can be traced to (1) electron or hole attachment to the molecular orbital above (below) Fermi level, (2) electron–hole pair attachment to a molecule decoupled from the metal substrate, and (3) formation of an excited molecular state, where an electron is excited from HOMO to LUMO state by inelastic scattering of tunneling electrons. These mechanisms can be differentiated to a certain degree by the polarity dependence of the reaction efficiency. Electron or hole attachment is a strongly unipolar process, Fig. 1.6a, b, determined by the position of the respective orbital relative to the Fermi level, and it will therefore exhibit strong bias dependence of the reaction rate. For example, in the author’s studies, the onset of dimethyldisulfide dissociation on Au(111) had a positive bias threshold at ~1.3 V and no observable reaction up to –2.0 V.

1 Excitation and Mechanisms of Single Molecule Reactions

13

Fig. 1.6 Mechanisms of inelastic excitation in the tunneling junction beyond vibrational heating: (a) electron attachment; (b) hole attachment; (c, d) electronic transitions; (e, f ) electron–hole pair attachment. The latter two mechanisms may occur if the participant electronic states are decoupled from a substrate by a thin insulating layer (oxide, NaCl) (reproduced with permission from Mayne et al. [20])

This roughly agrees with the simple estimate for the position of the LUMO as the difference of the metal’s work function [j, 5.1 eV for Au(111)], vertical affinity of the adsorbate (EA, 1.7 eV for dimethyldisulfide [74]), and image charge interaction energy, ϕ − EA − e2 / 4 Z . The image charge stabilization of 1.5–2 eV is consistent with 4–2 Å distance of the molecular anion to the surface. Similar logic was used to identify the LUMO of azobenzene as the orbital involved in its electronically induced isomerization by ~1.7 eV electrons above Fermi level [9]. Desorption of hydrogen from Si(100) is also asymmetric in bias, with the onset of the hole attachment to 5s resonance at <–7 V [60] and the onset of electron attachment to 6s* resonance at >4 V [31]. Electronic excitation via p → p* transition, Fig. 1.6c, d, was suggested to occur for an isomerization of polyaromatic 1,4″-paratriphenyldimethylacetone at a bias of ~4.0 V [75]. In general, one expects electronic excitation to have a weak

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P. Maksymovych

bias dependence by analogy with inelastic scattering. The process will, however, be complicated by the changes in the overall tunneling probability due to proximity of molecular resonances at high bias. A relatively high threshold coincident with weak bias dependence may also signal another electric-field induced excitation mechanism, which does not require inelastic electron scattering with a molecule. Only a few reactions of this type were suggested so far. They include reversible isomerization of Zn(II)-etioporphyrin [76], NO desorption from Si(111) [77], and isomerization of azobenzene on Au(111) [13]. In all these cases, the transformation involves a characteristic change in the shape of the molecule that changes its dipole moment, enabling efficient coupling to the electric field in the tip-surface junction. For example, from

Fig. 1.7 (a, b) STM images (3.3 × 3.3 nm2) along with the schematic structural models of ZnEtioI molecule on NiAl(110) surface. (c) Variations of threshold voltages for a ZnEtioI molecule to undergo the reversible conformational transitions from type I to type II and from type II to type I as a function of the change in tip–substrate separation (reproduced with permission from Qiu et al. [76])

1 Excitation and Mechanisms of Single Molecule Reactions

15

direct inspection of the STM images before and after the reaction, Qiu et al. [76] established that the porphyrin ring undergoes symmetry reduction from C4h to C2d, manifested as the symmetric “indentation” of the flat shape of the central ring (as in the gas phase) shape due to additional interaction with the NiAl surface (Fig. 1.7a, b). The involvement of the electric field was established for the reverse process (flattening of the molecular ring) from a roughly linear dependence of the excitation voltage on tip-sample distance, justified if the simplest V/d expression for the electric field in the tunnel junction (d is the tip-surface distance) is valid (Fig. 1.7c). The voltage threshold for the direct process (indentation) was found to be independent of the tip-sample separation and was therefore assigned to inelastic electron-induced excitation of the molecule.

Rate Constant of a Single-Molecule Reaction In a recent study, Rao et al. [78] have demonstrated that it is also possible to extract valuable information from the value of the rate constant k in R ~ k ·I N . The authors compared the rate of single-molecule dehydrogenation of a series of halogenosubstituted thiophenols on Cu(111). The goal was to build a parallel to a wellknown Hammet equation [79], log(kx / k0 ) = ρ·σ , which predicts the effect of substituents (X) in the phenyl ring (Ph) on the rate of a particular reaction of a PhR → PhR¢, where R is a reactive center. k0 and s refer to the dissociation of benzoic acid in water (Ph–COOH), making r a unique descriptor of the relative reactivity of a variety of substituents and reactive groups. The Hammet equation and its several derivatives, Swain–Lupton, Taft, and Yukawa–Tsuno equations [80], have predictive power not only for the chemical identity of the substituent, but also its position in the phenyl ring, in particular meta- and para-substitutions (orthosubstituents introduce additional steric effects). Using the single-molecule averaging statistics described above, Rao et al. [78] have measured the linear rate of dehydrogenation of thiophenol and its para- chlorine, bromine, and fluorine derivatives – identifying single-electron kinetics for RSH → RS + H reaction in each case (Fig. 1.8). The rate constants, extracted from the corresponding slopes of R = f(I), revealed a systematic dependence on the substituent, which was used to calibrate r in the Hammet equation against the literature values of s for the corresponding molecules. The extracted value of r (~1.4) was then combined with literature values for meta-halo substituted molecules as well as the meta-methylphenyl. The actual measured values for the meta-substituted thiolphenols were found to be in very good agreement with the predictions of the Hammet equation for the surface reactions. The positive values of r furthermore indicate that the electron-withdrawing substituents in the phenyl ring stabilize the transition state, which thus indicates a strong likelihood of an anionic transition state [80]. Overall, systematic analysis of similar reacting molecules enables using single-molecule rate constants to glimpse onto the quantitative aspects of the transition state to which the molecule is excited by tunneling electrons.

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Fig. 1.8 Linear free-energy relationship of one molecule at a time. STM images and singlemolecule dissociation rates of substituted thiophenols (TP). (a–c) STM images of para-methyl-TP, meta-F-TP and meta-Cl-TP on Cu(111) at 16 K. The images have a petal shape because of molecular rotation around the anchor bond. (d) The linear dependence of the reaction rate on the tunneling current indicates a one-electron process. The k-values were used to buildup a surface analogue of Hammet equation (reproduced with permission from Rao et al. [78])

Theoretical Aspects of Single- and Multiple-Electron Processes To overcome an activation barrier, the energy of the inelastically scattered electrons has to be routed to a vibrational mode corresponding to the RC. This process can happen directly, when the electrons excite the RC mode by coherent [15] or incoherent vibrational ladder climbing [16], or indirectly when the inelastic tunneling current excites some vibrational mode in the molecule and the energy is subsequently transferred into the RC mode via anharmonic coupling [17], as schematically shown in Fig. 1.9. The latter process has been observed in those experiments, where the excitation threshold is observed to correspond to a vibrational mode that does not actually correspond to the breaking bond, exemplified by S–S bond dissociation of dimethyldisulfide on Cu(111), excited by a C–H stretch [56], and translation of ammonia (NH3) on Cu(100), excited by the N–H stretch [58]. The latter case is particularly interesting, because molecular desorption from the surface (which intuitively has a large activation barrier) can be induced at a smaller tunneling bias than molecular diffusion (Fig. 1.10). The reason behind this mode selectivity is that desorption can be induced by a direct excitation of the umbrella N–H bending mode with an energy of ~270 mV via a multiple-electron-scattering incoherent vibrational ladder climbing, while translation anharmonically couples to an N–H stretch excited at >400 mV.

1 Excitation and Mechanisms of Single Molecule Reactions

17

Fig. 1.9 Schematics of a simple two-dimensional potential along the intramolecular stretch QA–B that is anharmonically coupled to the RC q. There are two ways to overcome EB: coherent or incoherent vibrational ladder climbing (green arrow) involving direct excitation of the RC mode by tunneling electrons; anharmonic coupling between the high-frequency A–B stretch mode excited by tunneling electrons (red arrow) and the low-frequency RC mode (reproduced with permission from Ueba et al. [88])

Very recently, Tikhodeev and Ueba [81] have analyzed an additional mechanism by which the vibrationally excited molecule can crossover the activation barrier for the reaction. Previous treatments considered only classical trajectories along the RC, whereby the molecule could overcome the reaction barrier only when the potential energy exceeded the barrier height. In light of recent experiments on the mobility of hydrogen atoms on Cu(100) [6], Co atoms on Cu(111) [59], and hydrogen exchange in water dimers on Cu(110) [82], quantum mechanical motion of the reacting species should also be considered if the effective mass of the particle is sufficiently light. Rotational tunneling (from the ground or excited rotational state) may also apply in a growing number of molecular rotors on the surface [83–85]. Tikhodeev and Ueba [81] modified the Pauli master equations (see below) to include the additional decay of the vibrational population via quantum mechanical tunneling across the potential barrier. The net result of their analysis is that the dependence of the reaction rate on the excitation bias may deviate from a power law (R ~ Vn) predicted by the models that consider only the classical trajectories. The power law itself originates from an Arrhenius type expression for the reaction rate, R ≈ Γ↑ exp( − EB / kTv ), Γ↑ rate of excitation into a given vibrational state, which is valid when “thermal” fluctuations in the presence of tunneling current (kTv) are much smaller than the vibrational quantum of the respective mode. Tv describes quasistationary population of the vibrational levels Γ↑ ≈ Γ↓ exp(− Ω / kTv ). The mechanism of vibrationally assisted tunneling was applied to hydrogen exchange in the water dimers [86], which was shown to be compatible with quantum tunneling from the lowest vibrational mode and first-excited vibrational mode (without the need for multiply excited states), while Co tunneling on Cu(111) [59] was reinterpreted as vibrationally assisted tunneling from many excited levels, which turns into classical behavior at large bias.

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Fig. 1.10 (a) STM image of three NH3 molecules before and (b) after injecting 0.5 nA of 420 meV electrons on top of each molecule. Two of the molecules moved to a new site, and the third was desorbed. Statistics of electron energy onset leading to molecular motion: (b) The distribution of threshold energies for the reactions at low tunneling currents (It < 0.5 nA) NH3 shows a threshold at 400 mV (a), which shifts down to 300 mV for ND3 (b), matching correspondingly the energy of N–H and N–D stretch modes (dashed lines). The insets show traces of tunneling current during tip-induced excitation. These events induce molecular translation with ratio 0.6. (c) For It > 1 nA, an additional threshold appears gradually at 270 mV in NH3, consistent with the energy of two umbrella mode quanta (dash-dotted lines). Now, desorption dominates with ratio 0.75. In ND3, the corresponding onset at 200 mV is observed (d) only after reaching tunneling currents higher than 10 nA (reproduced with permission from Pascual [12])

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Overall, the likelihood of electron-vibration coupling is determined by the properties of an adsorbate-induced resonance, its width (determined by coupling to the substrate and the tip), position relative to the Fermi level, the rate of vibrational deexcitation via electron–hole pair generation, and vibrational cooling down via the phonon bath of the surface. The involvement of the electronic resonance in the vibrational excitation is implicit in the theoretical models, where the tunneling process is mediated by the adsorbate-induced resonance [16, 40, 46, 50, 86]. Electron-vibration coupling is modeled by assuming that the energy of the resonance shifts linearly with the vibrational coordinate, typically within the first-order perturbative treatment, where the Hamiltonian operator is expanded in a Taylor series on the normal vectors and retaining only the first-order term [16]. The involvement of the electronic states in the electron-vibration coupling was recently directly confirmed by the Kawai’s group, in the analysis of CH3SSCH3 dissociation on Cu(111) using action spectroscopy [87]. The reaction was observed to have a threshold at the energy of the C–H stretch (~360 mV). However, the reaction rate was strongly bias dependent, being much faster at the positive sample bias. Furthermore, hopping of the CH3S fragments had symmetric thresholds at ±85 mV, but did not have a threshold at the energy of the C–H stretch. With the help of firstprinciples modeling, the interpretation of the bias asymmetry was that the dissociation reaction was assisted by the lowest unoccupied molecular orbital, which peaks at ~1.0 V above Fermi level (and is also very broad), and which stretches over the S–S and C–H bonds allowing the C–H stretch to be vibrationally excited. At the same time, the translation motion of the CH3S fragment was mediated by its LUMO and HOMO orbitals, which are spread relatively evenly around the Fermi level and coupled directly to the C–S bond. Neither of these orbitals had significant spatial concentration around the C–H bond, which served as a further indication that the excitation of the vibrational modes (and the corresponding reactions) did indeed depend on the availability of the adsorbate-induced resonance in the relevant tunneling window as well as its symmetry. Here we want to elaborate on one particular theoretical work that provides a kinetic framework for single- and multiple-electron excitation, and establishes the link between consecutive excitation processes and the power law of the reaction rate in current observed experimentally. Ueba et al. [88] considered the vibrational excitation scenario through the anharmonic coupling between a high-frequency mode (HF, levels labeled by n), directly excited by tunneling electrons, and reaction coordinate (RC, levels labeled by n), which may or may not be directly excitable, as schematically depicted in Fig. 1.11. For a one-electron process, the reaction can be symbolically represented as a sequence of two processes:

[υ : 0 → 1](υ :1 → 0 | RC : 0 → n) i.e., vibrational excitation of the ground into the first excited state, followed by simultaneous deexcitation and excitation of the RC from 0th to nth excited state.

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This representation clearly separates the excitation stage (accessible by STM) and the reaction rate. The reaction rate is by definition:

RRC =

∑τ

n > nth

1

N1,0

υ (1→ 0)RC(0 → n )

where τυ(1→0)RC(0→n ) is the rate of the anharmonic coupling, while N1,0 is the population of the first excited vibrational state. Note that the summation takes place over only those levels of the RC mode that are sufficient to overcome the activation barrier. N1,0 is subject to the following Pauli master equation reflecting the balance between inelastic scattering of tunneling current, vibrational relaxation, and reaction rate:

dN1,0 dt

= pν N 0,0 −

N1,0 τv

−∑ n

1 τv(1→0)RC(0→n )

N1,0

Here pν = ηv I is the inelastic fraction of the tunneling current (or the normalized probability of vibrationally exciting the HF mode), tv is the vibrational relaxation time of the HF mode and the third term is the reaction rate. The equation can be solved assuming a steady state condition of N1,0 = constant. The final expression for the reaction rate is RRC = τv,tot ηv I / τv,RC , where the total relaxation time tv,tot and the reaction time tv,RC are correspondingly:

1 τ v,tot

=

1 1 +∑ τv n τ v(1→ 0)RC(0 → n )

and

1 τv,RC

=

∑τ

n > nth

1 v(1→ 0)RC(0 → n )

The reaction rate of such a excitation/coupling mechanism is indeed directly proportional to the tunneling current, reflecting the one-electron excitation [u:0 → 1]. The analysis also reveals the terms that determine the total reaction rate: probability of inelastic scattering of the tunneling electron, total vibrational relaxation time of the HF mode, and the rate of the anharmonic coupling to the RC. Each one of these terms has been treated in several theoretical works over the years. For example, an earlier study by Persson and Ueba [89] has derived the analytical expression for the rate of anharmonic coupling of an HF mode to an RC frustrated translation/vibration mode, in competition with fast cooling of the HF mode by electron–hole pair excita2 3/ 2 −2 α tion in the metal: 1 / τ v,RC ≈ 1 / τ v (δω / EB ) α e , where dw is the anharmonic coupling, EB is activation barrier, hw is the energy of the RC mode, and alpha is α = EB / ω . Note that an increase of a will exponentially damp the reaction rate. To interpret this behavior, one needs to consider that upon reactive excitation the RC mode is promoted from its low-energy smooth ground-state wavefunction to a highly oscillatory wavefunction of the excited mode (which may even approach continuum), Fig. 1.11. The corresponding matrix element will be small, which ultimately affects the efficiency of anharmonic coupling and the total reaction rate [17]. The kinetic analysis applies also to multielectron excitation scenarios, although the final expression for the reaction rate becomes progressively more complex and

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Fig. 1.11 Schematic illustration of a single-electron process. The decay of the high-frequency mode with the energy excites the low-frequency RC mode from the ground state localized at the bottom of the RC potential well to the unbound state just above the top of the barrier. Electron–hole pair excitation is a competing channel for deexcitation of the low-frequency mode (reproduced with permission from Ueba et al. [88])

parameter rich as the number of excitation channels and coupling modes increases. For a two-electron process, Ueba et al. considered the following three parallel scenarios, depending on how the RC mode is excited. For pure anharmonic excitation one arrives at:

[ υ : 0 → 1]( υ :1 → 0 | RC : 0 → m)[ υ : 0 → 1]( υ :1 → 0 | RC : m → n) + [ υ : 0 → 1][ υ :1 → 2]( υ : 2 → 1| RC : 0 → m)( υ :1 → 0 | RC : m → n) + [ υ : 0 → 1][ υ :1 → 2]( υ : 2 → 0 | RC : 0 → n) which is a sum of three (exclusive) processes: vibrational pumping of the RC mode via consecutive excitation–deexcitation of the HF mode, initial vibrational pumping of the HF mode to its second vibrationally excited state followed by two- or onestep anharmonic coupling to the RC mode. The resulting rate equation is RRC = τ (ηv τ v I ) 2 , where τ characterizes the total effect of the rates of the anharmonic couplings along the RC, while 1 / τ v is the rate of vibrational relaxation of the HF mode. In case the RC mode can also be excited directly, two other scenarios are possible

[ υ : 0 → 1]( υ :1 → 0 | RC : 0 → m)[RC : m → n] + [RC : 0 → m][ υ : 0 → 1]( υ :1 → 0 | RC : m → n)  v η RC τv τRC I 2 and for only direct exciwith the reaction rate equation RRC = τη tation [RC:0→m][RC:m→n] with the corresponding reaction rate of 0→ →mm mm→ →nn 22 = ((ηη0RC ηRC RRRC RC = RC η RC II )) // ττRC RC. Because each one of the scenarios involves a product of two consecutive excitations, the rate expressions of all three are in the form R µ I2, justifying the experimental determination of the corresponding kinetics. For the case of NH3 desorption from Cu(100) surface [58], Fig. 1.10, the low-energy

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two-electron process at V < 300 mV is induced via the third process of directly pumping the reaction coordinated mode (umbrella breathing), while the highenergy two-electron desorption at 410 mV is the second process, which starts with the single-quanta population of the relatively long-lived RC mode at ~270 mV and followed by the anharmonic coupling between N–H stretch and umbrella mode in the second inelastic excitation.

Delocalized Excitation in the Molecular Junction In all the discussion so far, the position of the STM tip was implied to be exactly above the molecule during the chemical reaction, and the excitation to be localized only to that particular molecule. There is an exception to this rule, in a process we will refer to as delocalized excitation. Here, the tunneling position does not have to be above or even near the molecule, while the excitation (seen as chemical reactions or atomic diffusion) can occur as far as 100 nm away from the tunneling junction! Such a behavior has so far been observed in only a handful of experiments [90], but the phenomenon is likely to be general for a range of metal and semiconductor surfaces as well as excited molecules. The first account of delocalized excitation by Nakamura et al. [92, 93] reported the diffusion of chlorine atoms on a chlorinated Si(111) surface as far as 30 nm from the tunneling junction. Statistical analysis of the radial distribution of the diffusion probability revealed a standing wave-like pattern, and it was also angularly anisotropic. Based on this analysis, it was suggested that at a tunneling bias of ~4 V, hot electrons were injected into the s-derived surface band and ballistically transported toward the adsorbents where the inelastic scattering events occurred. Similar delocalized excitation was inferred to occur in the electron-induced modification of a close-packed C60 multilayer on Si(111) surface [93]. Upon injection of electrons at a bias of >3.3V, C60 films were observed to develop a beautiful ring structure (with diameters up to 20 nm) around the injection spot, Fig. 1.12, the size of which is dependent on the tunneling bias and duration of the excitation pulse. By analogy with chlorine diffusion, the formation of the ring structure was attributed to a reversible dimerization (and possibly partial polymerization) of the C60 molecules by hot-electrons, transported away from the tunneling junction via the surface band. The occurrence of the ring was explained as the initial dimerization of the molecules around the tip followed by a partial reverse monomerization in the immediate proximity of the tip. These reactions were previously established by the authors in local excitation experiments [94]. The barrier to polymerization and the reverse processes is different and so are the rates of the corresponding reactions. Because the surface current density of hot electrons scales as 1/r in purely two-dimensional transport (r, distance to the tip), after any excitation pulse, there will be slightly more polymerized than depolymerized molecules and they will be concentrated on the periphery of the excited region, where the net balance of polymerization–depolymerization is nonzero at the energy and current of hot electrons.

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Fig. 1.12 Asymmetric rings of C60 polymers formed by carrier injection into close-packed layers with grain boundaries (line across the image). STM images acquired after (a) electron injection and (b) hole injection at 3 V into the marked points (reproduced with permission from Nouchi et al. [97])

Silicon surfaces possess well-expressed surface bands, allowing significant fraction of electrons to flow along the surface [95], which explains why such reactions are feasible. However, similar excitation phenomena can also be observed on almost all low-index metal surfaces of Au, Ag, and Cu, although the lifetime of hot carriers at >1 eV above EF is almost two orders of magnitude smaller (10–50 fs) [96, 97] than on silicon surfaces. Here, the surface state in the projected band gap of the surface and its derivative surface resonance play the role of the twodimensional transport medium [98, 99]. Delocalized electron excitation on metal surfaces was first suggested by Stipe et al. for oxygen dissociation on Pt(111) in a ~10-nm radius [100]. The author has observed delocalized excitation with the aforementioned model system of the electron-induced dissociation of CH3SSCH3, Fig. 1.13a–c, as well as CH3SH and C6H5I on Au(111), Au(100), and Cu(110),_ all with distinct thresholds. Notably, the projected _ band gap is centered at the X -point in the Brillouin zone for 100 [101] and at Y for 110-terminations of the fcc metals [99], which strongly reduces the fraction of tunneling electrons injected in these states. Nevertheless, delocalized excitation is very pronounced on these surfaces, because of the efficient electron attachment to a wide range of molecules, which have been traditionally studied in the framework of surface photochemistry. A particularly pronounced reaction yield was observed by Maksymovych et al. [102] with CH3SSCH3 molecules on Au(111) efficiently dissociating in a rather striking radius of up to 100 nm (Fig. 1.13 shows dissociation at a distance of ~50 nm) when the tunneling bias exceeds 1.4 eV (Fig. 1.13e). The effect was very reproducible between adjacent surface areas (Fig. 1.13d)

Fig. 1.13 STM images and structural models of electron-induced dissociation of CH3SSCH3 molecule (a) on the Au(111) surface producing CH3S fragments (b). Delocalized CH3SSCH3 dissociation induced by a single 2.5 V = 1.0 nA = 200-ms pulse at the (blue) point. (c) The inset is a surface area ( yellow square) located ~46 nm away from the pulse position. u marks unreacted and r marks reacted CH3SSCH3 molecules. (d) Total number of dissociation events per pulse obtained from 18 pulses of the same magnitude. (e) Total number of dissociation events per pulse (1.0 nA = 200 ms) as a function of pulse voltage (reproduced with permission from Maksymovych et al. [102])

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Fig. 1.14 (a) Attenuation function of the delocalized CH3SSCH3 dissociation on Au(111) from two experiments using pulses at 1.8 V (average of 15 measurements) and 2.2 V (average of five measurements). Solid (red ) lines are exponential decay fits to the data (reproduced with permission from Maksymovych et al. [102]). (b) The radial probability g(r) of the delocalized electroninduced chlorine hopping on Si(111) at radial distance r from the tip position along the < 1 12 > direction. The oscillations are due to standing waves of the surface state on Si(111) at the excitation energy (reproduced with permission from Nakamura et al. [92])

and its hot-electron origin was established in a series of experiments, involving bias dependence of the effect, current injection through artificially created local nanoclusters, and a quantitative analysis of the spatial kinetics of the reaction events. In particular, the total reaction rate was derived to follow a kinetic equation:

−∑ ln(1 − Pr / N 0r ) = ktI 0g ∑ [ f (r )]g , r

r

where Pr and N0r is the reacted and initial numbers of molecules at a distance r from the tunneling junction, k is the rate constant, I0 is the tunneling current, g is the electron kinetic exponent, and f(r) is the intrinsic attenuation function. g was determined to be unity from experiments conducted at total constant tunneling charge (rather than current), further confirming the electron-induced kinetics. f(r) was closely fit by a simple monoexponential decay (Fig. 1.14a). Hot electrons emitted from the tunneling junction persist in the surface state (or resonance) of Au(111) surface with a lifetime less than 50 fs at energies exceeding 1.0 eV. The electron population will therefore decay exponentially in time and (correspondingly) distance. The decay exponent, Fig. 1.14a, extracted directly from properly normalized reaction probabilities is, in the first approximation, formally equal to twice the mean free path of the hot electrons at that energy [102]. By estimating the group velocity from the known band structure of Au(111), the relaxation times were estimated to be indeed on the order of 20–30 fs, in good qualitative

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agreement of spectroscopic measurements [102]. Similar statistics derived for chlorine jumps on Si(111), Fig. 1.14b, has revealed an oscillatory probability of a delocalized reaction which was assigned to the standing waves of the surface state on Si(111) at the excitation energy. Such measurements can therefore reveal the dynamics of hot electrons in a small region of the adsorbate-covered surface (in direct relevance to photochemistry), which is practically impossible to do using surface-resonance scattering techniques [97] at very high coverage due to the complexity of the scattering patterns and reactivity of the adsorbates. The advantage over averaging spectroscopic techniques is that the information can be extracted for a relatively localized region of the surface, which may contain a single-atom step, engineered defects, adlayers, etc. The application of the delocalized excitation for the characterization of chemical reactions allows one to remove tip artifacts (forces and fields) and has the potential to measure fast excitation kinetics for which single-molecule measurements would be limited by the measurement bandwidth of the STM.

Tip Effects and Field-Induced Manipulation Given the striking complexity and intricacy of the processes involved in the singlemolecule reactions, it is not at all surprising that they can be strongly influenced by the shape, electronic states and electric field of the STM tip, and other parameters that cannot be easily controlled. The tip influence can be manifested in the excitation rate (change of vibrational relaxation and intermode coupling), excitation mechanism (kinetics of vibrational ladder climbing) and, ultimately, the potential energy landscape of the transition state. The tip effects were directly demonstrated by Sloan and Palmer [30] using dissociation of chlorobenzene on Si(111) as a model system and two different tip states to excite the system. While, the measurement of reaction kinetics was not affected by the tip state, the absolute measured rates differed by more than an order of magnitude. Furthermore, the ratio of molecular desorption to dissociation also revealed about an order of magnitude difference. An arguably good experimental test system to reveal the tip effects on the transition state of the molecule is the so-called localized or patterned reaction, the term first introduced by the Polanyi group [71] to describe a reaction which imprints the spatial configuration of the parent molecule onto the surface arrangement of the reaction products. Such chemical reactions, analogous to those in the field of surface-aligned photochemistry [104, 105], proceed through a tightly bound transition state involving surface atoms. Patterned reactions have been observed for a number of alkyl- and arylhalides adsorbed on silicon surfaces [71], while the author has found a similar behavior in the dissociation of dimethyldisulfide (CH3SSCH3) on Au(111) and Au(100) surfaces.

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CH3SSCH3 is adsorbed on the gold surfaces in a characteristic trans-geometry [22], Fig. 1.13a, b, where the methyl groups are located on either side of the S–S bond, while the S–S bond itself is almost parallel to one of the three close-packed directions. The molecule has therefore six equivalent orientations, and each of these can be “imprinted” on the relative orientation of the reaction products (CH3S fragments). Substantial tip influence could then be inferred by observing the probability of “imprinting.” Upon single-molecule excitation, the “imprinted” reaction yield, Fig. 1.13b, was observed to be only ~10%, making it statistically equivalent to any other reaction scenario where the CH3S fragments rotate about the S–S bond during the reaction and diffuse on the surface around the location of the original molecule [22]. By decreasing the magnitude of the tunneling current, the “imprinted” scenario becomes dominant (~50%) and reaches a maximum fraction of 75% of the reaction events when the delocalized excitation is involved. The changes in the relative orientation of the CH3S fragments may originate from either the direct tip effect on the transition state of the dissociating molecule or a postfactum electron-induced rotation or diffusion of the CH3S fragments, provided the latter is efficient under the excitation conditions. Regardless of the details, the fact that a relatively high yield of the imprinted scenario can be recovered by tuning the excitation condition suggests that this scenario is representative of the transition state. Its symmetry is actually the same as the symmetry of the lowest unoccupied molecular orbital involved in the dissociation of the parent molecule via dissociative electron attachment mechanism [32]. It is tempting to suggest that a statistical analysis of the preferential scenarios in the remaining 25% of the reaction events where the transimprint is not observed will effectively produce a probabilistic description of the potential energy surface, experienced by hot CH3S fragments upon dissociation. To achieve this goal further theoretical analysis of this reaction is required, ideally in terms of wave packet propagation analysis that is similar to the decay of a charged Cs atom on Cu(111) surface [106]. In another study involving formation of a transient anion, Henningsen et al. [9] suggested that the electric field of the STM tip not only influences the transition state, but may also be necessary to direct the transient molecular anion of 3,3¢-dicyanoazobenzene on Au(111) toward the rotation of the phenyl ring about the N–N bond, which ultimately leads to trans–trans (3,3¢ to 3,5¢) isomerization of the molecule. While the tip effects may be undesirable to derive information about the excitation mechanism or the transition state of the dissociating molecule, they can be useful in molecular manipulation. Ohara et al. [107] have demonstrated that the direction of CH3S translation on the Au(111) surface, excited by inelastic electron scattering at 85 meV, can be controlled by positioning the tip slightly off-center on the molecule during excitation. The direction of motion could also be switched between away from and toward the tip by reversing the sign of the tip-sample bias, thereby allowing one to record arbitrary patterns on the surface Fig. 1.15. Notably, this is not a conventional scenario for single atom or molecule manipulation, which involves formation of a significantly strong tip-molecule bond and subsequent translation of the tip to “drag” the molecule or the atom along the surface.

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Fig. 1.15 STM images showing snapshots of construction of three letters, S, T and M using electric-field directed manipulation of vibrationally excited CH3S moieties on Cu(111) surface. Each image is 6x6 nm2 (reproduced with permission from M. Ohara et al. [107]). Copyright (2008) by the American Physical Society

Collective Reactivity of Molecular Aggregates Before concluding this chapter, a few comments are appropriate on several recent examples of collective chemical reactions observed in molecular assemblies or twodimensional polymers, where the energy of a localized inelastic excitation is channeled into a collective RC, rather than dissipating it into the electron–hole pair excitations and the phonon bath of the underlying substrate. The first example of such a reaction was reported by Okawa and Aono [108] using self-assembled diacetylene compound (10,12-nonacosadiynoic acid) on HOPG (Fig. 1.15). Application of a relatively high voltage/current pulse (~5 V) induced unidimensional polymerization of the molecules (RC≡CR–C≡CR→=(RC–CR=CR–C)= along the self-assembled rows, with the reaction surprisingly extending by tens of nanometers away from the excited position (Fig. 1.16b, c). Subsequently, the polymerized diacetylene rows were found to possess a delocalized electronic state, implying that this procedure can be used to make one-dimensional wires on the surface [109, 110]. The reaction of this type is not unexpected because polydiacetylene is a very well-known monomer [111], and formation of polydiacetylene moiety is thermodynamically very profitable. As a result, once initiated the reaction will self-propel. Furthermore, graphite has a zero-energy gap at the Fermi level [112], thus a low density of states and suppressed energy dissipation via electron–hole pair generation. Nevertheless, the degree of localization of this reaction is striking as is the accuracy and relative ease of initiation and propagation. A second example of an unconventional delocalized reaction was recently reported by Maksymovych et al. [32]. This time, the reaction took place on metal surfaces of Au(111) and Au(100), involving linear chains of CH3SSCH3 molecules

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Fig. 1.16 (a) STM image of self-assembled monolayer of diacetylene on HOPG (Vs = −1.0 V, It = 0.07 nA). (b) A bias pulse applied at a point shown by an arrow triggers one-dimensional polymerization of the diacetylene precursor. (c) STM image of the same area recorded immediately after (b). (d, e) Diagrams illustrating the initiation of chain polymerization with an STM tip (reproduced with permission from Okawa and Aono. J. Chem. Phys. 115 (2001) 2317)

that spontaneously form at T > 70 K. Upon electron attachment to any molecule in the chain of up to five molecules on Au(111) and up to ten molecules on Au(100), the S–S bonds of all the molecules were found to first break, and then all but one to reform, making new CH3SSCH3 molecules which relate as the offset mirror images to the initial molecules (ignoring the exact orientation of the CH3 groups), Fig. 1.17. Such a chain reaction can be therefore be visualized as a sequence of elementary steps, CH3S + CH3SSCH3 → CH3SSCH3 + CH3S, each of which is

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Fig. 1.17 STM images before and after electron-induced dissociation of CH3SSCH3 molecule chains self-assembled on Au(111) and Au(100) surfaces. (a) Chain reaction of CH3SSCH3 dimer and tetramer assemblies on Au(111) induced by a voltage pulse on top of the terminal molecule (blue dot, pulse voltage 0.9 V) leading to the synthesis of CH3SSCH3 molecules of opposite conformation. Schematic ball models of the selected structures are shown aside their STM images. (b) On Au(100) surface the reaction may involve as many as ten molecules in a row (reproduced with permission from Maksymovych et al. [32])

closely reminiscent of the photo-induced free-radical substitution reactions in the gas phase involving the thyil radical (RS·) [113]. The terminal CH3S fragments were found to bind to the twofold coordinated sites on the Au(111) surface. Unlike polymerization of the diacetylene cores, this reaction does not create stronger chemical bonds at each of its elementary steps (assuming a consecutive scenario), and its propagation is therefore quite intriguing in light of the presence of the Au(111) surface, which acts as an energy sink. The enabling chemical interactions in the interior of the chain were identified with the help of nudged elastic band density functional calculations. We have found that the energy to break an S–S bond in a single CH3SSCH3 molecule is nearly identical to the total energy required to react a trimer assembly of these molecules in a consecutive propagation scenario [32]. The main reason for the nearly activation-less reactions in the chains interior is the formation of a metastable complex where an intermediate atop-bonded CH3S species is stabilized by the neighbor CH3SSCH3 molecule. Experimental kinetic measurements have also revealed that the reaction is excited by relatively highenergy electrons, either two electrons at >0.8 eV or one electron at >1.3 eV. This high threshold and first-principles calculations of the electronic structure suggested that the chain reaction is initiated by electron attachment to the LUMO of the CH3SSCH3 molecules. A curious detail is that the LUMOs of the individual molecules form a delocalized state upon self-assembly of the chain reaction. Experimentally, the evidence for this state comes from a lower energy threshold for

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Fig. 1.18 Hydrogen exchange in a water dimer triggered by inelastic electron tunneling. (a) STM image of H2O and D2O dimers (5.5 × 5.5 nm2). (b) An enlarged image of the D2O dimer (2.4 × 2.0 nm2). The grid lines represent the lattice of Cu(110) that is depicted through nearby monomers centered on top of Cu atoms. The (H2O)2 dimer is observed to flip numerous times as the tip scans across. (b–d) represents step-by-step flipping of the (D2O)2 dimer. (c) The (D2O)2 dimer flipped between the two orientations as the tip was scanned horizontally across it. [Reproduced with permission from Kumagai et al. [82])

a chain reaction compared to an isolated molecule, which then poses a question of how the delocalized state participates in the initial excitation and whether more than one S–S bond in the chain is weakened by initial electron attachment. A third example of collective reactions is a hydrogen bond exchange within a water dimer formed by the tip-induced manipulation of the monomers on Cu(110) [82] (Fig. 1.18). The hydrogen exchange constituted a collective reorientation of the (H2O)2 and (D2O)2 molecules, so that the hydrogen bonding (O–H…O) would first be formed by the H atom of one molecule, and upon rearrangement switched for the H atom of the other. This is a surface analogue of a similar behavior of a free water dimer in the gas phase. What is peculiar is that much like the gas-phase reaction, the surface reaction is dominated by quantum tunneling, as evidenced from a large isotope effect. From the analysis of a bias dependence of the reaction rate, Tikhodeev and Ueba [81] concluded that this reaction proceeds through vibrationally

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assisted tunneling from the first vibrationally excited state. Overall, the growing experience in the field of atomically resolved imaging and single-molecule manipulation using scanning probe microscopy stimulates the studies of molecules and surfaces with increasing complexity, which is why we believe that collective reactivity will be observed more and more frequently in the future studies.

Conclusions and Outlook Single-molecule excitation in the tunneling junction is a versatile approach to chemical transformations that reveals the fundamental mechanisms of molecular reactivity on solid surfaces and the physical phenomena that govern the energy flow of a dynamic molecule-surface interaction. Statistical analysis of such reactions can address the electronic and vibrational interactions of adsorbed molecules in a way that is complementary to tunneling spectroscopy and, in some cases, is entirely exclusive (e.g., in the action spectroscopy of the vibrational modes). As such, it is poised to make many future discoveries that one day will enable us to tune chemical reactions by molecule-surface and inter-molecular interactions, harness singlemolecule machines and electronic circuitry and exploit molecules as a means to control the electronic properties of surfaces or thin films, including correlated electron behaviors. Many important steps have already been made, and we now have the statistical framework to analyze single-molecule reactions and a rather comprehensive understanding of the underlying mechanisms. The field is well positioned to take full advantage of the rapidly ongoing experimental developments in scanning probe microscopy, such as force sensitivity and mechanical energy dissipation on the single-molecule level and coupling to femto-second laser techniques. The increasing scope and scalability of ab initio theoretical methods enables complex systems and dynamic phenomena to be explored. We can thus proceed onto larger molecules, metal surfaces functionalized with insulating or polar thin films and molecular self-assemblies, all of which will present new dynamic functionality and new mechanisms for energy flow and dissipation that can eventually be resolved on the real timescale of single-molecule transformations. Acknowledgment The writing of this review was done at the Center for Nanophase Materials Sciences, Office of Basic Energy Sciences, U.S. Department of Energy.

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Chapter 2

High-Resolution Architecture and Structural Dynamics of Microbial and Cellular Systems: Insights from in Vitro Atomic Force Microscopy Alexander J. Malkin and Marco Plomp

Introduction One of the great scientific challenges at the intersection of chemistry, biology and materials science is to define the biophysical pathways of cellular life, and in particular, to elucidate the complex molecular machines that carry out cellular and microbial function and propagate the disease. To study this in a comprehensive way, the fundamental understanding of the principal mechanisms by which cellular systems are ultimately linked with their chemical, physical, and biological environment are required. Complete genome sequences are often available for understanding biotransformation, environmental resistance, and pathogenesis of microbial and cellular systems. The present technological and scientific challenges are to unravel the relationships between the organization and function of protein complexes at cell, microbial, and pathogen surfaces, to understand how these complexes evolve during the bacterial, cellular, and pathogen life cycles, and how they respond to environmental changes, chemical stimulants, and therapeutics. Development of atomic force microscopy (AFM) for probing the architecture and assembly of single microbial surfaces at a nanometer scale under native conditions, and unraveling of its structural dynamics in response to changes in the environment has the capacity to significantly enhance the current insight into molecular architecture, structural and environmental variability of cellular and microbial systems as a function of spatial, developmental, and temporal organizational scales. In this chapter we will demonstrate, focusing on the work conducted in our group in the past several years, the capabilities of AFM in probing the architecture and assembly of bacterial surfaces and integument structures and their evolvement during bacterial life cycles, as well as in response to environmental changes. We have used AFM to investigate spore coat architecture and assembly, structural dynamics, and germination of several species of Bacillus [1–7] and Clostridium [8] A.J. Malkin and M. Plomp (*) Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory, Livermore, CA, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_2, © Springer Science+Business Media, LLC 2010

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spores. These include Bacillus thuringiensis, Bacillus anthracis, Bacillus cereus, Bacillus atrophaeus and Clostridium novyi-NT species spores. B. anthracis, the causative agent of anthrax, is a gram-positive spore-forming bacterium [9, 10]. The B. cereus species is environmentally ubiquitous and can cause bacteriemia and septicemia, central nervous and respiratory system infections, endocarditis and food poisoning [11]. B. atrophaeus spores have been used as a biological stimulant for decontamination and sterilization processes [12, 13], and bioaerosol detection development [14, 15]. B. thuringiensis spores are insect pathogenic species [16]. Clostridium novyi (C. novyi-NT) is a motile, spore-forming, gram-variable anaerobic bacterium. C. novyi can cause infections leading to gas gangrene in humans, particularly after traumatic wounds or illicit drug use [17]. The pathology of C. novyi is attributed to the lethal alpha-toxin [18], which is absent in an attenuated strain, C. novyi-NT [19]. C. novyi-NT is one of the most promising bacterial agents for cancer therapeutics [20]. Intravenous injection of C. novyi-NT spores into tumor-bearing mice was found to successfully eradicate large tumors, either in combination with radiation therapy [19] and chemotherapy [21], or by itself, as it can induce a potent immune response [22]. Here, we will also describe the development of AFM for immunolabeling of the proteomic structures of bacterial spore surfaces [5]. Finally, we will present data on the elucidation of bioremediation mechanisms of Arthrobacter oxydans. A. oxydans is a gram-positive and chromium (VI)-resistant bacterium, which can reduce highly mobile, carcinogenic, mutagenic, and toxic hexavalent chromium to less mobile and much less toxic trivalent chromium [23].

AFM Investigations of Spore Morphology, Structural Dynamics and Spore Coat Architecture When starved for nutrients, Bacillus and Clostridium cells initiate a series of genetic, biochemical, and structural events that results in the formation of a metabolically dormant endospore [24]. Bacterial spores can remain dormant for extended time periods and possess a remarkable resistance to environmental insults, including heat, radiation, pH extremes, and toxic chemicals [24]. Their unique structure, including a set of protective outer layers, plays a major role in the maintenance of spore environmental resistance and dormancy [24–26]. The Bacillus bacterial spore structure (Fig. 2.1) consists [25], starting from the center, of an inner core surrounded by the inner cytomembrane, a cortex, outer membrane and an exterior spore coat. In some bacterial species, including Bacillus thuringiensis and Bacillus anthracis, the coat is surrounded by a loosely attached exosporium. The spore core contains DNA and dipicolinic acid, which is associated predominantly with Ca2+. The major role of the spore cortex, which consists of a thick layer of species-dependent peptidoglycan, is to maintain spore heat resistance and dormancy [25].

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Fig. 2.1 Structure of a Bacillus spore: spore core (1), inner membrane (2), cortex (3), outer membrane (4), spore coat (5), exosporium (6), and appendages (7). Inset: spore coat with two crystalline layers of the outer spore coat

Bacillus and Clostridium Spore Morphology AFM images of various species of air-dried Bacillus and Clostridium-NT spores are presented in Fig. 2.2. As illustrated in Fig. 2.2, B. thuringiensis, B. cereus and B. anthracis native spores are enclosed within an exosporium sacculus (indicated with the letter E in Fig. 2.2a–d), which is larger than the dimensions of the spore body for two former species (Fig. 2.1a,c) and often tightly attached to the spore coat in the case of B. anthracis spores (Fig. 2.2d). The major part of the exposporium is reported to consist of two glycoprotein components [27, 28] and significant amounts of lipid (18%) and carbohydrate (20%), and smaller amounts of other components including amino sugars [29, 30]. It is constructed out of an inner part with three to four thin hexagonal crystalline layers, and an amorphous, hirsute layer on the outside [31, 32]. For B. anthracis spores, the exosporium is composed of a paracrystal basal layer and an external hair-like nap layer extending up to 600 nm in length [33] with approximately 20 exosporium-associated protein and glycoprotein species being identified [33–36]. The thickness of substrate-bound exosporium patches and of the hirsute layer as measured from the AFM height data (Fig. 2.2b) varied in the range 15–25 and 30–35 nm, respectively [1, 2], which corresponds well with earlier EM measurements [31, 32]. Apart from the 30–35 nm hirsute layer, the exosporium surface is frequently decorated with surface appendages [37–39]. These structures are most often described as hollow tubular filamentous extensions similar to pili of enterobacteria. The reported dimensions are variable but are in the range of 0.03–0.6 mm in width and 1.5–3.0 mm in length. These structures are most frequently described in the

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Fig. 2.2 AFM images of air-dried bacterial spores. (a, b) B. thuringiensis, (c) B. cereus, (d) B. anthracis, (e) B. atrophaes, and (f) C. novyyi-NT spores. In (b) AFM image of the B. thuringiensis exosporium showing a footstep (F) with numerous hair-like appendages (A2) and longer and thicker tubular appendage (A1). Surface ridges, extending along the entire spore length are indicated with white arrows in (a) and (c–e). Exosporium is indicated with letter E in (a–d). “Shell tail” in (f) is indicated with S. Images (b) and (c) reproduced, with permission from Ref. [1]. © (2005) Biophysical Society, USA

closely related B. cereus, B. anthracis and B. thuringiensis group. The number of appendages per spore varies from 3 to more than 20. It is not clear in the case of Bacillus spores if appendages originate on the spore coat or exosporium. These filaments could be of importance for spore attachment to surfaces or ligands. As illustrated in Fig. 2.2b, the filamentous exosporial appendages appeared to be attached to the outer surface of the exosporium. Two types of appendages were typically observed. The first type (indicated as A1) appeared to be tubular with a diameter and length in the range of 8–12 and 400–1,200 nm, respectively. Other appendages, such as those indicated as A2 in Fig. 2.1b, were 2.5–3.5 nm thick and typically 200–1,600 nm long. Six to ten appendages were seen on the exosporium surface. B. atrophaeus spores (Fig. 2.2e) do not possess exosporia, with the outer spore coat being the outermost surface. As illustrated in Fig. 2.2f, C. novyi-NT spores were encased in amorphous shells. Many spores exhibited ~200 nm thick shell “tails” at their poles (Fig. 2.2f) similar to “tails” visualized by EM [8]. Highresolution AFM images reveal that the outer shell surface typically consists of irregular amorphous material [8]. The most pronounced morphological features seen on the surfaces of air-dried Bacillus spores are ridges (indicated on several spores with arrows in Fig. 2.2),

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which typically extend along the long axis. Thickness of the ridges typically varies from 20 to 60 nm. Similar ridges were previously reported by electron microscopy [40] and AFM studies of various species of bacterial spores [1–3, 41].

Spore Size Distributions The interspecies distributions of spore length and width were determined for four species of Bacillus spores in aqueous and aerial phases (Fig. 2.3). It was found [1] that the dimensions of individual spores differ significantly depending upon species, growth regimes, and environmental conditions. Spores of B. thuringiensis are substantially larger (~50% higher and ~20% longer) than B. atrophaeus and

Fig. 2.3 Distribution of spore width (a) and length (b) for plate-grown (pg) and solution-grown (sg) B. atrophaeus and B. thuringiensis spores, and solution-grown B. subtilis spores. Images reproduced, with permission from Ref. [1]. © (2005) Biophysical Society, USA

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B. subtilis spores. The difference in average width and length between plate-grown and solution-grown spores of B. atrophaeus and B. thuringiensis suggest that environmental/physiological factors can have significant effects on spore dimensions. These findings could be useful in the reconstruction of environmental/preparation conditions during spore formation and for modeling the inhalation and dispersal of spores.

Spore Response to a Change in the Environment from Fully Hydrated to Air-Dried State AFM allows, for the first time, a direct comparison of fully hydrated and air-dried native spores visualized under water and in air, respectively [1]. Images of fully hydrated B. atrophaeus spores are presented in Fig. 2.4. Surface ridges, the prominent structural features of air-dried spores (Fig. 2.2), are typically absent from the surface of fully hydrated spores (Fig. 2.4). The surface morphology of fully hydrated

Fig. 2.4 AFM images showing the dynamic response of B. atrophaeus spores to dehydration. (a) Spore coat surface morphology of a fully hydrated spore. The area indicated with a square in (a) is shown at high resolution in (b). A shallow wrinkle on the surface is indicated in (b) with an arrow. In (c) the same area is shown after air-drying. The wrinkle seen in (b) developed into a fold/surface ridge (indicated with white arrow) with increased height and length. Structural alterations are seen in the surfaces morphology of a hydrated spore (d) and after air-drying (e). Formation of a surface ridge (indicated with an arrow) is seen in (e) and at higher resolution in (f) along with the emergence of a number of smaller folds

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and air-dried spores spans a wide range of folding motifs (Figs. 2.1 and 2.4). Therefore, direct visualization of individual spores is required in order to probe the dynamic response of this aqueous to aerial phase transition. This analysis was performed for 35 individual spores [1]. Spores were visualized under water, then air-dried for ~40 h, imaged in air (65% relative humidity), and reimaged after rehydration. Typical examples of hydration/dehydration ultrastructural transitions are presented in Fig. 2.4. As illustrated in Fig. 2.4a, the spore coat of a fully hydrated spore appears to be tightly attached to the cortex, excluding an area with an ~600 nm long wrinkle having a height of ~15–20 nm (Fig. 2.4a, b). Upon dehydration (Fig. 2.4c) this wrinkle becomes a surface ridge/fold extending along the entire length of the spore, accompanied by a height increase of ~50 nm. Similarly, the surface of another fully hydrated spore (Fig. 2.1d), exhibits an ~500 nm long fold (indicated with an arrow) having a height of ~30 nm. Upon dehydration, a new ~800 nm long surface ridge/fold (indicated with white arrow in Fig. 2.4e), having a height of ~40–60 nm and a number of smaller folds with a height range of ~10–30 nm, forms on the spore coat. These profound dehydrationinduced changes in spore surface architecture were found to be accompanied by a pronounced decrease in spore size. As illustrated in Fig. 2.5, the average width of 35 individual air-dried spores was reduced to 88% of the size measured for fully hydrated spores. Upon re-hydration of air-dried spores, they returned to 97% of their original size after 2 h in water, establishing the reversibility of the size transition (Fig. 2.5). The results from these individual spore measurements were confirmed by independent experiments (not presented here) where B. atrophaeus spore width was measured for two independent sets of ~200 spores in water and air, respectively. The width of the air-dried spore was again reduced to ~88% when compared with spores imaged in water. The observed decrease in the width of bacterial spores upon dehydration is apparently due to the contraction of the spore core and/or cortex. The ability of the coat to fold and unfold concomitant with changes in spore size was suggested [1, 41, 42] based on measurements of B. thuringiensis spore dimensions induced by humidity transients [43]. First direct visualization [1] of the response of native spores to dehydration/rehydration described above clearly demonstrates that the spore coat is itself does not shrink/expand but is flexible enough to compensate for the internal volume decrease of core/cortex compartments by surface folding and formation of ridges. These studies establish that the dormant spore is a dynamic physical structure. The observed folding could involve either the outer coat layer or the entire ensemble of inner and outer coat layers. In case of B. atrophaeus, if only the outermost rodlet layer (Fig. 2.4) was exclusively involved in folding, then due to the highly anisotropic rodlet structures, folding would most likely take place preferentially along the orientation of the rodlets. However, as seen in Fig. 2.4 folding of the coat takes place in arbitrary orientations with respect to the rodlet structures. This suggests that the whole spore coat folds with the inner surface of the coat disconnecting locally from the outer surface of the spore cortex. A number of ridges

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Fig. 2.5 Spore width variations of 35 individual B. atrophaeus spores, as a function of the size of the originally hydrated spore, following dehydration (24 h) (diamonds, dashed trend line) and rehydration (2 h) (triangles, dotted trend line). For ease of comparison, the original hydrated spore width is (redundantly) depicted as circles, which by definition lie on the solid y = x line. Thus, the three data points for one individual spore, depicted with the same color, are all on the same vertical line. Several spores detached from a substrate during rehydration experiments resulting in a smaller amount of experimental rehydration points (triangles). Solution-grown B. subtilis spores. Images reproduced, with permission from Ref. [1]. © (2005) Biophysical Society, USA

vary broadly for different Bacillus species. Thus the outer coat of B. thuringiensis spores typically exhibited less ridge formation when compared with B. atrophaeus and B. cereus spores. This finding suggests that the outer spore coat elastic properties may vary among spore-forming species.

High-Resolution Structure and Assembly of the Spore Coat The multilayer spore coat (Fig. 2.1) consists of structural proteins and small amounts of carbohydrate [24–26]. The spore coat plays an important role in spore protection and germination. Approximately 50 Bacillus spore coat proteins or coat protein orthologs have been identified by genomic and proteomic analysis [24–26, 44]. Despite the recent advances in biochemical and genetic studies [44], spore coat morphogenesis, which includes self-assembly of crystalline layers of the spore coat, is still poorly understood. In particular, it is not clear which spore coat

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proteins form the various spore coat layers, what their roles are in the coat assembly, and, finally, which proteins are surface-exposed and which ones are embedded beneath the surface. The elucidation of bacterial spore surface architecture and structure–function relationships is critical to determining mechanisms of pathogenesis, environmental resistance, immune response, and associated physicochemical properties. Thus, the development and application of high-resolution imaging techniques, which could address spatially explicit bacterial spore coat protein architecture at nanometer resolution under physiological conditions, is of considerable importance. We have directly visualized species-specific high-resolution native spore coat structures of bacterial spores including the exosporium and crystalline layers of the spore coat (Fig. 2.6) of various Bacillus [1–4, 6, 7] and Clostridium novyi-NT [8] species in their natural environment, namely air and fluid. For B. atrophaeus (Fig. 2.6a, b), and B. subtilis spores (data not shown here), the outer spore coat was composed of a crystalline rodlet layer with a periodicity of ~8 nm. Removal of the B. cereus and B. thuringiensis exosporium by sonication [1] or single cell French Press treatment [8] revealed crystalline rodlet (Fig. 2.6c) and hexagonal honeycomb (Fig. 2.6d) outer spore coat structures, respectively.

Fig. 2.6 High-resolution spore coat structures of Bacillus spores. The outer spore coats of B. atrophaeus (a, b), B. cereus (c), and B. thuringiensis (d) consist of crystalline layers rodlet and honeycomb structures. B. cereus spores contain a crystalline honeycomb structure (e) beneath the exterior rodlet layer (c). B. thuringiensis spore coats do not contain rodlet structures. Rodlet assemblies can be seen adsorbed to the substrate (f). Images reproduced, with permission from Ref. [1]. © (2005) Biophysical Society, USA

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The ~10 nm thick rodlet layer of B. cereus spores (Fig. 2.6c) is formed by multiple randomly oriented domains, comprised of parallel subunits with a periodicity of ~8 nm. The size of the domains is typically 100–200 nm. In contrast to the multi-domain rodlet structure of the B. cereus spore coat, typically only a single continuous domain was found to be present on the outer coat of B. atrophaeus (Fig. 2.6b) and B. subtilis spores. Multiple rodlet domains are less common. Generally, the main domain covers 60–100% of the spore surface, while 0–40% is covered by 0–10 smaller domains [3]. Complete removal of the exterior B. cereus rodlet layer by sonication revealed an underlying honeycomb structure (Fig. 2.6e) similar to the exterior spore coat layer of B. thuringiensis (Fig. 2.6d). For both species, the lattice parameter for the honeycomb structure is ~9 nm, with ~5–6 nm holes/pits (Fig. 2.6d, e). In case of B. thuringiensis spores, rodlet structures were not observed as an integral component of the spore coat [1, 3]; however, as illustrated in Fig. 2.6f, patches of extrasporal rodlet structures were observed adsorbed to the substrate [1, 3]. Rodlet width and thickness (Fig. 2.6f) were similar to those observed for B. atrophaeus, B. subtilis and B. cereus spore coat structures (Fig. 2.6a–c), which indicates that the similar rodlet proteins could be present during the sporulation in these three species of Bacillus spores. Similar rodlet and honeycomb crystalline structures to those seen in Fig. 2.6 were observed in freeze-etching EM studies of several species of Bacillus spores [32, 40] and AFM studies of fungal spores [45]. Note that in the case of B. thuringiensis, spore coat rodlet structures were not observed in freeze-etching EM [32, 40]. Little is known about the assembly, physical properties, and proteomic nature of these bacterial spore rodlets. The closest structural and functional orthologs to the Bacillus species rodlet structure (not its protein sequence) are found outside the Bacillus genus. Several classes of proteins, with divergent primary sequences, were found to form similar rodlet structures on the surfaces of cells of gram-negative Escherichia coli and Salmonella enterica, as well as on spores of gram-positive streptomycetes and various fungi (for the review see Ref. [46]). Hydrophobins, a new class of structural proteins [47], were shown to be necessary for and an integral component of rodlet fungal spore surface structures. Hydrophobins can self-assemble and produce layers of rodlet structures at water–air interfaces [47]. Fungal hydrophobin rodlet layers cause hyphal fragments and spores to become water-repellant, which enables escape from the aqueous environment and stimulates aerial release, dispersal and attachment to hydrophobic host surfaces [46]. However, while hydrophobin-like proteins are found in fungal spores, it has not been possible to identify orthologs of these proteins in bacterial spores [48]. These similarities in crystalline outer coat layer motifs found in prokaryotic and eukaryotic spore types are a striking and unexpected example of the convergent evolution of critical biological structures. Further investigation is required to determine the molecular composition of prokaryotic endospore rodlets and their evolutionary relationship to eukaryotic rodlet structures. To observe the structure of the C. novyi-NT spore coat beneath the amorphous shell, we developed procedures to remove the shells by chemical treatment with various reducing agents and detergents or by physical treatment using a French

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Press [8]. When either French Press or chemical treatments were used, the majority of the exposed spore coat surface is formed by an ~8–10 nm thick honeycomb layer with a periodicity of 8.7 ± 1 nm (Fig. 2.7a). We have found that during the germination, the spores often had partly or completely lost their honeycomb layers, revealing the underlying layers (Fig. 2.7b). Note that a number of spores, as judged by AFM observations, do not possess the amorphous layer. Furthermore, the presence of the amorphous material sandwiched between the spore coat and coat-associated honeycomb layers combined with the quick disappearance of these honeycomb layers during AFM germination experiments (Fig. 2.7) [8], all indicate that these C. novyi-NT honeycomb layers are not an integral part of the spore coat, as the “rodlet layer” or “honeycomb layer” is for Bacillus species (Fig. 2.6) [1–3], but rather are a parasporal layer with an increased affinity for the spore coat. As seen in Fig. 2.7b–e, the removal of the honeycomb layer revealed a multilayer structure formed by ~6 nm thick smooth layers. Typically, there were three to six layers exposed on the spore surface. The spore coat surface patterns (Fig. 2.7b–e) were very similar to ones observed on the surfaces of inorganic, organic, and macromolecular crystals [49–52]. As seen in Fig. 2.7b–e, these crystalline spore coat layers exhibited growth patterns typically observed on inorganic and macromolecular crystals. These patterns include steps and growth spirals originating from screw dislocations, such as those previously described in studies of the crystallization of semiconductors [53], salts [54], and biological macromolecules [51, 52]. In the middle of the growth centers, the dislocations cause depressions, typically <15 nm, which are known as hollow cores in crystal growth theory and are formed by the stress associated with the dislocations [55, 56].

Fig. 2.7 C. novyi-NT spore coats – high-resolution AFM height images. (a) Removal of the amorphous shell by physico-chemical treatments reveals the underlying honeycomb layer with ~8.7 nm periodicity. (b) Most of the honeycomb layers disappeared from the spores within ~1 h during the germination process. Remaining honeycomb patches (left, lower sides) could be easily removed by scanning with increased force. Below the honeycomb layer several underlying coat layers (upper right) are revealed. (c–e) Typical growth patterns seen on C. novyi-NT spore surface after removing the honeycomb layers. (c) Whole spore with several ~6 nm thick layers exposed on the surface. (d) Zoom-in of the center of (c) showing that spore coat layers originate at screw dislocations. (e) Zoom-in of the area indicated in (c). The circle in (e) denotes a fourfold screw axis. Many dislocation centers show depressions reminiscent of hollow cores (arrows), which are found in a wide range of crystals. Images reproduced, with permission from Ref. [8]. © (2007) American Society for Microbiology

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Thus the presence of the above-mentioned growth patterns confirms crystalline nature of the coat layers. However, while AFM resolution is typically sufficient for visualization of crystal lattices on a molecular scale for a wide range of protein crystals [52, 57, 58], we were not able to resolve a regular, crystalline lattice on the C. novyi-NT spore coat layers. Hence, the lattice periodicity is assumed to be smaller than ~1 nm, which is the resolution associated with the sharpest AFM tips used. Such a periodicity would be small compared to the 6 nm thickness of individual spore coat layers. In the case of globular proteins, lateral lattice parameters typically do not differ to such an extent from the height of growth layers, which is reflected in relatively small differences between lateral and perpendicular crystallographic unit cell parameters [57]. Thus, the proteins forming the C. novyi-NT crystalline spore coat layers are likely not globular, but rather may be stretched peptides “standing upright” in the layers. This construction, which is found in paraffin [50] and fat crystals [58], results in a crystal class with relatively strong, hydrophobic interaction forces between the long neighboring units (here peptides) and weak interaction forces between the different crystalline layers. This generally leads to wide, thin crystals that mainly grow laterally and can grow perpendicular only via the screw dislocation spiral mechanism (as was indeed seen for the C. novyi-NT coat layers). Such a crystal type, with tightly packed, strongly interacting longitudinal peptides within a layer, would help explain the toughness associated with bacterial spore coats [24, 40]. It may also explain why spore coat proteins are difficult to dissolve [24, 40, 59], as this type of packing involves hydrophobic interactions, and hence a high proportion of hydrophobic amino acids. In addition to enabling the nucleation and growth of new coat layers during sporulation, the screw dislocations also pin several of these layers together, thereby making the spore coat an interconnected, cohesive entity, rather than a set of separate layers loosely deposited on top of each other. This, combined with the strong in-layer bonds, and possible cross-linking between the coat proteins, likely contributes to the resilient nature of the spore coat. In biology, crystallization is most often associated with biomineralization, where protein-directed crystallization leads to calcious bone [60] and shell formation [61, 62]. Screw dislocations and ensuing spiral growth have been observed for shell formation [63, 64]. High-resolution scanning electron probe X-ray microanalysis [65, 66] and nanometer-scale secondary ion mass spectrometry [67] studies have demonstrated that the proteinaceous coat of several bacterial spore species is essentially devoid of divalent mineral cations such as calcium, magnesium, and manganese. This indicates that C. novyi-NT spores could present the first case of non-mineral crystal growth patterns being revealed for a biological organism. The implication of observed crystalline nature of Bacillus and Clostridium spore coat layers for bacterial spore coat assembly is that, while the proteineous building blocks are produced via biochemical pathways directed by various enzymes and factors [24], the actual construction of these building blocks into spore coat layers is a self-assembly crystallization process. Similarly, the striking differences in native rodlet motifs seen in B. atrophaeus (one major domain for each spore), B. cereus (a patchy multi-domain motif) and B. thuringiensis (extrasporal rodlets) appear to be

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a consequence of species-specific nucleation and crystallization mechanisms which regulate the assembly of the outer spore coat. In the case of B. cereus outer coat assembly, the surface free energy [68] for crystalline phase nucleation appears to be low enough to allow the formation of multiple rodlet domains resulting in crosspatched and layered assemblies. During the assembly of the outer coat of B. atrophaeus spores, the surface free energy may be considerably higher; reducing nucleation to the point that only one major domain is formed covering the entire spore surface. In addition to the possible differences in the surface-free energies of the underlying inner coat, the pronounced difference in the nucleation rate of the outer coat rodlet layers for different Bacillus species could be caused by different supersaturation levels of the sporulation media during rodlet self-assembly. Since the molecular mechanisms of self-assembly of spore coat structural layers appear to be very similar to those described for nucleation and crystallization of inorganic and macromolecular single crystals [51, 52, 68], fundamental and applied concepts developed for the nucleation and growth of inorganic and protein crystals can be applied successfully to understanding the assembly of the spore coat. Thus based on experimentally observed rodlet structural properties, we have developed a model for rodlet spore surface assembly, which was derived from well-developed molecular-scale crystallization/self-assembly mechanisms [3]. The consequence of spore coat crystalline assembly process is that the spore coat is not only influenced by the biochemical pathways leading to the production of spore coat proteins, but also by the crystallization conditions during which these proteins assemble themselves. By analogy to “regular” protein crystallization, conditions during sporulation such as salt concentration, pH, the presence of impurities, nucleation rates of crystalline self-assembly of spore coat layers, and random variations in the number of screw dislocations on spores could change the growth rate and hence the thickness of the spore coat. This in turn could influence characteristics such as the resilience of spores, their lifetime, and their germination capacity. Furthermore, these observations suggest that spore coat architecture and assembly are not purely genetically determined, but could be also strongly influenced by the modifications of sporulation media, which in turn could affect spore germination competence and physicochemical properties. However, the effects of environmental and chemical perturbations on spore coat structure have not been investigated before. Thus, under different sporulation conditions, it is possible that rodlets could nucleate and assemble on the outer coat of B. thuringiensis spores instead of selfassembling in bulk media, as described above (Fig. 2.6f). By observing spore coat high-resolution structures, AFM analysis could be utilized to reconstruct the environmental conditions that were present during spore formation. In order to explore our hypothesis on the production-specific self-assembly of the spore coat, we have conducted AFM experiments on the characterization of the spore coat of B. thuringiensis spores grown under different sporulation conditions. Spore preparations grown in Nutrient Broth (NB) and G media were analyzed. For spores grown in NB media, only honeycomb crystalline layer was seen (Fig. 2.8a, b) on the spore coat of B. thuringiensis, similar to our previous results. In our previous experiments only extrasporal rodlets were found in preparations of

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Fig. 2.8 AFM images showing the outer coat structure of B. thuringiensis spores grown in NB media (a, b) and G media (c, d). Honeycomb and rodlet crystalline structures are indicated with hexagons (a, b) and circles (d), respectively

B. thuringiensis spores [2]. We have suggested [2] that variations in sporulation conditions could allow rodlets to nucleate, assemble, and attach to the outer coat of B. thuringiensis spores instead of being present in bulk media as extrasporal rodlets. This case was observed for spore samples grown in G medium (Fig. 2.8c, d). Both for solution- and agar-grown (G medium) spores, patches of rodlet structure were visualized on the spore coat of air-dried and fully hydrated spores. It appears that these rodlet structures are directly correlated to differences in the medium conditions during sporulation. Through systematic sampling (data not shown here), it was found that rodlet patches are present on all of the spores grown in G medium. On the other hand, none of the spores grown in NB medium contained rodlet structure patches. These data establish that outer coat structural motifs (patches of rodlet crystalline structures) are directly correlated to differences in the medium conditions during sporulation. These findings validate that AFM can identify formulation-specific structural attributes that could be used in bioforensics to reconstruct spore formulation conditions.

Unraveling of the Spore Coat Assembly with AFM-Based Immunolabeling AFM provides high-resolution topographical information about the spatial and temporal distribution of macromolecules in biological samples. However, simultaneous near molecular-resolution topographical imaging of biological structures and specific recognition of the proteins forming these structures is currently lacking. Of particular importance is the identification of the protein composition of pathogen

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surfaces. Pathogen outer surface structures (e.g. virus membranes and capsids, as well as bacterial cell walls, spore coats and exosporia) typically contain multiple proteins. While it is known to a certain degree which proteins are expressed for these surface structures, it is often unknown which of these are exposed on the outside of these structures and which are embedded within the structures. Detection of surface-exposed proteins is paramount for improving the fundamental understanding of their functional properties as well as for the development of detection, attribution, and medical countermeasures against these pathogens. In the past several years, considerable progress, in particular toward probing of microbial and cellular systems, has been made in identification and mapping of specific receptors and ligands on the biological surfaces using adhesion force mapping and dynamic recognition force mapping (for reviews, see Refs [69, 70]). Electron microscopy (EM)-based immunolabeling techniques have become an important tool for the elucidation of biological structure and function [71, 72]. AFM immunogold markers were utilized in the past for imaging of proteins, macromolecular ensembles, and protein–protein interactions [73–79]. We have recently utilized [5] AFM-based immunochemical labeling procedures for visualization and mapping of the binding of antibodies, conjugated with nanogold particles, to specific epitopes on the surfaces of Bacillus anthracis and Bacillus atrophaeus spores. In this study, we have established the validity (strength of antigen– antibody binding and avidity) of immunochemical labeling of the exosporium of Bacillus anthracis and the spore coat of Bacillus atrophaeus spores through various control experiments. We have further established the immunospecificity of labeling, through the utilization of specific anti-B. atrophaeus and B. anthracis polyclonal and monoclonal antibodies, which were targeted to spore coat and exosporium epitopes (Fig. 2.9). In particular, we have confirmed that bclA glycoprotein is the immuno-dominant epitope on the surface of B. anthracis spores [5].

Fig. 2.9 AFM images of specific binding of anti-B. anthracis gold-labeled polyclonal antibodies to the B. anthracis spore exosporia. Images reproduced, with permission from Ref. [5]. © (2009) American Chemical Society

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Spore Coat Assembly Proper assembly of a multilayer spore coat of Bacillus spores is dependent on a number of coat proteins [24]. Loss of any of those proteins could alter significantly the mechanisms of the spore coat assembly and the final spore coat structure. Indeed, as demonstrated in Fig. 2.10, deletion of a single spore coat protein could result in pronounced changes in the spore coat architecture [7]. Thus, the AFM analysis showed [6] that intact wild-type B. subtilis spores are completely or partially covered by a thin amorphous layer lacking defined structure (Fig. 2.10a). Directly below the amorphous layer is a rodlet crystalline layer (Fig. 2.10a), which has parameters similar to the B. atrophaeus rodlet spore coat layer [3]. CotE is a major spore coat morphogenetic protein, and in its absence the outer coat fails to assemble properly [80]. Indeed, we have demonstrated that for most (>90%) cotE spores, the outermost structure is formed by three to five crystalline layers, each of which is ~6 nm thick (Fig. 2.10b), which likely correspond to the inner coat layers, as is the case for the coats of Clostridium novyi (Fig. 2.7) spores [8]. Note that remnants of small patches of rodlet structures or groups of several individual rodlets were seen on the majority of cotE spores (Fig. 2.10b). Furthermore, surfaces of some cotE mutant spores exhibit patches or large regions covering the spore of a hexagonal crystalline layer (located between the rodlet layer and the inner coat multilayer structure) (Fig. 2.10b). Surfaces of gerE spores were found to lack completely both amorphous and rodlet structures, being encased in several inner spore coat layers [7]. Note that the number of inner coat layers was found to be less on gerE spores compared with cotE spores. Finally, spores lacking both CotE and gerE proteins (cotEgerE spores) were found to lack all outer and inner coat structures [7]. The outer surface of CotEgerE spores was found to be quite smooth and corresponds likely to the spore cortex, which is typically located in wild type spores under the multilayer spore coat structure (Fig. 2.1). Our recent comprehensive analysis of a wide range of B. subtilis mutants, which lack various spore coat proteins (data are not presented here) here, have provided improved understanding of the spore coat architecture and assembly.

Fig. 2.10 AFM images of B. subtilis spores of different strains. The spores analyzed were wildtype (a), cotE (b), gerE (c), and cotEgerE (d). Images reproduced, with permission from Refs. [6, 7]. © (2008) American Society for Microbiology

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Mechanisms of Spore Germination Upon exposure to favorable conditions, metabolically dormant Bacillus and Clostridium spores break dormancy through the process of germination [81–83] and eventually reenter the vegetative mode of replication. A comprehensive understanding of the mechanisms controlling spore germination is of fundamental importance both for practical applications related to the prevention of a wide range of diseases by spore-forming bacteria (including food poisoning and pulmonary anthrax), as well as for fundamental studies of cell development. Germination involves an ordered sequence of chemical, degradative, biosynthetic, and genetic events [81, 83]. While significant progress has been made in understanding the biochemical and genetic bases for the germination process [81], the role of the spore coat in the germination remains unclear [24, 81]. Spore coat structure regulates the permeation of germinant molecules [84, 85]. It is believed that penetration of germinants proceeds through pores in the coat structure and may involve GerP proteins [85]. We have recently developed [4, 8] in vitro AFM methods for molecular-scale examination of spore coat and germ cell wall dynamics during spore germination and outgrowth. To obtain a comprehensive understanding of the role of the spore coat in germination, AFM imaging on a nanometer scale is required. At this scale, the outer layer of the B. atrophaeus spore coat is composed of a crystalline rodlet array (Figs. 2.6a, b and 2.11a) containing a small number of point and planar (stacking fault) defects [3]. Upon exposure to the germination solution, disassembly of the rodlet structures was observed [4]. During the initial stages of germination, the disassembly was initiated through the formation of 2–3 nm wide micro etch pits in the rodlet layer (Fig. 2.11b). Subsequently, the etch pits formed fissures (Fig. 2.11b–d) that were, in all cases, oriented perpendicular to the rodlet direction. Simultaneously, etching commenced on the stacking faults (Fig. 2.11e,f) revealing an underlying hexagonal inner spore coat layer (Fig. 2.11g). Note that the hexagonal layer was previously observed [6] beneath the rodlet layer on B. subtilis spores (Fig. 2.10b). During later stages of germination, further disintegration of the rodlet layer (Fig. 2.11e, f) proceeded by coalescence of existing fissures, their autonomous elongation (at a rate of ~10–15 nm/h) and widening (at ~5 nm/h), and by continued formation of new fissures. Disassembly of the higher-order rodlet structure began prior to the outgrowth stage of germination (Fig. 2.12). Disaggregation of the rodlet layer occurred perpendicular to the orientation of individual rodlets resulting in the formation of banded remnants (Fig. 2.12). Further structural disruption led to the formation of extended, 2–3 nm wide, fibrils (indicated with arrows in Fig. 2.12e) which were also oriented perpendicular to the rodlet direction. The AFM studies presented here elucidate the time-dependent structural dynamics of individual germinating spores and reveal previously unrecognized nano-structural alterations of the outer spore coat. Disassembly of the higher-order rodlet structure initiates at micro-etch pits, and proceeds by the expansion of the pits to form

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Fig. 2.11 Disintegration of the spore coat layer. (a) The intact rodlet layer covering the outer coat of dormant B. atrophaeus spores is ~11 nm thick, and has a periodicity of ~8 nm (1–3). (b–d) Series of AFM images tracking the initial changes of the rodlet layer after (b) 13 min, (c) 113 min, and (d) 295 min of exposure to germination solution. Small etch pits (indicated with arrows in b) evolve into fissures (indicated with an arrow in c) perpendicular to the rodlet direction. The fissures expand both in length and width. (e, f) Series of AFM images showing another germinating spore. The spore long axis, as well as major rodlet orientation is left–right. Enhanced etching at stacking faults (running from left to center and indicated with an arrow in e), as well as increased etching at the perpendicular fissures were visible following (e) 135 min and (f) 240 min of germination. Fissure width and length increased from 10–15 and 100–200 nm (135 min) to 15–30 and 125–250 nm (240 min), respectively. (g) Etching and/or fracture of the rodlet layer at a stacking fault revealed the underlying hexagonal layer of particles with a 10–13 nm lattice period. Images reproduced, with permission from Ref. [4]. © (2007) National Academy of Sciences, USA

fissures perpendicular to the rodlet direction. What causes this breakdown of the rodlet layer? We proposed [4] that rodlet structure degradation is caused by specific hydrolytic enzyme(s), located within the spore integument and activated during the early stages of germination. The highly directional rodlet disassembly process suggests that coat degrading enzymes could be localized at the etch pits, and either recognize their structural features, or that the etch pits are predisposed to structural deformation during early stages of spore coat disassembly.

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Fig. 2.12 (a–d) Series of AFM height images showing the progress of rodlet disassembly. In the circled regions, banded remnants of rodlet structure (a) disassemble into thinner fibrous structures (d). Time between images was 36 min (a, b); 3 min (b, c) and 6 min (c, d), for a total time between (a, d) of 45 min. In (b), the area imaged in (c) is indicated with a light grey box. In (b, c) the area imaged in (a, d) is indicated with a dark grey box. In (e), which is an enlarged part of (d), arrows indicate the end point of rodlet disruption, i.e. fibrils with a diameter of 2–3 nm, oriented roughly perpendicular to the rodlets. Images reproduced, with permission from Ref. [4]. © (2007) National Academy of Sciences, USA

The gradual elongation of the fissures suggests that once hydrolysis is initiated at an etch pit, processive hydrolysis propagates perpendicular to the rodlet direction and to neighboring rodlets. The locations of the small etch pits may coincide with point defects in the rodlet structure. These point defects could be caused by misoriented rodlet monomers or by the incorporation of impurities into the crystalline structure. In both cases, point defects could facilitate access of degradative enzymes to their substrate in an otherwise tightly packed structure. Recent proteomic and genetic studies suggest that the inner and outer spore coats of Bacillus subtilis, which is closely related to Bacillus atrophaeus, are composed of over 50 polypeptide species [24]. However, it is unknown which of these proteins form the surface rodlet layer of the spore coat or how this outer spore coat layer is assembled. We have shown previously for B. cereus [1] and B. subtilis [7] spores and here for Bacillus atrophaeus spores (Fig. 2.11g) that the outer spore coat rodlet layer is underlain by a crystalline honeycomb structure.

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Several classes of proteins, with divergent primary sequences, were found to form similar rodlet structures on the surfaces of cells of gram-negative Escherichia coli and Salmonella enterica as well as spores of gram-positive streptomycetes and various fungi [46]. These rodlets were shown to be structurally highly similar to amyloid fibrils [46]. Amyloids possess a characteristic cross b-structure and have been associated with neural degenerative diseases (i.e., Alzheimer’s and prion diseases) [86]. Amyloid fibrils or rodlets form microbial surface layers [46], which play important roles in microbial attachment, dispersal, and pathogenesis. We have proposed [4] that the structural similarity of B. atrophaeus spore coat rodlets and the amyloid rodlets found on other bacterial and fungal spores suggests that Bacillus rodlets have an amyloid structure. AFM characterization of the nanoscale properties of individual amyloid fibrils has revealed that these self-assembled structures can have a strength and stiffness comparable to structural steel [87]. The extreme physical, chemical, and thermal resistance of Bacillus endospores suggests that evolutionary forces have captured the mechanical rigidity and resistance of these amyloid selfassembling biomaterials to structure the protective outer spore surface. Structural studies of amyloids have identified an array of possible rodlet assemblies, each consisting of several (two or four) individual cross-b-sheet fibrils, which are often helically intertwined [46]. The number of fibrils determines the diameter of the rodlet. Most amyloids resulting from protein-folding diseases, and some naturally occurring amyloids, form individual fibrils or disorganized rodlets networks. In spore coats of B. atrophaeus, the higher-order rodlet structure is organized as one major domain of parallel rodlets covering the entire spore surface [3]. Rodlet domain formation requires there must be periodic bonds in the rodlet direction (“parallel bonds”) as well in the direction perpendicular to it (“perpendicular bonds”) [88]. In the case of amyloid-like rodlets, the intra-rodlet, parallel bonds are known and consist primarily of hydrogen bonds associated with the cross-b sheets that form the backbone of the rodlet fibrils. However, the nature of the perpendicular bonds, i.e., the inter-rodlet bonds that keep the rodlets tightly packed, is unknown. Interestingly, for B. atrophaeus the ratio of length (parallel to rodlet direction) and width (perpendicular to rodlet direction) of the rodlet domains is on average ~1, indicating that during formation of these domains, growth velocity was similar in both directions, and hence parallel and perpendicular bonds were similar in strength. Based on these rodlet features, one might expect that during germination individual rodlets would detach or erode, leaving a striated pattern parallel to the rodlet direction. Surprisingly, striations perpendicular to the rodlet direction were observed (Fig. 2.12), and 2–3 nm wide fibrils perpendicular to the rodlet direction (Fig. 2.12e) were the culmination product of coat degradation. This result indicates that during germination, perpendicular rodlet bonds are stronger, or are more resistant to hydrolysis, than bonds parallel to the rodlet direction. Second, and most surprisingly, these perpendicular structures facilitate the formation of 200–300 nm long fibers perpendicular to the rodlet direction. It is unclear how microbial amyloid fibers form these perpendicular structures. One possibility is that during formation of the rodlet layer, both intra-rodlet parallel

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Fig. 2.13 (a–d) Series of sequential AFM amplitude images showing passage of sub-surface steps during the early stages of germination. The central step visible in (a, b) has a height of 18 nm and a velocity of ~10 nm/s. Scan direction is down (a, c), up (b, d); time between images is 36 s. (e) AFM amplitude image showing sub-surface permanent ring structures, corresponding to a height difference of 4–8 nm, at 65 min into germination. In general, observed step heights range from –10 to 0 to 25 nm, and velocities range from 0 to 100 nm/s

bonds and inter-rodlet perpendicular bonds form, similar in strength and leading to tern. During germination, the intra-rodlet parallel bonds are hydrolyzed, while the inter-rodlet perpendicular bonds remain intact over longer time periods. Spore coat hydrolytic enzymes could target a specific residue or structure (in this case, that of the cross-b sheets) and leave other (here, perpendicular) residues or structures intact. Identification of the gene(s) encoding the rodlet structure and the enzymes responsible for rodlet degradation are important areas for future research. Although in vitro AFM is a surface imaging technique, internal structural changes that are physically separated from the probe tip by the thin rodlet layer could be visualized, much as the arrangement of ribs beneath the skin is apparent to a finger passing over them, or, as in other AFM applications, the visualization of the cytoskeleton network beneath the surface of the cell [89]. During early stages of the germination process, movement of 2–10 nm steps beneath the rodlet structure was consistently recorded (Fig. 2.13a–d), which resulted in the formation of a pronounced ring structure beneath the outer spore coat surface (Fig. 2.2e). Moving steps were either ascending or descending with step velocities up to ~100 nm/s.

Emergence of Vegetative Cells Etch pits were the initiation sites for early germination-induced spore coat fissure formation. During intermediate stages of germination, small spore coat apertures developed that were up to 70 nm in depth (Fig. 2.14b). During late stages of germination these apertures dilated (Fig. 2.14c–e) allowing vegetative cell emergence (data not shown).

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Fig. 2.14 Emergence of vegetative cells. (a–g) Series of AFM height images showing 60–70 nm deep apertures in the rodlet layer (indicated with arrows in (b)) that gradually enlarged (c, d), and subsequently eroded the entire spore coat (e). Germ cells emerged from these apertures. (e) Prior to germ emergence from the spore coat, the peptidoglycan cell wall structure was evident. (f) At an early stage of emergence, the cell wall was still partly covered by spore remnants, while (g) immediately prior to cell emergence, the cell wall was free of spore integument debris. The germ cell surface contained 1–6 nm fibers forming a fibrous network enclosing pores of 5–100 nm. Images in (a–g) were collected on the same spore as those shown in Fig. 11e,f. Elapsed germination time (in h:min) was (a) 3:40, (b) 5:45, (c) 7:05, (d) 7:30, (e) 7:45, (f) 7:15, (g) 7:50. (h) In separate experiments, cultured vegetative B. atrophaeus cells were adhered to gelatin surfaces and imaged in water. AFM height images show a slightly denser, similar fibrous network compared with the germ cell network structure (g), with 5–50 nm pores. In the inset, the imaged part (h) of the entire cell is indicated with a white rectangle. Images reproduced, with permission from Ref. [4]. © (2007) National Academy of Sciences, USA

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In vitro AFM visualization of germling emergence allowed high-resolution visualization of nascent vegetative cell surface structure (Fig. 2.14e–g). Vegetative cell wall structure could be recognized through the apertures approximately 30–60 min prior to germ cell emergence. The emerging germ cell surface was initially partially covered with residual patches of spore integument (Fig. 2.14f). During the release of vegetative cells from the spore integument, the entire cell surface consisted of a porous fibrous network (Fig. 2.14g). In order to compare the cell wall structure of germling and mature vegetative cells, we carried out separate experiments in which cultured vegetative B. atrophaeus cells were adhered to a gelatin-coated surface [4], and imaged with AFM in water. As seen in Fig. 2.14h, the cell wall of mature vegetative cells contained a porous, fibrous structure similar to the structure observed on the surface of germling cells (Fig. 2.14g). The bacterial cell wall consists of long chains of peptidoglycan that are crosslinked via flexible peptide bridges [91, 92]. While the composition and chemical structure of the peptidoglycan layer vary among bacteria, its conserved function is to allow bacteria to withstand high internal osmotic pressure [91]. The length of peptidoglycan strands varies from 3 to 10 disaccharide units in S. aureus to ~100 disaccharide units in B. subtilis, with each unit having a length and diameter of ~1 nm [92]. The fibrous network observed on the germ cell surface with 5–100 nm pores (Fig. 2.14e, g), and the fibrous network observed on mature vegetative cells with 5–50 nm pores (Fig. 2.14h) appear to represent the nascent peptidoglycan architecture of newly formed and mature cell wall, respectively, and is comprised of either individual or several intertwined peptidoglycan strands. The cell wall density of mature cells appears to be higher with, on average, smaller pores and more fibrous material, as compared to the germ cells. These results are consistent with murein growth models whereby new peptidoglycan is inserted as single strands and subsequently cross-linked with preexisting murein [93]. The AFMresolved pore structure of the nascent B. atrophaeus germ and vegetative cell surfaces is similar to the honeycomb structure of peptidoglycan oligomers determined by NMR [91]. Note, the AFM data presented here suggests that peptidoglycan structure rearrangement may occur prior to the formation of the fibrous cell wall network. As seen in Fig. 2.13, during the early stages of germination, movement of 0–25 nm steps beneath the outer spore coat and formation of pronounced ring structures occurred. Similar concentric rings were observed by EM [94] and AFM [95] in newly divided gram-positive cell walls. It was suggested [95] that these rings were caused by peptidoglycan structural rearrangements during cell division. The rings and steps observed during the early stages of germination could be caused by peptidoglycan restructuring that accompanies maturation of the nascent sacculus [96]. The structural dynamics of C. novyi-NT [8] and B. atrophaeus germinating spores appears to be similar. Thus, at later stages of the germination process, the C. novyi-NT spore coat layers seen in Fig. 2.15, which are exposed at early stages of germination start to dissolve (Fig. 2.15). Thus this process was initiated by the formation of fissures (Fig. 2.15a), which subsequently widened and elongated

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Fig. 2.15 Dynamic AFM height imaging of degrading C. novyi-NT spore coat layers. Fissures first appeared (a, b), then laterally expanded into wide gaps (c–e) and eventually resulted in the removal of whole layers, exposing the underlying layer (e, f, arrows in (e)). One expanding fissure is indicated with a white oval in (a–f). Time in germination medium in h:min was 0:45 (a), 0:50 (b), 0:55 (c), 1:00 (d), 1:05 (e), 1:10 (f). Images reproduced, with permission from Ref [8]. © (2007) American Society for Microbiology

(Fig. 2.15b–e), resulting in isolated islands of remnant coat layers (Fig. 2.15e, f). The dissolution of coat layers revealed an underlying undercoat layer (marked with arrows in Fig. 2.15e). Similarly to B. atrophaeus spore germination mechanisms described above, coat degradation likely occurs under the influence of germination-activated lytic enzymes. In fact, such lytic enzymes are known to be encoded within the C. novyiNT genome [97]. Interestingly, C. novyi-NT spores contain mRNA, and these mRNA molecules are enriched in proteins that could assist with cortex and other degradation [97]. At the final stages of germination, the coat layers dissolved completely (Fig. 2.16a), fully exposing the ~20–25 nm thick undercoat layer. In the following stage of germination this layer also disintegrated. This proceeded through the formation and slow expansion of ~25 nm deep flat-bottomed apertures (Fig. 2.16a–f). The cortex was fully lysed by the time spore coat layers dissolved. Hence, the flatbottomed apertures in this undercoat layer show the underlying cell wall of the emerging C. novyi-NT vegetative cell, which, based on its lighter AFM phase contrast (Fig. 2.16f), has different physicochemical properties or/and hence, composition than the surrounding coat remnants. The nascent surface of the emerging germ cell appears to be formed by a porous network (Fig. 2.16e–f) of peptidoglycan fibers, similar to one described above for B. atrophaeus vegetative cells (Fig. 2.14g, h).

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Fig. 2.16 (a–e) AFM height images of the final outgrowth stage. (a) After the ~6 spore coat layers were largely dissolved, the underlying structural layer was exposed. (b–e) In this layer, 25 nm deep apertures appeared and grew laterally. (f) Phase image zoom-in of the largest aperture depicted in (c–e), showing the pronounced phase contrast, indicating the different material properties of the emerging cell wall (light) and remaining spore layer (dark). Inset in (f) is the concurrent height image, showing the 25 nm deeper position of the cell wall with respect to the surrounding spore layer. Time in germination medium in h:min was 1:40 (a), 2:15 (b), 2:50 (c), 3:35 (d), 3:50 (e), 3:55 (f). Images reproduced, with permission from Ref. [8]. © (2007) American Society for Microbiology

Note that the spore coat degradation process presented in Figs. 2.11–2.16 appears not to be affected by the scanning AFM tip [48]. The shapes of fissures and apertures remained unaltered after repeated scanning. Furthermore, when we zoomed out to a larger previously non-scanned area after prolonged scanning on a smaller spore area, the initially scanned area did not display any tip-induced alterations (such as a larger degree of coat degradation). Finally, when we did not image spores for more than an hour between two scans, the coat degradation pattern had developed similarly when compared with spores that were scanned continuously. Spore germination provides an attractive experimental model system for investigating the genesis of the bacterial peptidoglycan structure. Dormant spore populations can be chemically cued to germinate with high synchrony [81], allowing the generation of homogenous populations of emergent vegetative cells suitable for structural analysis. Proposed models for the bacterial cell wall structure posit that peptidoglycan strands are arranged either parallel (planar model) or orthogonal (scaffold model) to the cell membrane [90, 91]. Existing experimental techniques are unable to confirm either the planar or the orthogonal model. The experiments described here do not contain sufficient high-resolution data, in particular of individual peptidoglycan

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strands, to deduce with certainty the tertiary three-dimensional peptidoglycan structure. The pore structures (Figs. 2.14 and 2.16) of the emergent germ and mature vegetative cell wall – an array of pores – suggest a parallel orientation of glycan strands with peptide stems positioned in stacked orthogonal planes [91]. More detailed studies of germ cell surface architecture and morphogenesis will be required to confirm this peptidoglycan architecture and to investigate whether glycan biosynthesis precedes peptide cross-linking. The results presented here demonstrate that in vitro AFM has the capacity to provide important insight into peptidoglycan architecture and the biological role of the cell wall in critical cellular processes and antibiotic resistance.

Bacteria–Mineral Interactions on the Surfaces of Metal-Resistant Bacteria We are currently conducting studies on the elucidation of bioremediation mechanisms of Arthrobacter oxydans metal-resistant bacteria. A. oxydans is a gram- positive and chromium (VI)-resistant bacterium, which can reduce highly mobile, carcinogenic, mutagenic, and toxic hexavalent chromium to less mobile and much less toxic trivalent chromium. Toxic compounds and heavy metals can be removed from contaminated cites or waste by chemical and physical techniques, which are both difficult and expensive. The extraordinary ability of indigenous microorganisms, like metal-resistant bacteria, for biotransformation of toxic compounds is of considerable interest for the emerging area of environmental bioremediation. However, the underlying mechanisms by which metal-resistant bacteria transform toxic compounds are currently unknown and await elucidation. Stress response pathways are sure to play an important role in the niche definition of metal-resistant bacteria and their effect on the biogeochemistry of many contaminated environments. The present technological and scientific challenges are to elucidate the relationships between the stress-induced organization and function of protein and polymer complexes at bacterial cell wall surfaces, to understand how these complexes respond to environmental changes and chemical stimulants, and to predict how they guide the formation of biogenic metal phases on the cell surface. We have visualized air-dried A. oxydans bacteria and revealed the differences in surface morphology and flagella arrangements during different stages of bacterial growth. Thus, bacteria during the exponential stage growth (Fig. 2.17a) appear to have a rather smooth surface and show a peritrichous flagellar arrangement with flagella seen over the entire cellular surface. The surface of air-dried bacteria grown during the stationary phase (Fig. 2.17b) appears to be tubular (Fig. 2.17b, inset) and these bacteria show the lophotrichous flagellar arrangement with several flagella seen only at one pole of the cell. We have further visualized for the first timestress responses of A. oxydans bacteria in response to the exposure to the toxic environment. Thus, as illustrated in Fig. 2.17c, the formation of a supramolecular crystalline hexagonal structure on the

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Fig. 2.17 AFM and SEM images of A. oxydans bacteria. (a, b) Growth-dependent morphologies; (c) stress-induced supramolecular crystalline hexagonal layer on the bacterial surface; (d) stressinduced microbial extracellular polymers (MEP) layer covering a microbial colony; (e–f) SEM images of bacteria as a function of exposure time to Cr (VI). (e) 18 days; (d) 1 year

surface of A. oxydans bacteria exposed to 35–50 ppm Cr (VI) was observed. Since similar crystalline layers are not seen on control samples, this structure appears to be stress-induced in response to Cr (VI) exposure. At higher Cr (VI) concentrations, we have observed the formation of microbial extracellular polymers (MEP), which are seen in Fig. 2.17d, to cover a small microbial colony. Our AFM observations of the appearance of stress-induced layers on the surfaces of A. oxydans bacteria exposed to Cr (VI) are consistent with biochemical and electron microscopy (EM) studies of stress responses of A. oxydans bacteria. Thus, it was reported that A. oxydans grown with chromate concentrations above 40 mg/L significantly increased the production of a cell wall protein that had an apparent molecular mass of 60 kDa [98]. Presumably, this protein could form, as seen in Fig. 2.17c, a highly organized particulate layer on the surface of A. oxydans bacteria exposed to Cr (VI). The hexagonal stress-induced structure (Fig. 2.17c) is formed by a protein with the size of ~10–11 nm. High-resolution images (Fig. 2.17c, inset) reveal that these particles are oligomers, composed of monomers with a size of ~5 nm. Assuming the globular shape of the protein, this size corresponds well to the molecular mass of ~60 kDa. This 60 kDa protein was found to be positively charged and to act as an ion trap to bind negatively charged ions such as the soluble chromium anion (CrO4)2- [98]. Indeed, scanning electron microscopy (SEM) images of A. oxydans exposed to 40–50 ppm of chromate clearly showed that the surface of A. oxydans grown

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with Cr(VI) was coated with an array of particles (Fig. 2.17e). These nanoparticles remained on the surface of A. oxydans for months before they were released into the immediate environment either individually or with other cell envelope materials as a shell (Fig. 2.17f), thereby maintaining basic vital processes such as growth and division, as well as nutrient transport. We are currently developing procedures for in vitro high-resolution AFM characterization of the surface architecture, and structural dynamics of metal-resistant bacteria in response to changes in the environment and various chemical stimuli. It is expected that these experiments will improve the fundamental understanding of bioremediation mechanisms. The results presented here demonstrate that in vitro AFM is a powerful tool for revealing the structural dynamics and architectural topography of the microbial and cellular systems. AFM allows new approaches to high-resolution real-time dynamic studies of single microbial cells under native conditions. Environmental parameters (e.g., temperature, chemistry or gas phase) can be easily changed during the course of AFM experiments, allowing dynamic environmental and chemical probing of microbial surface reactions. Further incorporation of AFM-based immunolabeling techniques could allow the identification of spore coat proteins that play a role in spore germination, and provide a structural understanding of how these proteins regulate spore survival, germination, and disease. Acknowledgments The authors thank B. Vogelstein, T.J. Leighton, P. Setlow and H.-Y. Holman for spore and bacteria preparations, and helpful discussions. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract number DE-AC52–07NA27344. This work was supported by the Lawrence Livermore National Laboratory through Laboratory Directed Research and Development Grants 04-ERD002 and 08-LW-027, and was funded in part by the Federal Bureau of Investigation.

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Part II

Dynamic Spectroscopic SPM

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Chapter 3

Dynamic Force Microscopy and Spectroscopy in Ambient Conditions: Theory and Applications Hendrik Hölscher, Jan-Erik Schmutz, and Udo D. Schwarz

Introduction Since its introduction in 1986 [1] the atomic force microscope become a standard tool in surface physics. In early experimental setups, a sharp tip located at or near the end of a microstructured cantilever profiled the sample surface in direct mechanical contact (contact-mode) to measure the force acting between tip and sample. Maps of constant tip–sample interaction force, which are usually regarded as representing the sample’s “topography,” were then recovered by keeping the deflection of the cantilever constant. This is achieved by means of a feedback loop that continuously adjusts the z-position of the sample during the scan process so that the output of the deflection sensor remains unchanged at a preselected value (setpoint value). Despite the widespread success of contact-mode AFM in various applications, the resolution was found to be limited in many cases (in particular for soft samples) by lateral forces acting between tip and sample. In order to avoid this effect, it has been proven to be advantageous to vibrate the cantilever in vertical direction near the sample surface. AFM imaging with oscillating cantilever is often denoted as dynamic force microscopy (DFM). In most cases, short-range tip–sample forces, which might be repulsive as well as attractive in nature, are responsible for the topographic contrast observed in AFM or DFM. Since the oscillation amplitudes of the oscillating cantilever are typically much higher than the interaction range of these forces, the tip “feels” the influence of the surface only during a short period of an individual oscillation, making nanoscale cantilever dynamics in atomic force microscopes inherently nonlinear.

H. Hölscher (*) Institute for Microstructure Technology, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_3, © Springer Science+Business Media, LLC 2010

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To further complicate matters, cantilever dynamics is additionally governed by the specifics of how the oscillation is performed, as several distinct methods to drive the cantilever exist. For instance, the historically oldest scheme of cantilever excitation in DFM imaging is the external driving of the cantilever at a fixed excitation frequency chosen to be exactly at or very close to the cantilever’s first resonance [2–4]. For this driving mechanism, different detection schemes measuring either the change of the oscillation amplitude or the phase shift between driving signal and resulting cantilever motion were proposed. Over the years, the amplitude modulation (AM) mode, where the actual value of the oscillation amplitude is employed as a measure of the tip–sample distance, has been established as the most widely applied technique for ambient conditions and liquids. Another method to oscillate the cantilever is the frequency-modulation (FM) mode. It was primarily developed for application in vacuum where standard AFM cantilevers made from silicon or silicon nitride exhibit very high quality factors Q, what makes the response of the system slow if driven in AM mode. Therefore, Albrecht et al. [5] introduced in 1991 the FM mode, which works well for high-Q systems and consequently developed into the dominating driving scheme for DFM experiments in ultrahigh vacuum (UHV) [6–9]. In contrast to the AM mode, this approach features a so-called self-driven oscillator [10, 11], which uses the cantilever deflection itself as drive signal, thus ensuring that the cantilever instantaneously adapts to changes in the resonance frequency. In this review, we present a detailed theoretical analysis of the basic features of DFM driven in any of these two modes. For this purpose, we first highlight the similarities and differences between externally driven and self-driven cantilevers before we explicitly include tip–sample interactions. We close our overview by some experimental examples demonstrating the features of dynamic force spectroscopy (DFS) using the FM mode.

General Theory of Dynamic Force Microscopy Formulation of the Problem and Basic Equation of Motion A sketch of the experimental setup of a dynamic force microscope utilizing the amplitude-modulation technique is shown in Fig. 3.1. The deflection of the cantilever is typically measured with the laser beam deflection method as indicated [12, 13], but other displacement sensors such as interferometric sensors [14–16] can be applied as well. During operation in conventional AM mode, the cantilever is driven at a fixed frequency by a constant-amplitude signal originating from an external function generator, while the resulting oscillation amplitude and/or the phase shift are detected by a lock-in amplifier. The function generator supplies not only the signal for the dither piezo; its signal serves simultaneously as a reference for the lock-in amplifier in the analyzer electronics.

3 Dynamic Force Microscopy and Spectroscopy in Ambient Conditions photo diode

laser

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analyzer phase or frequency amplitude amplifier

z

piezo

D+2A

aexc

d D

θ0

sample

PID

function generator

amplitude

phase shifter

setpoint

x−y−z−scanner

Fig. 3.1 Schematic drawing of the experimental setup of a dynamic force microscope where the driving of the cantilever can be switched between amplitude-modulation (AM) mode (solid lines) or frequency-modulation (FM) mode (dashed lines). While the cantilever in the AM mode is externally driven with a frequency generator producing a fixed frequency, the FM mode exhibits a feedback loop consisting of a time (“phase”) shifter and an amplifier. In both cases we assume that the laser beam deflection method is used to measure the oscillation of the tip which oscillates between the nearest tip–sample position D and D + 2A. The equilibrium position of the tip is denoted as d

In contrast, a dynamic force microscope driven in the FM mode has no external driving imposing a fixed frequency, but a feedback circuit consisting of an amplifier and a phase shifter (dashed lines in Fig. 3.1). The properly amplified and time- (i.e., phase-)shifted displacement sensor signal is then used to excite the dither piezo driving the cantilever.1 Two different driving techniques have been established for use with the FM mode: The original constant-amplitude (CA) driving scheme, where the oscillation amplitude of the cantilever is held constant by an automatic gain control (AGC) [5] and the constant-excitation (CE) driving scheme [18, 19], where the excitation amplitude of the cantilever driving is kept constant. Both the CE driving scheme [20–24] as well as the CA driving scheme [25–28] are frequently used in air and liquids. However, since the amplitude can be used as a feedback signal for scanning in the CE driving scheme, its implementation is easily possible for an existing DFM build for AM mode applications in air and liquid (cf. Fig. 3.1). Therefore, a comparison of both modes is especially straightforward by focussing just on the constant-excitation driving scheme if operating in FM mode. It is also possible to implement the FM-mode with the help of a phase-looked loop (PLL) [17].

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Based on the above description of the experimental setup, we can formulate the basic equation of motion describing the cantilever dynamics in DFM [29, 30]:

mz(t ) +

2 π f0 m z(t ) + cz ( z(t ) − d ) = Fts [z(t ), z(t )]    Q0 tip − sample force

 aexc cz cos(2πfd t )  +  aexc  − A cz z(t − t0 )

for AM mode,

(3.1)

for FM mode.

Here, z(t) is the position of the tip at the time t; cz, m, and f 0 = (cz / m) / (2π) are the spring constant, the effective mass, and the eigenfrequency of the cantilever, respectively. Somewhat simplifying, it is assumed that the quality factor Q0 unites the intrinsic damping of the cantilever and all influences from surrounding media such as air or liquid, if present, in a single overall value. The equilibrium position of the tip is denoted as d. The (nonlinear) tip–sample interaction force Fts is introduced by the first term on the right side of the equation. The two operational modes (AM or FM) are considered by the distinction on the right side of the equation. For the AM mode, an external driving force of the cantilever is used, where the driving signal is modulated with the constantexcitation amplitude aexc at a fixed frequency fd. The self-excitation of the cantilever used in the FM mode is described by the retarded amplification of the displacement signal, i.e., the tip position z is measured at the retarded time t − t0. Nonetheless, a consideration of the time shift by a phase difference q0 is also possible, giving equivalent results. Therefore, we use “time shift” and “phase shift” as synonyms throughout this review and notice that both parameters are scaled by q0 = 2pfdt0. Before finishing this introductory paragraph, we would like to add some words of caution regarding the validity of the equation of motion (3.1), as it disregards two effects that might become of importance under specific circumstances. First, we describe the cantilever by a spring-mass-model and neglect in this way higher modes of the cantilever. This is justified in most cases as the first eigenfrequency is by far dominant in typical AM-AFM experiments (see, for example, Fig. 1 in [31]). Thus, a mathematical treatment that ignores higher modes is still able to describe and explain all major general features experimentally observed in standard DFM imaging, which is the limited goal of this review. Comparison with studies that include higher harmonics by numerical means (see, e.g., [32–35]) confirms this statement. It might, however, not apply if advanced signal analysis in certain DFM spectroscopy modes is intended. Second, we assume in our model equation of motion that the dither piezo applies a sinusoidal force to the spring, but do not consider that the movement of the dither piezo simultaneously also changes the effective position of the tip at the cantilever end by the current value of the excitation aexccos(2pfdt) [34, 36, 37]. This effect

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becomes important when aexc is in the range of the cantilever oscillation amplitude. Fortunately, for conditions characterized by sufficiently high quality factors, this effect can be neglected. This is usually safely the case for measurements in air, where oscillation amplitudes typically exceed excitation amplitudes by several hundred times. During operation in liquids, however, the Q factor is low, and the oscillation amplitudes might be comparable with aexc [37]. Finally, in order to avoid confusion with other literature we would like to mention some words regarding the terminology used throughout this review. Due to the frequently occurring intermitted contact between tip and sample at the lowest point of the oscillation, the amplitude modulation mode introduced above has often been denoted as tapping mode [3]. Over the years, use of the term “tapping mode” has then evolved into a synonym for the AM mode in many publications, disregarding whether the tip is actually making intermitted contact or not. On the other hand, if it is the operator’s firm believe that no contact is established during the oscillations, the AM mode is sometimes also referred to as “noncontact” mode. Please note, however, that the term “noncontact atomic force microscopy (NC-AFM)” is often employed in connection with the frequency-modulation mode, which is mostly applied in UHV (see, e.g., [7]). This simple example shows how to different driving modes might be mixed up if we use the assumed type of tip–sample interaction to define a DFM technique. Therefore, in order to avoid confusion in this review, we will always use the suitable expressions for the driving technique (AM or FM mode) to describe the applied technique.

Driven and Self-Driven Cantilevers in Dynamic Force Microscopy In order to analyze the specific features of the AM and FM mode, it is instructive to first examine the difference of both driving terms. For simplicity, we assume in these preparatory considerations that the cantilever vibrates far away from the sample surface. Consequently, we can neglect tip–sample forces (Fts º 0), resulting in a greatly simplified equation of motion (3.1). This restriction will be abandoned later. First, we consider the situation where the DFM is driven in the AM mode. Under these circumstances, the equation motion reduces to the well-known case of a driven and damped harmonic oscillator:

mz(t ) +

2 π f0 m z(t ) + cz ( z(t ) − d ) = aexc cz cos(2 π fd t ) . Q0

(3.2)

The external driving forces the cantilever to oscillate exactly at the driving frequency fd. Therefore, the steady-state solution is given by the ansatz

z(t  0) = d + A cos(2 π fd t + φ) ,

(3.3)

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where f is the phase difference between the excitation and the oscillation of the cantilever. Introducing this ansatz, we obtain two functions for the amplitude and phase curves: A=

aexc 2

 fd 2   1 fd  1 − +  f0 2   Q0 f0 

tan φ =

2

,

fd / f0 1 . Q0 1 − fd 2 / f0 2

(3.4a)

(3.4b)

The features of such an oscillator are well-known from introductory physics courses, and we will thus skip their further discussion at this point. In contrast, the case where the cantilever is entirely self-driven is much less discussed in the literature. Here the corresponding equation of motion reduces to

mz(t ) +

2 π f0 m a z(t ) + cz ( z(t ) − d ) = − exc cz z(t − t0 ) . Q0 A

(3.5)

As the cantilever is not excited with a specific externally set frequency, the cantilever itself serves as the frequency determining element. Therefore, we make the ansatz [10, 38]

z(t  0) = d + A cos(2 π ft ) ,

(3.6)

and introduce it into (3.6) into (3.5). As a result we obtain a set of two coupled trigonometric equations:

aexc cos(2 π ft0 ) =

f02 − f 2 , f02

aexc 1 f sin(2 π ft0 ) = . A Q0 f0

(3.7a)

(3.7b)

The two equations can be decoupled with the assumption that the time shift t0 is set to a value corresponding to t0 = 1 / (4 f 0 ) ( = 90 ˚ ), which simultaneous corresponds to the by far most common choice for t0. For this value, the solution of (3.7) is given by

f = f0 ,

(3.8a)

A = aexc Q0 .

(3.8b)

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This simple calculation demonstrates the very specific behavior of a self-driven oscillator if the phase (or time) shift is set to 90 ˚ . In this case, the cantilever oscillates exactly with its eigenfrequency f0. Due to this specific feature revealed by (3.8a), we define that the cantilever is in resonance if this condition is fulfilled. The linear relationship between the oscillation and excitation amplitude is described by (3.8b).

Tip–Sample Interaction Force in Air During surface imaging, the vibrating cantilever is brought closely to the sample in order to monitor changes in the oscillation behavior induced by the tip–sample interaction. Therefore, we have to include a suitable model for the tip–sample interaction into our analysis. Assuming that the tip experiences long-range attractive forces described by a van der Waals term and short-range repulsive forces upon contact has been regarded by many authors as an adequate approximative description for the tip–sample interaction at ambient conditions [34, 39–44]. A geometrical characterization of the tip–sample system by a sphere of radius R over a flat surface then results in reasonably simple equations, which allow to reproduce most features of the amplitude-modulation DFM applied in ambient conditions. In this approach [34, 40–43], the long-range van der Waals force is described by

FvdW ( z ) = −

AH R , 6z2

(3.9)

where AH is the Hamaker constant. If the tip comes very close to the sample surface, the repulsive forces between tip and sample become significant. For simplicity, we assume that the geometrical shape of tip and sample does not change until contact has been established at z = z0 and that afterwards, the tip–sample forces are given by the DMT-M model, denoting Maugis’ approximation to the earlier Derjaguin– Muller–Toporov model [45]. In this approach, an offset FvdW(z0) is added to the well-known Hertz model, which considers the adhesion force between tip and sample surface. Therefore, the DMT-M model is often also referred to as Hertzplus-offset model [45]. The resulting overall force law is given by

 FvdW ( z ) Fair ( z) =  4 E∗ R ( z0 − z )3/2 + FvdW ( z0 )  3

for z ≥ z0 , for z < z0 .

  

(3.10)

The effective elastic modulus E *

(1 − m t2 ) (1 − ms2 ) 1 = + E∗ Et Es

(3.11)

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tip-sample force (nN)

6 4 2 0 −2 −4 −1.5

−1.0

−0.5

0.0

z0 0.5

1.0

1.5

2.0

tip-sample distance (nm)

Fig. 3.2 Tip–sample model force for air (3.10) using the parameters given in the text. The dashed line marks the position z0 where the tip touches the surface

depends on the Young’s moduli Et,s and the Poisson ratios mt,s of tip and sample, respectively. Figure 3.2 displays the assumed tip–sample model force. For the plot, the following parameters were used, which are typical for DFM measurements in air: AH = 0. 2 aJ, R = 10 nm, z0 = 0. 3 nm, µ t = µs = 0.3 , Et = 130 GPa, and Es = 1 GPa. The eigenfrequency, the quality factor, and the spring constant of the cantilever were chosen to be f0 = 300 kHz, Q0 = 300, and cz = 40 N/m, respectively.

Theory of AM Mode Including Tip–Sample Forces The mathematical form of realistic tip–sample forces is highly nonlinear for almost all cases of tip–sample systems. This fact complicates the analytical solution of the equation of motion (3.1) even for the assumed simplified model introduced in (3.10). However, for the analysis of DFM experiments, we need to focus on steadystate solutions of the equation of motion with sinusoidal cantilever oscillation. Therefore, it is advantageous to expand the tip–sample force into a Fourier series Fts [ z(t ), z(t )] ≈ fd Ú

1/ fd 0

+ 2 fd Ú + 2 fd Ú +…,

Fts [ z(t ), z(t )]dt

1/ fd 0 1/ fd 0

Fts [ z(t ), z(t )]cos(2p fd t + φ)dt × cos(2p fd t + φ) Fts [ z(t ), z(t )]sin(2p fd t + φ)dt × sin(2p fd t + φ) (3.12)

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where z(t) is given by (3.3). In the following, we assume that the tip–sample force is so small and the Q-factor so high that, as a consequence, higher harmonics can be neglected. It has been shown by Cleveland et al. [31] and Rodriguez and García [33] that this condition is well fulfilled in many practical cases. The first term in the Fourier series reflects the averaged tip–sample force over one full oscillation cycle, which shifts the equilibrium point of the oscillation by a small offset Dd from d to d0. Actual values for Dd, however, are typically small. For amplitudes commonly used in AM-AFM in air (some nm to some tens of nm), the averaged tip–sample force is in the range of some pN. The resulting offset Dd is less than 1 pm for typical sets of parameters [46]. Since this is well beyond the resolution limit of an AM-DFM experiment in air, we neglect this effect in the following and assume d » d0 and D = d − A. For the further analysis, we now insert the first harmonics of the Fourier series (3.12) into the equation of motion (3.1), obtaining two coupled equations [44, 47]

−

f02 − fd2 a = I + (d , A) + exc cos φ , A f02

(3.13a)

1 fd Q0 f0

(3.13b)

= I − (d , A) +

aexc sin φ , A

where the following integrals have been defined:

2 fd 1/ fd Fts [ z(t ), z(t )]cos(2p fd t + φ)dt cz A ∫0

I + (d , A) = =

1 p cz A2

I − (d, A) =

∫

d+ A

d−A

( F↓ + F↑ )

z−d A − ( z − d )2 2

dz ,

(3.14a)

2 fd 1/ fd Fts [ z(t ), z(t )]sin(2p fd t + φ)dt cz A ∫0

=

1 p cz A2

=

1 DE (d, A) . p cz A2

∫

d+ A

d− A

( F↓ − F↑ )dz

(3.14b)

Both integrals are functions of the actual oscillation amplitude A and cantileversample distance d. Furthermore, they depend on the sum and the difference of the tip–sample forces during approach (F↓) and retraction (F↑) as manifested by the labels “ + ” and “ − ” for easy distinction. The integral I + is a weighted average of

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these tip–sample forces ( Fts = ( F↓ + F↑ ) / 2 ). On the other hand, the integral I − is directly connected to DE, which reflects the energy dissipated during an individual oscillation cycle. Consequently, this integral vanishes for purely conservative tip– sample forces, where F↓ and F↑ are identical. A more detailed discussion of these integrals can be found in [48] and [49]. Equations (3.13) can be used to calculate the resonance curves of a dynamic force microscope driven in AM mode including tip–sample forces. The results are ad A= , 2 2    1 fd  fd 2 (3.15a)  1 − f 2 − I + (d , A) +  Q f + I − (d , A) 0 0 0 1 fd + I − ( d , A) Q0 f0 tan φ = . fd 2 1 − 2 − I + (d , A) f0

(3.15b)

Equation (3.15a) describes the shape of the resonance curve, but it is an implicit function of the oscillation amplitude A and cannot be plotted in a direct way. Nonetheless, the equation can be simplified for the case we consider here. Since the assumed tip–sample force is conservative, the integral function I − vanishes. Due to this simplification, (3.15a) can be written as a fourth-order equation, and the relationship between driving frequency and oscillation amplitude can be obtained from the relationship [50] 2

 fd  1 − I + ad2 1 1 I = 1 − − ± − + 2 . + 2 4 2  f  2Qeff 4Qeff Qeff A 0 ±

(3.16)

Figure 3.3 contrasts this equation (solid lines) with the numerical solution (symbols). As pointed out by various authors (see, e.g., [34, 40, 50–55]), the amplitude vs. frequency curves are multi-valued within certain parameter ranges. Moreover, as the gradient of the analytical curve increases to infinity at specific positions, some branches are unstable. The resulting instabilities are reflected by the “jumps” in the simulated curves (marked by arrows in Fig. 3.3), where only stable oscillation states are obtained. Obviously, they are different for increasing and decreasing the driving frequency. This is a well-known effect frequently observed in nonlinear oscillators (see, e.g., [56, 57]). The analysis of the resonance curve, however, gives no direct insight into the characteristics of DFM imaging, as during scanning, the driving frequency is set to a fixed value (and not swept) and the oscillation amplitude (or the phase, respectively) is used to control the cantilever-sample distance. Therefore, we need to examine the change of the oscillation amplitude as a function of the cantileversample distance.

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11 amplitude (nm)

10

AM-mode (Q = 300)

9 8 7 6 299.2

299.6

300.0

300.4

300.8

driving frequency (kHz)

Fig. 3.3 Resonance curve if the cantilever oscillates near the sample surface with a distance of d = 8. 5 nm and a free amplitude of A0 = 10 nm, thereby experiencing the tip–sample force given by (3.10). The solid line represents the analytical result of (3.15a), while the symbols are obtained from the numerical solution of the equation of motion (3.1). The dashed line reflects the resonance curve without tip–sample force and is shown purely for comparison. The resonance curve including the tip–sample force exhibits instabilities (“jumps”) during a frequency sweep. These jumps take place at different positions (marked by arrows) depending on whether the driving frequency is increased or decreased

In amplitude-modulation DFM, the cantilever might be oscillated at any frequency located well within the resonance peak. However, for first analysis we restrict ourselves to a single well-defined situation and assume that the driving frequency is set exactly to the eigenfrequency of the cantilever (fd = f0). With this choice, we have defined imaging conditions, which lead to a handy formula suitable for further analysis. From (3.15a), we obtain the following relationship between the free oscillation amplitude A0, the actual amplitude A, and the equilibrium tip position d:

A0 = A 1 + (Q0 I + [ d , A]) . 2

(3.17)

For the derivation of this formula, we used the approximation that the maximal value of the free oscillation amplitude at resonance is given by A0 » aexcQ0. Solving this equation allows us to calculate an amplitude vs. distance curve as shown in Fig. 3.4. As observed before in the resonance curves stable and unstable branches manifest, which can unambiguously be identified by a comparison with numerical results (symbols). Most noticeably, the AM mode curve exhibits jumps between unstable branches, which occur at different locations for approach and retraction. The resulting bistable regime then causes a hysteresis between approach and retraction, which has been well examined in numerous experimental and theoretical studies (see, e.g., [34, 42, 50–62]). This instability divides the tip–sample interaction into two regimes [34, 42, 50, 62]. Before the instability occurs, the tip interacts during an individual oscillation exclusively with the attractive part of the tip–sample force. After jumping to the higher branch, however, the tip senses also the repulsive part of the tip–sample interaction.

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10 bistable regime amplitude (nm)

8 6 AM-mode (Q0 = 300)

4

A(d)-curve approach retraction

2 0

0

2

4 6 8 distance d (nm)

10

12

Fig. 3.4 Amplitude vs. distance curves for conventional AM mode (“tapping mode”) with the parameters A0 = 10 nm, f0 = 300 kHz, and a tip–sample interaction force as given in Fig. 3.2. The dashed lines represent the analytical result, while the symbols are obtained from the numerical solution of the equation of motion (3.1). The overall amplitudes decrease during an approach towards the sample surface, but instabilities (indicated by arrows) occur during approach and retraction leading to a hysteresis

Our analysis dealt only with the case where the cantilever is driven exactly at resonance frequency, as this leads to clearly defined imaging conditions. In practice, however, experimentalists often drive the cantilever not exactly at, but somewhere else within the resonance peak (“detuned”), hoping for benefits while imaging the sample surface (see [43, 46, 59] for a more detailed discussion of this issue).

Measuring the Tip–Sample Interaction Force In the above subsections, we have outlined the influence of the tip–sample interaction on the cantilever oscillation based on the assumption of a specific model force. However, in practical imaging, the tip–sample interaction is not a priori known. In contrast, the ability to measure the continuous tip–sample interaction force as a function of the tip–sample distance would add a tool of great value to the force microscopist’s toolbox. Since the cantilever reacts to the interaction between tip and sample in a highly nonlinear way, one might wonder how that could be done. Surprisingly, despite the over 15 years that AM-AFM is used, it was only recently that solutions to this inversion problem have been suggested [63–65]. We start our analysis by applying the transformation D = d − A to the integral I+ (3.14a), where D corresponds to the nearest tip–sample distance as defined in Fig. 3.1. Next, we note that due to the cantilever oscillation, the current method intrinsically recovers the values of the force that the tip experiences at its lower turning point, where F↓ necessarily equals F↑. We thus define Fts = ( F↓ + F↑ ) / 2 , and (3.14a) subsequently reads as

3 Dynamic Force Microscopy and Spectroscopy in Ambient Conditions

I+ =

2 p cz A2

D+2 A

∫

Fts

D

z−D−A A − ( z − D − A)2 2

83

dz .

(3.18)

The amplitudes commonly used in AM mode are considerably larger than the interaction range of the tip–sample force. Consequently, tip–sample forces in the integration range between D + A and D + 2A are practicably insignificant. For this so-called “large amplitude approximation” [66, 67], the last term can be expanded at z ® D to (z − D − A) / A2 − ( z − D − A)2 ≈ − A / 2( z − D) , resulting in

I+ ≈ −

2 p cz A3/ 2

D+2 A

∫

D

Fts z−D

dz .

(3.19)

Introducing this equation into (3.13a), we obtain the following integral equation:

cz A3/ 2  ad cos(f ) f02 − fd2  1 −  = A f02  p 2    

D+2 A

∫

D

Fts z−D

dz .

(3.20)

κ

The left hand side of this equation contains only experimentally accessible data, and we denote this term as k. The benefit of these transformations is that the integral equation can be inverted [67] and, as a final result, we find

Fts ( D ) = −

∂ ∂D

D+2 A

∫

D

k (z) z−D

dz .

(3.21)

It is now straightforward to recover the tip–sample force using (3.21) from a “spectroscopy experiment,” i.e., an experiment where the amplitude and the phase are continuously measured as a function of the actual tip–sample distance D = d − A at a fixed location. With this input, one first calculates k as a function of D. In a second step, the tip–sample force is computed solving the integral in (3.21) numerically. Additional information about the tip–sample interaction can be obtained noticing that the integral I− is directly connected to the energy dissipation DE. By simply combining (3.13b) and (3.14b), we get

 1 fd aexc  DE =  + sin φ p cz A2 . Q f A  0 0 

(3.22)

The same result has been found earlier by Cleveland et al. [31] using the conservation of energy principle. However, exceeding Cleveland’s work, we suggest to plot the energy dissipation as a function of the nearest tip–sample distance

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D = d − A in order to have the same scaling as for the tip–sample force (see Fig. 3.5d). A verification of the algorithm is shown in Fig. 3.5, where we present computer simulations of the method by calculating numerical solutions of the equation of motion with a fourth-order Runge-Kutta method [68]. In order to be able to check both (3.21) and (3.22), we need to add a dissipative component to our original model interaction force Fair (cf. (3.10)). Instead of exploring elaborate energy dissipation mechanisms, it is sufficient for principle demonstration to simply add an additional dissipative force term Fdiss, i.e., Fts = Fair + Fdiss. To characterize Fdiss,

phase (degree)

b

c

−40 −50 −60 −70 −80 −90 −100 −110 −120

8 force (nN)

10 9 8 7 6 5

d

6 8 10 12 cantilever position (nm)

4

Fts reconstructed

0 −4 −0.5 0.0 0.5 1.0 1.5 2.0 tip-sample distance (nm)

6 8 10 12 cantilever position (nm)

dissipation (eV)

amplitude (nm)

a

40 30 20

∆E reconstructed

10 0 −0.5 0.0 0.5 1.0 1.5 2.0 tip-sample distance (nm)

kappa (Nm1/2x10−15)

e 40 20 0 −20 −40 −0.5 0.0 0.5 1.0 1.5 2.0 tip-sample distance (nm)

Fig. 3.5 A numerical verification of the proposed algorithm. Based on the equation of motion (3.1), the amplitude (a) and phase (b) vs. distance curves during the approach toward the sample surface have been numerically calculated. Both curves show the instability that is typical for AM-DFM operation in ambient conditions. As described in the text, the data is used for the reconstruction of the tip–sample force (c) and the energy dissipation (d). The assumed tip–sample model interactions according to (3.10) (c) and (3.23) (d) are plotted by solid lines. Finally, (e) reflects the k(D)-values that can be computed from the amplitude and phase values given in (a) and (b)

3 Dynamic Force Microscopy and Spectroscopy in Ambient Conditions

85

we chose viscous damping with an exponential distance-dependence: Fdiss = F0 exp(− z / ζ 0 ) z . The energy dissipation caused by this type of dissipation is given by [69]

 D + A  A  DE = 4p 2 fd AF0z 0 exp  − I1 ,  z 0   z 0 

(3.23)

where I1 is the modified Bessel function of the first kind. Figure 3.5 displays the resulting amplitude and phase vs. distance curves during approach, respectively. The assumed parameters and the conservative force are the same as already given in section “Mapping of Tip-Sample Interactions in Ambient Conditions” while the following parameters have been used for the dissipative force: F0 = 10−6 Ns/m and z0 = 0. 5 nm. Again the amplitude curve shows the previously discussed discontinuity caused by an instability. The subsequent reconstruction of Fts and DE based on the data provided by the amplitude and phase vs. distance curves is presented in Fig. 3.5c, d. The assumed tip–sample force and energy dissipation are plotted by solid lines, while the reconstructed data is indicated by symbols; the excellent agreement demonstrates the reliability of the method. Nonetheless, it is important to recognize that the often observed instability in amplitude and phase vs. distance curves affects the reconstruction of the tip–sample force. If such an instability occurs, experimentally accessible k(D)-values will feature a “gap” at a specific range of tip–sample distances D. This issue is illustrated in Fig. 3.5e, where the gap is indicated by an arrow. As a consequence, one might be tempted to interpolate the missing k-values in the gap. This is a workable solution if, as in our example, the accessible k-values appear smooth and, in particular, the lower turning point of the k(D)-values is clearly visible. In most realistic cases, however, the k(D)-values are unlikely to look so smooth as in our simulation and/or the lower turning point might not be reached, and we thus advise to uttermost caution in applying any inter- or extrapolation. Finally, let us note two more issues: (1) The reconstruction of the energy dissipation does not require the continuous knowledge of k-values. Thus, it is not influenced by the instability and gives reliable values also after the jump Fig. 3.5d. (2) The “large amplitude approximation” is not a prerequisite for the inversion of the tip–sample forces from the amplitude and phase data. The application of other numerical methods where the amplitude is not restricted to large values is described in [64, 65, 70].

Theory of FM Mode Including Tip–Sample Forces Now we focus on the solution of (3.1) for the FM mode. In this case, the cantilever is self-oscillated by the feedback loop described in section “Formulation of the Problem and Basic Equation of Motion.” As before, we make the ansatz (3.6) assuming a sinusoidal cantilever oscillation. Consequently, we use the same mathematical

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treatment as in section “Theory of AM Mode Including Tip–Sample Forces” and develop the tip–sample force into a Fourier series. This leads to a set of two coupled trigonometric equations

aexc f2 − f2 cos(2p ft0 ) = 0 2 − I + (d , A) , A f0

(3.24a)

aexc 1 f + I - (d , A) , sin(2p ft0 ) = A Q f0

(3.24b)

which again contain the two integrals already defined in (3.14a) and (3.14b). Both equations can be simplified for the conditions typically found in DFM experiments where the FM mode is applied. First, we assume that the frequency shift ∆f := f − f 0 caused by the tip–sample interaction and the damping is small compared to the resonance frequency of the free cantilever ( Þ f ∕ f0 » 1 and f 2 − f 02 ≈ −2∆ff 0 ). Second, we consider that the phase shift is typically set to 90 ˚ in the FM mode. In this case the terms on the left side are given by cos(2pft0) = 0 and sin(2pft0) = 1. Due to these simplifications, the frequency shift and the driving amplitude can be calculated from

D f (d, A) = −

f0 f0 I+ = − 2 p cz A2 aexc =

d+ A

∫

Fts

d−A

z−d A − ( z − d )2 2

A DE (d , A) + . Q p cz A

dz ,

(3.25a)

(3.25b)

It is worthwhile to note that we got an explicit equation for the frequency shift while we got implicit equations (3.15a) and (3.15b) for the AM mode. Therefore, the frequency shift is continuous and shows no jumps caused by instabilities during approach and retraction. Both equations are valid for every type of interaction as long as the resulting cantilever oscillations are nearly sinusoidal. Equation (3.25a) coincides with the well-known result for the FM mode with constant-oscillation amplitude (see, e.g., [66, 67, 71]), but it is coupled with (3.25b) through the oscillation amplitude. Nonetheless, as shown previously for the AM mode, we can invert (3.25a) using the same mathematical treatment. For the inversion we apply again the “large amplitude approximation” for the integral I+ (3.21) and get

D f ( D, A) ≈ −

D+2 A

f0 2p cz A

3/ 2

∫

D

Fts ( z) z−D

dz .

(3.26)

The inversion now leads to the following formula for the tip–sample interaction potential [67, 72]

3 Dynamic Force Microscopy and Spectroscopy in Ambient Conditions

Vts ( D ) = − 2

D+2 A

∫

D

cz A( z ′ )3/ 2 D f ( z ′ ) dz′ . f0 z′ − D

87

(3.27)

Consequently, the tip–sample force is given by [67, 72]

Fts ( D) = 2

∂ ∂D

D+2 A

∫

D

Dg ( z ′) z′ − D

dz ′ ,

(3.28)

where we defined the so-called normalized frequency shift [66]

g ( D) :=

cz A3/ 2 D f ( D) , f0

(3.29)

which is independent on the oscillation amplitude, but a function of the nearest tip–sample distance D. In order to recover the tip–sample interaction force from a spectroscopy experiment, we have to measure the frequency shift and the driving amplitude as a function of the tip–sample distance before we calculate the normalized frequency shift. After that, we introduce this data into (3.28). The calculation of the energy dissipation DE is straightforward using (3.25b)

 A2  DE ( A) ≅ p cz  Aaexc −  . Q 

(3.30)

This equation is equivalent to (3.22) and follows also from the conservation of energy [31, 69].

Mapping of Tip–Sample Interactions in Ambient Conditions Experimental Comparison of the AM- and FM mode We realized the set-up presented in Fig. 3.1 by combining a commercial dynamic force microscope (MultiMode AFM with NanoScope IIIa Controller, Veeco Instru ments Inc.) with an additional electronics dedicated for the constant-excitation mode (QFM-Module, nanoAnalytics GmbH). In this way, it is easily possible to switch between AM and FM mode using the same cantilever and sample. Rectangular silicon cantilevers (provided by Nanosensors) were used as sensors. Their spring constants cz were determined via the resonant frequency f0 of the freely oscillating cantilever [73] and their quality factors Q were measured from resonance curves [74].

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To illustrate the main differences between the “conventional” AM mode and the presently much less used FM mode in air, we present two “spectroscopy experiments” in Fig. 3.6, where the oscillating cantilever was approached to and retracted from a mica surface in both modes. The corresponding “spectroscopy curves” are presented in Fig. 3.6a, b. The measured quantities in the AM mode are amplitude and phase shift, whereas the amplitude and frequency shift are recorded in FM mode. The parameters of the cantilever were f0 = 167 224 Hz, cz = 37. 5 N/m, and Q = 465. As highlighted before (see section “Theory of AM Mode Including Tip–Sample Forces”), the amplitude and phase shift curves recorded in AM mode show a significant hysteresis during approach and retraction. At specific positions (marked by arrows in Fig. 3.6a), the oscillation becomes unstable and the cantilever jumps into another stable oscillation state. However, such a hysteresis is not present in the spectroscopy curves measured in the FM mode due to the specific self-oscillation technique [11]. As shown in Fig. 3.6b, the particular amplitude and frequency shift curves are identical within the noise limit for approach and retraction. The amplitude is constant until the tip senses the interaction with the sample surface and decreases continuously during further approach. The frequency shift curves show a decrease and increase of the resonant frequency with a distinct minimum. As detailed in section “Theory of FM Mode Including Tip–Sample Forces” the continuous approach and retraction curves of the frequency modulation mode allow

20 10 0 −10 −20 −30 −40

0

2

4

6

8 10 12 14

piezo movement (nm)

20 19 18 17

approach retraction

16 15 0

0

−20 −40 −60

approach retraction

4

−50 −100 −150 −200

potential (eV)

approach retraction

c

21

force (nN)

b

20 19 18 17 16 15 14 13

frequency shift (Hz) amplitude (nm)

phase (degree)

amplitude (nm)

a

8.5 N/m

2 0 −2 −4 −6

0

2

4

6

8 10 12 14

piezo movement (nm)

−8

-7 nN

0

2

4

6

8

rel. distance (nm)

Fig. 3.6 Examples of “spectroscopy measurements” obtained on mica in ambient conditions. (a) Amplitude and phase vs. distance curves in the tapping mode. The instabilities during approach and retraction cause a hysteresis. (b) Such a behavior is not observed in the constant-excitation mode where the approach and retraction curves of the amplitude and frequency shift are identical within the noise limit. (c) Using the algorithm described in the text, we reconstruct the tip–sample potential and force from the data sets shown in (b). The interaction force decreases until it reaches a minimum of − 7 nN, and increases again with a slope of 8.5 N/m. The origin of all x-axes has been arbitrarily set to the left of the graphs

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the reconstruction of the tip–sample interaction by an inversion algorithm. An application of this procedure to the spectroscopy data is plotted in Fig. 3.6c and reveals the tip–sample potential and force. Both the tip–sample potential as well as the force show a distinct minimum of − 70 eV and − 7 nN, respectively. The minimum of the force curve is the minimal force needed to retract the tip from the sample surface. Therefore, we denote it as adhesion force in the following. During further approach the tip–sample force increases with a slope of 8.5 N/m as shown by a linear fit (solid line). This linear increase in the tip–sample force is caused by the contact of tip and sample. Justified by the almost linear increase, we use the term contact stiffness for the slope obtained by this linear fitting procedure.

Mapping the Tip–Sample Interactions on Biological Samples The frequency-modulation technique can also be used to record the tip–sample interaction as contour maps perpendicular to the sample surface [75]. In order to examine the possible resolution of this approach under ambient conditions, we recorded sets of spectroscopy curves along predefined scan lines on DPPC (l-adipalmitoyl-phosphatidycholine, Fluka), which frequently serves as a model for membranes [76]. The tip–sample interaction was subsequently calculated from the measured amplitude and frequency shift vs. distance curves with respect to the actual scan position. The obtained curves were then plotted in a color-coded contour map showing the potential of the tip–sample interaction. Monolayers of DPPC were prepared with the Langmuir–Blodgett technique. As shown in the topography image in Fig. 3.7, the monolayers have a lateral structure of alternating stripes and channels. This specific structure is obtained by rapidly withdrawing the mica substrate at a low monolayer surface pressure and constant temperature as described by Gleiche et al. [77]. The stripes consist of DPPC in a liquid condensed phase (LC-phase), whereas the channels between the stripes are filled with DPPC in the liquid expanded phase (LE-phase) [78]. The lateral periodicity of stripes and channels depends on the parameters used during the preparation of the sample. We imaged the sample using the oscillation amplitude as a feedback signal in the FM mode using constant-excitation before we recorded 50 spectroscopy curves along a predefined direction marked in Fig. 3.7a. The parameters of the cantilever were f0 = 170 460 Hz, cz = 39. 6 N/m, and Q = 492. All data sets were than transformed into tip–sample potential curves using the mathematical method described in section “Theory of FM Mode Including Tip–Sample Forces.” Finally, we computed the corresponding contour map as shown in Fig. 3.7. The complete procedure was done by a computer script using IGOR Pro software (Wavemetrics Inc.). The resulting color-coded image reveals the different tip–sample interaction on the stripes (LC-phase) and in the channels (LE-phase). The potential is significantly larger above the stripes ( » − 100 eV) compared to the channels ( » − 150 eV), as it can be seen by the color coding in Fig. 3.7.

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a

12 10

0

8

−50

6

−100

4

−150

potential (eV)

rel. distance (nm)

b

2 0 adhesion force (nN)

c −4 −5

LC phase

−6 −7 LE phase

−8 −9

0

2

4 6 8 10 12 14 scan position (µm)

Fig. 3.7 (a) Surface plot (scan size: 14 × 14 mm2) of the topography of the DPPC film prepared by the Langmuir–Blogdett technique. The monolayer shows alternating stripes and channels which consists of DPPC adsorbed in the LC- and LE-phase, respectively. The white line marks the position where we recorded the frequency shift and amplitude vs. distance curves for the construction of the contour map of the tip–sample interaction potential shown in (b). The graph in (c) displays the corresponding adhesion force obtained from the data shown in (b)

The local stiffness and adhesion force can be determined from the force curves plotted in Fig. 3.8. Here we show amplitude and frequency shift vs. distance curves and the resulting tip–sample potential and force measured at the positions marked in Fig. 3.7. The force curves reveal an adhesion force of − 6 and − 7. 5 nN on the stripes and channels, respectively. However, the local stiffness is about 9 N/m for both positions, and we could not determine a significant difference in the stiffness between the two phases of the DPPC monolayers. We attribute this outcome to the

3 Dynamic Force Microscopy and Spectroscopy in Ambient Conditions

b

a

150

8 6 4 2 0 −2 −4 −6 −8

100

0

35 force (nN)

34 33 32

LC phase LE phase

31

50

potenial (eV)

amplitude (nm)

36

frequency shift (Hz)

91

0 −50

9 N/m

LC phase LE phase

−6 nN −7.5 nN

−50 −100 −150

−100 0

5

10

15

piezo movement (nm)

20

0

2

4 6 8 10 rel. distance (nm)

12

Fig. 3.8 (a) Spectroscopy curves obtained on the stripes (circles) and the channels (crosses). The tip–sample interaction is calculated from this amplitude and frequency shift vs. distance curves. (b) Using the numerical procedure described in the text we calculated the corresponding tip–sample interaction force and potential. A significant difference between the curves is observed for distances between 2 and 6 nm

fact that the repulsive interaction forces for thin film depend strongly on the actual film thickness. This effect has been already examined by Domke and Radmacher [79] for thin polymer films. They observed that the underlying stiff substrate influences the measured forces for film thicknesses up to hundreds of nanometer. Consequently, it is evident that the contact stiffness measured on the two phases of the DPPC monolayer is dominated by the mica substrate.

Conclusion In this article, we reviewed the theoretical framework of amplitude-modulation and frequency-modulation DFM. First, we clarified the differences between both modes by analyzing their cantilever driving mechanisms. Subsequently, we explicitly included the nonlinear tip–sample forces into our analysis and developed a theory for AM-DFM. We then introduced a spectroscopy method for the measurement of conservative and dissipative tip–sample interactions using the AM mode. In a next step, we used the same mathematical treatment to get a theory describing the FM mode with constant-excitation.

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Finally, we presented some exemplary applications of the introduced methods for ambient conditions. A comparison between spectroscopy curves measured in AM and FM mode demonstrates that FM mode operation avoids instabilities as they are typically present in the amplitude and phase curves in the AM mode. Thus, it is possible to reconstructed the tip–sample interaction over the complete interaction range without jumps. The mapping technique enabling the measurement of two-dimensional “contour maps” of the tip–sample interaction potential was introduced by the example of a DPPC film. Acknowledgments The authors would like to thank X. Chen and L. Chi for the preparation of the sample used in section “Mapping the Tip–Sample Interactions on Biological Samples.” Furthermore, we acknowledge support from and many useful discussions with Boris Anczykowski and Marcus Schäfer (nanoAnalytics GmbH) as well as Daniel Ebeling, André Schirmeisen, and Harald Fuchs (University of Münster). U. S. acknowledges financial support from the National Science Foundation (grant No. MRSEC DMR 0520495).

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Chapter 4

Measuring Mechanical Properties on the Nanoscale with Contact Resonance Force Microscopy Methods* D.C. Hurley

Introduction Nanomechanics, the study of mechanical properties at the nanoscale, provides one of the scientific underpinnings to the rapidly expanding field of nanotechnology. In many systems, the mechanical properties of materials deviate from their macroscopic behavior when relevant length scales reach the nanoscale. Such changes often result from the greatly increased surface-to-volume ratio. As rapidly as materials with novel nanomechanical properties are discovered, they are developed for new products and applications. For example, structural components made with nanocomposites can be lighter, stronger, and tougher; nanostructured coatings possess superior resistance to scratch and abrasion; and ceramic nanomaterials may provide greater flexibility or improved insulating properties than their bulk counterparts. However, shrinking length scales mean that mechanical property information is now needed with nanoscale spatial resolution. For instance, accurate property data enable predictive modeling of complex systems in order to reduce the time and cost of development. In addition, it is increasingly necessary to image or visualize the spatial distribution in properties, rather than relying on a single “average” value. One reason is that new systems frequently involve several disparate materials integrated on the micro- or nanoscale. For these systems, nanoscale information is essential to differentiate the properties of the various components. Furthermore, failure in such heterogeneous systems often occurs due to a localized variation or divergence in properties (void formation, fracture, etc.). Engineering new complex systems therefore requires quantitative nanomechanical imaging to better predict reliability and performance. A variety of techniques to measure small-scale mechanical properties have been demonstrated. Among these are approaches based on indentation [1–3], ultrasonics [4, 5], and other physical phenomena [6, 7]. However, these methods are often not

*

Contribution of NIST, an agency of the US government; not subject to copyright.

D.C. Hurley (*) National Institute of Standards & Technology, 325 Broadway, Boulder, CO 80305, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_4, © Springer Science+Business Media, LLC 2010

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optimal in some way, particularly in their spatial resolution. Other drawbacks to some methods include lack of imaging capability and constraints regarding specimen size or shape. For these reasons, methods based on atomic force microscopy (AFM) present an attractive alternative. AFM was originally invented to measure surface topography with atomic spatial resolution [8], but its utility in other applications was quickly demonstrated. One reason for AFM’s appeal is its ability to provide nanoscale spatial resolution, due to the small radius (~5–50nm) of the cantilever tip. Furthermore, the scanning capabilities of the AFM instrument enable rapid, in situ imaging. Several AFM-based methods have been demonstrated to sense mechanical properties such as elasticity [9–12]. In force modulation microscopy [9], either the tip or the sample is oscillated and the resulting amplitude of the cantilever deflection is detected as the tip is scanned. Because the deflection amplitude depends on the sample’s local modulus, an image of elastic stiffness can be obtained. However, such images are qualitative, indicating only relative stiffness at each position. Methods based on force–displacement curves have also been extensively developed (for a review, see Ref. [13]). Force–displacement techniques work best when the compliance of the cantilever is roughly comparable to that of the test material. They are therefore better suited to very compliant (“soft”) materials, and lose effectiveness as the material stiffness increases. Although force–distance curves can provide quantitative information, the point-by-point nature of this method severely restricts its imaging abilities. The most promising AFM methods for quantitative measurements of relatively stiff materials are dynamic approaches, in which the cantilever is vibrated at or near its resonant frequencies [14]. Such methods often include the words “acoustic” or “ultrasonic” in their name, identifying the frequency of vibration involved (~100kHz to 3MHz). Among these methods are ultrasonic force microscopy (UFM) [15, 16], heterodyne force microscopy [17], ultrasonic atomic force microscopy (UAFM) [18], and atomic force acoustic microscopy (AFAM) [19, 20]. Because AFM was a relatively new field when these techniques were first devised, research focused on exploiting the features of commercial AFM instruments. Very recently, new techniques are emerging that involve customized sensors to replace the standard AFM cantilever [21, 22]. These methods leverage advances made over the last decade both in micromachining and in AFM instrumentation. Of these methods, the AFAM approach has achieved the most progress in quantitative measurements. A general name that encompasses AFAM, UAFM, and related methods is “contact resonance AFM” or simply “contact resonance force microscopy” (CR-FM) [23]. In this chapter, we describe how CR-FM methods can be used for quantitative measurements and imaging of nanoscale mechanical properties. We present the basic physical concepts and explain how they are implemented experimentally. Single-point methods to measure both elastic and viscoelastic properties are described, as well as quantitative imaging techniques to obtain property maps. Results are shown for specific material systems in order to demonstrate the potential of the methods for materials characterization. In this way, we hope to encourage further use of CR-FM as a tool for research and development in nanotechnology.

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Single-Point Measurements of Elastic Modulus Basic Concepts Cantilevers used in CR-FM experiments are micromachined from single-crystal silicon (Si) and have a rectangular shape. The long axis of the cantilever is usually oriented in the <110> crystalline direction, and the axis of the tip is <001>-oriented. For measurements on stiff materials (modulus greater than ~50GPa), cantilevers with a spring constant klever of approximately 30–50N/m are typically used. A variety of such cantilevers are available commercially and are usually designated as “noncontact” or “intermittent contact” probes. As described below, experiments are performed with a static force FN applied to the tip. FN is determined through the relation FN = kleverd, where d is the deflection of the cantilever measured by the AFM photodiode. In experiments on stiff materials, FN is typically in the range of several hundred nanonewtons to a few micronewtons. Such forces ensure that the tip–sample contact is predominantly elastic. In recent experiments on more compliant materials such as polymers, we have used more compliant cantilevers with spring constants of approximately 1N/m, with corresponding values of FN less than 100nN. An essential concept of CR-FM methods is that the cantilever can be considered as a microscale beam possessing mechanical or vibrational resonant modes. This idea is illustrated in Fig. 4.1. The resonant modes of the cantilever are excited either by an external actuator, as shown in the figure, or by an actuator attached to the cantilever holder. In Fig. 4.1a, the tip of the cantilever is out of contact with the sample. In this case, the resonant modes of the cantilever occur at the free or natural frequencies that depend on its geometry and material properties. When the tip is placed in contact with a sample, as in Fig. 4.1b, the resonant frequencies increase due to the interaction forces between the tip and the sample. Typically, experiments involve the lowest-order flexural (bending) resonant modes of the cantilever. For cantilevers such as those described above, the frequencies of the flexural modes

b

c

specimen

specimen

piezoelectric actuator

piezoelectric actuator

amplitude

a

2nd free resonance 1stc ontact 1st free resonance resonance

frequenc y

Fig. 4.1 Concepts of contact resonance force microscopy (CR-FM). Resonant modes of the cantilever are excited when the tip is (a) in free space and (b) in contact with a specimen under an applied static force. (c) Resonant spectra. The lowest-order contact resonance occurs at a higher frequency than the first free-space resonance, but is lower than the second free-space resonance

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occur at a few hundred kilohertz for the tip in free space and a few megahertz for the tip in contact. Instead of monitoring the amplitude or phase of the cantilever motion, as in some other AFM methods, CR-FM measurements consist of determining the frequencies at which the free and contact resonances occur. The mechanical properties of the sample are then deduced from the measured frequencies with the help of two models: one for the dynamic motion of the cantilever, and another for the contact mechanics between the tip and the sample. A practical advantage to this approach is that frequency shifts are easier to measure accurately than absolute amplitudes or phase.

Models for Data Analysis In the data analysis, the measured frequencies are first related to the tip–sample interaction force by means of a model for the dynamic motion of the cantilever. Here, we present the basic equations for an analytical model. This model is described in detail in Ref. [20]. Finite-element analysis methods have also been used to describe the cantilever dynamics [24–26]. Figure 4.2 shows several models to describe the cantilever’s vibrations. Note that a distributed-mass model is used; the point-mass (harmonic oscillator) approximation is not accurate under the conditions described here [27, 28]. The cantilever is modeled as an elastically isotropic beam of uniform cross section with length L, width w, thickness t, density r, and Young’s modulus E. The tip is located at a distance L1 < L from the clamped end of the cantilever. The remaining distance to the free end of the cantilever is L¢ so that L = L1+L¢. The flexural spring constant of the cantilever is klever = Et3w/(4L13). Figure 4.2a describes the cantilever when the tip is out of contact. The frequency fn0 of the nth free flexural resonance is related to its wavenumber xn0 by [20]

(x L) = 4πf 0 n

2

0 n

L2 τ

3r 2 = f n0 (cB L ) . E

(4.1)

However, xn0L is also a root of

1 + cos xn0 L cosh xn0 L = 0.

(4.2)

The first three roots of (4.2) are [x10L, x20L, x30L]=[1.8751, 4.6941, 7.8548]. Therefore, experimental values for the cantilever parameter cBL for each mode can be calculated from the measured values of the free frequencies fn0, and cantilever properties such as L and E do not need to be determined directly. Figure 4.2b depicts the simplest model for cantilever dynamics if the tip is in contact. In this case, the tip–sample interaction is entirely elastic and acts in a direction normal (vertical) to the sample surface. The tip–sample interaction is represented by a spring with spring constant k, also known as the contact stiffness.

4 Measuring Mechanical Properties on the Nanoscale

a

b

c

L L1

99

L L'

L'

L1

α

k

L1

d

L L' h

L L1

L'

κ k

k

σ

Fig. 4.2 Models for cantilever dynamics. (a) Tip out of contact. (b) Tip in contact, with only normal (vertical) elastic forces included. (c) Tip in contact, with both normal and tangential elastic forces included. (d) Tip in contact, with normal elastic and dissipative (damping) forces included

By considering the dynamics of this system [27], it is found that the normalized contact stiffness k/klever can be expressed in terms of the wavenumber xn of the nth flexural contact resonance and the relative tip position ratio g=L1/L as

k

klever

=

2 3 (1 + cos xn L cosh xn L ) , (xn Lγ ) 3 D

(4.3)

where D = sin xn L(1 − γ ) cosh xn L(1 − γ ) − cos xn L (1 − γ )sinh xn L (1 − γ ) [1 − cos xn Lγ cosh xn Lγ ] − [sin xn Lγ cosh xn Lγ − cos xn Lγ sinh xn Lγ ][1 + cos xn L(1 − γ ) cosh xn L(1 − γ ) ].

Equation (4.3) is used to calculate k/klever for each measured contact resonance frequency fn. First, the values of fn are used to calculate xnL:

xn L = cB L f n = xn0 L

fn , f n0

(4.4)

where (4.1) has been used. From the experimental values of xnL, (4.4) is then used to plot k/klever as a function of the relative tip position g for each mode n. Typically, the values of k/klever are the same for two modes at only one physically realistic value of g. The value of k/klever where the two modes intersect or “cross” is taken as the solution. After values of the normalized contact stiffness k/klever have been calculated from the experimental frequency data, the elastic properties of the sample can be determined. This step of the analysis requires a second model, namely a model for the contact mechanics between the tip and the sample. A complete discussion of contact mechanics is given elsewhere [29]. In many respects, this analysis is similar to that used for instrumented (nano-) indentation methods [1, 30]. Figure 4.3 depicts the parameters of the contact mechanics model. The indentation moduli of the tip and the sample are Mtip and Ms, respectively. Figure 4.3a

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D.C. Hurley

a

b

FN

R

MS

FN

M tip

M tip

a

MS

a

Fig. 4.3 Schematics for contact mechanics: (a) Hertzian contact between a hemispherical tip and a flat sample and (b) flat-punch contact between a flat tip and a flat sample

represents Hertzian contact between a hemispherical tip with radius of curvature R and a flat sample. A vertical (normal) static load FN is applied to the tip. For isotropic materials, the resulting contact between the tip and the sample is a circle with radius a. Figure 4.3b shows flat-punch contact between a flat tip and a flat sample. In this case, the contact area depends on the tip radius a. In either case, the normal contact stiffness k between the tip and the sample can be expressed as

k = 2aE * ,

(4.5)

where E* is the reduced system modulus between the tip and the sample:

1 1 1 = + . * E M tip M s

(4.6)

For elastically isotropic materials, the indentation or plane strain modulus M=E/ (1–n2), where E is Young’s modulus E and n is Poisson’s ratio. Then

2 1 1 − ν tip 1 − νs2 = + . E* Etip Es

(4.7)

Above, we discussed how the contact stiffness k is determined from the contact resonance frequencies. To determine E* from (4.5), it is necessary to know the contact radius a in addition to k. For a flat indenter, a is constant; for Hertzian contact, it is given by

a=

3

3RFN . 4E*

(4.8)

In principle, one could measure a directly (for a flat punch) or determine R and FN for a given experiment and calculate a. Equations (4.5) or (4.8) could then be used to determine E*. In actual measurements, it is more practical to use a referencing or

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101

comparison approach [25, 31]. This approach is described in more detail in the next section.

Experimental Techniques CR-FM experiments are performed with an apparatus such as the one shown schematically in Fig.4.4a. As seen in the figure, the apparatus contains a commercially available AFM instrument and a few other off-the-shelf components. The typical frequency range involved (~100kHz to 3MHz) means that access to the unfiltered (high-frequency) photodiode signal of the AFM instrument is required. The specimen is bonded to a piezoelectric actuator mounted on the translation stage of the AFM instrument. The actuator is excited with a continuous sine wave voltage by a function generator. The excitation voltage is kept sufficiently low that the tip remains in contact with the sample. This ensures a linear interaction between the tip and sample. The amplitude of the cantilever deflection is monitored by the AFM’s internal position-sensitive photodiode with a laser beam-bounce technique. A lock-in amplifier is used to isolate the component of the photodiode signal at the excitation frequency. By sweeping the transducer excitation frequency and recording the output signal of the lock-in amplifier, a spectrum of the cantilever response versus frequency can be obtained. In CR-FM measurements of modulus at a single sample position, this apparatus is used to acquire frequency spectra for two different resonant modes. From the measured contact resonance frequencies, values for the contact stiffness k and the tip position L1/L are determined as described above. Spectra are obtained on two samples in alternation: (1) the test or unknown sample and (2) a reference or calibration specimen whose elastic properties are known. Elastic properties of reference specimens can be determined by various means, including pulse-echo ultrasonics [32] and instrumented (nano-) indentation [1, 30]. For accurate measurements, the elastic properties of the reference specimen should be similar to those of the test specimen [25, 31, 33]. If measurements are performed on the test (subscript s) and reference (subscript ref) samples at the same values of FN, it can be shown [31] that m

 k  E =E  s  ,  kref  * s

* ref

(4.9)

where m=3/2 for Hertzian contact and m=1 for a flat punch. The indentation modulus Ms of the test specimen is then determined from Es* with (4.6) and knowledge of Mtip. This approach eliminates the need for precise values of R, FN, and a, which are difficult to determine accurately. Because the true shape of the tip is usually intermediate between a hemisphere and a flat [34], the values calculated with m=3/2 and m=1 set upper and lower limits on M.

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a

commercial AFM feedback loop & topography

unfiltered photodiode signal

laser

cantilever

photodiode detector sample piezoelectric actuator

ref

function generator

lock-in amplifier computer CR-FM electronics

b

commercial AFM feedback loop & topography

unfiltered photodiode signal

laser

cantilever

photodiode detector x-y scanning

sample piezoelectric actuator

sweep VCO

DSP

center D/A converter

A/D converter

RMS to DC

frequency-tracking electronics to AFM image acquisition

computer

Fig. 4.4 Schematics of CR-FM experimental apparatus. (a) Apparatus for stationary point measurements and qualitative imaging. (b) Frequency-tracking apparatus for quantitative imaging

In a typical experiment, measurements are made at several sample locations. At each location, data are acquired at several different values of the applied static force FN. Measurements are performed on the reference material immediately before and after each set of measurements on the unknown material. Multiple data sets are obtained by analyzing each set of measurements on the unknown sample with both sets of reference

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103

measurements using (4.9). This procedure also reduces the effects of tip wear [23]. In a typical experiment, approximately 20–40 individual values for the indentation modulus Ms of the unknown sample are obtained with this approach. Combining all of the data yields a single average value and standard uncertainty for Ms.

Results for Elastic Modulus CR-FM measurements of the indentation modulus M for several materials are shown in Fig.4.5 [25, 35]. The samples consisted of supported thin films that ranged in thickness from a few hundred nanometers to a few micrometers, with modulus values from about 50GPa to over 200GPa. Also shown in Fig.4.5 are the values of M obtained by instrumented (nano-) indentation (NI) and surface acoustic wave spectroscopy (SAWS) for the same samples. NI is typically destructive to the sample and has poorer spatial resolution than AFM methods, but is widely used in industry. The SAWS method [4] is nondestructive and is only now emerging from the research laboratory into industrial use. The values obtained by SAWS represent the average properties over a few square centimeters of the surface. The figure shows that the values obtained by the various methods are in good agreement, with differences of less than 10% for all samples. Such agreement establishes confidence in CR-FM methods for quantitative modulus measurements. Many more materials than those shown in Fig.4.5 have been evaluated with CR-FM methods. More detailed reviews of results can be found elsewhere [20, 23]. Applications to bulk or macroscopic systems include semiconductors

modulus M (GPa)

250

CR-FM Ni

200 150 100

NI SAWS

Nb Al

a -Si1-xCx:H

SiO:F

50 0

0.28 ± 0.03 0.77 ± 0.01 1.08 ± 0.01 1.09 ± 0.02 3.08 ± 0.01 film thickness t (µm)

Fig. 4.5 Indentation modulus M of thin supported films obtained by contact resonance force microscopy (CR-FM), instrumented (nano-) indentation (NI), and surface acoustic wave spectroscopy (SAWS). The thickness t of each film was determined by cross-sectional SEM analysis or by stylus profilometry methods. Film materials include fluorinated silica glass (FSG), amorphous hydrogenated silicon carbide (a-Si1-xCx:H), aluminum (Al), niobium (Nb), and nickel (Ni). The error bars represent 1 standard deviation of the individual measurements

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(InP and GaAs [36]), WC-Co cermets [37], piezoelectrics (PZT [31]), and temperature studies of martensitic phase transformations in nickel-titanium alloys [38]. The spatial resolution afforded by CR-FM has also enabled studies of micro- to nanoscale features such as piezoelectric domains and boundaries in BaTiO3 [39] and PZT [40]. Thin films are especially suited to characterization by CR-FM. Examples of films that have been studied include diamondlike carbon [41, 42], nanocrystalline ferrite with spinel structures [43], nanocrystalline SnSe [44], and nanocrystalline CrN [45]. Nanocrystalline effects as well as film thickness effects were examined in Ni films [46]. Various nanostructures have also been examined with CR-FM methods, including dickite clay booklets [47], polypyrrole polymer nanotubes [48], SnO2 nanobelts [49], epoxy/silica nanocomposites [50], and ZnO nanowires [51].

Beyond the Basics: Further CR-FM Techniques Measuring Shear Elastic Properties As discussed above, measurements of the cantilever’s flexural modes yield values of the modulus M. However, in isotropic materials, M depends on two distinct elastic quantities: Young’s modulus E and Poisson’s ratio n. In some cases, it may be preferable to know these individual elastic properties separately. For instance, properties such as n or the shear modulus G are valuable in understanding thin-film behavior [52]. It has been observed theoretically [53, 54] that torsional cantilever modes could assist in this matter. By simultaneously measuring the flexural and torsional contact resonance frequencies, it is possible to determine shear elastic properties such as n or G separately from E. Here, the physical and mathematical principles involved in such measurements are summarized. If lateral elastic forces are included, the cantilever model shown in Fig.4.2c is used. The cantilever is tilted by an angle a with respect to the sample surface. The tip has height h. The elastic interaction between the tip and the sample is represented by two springs: a vertical spring with stiffness k and a horizontal (tangential) spring with stiffness k. The vertical contact stiffness k normalized by the flexural cantilever stiffness klever is given by [55–57] the positive root of

k

klever

=

− B ± B 2 − 4 AC , 6A

(4.10)

where 2

 κ  h  A =     (1 − cos xn Lγ cosh xn Lγ )1 + cos xn L (1 − γ )cosh xn L (1 − γ ) ,  k   Lγ 

4 Measuring Mechanical Properties on the Nanoscale

105

B = B1 + B2 + B3 , C = 2(xn Lγ ) (1 + cos xn L cosh xn L ), 4

with  h B1 =    Lγ 

2

3

(xn Lγ )

 sin

2

α+

κ k

2  cos α



{[1 + cos xn L(1 − γ ) cosh xn L(1 − γ )][sin xn Lγ cosh xn Lγ + cos xn Lγ sinh xn Lγ ]

×

[

][

− 1 − cos xn Lγ cosh xn Lγ sin xn L (1 − γ ) cosh xn L (1 − γ ) + cos xn L (1 − γ ) sinh xn L (1 − γ )



B2 = 2  

2  κ  xn L γ )  (   k Lγ   h

]},

 

 

− 1 cos α sin α 

{[1 + cos xn L (1 − γ )cosh xn L (1 − γ )][sin xn Lγ sinh xn Lγ ] + (1 − cos xn L γ cosh xn L γ )[sin xn L (1 − γ )sinh xn L (1 − γ )]}, ×

(

B3 = xn Lγ ×

) cos2 α + k sin2 α κ

{[1 + cos xn L(1 − γ ) cosh xn L(1 − γ )][sin xn Lγ cosh xn Lγ − cos xn Lγ sinh xn Lγ ]

[

− 1 − cos xn Lγ cosh xn Lγ

][sin xn L (1 − γ ) cosh xn L(1 − γ ) − cos xn L(1 − γ ) sinh xn L (1 − γ )]}.

Considering the torsional motion of the cantilever, the tangential contact stiffness k is given by [54]

κ κlever

=−

yn L cos yn L , sin yn L γ cos yn L (1 − γ )

(4.11)

where yn is the torsional wavenumber and klever = wt3G/(3Lh2) is the lateral spring constant of the cantilever. In analogy to the flexural case, direct measurements of cantilever properties are avoided by relating the frequency tn of the nth torsional contact resonance tn to tn0, its corresponding free-space frequency. The result is an expression analogous to (4.4):

 (2n − 1)π  tn yn L =   0 . 2   tn

(4.12)

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The vertical and tangential contact stiffnesses are interpreted with a model for the tip–sample contact mechanics, in analogy to the discussion in section “Models for Data Analysis”. The tangential contact stiffness k is given by [54, 58]

κ = 8aG * ,

(4.13)

where the reduced modulus G* is given by

2 − n tip 2 − n s 1 = + . * G Gtip Gs

(4.14)

For isotropic materials, the shear modulus G is defined by G = E/(2+2n). We define a quantity N ≡ G/(2–n) so that

1 1 1 = + . * G N tip N s

(4.15)

Combining (4.5), (4.9), and (4.13) yields an expression for the reduced modulus Gs* of the unknown sample:

*  κ s  ks  Gs* = Gref     κ ref  kref 

m −1

.

(4.16)

With these relationships and their counterparts in the section “Single-Point Measurements of Elastic Modulus”, individual elastic properties can be determined from CR-FM experiments. In the experiments, the contact resonance frequencies of the flexural and torsional modes are measured for each sample position under the same experimental conditions. The frequencies fn of the flexural contact resonances are used to determine the normalized vertical contact stiffness k/klever. The measured torsional frequencies tn yield values for the normalized lateral contact stiffness k/klever. From the contact stiffnesses, values for E* and G* and subsequently for M and N are determined. Given M and N, it is straightforward to determine the individual elastic properties n = (M–4N)/(M–2N), E = M (1–n2) = (12M2N–4N2M)/ (M–2N)2, and G = ½M(1–n) = MN/(M–2N). The above theoretical concepts were demonstrated experimentally using materials with known properties [57]. A plate of fused silica (SiO2) was considered to be the reference material, and a disk of borosilicate crown glass was the unknown material. The “true” values of the materials’ elastic properties, based on pulse-echo ultrasonic measurements and literature values, are shown in the first two rows of Table 4.1. The cantilever used in these experiments had a nominal flexural spring constant klever = 3.5±2N/m. This cantilever was chosen because it provided more sensitivity to changes in the torsional frequency with contact stiffness than the stiffer (klever ~ 45N/m) ones typically used [57, 59]. To excite the torsional resonant cantilever vibrations, shear (transverse) piezoelectric actuators were used that

4 Measuring Mechanical Properties on the Nanoscale Table 4.1 Material SiO2 Glass

107

Results of combined flexural and torsional CR-FM experiments Source M N n G E Literature 74.9 17.0 0.171 31.1 72.7 Literature 84.7 18.7 0.206 33.6 81.1 expt. m=1 81±5 18±2 0.21±0.11 32±5 76±6 19±3 0.17±0.16 35±8 79±10 expt. M=3/2 85±8

All quantities except Poisson’s ratio (n) are given in GPa. The first two rows contain the values assumed for the two materials based on pulse-echo ultrasonic experiments and literature surveys. The last two rows show experimental CR-FM results for the glass specimen with the fused silica (SiO2) specimen used as the reference material. Values are given for flat-punch (m = 1) and Hertzian (m = 3/2) contact

produced in-plane displacements. The actuators were positioned beneath the samples such that the displacement was perpendicular to the long axis of the cantilever. Note that access to the unfiltered, high-frequency signal for both the vertical and horizontal channels of the AFM photodiode detector is needed to perform these experiments. The vertical photodiode signal was monitored to detect the flexural vibrations, while the horizontal photodiode signal was used to detect the torsional modes. The last two rows of Table 4.1 show the experimental results assuming flatpunch (m = 1) and Hertzian (m = 3/2) contact mechanics. The values obtained with these two models represent the lower and upper limits of the measured properties. Shown are the average values of the indentation modulus M and the quantity N obtained directly from the CR-FM experiments. Also included are the values of Young’s modulus E, Poisson’s ratio n, and the shear modulus G calculated from M and N. The measurement uncertainties represent one standard deviation due to scatter in 20 individual data points. Table 4.1 reveals that the CR-FM experimental results agree well with the assumed values within measurement uncertainty. All of the glass sample’s true properties except E are bracketed by the measured values calculated for m = 1 and m = 3/2. For the quantities M and N that are directly determined from the measurements, the measurement uncertainty is approximately 5–15%. In comparison, the typical uncertainty for CR-FM measurements of M using stiffer cantilevers is approximately 5–10%. The uncertainties for the other properties (E, n, and G) are larger because they are calculated from combinations of M and N. The uncertainty might be reduced through refinements to either the experimental approach or the analysis procedure. Nonetheless, these experiments provide a proof of concept for combining torsional and flexural mode measurements in order to determine individual elastic properties, especially shear properties such as n or G.

Measuring Viscoelastic Properties Many new nanotechnology applications contain components fabricated from polymers. Successful development of these products requires knowledge not only

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D.C. Hurley

of the elastic properties, but also the viscoelastic properties. However, established methods to measure viscoelastic properties typically do not possess sufficient spatial resolution for nanoscale materials. Various AFM-based methods have been demonstrated for evaluation of nanoscale viscoelastic properties (for a review, see Ref. [60]). Although these methods provide valuable information, it is often difficult, if not impossible, to obtain quantitative data with them. We have recently developed new CR-FM methods to enable quantitative measurement of nanoscale viscoelastic properties. A complete theoretical development is given elsewhere [61]; here, we summarize the analysis. The revised model for the cantilever dynamics is shown in Fig. 4.2d. The effect of the specimen’s viscoelastic properties is included by adding a dashpot with damping s in parallel with the elastic spring of stiffness k. This addition leads to a characteristic equation [61, 62] analogous to (4.3) for the relation between the wavenumbers ln of the resonant system and the quantities k and s:

α + iβ(λ n Lγ ) = 2

2 3 (1 + cos λ n L γ cosh λ n L γ ) , (λ n Lγ ) 3 ∆

(4.17)

where ∆ = [sin λ n L(1 − γ ) cosh λ n L(1 − γ ) − cos λ n L(1 − γ )sinh λ n L(1 − γ ) ][1 − cos λ n Lγ cosh λ n Lγ ] − [sin λ n Lγ cosh λ n Lγ − cos λ n Lγ sinh λ n Lγ ][1 + cos λ n L(1 − γ ) cosh λ n L(1 − γ )].

In (4.17), a = k/klever and β = σ L / (9 EI ρA ) . The quantities L, E, I = wt3/12, r, and A = wt refer to the cantilever’s length, Young’s modulus, bending moment of inertia, density, and cross-sectional area, respectively. The parameter g = L1/L denotes the relative tip position. Whereas the wavenumber xn is real in the elastic case [Fig. 4.2b and (4.3)], the wavenumber ln for the viscoelastic equation is complex. The quantity λ n Lγ = an + ibn contains a real term an that represents the elastic tip– sample stiffness, and an imaginary term bn that represents the sample’s viscous damping behavior. In the case of elastic interaction discussed earlier, the characteristic equation relates a single measured quantity, namely the contact resonance frequency fn, to a single unknown, the normalized contact stiffness k/klever. In contrast, the revised characteristic equation for viscoelastic behavior contains two unknowns: k/klever and s. Two types of experimental data are therefore needed to determine both parameters. To address this issue, contact resonance experiments for viscoelastic properties include measurements of the phase response versus frequency in addition to the usual amplitude spectrum. The experimental contact resonance amplitude and phase spectra are analyzed with a model for the response of the cantilever to harmonic excitation [61]. With this model, equations are derived to predict the phase and amplitude spectra for a given flexural mode n as a function of the quantities an and bn. A nonlinear fit of the data to these equations is performed to determine experimental values of an and bn. Equation (4.17) is then solved numerically for a and b with the values of an and bn. In analogy to (4.4) for the elastic case, measurements 2

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of the free-space amplitude and phase response when the tip is out of contact are used to characterize the cantilever’s elastic and damping behavior. Given the values of a and b, a revised contact mechanics model is used to determine the viscoelastic properties of the specimen. The reduced complex modulus for linear viscoelastic materials is given by E * ( f ) = E ′* + iE ′′* ( f ) , where E ′* is the * reduced storage modulus, E ′′ is the reduced loss modulus, and f is the measurement frequency. As in the elastic case, a referencing approach can be used to relate the properties of the unknown specimen (subscript s) to those of a reference material with known properties (subscript ref): m

m

 k   α  E′ = E′  s  = E′*ref  s  ,  kref   α ref  * s

* ref

(4.18)

and m

m

 fσ   fb  E ′′*s ( f ) = E ′′*ref  s s  = E ′′*ref  s s  .  f ref σ ref   f ref b ref 

(4.19)

As before, the exponent m refers to the tip shape: m=1 for a flat punch and m=3/2 for a hemisphere (Hertzian contact). Experiments to validate this approach were performed [61] with two polymer materials whose properties have been reported in the literature. A film of poly(methyl methacrylate) (PMMA) approximately 890nm thick was used as the test or unknown specimen. It was created by spin-coating a PMMA solution onto a (001) Si wafer and then annealing at 150°C. The reference specimen was a * polystyrene (PS) plate. Reference values for the reduced storage modulus E ′ and * the reduced loss modulus E ′′ of the PS specimen were determined by an oscillatory nanoindentation technique (Bio Ubi VII; Hysitron Inc., Minneapolis, MN, USA).1 In the measurements, a frequency sweep from 10 to 250Hz was performed with a quasistatic load of 1,000mN and a dynamic load of 20mN. The * measurements provide values for the reduced moduli E ′ and E ′′* defined above. Expressions analogous to (4.6) can be obtained between the reduced modulus E′* (or E′′* ) and the corresponding indentation modulus M ′s (or M ′′s ) of the sample. With Mtip=1,145.6GPa for the diamond indenter tip, the values M ′=(5.1±0.1)GPa for the storage modulus and M ′′=(117±35)MPa for the loss modulus of the PS reference were obtained. These values are consistent with literature values for PS [63–65]. As with all CR-FM methods, the accuracy of the approach relies on accurate information about the reference sample’s properties.

ommercial equipment and materials are identified only in order to adequately specify certain C procedures. In no case does such identification imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best for the purpose.

1

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a

b

phase (°)

amplitude (a.u.)

d = 70nm d =30nm

frequency (kHz)

d = 70nm d = 30nm

frequency (kHz)

Fig. 4.6 Example results for CR-FM measurements of viscoelastic properties. Shown are contact resonance spectra for the (a) amplitude and (b) phase response versus frequency for the lowest flexural mode of a cantilever on PMMA. The two data curves in each graph correspond to two different values of the cantilever deflection d (i.e., two different applied static loads). The symbols indicate the experimental values, while the lines represent the fit to a harmonic-excitation model

Contact resonance spectra were acquired with a cantilever that had a nominal spring constant klever=1N/m. Data were acquired for both the first- and second-order flexural resonant modes. Examples of amplitude and phase spectra are shown in Fig.4.6 for the lowest flexural mode. The symbols represent the experimental data, while the solid lines indicate the response predicted by the harmonic excitation model. Curves are shown for two values of the cantilever deflection d (i.e., two applied loads). The excellent agreement between the predicted and measured response for both amplitude and phase indicates that the model captures the essential elements of the system. The preliminary experiments involved a minimal number of measurements. Spectra were acquired at three different values for the cantilever deflection d at a single position on the unknown specimen. Due to difficulties in obtaining consistent results when a variable tip position was included, a constant value L1/L=g=1 was used and each mode was analyzed separately. Two data sets for the reference material acquired immediately before and after the measurements on the unknown specimen were used with (4.18) and (4.19). This yielded a total of six values for each flexural mode, for a grand total of 12 separate values for each of the quantities M ′ and M ′′ . All of the results were combined to obtain a single average value and standard deviation for each quantity. With this approach, for the storage modulus of PMMA film we obtained M ′ =(7.0±0.5) GPa for flat-punch contact (m=1) and (8.1±0.9)GPa for Hertzian contact (m=3/2). For the loss modulus of the PMMA film, we obtained M ′′ =(160±16) MPa (m=1) and (190±30)MPa (m=3/2). These results are roughly consistent with literature values [63, 65, 66]. Exact comparison is difficult due to potential effects such as measurement frequency and tip size. Although further work is

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necessary to completely validate and refine the method, these results demonstrate the feasibility of measuring viscoelastic properties on the nanoscale with CR-FM methods.

Imaging with CR-FM Techniques Qualitative Stiffness Imaging The above discussion describes how CR-FM techniques can be used to determine elastic properties at a single position on the sample. However, as mentioned earlier, many new applications require images of the spatial distribution in mechanical properties. An apparatus like that in Fig. 4.4a for point measurements can also be used for imaging [19, 67]. The resulting images provide only qualitative information, but they are easy to acquire and are useful in many applications. In this approach, the excitation frequency of the actuator is set at a frequency near the contact resonance. The output amplitude signal of the lock-in amplifier is connected to an input channel of the AFM instrument. A scan is performed with the tip in contact while the frequency remains constant, and an image of the lock-in output signal is acquired. In the resulting “amplitude image,” the intensity corresponds to the relative amplitude of the cantilever vibration at the excitation frequency. Signal contrast in amplitude images depends on the variation in cantilever vibration amplitude at different positions on the sample [31], which in turn depends on the relative elastic stiffness at those positions. Therefore, image contrast may be enhanced or suppressed by the choice of excitation frequency. In images acquired at relatively low frequencies, the cantilever vibration amplitude will be higher for regions with lower elastic modulus. More compliant regions will therefore appear brighter in the image. If the excitation frequency is close to the contact resonance frequency of

Fig. 4.7 Example of qualitative CR-FM imaging. The sample contained copper (Cu) lines deposited in trenches in an organosilicate glass (SiOC) blanket film. (a) Topography image. The image was acquired in contact mode at the same time as the image in (c). (b) Amplitude image of the cantilever vibration for an excitation frequency f = 550 kHz. (c) Amplitude image for f = 630 kHz. The images were acquired with a cantilever with a free-space frequency f10 = 155.1 kHz

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regions with higher modulus, brighter areas will correspond to stiffer sample components. In this way, one can observe a reversal or “inversion” in image contrast by changing the excitation frequency. An example of these effects is shown in Fig. 4.7. The sample contained a blanket film of an organosilicate glass (denoted SiOC) approximately 280nm thick. Inlaid in the film were copper (Cu) interconnect lines. The sample was etched briefly in a hydrofluoric acid solution to remove a protective surface layer. The topography image of the sample in Fig. 4.7a reveals that the sample is very flat, with height variations of less than 10nm. Figure 4.7b, c shows amplitude images for the same region of the sample acquired at two different excitation frequencies. In addition to the SiOC film and the Cu lines, bright regions can be identified at the SiOC/Cu interfaces. This feature corresponds to a thin barrier layer and is not apparent in the topography image. In Fig. 4.7b, the SiOC regions are brighter than those composed of Cu. In Fig. 4.7c, which was acquired at a higher excitation frequency, the Cu regions are brighter. This information suggests that the contact resonance frequency of the Cu regions is generally higher than that of the SiOC regions. If the contact area remains constant, higher contact resonance frequencies usually imply greater elastic modulus. Thus it can be inferred, but not directly measured, that the modulus of the Cu lines is greater than that of the SiOC film. Qualitative contact resonance imaging techniques have been used to investigate the properties of various systems with micro- to nanoscale features. Reported results include carbon-fiber-reinforced polymers [53], dislocations in graphite [68], silica-epoxy nanocomposites [42], and human teeth [69]. A number of studies have been performed on different piezoelectric ceramics, including PIC 151 [31], PZT or Pb(Zr,Ti)O3 [39, 69, 70], and BaTiO3 [39, 69]. Another study examined the effect of annealing conditions on the properties of nanocrystalline piezoelectric leadcalcium titanate films [71]. Qualitative imaging techniques have also been applied to other nanocrystalline materials such as CrN hardness coatings [45].

Quantitative Imaging: Modulus Mapping Regions or components of different elastic stiffness can be easily identified from amplitude images such as those in Fig. 4.7. However, the qualitative nature of amplitude images means that further analysis is rather challenging, for example determining the relative stiffness of various components. For detailed analysis of nanoscale elastic properties, quantitative imaging – mapping – is required. To obtain quantitative images, the contact resonance frequency must be detected at each position as the tip is scanned. A single such frequency image could provide more information than an entire series of amplitude images. However, the contact resonance frequency can vary significantly from position to position in an image, depending on the stiffness range of different sample components. A straightforward imaging approach is to perform a frequency sweep over the same frequency range at each image pixel. The frequency range of the sweep

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must be wide enough to encompass all possible peaks. Depending on the frequency resolution desired, however, the acquisition speed of this approach is not practical. If acquisition rates similar to those of point-measurement techniques are assumed, a single image could take a few days to obtain. Therefore, new imaging techniques are needed for practical CR-FM quantitative imaging. This challenge has been approached in various ways [16, 33, 39, 72–75]. Applications studied with these methods include carbon-fiber-reinforced polymer composites [72] and epoxy/silica nanocomposites [50], piezoelectric domains in BaTiO3 [39], and films of diamondlike carbon [42] and nanocrystalline SnSe [44]. We have also developed contact resonance frequency imaging techniques for nanomechanical mapping [23, 35, 76, 77]. Our approach differs conceptually from other implementations. Using feedback techniques, we continuosly adjust the starting frequency of the frequency sweep window in order to follow or “track” the peak contact resonance frequency as it changes. In this way, a high-resolution spectrum is acquired with a relatively small number of data points, even if the contact resonance frequency shifts significantly across the imaged region. With this feedback or frequency-tracking approach, imaging time is reduced without sacrificing frequency resolution. Our approach utilizes a digital signal processor (DSP) architecture. One advantage of a DSP approach is that it facilitates future upgrades, because changes can be made in software instead of hardware. A schematic of the frequency-tracking apparatus is shown in Fig.4.4b. Additional details of the circuit electronics are given elsewhere [77]. In brief, an adjustableamplitude, swept-frequency sinusoidal voltage is applied to the piezoelectric actuator beneath the specimen. As the cantilever is swept through its resonant frequency by the piezoelectric actuator, the photodiode detects the cantilever’s vibration amplitude and sends this signal to the DSP circuit. Inside the circuit, the signal is converted to a voltage proportional to the root-mean-square (rms) amplitude of vibration and sent to an analog-to-digital (A/D) converter. The DSP reads the A/D converter output signal and constructs a complete resonance curve as each sweep completes. It then finds the peak in the resonance curve and uses this information in a feedback-control loop. The control loop adjusts a voltage-controlled oscillator (VCO) to tune the center frequency of vibration to maintain the cantilever response curve centered on resonance. The control voltage is also sent to an input port of the AFM instrument and used to acquire an image. Each pixel in the resulting image thus contains a value proportional to the peak or resonant frequency at that position. For example, if the detected resonant frequency is 1.25MHz, the output voltage is 1.25V. A frequency range can be user-specified to include only the cantilever mode of interest. Each frequency sweep consists of 128 data points, which are acquired at a rate of 375Hz per sweep. The AFM scan speed must be adjusted to ensure that several spectrum sweeps are made at each image position. For scan lengths up to several micrometers, an image with 256×256 pixels is usually acquired in less than 25min. Quantitative imaging with our frequency-tracking electronics is shown in Fig.4.8. The images correspond to approximately the same region of the SiOC/Cu structure as shown in Fig.4.7. The topography image in Fig.4.8a indicates very little

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Fig. 4.8 Example of quantitative CR-FM imaging with frequency-tracking techniques. (a) Topography image acquired in contact mode. The free-space frequencies of the cantilever’s lowest two flexural modes were f10=151.3 kHz and f20=938.0 kHz. (b, c) Contact resonance frequency images of the first (f1) and second (f2) flexural modes, respectively. (d) Image of the normalized contact stiffness k/klever calculated from the frequency images in (b, c). (e) Map of the indentation modulus M calculated from (d), as described in the text

difference in height between the SiOC blanket film and the Cu lines. The contact resonance-frequency images for the two lowest flexural modes of the cantilever are shown in Fig.4.8b, c, respectively. The frequency images reveal directly that the contact resonance frequency in the Cu regions is higher than that in the SiOC regions. The images also directly show that the contact resonance frequency in the barrier layer regions is intermediate between that in the SiOC and Cu regions. Small features inside the Cu lines are also apparent. These relatively sharp changes in frequency are most likely due to shifts in the contact area that arise from small topographical features (e.g., pores, polishing effects). A map of the normalized contact stiffness k/klever can be calculated from the two frequency images in Fig.4.8b, c. The calculation is performed on a pixel-by-pixel basis with the analysis approach of section “Single-Point Measurements of Elastic Modulus”. The resulting map of k/klever is shown in Fig.4.8d. Depending on the application, it may be sufficient to evaluate the contact stiffness map alone. In other cases, a map of the indentation modulus M may be preferred. Calculating a modulus map from the contact stiffness image involves the same models and assumptions used for point measurements. For instance, a specific contact mechanics model must be chosen. Reference values of E* and k/klever are also needed. Here, the modulus

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map was calculated from the contact stiffness map assuming that the tip was flat (m=1). We also assumed that the mean value of E* in the SiOC region corresponded to MSiOC=44.3GPa. This value was obtained from point measurements directly on the SiOC film, with a borosilicate crown glass specimen as reference. The average value of k/klever in the SiOC region of the image was used as the reference value of k/klever. The resulting modulus map is shown in Fig.4.8e. It can be seen that the modulus of the Cu interconnect regions is higher than that of the SiOC film, as expected from the modulus values for bulk copper and silica. We have also used frequency-tracking techniques to examine the mechanical properties of polymer composites containing wood fibers [78]. A widely accepted concept in many composites is the existence of an interphase region that surrounds the fiber but possesses properties different from that of the bulk matrix. The mechanical properties of this nanoscale region, for instance modulus and ability to transfer stress, are critical to the macroscale structural performance of the composite. Successful development of wood–polymer composites is currently hindered by the fact that the hydrophilic wood fiber and the hydrophobic polymer matrix typically form a weak interphase with poor adhesion. Various additives have been shown to help this situation, but a clear understanding of the relation between processing chemistry and interphase mechanical properties is not well understood. Unfortunately, many techniques lack the spatial resolution needed to directly characterize the interphase [79]. AFM-based approaches possess sufficient spatial resolution, and have been used on several composite systems [80–82]. However, these methods typically cannot provide the quantitative data needed for detailed modeling. In contrast, CR-FM methods promise quantitative measurements with nanoscale resolution. Preliminary CR-FM experiments were performed on a wood–polymer composite. The specimen consisted of a polypropylene matrix with several lyocell (cellulose) microfibers aligned in the same direction. A smooth surface was cut perpendicular to the long axis of the fibers by microtoming. In this way, the surface exposed the fibers in cross section. The cantilever had a spring constant klever=1.5±0.4N/m as determined by the Sader method [83]. The free-space frequency of the lowest flexural mode was measured to be 66.92±0.03kHz. The applied static force FN=kleverd was approximately 30nN. With this value of FN, an estimated tip radius of curvature R=25–35nm, and values of the sample modulus M between approximately 5 and 15GPa, (4.8) predicts that the contact radius a is between 3.5 and 5.5nm for Hertzian contact. With a conservative estimate of 3a for the lateral spatial resolution [23], this means that the spatial resolution was approximately 10–16nm in this case. A 10nm spacing between individual data points (pixels) was therefore chosen for scanning. Contact resonance frequency images were acquired at the intersection of the matrix material and a fiber. Regions as flat as possible (height ~15nm or less across the fiber–matrix interface) were examined in order to avoid potential signal artifacts due to topography effects. The CR-FM methods described above need frequency values from two resonant modes to calculate the normalized contact stiffness k/klever and the relative tip position L1/L. This means that two frequency images must be

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acquired in the same sample region. However, subsequent images do not always correspond to exactly the same area due to drift and hysteresis in the scanners of the AFM instrument. To avoid this issue, a new approach was used. A single frequency image was acquired for each sample region. The second flexural mode was determined to provide the greatest sensitivity for the given experimental conditions [23, 59]. Contact stiffness images were calculated from the frequency image assuming a constant value of L1/L. The value L1/L=0.97 was determined from preliminary measurements with the cantilever. The resulting inaccuracy is estimated to be less than 5% for k/klever and less than 0.5% for M in this case. The reference values Mmatrix=(13.7±2.0)GPa and Mfiber=(5.9±0.1)GPa were obtained by instrumented (nano-) indentation for the matrix and fiber, respectively. A dual-reference calibration [33] was performed with these values and the average values of k/klever in the corresponding image regions. Hertzian contact mechanics were assumed. The resulting map of the indentation modulus M is shown in Fig.4.9a. Figure 4.9b contains the values of M as a function of position for the radial line segment indicated in the map. The dots in Fig.4.9b correspond to individual data points. It can be seen from the figure that a small but finite transition region exists between the fiber and the matrix. We applied a thresholding technique to crudely quantify the width of this region for 30 radial profiles selected randomly throughout the map. The threshold used to define the interphase corresponded to modulus values M between 1.1Mmatrix and 0.9Mfiber. The average number of pixels in each profile that met this criterion was multiplied by the pixel size (10nm) to estimate the spatial extent of the transition. With this procedure, the average width of the transition was found to be 70±30nm. Refinement of the experimental methods could improve the spatial resolution. Nonetheless, these preliminary results suggest that CR-FM

Fig. 4.9 CR-FM modulus mapping of a lyocell fiber-polyproplyene (PP) matrix composite. (a) Map of the indentation modulus M. (b) Modulus values for the line segment indicated in (a). The dots correspond to individual data points (image pixels)

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methods may be useful for investigating the nanoscale interphase in composites. In this way, we may begin to clarify the relationship between processing chemistry and the resulting mechanical properties of composites.

Application to Buried Interfaces Properties besides elastic modulus and other sample features can be imaged with CR-FM methods, if they influence the contact stiffness. For example, subsurface features such as defects or voids can be detected. A better understanding of this phenomenon is obtained by realizing that all contact methods generate a stress field that extends a finite depth into the sample. For Hertzian contact, the compressional stress sz directly beneath the tip as a function of depth z into the sample is given by [29] −1

 z2  σ z = p0 1 + 2  ,  a 

(4.20)

where p0=3FN/(2pa2) is the maximum stress applied at the surface z=0. Equation (4.20) shows that sz decreases rapidly with increasing distance into the sample. Because sz=0.1p0 at z=3a, sz is considered negligible for z>3a. This leads to the general guideline that measurements probe to a depth z»3a [29, 84]. For Hertzian contact, a is given by (4.8) and depends on the applied force FN, the tip radius R, and the reduced modulus E* between the tip and the sample. Typical values of these parameters for experiments on stiff materials yield values of z between approximately 10 and 60nm. Such concepts have been exploited experimentally to sense variations in near-surface mechanical properties. For example, reversible displacement of dislocations approximately 10nm beneath the surface in highly oriented pyrolytic graphite was observed with UAFM [68]. UFM methods were used to study cracking during tensile loading at a buried interface approximately 15nm deep between a brittle glass film and a ductile polyethylene terephthalate (PET) substrate [85]. Another subsurface mechanical property of industrial interest is the relative bonding or adhesion between a film and substrate. We have performed CR-FM experiments to investigate variations in adhesion at a buried interface [86]. As shown in Fig. 4.10a, a model system of gold (Au) and titanium (Ti) films on (001) Si was created. A thin Ti interlayer grid with 5mm×5mm holes and a blanket film of Au were deposited with standard microfabrication techniques. This design was intended to contain variations in the adhesion of a buried interface, yet minimize variations in topography and composition at the surface. A very thin Ti topcoat was included merely to prevent contamination of the AFM tip by the soft Au film. A crude scratch test was performed by pulling one end of a tweezer across the sample. Optical micrographs showed that this treatment had removed the film in the scratched regions without the Ti interlayer (square holes) and left the gold intact in the scratched regions containing the Ti interlayer (grid). This result confirmed the

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a

b

2 nm Ti

20 nm

Au

10 µm

1 nm

c k/klever

5 µm Si

Fig. 4.10 CR-FM mapping of film/substrate adhesion. (a) Schematic of sample in cross section. (b) Map of the normalized contact stiffness k/klever calculated from contact resonance frequency images. (c) Normalized contact stiffness versus position. The line scan is the average of 40 lines across the center of the image in (b), as indicated by the dotted lines

premise that the film adhesion was much stronger in regions containing the Ti interlayer. Contact resonance frequency images of the two lowest flexural cantilever modes were acquired for this sample. A map of the normalized contact stiffness k/klever calculated from the frequency images is shown in Fig.4.10b. The map clearly reveals that the contact stiffness is lower in the square region with poor adhesion (no Ti interlayer). A line scan of k/klever versus position is shown in Fig.4.10c. The scan represents the average value of k/klever for 40 lines in the map indicated by the dashed lines. Analysis of the entire contact stiffness map gives a mean value for k/klever of 39.1±0.6 in the grid regions and 37.1±0.5 in the square, a difference of 5%. Other contact stiffness images acquired at different sample positions consistently showed a decrease of 4–5% in k/klever for the regions of poor adhesion that lacked a Ti interlayer. With (4.8), we estimate that a=6nm to 8.5nm in this case, sufficient to probe the film interface (z»3a=22–24nm). The observed results are consistent with theoretical predictions for layered systems with disbonds [87]. In that work, an impedanceradiation theory was used to model the disbonded substrate/film interface as a change in boundary conditions (i.e., zero shear stress at the interface). For a disbond in a 20-nm aluminum film on (001) silicon, a reduction of ~4% in the contact stiffness was predicted, very similar to our results. The parameters used for modeling (E*, FN, etc.) differed from those in our experiments, but the overall combination of conditions was sufficiently similar that a comparison between the two results is valid. Therefore, we believe that weak adhesion is responsible for our experimental

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results. These results represent progress toward nanoscale mapping of adhesion, a goal with significant implications for development of thin-film devices in many technological applications. The above contact mechanics arguments imply that typical CR-FM experiments are not sensitive to features deeper than a few tens of nanometers beneath the surface. In contrast, several CR-FM experiments have demonstrated the ability to detect voids several hundred nanometers beneath the surface [88–90]. Simulations and modeling of this behavior indicate that even relatively deep defects can be detected if they are large enough [88, 90–92]. Quantitative values for “large enough” and “relatively deep” depend on the exact experimental configuration. However, studies suggest that as a rough guide, typical CR-FM methods can sense defects buried a few hundred nanometers deep if they are at least a few hundred nanometers in diameter [90]. With this in mind, we have begun CR-FM studies of stress-induced buckling mechanics in thin films [93]. Deposition of thin films can lead to residual stress in the finished film. If the stress is sufficiently large, a variety of stress-relief mechanisms can occur [94–96]. For compressive stress, these include wrinkling, in which the film and the substrate deform together, and buckling or blistering, in which the film delaminates from the substrate. A better understanding of the relationship between the film deposition conditions and the resulting mechanical response of the film is needed. Such information would enable improved modeling of device performance for many applications including flexible electronics and microelectronic interconnect. The sample used for preliminary experiments contained a 300-nm thick gold film on a Kapton polyimide substrate (see footnote 1). Buckling was induced after the film was deposited by clamping the sample. Figure 4.11a shows an optical micrograph of a region of the sample containing “telephone cord” buckles. CR-FM images were acquired for several different regions at the edge of several different buckles. The dotted lines in Fig. 4.11a indicate the approximate scan size and relative location of the images. The frequency images were acquired with a cantilever with nominal spring constant klever=30N/m and an estimated applied static force FN=600nN. The nominal tip radius of curvature was R=30nm when new; little or no wear was observed during scanning. With Hertzian contact mechanics, the estimated contact radius a=5nm, so that the lateral spatial resolution ~3a=15nm. We found that it was sufficient to obtain images of the lowest flexural mode frequency f1 only. In the data analysis, it was important to examine the sample topography and the corresponding CR-FM response simultaneously. An example is given in Fig. 4.11b, which shows the contact resonance frequency f1 as a function of position (solid line) and the corresponding topography (dashed line). Four regions with different behavior are indicated by the vertical dotted lines and the labels A–D. In region A, the height is low, indicating that the tip is located away from the buckle. The contact resonance frequency f1 is high. As mentioned above, higher values of f1 usually imply higher contact stiffness. Therefore, the CR-FM response suggests that the film is adhered to the substrate in this region. In region B, the high height values indicate that the tip is on the side of the buckle. The contact resonance frequency is significantly lower than that in region A. Therefore, the film appears to be delaminated.

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Fig. 4.11 CR-FM results for stress-induced buckling. (a) Optical micrograph of telephone cord buckles in a gold film on a polyimide substrate. The dotted lines indicate the approximate size and relative position at which CR-FM frequency images were acquired. (b) Plot of frequency and topography versus position at the edge of a buckle. The solid line shows the contact resonance frequency f1 of the lowest flexural mode. The dashed line shows the corresponding topography. The topography data have been processed to remove a linear tilt and constant offset from each line. The vertical lines and the labels A–D indicate four regions with different types of behavior, as discussed in the text

The observed behavior in regions A and B is consistent with that expected for a buckle. However, regions C and D exhibit behavior that is less expected. Region C represents a zone at the very edge of the buckle over which the contact resonance frequency f1 is consistently high. This implies that the film is adhered to the substrate at the outermost edge of the buckle. The behavior is consistent with a hybrid wrinkling–buckling mechanism or “meniscus” effect predicted by nonlinear finiteelement analysis for nickel films on polycarbonate substrates [97]. A meniscus effect was observed in all of our experimental images and ranged in width from roughly 1 to 3mm. Finally, region D represents a transition zone several micrometers wide, in which the contact stiffness gradually decreases. This effect represents a decrease in the structural compliance with increasing distance from the clamped edges of the buckles, similar to that observed in suspended nanowires [48]. These results suggest that CR-FM can serve as a nondestructive tool for better understanding the mechanics of thin-film buckling.

Summary and Conclusions In this chapter, we have presented contact resonance force microscopy (CR-FM) measurement methods and their use for materials characterization on the nanoscale. The basic physical principles of CR-FM and the equipment needed to

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implement them experimentally have been discussed. CR-FM experiments involve measuring the resonant frequencies of a vibrating cantilever when its tip is in contact with a material. Models for the cantilever dynamics and for the tip–sample contact mechanics are then used to relate the contact resonance frequencies to the near-surface elastic properties. CR-FM methods to nondestructively measure quantitative elastic properties on the nanoscale were described. Extensions of the original approach that provide further information about mechanical properties were also presented, including the use of torsional modes to measure Poisson’s ratio and a method to determine viscoelastic properties. The use of CR-FM methods for imaging applications was also discussed. In particular, frequency-tracking tools to map the contact resonance frequencies were described. Examples showed how CR-FM frequency-tracking methods provide quantitative images or maps of properties such as elastic modulus and thin-film adhesion at a buried interface. The information provided here is intended to serve merely as an introduction to CR-FM methods. A growing body of work by groups worldwide point to the utility of these methods for a wide range of applications. It is hoped that this discussion stimulates readers to envision further uses in their own research. By providing quantitative nanomechanical information for a variety of material systems, CR-FM techniques will contribute to the rapid growth of nanoscale materials science and will play a significant role in future nanotechnology efforts. Acknowledgements Many current and former NIST coworkers contributed to this work, including R. Geiss, M. Kopycinska-Müller, A. Kos, C. Stafford, and G. Stan. I value interactions with researchers from several institutes, especially J. Turner (University of Nebraska-Lincoln) and W. Arnold, U. Rabe, and S. Hirsekorn (Fraunhofer Institute for Nondestructive Testing IZFP, Germany). The results for viscoelastic properties represent a collaboration with J. Turner and P. Yuya (University of Nebraska-Lincoln). The lyocell-polypropylene composite sample was provided by S. Wang and S. Sudhakaran Nair (University of Tennessee Forest Products Center). The buckling results represent a collaboration with M. Kennedy (Clemson University) and N. Moody (Sandia National Laboratories). I also appreciate interactions with R. Geer (State University of New York at Albany), B. Huey (University of Connecticut-Storrs), and N. Jennett (National Physical Laboratory, UK).

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Chapter 5

Multi-Frequency Atomic Force Microscopy Roger Proksch

The atomic force microscope (AFM) was invented in 1986 [1], a close relative of another instrument, the scanning tunneling microscope (STM), invented in 1981 [2]. Both fall under the umbrella of techniques and instruments referred to as scanning probe microscopes (SPMs), with the common thread being that a sharp probe is scanned in a regular pattern to map some sample characteristic. Unlike the STM, the AFM can readily image insulating surfaces. Combined with the ability to study a wide variety of samples and sample environments – ambient, liquid, and vacuum – has made AFM the technique of choice for many high resolution surface imaging applications, including imaging with atomic resolution. Since those early days, AFM techniques have become the mainstay of nanoscience and nanotechnology by providing the capability for structural imaging and manipulation on the nanometer and atomic scales. Beyond simple topographic imaging, AFMs have found an extremely broad range of applications for probing electrical, magnetic, and mechanical properties – often at the level of several tens of nanometers. One ongoing “holy grail” quest of AFM, since very nearly the beginning [3] has been compositional mapping where materials differences are mapped out with the same nanometer resolution as topographic images. There are many forces acting between an AFM tip and a sample, long-ranged van der Waals, electrostatic and magnetic forces, short-ranged forces stemming from the elasticity of the tip and sample, and dissipative forces associated with adhesion, plasticity, phonon gene ration, and eddy currents, to name a few. Many if not all of these interactions carry compositional information about the sample. However, the forces also depend on the geometry of both the tip and the sample. The sample topography conspires with the tip geometry to make unraveling of the specific contributions to the net forces very difficult. For example, a situation where there is zero net force on the cantilever tip might mean that there are no forces acting on the tip or that there is a very large adhesive (attractive) force being balanced by a very large elastic (repulsive) force. Much of the theoretical and experimental work being done in AFM originates from R. Proksch (*) Asylum Research, Santa Barbara, CA 93117, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_5, © Springer Science+Business Media, LLC 2010

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the desire to separate and understand these different forces and, ultimately, to use them to identify specific materials and material properties at the nanoscale. Multiple frequency techniques, where the cantilever motion is measured (and sometimes driven) at multiple resonant frequencies, have become an active research topic, in part because of their ability to differentiate material properties from the topography of the sample. In this chapter, we will review some of the emerging techniques using AFM cantilevers at multiple frequencies. The scope will be limi ted to techniques where the multiple frequencies are each “active,” or driven, either by the cantilever excitation, as in the case of non-contact and intermittent contact mode techniques, or by a modulated tip–sample interaction, as in the case of electric or piezoresponse force microscopy (PFM). In addition, there is a broad range of “passive” techniques that will not be considered here [4–6], where harmonics of a drive frequency can be monitored and used to glean information regarding the tip–sample interaction [4, 6–8]. Also not reviewed chapter, but included elsewhere in this volume are some very promising techniques where a single frequency is swept, or “chirped” over a range of values to extract more detailed information about the cantilever resonances in contact with the surface [9–13]. A new, related technique called band excitation [14–17] allows rapid acquisition of spectra, was originally developed in the context of PFM but has wide application in a variety of scanning probe techniques and application areas. As we discuss below, simulations and experimental AFM data have shown intriguing contrast on a variety of samples, both in air and liquid, for multi- frequency techniques. However, a major challenge that remains in this area – as indeed in any AFM technique that aspires to quantify material properties – is to unambiguously separate compositional differences in the tip–sample forces from geometrical effects, more specifically, tip-shapes. While straightforward in simulations, real samples have topography that is not known a priori, and real tips have a great deal of shape and size variability, especially as the tip size drops below 100nm [18–20]. This has proven to be a thorny problem to solve. Tips are difficult to characterize, requiring either scanning some sort of calibration sample or characterization with some other kind of microscopy, usually high resolution SEM. To confound the problem further, tips are typically not stable creatures that are static over their lifetime. They can both wear and pick up contaminants during the course of imaging. Both issues render initial characterizations moot. For this reason, many of the most robust results end up being relative comparisons of properties over the course of single images rather than quantitative absolute measurements. Despite these quantification difficulties, relative property mapping on length scales of nanometers can be very powerful, intriguing, and useful.

Multi-Frequency Motivation Using higher modes can be motivated by a number of physical arguments. In pioneering early work in non-contact microscopy, Rodriguez and Garcia pointed out that the second resonant mode oscillations are sensitive to weak, long-ranged van der Waals

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interactions due to the nonlinear coupling of the two oscillating modes (coupled through the tip–surface interactions) combined with their higher Q-factors [21]. In addition, while operating the distance control feedback on the first mode necessarily constrains it to the “set-point,” the second mode is then free to explore a larger part of phase space [22]. In contact mechanical measurements, the dependence of contact resonance frequencies on the contact stiffness for various modes is shown in Fig. 5.2 [23]. This dynamic stiffness of successively higher modes makes them sensitive to greater tip–sample stiffness. Finally, and perhaps one of the greatest motivations for multiple frequency methods, is the large number of unknown parameters in even the simplest models of AFM tip–sample interactions [17]. Conventional single frequency AFM measurements typically produce two measured quantities at each point in space – the amplitude and phase. Even the simple harmonic oscillator (SHO) model contains more than two parameters. The resonance frequency, quality factor, drive amplitude, and drive phase are all required to unambiguously calculate the response. In some cases, separate measurements of the drive sensitivity of the excitation piezo can be made, allowing quantification of the SHO response, including quantities such as the power dissipated between the tip and the sample [24]. However, as discussed in greater detail below, this assumption can break down if the transfer function of the cantilever drive mechanism is not well understood. In addition, applications such as PFM [17, 25], the drive amplitude and phase originate from the sample being measured and are inherently unknown. In this particular case, the SHO model necessarily includes four unknown parameters (resonant frequency, quality factor, drive amplitude, and drive phase), implying that single frequency measurements returning only two independent parameters will always leave the SHO model under-determined. Finally, there are many examples where the SHO model is clearly insufficient to extract details of the nonlinear tip– surface interactions, requiring even more parameters in any model to describe the dynamics [26].

Cantilever Resonant Modes and Boundary Conditions All of the results presented here, and indeed in AFM in general, were made possible by the batch fabrication of disposable, reliable mechanical probes, typically in the form of cantilevers with sharp tips on the end. Because cantilevers are extended mechanical objects, they have many different flexural resonant modes. Historically, atomic force microscopes have typically excited one of those modes, usually the lowest frequency, or “fundamental” mode, and then used the amplitude or frequency of that mode as the input for a feedback system that controls the tip–sample separation. When the amplitude of the fundamental motion is used, the term “amplitude modulated AFM” (AM-AFM) has been employed [27]. Included under this umbrella are labels such as “tapping,” “intermittent contact” and “AC” modes.

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Fig. 5.1 Theoretical resonant frequencies of diving board shaped cantilevers in terms of the f undamental, f1 as well as actual resonant frequencies for the Olympus AC240 and Bio-Levers. In these cases, the theoretical predictions typically agree with the measured results to well within 10%

Fig. 5.2 Calculated local vibration amplitudes along a cantilever with rectangular cross section for modes 1–3 (from left to right) and for different values of k*/kc (increasing from top to bottom). Because of space, in the second column, the spring was drawn with its fixed end on top. The springs are always in their un-deflected rest positions when the end of the cantilever is at zero. (a) In the clamped-free case the maximum of the vibration amplitudes is at the free end. (b, c) show the effect of a spring with increasing stiffness. (d) For an infinitely stiff contact, the end of the cantilever is pinned but free to rotate. At this end, the vibration amplitude of the cantilever becomes zero. From [23]

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Because of their shape, the relationship between the resonant modes is somewhat more complicated than most “Physics 101” treatments of resonant or normal modes where the normal modes are also harmonics of the fundamental. For example, a plucked guitar string has higher flexural resonances that are harmonic, meaning that if the fundamental mode is at a frequency f1, the next resonance is at 2f1, the next is at 3f1, and so on. However, for all but a few very specialized types, cantilever resonant frequencies are non-harmonic. Figure 5.1 gives the theoretical resonant frequencies for diving board-shaped cantilevers for the first few resonant frequencies. The measured resonant frequencies for two representative cantilevers are also shown: a 240-µm long Si cantilever primarily used for imaging in air (AC-240, Olympus), and a 60-µm long SiN cantilever primarily used for imaging in fluid (BioLever, Olympus). There are a large variety of dynamic techniques where the probe is essentially in continuous contact with the surface while it is oscillated, subject to the tip–sample boundary condition. Generally, these techniques are especially useful for samples where the stiffness is larger than the cantilever spring constant, a condition that makes conventional AFM-based force curve measurements more difficult. Of these techniques, the most promising one make use of one or more of the contact resonance frequencies. These techniques include so-called acoustic and ultrasonic force microscopies [28]. These same techniques are also relevant to techniques such as PFM where the contact resonance is very useful for amplifying the small displacements from the piezoelectric properties of a sample (typically of order 2–500pm of displacement for a volt of applied potential). Figure 5.2 shows a simplified view of the changes in the cantilever boundary conditions as it comes into contact with a surface. This causes the resonances to shift in a manner strongly dependent on the contact mechanics [23]. The general idea of these contact resonances is shown in Fig. 5.2, where the tip of a cantilever is placed in contact with a sample surface. Resonant vibrational modes of the cantilever are excited by an external actuator attached either to the sample (dubbed atomic force acoustic microscopy [AFAM]) [23] or to the AFM cantilever holder (ultrasonic force microscopy [UFM]) [29–32]. Oscillations can also be induced by other means, as in PFM [33, 34] where localized motion of the sample is induced by an oscillating tip potential, or as in a variation of localized thermal analysis where motion is induced by localized heating of the sample [35, 36]. Whatever the drive mechanism, the tip–sample interactions can be sensitively probed by measuring the contact resonance frequency. When the cantilever approaches the surface and the tip changes from free to contact, the boundary conditions alter the normal cantilever flexural modes. Generally, the frequencies of the resonant modes increase, and are strongly dependent on the tip–sample interactions. CR AFM involves measuring the frequencies at which the free and contact resonances occur. The mechanical properties of the sample are then deduced from these frequencies with the help of appropriate models, as is discussed in detail in several review articles [10, 13, 37]. A complication with this picture is that most surfaces have some roughness at the length scales of nanometers. This roughness will affect the contact area of a tip

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as it scans over the surface. To explore this effect, consider a simplified Hertz model consisting of two spheres having radii Rt and Rs, the tip and sample radii, respectively, in contact with each other and loaded by a force Fn. They will have a contact area given by [38] 1/3

 Fn Rt Rs  a=  E (R + R )  t s  

(5.1)

In this expression, the elastic modulus of the two spheres, E is assumed to be the same for the two materials, as is the Poisson ratio v = 0.3. The stiffness of this contact is given by kts=aE and the resulting resonance frequency of the cantilever in contact with the surface, including the first linear term then becomes

 k  f res = 1 − 5.8 c  f res,bound k ts  

(5.2)

Here, fres,bound is the resonance frequency of the cantilever pressed into an infinitely stiff sample material and kc is the spring constant of the cantilever. This equation can be used to model the dynamics of a spherical tip as it scans along a rough surface. The results of such a simulation are shown in Fig. 5.3. The phase and amplitude curves in Fig. 5.3 show the results of a cantilever scanning in contact across a rough, bit otherwise homogeneous surface. Despite the intrinsic parameters (Young’s modulus, dissipation) remaining constant over the

Fig. 5.3 The phase (a) and amplitude (b) of a modeled cantilever in contact with a rough surface (c) is plotted as a function of lateral position. The cantilever response is calculated assuming that the resonance frequency is dependent on only the contact area, a quantity estimated from the localized curvature of the surface near the tip. The resonance frequency drops when the tip is on top of a sharp feature because the contact area is small (highlighted in blue), causing the amplitude (blue curve in (e)) to drop as the peak moves away from the drive frequency. The phase increases at the drive frequency (blue curve in (f)). Similarly, the resonance frequency increases when the tip is in a valley and the contact area is large (red highlight and curves), and is intermediate over a flat part of the sample (green highlight and curves)

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surface, the amplitude, and phase show a great deal of contrast. This contrast results since the single drive frequency was chosen to be close to the contact resonance. As the tip scans over the surface, the contact resonance frequency varies, resulting in large, difficult to interpret changes in the cantilever amplitude and phase.

Methods The most direct method for using higher modes is to simply drive the cantilever at a single frequency in the vicinity of the higher mode resonance. This can be accomplished using the same experimental setup used for imaging at the fundamental mode. As with imaging using the first resonant frequency, the amplitude of the cantilever measured at the drive frequency is used as the error signal in a feedback loop. There have been a number of groups who have reported enhanced phase contrast when operating at a higher resonance mode and who have reported contrast presumably associated with variations in sample elasticity, adhesion or dissipation [4, 6, 39]. A major issue with operating at a higher mode is the imaging stability. Operating at a higher mode is not as stable as imaging at the fundamental. This has limited the applications of single frequency higher mode imaging.

Attractive and Repulsive Mode During a single AM-AFM oscillation cycle, the tip typically samples a range of forces, from the long-range attractive to the short-range repulsive. These forces in turn affect the dynamics of the cantilever, specifically the amplitude and phase shift of the cantilever oscillation relative to the driving force [27]. If the time-averaged phase shift of the cantilever interacting with the surface is positive relative to the phase well above the surface, it is customary to refer to the imaging mode as “net attractive” or simply “attractive.” If the phase shift is negative, the mode is referred to as “net-repulsive” or simply “repulsive.” In simulations of attractive mode operation, the phase of the second mode has a strong dependence on the Hamaker constant of the material being imaged, implying that this technique can be used to extract compositional information about the surface. Note that since the van der Waal’s interaction is usually attractive and relatively long-range, attractive mode imaging is likely to be more non-invasive and gentle than imaging in repulsive mode. Figure 5.4 shows the basic idea of bimodal imaging mode using two (non-harmonic) resonant frequencies of a cantilever. The cantilever is driven with a linear combination of sinusoidal voltages at or near two resonant frequencies, f1 and f2. This signal is used to drive the base of a cantilever with a piezo shaking the base of the cantilever. Similar results can be obtained with magnetically activated cantilever with similar results. It is expected that other actuation methods where two drive

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Fig. 5.4 In bimodal dual AC, the cantilever is both driven and measured at two (or more) frequencies. The sinusoidal “shake” voltage is the sum of voltages at frequencies f1and f2. The cantilever deflection then contains information at both of those frequencies, as shown in the red curve. The amplitude and phase at the two frequencies are then separated again by the two lock-ins and passed on to the controller. The controller can use one or both of the resonant frequencies to operate a feedback loop

waveforms can be summed will prove as effective. The resulting motion of the cantilever is measured with the standard position-sensitive detector. This signal in turn is used as the input for two lock-in amplifiers, where the sinusoidal drive at f1 is used as a reference for one lock-in and f2 is used as a reference for the other. The output of the lock-in amplifiers, including the Cartesian in-phase and quadrature pairs (x1, y1, x2, y2) and polar amplitude and phase (A1, j1, A2, j2) representations of the cantilever motion at the two or more frequencies, can then be passed on to the controller where they can be displayed, saved, combined with other signals, and used in feedback loops [40].

Feedback As with conventional AC imaging, the amplitude of the cantilever is used as the feedback error signal. There is a difference here, however, since there are two amplitudes – one at each drive frequency, either of which could be used as a feedback signal. The initial results we present use the amplitude of the fundamental frequency A1 as the feedback error signal and the fundamental phase ϕ1 , the second resonant frequency amplitude A2, and phase ϕ2 as “carry-along” signals. Reversing this and using the higher resonant frequency amplitude as a feedback, and carrying the fundamental amplitude and phase along, can also yield interesting results. The sum of all of the amplitudes as the error signal also allows stable imaging.

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An interesting feature of this measurement is that the signal processing can be performed on the same cantilever deflection data stream for each flexural mode. With a digital lock-in implementation, for example, this implies that the same position-sensitive detector and analog-to-digital converter (as long as it has sufficient bandwidth for the higher mode) can be used to extract information regarding the various distinct resonant frequencies.

Intermittent- and Non-contact Bimodal Experimental Results Figure 5.5 shows a 30-µm image of a highly oriented pyrolitic graphite (HOPG Graphite, SPI Incorporated, West Chester, PA, USA) surface. The cantilever was a silicon AC-240 driven at the fundamental frequency (f1~69.5kHz, A1~8nm) and second resonant frequency (f2~405kHz, A2~8nm). The Z-feedback loop was operated using the fundamental amplitude A1 as the error signal. The topography (a) shows the expected terraces separated by single or multiple atomic steps. The first mode amplitude (b) channel resembles a high-pass filtered image of the topography, typical for the error signal in AFM. The fundamental phase image (c) shows an average phase lag of ~34° and very little variation (£1° standard deviation), implying that the cantilever was consistently in repulsive mode. Again, there is very little contrast in the fundamental phase image. However, the second mode amplitude image (d) has significant contrast, with broad patches showing regions where A2, the second mode amplitude, was reduced by tip–sample interactions. A threedimensional rendering of the surface topography (a) with the second resonant frequency amplitude (d) “painted” onto the rendered surface (e) allows the high contrast second mode data to be correlated with the topography. Although (e)

Fig. 5.5 (a) Topography, (b) fundamental amplitude, (c) fundamental phase, and (d) bimodal second mode amplitude of HOPG, 30 µm scan. (e) Second mode amplitude data overlaid on rendered AFM topography. From [22]

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makes it clear there is a high degree of correlation, there are also boundaries in the second mode amplitude that seem to have no connection to topographical features. As another example, the second resonance mode phase of an epoxied natural rubber and polybutadiene rubber blend with embedded Si particles is shown in Fig. 5.6a. In this case, bimodal imaging allows the three components of the polymer blend to be clearly identified. To verify this interpretation, more conventional force–distance or “force” curves were performed over different regions. Colorcoded force curves made over three circled regions in Fig. 5.6a are shown in Fig. 5.6b. The curves were each offset along the vertical axis for visibility. In general, the slope of the deflection curve versus piezo extension provides a measure of the sample stiffness while the area enclosed by the loop provides a measure of the energy dissipated during an approach–retraction cycle. From the force curve data, the black dots (the most repulsive phase regions) can be identified as likely belonging to the embedded Si particles since the force curve (red) shows the steepest slope and the smallest dissipation. Similarly, the relatively stiff epoxied natural rubber ENR component is identified with the intermediate stiffness (blue), appearing as dark purple patches in Fig. 5.6a, while the softest polybutadiene rubber BR component (green) is associated with the yellow regions. Bimodal imaging can also be successfully applied for AM-AFM imaging in fluids. A high-density lambda-digest deoxyribonucleic acid (DNA Sigma 50µg/ml, imaged in 40mM HEPES, 5mM NiCl2 with a pH of 6.6–6.8) sample was prepared in a dense mat on freshly cleaved mica. Figure 5.7 shows the response of a 60-µm long Olympus Bio-Lever in fluid being driven at its fundamental resonance (f1~8.5kHz, A1~8nm) and at its second mode (f2~55kHz, A2~5nm) in the DNA buffer solution. The topography in Fig. 5.7a shows a dense mat of material on the surface with no clear strands of DNA visible. Similarly, the fundamental amplitude in Fig. 5.7b, the channel used for the feedback error signal, shows no particular structure. The fundamental phase channel

Fig. 5.6 (a) Second resonance mode phase of epoxied natural rubber (ENR) and polybutadiene rubber (BR) blend with embedded Si particles, 3 µm scan. (b) A series of force curves made over the color-coded locations on the sample, shown in (a) as circles. Sample courtesy of S. Cook, TARRC, UK

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Fig. 5.7 (a) Topography, (b) fundamental amplitude, (c) fundamental phase, (d) bimodal second mode amplitude of DNA, 750 nm scan. (e) Second mode amplitude data overlaid on rendered AFM topography. From [22]

(Fig. 5.7c) shows subtle contrast between the background and a structure that shows hints of being strands of DNA molecules. The second mode amplitude (Fig. 5.7d) shows clear, high contrast images of the same DNA strands. The strands appear dark, corresponding to an increased dissipation. This is consistent with the DNA strands being slightly less bound to the sample and thus able to absorb some of the second resonant frequency energy. Again, rendering the topography in three dimensions and painting the second mode amplitude on top in Fig. 5.7e allowed the topography and second mode amplitude to be spatially correlated.

The Energy Viewpoint Energy analysis is a useful way to look at bimodal imaging. The power dissipated by the ith resonant mode of the cantilever, Pi is given by the expression

Pi =

πkA2 f Q

 QAdrive sin ϕ f  −   A f0  

(5.3)

where the cantilever spring constant k, the piezo drive amplitude Ad, the cantilever quality factor Q, and the resonant frequency fn are all measured at a reference position above the sample surface (i.e., where the dissipative interactions are assumed absent). The amplitude A and phase j are used in (5.3) to yield the dissipated power relative to the reference position. Note that depending on the reference point, the dissipated power can be positive or negative. If the power dissipated by the tip due to the tip–surface interactions is zero, (5.3) can be solved to give a characteristic relationship between the amplitude and the phase of the cantilever,

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A−

ω 0 QAdrive sin ϕ w

(5.4)

This characteristic arcsine behavior of the phase on the amplitude is illustrated by the red curves in Fig. 5.10. During typical imaging, a feedback loop constrains the amplitude of the first mode to be constant. In this case, (5.3) implies that any phase contrast requires dissipative interactions between the tip and the sample. Purely conservative forces will not change the phase, they will only change apparent height of the sample. This is a significant limitation on compositional contrast using single frequency AM-AFM. However, if a second resonance is also excited, without a feedback loop, there is no such constraint on the dynamics. The second mode amplitude and phase are able to freely evolve in response to both conservative and dissipative forces. An example of this is shown in Fig. 5.8, where we imaged collagen fibers extracted from a rat tail tendon. The tail was mechanically dissected in PBS buffer. The extracted collagen fibers were torn apart and deposited on a mica surface. After rinsing with de-ionized water, the fibers were imaged. An AC240 cantilever (Olympus) was bimodally driven. The first resonance was at 72.1kHz and the amplitude set-point was ~60nm. The second resonance was at 437.5kHz and the amplitude was nominally ~6nm. The topographic data show the very typical 67nm banding pattern. Note that the fundamental resonance amplitude signal is relatively

Fig. 5.8 (a) Topography, (b) fundamental amplitude, (c) fundamental phase, (d) second mode amplitude and (e) second mode phase, 750 nm scan. Clear contrast differences between the background glass substrate, the collagen fibers and various contaminants are visible in the second mode phase channel (e)

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featureless as might be expected – this was the feedback error channel. The fundamental phase in Fig. 5.8c shows contrast at the edges of features, becoming more positive at these points. This is consistent with the overall interaction becoming more attractive at the edges. The second mode amplitude (Fig. 5.8d shows a similar contrast. The second mode phase (Fig. 5.8e) clearly differentiates the glass substrate, the fibers, and contaminants. In addition, the second mode phase is reduced over lengths of fibers that are suspended. Figure 5.9a shows image (Fig. 5.8e) with blue ovals indicate suspended fibers that appear dark in the second mode phase. In a manner similar to that discussed for Fig. 5.6, force curves were performed over the glass substrate (red) and on a suspended fiber (black). The adhesive portions of the curves are quite similar while the contact portions show distinctly different slopes, indicative of different stiffness values. In particular, the smaller slope indicates, not surprisingly, that the suspended fiber is significantly softer than the glass substrate. This in turn is clear evidence that the second mode phase is sensitive to variations in the sample stiffness. Figure 5.10 show two-dimensional histograms of the amplitude and phase values for the images shown in Fig. 5.8. In Fig. 5.10a, the fundamental phase and amplitude are plotted, showing the effects of the feedback loop. The amplitude is maintained at a more or less constant value of ~65nm with a narrow distribution of a few nanometers. The phase shows a wider distribution, with dissipation causing phase excursions towards 90°, as described by (5.3). Figure 5.10b shows very different behavior, with the second mode dynamics ranging over a much larger range of values. Since there is no feedback loop constraining the dynamics, the second mode is able to sample a wide range of both conservative and dissipative interactions as it scans over the surface.

Fig. 5.9 (a) Second mode phase image from Fig. 5.9. Two suspended fibers are indicated with blue ovals where the second mode phase shows a clear localized decrease. (b) Shows force– distance curves made over a supported (red) and a suspended (black) part of a collagen fiber. The adhesion over the two regions is quite similar while the slope of the contact portions of the curves are quite different, suggesting that the difference between the tip–sample interactions in the two areas is primarily elastic

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Fig. 5.10 Two-dimensional histograms constructed from the collagen data shown in Fig. 5.9. (a) Shows the distribution of the fundamental phase and amplitudes while (b) shows the distribution for the second resonance amplitude and phase. The red curves in each figure are the zero dissipation phase versus amplitude curves calculated from (5.4)

High Resolution, Low Force Bimodal Imaging One significant benefit of bimodal attractive mode imaging is that the forces between the tip and the sample are extremely small, allowing clear and nondestructive differentiation of composition, even on soft biological samples. As an example, type I collagen molecules form tensile-bearing structural fibers that are very common in connective tissue and the extracellular matrix. The fibril packing structure of collagen has been debated for some time. The most commonly accepted structure of the fibers corresponds to the model where the molecules of collagen are

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arranged in a staggered manner, leading to a typical 68nm pattern [41]. However this staggered arrangement has never been clearly demonstrated, and other models have been proposed. More recently AFM has been used to image collagen fibers, and additional models have been proposed [42]. A few AFM studies have also followed the assembly of molecules dynamically, but no insight was given on the molecular level. In Fig. 5.11, bimodal imaging was used to probe the ultra-structure of the same collagen sample shown in Fig. 5.8. In this case, we are imaging at very small amplitudes, keeping the cantilever in a net-attractive state. The first resonance was at 73.4 kHz and the set-point was 5nm. The second resonance was at 444.7 kHz and the amplitude was nominally ~1nm. The topographic data show the typical 67nm banding pattern, while the second mode amplitude shows detailed features at the surface of the fibers on the nanometer scale. Note that the fundamental resonance phase signal is relatively featureless. The small elongated structures visible in the second mode amplitude channel are on a length scale consistent with individual molecules inside the fibers. The round features could correspond to the terminal parts of the molecules forming the top layer of the fiber.

Fig. 5.11 Bimodal images (300 nm scan) of the (a) topography obtained from feeding back on the first mode amplitude, (b) first mode phase and (c) second mode amplitude of a collagen fiber. Image (d) shows a zoom into a region of the second mode amplitude image and (e) shows a section taken along the red line. The first mode phase is relatively featureless. The second mode amplitude shows a fine structure with a resolution of 2–3 nm. The white bar in images (a) and (d) is 50 nm long

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Separating Long- and Short-Range Forces: Bimodal Magnetic Force Microscopy Magnetic samples provide a nearly ideal situation for examining the relative role of short- and long-ranged forces because the long-range magnetic forces can change sign while other sample properties that affect short-ranged forces remain the same. Furthermore, the interaction length scales are set by the size of the domains or recorded bits, and more of the sample volume participates in the interaction than in the case of van der Waals forces. This means the long-range magnetic forces are much more independent of topographic effects like surface roughness. This allows single-pass imaging of the topography and magnetic forces between the tip and sample at high spatial resolution. Furthermore, magnetic samples are ideal test cases for determining contrast mechanisms in bimodal imaging theory (Fig. 5.12). In the above examples, the second cantilever mode was driven with the same excitation piezo. It is also possible to drive different modes with different mechanisms, including the tip–sample interaction, in cases where this interaction can be modulated. One example is the electrical force between the tip and the sample [44]. In a manner similar to the MFM results above, this allows the short-range topographic interactions to be separated from the long range, in this case, electrical interactions. Rodriguez et al. have recently shown that the second resonance can be used to image the ferroelectric domain structure in a number of samples in fluid [45]. Finally, bimodal drive techniques can also be used for contact resonance measurements, where the tip is pressed into the surface with a preset load and the resonances of two modes are excited [46].

Multiple Frequencies at the Same Resonance Peak Revisiting Past Assumptions: More Than Two Independent Variables As mentioned in the introduction, a common and useful approximation for the dynamics of a cantilever is the single simple harmonic oscillator. If we explicitly include the drive amplitude Adrive and phase j drive along with the cantilever resonance frequency w 0 and quality factor Q as unknown parameters in the steady-state amplitude can be written as [47]:

A (w ) =

Adrivew 02 (w 02 − w 2 ) 2 − (w 0w / Q )

2

,

(5.5)

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Fig. 5.12 A bimodally driven cantilever was used to image and then locally probe a hard disk sample, shown in (a) and (b). An attractive and a repulsive region were individually probed as indicated by the circular and triangular markers. Separate plots of phase corresponding to the first (c) and second modes (d) are shown as functions of Z position as the tip approaches the surface. The marker used in each plot also corresponds to the location where that particular data was acquired. After the point indicated by the dashed line when the short-range forces begin to dominate the tip–sample interaction, there is no longer any difference in the first mode phase signals. In contrast, there still remains a distinct difference in the second mode phase signals. From [43]

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and the steady-state phase as

  w 0w + j drive (w ). j (w ) = j cant (w ) + j drive (w ) = tan −1  2 2   Q(w 0 − w ) 

(5.6)

For a given imaging technique, these quantities carry different information about the tip–sample interactions. In the case of PFM for example, Adrive can be linked to the local piezo coefficient of the sample, jdrive depends on the polarization orientation and is assumed to be frequency independent, w0 depends on the contact stiffness which in turn depends on the contact mechanics and elastic moduli or the tip and sample, and variations in Q give information about the localized damping. This damping can be related to contact mechanics or polarization-lattice dissipation. In conventional single frequency measurements, information on two of the above quantities can be extracted based around the assumption that the other two quantities remain constant during the measurement. One example of this is dissipation imaging [24, 48]. In this formalism, the dissipated power Ptip is related to the phase shift and other cantilever parameters through the expression

Ptip =

πkA2 f  QAdrive sin (j − j drive ) f  −   Qref  A f0 

(5.7)

In this expression, we have included an unknown variable drive phase jdrive, which Cleveland et al. assumed to be zero. The spring constant of the cantilever k, the drive amplitude Adrive (usually provided by a vibrating piezo element) and the quality factor Qref are usually only calibrated once, before or after the dissipation and at a reference position. They are then assumed to remain constant over the duration of the dissipation measurements. For applications where the drive is supplied by a separate piezo or other actuator and, most importantly, the drive frequency is kept constant, the assumption is likely reasonably valid. If it does change, it is unlikely to correlate with other image or measurement features, allowing the artifact to be diagnosed. However, this is manifestly not the case for frequency tracking measurements of the type discussed here. In this case, we expect the transfer function to introduce artifacts into the measurement.

Frequency Tracking There are many examples of frequency tracking applications in scanning probe microscopy, most of which originate from the seminal frequency-modulated AFM (FM-AFM) work of Albrecht et al. [49]. This technique has garnered a great deal of attention because of its high spatial resolution in air, vacuum, and even fluid [50–55]. Of the many successes of FM-AFM, a few of note include single atomic defect resolution [56] and even sub-atomic resolution, allowing individual chemical

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bonds between surface atoms to be imaged [57] and used to directly measure the force as a single atom was moved across a surface [58]. FM-AFMs typically use phase-locked loops (PLLs) to maintain the drive frequency of the oscillator at the cantilever resonance. Briefly, PLLs involve circuitry or logic that utilizes the measured phase lag between excitation and response signals as the error signal for a feedback loop that maintains the cantilever phase at a constant value (typically 90°, the resonance point of the oscillator) by adjusting the frequency of the cantilever drive oscillator. In addition to the non-contact applications, PLLs can also be used to track contact resonance frequencies [59, 60].

Dual AC Resonance Tracking In the last several years, measuring dissipation has attracted increasing interest as it provides information on energy losses and hysteretic phenomena associated with magnetic, electrical, and structural transformations at the tip–surface junction. Traditional heterodyne detection schemes for the amplitude and phase dispersion in the cantilever drive can change the quantitative and even the qualitative dissipation [61]. Multiple frequency measurement including bimodal [25] and band- excitation [17, 62] techniques offer the possibility of independent correction of transfer function dispersion. Dual AC resonance tracking [25] (DART) technique provides much higher sensitivity to mechanical changes at the tip–sample junction than does the cantilever deflection. In addition to tracking the resonant frequency, this technique allows the additional model parameters such as the tip–sample dissipation to be measured. Figure 5.13 shows the basic idea behind DART. The cantilever is driven with a linear combination of sinusoidal voltages, one above (f2) and one below (f1) the resonant frequency f0. This dual frequency signal is used to modulate the tip–sample interaction. This modulation could be of the tip–sample distance as in UFM and AFAM or the tip potential as in PFM or even the tip–sample temperature if a heated cantilever is used. The resulting motion of the cantilever is measured with the standard position-sensitive detector. This signal in turn is used as the input for two lock-in amplifiers, where the sinusoidal drive at f1 and f2 is used as references for each lock-in. The output of the lock-in amplifiers consists of the two amplitude and phase pairs measured at each frequency A1, j1, A2, j2. To implement resonance tracking, the difference between the two amplitudes, A2–A1, is used as the error signal for frequency feedback. As noted above, since there are four independent measured quantities in DART, we can solve for four model parameters. By measuring the amplitudes and phases at the two drive frequencies, one can solve for the unknown model parameters Adrive, jdrive, w0 and Q as described below [63]. From (5.5) and noticing that A1»A2 we have

(ω 02 − ω12 )2 + (ω 0 ω1 / Q)2 = (ω 02 − ω 22 )2 + (ω 0 ω 2 / Q)2

(5.8)

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Fig. 5.13 In dual AC resonance tracking (DART), the cantilever is both driven and measured at a frequency below and a frequency above resonance. The total drive voltage is the sum of voltages at frequencies f1 (blue) and f2 (red). The cantilever deflection then contains information at both of those frequencies (purple). The amplitude and phase at the two frequencies are then separated again by the two lock-ins and passed on to the controller. Resonance tracking uses the difference of the two amplitude measurements as the error signal input for a frequency feedback loop

which, on simplification results in

ω 02 =

ω 22 + ω12 2(1 − 1 / 2Q 2 )

(5.9)

The parameters Q, w0 and jdrive can be solved iteratively using the following relations in addition to (5.3)

Q−

ω 0 ω 2 / (ω 02 ω 22 ) tan (ϕ '2 − ϕ drive )

ω ω /Q ϕ drive = ϕ '1 − tan −1  02 1 2  ω 0 − ω1

(5.10)

 . 

(5.11)

An example of the results of this process is shown in Fig. 5.14 where the topography, contact resonance and Q are plotted on a sample from a section of a

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Fig. 5.14 (a) Topography, (b) contact resonance frequency, and (c) quality factor for a section of a printed circuit board at the border between the copper conductive layer and polymer support. The polymer support shows both a greatly reduced stiffness and increased dissipation (decreased cantilever quality factor, Q) compared with the Cu. Sample courtesy of Junhee Hahn, Korea Research Institute of Standards and Science (KRISS), South Korea

printed circuit board at the border between the copper conductive layer and polymer support. The topography image in Fig. 5.14a shows some increased roughness on the right side over the polymer. The derived contact resonance (b) shows a distinct drop over the polymer (roughly 100 kHz) and the quality factor

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(c) shows a significant lowering, consistent with increased dissipation over the polymer region.

Intermodulation AFM As discussed above, nonlinearities in the tip–sample interaction can complicate the dynamics and interpretation of the cantilever response. A technique that makes direct use of this is dubbed Intermodulation AFM (IMAFM) [64]. IMAFM uses two pure harmonic drive tones at frequencies f1 and f2 to excite the cantilever. The nonlinear tip–surface force will cause a mixing of these drive tones, so that response is generated at frequencies called intermodulation products, fIMP=nf1+mf2, where n and m are integers (see Fig. 5.15). The response at each intermodulation frequency contains both amplitude and phase, each carrying information about the cantilever dynamics. The technique allows much information to be gathered in the spectral

Fig. 5.15 (a) The amplitude of the response of a free cantilever, far away from the surface, when driven with two pure harmonic tones, f1 and f2, both placed near resonance. (b) When the cantilever engages the surface, the nonlinear tip–surface force causes intermodulation response of odd order near resonance (e.g., 3H = 2f2 - f1, 3L = 2f1 - f2, 5H = 3f2 - 2f1, 5L = 3f1 - 2f2, etc.)

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Fig. 5.16 The amplitude of intermodulation response at frequencies 3L, 5L, 7L, and 9L, plotted as a function of the approach distance. The zero of approach distance is taken to be the start of the approach ramp, and the cantilever begins to engage the surface at about 2 nm. The intermodulation amplitudes are plotted on a log scale, where the curves 3L, 5L, and 7L have been offset by 1.0, 0.4, and 0.2 nm, respectively, for clarity

response near resonance, which is highly advantageous because the sensitivity of the oscillating cantilever as a force transducer is largest near resonance. With intermodulation AFM one can generate response of high order, l = |m|+|n|, efficiently using the system bandwidth in comparison to measuring harmonics. Harmonics requires a second torsional oscillator to get an appreciable transfer gain, and a system bandwidth of l times the resonant frequency to acquire the lth harmonic. Each intermodulation product shows a unique and complex dependence on approach distance (see Fig. 5.16), and detailed images can be constructed by recording the amplitude and phase at each intermodulation frequency while scanning. Research is underway on inversion algorithms to extract the tip–surface force from the measured intermodulation spectrum at one distance, thus enabling determination of the force curve at each measurement point.

Conclusions, Future Challenges and Opportunities There have been many promising developments in multi-frequency nano-scale measurements. In the case of non- and intermittent-contact imaging applications, the signals at higher resonance modes provides higher compositional contrast and higher spatial resolution while at the same time maintaining very low-force, gentle

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imaging conditions. For contact resonance measurements, multiple measurement frequencies opens up many opportunities to better constrain various tip–sample interaction model parameters. Compositional mapping using the AFM still has many challenges. Qualitatively, the methods described here and others in the literature have been quite successful in mapping relative differences between various regions of a multi-component sample. True quantitative mapping is a much more serious challenge that can also be viewed as opportunities for future work. These barriers include at least the following: (1) quantifying the forces depends on quantifying spring constants, detector sensitivities, effective masses, and damping of the various cantilever resonant modes; (2) interpreting and controlling the sometimes complex dynamics of the multi-modal cantilever interacting with the highly nonlinear and dissipative sample surface, and perhaps most significantly (3) ill-defined contact mechanics that strongly depend on the shape of the cantilever probe. The last issue may be the most difficult to overcome because the relevant length scale of the mechanical contact is extremely small (this is where the high resolution of AFM comes from after all) and even if a tip is well characterized before and/or after making measurements, tips clearly wear and pick up contaminants over time, making those shape measurements only of transient relevance.

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35. A. Hammiche, M. Reading, H.M. Pollock et al., “Localized thermal analysis using a miniaturized resistive probe,” Review of Scientific Instruments 67 (12), 4268–4274 (1996). 36. A. Hammiche, D.J. Hourston, H.M. Pollock et al., 1996 (unpublished). 37. D. Hurley, in Applied Scanning Probe Methods, edited by B. Bushan, H. Fuchs, and H. Yamada (Springer, Berlin, 2009), Vol. XI. 38. J.N. Goodier S.P. Timoshenko, Theory of Elasticity. (McGraw-Hill, London, 1970). 39. R. Hillenbrand, M. Stark, and R. Guckenberger, “Higher-harmonics generation in tappingmode atomic-force microscopy: Insights into the tip–sample interaction,” Applied Physics Letters 76 (23), 3478–3480 (2000). 40. R. Proksch, Patent No. 7,603,891 B2 (2009). 41. D.J.S. Hulmes, A. Miller, D.A.D. Parry et al., “Analysis of primary structure of collagen for origins of molecular packing,” Journal of Molecular Biology 79 (1), 137–148 (1973). 42. A.V. Kajava, “Molecular packing in type-I collagen fibrils – a model with neighboring collagen molecules aligned in axial register,” Journal of Molecular Biology 218 (4), 815–823 (1991). 43. J.W. Li, J.P. Cleveland, and R. Proksch, “Bimodal magnetic force microscopy: Separation of short and long range forces,” Applied Physics Letters 94 (16) (2009). 44. D. Ziegler, J. Rychen, N. Naujoks et al., “Compensating electrostatic forces by single-scan Kelvin probe force microscopy,” Nanotechnology 18 (22) (2007). 45. B.J. Rodriguez, S. Jesse, S. Habelitz et al., “Intermittent contact mode piezoresponse force microscopy in a liquid environment,” Nanotechnology 20 (19) (2009). 46. D. Passeri, A. Bettucci, M. Germano et al., “Local indentation modulus characterization of diamondlike carbon films by atomic force acoustic microscopy two contact resonance frequencies imaging technique,” Applied Physics Letters 88 (12) (2006). 47. A.P. French, Vibrations and Waves. (CRC, Florida, 1971). 48. J. Tamayo and R. Garcia, “Relationship between phase shift and energy dissipation in tappingmode scanning force microscopy,” Applied Physics Letters 73 (20), 2926–2928 (1998). 49. T.R. Albrecht, P. Grutter, D. Horne et al., “Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity,” Journal of Applied Physics 69 (2), 668–673 (1991). 50. F.J. Giessibl, “Atomic-force microscopy in ultrahigh-vacuum,” Japanese Journal of Applied Physics Part 1: Regular Papers Short Notes & Review Papers 33 (6B), 3726–3734 (1994). 51. E. Meyer, L. Howald, R. Luthi et al., “Scanning probe microscopy on the surface of SI(111),” Journal of Vacuum Science & Technology B 12 (3), 2060–2063 (1994). 52. H. Yamada, K. Kobayashi, T. Fukuma et al., “Molecular resolution imaging of protein molecules in liquid using frequency modulation atomic force microscopy,” Applied Physics Express 2 (9) (2009). 53. J.I. Kilpatrick, A. Gannepalli, J.P. Cleveland et al., “Frequency modulation atomic force microscopy in ambient environments utilizing robust feedback tuning,” Review of Scientific Instruments 80 (2) (2009). 54. T. Fukuma, K. Kobayashi, K. Matsushige et al., “True atomic resolution in liquid by frequency-modulation atomic force microscopy,” Applied Physics Letters 87 (3) (2005) 55. T. Fukuma, M. Kimura, K. Kobayashi et al., “Development of low noise cantilever deflection sensor for multienvironment frequency-modulation atomic force microscopy,” Review of Scientific Instruments 76 (5) (2005). 56. F.J. Giessibl, “Advances in atomic force microscopy,” Reviews of Modern Physics 75 (3), 949–983 (2003). 57. F.J. Giessibl, S. Hembacher, H. Bielefeldt et al., “Subatomic features on the silicon (111)(7×7) surface observed by atomic force microscopy,” Science 289 (5478), 422–425 (2000). 58. M. Ternes, C.P. Lutz, C.F. Hirjibehedin et al., “The force needed to move an atom on a surface,” Science 319 (5866), 1066–1069 (2008). 59. K. Yamanaka, Y. Maruyama, T. Tsuji et al., “Resonance frequency and Q factor mapping by ultrasonic atomic force microscopy,” Applied Physics Letters 78 (13), 1939–1941 (2001).

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60. K. Kobayashi, H. Yamada, and K. Matsushige, “Resonance tracking ultrasonic atomic force microscopy,” Surface and Interface Analysis 33 (2), 89–91 (2002). 61. R. Proksch and S. Kalinin, “Energy dissipation measurements in frequency modulated scanning probe microscopy,” Nanotechnology, submitted (2009). 62. S. Jesse, P. Maksymovych, and S.V. Kalinina, “Rapid multidimensional data acquisition in scanning probe microscopy applied to local polarization dynamics and voltage dependent contact mechanics,” Applied Physics Letters 93 (11) (2008). 63. A. Gannepalli and R. Proksch, “Submitted,” (2009). 64. D. Platz, E.A. Tholen, D. Pesen et al., “Intermodulation atomic force microscopy,” Applied Physics Letters 92. (15) (2008).

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Chapter 6

Dynamic Nanomechanical Characterization Using Multiple-Frequency Method Ozgur Sahin

Abstract Macroscopic behavior of materials, whether synthetic or biological, depends on the morphology and characteristics of their microscopic constituents. Improving the performance of engineered materials and understanding the design principles of biomaterials demand tools that can characterize material properties with nanoscale resolution. What is the spatial arrangement of the components of a heterogeneous material? Are the material properties of those components different from their respective bulk properties? How do material properties change near the interfaces? What is the influence of temperature, electric or magnetic fields, or solvents? Answering these questions is of critical importance to the rational design of advanced materials and to the analysis of biological materials. In this chapter, we focus on the recent advances in the measurement and characterization of dynamic nanomechanical properties with high spatial resolution using specially designed atomic force microscope cantilevers. We will first describe the basic operation principles of this method and present results to judge its performance on various material systems. Functional materials generally consist of multiple components with different material properties and feature dimensions. The level of complexity in these materials demands sophisticated characterization tools. Before going into details of the nanomechanical measurements, we list some of the desired attributes of a good nanomechanical characterization instrument below: Spatial resolution. A characterization tool should be able to provide a spatial resolution better than the smallest feature size of the sample, which approximately corresponds to 1–10 nm. In addition, the material properties should be mapped across a surface area wide enough to capture the properties of different components. The scan area can range between 100 nm and 100 µm or even more. Sensitivity and dynamic range. The properties of different components of a material can vary by several orders of magnitude and in some cases they can vary only slightly. The characterization tool should be able to differentiate both small and large variations in the properties being mapped. O. Sahin (*) The Rowland Institute at Harvard, Cambridge, MA, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_6, © Springer Science+Business Media, LLC 2010

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Measurement speed. In research and development of functional materials, the characterization step should be ideally faster than the preparation of the samples so that it does not become a bottleneck in the development process. While faster measurements offer practical advantages, it should be noted that mechanical properties of materials are frequency dependent, i.e., faster measurements will necessarily reflect high frequency mechanical response. Non-destructive measurements. Mechanical interactions can damage the sample under test as well as the probe, such as a sharp indenting tip. Non-destructive measurements permit repeated analysis of the same sample region and also permit additional analysis with other methods. In some cases, non-destructive measurements enable characterization of the sample while it is functioning. In order to measure mechanical properties of samples with high spatial resolution, one has to bring a sharp object, such as an atomic force microscope tip, in contact with the sample surface. If the sharp tip is pressed against the surface, the sample will deform under the influence of localized interaction forces. The amount of stored or dissipated energy in this process, then, allows to measure local mechanical properties. In principle, larger forces provide better sensitivity and measurement speed. However, larger forces also result in increased contact areas that compromise spatial resolution and potentially damage the sample or the probe tip. In these respects, the attributes of a good nanomechanical characterization instrument, as listed above, have contradicting requirements on the forces involved in the mechanical measurements. Non-destructive operation and high spatial resolution demand lower forces, whereas sensitivity, dynamic range, and measurement speed demand higher forces. In general, spatial resolution and non-destructive operation are given higher priorities over measurement speed and dynamic range; therefore, researchers often spend more time characterizing their samples and they also use different probes or instruments to extend the dynamic range of the measurements. In this chapter, we will describe a non-destructive nanomechanical measurement technique that promises high spatial resolution and high speed characterization of materials with large dynamic range. This method is based on tapping-mode atomic force microscopy, where an oscillating cantilever periodically and intermittently interacts with the sample. A special geometrical design of the tapping cantilever turns this widely used nanoimaging method into a quantitative material analysis tool. We will first describe the technical details of this approach, which will allow us to judge its limitations and potential. Subsequently, we will give examples of measurements obtained on a wide range of material systems that illustrate the speed, spatial resolution, and dynamic range of this technique. Finally, an application of this technique in observing the glass transition of polymer blends with sub-micron domains will be presented.

Measurement Basics Measurement of mechanical properties of materials primarily relies on the observation of the deformation of a sample under a known external load. Two physical quantities, position and force, have to be determined. For example, in the case of a

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Hookian spring, the ratio of force and displacement corresponds to the spring constant that fully characterizes the mechanical properties of the spring. When a sharp object is indenting a sample, such as nanoindenters or atomic force microscopes, the forces and displacements do not necessarily follow a linear relationship. This is not because the sample has a non-linear mechanical behavior, but it is merely a consequence of non-linear evolution of the contact geometry as the sharp object indents the sample. In this more general case, measurement of a force– distance curve is needed to account for the geometrical effects and fully characterize the mechanical properties of the sample. To obtain high spatial resolution in mechanical measurements, one has to keep the size of the contact region small. Sharper probes and lower interaction forces are the two important factors in improving spatial resolution. As a result of lower interaction forces, the deformation of the sample also gets smaller. Note that the magnitudes of the two quantities necessary for mechanical characterization have to be reduced in order to improve spatial resolution. Therefore, the sensitivity and accuracy of the displacement and force measurements become even more critical in high spatial resolution mechanical measurements. In this section, we will review the displacement and forces in tapping-mode AFM. We will show that a specially designed cantilever can perform accurate displacement and force measurements while the tip is intermittently striking and deforming the sample.

The Tapping Cantilever Moves in a Sinusoidal Trajectory In tapping-mode AFM, the force probe, a flexible cantilever beam with a sharp tip, is vibrated at its resonance frequency near the surface [1]. The amplitude of the vibrations is sufficiently large so that the sharp tip interacts with the sample only at the bottom of its trajectory and does not stick to the surface. The amplitude of the vibrations is held constant by a feedback mechanism that adjusts the vertical position of the cantilever base with respect to the sample. When the cantilever is scanned across the sample, the feedback signal corresponds to the surface topography. The intermittent contact of the tip minimizes lateral interactions and prevents damage to the sample and the tip. The resonant operation minimizes the vertical forces because on resonance it takes much less force to generate or dampen the motion of the cantilever. The reduction in forces is mainly determined by the qualityfactor of the cantilever and it is on the order of 100 in ambient conditions. The resonant operation has another important advantage in mechanical measurements – the tip moves in a sinusoidal trajectory. By measuring the vibration amplitude and phase, one can accurately determine the position of the tip with respect to time. The reason why the tip is moving in a sinusoidal trajectory can be understood by considering the speed of the response of the cantilever to rapid changes in external forces on the tip. When the external forces change at rates below the resonance frequency of the cantilever, the cantilever can respond to the forces.

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If the forces change at rates beyond the resonance frequency, however, the cantilever cannot respond to the forces effectively. In the special case of tappingmode operation, the cantilever is driven at the resonance frequency, so the changes in forces due to the intermittently striking tip are at much higher rates than the resonance frequency. Therefore, the sinusoidal motion of the cantilever remains largely unaffected by the intermittent force pulses acting on the tip. As a result, the resonant excitation in the tapping-mode allows accurate determination of the tip position in time. Mechanical measurements require determination of both forces and displacements. As discussed above, the tapping cantilever provides a good way of determining the tip position in time. If one can also determine the temporal variation of forces acting on the tip, one can extract force–distance relationships and characterize mechanical properties of the samples. In principle, forces acting on the tip can be detected from the bending or vibrations of the cantilever. Unfortunately, the very reason that moves the cantilever in a sinusoidal trajectory makes it insensitive to the tip–sample interaction forces, because the forces are changing faster than the resonance frequency of the cantilever and their effect on the cantilever motion is negligible. This dilemma can be resolved by using special cantilevers that allow access to multiple mechanical modes with different resonance frequencies. In this scheme, the mechanical mode with the lower resonance frequency can be used to determine position by using the tapping-mode principles described above and the mode with the higher resonance frequency can be used to determine temporal variation of interaction forces. Surprisingly, the twisting and bending modes of rectangular AFM cantilevers provide the mechanical resonances with low and high resonance frequencies. To gain better insight on the dilemma of making simultaneous displacement and force measurements and how it is resolved by specially designed cantilevers, we are going to investigate cantilever motion and tip–sample interaction forces in tapping-mode AFM.

Information About the Mechanical Properties Are Contained in Higher Harmonic Forces The vibrating cantilever in tapping-mode AFM can be modeled with a damped harmonic oscillator driven by the external excitation force and the tip–sample forces. Both of these force components are time depended, and so does the cantilever motion. The equation of motion for the cantilever depends on its spring constant, quality factor (a measure of damping), resonance frequency, and the external forces. Under typical imaging conditions, it is observed that the cantilever motion is mainly sinusoidal, i.e., single frequency. This has been justified by the qualitative arguments in the previous section and by the numerical simulations of the cantilever dynamics reported extensively in the literature. Here, we will summarize the results of numerical simulations to the extent they relate to the measurement of the mechanical properties of the sample.

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Our analysis of the tip–sample interaction forces begins by modeling the mechanicsof the tip–sample contact. We use the Derjaguin–Muller–Toporov model [2], which predicts the forces between a spherical tip and a flat surface as follows:

4 Ftip - sample = E * R (d - d 0 )3/ 2 + Fadhesion . 3

(6.1)

Here Ftip-sample is the interaction force between the tip and the sample, E* is the reduced elastic modulus of the tip–sample ensemble, R is the radius of the tip, d0 is the rest position of the surface, d-d0 is the depth of indentation, and Fadhesion is a constant attractive adhesion force between the tip and the sample that is present during the contact. The conclusions that will be derived in this section are independent from the specific functional form of (6.1). The key attribute of the contact mechanics that we intend to capture is the increase in the forces with higher elastic modulus or with larger indentations. Nevertheless, experimental measurements of mechanical properties that will be presented later in this chapter suggest that (6.1) provides a good description of the tip–sample forces. We have carried out numerical simulations [3] of the equation of motion for a cantilever tapping on samples with three different elastic modulus values: 1GPa, 100MPa, and 10MPa. The cantilever’s spring constant and quality factor are assumed to be 2N/m and 50, respectively. The amplitude set-point is assumed to be 30 nm and the driving force is selected such that the free vibration amplitude of the cantilever is equal to 40 nm. In addition, the adhesion force Fadhesion is set to 1.5 nN. Figure 6.1 presents the spatial, temporal, and frequency domain characteristics of tip–sample interaction. Each row of graphs belongs to a different elastic modulus value for the sample. The force–distance relationships given in the left column extends 60 nm, twice the set-point amplitude. The rest position of the surface is set to 0nm in each plot. The slopes of the forces in the interaction region (negative distances) are proportional to the elastic modus of the sample as determined by (6.1). Note that the peak forces and indentation depths are different for each sample. While the indentation depths increase toward the more compliant samples, the peak forces reduce. The reason why peak forces and indentation depths vary with the elastic modulus is better understood by analyzing the temporal characteristics of the tip–sample forces. During the vertical oscillations of the cantilever, the position of the tip remains in a sinusoidal orbit. So the force–distance relationships on the left column in Fig. 6.1 can be plotted against time to obtain force–time waveforms seen in the middle column. In addition to the observed reduction in peak forces toward the more compliant samples, these force–time waveforms show that the duration of contact is also increasing with decreased elastic modulus. This behavior is predicted by the analytical solutions derived to relate cantilever vibration amplitude and phase to the driving force and the mechanical properties of the sample. It is found that under the same driving force and set-point amplitude, the time-average forces remain approximately constant regardless of the elastic modulus of the sample. So with larger peak forces, the duration of contact gets smaller.

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Fig. 6.1 Simulated tip–sample forces in tapping-mode AFM. Three samples with different elastic modulus values are used: (a) 1 GPa, (b) 100 MPa, and (c) 10 MPa. Force–distance, force–time, and force–frequency relationships calculated on each sample are given in separate columns (cf. [3])

In the above analysis of the temporal characteristics of the interaction forces, we see that despite large variations in the tip–sample force waveform, the time-average forces remain approximately constant on samples with different elastic modulus values. Unfortunately, the two measured quantities in the tapping mode, vibration amplitude and phase, tell us only about the average forces. Therefore, much of the information about the elastic properties of the sample is lost. The following analysis of the frequency domain characteristics of the tip–sample forces will explain where the lost information is and what is needed to recover it back. The cantilever vibrations and the tip–sample forces are periodic in time. The force–time waveforms given in the middle column of Fig. 6.1, therefore, represent single cycles of the periodic tip–sample forces. Each of these waveforms can be seen as a superposition of forces at different frequencies, which corresponds to the Fourier transform of these waveforms [4]. In the specific case of periodic waveforms of the tapping mode, the force components are at exact integer multiples of the fundamental frequency, i.e., the driving frequency. These force components are referred as harmonic forces. Their magnitudes are plotted with respect to frequency on the rightmost column in Fig. 6.1. When we compare the spectrum of harmonics on each sample, we see that the magnitudes of the first few harmonics are approximately the same at different elastic modulus values. The differences between the samples are mainly at higher order harmonics. This tells us that the information regarding the elastic properties of the samples is at the higher harmonics. Moreover,

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it is seen that the magnitudes of the higher harmonics are larger on the stiffer samples. The harmonic index where the first local minimum of the spectrum is a bigger number for stiffer samples compared to the compliant samples. This is a consequence of the Fourier transform, in which temporally narrower pulses lead to wider frequency spectra. From the analysis of the higher harmonics of the tip–sample forces, we learn that measuring high frequency force components is necessary to reconstruct tip–sample force waveforms. If recovered, this waveform is sufficient to obtain the force– distance relationship for a given sample, because the position of the tip is known to be a sinusoid in time. Then, by assuming a contact mechanics model, such as (6.1), one can estimate material properties like elastic modulus. The spectra of higher harmonics in Fig. 6.1 illustrate the challenge of measuring the mechanical properties of samples in tapping-mode AFM. The two experimentally accessible parameters, vibration amplitude and phase, are measured at the driving frequency, i.e., the first harmonic. The high frequency force components of the tip–sample interactions are not accessed by the conventional amplitude and phase measurements; hence the information about the elastic properties of the sample is lost. Just like the tip–sample force waveforms, the vibrations of the cantilever also contain frequency components. Therefore, in principle, the high frequency vibrations can be used to recover the high frequency force components [5]. However, the amplitude of those vibrations is small and therefore difficult to detect. Indeed, the statement that the cantilever motion is mainly a sinusoid is equivalent to the statement that the higher harmonic vibrations have small amplitudes. This brings us to the conclusion that accessing higher harmonic forces, and therefore the mechanical properties, demands improved performance of the force-sensing cantilever at higher frequencies without perturbing the tapping-mode operation.

There Are Multiple Force Sensors in a Single Cantilever The vibrations of the AFM cantilever are governed by continuum mechanics. The cantilever base is fixed to a supporting chip and the sharp tip is near the free end. This configuration leads to multiple vibration modes whose resonance frequencies lay near the frequencies of the higher harmonics forces discussed in the previous section [6]. The response of a given vibration mode to the forces acting on the tip can be represented with an effective spring constant, resonance frequency, and quality factor. In principle, any of these modes can be actuated to displace the tip or more importantly they can be used as independent force sensors. In AFM cantilevers the fundamental mode, i.e., the mode with the lowest resonance frequency, has an effective spring constant more than an order of magnitude smaller than any of the higher order modes. In the special case of tapping-mode AFM, the cantilever is driven at the resonance frequency of the fundamental mode where the displacement response of this mode is further enhanced by the sharp resonance. As a result,

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the response of the fundamental mode dominates the response of the vibrating cantilever in the tapping mode. For practical purposes, a single mode description of the cantilever is sufficient to predict the vibration amplitude and phase response of the tapping cantilever. However, for the measurement of higher harmonic forces the response of those vibration modes whose resonance frequencies fall within the ranges of harmonic frequencies of interest should be considered. The tapping cantilever is subject to tip–sample forces whose pulse-shaped temporal characteristics are depicted in Fig. 6.1. The width and the height of the pulses depend on the mechanical properties of the sample [7]. The overall motion of the cantilever is a superposition of the response of each mode to these forces. Unfortunately, this superposition makes it difficult to access the individual responses of these modes. The motion of the cantilever is detected by a laser beam reflected from the cantilever falling onto a position-sensitive photo detector. The signal at the detector is proportional to the slope of the cantilever at the region where the laser spot is focused [8]. Because the slope of the cantilever results from a summation of the slopes originated in each mode, one cannot determine the contribution of each mode separately. However, there is one important exception, and that enables accessing the temporal characteristics of the tip–sample force waveform. The photo detectors used in AFM instruments are split into four quadrants to detect lateral and vertical motion of the laser spot simultaneously. This allows distinguishing flexural (bending) and torsional (twisting) motion of the cantilever; since flexural deflections result in vertical motion of the laser spot and torsional deflections result in lateral motion. In typical AFM cantilevers, the tip–sample forces do not excite torsional vibrations of the cantilever because the sharp tip is conventionally placed at the center of the cantilever so that no torque is generated around the symmetry axis of the cantilever. However, if the sharp tip is placed offcentered, the interaction forces will twist the cantilever. It turns out that the mechanical properties of the first torsional mode make it suitable for measuring the tip–sample forces. This configuration is illustrated in Fig. 6.2. The T-shaped cantilever in Fig. 6.2 is named as the torsional harmonic cantilever, since it allows the measurement of higher harmonic forces through the torsional oscillations of the cantilever [9, 10]. When this cantilever is oscillated vertically as in conventional tapping-mode AFM, the tip–sample forces generate a twisting motion that can be independently detected by the quadrant photo detector. The twisting motion has much smaller amplitude compared to the vertical oscillations of the cantilever and even compared to the typical indentation depths of samples; therefore its effect on the overall tapping-mode operation is negligible. On the other hand, the first torsional mode has a much higher resonance frequency compared to the fundamental flexural resonance frequency so that it can follow the rapidly changing tip–sample forces. In addition, the sensitivity of the torsional modes is geometrically enhanced by the short torsion arm, i.e., the tip offset distance, because smaller tip displacements resulting from a twist in the cantilever produce larger slopes compared to flexural modes. Figure 6.3 shows calculated frequency responses of flexural and torsional vibration modes of a torsional harmonic cantilever and experimentally recorded vibration spectra

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Fig. 6.2 Diagram of the torsional harmonic cantilever operation. The T-shaped cantilever with an offset tip is vibrated near the surface vertically at its resonance frequency. The cantilever is twisted by the interaction forces. The resulting torsional motion is used to generate high-speed force– distance curves (cf. [10])

of a torsional harmonic cantilever tapping on a polystyrene sample. The frequency response calculations are performed for rectangular cantilever geometry, 30 µm wide and 300 µm long, with a tip 15 µm offset from the longitudinal symmetry axis of the cantilever. The response curves are plotted for frequencies up to 20 times the fundamental flexural resonance frequency. In this frequency range we see three resonance peaks in the flexural response and one resonance peak in the torsional response, which has a frequency comparable to the third flexural resonance. Furthermore, the response of the torsional mode is larger than the flexural modes at higher frequencies. This means the torsional mode combines high bandwidth and sensitivity, which is essential for sensitive and accurate measurements of tip–sample forces. The higher harmonic force components of the tip–sample interactions drive flexural and torsional vibrations in proportion to their response at each harmonic frequency. The vibration spectra shown in Fig. 6.3b, c show the measured amplitudes of the flexural and torsional vibrations at each frequency. The peaks in these spectra correspond to the harmonics, i.e., integer multiples of the drive frequency. The peak with the largest amplitude is in the flexural vibrations and it is exactly at the drive frequency. All other peaks in flexural and torsional vibrations have amplitudes that are orders of magnitude smaller than the large peak at the drive frequency. This means, for practical purposes, the cantilever motion can be considered to be sinusoidal in time at the drive frequency. Other peaks, while good for mechanical property measurements, do not have significant effects on the tip position.

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Fig. 6.3 Frequency response (a) and vibration spectra (b, c) of a torsional harmonic cantilever. The frequency axis is normalized to the first flexural resonance frequency. Flexural (b) and torsional (c) vibration spectra is recorded on a polystyrene sample. The torsional peaks show enhanced signal levels at higher frequencies (cf. [9])

The spectrum of torsional vibrations does not attenuate up to frequencies 20 times as much as the driving frequency. Therefore, it recovers the information in the higher harmonic tip–sample forces. However, the relationship between the forces and torsional vibrations is not trivial. The harmonic forces that are closer to the torsional resonance frequency are translated into larger vibrations compared to the harmonics that are away from the torsional resonance. This feature of the torsional response is evident in the torsional vibration spectrum in Fig. 6.3b as peaks around the 16th harmonic exhibit larger magnitudes compared to the other peaks. The effect of the torsional resonance can be approximated by the frequency response of a simple damped harmonic oscillator that matches the resonance frequency and

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q uality factor of the first torsional vibration mode. Then, one can numerically correct the torsional resonance effects. Given the periodic motion of the tapping cantilever, it is easier to carry out these numerical calculations in frequency domain where the frequency response of the torsional mode can be approximated as:

H T (w ) =

1 . w - w + iww T / Q 2 T

2

(6.2)

Here w is the angular frequency, wT is the torsional resonance frequency, and Q is the quality factor of the torsional resonance. Note that HT is a complex number. This is because there is a time delay between the higher harmonic forces and the response of the cantilever to those forces, which translates to a phase shift in the frequency domain. In practical implementations, the numerical calculations are carried out with a computer in a semi-automatic fashion. The user needs to input calibration parameters like the force sensitivity of the torsional mode and photo-detector sensitivity to tip displacements. Then, the computer program transforms the torsional vibrations into the frequency domain by taking its Fourier transform and then corrects for the torsional resonance effects by dividing it to (6.2). The resulting signal is transformed back to the time domain, by taking its inverse Fourier transform, so that a tip–sample force waveform is obtained. Figure 6.4 depicts detector signals and calculated tip–sample forces recorded on a highly oriented pyrolytic graphite sample. The cantilever used in this experiment is driven at its flexural resonance frequency at 47.4 kHz. The periodic vibration signals in both vertical and lateral deflection signals are averaged over 12 consecutive oscillation cycles to reduce noise. The resulting waveforms are given in Fig. 6.4a. Note that the flexural response is mainly a sinusoid; however, the torsional response is complicated. The most important component of the torsional response is marked with an orange arrow. This is where the tip hits the surface at the lowest point in its trajectory. As a result of the impact, the cantilever twists, and the peak at the arrow is generated. After this impulse, the torsional mode exhibits oscillatory response that is natural to weakly damped harmonic oscillators. Because the tip swings away and hits back to the surface in the following oscillation cycle, the high frequency torsional oscillations do not decay completely. Another feature of the torsional vibration signal is the crosstalk from the large vertical oscillation signal. It is seen that the impact at the arrow and the oscillatory response are superimposed on a low frequency sinusoid. That low frequency signal is exactly at the drive frequency and it is mainly due to a crosstalk originating from misalignments in the cantilever and photo detector alignment. Even a small angular misalignment can result in considerable crosstalk because vertical oscillation has much larger signal amplitude compared to the torsional vibrations. The effects of the torsional resonance can be corrected numerically by using (6.2). However, this procedure will not eliminate the crosstalk-related sinusoidal component. The crosstalk component can be eliminated by subtracting a sinusoid such that the value of the resulting waveform is zero when the tip is away from the surface, i.e., outside the pulse marked with the arrow in Fig. 6.4a. The amplitude

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Fig. 6.4 Reconstructing the tip–sample force waveform. (a) Oscilloscope traces of the periodic flexural (blue) and torsional (orange) vibration signals at the position-sensitive detector, obtained on graphite. (b) Time-resolved tip–sample force measurements calculated on graphite. (c) The same data as in (b), plotted against tip–sample distance. Negative distances mean that the sample is indented. Arrows indicate the direction of motion. The solid part of the curve marks the points between the largest sample indentation and breaking of the contact on the retraction portion of the curve (cf. [9])

and phase of the sinusoid to be subtracted is determined by linear curve fitting. Note that the complete numerical procedure takes about 10 µs for a typical desktop computer (2008). This is sufficient to carry out real-time calculations for a cantilever tapping at 100 kHz. Figure 6.4b shows the resulting tip–sample force waveform after calculations. The pulse-shaped force waveform and attractive and repulsive force components are visible in this waveform. Once the tip–sample force waveform is recovered, it can be plotted against the tip–sample distance to obtain the force–distance relationship. The tip sample distance

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can be obtained from the vertical signals in Fig. 6.4a; however that waveform does not directly correspond to tip position. Instead, we use a computer-generated sinusoid that has the same amplitude with the cantilever vibrations. This change also eliminates the noise in instantaneous position measurements. Figure 6.4c plots the resulting force–distance relationship on a graphite sample. The origin of the position is arbitrarily referenced to the lowest force measurement. This point approximately corresponds to the rest position of the surface. Note that there are two force values recorded for a given position of the tip; one for the approach and one for the retraction. The differences between the approach and retraction become more prominent at the first 2nm above the rest position of the surface. The resulting hysteresis dissipates energy at each cycle of the oscillating cantilever. The force–distance relationship also displays increasing forces at positions below the rest position of the surface. As the tip pushes into the sample, mechanical restoration forces rise up. For spherical tip geometry, the relationship between the forces and indentations are approximately given by (6.1). With the knowledge of tip size, one can estimate the elastic modulus of the sample by analyzing the measured force–distance curve. To demonstrate the effects of sample elastic modulus on the tip–sample force waveform and force–distance curves, we present measurements on samples with different stiffness. The force–time and force–distance relationships given in Fig. 6.5a, b belong to high- and low-density polyethylene samples. These samples are

Fig. 6.5 Experimental measurements of force–time and force–distance relationships on (a) highand (b) low-density polyethylene samples under identical tapping-mode feedback conditions. The shapes of the curves are determined by the mechanical properties of each sample (cf. [3])

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known to have distinct mechanical properties with typical elastic modulus values of 750 and 50 MPa, respectively. The tip–sample force waveform recorded on highdensity polyethylene shows larger peak force and smaller pulse duration compared to the waveform recorded on low-density polyethylene. As previously discussed, this leads to constant time-average forces in tapping mode. The force waveforms also exhibit non-zero values during the times when the tip is away from the surface. Therefore, these forces reflect the noise in force measurements. The respective force–distance curves in Fig. 6.5 reveal the differences in the samples more clearly. The slopes of the curves in the region where the tip is pushing into the sample are largely different. The tip radius for the cantilever used in this experiment is approximately 7 nm. When we use this number with (6.1) and perform a curve fitting to the force–distance data, we estimate the reduced elastic modulus values as 588 MPa for the high-density polyethylene and 56 MPa for the low-density polyethylene. These numbers are fairly close to the typical values for these samples. Note that unlike the theoretical models, the exact position of the sample surface is not well known. This poses a challenge while fitting (6.1) to the force–distance curves. Instead of assuming a fixed position for the surface, the fitting process finds the surface position that provides the best fit together with the elastic modulus value of the sample.

High-Speed and High Spatial Resolution Nanomechanical Analysis with Large Dynamic Range At the beginning of this chapter, we have listed a number of key aspects of nanomechanical sensors: spatial-resolution, sensitivity and dynamic range, measurement speed, and non-destructiveness. In this section, we will present experimental results that demonstrate the performance of the torsional harmonic technique according to the listed measures. We will first investigate the accuracy, sensitivity, and dynamic range of the measurements. Second, we will provide mechanical property maps of heterogeneous materials with feature sizes ranging from 1 to 500 nm. Lastly, we will present an application of this technique in characterizing the phase transition behavior of a polymer blend with sub-micron features.

Torsional Harmonics Provide a Large Dynamic Range in Mechanical Measurements Any measurement instrument or sensor has upper and lower limits on the magnitude of the quantity that it can measure. Beyond a certain value of the measured signal, the output of the sensor will saturate and below a certain value of the measured signal, the output will display noise. The range of values in between these two limits is the dynamic range of the sensor. In the case of elastic modulus measurements, there is a lower and an upper limit of elastic modulus. To measure

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a sample that has an elastic modulus value that falls outside its dynamic range, one has to change the sensor device, i.e., the cantilever in AFMs. In some cases, using different settings for the sensor device can also shift the measurement range slightly. To assess the dynamic range of the measurements with the torsional harmonics, it is therefore important to carry out the measurements with the same cantilever and under identical feedback conditions. These circumstances also reflect the imaging conditions. Furthermore, the measurements bandwidth has to be constant and it should be equal to the bandwidth that is relevant to the experimental timescale. For typical tapping-mode imaging conditions, a bandwidth of 1kHz is sufficient to generate high resolution mechanical maps in less than 10 min. Therefore, in the following analysis of the dynamic range of the torsional harmonic measurements, the bandwidth was set to 1kHz. Figure 6.6 presents histograms of measurements on ten different samples with elastic modulus values ranging from 1 MPa to 50 GPa. This analysis will not only determine the dynamic range of the measurements, but it will also demonstrate the sensitivity and accuracy of the measurements. These samples are chosen because of their availability and knowledge about their approximate elastic modulus. On a given sample, 5,000 consecutive measurements of the tip–sample force waveform are performed and the resulting values are plotted in a histogram given in Fig. 6.6a. The spread of these histograms reflect the measurement noise. The width of the histograms gets narrower toward samples with increasing elastic modulus. Note that the histograms are plotted on a logarithmic scale. This means that the relative errors reduce toward stiffer samples. At lower elastic modulus, around 1MPa, the relative error reaches to 50%. This is approximately where the signal-to-noise ratio of the measurement becomes unity. Measurements below this level of elastic modulus will be dominated by noise. The origin of noise in elastic modulus measurements can be best understood by analyzing the tip–sample force waveforms in Fig. 6.5. We see that the measured forces outside the duration of contact are non-zero due to the noise in the detection of cantilever deflections. As the sample elastic modulus reduces, the peak forces drop and contact durations increase. At a certain level of peak forces, the interactions will become difficult to distinguish from the noisy background forces, therefore the elastic modulus estimate based on the waveform will reflect the force noise. Toward increased elastic modulus values, the histograms point out another mechanism of measurement error besides noise. The measured values on Mica underestimate the elastic modulus of this material. As we discussed in the “Measurement Basics” section, the contact duration becomes narrower and peak forces larger toward stiffer samples. Beyond a certain level of stiffness, the bandwidth of the torsional mode becomes insufficient to follow the rapidly changing forces. As a result, the contact duration in the measured force waveform appears wider than the actual interactions. The elastic modulus estimate based on the measured waveform, therefore, reflects a lower value than the actual elastic modulus of the sample. So, this phenomenon presents a saturation mechanism for the mechanicalmeasurements with torsional harmonics. The analysis of measurement noise and saturation based on the histograms in Fig. 6.6a tells that the lower limit of measurements is around 1MPa and the upper

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Fig. 6.6 Nanomechanical measurements on different materials. (a) Histograms of elastic modulus values calculated from the tip–sample forces. The histograms represent 5,000 consecutive measurements, 1 ms per data point. (b) Mean values and standard deviations of the histograms are plotted against nominal elastic modulus values of the samples (cf. [10])

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limit of measurement is around 10 GPa. A dynamic range of four orders of magnitudecan be especially helpful when analyzing heterogeneous materials with largely different components. Furthermore, in principle, this range can be shifted toward larger or lower values of elastic modulus by using cantilevers with different spring constants. The histograms in Fig. 6.6a also allow judging the accuracy of the measurements. In Fig. 6.6b, the mean values of elastic modulus measurements are plotted against the expected elastic modulus of each sample. The widths of the histograms in Fig. 6.6a are also shown as error bars. A perfect agreement would bring all the data points onto the dashed diagonal line. However, due to deviations between the bulk and surface values of elastic modulus and unknown thermal history, cross-linking density, frequency dependence of mechanical properties, and anisotropic mechanical response of the samples, a perfect agreement is not expected. While there is a fair degree of agreement between the measured values and the expected ones, the deviations could also be attributed to the inaccurate modeling of the contact mechanics. In this analysis, (6.1) is used to estimate the elastic modulus values, however on more compliant samples, around 1–10 MPa, adhesive forces have a larger contribution to the contact geometry and the contact mechanics model becomes more complicated. In addition, viscosity of the samples can also play a larger role in contact mechanics.

High Resolution Maps of Stiffness, Adhesion, and Dissipation Can Be Obtained in a Single Tapping-Mode Scan The analysis of the range of elastic modulus values that can be measured by the torsional harmonics showed that an elastic modulus value can be obtain in 1ms with good sensitivity and dynamic range. This speed is sufficient to generate images of elastic modulus of the samples by scanning the tapping cantilever across the sample surface. In addition to the stiffness of the sample, tip–sample force waveforms also allow to characterize adhesive properties of the sample. The peak attractive force observed during the retraction of the tip provides a measure of adhesion. This value can also be mapped simultaneously with the elastic modulus measurements. Furthermore, the tapping-mode operation readily provides conventional topography and phase image, the latter of the two is a map of energy dissipation [11]. So, in a single scan one can obtain four different and complementary images of the surface. In this section, we will present images on two samples: thermoplastic vulcanizate (TPV) and a triblock copolymer (polymethylmethacrylate-polyisobutylenepolymethylmethacrylate, PMMA-PIB-PMMA). These samples contain stiff and compliant regions, however their morphological organization is different. TPV is composed of micron-sized rubbery domains of ethylene-propylene-diene monomer elastomer (EPDM) and a stiff polypropylene matrix (PP). The triblock copolymer exhibits lamellar phase-separated morphology. Figure 6.7a shows the reduced elastic modulus map of TPV across a 20-µm-sized region. The image shows the rubbery domains of EPDM and the stiffer PP matrix.

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Fig. 6.7 High resolution nanomechanical mapping. Images of thermoplastic vulcanizate (left column) and PMMA-PIB-PMMA block-copolymer film (right column) are obtained in the tapping mode by using torsional harmonic cantilevers. Calculated local elastic modulus are given in (a) and (b).

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The image is colored according to the logarithm of the elastic modulus values and brighter colors correspond to larger values. The numerical values of elastic modulus across a section indicated by the dashed line are plotted right below the elastic modulus map. We see that the EPDM regions are measured to be 5–10 MPa and the PP regions are about 2 GPa. These values are close to the bulk values reported in Fig. 6.6. Note that there is more than two orders of magnitude difference between the two components. The large dynamic range in measurements allowed us to characterize the two components with the same cantilever under imaging conditions. Figure 6.7b shows the map of reduced elastic modulus recorded on a 1-µm-wide region across the triblock-copolymer sample. The two chemical blocks are differentiated by their elastic modulus. The numeric values of elastic modulus across the dashed line given below the image shows that the stiffer region is approximately 1.5 GPa and the compliant region is about 0.5 GPa. While the stiffer component has a value that is close to the bulk value recorded on PP, the value of the compliant region is much higher than that of rubbery materials. The frequency-dependent mechanical response of rubbery materials can account for an increase in the values measured by this technique; however geometrical confinement can be another mechanism that is particularly important in the design of materials with nanoscale feature sizes. Geometrical confinement can enhance molecular or supramolecular ordering. It can also increase the roles of surface tension and surface stress. Therefore, the effects of confinement on the mechanical properties of materials can be dramatic. A high resolution mechanical analysis technique is therefore invaluable for the analysis of confinement effects. Simultaneously with the elastic modulus measurements on each of the two samples, the peak attractive forces are recorded as a measure of adhesion. The resulting images are presented in Fig. 6.7c, d. The respective cross section of the numerical values of peak attractive forces across the same dashed lines shown in Fig. 6.7a, b are given below each image. The images are colored such that attractive forces with larger magnitudes are represented with brighter colors. The image in Fig. 6.7c shows that the stiff PP regions are more adhesive compared to the rubbery EPDM regions. On the contrary, the image in Fig. 6.7d shows that the stiffer PMMA blocks show lower adhesive forces. The noise in force measurements is approximately 0.25 nN rms. However, the cross section data on the TPV sample shows much larger fluctuation on the PP regions. A closer look at the image data reveals that the fluctuations are not due to noise, but there is substantial local variation in adhesive forces. A possible reason for this variation might be the surface roughness. The preparation of the TPV sample for imaging involves cutting process that can introduce surface roughness. Figure 6.7e, f show the phase image recorded on each of the two polymer samples. Phase contrast reflects the energy dissipated at the tip–sample interaction. A lower value of phase corresponds to a larger amount of energy dissipated. There are

Fig. 6.7 (continued) Peak adhesion forces at each location are mapped in (c) and (d). Conventional phase images are given in (e) and (f). Numeric values of each data type across the sections indicated by the dashed lines in (a) and (b) are given below their respective images. Scale bars are (a) 2 mm, (b) 100 nm. Images in (a) and (b) are colored according to logarithms of elastic moduli (cf. [10])

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several physical mechanisms that contribute to the tip–sample energy dissipation. Hysteresis in adhesive forces, capillary forces [12], and viscosity of the sample are among the major contributors. The amount of energy dissipation is also influenced indirectly by the elastic modulus of the sample. For example, during the retraction of the tip, the attractive forces can pull the surface and locally raise it above its equilibrium level. Once the contact is broken the energy stored in the sample is dissipated. In this mechanism, samples with lower elastic modulus and larger adhesion force will dissipate more energy. According to the elastic modulus maps given in Fig. 6.7a, b, there is a larger stiffness contrast on the TPV sample, which would favor increased energy dissipation contrast on this sample. However, the adhesion force maps in Fig. 6.7c, d show an inverted contrast, i.e., on the TPV sample the stiffer component is more adhesive and on the triblock-copolymer sample the compliant component is more adhesive. Therefore, the inversion in adhesive force contrast would increase the dissipation contrast on the triblock-copolymer sample and decrease the contrast on the TPV sample. Hence, the larger elastic modulus contrast on the TPV sample is balanced by the inverted adhesion force contrast so that the contrast in energy dissipation maps on each sample may end up being similar. The numerical values of phase recorded across the same section indicated by the dashed lines in Fig. 6.7a, b supports this hypothesis. The data show that there is approximately 40° of phase difference on the two material components of both samples. The images obtained on the two polymer samples illustrate the ability of torsional harmonic measurements to generate high resolution maps of elastic modulus and adhesion at the typical imaging speed of the tapping mode. The spatial resolutions of these maps are also comparable to that of the typical resolution obtained in the tapping mode, i.e., ~10 nm. In addition, the images exhibit a large dynamic range in elastic modulus measurements, both small and large variations can be mapped.

Sub-molecular Resolution Mechanical Measurements with Ultra Sharp Tips and Lower Forces The level of spatial resolution observed in Fig. 6.7a–d is sufficient to analyze a wide range of samples. However by carefully adjusting imaging forces and using ultra sharp tips, one can improve the spatial resolution further. Figure 6.8a shows transmission electron microscope image of carbon spikes grown on top of the silicon tip of a torsional harmonic cantilever [13]. These carbon spike tips provide tip apex radii of ~1nm. While this improvement is helping to achieve better spatial resolution, the reduction in contact area leads to substantial increases in contact pressure. If one employs the imaging forces used for mechanical analysis in Fig. 6.7, the resulting

Fig. 6.8 (continued) An example of tip–sample force waveforms recorded with this tip during the experiments is given to the right. Scale bar in (a) is 100 nm. Topography (b), elastic modulus (c), and phase images (d) are given with section plots on the right of respective images. Scale bar in (b) 10 nm. All images are recorded simultaneously (cf. [10])

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Fig. 6.8 Sub-molecular resolution mechanical mapping. Images across the molecular self-assembly of C36H74 alkane layers on graphite are obtained using a torsional harmonic cantilever with an ultra sharp carbon spike grown on its tip. (a) TEM picture of the carbon spikes grown on the THC.

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pressure will damage the carbon spike. Therefore, the forces have to be kept as small as possible. Figure 6.8b shows a tip–sample force waveform with a peak tapping force less than 1nN obtained with the carbon spike tip. Under these conditions, images of the surface formed by self-assembling ultra-thin layers of C36H74 alkane chains on highly oriented pyrolytic graphite were obtained. These molecules assemble into lamellar ribbons with a periodicity of 4.7 nm, close to the fully extended conformation of the alkane chain. The topography image in Fig. 6.8c reproduces the ribbon pattern. Scanning tunneling microscope studies on this molecular system showed that the carbon backbone of the alkanes are aligned parallel to the substrate and perpendicular to the ribbon edges [14]. The height profile across the dashed line shows that there is approximately 3 Å of height variation across the ribbons. The simultaneously recorded elastic modulus map of the ribbons shows that topographically lower regions also exhibit lower elastic modulus. The regions with lower stiffness are likely to be near the –CH3 end groups because they are covalently connected to the carbon backbone only on one side and therefore they are more flexible. The numerical values of elastic modulus show that the reduced elastic modulus values vary between 200 and 450 MPa. These values are calculated based on (6.1) and further assuming a tip radius of 1nm. However, that formula is derived for the contact of macroscopic objects. Therefore, the numerical values provide a measure of stiffness, but they do not correspond to true elastic modulus values. The phase image recorded on the self-assembled alkanes is given in Fig. 6.8e together with its cross section data. The ribbon structure is also visible in the phase image; however, the contrast is about 0.5°. This is much smaller than typical phase contrasts observed in tapping mode. Smaller peak forces and contact also reduce the energy that can be dissipated in the contact region. The analysis on the self-assembled alkane chains with the ultra sharp carbon spike tips demonstrates the possibility to obtain sub-molecular resolution in mechanical measurements. These kinds of measurements can be helpful in the analysis of intermolecular interactions and changes in material properties across abrupt interfaces.

High Resolution Thermo-Mechanical Characterization of Polymer Blends In this section, we will conclude our discussions on the dynamic nanomechanical analysis with torsional harmonic cantilevers with an investigation on the thermal behavior of polymer blends with sub-micron segregation features. If the components of blended polymer have different glass transition temperatures, it becomes difficult to understand the thermal behavior of the blend via bulk mechanical measurements. Here, we present high resolution mechanical analysis of a binary polymer film near the glass transition temperatures of its components; polystyrene (PS) and polymethylmethacrylate (PMMA). The thickness of the film is approximately 50 nm and the two polymer components form sub-micron domains in the film.

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Below the glass transition temperatures, these polymers are in a stiff glassy state and above the glass transition temperature they are in a rubbery phase. PS has a glass transition temperature of around 100°C and PMMA has a glass transition temperature around 130°C. As a result, there is a temperature range where one component is in the rubbery phase and the other one is in a glassy phase. Mechanical measurements can differentiate the two components in that temperature range. To study the mechanical response of the binary polymer film, topography and higher harmonic images (magnitude of the tenth harmonic of torsional vibrations) were recorded at temperatures between 85 and 215°C. These measurements had been performed before the real-time calculations of tip–sample force waveforms were possible. A lock-in amplifier was used to determine the amplitude of the tenth harmonic as a qualitative measure of stiffness. Torsional vibration signals were also recorded during the image process and calculations of the tip–sample force waveforms on each component were carried out offline. Those results will be presented later. First, we will discuss the topography and harmonic images presented in Fig. 6.9. Images recorded at different temperatures in Fig. 6.9 belong to approximately the same region on the sample surface. The topography images show that there are height variations between the two components. Especially at low temperatures, the boundaries are clear. PMMA forms the round features within the PS matrix. At elevated temperatures, the rubbery material forms round, droplet-like, features on the surface. Note that the height ranges of the images are larger at higher temperatures. When the polymers go through glass transition, they acquire a liquid-like mobility so that they can flow and rearrange themselves. As a result topographical variations are also getting smoother at higher temperatures. The qualitative stiffness maps in Fig. 6.9 show that the domains of PS and PMMA have different stiffness even at 85°C. However, the contrast between the two regions increases at 160 and 175°C, mainly because of a reduction in the harmonic amplitude on the PS regions. The contrast between the two regions reduces around 190°C and almost completely disappears at 215°C. Beyond 190°C, the amplitudes of the tenth harmonic are small, because both materials are in rubbery phase. The increased contrast around 160 and 175°C suggests that this is the temperature range where one material is in the glassy phase and the other is in the rubbery phase. The changes in the mechanical behavior of the two polymer components can be better understood by analyzing the tip–sample force waveforms recorded during the imaging process. Figure 6.10a, b shows the unloading portions of the tip–sample force–distance relationships on the two regions at each temperature including the room temperature. It is seen that the slopes of the curves gradually reduce with temperature. According to (6.1), this indicates a lowering of stiffness. After fitting (6.1) to the recorded curves, the reduced elastic modulus values are obtained. Figure 6.10c plots the resulting values against temperature. The values of PS and PMMA at room temperature, 2.3 and 3.7 GPa, are in the range of typical bulk values for these materials. The stiffness of both components reduce by two orders of magnitude at high temperatures, however, the transitions are located at different temperatures for the two materials. The stiffness of PS reduces around 160°C and the

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Fig. 6.9 Observing the glass transition of a polymer blend through nanomechanical measurements. A thin polymer film composed of PS and PMMA is investigated. Topography and tenth-harmonic images are recorded at different temperatures. The circular features are PMMA, and the matrix is PS.

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Fig. 6.10 Indentation forces measured as a function of temperature in a polymer blend. Time-varying force measurements are plotted against the depth of indentation in the sample for polystyrene (a) and PMMA (b). Corresponding effective reduced elastic modulus values are plotted in (c) (cf. [9])

Fig. 6.9 (continued) The scan area is 2.5 × 5 µm. The color bar represents different height ranges at each temperature where the range is given in the top left corner of each panel. The color bar represents a 10-V lock-in output signal at all temperatures for the harmonic images. Note that height contrast increases with temperature, however the contrast in the harmonic image is first increasing and then decreasing (cf. [9])

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stiffness of PMMA reduces around 180°C. This range approximately corresponds to the temperature range where the harmonic images in Fig. 6.9 exhibit the largest contrast. Note that the apparent transition temperatures are approximately 60°C higher than the typical transition temperatures for PS and PMMA. The glass transition temperatures are expected to rise by 5–10°C for an order of magnitude increase in the measurement frequency [15]. The tapping cantilever interacts with the sample at the drive frequency, which is around 50 kHz. This is several orders of magnitude faster than the frequencies of conventional dynamic nanomechanical measurements. Therefore, the observed increase in the glass transition temperatures in high-speed nanomechanical measurements is an expected consequence of the frequency dependence of glass transition.

References 1. R. Garcia and R. Perez, “Dynamic atomic force microscopy methods,” Surf. Sci. Rep. 47 197–301 (2002). 2. J. Israelachvili, Intermolecular and Surface Forces. (Academic Press, London, 2003). 3. O. Sahin, “Accessing time-varying forces on the vibrating tip of the dynamic atomic force microscope to map material composition,” Israel J. Chem. 48 55–63 (2008). 4. R. W. Stark and W. M. Heckl, “Fourier transformed atomic force microscopy: tapping mode atomic force microscopy beyond the Hookian approximation,” Surf. Sci. 457 219–228 (2000). 5. M. Stark, R. W. Stark, W. M. Heckl et al., “Inverting dynamic force microscopy: From signals to time-resolved interaction forces,” Proc. Natl. Acad. Sci. U.S.A. 99 8473–8478 (2002). 6. U. Rabe, K. Janser, and W. Arnold, “Vibrations of free and surface-coupled atomic force microscope cantilevers: Theory and experiment,” Rev. Sci. Instrum. 67 3281–3293 (1996). 7. J. Tamayo and R. Garcia, “Deformation, contact time, and phase contrast in tapping mode scanning force microscopy,” Langmuir 12 4430–4435 (1996). 8. R. W. Stark, “Optical lever detection in higher eigenmode dynamic atomic force microscopy,” Rev. Sci. Instrum. 75 5053–5055 (2004). 9. O. Sahin, S. Magonov, C. Su et al., “An atomic force microscope tip designed to measure time-varying nanomechanical forces,” Nat. Nanotechnol. 2 507–514 (2007). 10. O. Sahin and N. Erina, “High-resolution and large dynamic range nanomechanical mapping in tapping-mode atomic force microscopy,” Nanotechnology 19 445717 9 (2008). 11. J. P. Cleveland, B. Anczykowski, A. E. Schmid et al., “Energy dissipation in tapping-mode atomic force microscopy,” Appl. Phys. Lett. 72 2613–2615 (1998). 12. L. Zitzler, S. Herminghaus, and F. Mugele, “Capillary forces in tapping mode atomic force microscopy,” Phys. Rev. B 66 155436 8 (2002). 13. D. Klinov and S. Magonov, “True molecular resolution in tapping-mode atomic force microscopy with high-resolution probes,” Appl. Phys. Lett. 84 2697–2699 (2004). 14. S. De Feyter and F. C. De Schryver, “Two-dimensional supramolecular self-assembly probed by scanning tunneling microscopy,” Chem. Soc. Rev. 32 139–150 (2003). 15. I. M. Ward, An Introduction to the Mechanical Properties of Solid Polymers. (Wiley, Chichester, UK, 2004).

Part III

Thermal Characterization by SPM

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Chapter 7

Toward Nanoscale Chemical Imaging: The Intersection of Scanning Probe Microscopy and Mass Spectrometry Olga S. Ovchinnikova

Introduction There exists a clear need to extend the limits of our understanding of chemical and physical phenomena of materials and biosystems to the nanoscale. Chemical analysis on the nanometer scale has become necessary in the fields of biology, medicine, and material science. Understanding the processes such as signal transduction in a single cell [1, 2], elemental distribution in single cells and organelles [3], characterization of nanoelectronics to trace impurities [4], and the study of pharmaceuticals [5] requires nanometer-resolved chemical analysis. The chemical analysis of compounds in complex mixtures or specimens such as biological tissue requires highly specific and sensitive analytical methods that require little or no sample preparation and can be studied in vivo. Analyzing with an ability to chemically image fragile biological systems at atmospheric pressure (AP) is a growing field of interest because preparation of biological samples for analysis in vacuum is often difficult, time-consuming, and even at times not possible. An analytical tool with high sensitivity that can provide detailed molecular information with high spatial resolution has been the goal behind the development of different chemical imaging techniques. In mass spectrometry, the desorption and ablation of material from surfaces in vacuum is a well-established field. Techniques like secondary ion mass spectrometry (SIMS) [6] and matrix-assisted laser desorption ionization (MALDI) [7, 8] have become common tools for surface sampling and imaging. Each approach has advantages and disadvantages depending on the particular application. In SIMS, surface species are desorbed by keV particle bombardment [9] while the ionization of the particles occurs during their desorption by intrinsic processes. SIMS can yield images with lateral resolution on the order of 100nm [10]. Laser-based desorption systems like MALDI are capable of routinely achieving 20mm spatial resolution and have the advantage of not fragmenting the molecules of O.S. Ovchinnikova (*) Department of Physics and Astronomy, University of Tennessee, Knoxville 401 Nielsen Physics Building, 1408 Circle Drive, Knoxville, TN 37996-1200 e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_7, © Springer Science+Business Media, LLC 2010

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interest [3]. However, both of these imaging techniques have the drawback that the imaging has to be carried out in vacuum, which limits the type of sample that can be studied because of the constraints of placing the sample into a vacuum system. Atmospheric pressure mass spectrometry (MS) is especially suited for this with its high degree of chemical specificity and its minimal or no sample preparation. The ability to carry out AP-MS experiments largely depends on the desorption and ionization mechanisms. Techniques such as atmospheric pressure chemical ionization (APCI) [11], electrospray ionization (ESI) [12] and inductively coupled plasma ionization (ICP) [13] that have the ability to ionize samples at atmospheric or reduced pressures in combination with highly localized sample desorption techniques such as near-field laser desorption [14] and micro-thermal desorption [15] are opening the door for mass spectrometry to carry out nanometer scale chemical imaging at AP. Combining AFM and mass spectrometry has been approached from two directions: using laser desorption/ablation as the means to sample the surface [16] and using AFM heated probes to thermally desorb the sample [15]. The initial laser desorption/ablation work developed by Zenobi et al. in 2001 used a 355-nm Nd:YAG laser with up to 250mJ of energy per pulse coupled into the back of an NSOM fiber with a 170-nm diameter aperture to create laser ablation craters 200 by 20-nm [16] in size on an anthracene crystal surface. This volume of material ablated corresponds to a quantity of 1.7amol of material desorbed from each individual crater. Using a heated stainless steel capillary, the ablated material was then directed at an electron impact ionizer (EI) source of a quadruple mass spectrometer. A quadruple mass spectrometer was used in the experiment because it allows for material from several laser shots to be accumulated in the trap to allow for better signal to noise. It should also be noted that EI requires a vacuum environment to operate and is only capable of producing singly charged negative ions, thereby drastically limiting the samples that can be studied. After ablation of the surface, the NSOM tip was used in a force feedback arrangement to scan the surface to acquire a topographic image. More recently in 2008 Zenobi et al. modified their experiment to operate using a combined quadruple/time-of-flight (TOF) mass spectrometer to increase the detection sensitivity [17] as well increased the NSOM tip size to be in the range of 500–800nm in order to produce a larger ablation area. Using the ability of a quadruple to accumulate ions with the sensitivity afforded by a time-of-flight mass spectrometer Zenobi et al. were able to acquire mass spectra from ~1.7mm wide × 0.8mm deep craters in anthracene and 2,5-dihydrobenoic acid (DHB) [17]. This set-up allowed them to acquire multiple laser shots in the quadruple trap prior to analysis with the time-offlight. This experiment demonstrated the feasibility of extracting chemically specific information as well as topographic information in tandem using an AFM/MS set-up. Work has also been done by Goeringer et al. toward AP apertureless near-field laser ablation mass spectrometry [18]. Here a frequency doubled 532nm Nd:YAG laser was focused at the tip of a gold-coated AFM probe that was positioned ~10nm from the surface to create ~50nm craters in a rhodamine 6G thin film. More recently Goeringer et al. used a quadruple trap mass spectrometer and a focused laser beam to create 1-mm spatially resolved chemical images with corresponding AFM topographic images (Fig. 7.1) [19]. These works demonstrated a clear application of combined AFM/MS technique for chemical imaging.

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Fig. 7.1 Combined AFM and MS chemical imaging of an “X” etched out from a rhodamine 6G thin film surface (used with permission from [19])

Another laser desorption technique that shows promise for its application to the set of techniques already incorporating AFM and MS is laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS). Becker et al. have shown the possibility of using LA-ICP-MS to perform elemental analysis of Cu, Zn, Au, Si, and U from ~200nm craters using near-field enhancement from a frequency doubled 532-nm Nd:YAG laser focused on the end of a sharp silver needle that is positioned 200nm above the surface [20, 21]. Their experiment demonstrated the possibility of nano-local analysis by near-field (NF) laser ablation ICP-MS, and opens the door for coupling NF-ICP-MS with current AFM techniques for an apertureless NF-ICP-MS experiment using AFM probes as the source for the near-field enhancement. The details and applicability of the ICP-MS technique are described in more detail later on in this chapter. The other successful avenue for coupling AFM imaging with MS identification was developed using thermal desorption via micro-thermal (micro-TA) AFM probes coupled to an electron impact (EI) gas chromatogram mass spectrometer (GC-MS) [22]. In this work Wollaston wire micro-TA AFM probes were used to desorb craters with ~6mm diameter in feverfew leaves. The evolved gas was extracted via a heated capillary packed with a suitable sorbent such as Tenax and sent to an EI GC-MS where the camphor molecule extracted from the leaf by means of thermal desorption was monitored on the GC-MS. The same micro-TA/GC-MS analysis technique has also been applied to the study of pharmaceuticals [23] where the precise distribution of chemicals can be correlated to an AFM topographic image. Recently, the use of thermal desorption combined with AFM as a means of studying samples at the nanoscale has become possible with the invention of silicon-based heated AFM probes that have a 50-nm radius of curvature and are able to rapidly heat and cool up to 400°C. Using these new nano-TA AFM probes for thermal desorption combined with mass spectrometry opens up a new avenue for analyzing samples. This technique alleviates tedious experimental set-up that is associated with near-field laser desorption/AFM experiments. Using thermal desorption also minimizes the tip-to-tip fluctuation in enhancement that is found when using near-field tip enhancement technique.

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The goal of true nanoscale chemical imaging is now becoming a reality due to recent advances in both mass-spectrometry and scanning probe microscopy. The availability of commercial mass spectrometers that are able to efficiently detect an attomole of material transferred from AP as well as the development of online AP secondary ionization techniques have increased sensitivity to the point needed for nanoscale sampling. Closed loop stages that reduce drift making long dwell times needed for spectroscopy possible and the development of new AFM probe technology that makes thermal desorption possible on the nanoscale have brought about the successfully marrying of AFM and MS for nanoscale chemical imaging. In this chapter the AP MS methods that are most appropriate for coupling with scanning probe systems are described in detail.

Atmospheric Pressure Mass Spectrometry Techniques Thermal Desorption with Secondary Ionization Mass Spectrometry Ambient surface sampling techniques remove the constraints that are associated with vacuum-based sampling/imaging techniques. Using ambient techniques allows for the study of living samples, wet samples, large samples, and allows for fast and easy interchange of samples and often removes the need for extensive sample preparation. Literature reports suggest that scanning probe techniques using atmospheric pressure thermal desorption (AP-TD) might be an alternative means to achieve nanometer-scale surface sampling resolution for mass spectrometry. Atmospheric pressure thermal desorption (AP-TD) is a well-established surface sampling technique in mass spectrometry. In this approach to surface sampling, heat is used to liberate the sample intact from the condensed phase to the vapor phase. Typically, this heating is accomplished through the use of a heated gas passingover the sample and/or a resistively heated sample surface. Given the nature of the desorption process, this approach is limited to relatively low mass species (ca. 2,000Da or less) that can be liberated intact into the gas phase by heat; thermally labile, highly polar, and high molecular-mass species are typically not amenable to vaporization by heating. Once in the gas phase, the sample can be ionized by any number of ion/molecule chemistries. An example of an emerging AP-TD/MS technique is atmospheric pressure chemical ionization (DAPCI) [24]. As defined and used by Cooks’ group [25], DAPCI consists of a capillary with a taper-tip stainless-steel electrode aimed at the surface. An inert sheath gas, into which a solvent vapor is in some cases introduced, is supplied to the capillary and flows through the emitter at high velocity. A highvoltage power supply is used to apply a voltage (typically ±3–6kV) to the electrode; this induces a corona discharge at the tip of the electrode, ionizing the introduced solvent vapor. The sheath gas in some cases may be heated. Ionization mechanisms

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in DAPCI are similar to other APCI ionization sources. As in all APCI systems, the reagent ion species formed can be influenced by means of initiating the reagent ion plasma, and by the particular solvent vapor and sheath gas used [26]. The desorption mechanisms in DAPCI are in some cases unclear. When the sheath gas is heated, TD is certainly a dominant process. In other cases, the high velocity gas might actually liberate minute particles from the surface that can then be ionized in the gas phase. Desorption atmospheric pressure photoionization (DAPPI) [27] is similar to DAPCI except that the reagent ion population is initiated by a photoionization process utilizing a UV lamp rather than a corona discharge. Currently, there are several commercially available AP-TD with secondary ionization sources for mass spectrometry (AP-TD/SI-MS) [28]. Direct analysis in real time (DART) [29] is one of the best known. In DART, He gas flowing through a probe is subjected to a discharge at a needle electrode, producing ions, electrons, and metastable species. Perforated electrodes downstream act to remove ions from the gas stream, while neutral metastable species are carried by the gas through a heated chamber, passing though a grid electrode before entering the ambient atmosphere. The grid electrode prevents ion–ion and ion–electron recombination, and also acts as a source of electrons, either through Penning ionization of a neutral species or through surface Penning ionization [30]. The exiting gas flow is directed at the entrance of the mass spectrometer and the sample surface to be analyzed is placed between the two. The ionization process in DART is a variation of APCI in which the reagent ion population originates from the gas phase reactions of the metastable He atoms (He* (1s2s3S1) produced in the discharge. Another commercially available device is atmospheric-pressure solids analysis probe (ASAP) [31, 32]. Here analyte deposited on the glass meltingpoint capillary is thermally desorbed by a hot gas, ionized by APCI, and analyzed by mass spectrometry (Fig. 7.2). This technique has similar merit to DART, allowing for fast analysis of individual samples. None of the above-mentioned AP-TD/SI-MS techniques have shown to be useful for chemical imaging of surfaces. Moreover, they are fundamentally limited in their ability to carry out chemical imaging because they sample from a relatively large area (many mm to cm scale). In order to overcome the limitation of available thermal desorption techniques for chemical imaging Reading et al. used a Wollaston wire AFM tip to carry out local thermal desorption in order to analyze the desorbed material using gaschromatography mass spectrometry (GC/MS) [23]. With their set-up they were able to desorb material from a PMMA surface by heating to 600°C the Wollaston probe in contact with the surface. The heated probe created craters that were on the order of 7mm wide by 2mm deep. Using a syringe co-located with the Wollaston probe they were able to suction out the desorbed vapor and analyze it with GC/MS. The sample was imaged afterward in an AFM to confirm the presence of ablation craters. Reading et al. [15] applied this technique for examination of polymers as well as small molecules found in pharmaceuticals [5]. Reading’s work demonstrates the potential for a nanometer spatially resolved TD-based mass spectrometry chemical imaging technique.

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Fig. 7.2 Cross-sectional drawing of an atmospheric pressure LC/MS ion source modified for ASAP analysis (used with permission from [23])

Laser Desorption (Ablation) Ionization Mass Spectrometry Laser desorption/ablation experiments that couple laser ionization with a mass spectrometer were introduced in the 1960s. In some of the first laser ionization experiments, Vastola et al. [33] used a 694.3nm ruby laser to ionize organic salts in vacuum. These early laser desorption experiments provided a new technique for desorbing material that decomposed from direct heating, like organic salts. Although, direct laser desorption ionization has been used to examine larger molecules like peptides, it is much better suited for small molecule analysis like organic dyes, porphyrins, organic salts, and UV-light absorbing synthetic polymers [34]. However, laser desorption produces a greater number neutrals over ions, and therefore can be successfully coupled with many different secondary ionization sources. For laser desorption and ionization at AP these techniques include: inductively coupled plasma (ICP), and various forms of APCI, ESI, as well as AP-MALDI.

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Introduced by Alan Gray [13] in 1985, Laser Ablation Inductively Coupled Plasma (LA-ICP) is one of the most established AP laser ablation techniques. In the LA-ICP-MS technique material is ablated from the surface by a laser; and then atomized and ionized in the high-temperature plasma (Fig. 7.3). In LA-ICP-MS experiments, the sample is placed in a closed ablation chamber which is flushed with argon or helium as a carrier gas and the laser beam is focused on the sample surface through a cell window [35]. When the laser irradiance is high enough the material is ablated and carried in the carrier gas to the plasma of the ICP-MS. The ICP is a separate excitation source where laser-generated particles are vaporized, atomized, and ionized. The ions are then extracted by a vacuum interface and guided into a mass analyzer. The advantages of LD-ICP-MS analysis are high sensitivity, large dynamic range, and simple spectra. The LA-ICP-MS technique’s biggest merit is its sensitivity, achieving a sub-ng/g detection limit for spot sizes that are above 100mm [20]. Combining high spatial resolution that is achieved with laser ablation with the high sensitivity in the ICP allow for trace-element determination at a micron scale. This makes the LA-ICP-MS a tool that is widely used in geology for the detection of trace elements in minerals, as a screening tool in geochemical studies of (U/Th/Pb), and for the elemental analysis of fluid inclusions for studies of ore-formation processes [36]. Forensics is another area where the LA-ICP-MS is heavily used; within this field fingerprinting of gemstones seems to be the fastest growing areas of LA-ICP-MS [20]. The LA-ICP technique has also recently been used by Becker et al. to image elements in thin cross section of human brain samples [37–39]. The tissue samples were ~20 mm thick, and an ablation of the sample was preformed with a 266-nm frequency quadrupled Nd:YAG laser. The ablated material was carried by argon gas into the inductively coupled plasma. Laboratory standards were created by spiking the brain tissue with known solutions of the selected elements such as Cu. The signal was then maximized to the maximum ion intensity of 63Cu+ using the laboratory standards.

Fig. 7.3 Schematic of LA-ICP-MS set-up (used with permission from [20])

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Using this LA-ICP-MS technique Becker et al. [22] were able to do a 50 line scan of a section of tissue in 6h. Chemical imaging of a human brain sample done using LA-ICP-MS by Becker et al. [22] can be seen in Fig. 7.4. Quantitative analysis on the amount of element in interest was performed only on the elements for which suitable standard solution could be created. This therefore creates a limitation on the ions that can be quantified in a mapping scan. The LA-ICP-MS is a therefore rather slow compared to other biological surface mapping techniques such as MALDI and DESI. As a tool for analytical application to biology, the LA-ICP-MS has been used for a variety of applications such as tissue imaging as discussed earlier as well as for proteomics with protein quantification by the detection of trace elements in proteins by means of polyacrylamide gels [40, 41]. Laser Desorption Atmospheric Pressure Chemical Ionization (LD-APCI) is another technique that uses laser desorption to produce a plume of neutrals that then undergo secondary ionization. AP chemical ionization uses an IR-laser to desorb neutral molecules from a surface and then uses a corona discharge to carry out the secondary ionization. The corona discharge is used as an electron emitter to ionize neutral molecules in the gas phase. The neutral molecules are created by laser desorption from a target that is near to the discharge. The ionized molecules are

Fig. 7.4 Cu images in part of (top, right) and the whole (bottom, right) human hemisphere measured by LA-ICP-MS compared with the light photograph of the thin tissue section (used with permission from [22])

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then sucked into a heated capillary into mass spectrometer. The LD-APCI apparatus schematic can be seen in Fig. 7.5. Currently Harrison et al. [42, 43] are developing LD-APCI as tool for detection small organic molecules. Harrison et al. use a focused IR 10.6-mm pulsed CO2 laser to desorb neutral molecules at AP followed by ionization in the gas phase with a corona discharge [44]. By separating out the laser desorption signal from the ionization by corona discharge Harrison et al. were able to increase their ion signal 150-fold [29]. To demonstrate an increase in the signal by carrying out secondary ionization of the laser desorbed molecules, Harrison et al. toggled the corona discharge on and off while the laser desorption was continuously running [29]. Using this method they were able to show that the 150-fold reduction of sepirerone still yielded a signal using the LD-APCI method and no signal where only direct laser desorption was used. Also, despite the 150-fold decrease in the amount of seiperone present in the sample, the ion signal only dropped by a one quarter of the original signal in the LD-APCI. In later work Harrison et al. [28] were able to use the LD-APCI to detect desorbed gas-phase neutral peptide molecules at AP. Using the LD-APCI technique provides benefits over using direct AP-MALDI for detecting peptides because in LD-APCI the matrix containing analyte does not need to assist in the ionization, this allows for the opportunity for examining analyte matrixes that include biological solutions, tissue, polyacrylamide gels [27, 45], and thin layer chromatography plates. Work has also been done with UV laser sources for the LD-APCI [46], in these experiments a 337-nm nitrogen laser was used to desorb the molecules from a sample. Using a nitrogen laser for desorption rather that an IR laser produced a larger number of neutral molecules, and therefore gave a better signal in the mass spectrometer.

Fig. 7.5 Schematic representation of LD-APCI source (used with permission from [28])

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Laser Desorption Electrospray Ionization (LD-ESI) was introduced by Shiea et al. [47] under the acronym ELDI, electrospray-assisted laser desorption/ionization. The LD-ESI technique works on the principle that a solid substrate is first desorbed by a pulsed nitrogen laser at AP, followed by secondary ionization of the desorbed material with an electrospray ionization source (ESI) [48] (Fig. 7.6). Combining laser desorption with the ESI technique allowed Shiea et al. to demonstrate the possibility of detecting intact protein spectra without the use of a matrix compound. The ELDI technique is able to overcome the low mass limit imposed by direct laser desorption by using an ESI source to post-ionize neutral molecules generated by laser desorption [32]. The ELDI set-up consists of a 337-nm nitrogen laser operating at 20 mJ per pulse and a focused laser spot size of 100 mm×150 mm. The incident angle of the laser beam in Shiea’s work is fixed at 45°[32]. The sample is placed on a translation stage so as to produce a continuous ion signal. The laser ablated material is then ionized by electrospraying. Using this technique Shiea et al. were able to perform direct characterization of chemical compounds like amines on thin layer chromatographic plates and detect intact proteins in dried biological fluids such as blood, tears, saliva, and serum. They were also able to estimate their detection limit using cytochrome c to be from 10-4 to 10-9M. With their laser desorption area ranging between 100. and 200 mm they were able to detect 50amol of protein molecules per each ELDI spectrum. Very similar to the ELDI technique that uses a UV laser source, Vertes et al. [49, 50] have developed the LAESI technique, laser ablation with electrospray ionization, which uses an infrared laser source instead of a UV source. Using a mid-infrared laser source instead of a UV laser source, Vertes et al. [35] were able to look at biological samples that have resonant frequencies in the infrared versus the near-UV. Vertes et al. used an infrared laser source because of the water that is

Fig. 7.6 Detailed schematic of ELDI set-up, (A) sampling skimmer, (B) laser beam, (C) electrospray capillary, (D) sample plate, (E) focusing lens, (F) reflecting lens, (G) syringe pump (used with permission from [33])

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intrinsic to biological samples has a resonance frequency in the infrared. Vertes’ approach allows for direct analysis of biological samples without any prior sample preparation such as drying. The LAESI technique depends on samples high water content and therefore it works very well in water-rich tissue, but is limited in analysis of tissues like bone, nail, or dry skin which has a low water concentration. Atmospheric Pressure Matrix Assisted Laser Desorption Ionization (AP-MALDI) was introduced by Dorshenko et al. in 2002 as an alternative to vacuum MALDI for carrying out MALDI experiments in situ [36]. AP-MALDI operates on the same principle as vacuum MALDI, where a low melting point substrate, the matrix, which is doped with the analyte is desorbed by laser irradiation. In most cases this is accomplished with a UV-laser source; however, IR lasers have also been used due to the strong absorption of water molecules in the IR region. In the AP-MALDI system, ions are transferred from the AP into vacuum via a heated capillary into a differential pressure region (Fig. 7.7). While ion loss is inevitable during the transfer, the total ion yield in the AP-MALDI is usually higher than in vacuum MALDI due to fast thermal stabilization at atmospheric conditions. The AP-MALDI introduces the possibility of direct coupling separation techniques like liquid chromatography (HPLC) with MS,

Fig. 7.7 Schematic of AP-MALDI set-up (used with permission from [36])

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which is not possible with vacuum MALDI where the samples have to be dried prior to analysis. The AP-MALDI has also demonstrated large tolerances to laser fluence variations and minimal fragmentation of molecular ions [51]. There are different AP-MALDI systems that use different laser sources, the most common laser is the 337nm UV nitrogen laser. The nitrogen laser is used in the AP-MALDI set-up because of the strong interaction of the matrix compounds in the UV regions. Doroshenko et al. [36] found the AP-MALDI source to be extremely practical for studies of small molecules because the AP-MALDI allows for low detection limits, and the limitation of mass range was mostly affected by the limits imposed by the mass spectrometer in question and not the AP-MALDI. Doroshenko et al. also reported that the AP-MALDI system allowed for softer ionization than vacuum MALDI because the analyte could also become thermally ionized in the air, subsequently requiring lower laser power for ionization and allowing more fragile samples can be examined. Recent work by Vertes et al. [35] has shown that 2,940-nm IR laser can also be used for carrying out AP-MALDI experiments. Using a wavelength that is in the IR region allows for sampling tissue in situ and using the water in the sample as an inherent matrix. Vertes et al. have demonstrated this technique on tissue samples of different fruit such as bananas and strawberries [34]. In Vertes’ experiments the sample plate sits on an x and y stage controller allowing for rapid analysis of sequential spots as well as opening the door for in situ surface sampling of tissue. The AP-MALDI has been successfully used by Vertes et al. to image sections of fruit [35] and plants [52] as well as for detecting peptides [35]. When it comes to imaging mass spectrometry spatial resolution plays a significant role in the ability to precisely map chemical distributions, therefore, the ability to lower the spatial resolution of in chemical imaging will provide more accurate determination of compounds in samples such as plant tissue, human, and animal tissues as well in materials. Currently for laser desorption experiments the spatial resolution is determined by first the laser spot size and the scanning spot size. Vertes et al. used a 250-mm laser spot size which they used to image an electron microscope grid with bar spacing of 92 mm [50]. Moving their stage in 40-mm increments they were able to generate a chemical image through over sampling where the resolution was determined by the spot size of the stage. The obtained chemical image can be seen in Fig. 7.8. Vertes’ work demonstrates the ability to create chemical images at micron resolutions, but at the same time leaves lots of room for improvement and generation of competing techniques. Advances in technology are required to push the limit of what we are able to study and examine. The push toward studying materials and systems on the nanoscale range creates the opportunity for new discoveries. However, the currently available techniques that allow nanometer resolution studies are limited in the amount of chemical information provided. Techniques such as electron microscopy and scanning probe microscopy (SPM) that allow for spatial imaging resolution of 1nm or better provide almost no chemical information about sample. In contrast, techniques such as mass spectrometry and RAMAN and IR imaging provide a vast supply of chemical information about a sample on the molecular level but are not

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Fig. 7.8 (a) Electron microscope grid. (b) Chemical image of toluidine blue O obtained through over sampling (used with permission from [50])

capable of imaging at a submicron resolution. Therefore, it seems that the coupling of two techniques like SPM and mass spectrometry would be the next stage in scientific progress toward the development of new methods for studying systems that require low spatial resolution as well as high chemical information such as cell walls, membrane proteins, material junctions, etc. In recent years Zenobi et al. have attempted to couple mass spectrometry with near field scanning optical microscopy (NSOM) in order provide chemical information at the nanometer spatial resolution [16] Zenobi’s technique relied on using tip-enchanced laser ablation to create submicron-sized ablation craters in a material and then sample the chemical composition of the material using mass spectrometry. Zenobi’s experiments showed promise for the technique but were limited in what type of systems they could study because the experiments were carried out under vacuum. More recently Zenobi et al. have carried out similar tip-enchanted NSOM experiments at AP [16]. This experimental set-up for coupling a mass spectrometer with an SNOM system in AP can be seen in Fig. 7.9. Their experiments show the feasibility of acquiring mass spectrometric data from nanometer-sized craters at AP as well as obtaining structural information about a surface through NSOM imaging. Aperture based near field optical microscopy works by overcoming the diffraction limit also known as the Abbe criterion

∆x = 0.61λ / NA

(7.1)

where Dx is the spatial resolution, l is the wavelength of the interacting radiation, and NA=n sina is the numerical aperture of the objective lens. In near-field optical microscopy, the resolution Dx no longer depends on l but on a characteristic length d, the aperture diameter or tip diameter, of a local probe (Fig. 7.10) [16]. NSOM relies on a confined photon flux between a local probe and a sample surface.

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Fig. 7.9 Instrument set-up for SNOM-MS [10]

Fig. 7.10 Comparison of diffraction-limited optical microscopy and near-field optical microscopy. (a) Schematic representation of the diffraction limit showing the minimum detectable separation of two light sources. (b) Schematic of aperture scanning near-field microscopy [7]

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The probe is scanned over a surface and an x–y detector acquires the position and optical information. This idea for overcoming the diffraction limit by confining light to an aperture smaller than to the wavelength of light was first proposed by Edward Synge in his paper in 1928 [16]. However, because of a lack of a coherent light source like a laser that would produce a sufficient photon flux as well as necessary electronic and detection equipment, Synge’s idea was not realized until the 1980 when researchers at the IBM Research Laboratory in Switzerland where able to produce the first near-field optical measurements. In the most common NSOM light is sent down a tapered optical fiber which is uniformly coated in a metal such as aluminum. The fiber is chemically etched to an aperture diameter of around 50 nm. Companies like Nanonics use this technique to create hybrid SNOM/AFM scanning probe equipment to image at the nanometer resolution. Zenobi’s technique relied on using tip-enchanced laser ablation to create submicronsized ablation craters in a material, sampling the ablated material into an electron ionization (EI) source of a mass spectrometer for ionization and detection. Zenobi’s experiments demonstrated proof of principle, but were limited in the types of systems they could study because the experiments were carried out under vacuum and because EI creates only singly charged positive ions as well as fragments of the original molecules. More recently, Zenobi et al. [17] have carried out similar tip-enchanced NSOM experiments at AP. Their experiments show the feasibility of acquiring mass spectrometric data from nanometer-sized craters at AP as well as obtaining structural surface information through NSOM imaging. Discussed in this chapter is research toward a novel set of techniques, using AP hybrid proximal probe topography chemical imaging, where a mass spectrometer is coupled with a state-of-the-art atomic force microscope (AFM). These techniques use AFM probes to desorb material from a surface and then analyze the desorbed gas phase material with a mass spectrometer. These techniques have broad application for studying systems such as cell walls, material junctions, etc. where enhanced or high spatial resolution chemical information is required. Laser desorption/ablation and thermal desorption with a secondary ionization are the two main routes for analyzing surfaces with mass-based chemical imaging that also allow for simultaneous topographical imaging.

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26. Song, Y.; Cooks, R. G. “Atmospheric pressure ion/molecule reactions for the selective detection of nitroaromatic explosives using acetonitrile and air as reagents.” Rapid Commun. Mass Spectrom. 2006, 20, 3130–3138. 27. Haapala, M.; Pol, J.; Saarela, V.; Arvola, V.; Kotiaho, T.; Ketola, R. A.; Franssila, S.; Kauppila, T. J.; Kostiainen, R. “Desorption atmospheric pressure photoionization.” Anal. Chem. 2008, 79, 7867–7872. 28. Van Berkel, G.; Pasilis, S.; Ovchinnikova, O. “Established and emerging atmospheric pressure surface sampling/ionization techniques for mass spectrometry.” J. Mass Spectrom. 2008, 43, 1161–1180. 29. Cody, R. B.; Laramée, J. A.; Durst, H. D. “Versatile new ion source for the analysis of materials in open air under ambient conditions.” Anal. Chem. 2005, 77, 2297–2302. 30. Bell, K. L.; Dalgarno, A.; Kingston, A. E. “Penning ionization by metastable helium atoms.” J. Phys. B (Proc. Phys. Soc.) 1968, 1, 18–22. 31. McEwen, C. N.; Gutteridge, S. “Analysis of the inhibition of the ergosterol pathway in fungi using the atmospheric solids analysis probe (ASAP) method.” J. Am. Soc. Mass Spectrom. 2007, 18, 1274–1278. 32. McEwen, C. N.; McKay, R. G.; Larsen, B. S. “Analysis of solids, liquids, and biological tissues using solids probe introduction at atmospheric pressure on commercial LC/MS instruments.” Anal. Chem. 2005, 77, 7826–7831. 33. Vastola, F. J.; Mumma, R. O.; Pirone, A. J. “Analysis of organic salts by laser ionization.” Org. Mass Spectrom. 1970, 3, 101–104. 34. Gross, J. “Matrix-assisted laser desorption/ionization.” In Mass Spectrometry a Textbook; Springer: Heidelberg, 2004, pp. 411–440. 35. Gunther, D.; Hattendorf, B. “Solid sample analysis using laser ablation inductively coupled plasma mass spectrometry.” Trends Analyt. Chem. 2005, 24, 255–265. 36. Mokgalaka, N. S.; Gardea-Torresdey, J. L. Laser ablation inductively coupled plasma mass spectrometry: principles and applications. Appl. Spectrosc. Rev. 2006, 41, 131–150. 37. Becker, J. S.; Zoriy, M.; Becker, J. S.; Dobrowolska, J.; Matusch, A. Laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) in elemental imaging of biological tissues and in proteomicsw. J. Analyt. Atom. Spectrom. 2007, 22, 736–744. 38. Zoriy, M. V.; Becker, J. S. “Imaging of elements in thin cross sections of human brain samples by LA-ICP-MS: A study on reproducibility.” Int. J. Mass Spectrom. 2007, 264, (2–3), 175–180. 39. Zoriy, M. V.; Dehnhardt, M.; Reifenberger, G.; Zilles, K.; Becker, J. S. “Imaging of Cu, Zn, Pb and U in human brain tumor resections by laser ablation inductively coupled plasma mass spectrometry.” Int. J. Mass Spectrom. 2006, 257, (1–3), 27–33. 40. Chery, C. C.; Gunther, D.; Cornelis, R.; Vanhaecke, F.; Moens, L. “Detection of metals in proteins by means of polyacrylamide gel electrophoresis and laser ablation-inductively coupled plasma-mass spectrometry: Application to selenium.” Electrophoresis 2003, 24, (19–20), 3305–3313. 41. Wind, M.; Feldmann, I.; Jakubowski, N.; Lehmann, W. D. “Spotting and quantification of phosphoproteins purified by gel electrophoresis and laser ablation-element mass spectrometry with phosphorus-31 detection.” Electrophoresis 2003, 24, (7–8), 1276–1280. 42. Coon, J. J.; Steele, H. A.; Laipis, P. J.; Harrison, W. W. “Laser desorption-atmospheric pressure chemical ionization: a novel ion source for the direct coupling of polyacrylamide gel electrophoresis to mass spectrometry.” J. Mass Spectrom. 2002, 37, (11), 1163–1167. 43. Coon, J. J.; Harrison, W. W. “Laser desorption-atmospheric pressure chemical ionization mass spectrometry for the analysis of peptides from aqueous solutions.” Analyt. Chem. 2002, 74, (21), 5600–5605. 44. Coon, J. J.; McHale, K. J.; Harrison, W. W. “Atmospheric pressure laser desorption/chemical ionization mass spectrometry: a new ionization method based on existing themes.” Rapid Commun. Mass Spectrom. 2002, 16, (7), 681–685. 45. Coon, J. J.; Steele, H. A.; Laipis, P. J.; Harrison, W. W. “Direct atmospheric pressure coupling of polyacrylamide gel electrophoresis to mass spectrometry for rapid protein sequence analysis.” J. Proteome Res. 2003, 2, (6), 610–617.

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Chapter 8

Dynamic SPM Methods for Local Analysis of Thermo-Mechanical Properties M.P. Nikiforov and Roger Proksch

The Need for Localized Mechanical Analysis: Industrial and Basic Science Perspective Thermo-mechanical properties of materials determine whether they will be useful or not. Measurement techniques for bulk mechanical parameters (Young’s modulus, loss modulus, Poisson ratio, etc.) are well established and regulated by ASTM standards [1–4]. As length scales shrink, Nanoindentation (NI) and Atomic Force Microscopy (AFM) techniques allow sub-1,000nm spatial resolution thermo-mechanical properties measurements. These techniques started becoming available less than two decades ago [5] and are evolving rapidly. Examples of nanoscale property measurements include subsurface delamination in thin films [6], the structure of the cytoskeleton of a single cell [7] and composition of polymer blends [8]. As the spatial resolution increases, the variability in the measurements also increases. Part of this increase in variability is real in the sense that as one approaches characteristic length scales of a material, one begins to sample domains with very different thermal and mechanical properties. Another reason for increased uncertainty comes from experimental noise associated with measuring small forces and displacements and uncertainties and instabilities in the contact areas of the probe interacting with the sample. Despite signal to noise and calibration issues associated with measuring small forces and distances, NI- and AFM-based techniques allow researchers to answer the questions which cannot be addressed with conventional bulk techniques. For example, measurements of the kinetics of the polymerization of automotive clear-coats was not possible with conventional dynamic mechanical analysis (DMA) because of the softness of the material in the first several hours after deposition and the small thickness of the film [9]. From the basic science point of view the change in thermo-mechanical properties of polymers is closely related to phase transitions. AFM/scanning probe-based techniques can ultimately provide 10-nm resolution or better for the measurements of mechanical parameters – changes

M.P. Nikiforov (*) Oak Ridge National Laboratory (ORNL), Oak Ridge, TN 37831, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_8, © Springer Science+Business Media, LLC 2010

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in which are intimately associated with phase transitions. In many cases, this resolution is approaching the length scale at which phase transformations occur. In polymeric materials this length scale corresponds to the length of the Kuhn’s segment (0.1–2 nm) [10–12]. Thus, studying temperature dependencies of the mechanical properties allows phase transformation in polymers to be understood at a physically relevant length scale. In this chapter the basic NI models and mechanical analysis with AFM are briefly introduced; however, this is not the main focus. Most attention is paid to experimental techniques which allow mapping of the thermo-mechanical properties at different temperatures. More details on NI and mechanical analysis with AFM at room temperature can be found in several excellent reviews [13–16].

Mechanical Characterization of the Materials with High Spatial Resolution: Lessons from Nanoindentation Mechanical properties measurement using indentation is a well-established technique for the characterization of hard surfaces [5]. This technique involves a rigid probe with a well-defined geometry that is pressed into the flat surface of a test sample. This forces the material to undergo elastic and usually plastic deformation at the probe location. Mechanical properties are determined from the values of indentation force for a measured indentation depth and other parameters. The methods developed for macroscopic indentation can be applied for tests at micron and sub-micron length scales on hard materials such as metals, ceramics, plastics, calcified biological tissues, composites etc. [17–21]. Because of their relevance in everyday life, among the most interesting objects for thermo-mechanical characterization are polymeric materials and biological samples including cells [22, 23] and various biological molecules [24]. When the indentation methodology is applied to soft materials, the physics of the indentation process is inherently more complex because soft materials undergo purely elastic deformations even at large indentation depths. This results in the following experimental issues: • Data analysis usually requires complicated models, which take into account contributions from interactive forces. This happens because tip–sample interactions (e.g., adhesion) are stronger in compliant samples. • The point of contact is difficult to identify, because applied forces for a given indentation depth are much smaller in soft materials than in hard ones. • The transition from linear to nonlinear stress–strain behavior may be ambiguous, because of the large elastic deformation range. During the last two decades the accuracy of NI measurements has been constantly improving. Models have been created to address the issues raised in the list above [25–30]. The main theoretical models of tip–surface contacts and experimental protocols used for data fitting based on these models are presented below. Theoretical models can be divided into three main classes based on the tip–surface

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interactions: (1) elastic indentation of an infinite half sphere without adhesion, (2) elastic indentation of an infinite half sphere with adhesion, and (3) repulsive tip–surface interactions (electrostatic forces, when indentation is performed in solution; electrosteric interactions). The Hertz model is the prototypical contact model. This model is based on the following assumptions: • The strains are small, i.e., rcont⪌R (and d⪌R), where rcont is the contact radius, R is the radius of the sphere, d is the indentation depth. • The indented solid is a linear elastic, infinite half space. • The surfaces are frictionless. In the simplest case of indentation with a sphere of radius R the force F=ER1/2d3/2, where E is Young’s modulus, and d is indentation depth. More complicated geometries are reviewed in [31]. Since the seminal paper by Hertz in 1881, it has become obvious that attractive interactions between indenter and surface cannot be neglected in many real world systems. In 1971 Johnson–Kendall–Roberts (JKR) [32] suggested a model where Hertz theory was modified by introducing an apparent Hertz load, or the equivalent load in the absence of adhesion. In the JKR model, the contact radius is typically increased compared to the Hertz model. Later, in 1975 Derjaguin–Muller–Toporov (DMT) [33] proposed a theory in which the adhesion forces are taken into account, but the profile is assumed to be Hertzian, neglecting the contribution of adhesion forces to the surface deformation profile. In 1976, Tabor identified the applicability of the two theories for a range of sample compliances and adhesive forces. The JKR theory was found to be valid for the indentation of relatively compliant materials with probes of relatively large radii and strong adhesive forces. In contrast, the DMT theory applies under conditions of stiff materials, small probe radii, and weak adhesive forces. In 1992 Maugis suggested a universal model, which more realistically describes adhesion between spheres and has both the JKR and DMT solutions as limiting cases [34]. When a probe approaches a surface, attractive interactions typically prevail. However, repulsive interactions between probe and surface can also be observed when the probe comes in the close proximity with the surface [35]. Those interactions are mainly of electrostatic nature. Various theories, such as Derjaguin– Landau–Verway–Overbeek (DLVO) [36, 37] and theories based on electrosteric and electrostatic interactions [38–40], were used to explain data on indentation obtained with AFMs. In soft materials and large deformations, the small strain approximation – one of the assumptions of the Hertz model – is no longer valid. The contact mechanics beyond the Hertzian regime is an interesting part of applied mathematics and it is well reviewed by Lin and Horkay [31]. Here we will discuss only one model which treats slight deviations from rubber–elastic behavior of materials. This model was proposed by Lin et al. [27] where the nonlinear force–indentation relationship was derived from an approximate equation based on the Mooney–Rivlin strain energy function [41]. Several groups used the theoretical models described above for determination of materials properties such as Young’s modulus from force distance curves measured

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Table 8.1 Comparison of the applicability ranges for different models Small force Mooney – NonRivlin interactive, Herzian Adhesive Repulsive elastic Lin et al. [27]     Jaasma et al. [44]     Guo and    Akhremitchev [43] Crick and Yin [42]    Oliver and Pharr   [5, 46–50] Sun et al. [45]

Large force JKR – DMT JKR transition DMT   





with AFM. The main challenges for the correct determination of Young’s modulus were tip shape, the determination of contact point, thermal (Brownian) fluctuations, and instrumental noise. Using elaborate data processing techniques and data acquisition protocols, six main models for the extraction of mechanical properties from NI and AFM data were proposed: Crick and Yin (AFM) [42], Guo and Akhremitchev (AFM) [43], Jaasma et al. (AFM) [44], Lin et al. (AFM) [27], Oliver and Pharr (NI) [16], Sun et al. (AFM) [45]. Table 8.1 shows what types of interactions are treated in each of the models. The model developed by Lin et al. provides the biggest versatility, albeit at a price of using multiple triggers to differentiate between different tip–surface interactions. Each of the methods presented in Table 8.1 is accompanied by a specific protocol for the analysis of force–distance data (see original articles for details).

Principles of Thermo-Mechanical Analysis Using AFM Platform: Examples of Thermo-Mechanical Properties Mapping The models and data analysis protocols discussed in the previous section provide a framework for the conversion of the data obtained in the experiments (force–distance curves) into mechanical properties of the materials. AFM is a common tool for obtaining force–distance curves with high spatial resolution, making it an excellent platform for high-resolution mechanical analysis of the materials. Extensions of this technique to measure temperature dependence of mechanical properties with high resolution provides an exciting opportunity to study phase transformations in polymeric materials. An AFM platform allows two implementations for this concept: variation of the entire sample temperature using a “bulk” heater, and high resolution variation of the sample temperature using a localized heater built into the AFM tip. Bulk variation of the sample temperature allows significantly more stable and precise temperature control while local heating achieves much faster heating rates of the material. Recently, several groups have implemented thermo-mechanical analysis

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using AFM platforms [51–53]. Thermo-mechanical analysis was implemented on various AFMs operated in contact mode (where the tip stays in contact with the surface all the time and applies constant force on the surface). The brief description of operational principles and hardware for each of the techniques is given below as well as examples of the results of mapping thermo-mechanical properties of the materials.

Transition Temperature Microscopy Transition Temperature Microscopy (TTM) uses the probe deflection as a function of the tip temperature to measure localized thermal expansion. When these probes are heated, the region under the tip usually expands, causing the cantilever to deflect upwards (Fig. 8.1a). When the material under the tip warms enough to soften, this process can begin to reverse as the tip penetrates the softer, but still expanding, material. In TTM the temperature of the transition is determined as the temperature maximum on the probe deflection versus temperature curve. Figure 8.1b shows a map of the glass transition temperature with 2.5-mm point-to-point resolution on a poly-styrene/poly(methyl metacrilate) sample [54, 55].

Fig. 8.1 (a) The red curve shows the cantilever deflection as a function of the heater sensor. Initially, the thermal expansion of the PET sample under the tip causes the deflection to increase. As the sample starts to soften under the tip, the deflection levels off and then decreases. To minimize damage to the tip and sample, the tip was manually pulled off the surface at the asterisk. The transition temperature can be estimated from the crossover between the expansion and melting portions of the curve. The inset image shows the topography of the indent created by this measurement. (b) Transition Temperature Microscopy (TTM) image of a polymer blend sample of Polystyrene and Poly(methyl methacrylate) taken with a scan size of 100 × 100 mm. These two materials are immiscible and so phase separated into micron size domains. The two materials can be easily differentiated by their different Tg values and TTM resolves the polystyrene domains as the blue roughly circular features (image is courtesy of Anasys Instruments, http://www.anasysinstruments.com)

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One complication of the variable deflection approach is that the load on the cantilever varies as the sample expands and then starts to soften or melt. This increasing and then decreasing load can in turn cause variations in the mechanical behavior of the tip–sample junction, as well as variations in the thermal coupling between the tip and the sample. For example, increasing the load will typically lower the measured transition temperature, while increasing the contact area will typically increase the heat transfer between the tip and the sample. Finally, the load can change the measured transition temperature. Multiple frequency techniques discussed in Section “Multiple Frequency Methods for ThermoMechanical Mapping (BE-NanoTA and Z-Therm)” solve some of the problems mentioned above by utilizing more sensitive oscillatory techniques for the detection changes of the mechanical properties of the sample as a function of temperature. As suggested by Nelson and King, one way to simplify these variations is to instead use a feedback loop to keep the deflection constant by varying the z-position of the base of the cantilever [56]. In the absence of thermally induced bending of the cantilever (discussed below), this approach provides much the same signal as simply measuring the deflection, with the advantage that the load on the sample is constant. Nelson and King reported better reproducibility using this technique. Not surprisingly, this allows smaller volumes of material to be probed. An example is shown in Fig. 8.2 below, where the indentations made into the sample were significantly smaller when deflection feedback was enabled. Note that this method is more quantitative when the signal from an independent position sensor on the z-axis is plotted rather than just the z-piezo drive voltage. Piezo motion is susceptible to hysteresis and creep, introducing significant error in the plotted position when the piezo drive voltage is used. Repeatability of the thermal curves of the type shown in Fig. 8.2 can be improved with automatic, high speed triggering of the pull-off. In both cases of varying deflection or deflection feedback, using a defined trigger point results in more repeatable melting curves. An example is shown in Fig. 8.2 where melting curves on a polyethylene-terephthalate (PET) surface were made using both variable cantilever deflection (open loop) and closed loop (feedback) methods, each with a nominal 50nm trigger pull-off. This trigger condition continuously monitors the signal and stores the maximum value. If the measured signal drops a specified amount below the recorded maximum value, the z-piezo is instructed to pull the cantilever away from the surface while it is still at a high temperature. Figure 8.2a shows four deflection curves made at different positions on a PET surface while Fig. 8.2b shows similar curves of the z-sensor output made with the deflection feedback enabled. Perhaps not surprisingly, there is a significant difference between the position of the maxima for the curves in (a) and (b). This is consistent with a higher loading force causing an earlier melting onset. Figure 8.2c shows an image taken after the melting curve measurements, with the resulting pits visible. The pit depths for the two methods are virtually identical. However, there are systematic pile-ups next to the open-loop pits. This may originate from the lateral motion of the tip as the load, and therefore the distance, between the base of the cantilever and the sample surface changes.

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Fig. 8.2 (a) Four deflection curves made at different positions on a PET surface. The pull-off trigger was set to nominally 50 nm below the maximum recorded value, based on the cantilever sensitivity calibration. (b) Similar curves of the z-sensor output made with the deflection feedback enabled. As with (a), the trigger value was chosen to be 50 nm below the maximum value. Notably, there is a significant difference between the position of the maxima for the curves in (a) and (b). This is consistent with a higher loading force causing an earlier melting onset. (c) An image taken after the melting curve measurements. The top four pits, circled in red correspond to the variable deflection measurements. In these any almost all other variable deflection measurements we observe a small pile-up besides the melted pits. This is consistent with the cantilever tip sliding laterally as it heats and is deflected by the expanding material. The bottom four pits, circled in blue, show the feedback enabled results. The depth of the different techniques using this rather large trigger value was not statistically different

One of the natural questions that arises with these measurements is what is the fundamental limit on the volume of probed material? To probe smaller volumes, there are some requirements. The first is to operate at smaller forces to minimize the contact area of the tip. Second, as the material starts to undergo a melting transition, we need to avoid having the cantilever tip rapidly plunge into the melted material. We can avoid this by having a low pull-off trigger threshold. Figure 8.3 shows the results of a short study where the two methods were tested using small trigger values. Respectively, the trigger values were 25, 10, and 5 nm

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Fig. 8.3 Cross sections of pits created with trigger values of (a) 25 nm, (b) 10 nm and (c) 5 nm. The solid curves were taken using the open-loop (variable deflection) method while the dashed lines show the results of the closed-loop technique. The color of the section curves are coded to the location on the image (d). Consistent with the load increasing because of the variable cantilever deflection, the width of the open-loop measurements systematically increases compared to the closed-loop pits. The arrow shows the location of an attempted 1 nm trigger value that resulted in an early pull-off for both techniques. The open loop deflection increased enough to leave an indentation in the sample

for the red, blue, and green color coded results. Probably due to the increasing load, the depth and width of the open-loop measurements systematically increases compared to the closed-loop pits. The arrow shows the location of an attempted 1nm trigger value that resulted in an early pull-off for both techniques. This is close to the thermal noise limit of the deflection detector in the AFM measurement and likely represents a fundamental limit on the trigger value. If this is indeed the case, the limit for this particular cantilever using the open-loop technique is a pit roughly 100 nm in diameter and 100 nm deep, while for the closed-loop technique, the limit is roughly 75 nm in diameter and 50 nm deep.

Scanning Thermal Expansion Microscopy One of the first measurements of thermo-mechanical properties with AFM was reported by Varesi and Majumdar [57] in 1998. These authors measured thermal expansion of Au wires on an SiO2/Si substrate during AC heating. The difference

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in thermal expansion for Au wires and SiO2 is clearly seen in Fig. 8.4b despite some topographical crosstalk (Fig. 8.4a). The amplitude of thermal expansion for Au wires was determined to be 24 pm. The sample was heated using AC current with a frequency on the order of tens of kilohertz. The fixture for the resistive heating of the sample was the add-on module used in this experiment. Periodic heating of the sample resulted in periodic expansion of the material, which was detected as displacement of the AFM tip. The amplitude of the tip oscillations in this case is proportional to the thermal expansion of the material. This method is quite simple in implementation and does not impose any special requirements on the AFM tip. However, the requirements for high electrical conductivity and large thermal expansion coefficient of the sample material significantly limited the applicability

Fig. 8.4 Thermal expansion maps of the Au stripes deposited on Si substrate (b) and PET/resin (d). The map (b) was acquired using AC heating of Au wires. AC heating of the tip was used for creation of map (d). Images (a, b) are courtesy of J. Varesi and A. Majumdar. Reproduced with permission from Appl. Phys. Lett. 72(1), 37 (1998). Images (c, d) are courtesy of A. Hammiche et al. Reproduced with permission from J. Microsc.-Oxf. 199, 180 (2000)

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of this method. Furthermore, this implementation of Scanning Thermal Expansion Microscopy (SThEM) provides accurate relative information about the sample’s mechanical properties. Unfortunately, quantification of the results is virtually impossible, because of the unknown temperature profile. In 2000, Hammiche et al. [51] used a similar approach to create a thermal expansion map of the material (Fig. 8.4c, d), but instead of periodic heating of the sample, he used periodic heating of the tip with AC current. This approach eliminates the sample’s electrical conductivity restriction and opens the pathway to measuring mechanical properties of non-conductive materials, such as polymers (PET/resin). The other benefit of localized heating of the tip is better lateral resolution demonstrated for this method compared to that demonstrated by the Varesi and Majumdar implementation of the SThEM technique. In Hammiche’s version of SThEM a specialized AFM tip (Wollastone probe or micro-machined probe) was used. Heating of the probe allowed researchers to heat the sample and also to know the precise temperature at the probe end. Quantification of the results and conversion of the thermal expansion amplitude to the thermal expansion coefficient of the material is possible with careful calibration of tip–surface contact properties, photodetector sensitivities, and other instrumental parameters. Without the calibration data Fig. 8.4c, d clearly show that the thermal expansion coefficient of PET is smaller than that of the resin, because the thermal expansion amplitude measured on PET is smaller than that measured on the resin.

Thermally Assisted Atomic Force Acoustic Microscopy Concurrently with SThEM development, Thermally Assisted Atomic Force Acoustic Microscopy (TA-AFAM) method was being developed by Oulevey and co-workers [58]. In this method, the stiffness of the sample was measured as a function of temperature using the same principle used in AFAM. They used a piezooscillator on top of the heater for the single frequency modulation of the normal force. PVC/PB polymer samples were measured. Several years later Oulevey, in collaboration with researchers from the UK, improved the resolution of this method and used TA-AFAM to determine softening transition of a PS/PMMA blend spincoated on a glass substrate. The softening transition temperature was determined by measuring stiffness maps at different temperatures. At 98°C, below the softening point of PS, the stiffness map has almost no contrast (Fig. 8.5b). At 103°C, above softening point of PS, some domain structure starts to appear (Fig. 8.5d). The dark domains in this figure were interpreted as the PS phase because of the lower stiffness of those domains at higher temperature [52]. In this experiment the contact area is not known and has not been controlled, thus, quantitative interpretation of the data is impossible. The TA-AFAM technique appears in publications under different names, such as variable temperature scanning local acceleration microscopy (T-SLAM) [52, 58] and dynamic mechano-thermal analysis by scanning microscopy (D-MASM). Since there is variation in the acronyms in the literature and the

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Fig. 8.5 The left-hand images (a, c) are topography of PS/PMMA on glass; the right-hand ones are simultaneously acquired DLT-MA amplitude (elasticity) images (b, d). The top set (a, b) were taken at 98°C, and the bottom at 103°C (c, d). Because of thermal drift, the three sets of images represent similar, but not the same, locations. The image is courtesy of Oulevey et al. Reproduced with permission and adapted from Polymer 41(8), 3081 (2000)

working principle of the technique is similar to Atomic Force Acoustic Microscopy AFAM [59], we will refer to this method as TA-AFAM. This method provides qualitative measurements of stiffness with sub-100nm spatial resolution [52]. Quantification of the results is possible in principle, however, the unknown properties of the tip–surface contact and cantilever stiffness makes this process quite challenging. Regular AFM probes are required for this technique.

Multiple Frequency Methods for Thermo-Mechanical Mapping (BE-NanoTA and Z-Therm) Both SThEM and TA-AFAM use single frequency detection techniques. In the case of SThEM, periodic heating of the sample is done at a single frequency and thermal

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expansion of the material is detected at the same frequency. In TA-AFAM, the transducer is excited with a single frequency and tip displacement is measured at the same frequency. In SThEM and TA-AFAM experiments, two oscillation parameters (amplitude and phase) are measured. The single frequency measurement approach for characterization of tip–sample contact does not provide all information about the tip–sample contact, because even the simplest model of tip–surface contact (Simple Harmonic Oscillator) has four independent fitting parameters (amplitude, resonance frequency, quality factor, and phase). In order to avoid this shortcoming multi-frequency measurement techniques are required. Contact mechanics can be probed with extremely high resolution by measuring the resonant frequency and quality factor rather than the DC deflection or feedback signal. That resonant frequency–domain measurements are more sensitive to tip–sample interactions is not news to the AFM community. AC methods were used by Binnig, Quate, and Gerber in their first AFM [60] to improve the sensitivity and noise rejection of the microscope over DC contact techniques. Recently, Band Excitation (BE) [61] techniques have been used on polymer samples to demonstrate the high sensitivity of the contact resonance and dissipation to thermal transitions [53]. Similarly, using the Dual AC Resonance Tracking [62] (DART™) technique provides much higher sensitivity to thermo-mechanical changes at the tip–sample junction than does the cantilever deflection (see Chaps. 5 and 17 for more details on DART technique). In addition to tracking the resonant frequency, this technique allows the tip–sample dissipation to be measured, a quantity that shows very strong temperature dependence in a wide variety of samples. In the examples (Figs. 8.6–8.8) shown here, we measure changes in the contact resonance behavior as a function of tip heating. To do this, we limit ourselves to the following experimental conditions: • The sample position is modulated at two frequencies, one above and one below the resonance to enable the resonance tracking (DART). • The temperature of the tip is ramped at a frequency significantly below the DART contact resonance modulation frequencies. PVDF and its copolymers are the most well-known polymer ferroelectrics with a wealth of potential applications in piezoelectric and pyroelectric devices, such as pyroelectric infrared sensors, shock sensors, energy conversion systems, etc. Although magnitudes of these effects in PVDF are much lower than those of ferroelectric ceramics, other factors make these materials very desirable for many applications. The PVDF polymers have low permittivity, low acoustic impedance and low thermal conductivity and are available in flexible large area sheets at a relatively low cost. Investigation of thermomechanic and thermoelectric responses in nanoscale PVDF structures is important from the viewpoint of their integration in nanoelectronic devices, as well as for advancing the fundamental understanding of the thermodynamics of phase transitions in ferroelectric polymers, where only rudimentary attempts at using SPM techniques have been undertaken so far. In the experiments below, PET samples were cut from a commercially available water bottle and glued to the top surface of the piezoelectric transducer. These

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Fig. 8.6 The effects of heating curves on PVDF sample. A 6 × 3 array of locations were heated to ~80°C while the tip was pressed into the sample with a preload of ~5 nN. (a) A topographic scan before the probing. (b) The same location afterwards. Selected heating curves were plotted in (c), color-coded by location in the images (a), (b), (d) and (e). Locations red, violet and dark blue were on the sample substrate and show large resonance frequencies that are independent of the heating. Locations green, yellow and light blue were made on the PVDF and show both lowered resonance frequencies because of the reduced modulus of the material and also clear temperature-dependent changes. The black location is a place where the localized heating resulted in a relatively large change in the sample topography. The black curve in (c) indicates that this change happened during the first heating cycle, where the resonance frequency initially increased significantly and then leveled out to behavior intermediate between the substrate and the on-PVDF curves. (d, e) Zooms before and after. In these images, the sample change at the black location is easy to see. (Sample courtesy of Alexei Gruverman, University of Nebraska–Lincoln)

samples were probed and imaged using silicon heated probes (see Section “Silicon Heater” below for more details). The probes were used in AC mode to produce high resolution topographic and phase images of the surfaces before and after thermal probing. The same probes were used in DART mode to locally probe the surface deformation, contact resonance and dissipation during thermal cycling [63].

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Fig. 8.7 Demonstration of mapping capabilities of Ztherm™. Thermo-mechanical properties of tip–surface junctions (resonance frequency, amplitude as a function of temperature) were measured on 6 × 3 grid on PET. Topography (a–d) and phase images ((a¢–d¢)) of PET sample after probing with decreasing peak heating voltages show that the decrease in peak probing voltage results in the decrease of sample damage. The temperature response curves are shown in (e), with each curve color-coded to the appropriate topographic and phase image. (f) The associated frequency shifts, (g) the associated local deformation of the sample surface and (h) the amplitude (related to the dissipation) of the cantilever during each cycle. The frequency (f) and amplitude (h) curves were each given arbitrary offsets to make them visible

PET, (C10H8O4)n), is a common thermoplastic polymer resin material used in a variety of products. The bulk Young’s modulus is ~3 GPa. The glass transition temperature is ~75°C while the melting temperature is ~260°C. As with most thermoplastics, PET has poor thermal conductivity (~0.24W/mK) and a large coefficient of thermal expansion (cte ~7×10-5/K). The above capabilities [64] allow us to take this technique to its logical conclusion – repeatable thermal probing of volumes of material so small that surface alterations are undetectable by normal AFM imaging (Fig. 8.7). Moreover, the improvements enable observations of significant and repeatable changes in the contact mechanics as a function of temperature – while afterwards being unable to locate the sites of these probed volumes with high resolution AC mode (tapping) images. Thermomechanical analyses of the amyloid fibrils were done using the Ztherm technique; tapping mode topography images before (Fig. 8.8a) and after (Fig. 8.8b) analysis show the degradation of the fibril after the measurements. Temperature dependencies of resonance frequency, amplitude and probe displacement were recorded. Figure 8.8c shows that temperature dependence of probe displacement shows a monotonic increase, while resonance frequency shows a significant jump, which probably corresponds to the transformation of the fibril. This observation proves that resonance frequency detection is much more sensitive than probe displacement to the change in mechanical properties of the sample. The Band Excitation-NanoTA method is the most versatile technique for contact characterization of all the techniques presented above. It allows direct determination

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Fig. 8.8 (a) An AC mode image of insulin fibers deposited on a mica surface. After imaging, a series of thermal bending compensated, low-temperature thermal cycles were performed in a 12 × 6 array of points. A small selection of those locations are indicated by the colored markers in both (a) and (b). (b) An AC image of the same region after the thermal cycling was completed. There are numerous gaps in the fibers where thermomechanical decomposition has occurred. (c) The local thermal expansion (top plots) and resonant frequency shifts (bottom plots) associated with the thermal cycles, color coded by location. Note the clear signal putatively associated with thermal decomposition of the fibers visible in the frequency shift curves. The deflection curves show no significant response at the same temperature. Some tip broadening has occurred during the thermal cycling that reduces the resolution between (a) and (b). Because the heating cycles were made at constant load, compensated for the thermally induced bending of the lever, the resonant shifts can be associated primarily with thermal decomposition, rather than a simple mechanical effect

of the mechanical properties of the tip–surface junction as a function of temperature by measuring mechanical response in the band of frequencies and comparing it with the excitation signal spanning the same band of frequencies. There are two implementations for this method. The first one is the modification of the atomic force acoustic microscopy (AFAM) [59, 65] technique by mechanical excitation and oscillation detection over a broad frequency range (band excitation (BE) technique) [66], and by using the heated tip probe [53] (BE-AFAM). The second one is the use of the heated tip and thermal expansion of the material combined in conjunction with the BE technique (BE-SThEM) [67]. The BE technique used in the BE-NanoTA method provides direct measurement of the four parameters of system resonance (resonance amplitude, resonance frequency, quality factor and phase) [67]. Furthermore, a protocol that maintains a constant tip/surface pressure and reproducible contact area during a temperature sweep has been reported, effectively extending the Oliver– Pharr method for NI to local thermal analysis [16, 53]. This method also provides measurements of mechanical properties as well as glass transition temperatures with sub-100 nm resolution. Specialized probes with a heater at the end are required for both implementations of this technique. The AFAM-based implementation of the method requires the use of the transducer to induce mechanical oscillations of the sample.

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Resonance frequency, Hz

a

b 420k

150 °C

165 °C

164 °C 160 °C

415k 410k

155 °C 405k

Region A 50

Region B

100 150 200 Temperature, °C

150 °C

Fig. 8.9 (a) Contact resonance frequency of tip–surface contact as a function of temperature. Red curve corresponds to the PMMA-rich region and extracted from the position (1) on the Tg map (b). Black curve corresponds to SAN-rich region and corresponds to the position (2) on the Tg map. (b) Spatial distribution of glass transition temperatures (Tg) determined as a temperature of maximum on contact resonance frequency curve. The point-to-point resolution is 50 nm. The image size is 3.75 × 3.75 mm. The image is courtesy of M.P. Nikiforov (unpublished)

Recently, Nikiforov, Jesse, and Kalinin mapped the glass transition temperature of SAN/PMMA phase separated polymers based on the measurement of the thermo-mechanical properties of this polymer mixture (Fig. 8.9b) [68]. The BE-SThEM technique was used. The point-to-point resolution of the glass transition temperature map was 50nm, which makes this technique applicable for the analysis of the internal structure of the polymeric materials. The glass transition temperature can be determined either as a temperature maximum on the resonance frequency versus temperature curve or as a temperature at which the quality factor falls below a certain value. Figure 8.9a shows the typical dependence of resonance frequency as a function of temperature. The initial increase in resonance frequency (region A) comes from the increase of the tip–surface contact area associated with initial sinking of the probe into the surface. The decrease of the contact resonance frequency (region B) as a function of temperature occurs because of the sharp decreases in the Young’s modulus of the sample after the polymer undergoes glass transition. Thus, the temperature maximum on the contact resonance frequency curve is a good measure of the glass transition temperature. To measure the Young’s modulus as a function of temperature, the “freeze-in” protocol was developed [53]. This protocol is similar to the Oliver–Pharr method used for NI when on the first temperature cycle, the AFM tip sinks into the surface and materials properties are measured on the subsequent cycles. This protocol allows maintaining contact area constant during the experiment and quantitative determination of Young’s modulus maps at different temperatures. Young’s modulus of PET as a function of temperature was measured using this technique. Temperature, resonance frequency, and quality factor as a function of time are presented in Fig. 8.10a. Using the mechanical model (Fig. 8.10c) and the Young’s modulus of

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b

Quality factor

210 180 150 120 90 60 30 0

340k

120

330k 90

320k

60

310k

30 0

300k 0

290k 50 100 150 200 250 300 Approximate probe temperature,∞C

Resonance frequency, Hz

240

Resonance frequency, Hz

360k 300∞C Probe Temperature 350k 340k 330k 20∞C 320k 310k 300k 290k 280k 270k 0 50 100 150 200 250 300 Time, sec

Quality factor

a

215

d

c

Young modulus, GPa Contact stiffness, k2(N/m)

k1 m

k2

c

A cos ωt

5

200 150

Glass Transition

Melting

1,8x10-5

4

1,6x10-5

3

1,4x10-5

2

1,2x10-5

100 50 0

1,0x10-5 50 100 150 200 250 300 Approximate probe temperature, ∞C

Damping coefficient, c (kg/s)

250

Fig. 8.10 Resonant frequency and quality factor of the tip–surface system as a function of time (a) and as a function of temperature (b). (c) Equivalent model for tip–surface contact. (d) The local Young’s modulus and damping coefficient for PET sample as a function of temperature. The image is courtesy of S. Jesse et al. Reproduced with permission and adapted from Appl. Phys. Lett. 93(7), 073104 (2008)

PET at room temperature, Young’s modulus of PET as a function of temperature can be calculated [53, 67]. The main characteristics of each of the techniques discussed above are listed in Table 8.2. Table 8.2 also shows that most of the work on local thermo-mechanical characterization was done on polymeric samples. The reasons for such behavior are discussed below. The small size of the heated probes limits the output heating power these probes can supply to the junction. This imposes restrictions on the thermal conductivity of the samples analyzed with those techniques. For example, the high thermal conductivity of metallic samples does not allow measurement of mechanical properties as a function of temperature. Samples with small thermal conductivity or small heat capacity are the most suitable for SThEM, TA-AFAM, Z-Therm, and BE-NanoTA. Thus, these methods are well positioned for the study of thermomechanical properties of polymeric materials. The samples with relatively small surface roughness (below 10–20nm RMS on the area 10×10µm2) provide the best measurements of the thermo-mechanical properties because of the minimization of the cross-talk between topography and mechanical measurements.

Table 8.2 The comparison of experimental implementations for TTM, SThEM, TA-AFAM, Z-Therm, BE-NanoTA TTM SThEM TA-AFAM Z-Therm Detection of Detection of Detection of thermal Working principle Detection of the changes the changes expansion in out-ofthe onset in contact in contact plane direction of material stiffness stiffness softening Spatially resolved Spatially resolved thermal Spatially resolved Measured sample Spatially resolved contact stiffness expansion map contact stiffness property glass transition map at constant map at constant maps temperature; temperature glass transition and softening temperature Samples PS/PMMA blend Au stripes on SiO2, Au PVC/PB blend PET, PVDF stripes coated with PMMA, ITO resistor Conditions Ambient Ambient Ambient Ambient Time of the Tens of minutes Tens of minutes Tens of minutes Tens of minutes experiment

Ambient Several hours

PET, SAN/PMMA, PETG

BE-NanoTA Detection of changes in contact resonance characteristics as a function of temperature Spatially resolved elastic and viscous properties; glass transition and softening temperature

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Types of Probes Used for Thermo-Mechanical Analysis The cantilever probe is a crucial component of every AFM-based technique. In the previous section we saw that two types of probes were mainly used for local thermal analysis using AFM: regular cantilevers and cantilevers with miniature heaters at the end. The description and specifications of various regular AFM cantilevers can be found on the web-sites of major AFM tip manufacturers, such as MikroMash [69], Nanosensors [70], and Olympus [71], as well as major AFM manufacturers (Agilent [72], Asylum Reseach [73], NT-MDT [74], Veeco [75], etc.). In this section we focus on the progress made in the manufacturing of thermally active probes. The development of thermally active probes, started in 1986 by Wickramasinghe et al. [76] continues today [77]. The main goal of this development is to manufacture the probe whose temperature can be independently measured and controlled. In the course of the probe development, several constraints need to be balanced: the fast heating/cooling rate necessary for fast probing; the heater power that determines which materials can be studied and what heating electronics hardware can be used; and the size of the contact area of the heater which determines spatial resolution of the mechanical properties mapping, etc. Multiple designs have been explored over the past two decades [78–85]. Nowadays, there are two widely accepted designs, which are described in detail below. Both designs utilize resistive heating of the probe. As a result of the efforts in tip development, the spatial resolution of the measurements has reached tens of nanometers and the temperature resolution has reached a few degrees Kelvin [86]. The outstanding problem is the batch processing and reproducibility of the properties of the thermal probes.

Wollaston Probe Resistive probes have one great advantage in applications requiring more than topographic and thermal imaging: they may be used in active mode, thereby acting as a self-contained device for the supply of heat as well as temperature measurement. Consequently, they may be used for localized thermal analysis as well as AFM and Scanning Thermal Microscopy. The type of sensing element often used in nearfield resistance thermometry consists of the apex of a fine V-shaped wire, known as the Wollaston probe after the type of silver-sheathed noble metal wire from which they are made (Fig. 8.11) [87]. The resistive probe is capable of performing three functions: it exerts a force on the sample surface; it acts as a highly localized heat source; and it measures heat flow. Probes can be used in either of two operating modes [88, 89]: • Constant temperature mode and variable temperature mode, where the probe acts as local heater. • Constant current mode – where a small current is passed through the probe which then acts as a thermometer.

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Fig. 8.11 Scanning electron microscopy image of the Wollaston probe. Image is courtesy of H.M. Pollock and A. Hammiche. Reprinted with permission from J. Phys. D: Appl. Phys 34 (2001) R23–R53

Constant temperature mode: The thermal element can be used as a resistive heater and forms part of a Wheatstone bridge circuit. When the probe contacts successive regions of the sample surface that may differ in their thermal properties, varying amounts of heat will flow from the probe to the sample. However, the probe heating circuit uses a feedback loop to adjust the current through the probe to keep the probe resistance, and hence the temperature, constant. The required feedback voltage is used to create contrast in the thermal image. The power involved may be calculated from the bridge output voltage and is affected by several factors including the probe/sample contact area, the temperature difference between probe and sample, and variations in the local thermal conductivity of the near-surface regions of the sample. For the samples with low thermal conductivity, such as polymeric samples, the constant power approach can be used. Tip temperature in contact with surface can be calibrated using the melting temperature of the polymer as a reference point. The calibration of the input voltage as a function of tip temperature works well for materials with similar thermal conductivities. The spatial resolution of Wollaston probes is on the order of 10µm [90]. Variable temperature mode: Similar hardware (Wheatstone bridge circuit) is used for both variable temperature mode and constant temperature mode. Variable temperature mode is used to measure local thermal expansion of the material under the tip, when amplitude (and phase) of the oscillations in the displacement is measured at the frequency of local heating [51]. Local thermal conductivity of the material can also be measured when the amplitude (and phase) of the oscillations in tip–surface temperature is measured at the third harmonic of the frequency of local heating [91, 92].

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Silicon Heater In 2006 King et al. developed and patented a process for manufacturing silicon heaters [93]. The SEM photograph of the silicon heater is presented in Fig. 8.12. The end of the probe is heated by passing current through the resistor (low doped Si region) located at the end of a degenerately doped U-shaped silicon cantilever. The functionality and performance of these heater probes, such as tip temperature as a function of sourced electrical power [86], mechanical behavior of the cantilever as function of time [94], etc. [90, 94] were extensively characterized. High heating rates (up to 25×106K/s) [94] achievable by these cantilevers make them suitable for imaging mechanical properties at high frequencies (up to 5MHz). The extremely sharp tip (~20nm radius) located at the end of the cantilever allows imaging of the mechanical properties with sub-100nm spatial resolution. Precise temperature calibration of these tips remains the main issue for these cantilevers. For free-standing cantilevers, various methods are available such as Stokes Raman peak shift [95], IR-thermometry [86], and others. However, when the tip is brought into contact with the surface, the temperature of the tip–surface junction changes depending on the thermal conductivity and heat capacity of the surface material. One of the possible solutions for this issue is the use of melting standards to calibrate the tip–surface temperature. The input voltage required to melt the tip into the polymer surface was measured for the polymers with different melting temperatures, providing a calibration for tip–surface contact temperature as a function of input voltage [67].

Fig. 8.12 Scanning electron microscopy image (left) and infrared microscope (right) of the heated AFM cantilevers. The image is courtesy of B.A. Nelson and W.P. King. Reproduced with permission from Sensors and Actuators A 140 (2007) 51–59

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Mathematical Models for Understanding Thermo-Mechanical Results Mathematical models for tip–surface heat transfer, mechanical response of the material to temperature gradients, and tip–surface contact mechanics are necessary for quantitative interpretation of the data obtained using the experimental techniques described above. Each of the topics listed above is broad and important enough to be a topic of a review paper, thus, only the basic concepts necessary to understand the techniques discussed above, will be presented here. Excellent papers and books have been written on the topics of tip–surface heat transfer [96–98], mechanical response of the material to temperature gradients [99], and tip–surface contact mechanics [100], and readers are encouraged to read those for better coverage of the specific topic. Several processes are happening at the same time when the heated probe is located in close proximity to the surface: • • • •

Radiative heat transfer according to the Stephan–Boltzmann law. Diffusive heat transfer. Heat transfer through solid–solid contact. Heat transfer through the water meniscus between probe and the surface.

Xu et al. determined [101] that under ambient conditions diffusive heat transfer by air is several orders of magnitude larger than radiative heat transfer. For probesurface distances smaller 100nm, heat flux transferred by air is ~4W/(Kcm2), while radiative heat flux transferred by irradiation is ~0.1W/(Kcm2) [101]. After that a group of researchers led by Majumdar started to investigate other mechanisms of heat transfer. The authors found that heat transfer rate does not depend on the indentation force and ruled out heat transfer through solid–solid contact as a possible mechanism [102]. However, the other group of researchers [103] found that for the warm Wollaston probe in contact with copper and Duralium, heat transfer through solid–solid contact is the main mechanism for heat transfer. The other hypothesis to be checked was a diffusive heat transfer, Majumdar et al. [102] varied the gas atmosphere during the experiments and found no significant change in heat transfer rate, despite the fact that the gases chosen had several orders of magnitude difference in thermal conductivity. Recently, Shen et al. found that heat transfer coefficients at nanoscale gaps (tens of nanometers) are three orders of magnitude larger than that of the blackbody radiation limit [104], which indicates the surface phonon polaritons dramatically enhance energy transfer between two surfaces at small gaps. The widely accepted model of heat transfer between probe and surface is transfer through the water meniscus. It is well known that when a sharp probe approaches the surface the water from air condenses around the contact area. The calculated heat resistance of such contact corresponds pretty well with the value measured by Majumdar et al. [102]. This leaves two main mechanisms for heat transfer: heat transfer through water meniscus and via solid–solid contact. Both of these mechanisms allow for high resolution imaging of thermo-mechanical properties, because the heating region is confined to the contact area in both cases.

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Contact Mechanics Model of Elastic Media in the Presence of a Heat Transfer Understanding of a material’s response to a temperature gradient requires knowledge about the temperature dependent contact mechanics of the tip–surface junction. From simple dimensionality arguments, the thermal response of a material, dl , is linear and related to temperature variation, dT, through the thermal expansion coefficient, a, and the characteristic tip–surface contact radius, R0, i.e., dl=adTR0. To establish the contact mechanics model, Morozovska et al. has used the decoupled approximation originally developed for Piezoresponse Force Microscopy (i.e., bias-induced phenomena), i.e., the equations of state for an isotropic elastic medium in the presence of a heat transfer. The mechanical stress tensor sij and elastic strain uij are linked as dij adT + sijkl s kl = uij , where dij is the Kroneker delta, and sijkl is the tensor of elastic compliances [105, 106]. The general equation of mechanical equilibrium, ¶s ij / ¶x j = 0, leads to the equation for the mechanical displacement vector, ui, as:

cijkl

¶ 2 uk ¶dT - cijkk a = 0. ¶x j ¶xl ¶x j

(8.1)

Here, we introduce the boundary conditions of the free surface S:

æ ö ¶uk - cijkk adT ÷ n j ç cijkl ¶xl è ø

= 0.

(8.2)

S

where cijkl is the tensor of elastic stiffness, and nj are the normal components. The maximal surface displacement (corresponding to the point x=0) below the tip is

u3 (0, t ) = -

x3 adT (ξ , t ) 1+ n dx1dx 2 dx3 . òòò p x3 > 0 (x 2 + x 2 + x2 )3/ 2 1

2

(8.3)

3

where Gijs is the appropriate tensorial Green function [105–107]. Temperature distribution is found as a solution of Laplace’s equation k 2 Ñ 2 dT (x, t ) = 0 , where the k 2 = c / cr is the thermal diffusivity of the media, c is thermal conductivity [W/(mK)], and c is the specific heat [J/(kgK)], and r is density [kg/m3] [108]. After integration, we derive

ui (0) » - (1 + n )adTR0

(8.4)

While the numerical prefactor in (8.4) depends on the choice of boundary conditions (zero temperature, zero flux, or mixed) on the free surface, the overall functional form is universal as dictated by dimensionality considerations. Physically, there are two limiting cases for boundary conditions: zero temperature outside of the contact (provides smallest values of strain) and zero flux outside of the contact (provides largest values of strain), which give the prefactors in (8.4) of 1 and p/2, respectively.

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For all other boundary conditions, the prefactors in (8.4) will lie in the 1-p/2 range. If the mean value of the prefactor range is chosen, the model error will not exceed 30%, which is comparable to the experimental error of ~10% (especially given the uncertainty in contact radius). The choice of a static (zero frequency) regime is justified by the fact that the temperature drops exponentially with distance from the contact [109]

T (r ) = T0

æ æ æ rcont w ö w öö exp çç - (r - rcont ) ÷÷ exp çç i çç wt - (r - rcont ) ÷ ÷ (8.5) r 2k ø 2k ø÷ ø÷ è è è

The characteristic thermal length is determined to be either d = 2k / w or the contact radius (whichever is smaller), where w is the frequency of incoming heat waves. For the typical polymer material (e.g., PET), the characteristic length at experimental frequencies (~500kHz) is about 500nm, much larger than the contact radius of the tip (~10–100nm). Thus, the major temperature drop happens at the distances between one and two contact radii from the center of the tip and the imaging is essentially near-field for thermal strain waves within the material. Hence, the measured thermo-mechanical response is directly related to contact radius which, in turn, is determined by the indentation force and Young’s modulus.

Contact Mechanical Model for Deconvolution of the Mechanical Properties of Samples from Parameters of Tip–Surface Contact Resonance Young’s and loss moduli as a function of temperature can be determined from temperature dependencies of quality factor and resonance frequency of the tip–surface junction [53]. The decrease in quality factor and resonance amplitude of the tip– surface junction is an indicative of polymer softening and subsequent melting. When the polymer melts, the Q factor for mechanical resonance decreases due to viscous damping in the molten polymer (Fig. 8.10a, b). A simplified model of the tip–surface junction is presented in Fig. 8.6c. The spring constant, k1 is the stiffness of free cantilever, m is the effective mass of the cantilever, k2 is the tip–surface contact stiffness, and c is the viscosity of the polymer. The change in resonant frequency can be attributed to either changes in the effective dissipation [110] or effective material properties. For a damped oscillator, the resonant frequency decreases with the decrease of Q as

fres = f0 1 - 1 / Q 2

(8.6)

Equation (8.6) predicts a 0.3% decrease in the resonant frequency during the heating cycle (taking Q values from Fig. 8.6b), as compared to the experimentally observed

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10% decrease in resonant frequency. Therefore, the decrease in resonant frequency is ascribed to a decrease in the contact resonance stiffness only. Using the approximate formulae [111, 112], different contact resonances can be used to calculate contact stiffness as a function of frequency:

æ k ö f resi @ ç1 - bi 1 ÷ f 0BOUND k2 ø è

(8.7)

where fres is the resonance frequency of tip–surface contact, i is the order of the resonance, f0BOUND is the resonance frequency of the tip in contact with an infinitely stiff surface (this parameter depends only on the cantilever), k1 is the spring constant of the cantilever, and k2 is contact stiffness. The coefficients for the first five contact resonances are calculated (b1=7.7, b2=36.5, b3=80.4, b4=144.9, b5=230.8, …) based on the numerical analysis of contact resonance developed by Mirman [113]. Estimating a constant area between the tip and surface (with the radius a=50nm) and taking the Young’s modulus for PET at room temperature as E=3GPa and k2=2aE, we extract the relative change of the contact stiffness for the tip–polymer contact as a function of temperature, as shown in Fig. 8.10d. In this analysis the assumption was that the spring constant of the cantilever was temperature independent. Notably, this assumption holds within 20% error, as shown by Lee and King [114] who performed a thermal actuation of the silicon heated cantilever and found that the resonant frequency of the oscillations changes <10% for temperatures <200°C. The damping coefficient, c, as a function of temperature can be extracted from the temperature dependence of the Q-factor (assuming Q>>1):

c@

m(k1 + k2 ) æ 3ö ç2 - ÷ Q Q è ø

(8.8)

Figure 8.10d shows the dependence of c as a function of temperature assuming tip mass m=2.2×10-9kg (re-calculated from w0BOUND ) and k1=1N/m. The two transitions at ~80°C and 250°C that are seen after the first heat/cool cycle are presumably associated with the re-crystallization and melting which occurs after first heating. Thus, temperature dependencies of Young’s modulus and the damping coefficient can be used for determination of local glass transition and melting temperatures of the polymeric materials.

Technique Development Prospects and Limitations In order to understand the positioning of the local thermal analysis technique within the group of techniques used for mechanical analysis, we compared the spatial resolution and sensitivity of the displacement measurements provided by various techniques

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used in mechanical analysis (local thermal analysis (LTA) with Wollason probe [51], silicon heater [90]; local thermal analysis with silicon heater and band excitation detection (NanoTA, current work); dynamic mechanical analysis (DMA) [115]) with the resolution and sensitivity required for the solution of practical problems in different industrial applications, such as analysis of pharmaceuticals, organic layers in OLEDS, lithography masks, and mechanical properties of surfaces (auto, optics, etc.), as well as for the understanding of basic scientific problems such as thermomechanical motion of single molecules (Fig. 8.13). The spatial resolution and sensitivity of the technique discussed in this paper are estimated below. Recently, it was shown [67] that the spatial resolution for the local thermo-mechanical measurements in general and Tg measurement in particular is limited by the tip–surface contact radius, which is usually ~10nm. The vertical sensitivity level in static AFM (e.g., contact mode) is estimated as ~0.1nm (experimental noise limit for conventional photodetectors). For ac-detection methods based on amplitude or frequency

Fig. 8.13 Comparison map for precision in displacement measurements achievable with different techniques of thermal analysis as a function of spatial resolution overlaid with precision in displacement measurements and spatial resolution desired by different industries. Precision in displacement measurements is often limited by thermal noise, thus, low temperature (4 K) diagram draws the limits for ultimate resolution of the local thermal analysis techniques

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detection, the vertical sensitivity as limited by the thermo-mechanical noise [116] is dltm » 2 kBTB / kw r , where kB – Boltzman constant, T is the temperature, B is the measurement bandwidth, K is the cantilever spring constant, and wr is the cantilever resonant frequency. This yields the relationship between resolution and sensitivity as

dl » - (1 + n )adT . For a typical polymer linear expansion coefficient (a) 5×10-4/K, R0 Poisson’s ratio is about 0.34. For the typical cantilever parameters, k~1N/m, wr ~ 2p 300kHz (contact resonance frequency), and B~1kHz (typical experimental bandwidth). At room temperature (300K), the thermo-mechanical noise will be on the order of 3pm, lowering temperature to 4K results in an order of magnitude decrease of thermo-mechanical noise (dl~0.3pm). We have reviewed several techniques for high spatial resolution analysis of the mechanical properties of samples as a function of temperature, with the main area of applications being the thermo-mechanical analysis of polymeric materials. Clearly, significant opportunities and challenges for development of the technology must be met before the field reaches maturity and widespread adoption. On the data analysis side, better mathematical models are required for modeling thermal expansion response to local heating, specifically frequency dependencies of thermal expansion for different indenters, better understanding of the heat transfer mechanism between tip and surface etc. On the experimental side, heated probes with better mechanical stability and better reproducibility of the mechanical properties would be highly desired. The probes currently available have significant drift in the mechanical properties during thermal cycling. Extension of thermo-mechanical measurements towards calorimetric measurements seems to be a logical direction for technique development. However, in ambient air, the main heat transfer mechanism between probe and the surface is heat transfer through the surface liquid film, which precludes these experiments being performed quantitatively in air. The shape and properties of the liquid meniscus forming between the tip and surface depend on many external conditions such as temperature and humidity, which are often difficult to control. This makes the contact thermal properties extremely irreproducible. Thus, under ambient conditions, scanning calorimetry is very tricky; however, in low vacuum, where the liquid meniscus between tip and surface is eliminated, scanning calorimetry might work. Finally, the resolution of thermo-mechanical techniques is limited to the tip size, which is about 10nm. This resolution allows scientist to study fundamental processes of phase transitions in polymers (glass transitions, melting transitions) at physically relevant length scales in the polymeric materials, i.e., the size of Kuhn’s segment. Acknowledgements Research at Oak Ridge National Laboratory’s Center for Nanophase Materials Sciences was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy.

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53. Jesse, S.; Nikiforov, M. P.; Germinario, L. T.; Kalinin, S. V. Applied Physics Letters 2008, 93, (7), 073104. 54. The TTM technique was recently developed by Anasys Instruments for thermo-mechanical property based imaging and the technique can be implemented either using an AFM or the VESTA tool from Anasys. The only difference between the TTM technique using an AFM and using the VESTA tool is the implementation of image rastering. AFM typically uses piezoelectric crystals, which allow single digit nm resolution during image rastering; motors are used in the VESTA, which allow 1um point to point resolution during image rastering. 55. http://www.anasysinstruments.com/. 56. Nelson, B. A.; King, W. P. Review of Scientific Instruments 2007, 78, (2). 57. Varesi, J.; Majumdar, A. Applied Physics Letters 1998, 72, (1), 37–39. 58. Oulevey, F.; Gremaud, G.; Semoroz, A.; Kulik, A. J.; Burnham, N. A.; Dupas, E.; Gourdon, D. Review of Scientific Instruments 1998, 69, (5), 2085–2094. 59. Rabe, U.; Arnold, W. Applied Physics Letters 1994, 64, (12), 1493–1495. 60. Binnig, G.; Quate, C. F.; Gerber, C. Physical Review Letters 1986, 56, (9), 930–933. 61. Jesse, S.; Kalinin, S. V.; Proksch, R.; Baddorf, A. P.; Rodriguez, B. J. Nanotechnology 2007, 18, (43), 435503. 62. Rodriguez, B. J.; Callahan, C.; Kalinin, S. V.; Proksch, R. Nanotechnology 2007, 18, (47), 6. 63. . Gannepalli, A.; Proksch, R. 2009. 64. . http://www.asylumresearch.com/Products/Ztherm/Ztherm.shtml. 65. Rabe, U.; Janser, K.; Arnold, W. Review of Scientific Instruments 1996, 67, (9), 3281–3293. 66. Jesse, S.; Kalinin, S. V.; Proksch, R.; Baddorf, A. P.; Rodriguez, B. J. Nanotechnology 2007, 18, (43), 435503. 67. Nikiforov, M. P.; Jesse, S.; Morozovska, A. N.; Eliseev, E. A.; Germinario, L. T.; Kalinin, S. V. Nanotechnology 2009, 20, (39), 395709. 68. Nikiforov, M. P.; Gah, S.; Jesse, S.; Composto, R. J.; Kalinin, S. V. unpublished. 69. http://www.spmtips.com/, In. 70. http://www.nanosensors.com/, In. 71. http://www.probe.olympus-global.com/en/index.html. 72. http://www.home.agilent.com/agilent/editorial.jspx?action=download&cc=US&lc=eng&cke y=914532. 73. http://www.asylumresearch.com/Products/Levers/LeverGuide.shtml. 74. http://www.ntmdt-tips.com/catalog/golden.html. 75. https://www.veecoprobes.com/. 76. Williams, C. C.; Wickramasinghe, H. K. Applied Physics Letters 1986, 49, (23), 1587–1589. 77. Lee, J.; King, W. P. IEEE Sensors Journal 2008, 8, (11–12), 1805–1806. 78. Chui, B. W.; Asheghi, M.; Ju, Y. S.; Goodson, K. E.; Kenny, T. W.; Mamin, H. J. Microscale Thermophysical Engineering 1999, 3, (3), 217–228. 79. Luo, K.; Shi, Z.; Lai, J.; Majumdar, A. Applied Physics Letters 1996, 68, (3), 325–327. 80. Majumdar, A.; Lai, J.; Chandrachood, M.; Nakabeppu, O.; Wu, Y.; Shi, Z. Review of Scientific Instruments 1995, 66, (6), 3584–3592. 81. Nakabeppu, O.; Chandrachood, M.; Wu, Y.; Lai, J.; Majumdar, A. Applied Physics Letters 1995, 66, (6), 694–696. 82. Gimzewski, J. K.; Gerber, C.; Meyer, E.; Schlittler, R. R. Chemical Physics Letters 1994, 217, (5–6), 589–594. 83. Majumdar, A.; Carrejo, J. P.; Lai, J. Applied Physics Letters 1993, 62, (20), 2501–2503. 84. Nonnenmacher, M.; Wickramasinghe, H. K. Applied Physics Letters 1992, 61, (2), 168–170. 85. Williams, C. C.; Wickramasinghe, H. K. Journal of Vacuum Science & Technology B 1991, 9, (2), 537–540. 86. Nelson, B. A.; King, W. P. Sensors and Actuators A: Physical 2007, 140, (1), 51–59. 87. Royall, P. G.; Kett, V. L.; Andrews, C. S.; Craig, D. Q. M. Journal of Physical Chemistry B 2001, 105, (29), 7021–7026. 88. Hammiche, A.; Pollock, H. M.; Song, M.; Hourston, D. J. Measurement Science & Technology 1996, 7, (2), 142–150.

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Part IV

Electrical and Electromechanical SPM

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Chapter 9

Advancing Characterization of Materials with Atomic Force Microscopy-Based Electric Techniques Sergei Magonov, John Alexander, and Shijie Wu

Scanning Probe Microscopy in its Development Modern materials science is focused on development of novel materials and functional structures for a number of fast progressing industries (semiconductors, data storage, biomaterials, health care, and others) where miniaturization plays the dominant role. This tendency influences not only synthesis of materials and technology of structure fabrication but also a development of analytical methods applied for materials characterization at small scales. In recent years an increasing number of analytical methods have been introduced for comprehensive analysis at the micron and nanometer dimensions. Of particular interest is scanning probe microscopy (SPM) – a family of methods allowing visualization of surface structures and examination of their mechanical, electromagnetic, optical, and other properties at such scales. The invention of scanning tunneling microscope in 1981 [1] was the starting point of SPM, which attracted many researchers by fascinating capabilities of visualization surface structures from the macroscopic down to the atomic-scale. The history of SPM is curious in itself. The revolutionary move was to use prior known technologies, such as profiler [2] and topographiner [3] in studies at the atomic scale in STM and its offspring – atomic force microscope [4]. The core of SPM is the measurement and control of current and force interactions between a minute probe and a sample surface. Actually, atomic force microscopy (AFM), which is based on the probe–sample force detection and is not limited by sample conductivity, became the leading SPM method. The exciting features of these methods are their applicability to a broad range of materials and operation in different environments: vacuum, air, gases, liquids, and at different temperatures. The introduction of AFM was accompanied by several important developments such as the use of optical deflection detection [5, 6], and oscillatory modes [7, 8], especially allowing the imaging of soft materials in air [9]. The batch fabrication of AFM probes made of Si3N4 and Si [10, 11] was also the important step in establishing this technique as a routine characterization tool. The progress of SPM S. Magonov (*) Agilent Technologies, 4330 W. Chandler Blvd, Chandler, AZ 85226, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_9, © Springer Science+Business Media, LLC 2010

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instrumentation and enthusiastic motivation of many researchers who joined this field have stimulated the broadening of AFM applications and its fast spread from academic labs to industrial environment. Through practical applications it became clear that true AFM value extends far beyond its basic capability of high-resolution visualization of surface structures. The AFM probe, which is a microfabricated cantilever with the sharp tip at its end, is actually a highly sensitive detector of mechanical, electromagnetic and other tip–sample interactions. The probe responds differently at sample locations or components with dissimilar properties (mechanical, electromagnetic, etc.) that provides the basis of AFM compositional mapping of heterogeneous and multicomponent materials. For many complex polymer materials the AFM phase images obtained in oscillatory amplitude modulation (AM) mode provide the best contrast differentiating individual components. Naturally, the probe sensitivity to different materials’ properties raises a question about its possible use for local quantitative measurements of these properties. This refers not only to the examination of local mechanical properties (e.g., by making AFM-based nanoindentation measurements), but also to the studies of various electric and magnetic properties. There is a large amount of data illustrating the AFM sensitivity to these properties, yet the rigorous approaches towards reliable quantitative measurements have not been fully developed. It happened that for a number of years the development of SPM instrumentation was in relative stagnation and the growth of practical applications was not adequately supported by instrumental innovations. Nowadays this trend is changing and the progress is noticeable in several areas. The ongoing improvement of noise characteristics of AFM electronics, the microscope overall performance (e.g., by minimizing thermal drift), and manufacturing of sharper probes will enhance the capabilities of high-resolution imaging in different environments. Particularly, the improved home-built microscope allows extending the frequency modulation (FM) mode, which is commonly used for atomic-scale imaging in UHV [12, 13], to measurements in air and under liquid, where imaging with molecular and atomic resolution was demonstrated [14]. Similar efforts are needed to make molecular-scale measurements in air and liquid the routine features of the commercial microscopes. When these capabilities become available to a large number of researchers we might expect a better understanding of the true atomic-resolution imaging in AFM. In the probing of local mechanical and electric properties, the multifrequency measurements offer new capabilities of characterization of materials in the broad frequency range and, particularly, for quantitative analysis of these properties. AFMbased mechanical studies are traditionally performed in the quasistatic regime when a probe is positioned at one location and is moved towards and into the sample. In this procedure, the cantilever deflects in response to elastic, plastic or viscoelastic tip–sample interactions [15] and such deflection-versus-distance (DvZ) curves – the analog of stress–strain dependences are used for the extraction of quantitative mechanical properties of the sample. The tip indentation is usually performed in the rate of 0.1–10 Hz. This measurement can be performed on many locations, and the resulted map of the mechanical response might have the nanometer spatial

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resolution if thermal drift of the instrument is low. Latest progress showed that reliable quantitative data can be obtained in this way [16] yet such measurements are relatively slow and cover only a relatively low frequency range. The expansion of mechanical measurements to higher frequencies has been recently demonstrated by fast detection of the DvZ curve practically during every cycle of the probe oscillation in the AM mode. This can be done either through the reconstruction of the probe response at large number of harmonics (multiples of the main cantilever resonance) [17] or directly by using the grating-type probes [18]. Therefore, the way for quantitative nanomechanical measurements in the broad frequency range (up to 100kHz) is demonstrated and the practical results are accumulating. Besides local mechanical measurements, AFM has excellent capabilities for measurements of electric and magnetic properties of samples and their mapping with high spatial resolution. The response of the AFM probe to the electrostatic tip–sample forces was examined shortly after the introduction of this technique [19]. The main issue of the sensing of the electrostatic forces by the probe is related to the fact that during AFM imaging the probe behavior is influenced by mechanical and electrostatic forces simultaneously. Therefore efficient separation of these effects is essential for reliable measurements of the local electric response. For this purpose the use of different frequencies was suggested long time ago and this operation was performed in the non-contact mode that limits the spatial resolution of such electrostatic measurements. When the electrostatic force and mechanical tip–sample interactions are detected at single frequency then the lift method [20] was suggested for the separation of their responses. The main limitation of this approach is a remote position of the probe while recording the electrostatic force response. As a result the spatial resolution of these measurements is rather restricted. With development of the multifrequency SPM, novel approaches in probing of local electric properties can be suggested and verified. Recent results obtained in the electric force microscopy (EFM), Kelvin force microscopy (KFM), and piezoresponse force microscopy (PFM) with a commercial microscope equipped with three lock-in amplifiers that allows simultaneous measurements at different frequencies are presented in this Chapter. Novel schemes of these techniques will be described below. The value of these methods will be illustrated by experimental results obtained on different samples. Such direct interplay between the instrumental developments and their practical evaluation is indispensable for AFM advancement. The chapter consists of three parts. The first deals with different AFM approaches in measurements of electrostatic and electromechanical surface properties, overview of EFM and KFM applications, and discussion of the outstanding issues of these techniques. In the second part a practical implementation of EFM and KFM measurements will be described and novel approaches to electrostatic measurements will be introduced. The use of PFM for examination of electromechanical sample properties will be described shortly. Several issues regarding experimental procedures and choice of the probes will be also discussed. The third part presents a number of recent results obtained with EFM, KFM, and PFM on semiconductor materials, metal surfaces, and organic adsorbates on different substrates.

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Electrostatic and Electromechanical Interactions in Atomic Force Microscopy Tip–sample force interactions are the basis of AFM and related modes, which are used for the examination of surface topography as well as local mechanical, electromagnetic, and other properties. These forces of different origin cause displacements of the same microfabricated AFM probe in quasistatic (contact) and dynamic modes applied in this method. The dynamic modes are more applicable to a broad range of materials including soft polymer and biological specimen. Studies of electric properties are performed using both types of techniques. The quasistatic or contact mode is preferable for measurements of electromechanical response of piezo-materials known as PFM, which will be discussed later. The contact mode and consequently, PFM is limited in applications to soft materials due to excessive lateral forces common to the tip scanning in a permanent contact with the sample. This drawback is overcome in the oscillatory techniques, which are also employed for studies of electrostatic tip–sample force interactions in EFM and KFM. In addition to the mentioned AFM-based modes, which are discussed in this chapter, there are also scanning capacitance microscopy, scanning current sensing microscopy and scanning microwave microscopy – all operating in the contact mode. The oscillatory techniques in AFM are implemented as the amplitude modulation (AM) [3] and frequency modulation (FM) [4] modes, which complement each other in study of samples in various environments. In AM the probe is brought into oscillation near or at its resonance frequency and the damping of amplitude, which is measured as difference between the probe oscillation A0 (before it interacts with a sample) and the set-point amplitude Asp is used for sensing the surface topography. Initially, this mode was implemented at small damping caused primarily by the attractive part of van der Waals tip–sample force interactions – often described as a non-contact operation. The breakthrough in AFM applications was made with the introduction of AM imaging in the intermittent contact [9], which is characterized by a steep decline of amplitude-versus-separation (AvZ) curves. Although the AM operation at set-point amplitudes (Asp) along the steep part of the AvZ curve, which is also known as the tapping mode, is characterized by higher tip–sample forces compared to the non-contact mode, this operation allows a non-destructing and high-resolution imaging of soft materials in ambient conditions. Furthermore, a reduction of Asp leads to stronger tip–sample forces and local sample deformation and energy dissipation. These effects, which depend on mechanical properties of surface locations, allow compositional mapping of heterogeneous samples best revealed in phase images [21]. Theoretical description of the probe behavior in dynamic modes was based on first principles and mechanical setting of the AFM [22, 23]. The probe motion was described by Euler–Bernoulli’s type of equation, and a relationship between the empirical “effective” parameters of mass–spring models and the properties of cantilever and mechanical characteristics of the AFM was formulated. Krylov– Bogoliubov–Mitropolsky (KBM) averaging method [24] was used to derive asymptotic dynamics with amplitude and phase as the state variables.

9 Advancing Characterization of Materials with Atomic Force

sin q =

cos q = −

1 p A  ∫ [Fa − Fr ](ZC + A cos y ) sin ydy +  A0  2 N  2g   0 1+    w1  1

1

237

(9.1)

1 p 2g A   ∫ [Fa − Fr ](ZC + A cos y ) cos ydy +  (9.2) w1 A0  2 N   0

 2g 1+    w1 

where N = πA0w12 m and A0 is the amplitude of the non-interacting probe, w1is the initial (usually resonant) frequency of the probe, and mis the mass of the probe. In these equations there are four unknowns: A is the amplitude during tip–sample interaction, q is the phase of the probe, g = (w − w1 ) / e = Q1 (w − w1 ) is the frequency shift, and Z C is the vertical position of the probe. Other parameters are either known or those that can be measured separated or reasonably approximated, e.g., Fa and Fr are the functions defining tip–sample interaction forces during the probe approach to the sample surface and its removal, respectively. The latter depend on the particular surface locations and can be approximated using different models such as van der Waals, Hertz, JKR or DMT. Keeping two of four unknown variables constant (and re-solving other two by the equations for steady state) determines six potential dynamic AFM modes, of which four are widely used. They are AM and FM with force spectroscopy and imaging operations. In AM mode, frequency shift g = Q1 (w − w1 ) is kept constant (usually 0) and the force spectroscopy curves AvZ and qvZ are calculated for solving the above equations for each surface location. In the imaging operation one should take into account that the tip–sample force interactions ( Fa and Fr ) might change in every surface location (XY). Therefore, the images in AM mode, which reflect the changes of Z (topography or height image) and the changes of q (phase images) at XY positions, are obtained by solving the equations for amplitude A fixed at the set-point level (Asp). In FM mode, the phase q is kept constant (usually as p/2) and the force spectroscopy curves AvZ and wvZ can be calculated by solving the equations for each surface location. The images in this mode (Z versus XY – topography image and A versus XY – amplitude image) are calculated for constant frequency shift dw = w − w1 . This theoretical consideration shows that the AM and FM modes are close to each other and they bring similar information about the sample. Actually, the presented approach is further applied in the AFM simulator that is very helpful for practitioners in verifying their experimental results by computer modeling of the images of structures with probes of different stiffness and various apex size [25]. Furthermore, the addition of the electrostatic tip–sample interactions can increase the value of this simulator. Parallel to the development and practical applications of the intermittent-contact AM mode at ambient conditions, its companion – FM mode [8] has been broadly

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applied in AFM studies in UHV and, particularly, for imaging of atomically smooth crystalline surfaces. Monitoring of the frequency shift caused by tip–sample forces is the winning procedure in UHV, where extremely high quality factor of the probe limits AM capabilities. FM imaging is usually performed in the overall attractive force range by keeping the frequency shifts below the resonant frequency of the non-interacting probe. The fine control of tip–sample interactions in FM mode helps reaching atomic-scale imaging with the “lattice-plus-defect” observations of crystalline surfaces [12, 13]. Later the similar observations were reported in FM studies in air and under liquid [14, 26] as well as in ambient AM measurements [27], yet the question regarding true atomic-resolution in AFM is still open [28].

Detection of Electrostatic Responses in AFM Already in the first AFM applications, AM was applied for detection of electrostatic tip–sample interactions, which are enhanced when a conducting probe is used [19]. A scheme of detection of electrostatic forces includes a conducting probe which is biased with respect to a back electrode or substrate carrying a sample on top, Fig. 9.1. U = U DC + U AC sin (wt )

(9.3)

In a simplified form of the capacitor-like set-up the contribution of electrostatic force is proportional to j 2 and ∂C / ∂Z, where j is the potential difference, C is the capacitance, and Z is the probe–sample separation. Felec (Z ) =

1 ∂C 2 j 2 ∂Z

(9.4)

When DC (UDC) voltage and AC (UAC) voltage at frequency w, are applied to the probe then the electrostatic force can be expressed as 2 1 ∂C (9.5) (j − U DC − U AC sin(w t )) 2 ∂Z  where j is the surface potential or contact potential difference between the probe and the sample. This equation can be separated into three components definingthe DC and frequency responses:

Felec (Z ) =

Fig. 9.1 A set-up for measurements of electrostatic forces in atomic force microscopy

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FDC (Z ) = Fw (Z ) = −

239

1 ∂C 1 2 (j − UDC )2 + 2 UAC 2 ∂Z

(9.6)

∂C  (j − U DC )U AC sin(w t )   ∂Z 

(9.7)

1 ∂C 2 U cos (2w t ) 4 ∂Z AC

(9.8)

(

F2w (Z ) = −

)

In the attempt to record surface topography and electrostatic forces simultaneously and independently the AFM measurements were performed at two different frequencies [19]. The AM detection at the probe resonant frequency (wmech) was employed for topography imaging. At the same time, an AC voltage was applied to the probe at welec, which was chosen well below wmech, and the cantilever deflections at this frequency were used for detection of the electrostatic tip–sample interactions. Additionally, the force variations were observed at 2welec. The experiments showed that the probe responses at welec and 2welec were in-line with a model representing the probe–sample geometry as a capacitor. In general, any of the probe responses (amplitude, frequency or phase) at welec can be measured in EFM and used for mapping of the electrostatic force variations. A quantitative analysis of EFM data in terms of individual electric properties (such as surface potential, capacitor gradient, charge, dielectric constant) is the very complex problem [29]. It can be simplified when surface potential is of the prime interest. In KFM, the electrostatic tip–sample forces are nullified by applying an appropriate voltage to the probe [30, 31]. This voltage equals to a difference of surface potentials of the tip and a nearby sample location. Though surface potential is usually used in studies of semiconductors and metals, a charge deposited on dielectric sample or presence of dipoles also influence surface potential of organic compounds, polymers, etc. Therefore, KFM is applicable to a broad range of materials. The value of EFM and KFM has been proved in measurements of surface potential, dielectric constants, and local surface charges [31–34]. Such measurements were conducted in non-contact FM mode in UHV, and the first flexural mode was applied for tracking a sample topography, whereas the second flexural mode [35] or a frequency much lower than wmech was used for KFM feedback [36]. There is also another possibility to map electrically different sample locations using an intermediate of EFM and KFM. Instead of nullifying the electrostatic force by applying the voltage that equals to surface potential, the feedback can be used to maintain the electrostatic force at some constant level. In this case, according to the (9.4), the map of UDC reflects spatial changes of j and ∂C / ∂Z . When surface potential does not change over the examined area then the UDC map represents the ∂C / ∂Z variations, see the example below. In the alternative case the subtraction of the surface potential data (obtained in the KFM operation) from the −1 UDC map will be proportional to the (∂C / ∂Z ) variations. Studies of the electrostatic force effects should not be limited by KFM, and they can be diversified by using the 2welec response for the feedback mechanism [37–39]. In such a way one can get information regarding local dielectric constant and its

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high-frequency dispersion [37]. Simultaneous measurements of sample topography (wmech=70kHz), surface potential (welec) and dielectric or polarization response (2welec) were performed while the probe was scanning ~30nm above the sample surface [39]. Important questions regarding a way of detection of the electrostatic forces, a choice of mapping procedure, and a cross-talk between electrostatic response and sample topography were raised already in first AFM-based electric studies. There was also a strong intention to use the non-contact regime where the probe response to electrostatic forces is not disturbed by tip–sample mechanical interactions [19]. Since 1993 the intermittent contact regime in AM in air became very popular due to highresolution imaging of soft samples. This operation, however, imposes definite limitations for simultaneous and independent measurements of surface topography and electrostatic responses, especially, when the measurements are performed at the same frequency. An easy solution of this problem was provided by the two-pass technique or lift mode [20], which was originally introduced to tackle the similar problem in magnetic force microscopy. In this approach, for each scan line, the height profile is recorded during the first pass with probe oscillating at wmech. At the end of the first pass the probe is lifted typically 5–50nm above the surface. After that the AM feedback is switched off, and the probe is moved along the just-learned profile but at the constant lift above the surface. During this second pass, the DC bias voltage is applied to the probe oscillating at wmech. The changes of the probe frequency or phase, which are caused by electrostatic probe–sample forces, are monitored and mapped in EFM. In KFM these changes are nullified to obtain surface potential images. Though the lift operation has been routinely applied for local electric measurements [40], a more strict consideration of this operation reveals several problems. Imaging of surface topography in the first scan even with a non-biased and a non-metallic probe can be already “contaminated” due to electrostatic interactions between the sample locations and Si probes, which are electrically conducting to some level. The use of the same wmech for detection of mechanical and electrostatic tip–sample interactions demands a finding of an appropriate lift height to avoid the cross-talk between topography and electrostatic force variations, especially for corrugated surfaces. The moving the probe away from the surface in the second pass reduces sensitivity and resolution of the electrostatic detection. The use of the two-pass approach not only increases the experiment time but also limits studies of dynamic processes related to electrostatic forces.

KFM Applications Despite a number of open questions in AFM-based electric modes, their applications have been fast developed and they include studies of semiconductor structures and devices, quantum nanodots, organic layers and biological objects. Visualization of doped pn-structures covered by 2-nm Si oxide layer [41] is instructing for KFM imaging of semiconductor structures. In measurements performed at 54% humidity the surface potential of p-regions was 18mV higher than that of n-regions. Yet this difference depends strongly on hydrophobicity of the oxide surface because

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of potential shielding by the surface water layer. In addition to this effect, a possible sample contamination during imaging in air influences quantitative surface potential measurements, which are more reliable in UHV studies or in the properly controlled environment. KFM studies of the carbon nanotube/Au junction in different environments showed that the interfacial dipole layer changes direction when measurements were transferred from ambient air to vacuum or oxygen-free media [42]. Highresolution surface potential images of individual quantum dots (QD) with spatial dimensions in the 22–46nm range were obtained in UHV [43]. The surface potential of QD with smaller height was larger than that of higher ones. This finding was interpreted in terms of the quantum size effect by which the amount of charges is determined through the confinement energy levels in the QD. KFM studies of organic thin-film transistors in different configurations led to mapping of surface potential in the accumulation layer [44]. Particularly, it was shown that the potential changes take place at the film interfaces between source and drain elements [45]. A correlation between surface photovoltage and polymer blend morphology has been examined in polyfluorene-based photodiodes [46]. In the bilayer geometry, two polymers, which serve as holes-rich and electrons-rich reservoirs, adopt a complex morphology with domains of different charges. These domains were observed with KFM images under illumination and in dark that indicates on steric hindrances to the charge recombination and lower photodiode efficiency. In further KFM approach to photovoltaic materials, this method was applied to the 100-nm film of an organic blend consisting of soluble fullerene derivative (acceptor) and paraphenylene-vinylene (PPV)-based polymer [47]. The 100-nm thick films of this blend, which were prepared by spin-casting of its solutions on ITO substrate, have been examined in dark and illuminated states. The differences of surface morphology and surface potential maps were found in the films cast from toluene and chlorobenzene solutions. A smooth morphology was found in the chlorobenzene-cast film and its surface potential decreases from 4.55 ± 0.06V to 4.22 ± 0.06V. The toluene-cast film was rougher with well-defined hills and valleys, whose surface potentials were 4.51 ± 0.04V and 4.43 ± 0.04V, respectively. There was a minor population of the hills with lower potential of 4.39 ± 0.05V. In response to illumination, these hills did not change their potential, whereas the surface potentials of other hills and valleys increased to 4.66 ± 0.05V and 4.46 ± 0.05V. The obtained results together with TEM findings indicate that the hills are composed of PCBM-rich material and the surrounding matrix consists of both components were useful for understanding the photovoltaic behavior of the films prepared from different solvents. Conducting polymers, which are important materials for molecular electronics and solar cells, are attractive objects for KFM studies. A correlation between surface potential and sample morphology has been examined at various doping levels of poly-3-methylthiophene [48]. The surface potential images of similar semicrystalline polymer – poly(3-octylthiophene) revealed different contrast not only between amorphous and crystalline components but also between sectors of crystallites [49]. The potential variations in the segments are in the 40–60mV range and, most likely, they reflect different molecular dipole orientation in these crystalline domains.

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In KFM studies of organic films and layers on various substrates one can observe surface locations with different surface potential, which are interpreted in terms of layer architecture and molecular dipole arrangement within the layers. KFM studies of fluoroalkanes, alkylthiols, and organosilanes are most relevant to our results presented below. Study of a mixed monolayer of perfluorodecanoic acid (FC) and arachidic acid (HC) polyion, which formed a complex with (poly(4-methylvinyl pyridinium)) iodide, revealed that the phase-separation in this system led to the “islands” and “sea” morphology with different surface potential in these locations [50]. The KFM contrast was assigned to differences of surface dipoles of Cd+–Fb- and C–H bonds of the FC and HC components. The same morphology was observed in molecular layers of C19H39COOH and C9F19COOH mixture on Si substrate [51]. For this system, the difference between surface potentials of the “islands” and “sea” was 0.057 V that is substantially smaller than surface potential (~0.5V) of the layers made of the individual components. Therefore, the presence of a low surface potential capping layer was suggested. Surface potential of 0.007V, 0.17V, and 0.25V was measured respectively for single, double, and triple bilayers in Langmuir–Blodgett layers of arachidic acid (AA) on hydrophobic Si surface. In these measurements surface potential of the substrate (~-0.98V) versus Au-coated probe was taken into account. The same measurement for an oxidized Si gave the value of -0.76V. Surface potential of the single, double and triple bilayers of AA on the oxidized Si surface was -0.17V, -0.16V, and -0.11V, respectively. In the convention used by the authors, a negative surface potential is consistent with dipole orientation having negative charge closer to the surface. Dissimilar types of the AA layers’ organization on hydrophobic and oxidized Si substrates were suggested on the base of the KFM data [52]. A micro-printed pattern with alternative domains of alkylsilane [H3C(CH2)17Si(OCH3)3] – ODS and fluoroalkylsilane [F3C(CF2)7(CH2)2Si(OCH3)3] – FAS on Si has been prepared as a test structure for KFM [53]. The highest-contrast surface potential images, which differentiate fluorinated material (DV=171mV), were obtained when high AC voltage (15V) at 25Hz was applied for electrostatic force detection and when recording was made at slow scanning rates (down to 0.1Hz). The origin of the surface potential difference between alkylsilane and fluoroalkylsilane domains was discussed in terms of dipole moments of the QDC and FAS molecules estimated from the molecular orbital calculations. An estimate of the surface potentials of the layers based on the dipoles of individual molecules (OD – 1.18D, FAS – 1.47D) and their orientation gives a much stronger DV compared to the measured one. Therefore, several other factors such as intermolecular interactions, screening, and depolarization effects should be considered for the rational interplay between the experimental and theoretical results. A condensation of dipalmitoylphosphatidylcholine (DPPC) monolayer on water sub-phase in Langmuir–Blodgett trough from an expanded state to a mixture of liquid expanded (LE) and liquid condensed (LC) phases and to solid condensed (SC) state was monitored with KFM [54]. The single layers, which were transferred to Al-coated glass substrate, exhibit surface potential which is 270mV higher than that of bare Al. Surface potential of the layers increases 50–100mV on transition from LE to LC phase and 300mV on transition from LC to the SC state. The increase of molecular

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density was adequate for explanation of the changes of LE and LC potentials whereas the steep potential raise in the SC phase resulted from a compression-induced change of the effective dipole near the polar head group of lipid molecules. Several research groups have examined surface potential of molecular layers of alkanethiols and their various derivatives and its dependence on length of molecular chain. With respect to Au-coated probe surface single layers of hexadecanethiol HS-(CH2)15-CH3 and mercaptohexadecanecanoic acid HS-(CH2)15-COOH chemisorbed on gold substrate exhibit potential of ~0.181V and 0.392V, respectively [55]. Variations in the chain length of both compounds (n=11-22) led to the potential increase of 14.1±3.1mV per (-CH2-) unit. These findings were further confirmed for alkanethiol couples C8/C12, C10/C12, and C12/C14 [56]. Furthermore, a sample of octadecanethiol monolayer partially covering a gold substrate was used for comparison of force- and frequency-based detection of tip– sample electrostatic interactions. Pt-coated Si probes were used in this study [57]. The frequency-based spectroscopy (d2C/dz2 versus piezo displacement) and surface potential images were more sensitive and clearly revealed the difference of surface potentials of the C18 layer (-680±10 mV) and the gold substrate (-540±10 mV). The correlation between the molecular structure and distribution of electron density leading to charge and dipole formation on one hand and the experimentally determined surface potential, dielectric properties, etc. – on other hand is the principal question determining the KFM value. Only initial efforts are made in this direction. Self-assembled molecular layers of various aromatic thiols on gold were examined with KFM and compared with calculated dipole moments of these molecules. Even these simplified estimates (without considering molecular packing and Au–S bonding) help to understand the surface potential data. First of all, the layers formed by aromatic thiols with symmetric molecular structure exhibit very small potential (few tens of millivolts) compared to that (150–240 mV) of the layers of non-symmetric molecules. This is consistent with difference of dipoles for these molecules (0 Debye for symmetric molecules and 0.9–1.0 for non-symmetric). Second, the fit between the experimental and theoretical data implies that Au–S bonding does not cause strong charge rearrangement in the attached molecules. Finally, an addition of electron acceptor to the layer formed by symmetric aromatic thiols induced strong negative potential due to a formation of charge–transfer complex.

Challenges and Solutions Despite the obvious progress in electrostatic AFM modes, the issues regarding the practical implementation of EFM and KFM, spatial resolution of these modes, sensitivity and reliability of surface potential studies are under intense scrutiny. As already mentioned the separation of the probe responses to mechanical and electrostatic interactions is the main problem of AFM-based electric modes and the measurements and control of these forces can be performed in different ways. We will follow the notation introduced in [58] where experimental KFM approaches were

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named according to the modes applied for detection of the mechanical and electrostatic responses. For example, the AM–AM approach means that both force responses were recorded using amplitude modulation mode. Actually, this approach, in which the mechanical interactions are recorded at (or near) the resonant frequency of the probe and the electrostatic forces, at lower frequency, is the most common in the applications discussed above. These measurements are typically conducted in the non-contact mode, and they efficiently separate the mechanical and electrostatic effects. However, the detection of electrostatic force on a non-resonant frequency is not very sensitive. The use of the second Eigen mode, which is ~6.3 times higher in frequency than the main flexural mode, might improve this instance. The situation is different in the single-frequency measurements that are performed in combination with the lift mode. The detection of the electrostatic response in the second pass is performed by monitoring the frequency or phase changes at the probe resonant frequency and this mode can be described as AM–FM. Yet the remote position of the probe from a sample during the electrostatic measurements is the serious drawback of this approach limiting its spatial resolution [59]. In the non-contact AFM, which was originally applied mostly in UHV, the topography control is performed in FM mode and the FM–AM and FM–FM approaches can be used for KFM studies. In the FM–AM mode, the use of the second Eigen mode for electrostatic measurements is preferable. In FM–FM method the electrostatic force is stimulated at a non-resonant frequency but in the bandwidth of FM detection. Therefore the electrostatic response is detected as a modulation of the resonant frequency. It is worth noting that KFM measurements in UHV on clean crystalline surface demonstrated the contrast variations at the atomic-scale [60] that is way better than the spatial resolution of tens of nanometers common for ambient studies. A thorough consideration of the imaging procedures, optimization of probe, and data interpretation in AFM-based electrostatic measurements was given in [61]. The authors estimated the contributions of a cantilever, tip cone, and tip apex to the electrostatic probe–sample force and force gradient and came to conclusion that high spatial resolution can be achieved when the tip–apex contribution will be dominating. This condition can be realized by using probes with special geometry (the probes with long and sharp tips) or by employment of the force gradient detection. The other possibility – the imaging at tip–sample distances <2 nm was expected to be practically difficult. Actually, the most KFM applications were performed in the non-contact mode and AM and FM detection schemes in vacuum were critically analyzed in [62]. The higher sensitivity and more accurate FM detection was shown in the surface potential images of KCl sub-monolayer on Au (111) and the obtained quantitative values were in agreement with ultraviolet photoelectron spectroscopy. In contrast to AM-detection, the surface potential measured with FM did not vary at the probe sample separations in the 30-nm range. The lateral resolution of these measurements (~50 nm) was not higher than that expected for the non-contact KFM. There is also the hope that sharper probes might lead to better results [63]. In both classical oscillating-plate Kelvin probe and KFM there are always tradeoffs in terms of spatial resolution and voltage resolution. (In some sense, this is a general situation because high-resolution visualization of surface structures is

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related to magnitude of surface corrugations and the microscope capability to sense them.) Factors such as a smaller size of the probe plate or a sharper tip will increase spatial resolution at the cost of the voltage resolution. A smaller separation between the electrodes will be in favor of both resolutions. Equation (9.2), which was discussed in [55], shows how the minimum detectable potential difference DU (U/DU “potential resolution”) of KFM is related to different parameters:

∆U =

2 kBTkB Z π3Qf0 e 0U s R

(9.9)

where kB is the Boltzmann constant, T is the temperature, k is the cantilever spring constant, B is the bandwith Q is the quality factor of the cantilever resonance, f0 is the resonant frequency of the cantilever, Z is the tip–sample separation, e0 is the dielectric constant (vacuum permittivity), Us is the amplitude of the applied modulation voltage, R is the radius of the tip. The estimate showed that in air the minimal detectable surface potential is ~1mV [55]. Please note that the increase of the tip apex size leads to higher sensitivity. From this model we can also see that increasing the tip–sample separation (as it happens in the lift mode) will decrease the “potential resolution.” We can also lower the bandwidth to increase the “potential resolution.” It is clear that there are tradeoffs in the choice of each operating parameter. It should be noted, however, that this equation is for AM detection at the resonant frequency and is not applicable for the force gradient KFM method. Recently, the relation between KFM sensitivity and spatial resolution was discussed in the analysis of the FM–FM studies of KBr islands on InSb surface, which were performed in UHV [64]. The difference of surface potential of the islands and substrate (~210 mV) was saturated as the size of KBr islands exceeded 100 nm. This difference on smaller islands decreases due to the increased sensing of the substrate’s potential, and it dropped by half when the island size became comparable with the tip dimensions (~20 nm). The interplay between these experimental data and simulations brought the authors to the limit of KFM lateral resolution of ~3 nm. This result is in line with the expectation of high-resolution of FM–FM mode outlined in [61]. In case of periodical structures such as crystalline surfaces the KFM images revealed atomicscale variations [64]. Yet as in case of atomic-scale AFM observations there is no clear understanding of the origin and local character of these changes. The extension of the computer modeling, which was applied to the simulation of the atomic-scale AFM images in the AM mode, might be useful in this respect.

Piezoresponse Force Microscopy: Background and Applications In the family of AFM-based techniques, PFM is applied for nanoscale characterization of ferroelectric and piezoelectric materials [65–67]. This method is based on the converse piezoelectric effect, i.e., piezoelectric materials change their dimensions when subjected to an external electric field. The schematic illustrating this technique

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Fig. 9.2 A sketch illustrating the operation of piezoresponse force microscopy

is presented in Fig. 9.2. It shows a general set-up and a detection of electromechanical response as well as an enlarged view on the tip–sample junction. In PFM, a conductive probe is scanned across the surface of a piezoelectric sample in the contact mode. The conductive cantilever serves as the top electrode to provide the polarization field to the sample. The sample is often connected to a bottom electrode and the electric bias voltage can be applied to the sample as well. The piezoelectric activity is stimulated by an AC voltage applied between the tip and the sample and the sample undergoes the periodical expansion or contraction. The changes of the sample dimensions cause the cantilever deflection or bending that are detected by analyzing the AC component signals of different segments of the quadrant photodetector. The strain Sj developed in a piezoelectric material by the applied electric field Ei is described by the following matrix equation [68]:

S j = dij Ei

(9.10)

where dij is the piezoelectric coefficient with the unit of m/V. The indices i=1, 2, 3 and j=1, 2, 3…6 denotes the direction of the electric field and the tensor component of the strain following the Voigt notation [69]. In general, indices 1–3 indicate components along the x, y, and z axis of an orthogonal coordinate system, and indices 4–6 indicate shear components of the strain tensor. The axes of the coordinate system are often conveniently aligned with the crystallographic axes of a crystal. In the case of ceramics and thin films, the z axis is usually aligned with the

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direction of the polarization, or the direction normal to the plane of the film. The piezoelectric coefficient, which is measured along the direction of the applied field, is also called the longitudinal piezoelectric constant. The other coefficient, which is determined in the direction perpendicular to the field, is called the transverse piezoelectric constant. Generally, the longitudinal piezoelectric constant, d33, can be determined by measuring the displacement (Dz) of the sample along the applied field (E3):

∆ z = d33V

(9.11)

assuming S3=Dz/z0 and E3=V/z0, where V is the applied voltage and z0 is the thickness of the sample. The measured displacement will be positive, i.e., expansion, if the polarization direction is parallel to the applied field, and it will be negative if the polarization direction is antiparallel to the applied field. In response to the AC voltage V = Vaccos(wt) applied to the tip, the cantilever displacements can be expressed as

∆z = d33 Vac cos(w t + f )

(9.12)

When f=0, polarization points down (P –), and when f=p, polarization points up (P+). Therefore, when a small AC modulation being added to the applied electric field, the piezoresponse will oscillate in-phase with the AC modulation if the polarization is parallel to the field, and out-phase if antiparallel. Consequently, a lock-in amplifier can be used to analyze the piezoresponse signal and to determine both the magnitude of the displacement and the polarization direction of the sample. The detection of longitudinal piezoresponse is also called vertical piezoresponse force microscopy (VPFM). Because of the presence of non-zero transverse piezoelectric constants, e.g., d15, and the possible misalignment of polarization to the applied field due to a random crystal orientation, an electric field normal to the surface can also cause in-plane shear deformation. This in-plane shear deformation of the sample will give rise to a torsion-like motion to the AFM cantilever, thus results in a change in the lateral signal of a photodetector. In a way similar to VPFM, a lock-in amplifier can be used to detect the in-plane piezoresponse signal DL induced by an AC voltage:

∆L = deff Vac cos(ωt + f )

(9.13)

where deff is the effective in-plane piezoresponse constant, and f is the phase angle related to polarization direction. For antiparallel domains the phase angle will be 180° in difference. The measurement of in-plane piezoresponse is often called lateral piezoresponse force microscopy (LPFM). In addition to imaging, studies of local electromechanical properties can be performed at individual surface locations. By sweeping the voltage between limits of different polarities one can learn about the sample piezo-hysteresis, detect changes of

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domain polarization, etc. One can also apply voltage to the tip and polarize the specific sample location or locations. The effect of polarization, its magnitude, and possible dissipation can be monitored in the PFM mode. This function, which was demonstrated even in first applications [66], can be employed for lithographic purposes. Sensitivity, calibration, spatial resolution, and frequency characteristics are important features of PFM measurements, which are addressed in various application of this technique. PFM has found major applications in the study of ferroelectric materials, particularly for high resolution imaging, domain switching, local hysteresis measurements, and switching of ferroelectric capacitors [70]. For example, ferroelectric materials are used in many field effect transistor applications due to their high permittivity. With the increasing demand of miniaturization of electronic devices, it becomes important to study the size effect of ferroelectric materials, i.e., to study the critical size range where significant deviation from bulk properties occurs. The high resolution imaging capability makes PFM suitable for direct measurement of the change of piezoelectric properties with the domain size. The PFM study of the piezoelectric properties of PbTiO3 nanoparticles revealed that the signal of particles >20 nm can be reliably detected [71]. Ferroelectric materials have potential application in ultrahigh-density rewritable data storage systems. A data storage system based on a thin film of ferroelectric single-crystal lithium tantalite has been demonstrated at a data density above 10.1Tbit/in.2 with 500ps domain switching speed [72]. The high spatial resolution of PFM provides a unique opportunity to study the fundamental process of domain switching, including the thermodynamics and kinetics of domain nucleation, growth, and relaxation [70]. PFM has also been used to study the switching dynamics of ferroelectric capacitors. The fast domain switching kinetics using a step-by-step switching approach was investigated in [73]. PFM visualization of the domain switching process revealed that for larger capacitors, two distinct stages of polarization reversal can be observed: a fast switching stage (<1ms) dominated by domain nucleation and a slower stage (>1ms) via lateral domain wall motion; while for small capacitors the process is dominated by domain wall motion. Consequently, large capacitors switch faster in low-field while small capacitors switch faster in high field. By analyzing local hysteresis loop measured by PFM, important information about the piezoelectric properties of small ferroelectric domains can be obtained, including coercive voltages, nucleation voltages, forward and reverse saturation and permanent responses, as well as the effective work of switching defined by the area included in the hysteresis loop. Simultaneously recorded VPFM and LPFM nanoscale hysteresis loops were used to differentiate 90° domain switching from 180° domain switching [74]. Furthermore, the dependence of PFM on crystallographic orientations has been examined in [75], and a novel approach for nanoscale imaging and characterization of the orientation dependence of electromechanical properties – vector piezoresponse force microscopy (Vector PFM) was introduced. The vertical and lateral electromechanical responses are analyzed to give a complete threedimensional (3D) reconstruction of the electromechanical response vector as functions of position. The approach can be applied to crystallographic orientation imaging in piezoelectric materials with a spatial resolution better than 10 nm.

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Similar to the force volume measurement in AFM, the spatial information about the local switching behavior of ferroelectric materials, particularly nanoscale domains and particles, can be obtained by performing switching spectroscopy at predefined locations that form the M×M point mesh on the scanned surface area [76, 77]. This approach, switching spectroscopy PFM (SS-PFM), has been applied for 2D mapping of the positive and negative coercive bias, imprint voltage, saturation response, and the work of switching. The spatial variations of these parameters reflect the change of local switching behavior of the surface location and nanostructures. Despite the wide application of PFM in the research of piezoelectric and ferroelectric materials, the imaging mechanism, particular the quantitative interpretation of PFM is yet to be fully understood. In the detailed analysis [78, 79] the additive contributions of the long-range electrostatic and the electromechanical interactions to the total PFM response have been considered as follows:

A = Ael + Apiezo + Anl ,

(9.14)

where A is the total PFM amplitude, and Ael, Apiezo and Anl are the amplitudes of electrostatic, electroelastic, cantilever-surface capacitive contributions, respectively. The presence of Anl results in a constant background to the measured piezoresponse, and it can be minimized by using tall and high aspect-ratio tips. High qualitypiezoresponse data requires the maximization of the electroelastic contribution and the minimization of electrostatic contribution. The electroelastic contribution depends strongly on the contact interface between the tip and the surface, the shape, size, and material of the tip itself. The presence of a “dielectric gap” between the tip and the surface below, particularly in the case of “week indentation” with a soft cantilever, can cause screening of the electric field introduced by the tip, and consequently reducing the piezoresponse amplitude. Therefore, the use of metalcoated, stiff cantilever with large force (strong indentation) is most desirable for PFM imaging. However, sample modification and other effects such as stressinduced suppression of piezoelectricity [80] may occur under high mechanical force due to the “contact” nature of PFM. Another concern for quantitative PFM measurement is the calibration of the PFM system. The general procedure is to calibrate the instrument with each cantilever mounted against a piezoelectric sample of known piezoelectric coefficient (PC) [70]. For example, the commercially available a-quartz has a precisely known PC of d11=2.3±0.05pm/V, and is often used as a standard for PFM calibration. The system coefficient is thus calibrated from the PFM signal (Pac), measured at a specific frequency and voltage, against the displacement calculated from the known PC, x=d11Vac /Pac. However, this process could be complicated by the presence of a frequency dependent background which is unique for the particular instrumental setup used for the measurement. Consequently, extra efforts that might be needed for accurate calibration and measurement of quantitative piezoelectric constants were proposed in [81, 82]. Furthermore, one should realize that PFM deals with the

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piezoresponse of a local, confined volume, which is effected by the tip–sample field. This response might be different from that of the same volume when it is subjected to a large field when both macroscopic surfaces of the sample serve as the electrodes.

Implementation of EFM, KFM and PFM AFM-based electric measurements summarized in this Chapter were performed with the commercial scanning probe microscope 5500 (Agilent Technologies) equipped with the MACIII accessory, which is designed for multifrequency measurements. The MACIII has three dual phase lock-in amplifiers (LIA) converting the AC inputs to amplitude and phase. These digitally controlled analog LIA have a broad bandwidth (up to 6 MHz) that covers the operation bandwidth of the photodetector employed in the microscope. The auxiliary inputs and drive outputs are accessible through the MACIII signal access box. The software, which is flexible in routing signals back to the controller, supports two servo systems related to these LIA. One LIA is used for AM tracking of sample topography with the probe peakto-peak amplitude or its X-, Y-vector components used for feedback. The other servo can be applied for electric or mechanical measurements. The third LIA can be used for tuning the operation parameters or for recording of various signals (lateral response, torsional signal, harmonics, etc.) during measurements. Voltages up to 20 V in DC or in different pulse regimes can be applied to the probe–sample junction as an external stimulus for lithography or other applications.

Electric Force Microscopy and Kelvin Force Microscopy: AM–AM approach The set-up for EFM and KFM measurements in the AM–AM mode is shown in Fig. 9.3. A sample on a microscope stage is grounded (or biased) and an electric signal is applied to a conducting probe. The probe oscillation, which is excited at or near its resonance wmech, is influenced by the tip–sample forces and it is monitored by a positional four-quadrant photodetector. The photodetector output carrying the AC probe amplitude is sensed in parallel by LIA-1 and LIA-2. The first LIA is tuned to w1=wmech, and it delivers the error amplitude signal (Ai-Asp, where Ai is measured amplitude in a new surface location) to the servo that controls the vertical tip–sample separation. This servo loop is used for topography imaging. The second LIA is tuned to w2=welec and from the input signal the X-component of amplitude at welec is selected for KFM servo, i.e., for finding the tip bias voltage that nullifies the incoming signal related to the electrostatic force. In preparation for KFM imaging the phase of the second LIA is tuned to maximize the X-component signal. The operation of the electric servo loop can be monitored and controlled with the

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Fig. 9.3 A scheme illustrating the general set-up and signal wiring of the AM–AM approach of KFM

third LIA, which can determine the magnitude of the electrostatically induced oscillation by sweeping the frequency around welec. Typical sweep curves in the “on” and “off” states of the electric servo loop are shown in Fig. 9.4a, b, where the scale is ~10× smaller in b. The detection of the amplitude signal at welec=10 kHz in the “off” state helps to choose the level of AC voltage applied to the probe in the second loop. The MACIII accessory can provide a voltage up to ±10 V, however the voltage should be chosen as small as possible to minimize its influence on the sample’s electronic states. We are mostly operating with the voltages in the 1–5 V range. In the “on” state one should minimize the remainder of the amplitude signal at welec=10 kHz (the error signal in the feedback operation documented in Fig. 9.4b) by optimizing the servo gain parameters. The experimental protocol for KFM also includes a compensation of the occasional contribution to surface potential from the probe and/or sample surroundings; which causes a dependence of surface potential on the probe–sample separation. This dependence is eliminated by finding the proper setpoint voltage for the KFM servo loop. The optimization procedures for KFM measurements are described in more detail elsewhere [83]. Here it is appropriate to mention about the relationship between the EFM and KFM operations. The electrostatic probe response is generally measured in EFM and its spatial variations are presented in the related image. In the capacitor-like set-up the electrostatic force depends on the tip–sample surface potential difference and capacitance vertical gradient [(9.7) and (9.8)] and its sensitive detection relies on dynamic (resonant and non-resonant) approaches. In the AM–AM approach one can use the amplitude response at 10kHz in the “off” servo state and use it in the EFM image.

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Fig. 9.4 (a, b) The amplitude-versus-frequency curves showing the electrostatic-force induced oscillation in the “off” and “on” state of the KFM servo in the AM–AM mode

The latter might bring a better contrast in studies of samples with electric heterogeneities but add more complexity in quantitative assignment of these changes to particular material property. This contrast will disappear on switching the servo to the “on” state and KFM contrast will outline only surface potential ( j ) variations leaving aside those of ∂C / ∂Z . Therefore, the relationship between EFM and KFM mode is not simple, because the contract of the first depends on more surface properties than the contrast of the second. In the AM–AM experiment there is a choice of frequencies for wmech and welec. A mechanical drive of the probe is typically done at wmech chosen near the first flexural resonance of the cantilever, whereas the electric servo loop is set either at much lower frequency (10kHz as was suggested above) or at the second or even third flexural mode. The following arguments are usually considered in the choice of welec. The electrostatic probe response is higher at the various resonant frequencies, yet this also increases the possibility of cross-talk between different force interactions. The cross-talk is less probable when welec<<wmech, but the probe response at non-resonant frequencies is also smaller. If the sensitivity is the real problem, then one can try to use the second flexural mode for wmech and first flexural mode for welec. The particular choice of the frequencies is also related to the type of AFM probes used in KFM studies. Several comments about the probes applied for AFM-based electric modes are given below.

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Electric Force Microscopy and Kelvin Force Microscopy: AM–FM Approach Surprisingly, in the description of different approaches in electrostatic force measurements in AFM [58] there is no mentioning of the AM–FM combination despite the above considerations suggesting the high value of FM detection of electrostatic forces [61, 62, 64]. We have implemented this capability in the Agilent 5500 microscope and critically evaluated this mode in studies of a variety of samples in the intermittent contact regime. The block scheme of the AM–FM mode is presented in Fig. 9.5. The principal difference of this set-up with the one used for the AM–AM approach is that the input of LIA-2 is connected to LIA-1 for measuring the phase data at w1=wmech. The phase changes are directly related to changes in the force gradient and this yields AM–FM type studies. Practically, the electric potential was applied to the probe at w2=welec and mechanical phase response at the mixed frequencies (wmech±welec), which needs to be within the bandwidth of the first LIA, is detected by second LIA and further used for feedback purposes in KFM. The amplitude versus frequency spectra in the “off” and “on” states of the electric servo loop are presented in Fig. 9.6a, b. The electrostatic forces between the cantilever and sample cause the cantilever to deflect at welec, and at twice that frequency. The voltage modulation also causes a modulation of the force gradient which is greatest between the tip and the sample. These changing force gradients cause the resonant frequency of the cantilever to shift thus giving rise to side bands on the mechanical resonance of the cantilever. After demodulation by the first LIA, the output shows modulation at welec and at twice that frequency.

Fig. 9.5 A scheme illustrating a general set-up and electric wiring of AM–FM approach in KFM

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Fig. 9.6 (a, b) The amplitude-versus-frequency curves showing the electrostatic-force induced oscillation in the “off” and “on” state of the KFM servo in AM–FM mode

There is another implementation of AM–FM combination in which the Y-vector component of amplitude can be used instead of the phase signal of the first LIA. When the phase of the first LIA is adjusted so the amplitude of the cantilever is aligned with the X-component then the tip–sample force interactions can be observed as the Y-component variations. The variations are similar to the phase signal, which is calculated using the X- and Y-component data. The use of the Y-component can be justified by improved signal-to-noise ratio compared with the phase signal. Practically, with the first LIA set to wmech one needs to maximize the X-component and make the Y-component close to zero. Then, the Y-component changes caused by the electrostatic force will be substantial and they will be directed to the second LIA and the KFM servo loop. The AM–AM and AM–FM modes of KFM are similar in that the DC bias has a servo to minimize the welec component from the input of the second LIA. The main differences between these modes are related to the choice of welec, which in case of AM–FM is limited to low frequency (say 5kHz), lower feedback gains of the electric servo loop and to lower AC voltages. The last difference is very positive remembering the possible voltage influence on sample electric properties.

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Experiments in PFM We performed PFM measurements with the same microscope as the EFM and KFM. The studies were carried out in the contact mode and one of three LIAs was used for detection of the electromechanical response. The AC-voltage, which stimulates the sample piezoresponse, is applied either to the probe or to the sample. This operation proceeds at a particular frequency, and the amplitude of X- or Y-vector components of the probe oscillation is monitored. The frequency capabilities of the microscope allow PFM measurements up to 6 MHz, and the choice of frequency can be assisted by checking the probe response spectrum. The amplitude-versus-frequency spectrum (0–100 kHz range), which was recorded for the NSC14Ti–Pt probe placed on LiNbO3 sample, is shown in Fig. 9.7. This spectrum is characterized by large (at few kilohertz) and small (at ~55 kHz) resonances and a relatively flat dependence in other regions of this bandwidth. It is worth noting that the spectrum depends also on the level of the tip–sample mechanical interaction. We usually performed the PFM measurements at frequencies between 5 and 20 kHz. The other choices of the operation frequency are also possible. They might be aimed towards either more sensitive detection of the piezo-mechanical response or towards studies of the sample behavior in the specific frequency range. In first case, one of the frequency resonances might be helpful yet such measurements will be more difficult to quantify. In the second case, the interest to the material fast response can be of importance. At the beginning of the PFM experiment, the phase of the oscillating probe was set to maximize one of the vector components. In the applied set-up the Y-component was maximized so the expansion of the sample will be aligned with this signal. It is also worth noting that one of the LIAs can be used for recording the normal

Fig. 9.7 The amplitude-versus-frequency response of NSC14Ti–Pt probe placed on LiNbO3 surface

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signal of the photodetector and the other one – for monitoring the lateral signal. This enables simultaneous studies of VPFM and LPFM. Such studies will be illustrated by the images of the ferroelectric SrBiTaO and PZT films presented in Figs. 9.8–9.10. These images were obtained with NSC14Ti–Pt probes (MikroMasch) and the piezoresponse was measured at 10kHz with voltage amplitudes in the 5–8V range. In our microscope the voltages up to 20V can be applied to the sample. (In case of a sample with weak piezoresponse an external supply can be employed for the high-voltage experiments.) The topography image of SrBiTaO sample (Fig. 9.8a) reveals the multi-grain morphology, and the differently oriented nanostructures are seen in the individual grains. The PFM responses recorded in the normal and lateral directions indicate different polarization of the domains, Fig 9.8b, c. The contrast of the domains marked with white circles and triangles are reversed in these images. The bright contrast of the domains, which are marked with white triangles, in Fig. 9.8b means their vertical expansion in response to the applied field. In contrast, the domains, which are marked with white circles, are brighter in Fig. 9.8c, and this suggests the lateral direction of their field-induced expansion. The polarization of the domains marked with white squares does not have a preferential orientation. A comprehensive PFM analysis should include the consideration of different contributions to the total piezoresponse amplitude. In this respect, the KFM imaging can provide the surface potential maps caused by electrostatic forces. For the example, the images in Fig. 9.9a–c demonstrate the topography and surface potential of SrBiTaO film. It will be quite useful to get the high-quality PFM and KFM data of the same surface location, however, this might be cumbersome because such the optimal measurements of these modes require probes of different types.

Fig. 9.8 The topography (a) and piezoresponse (b, c) images of SrBiTaO film. The images in (b) and (c) were obtained with the normal and lateral signals. In the topography image the contrast covers surface corrugations in the 0–35 nm range. Scans are 5 mm. The white squares, circles and triangles in (b) and (c) indicate the same crystalline domains. Sample – courtesy of Prof. A. Gruverman

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Fig. 9.9 (a, b) Topography and surface potential images of SrBiTaO film obtained in the KFM AM–AM mode. Scan size 6 mm. The surface potential was detected at the second Eigen mode. (c) The cross-section profile taken along the direction indicated with a white line in (b)

Fig. 9.10 Topography (a) and piezoresponse (b, c) images of PZT film. In the topography image the contrast covers surface corrugations in the 0–120 nm range. Scans are 5 mm. The amplitude contrast in the amplitude images is in relative units

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Fig. 9.11 The PFM hysteresis curves (amplitude-versus-voltage) obtained for normal and lateral components on one of the PZT domains shown in Fig. 9.10a. These curves were obtained in the single run and, therefore, exhibit a noticeable level of noise. Multiple runs and the averaging these responses provide higher quality curves

Another example of PFM studies is presented in Fig. 9.10a–c, where the p iezoresponse in normal and lateral directions is observed for large number of crystalline domains of PZT film. The domains with different piezoresponse can be identified in the images in Fig. 9.10b, c. There is also the obvious difference between topography and amplitude patterns that is related to the fact that multiple crystalline steps of the individual grains are more pronounced in the amplitude images. In addition to mapping of sample piezoresponse, one can also perform local studies of the voltage-dependent amplitude changes. For example, the amplitude changes in the ±10V voltage range (Fig. 9.11) reveal the hysteresis loop of the PZT film.

Probes for AFM-Based Electric Studies A use of conducting probe is essential for detection of the electrostatic forces and in the first applications a piece of tungsten wire, which was bend and sharpened at the end, has been used for the measurements [19]. Currently, Pt-coated Si probes are mostly applied for the EFM, KFM, PFM, and other AFM-based electric modes. In addition to the general probe parameters such as the cantilever stiffness, reflecting

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coating, tip geometry, and the tip apex size, which are essential for AFM applications, quality of conducting coating is quite important. The geometrical dimensions of most commercial Si probes are quite similar. They have a rectangular shape and depending on length (100–400 mm), width (3–40 mm) and, particularly, thickness (1–6 mm) their stiffness varies from 0.01 to 50 N/m. Softer probes (0.01–5 N/m) are usually employed for the contact mode and stiffer ones (0.2–50 N/m) are used for oscillatory modes. The tips being fixed at the free end of the cantilevers are of 5–8 mm in length and in best cases have a shape of a triangular pyramid with opening angle ~30°. The apex is typically <20 nm in diameter and many of sharper tips are around 5 nm in diameter. In extreme cases, the tips’ apex can approach the 1-nm size yet the use of such probes is extremely difficult because they can be easily damaged already in the engagement procedure. For electric AFM measurements, the probes are coated with Pt and the intermediate layer of Ti, which is used for improving the Pt adhesion to the Si probe. Because of the coating the tip dimensions of these probes are increased, and this circumstance should be considered in the applications. The probe characterization with SEM and TEM is most reliable for evaluation of geometry of the probe, the tip, and the dimensions of the tip apex. We have applied these techniques for examination of commercial Pt-coated Si probes with cantilever stiffness in the 2–5 N/m range, which are made by different manufacturers (Olympus, NanoSensors and MikroMasch). TEM and SEM micrographs in Fig. 9.12a–c demonstrate the tips whose apex has a diameter in the 20–35nm range. Our experience shows that the Pt-coated probes made by Olympus are consistently sharp (<20 nm in diameter). The better sharpness of these probes might be a consequence of fine dimensions of the bare Si probes and finesse of Pt coating. These features are seen in the SEM micrograph of the Olympus probe (Fig. 9.12b) whose apex was broken during imaging. In most experiments described in this Chapter we used Olympus probes, which have a spring constant of 3–5 N/m and the resonance of the first flexural mode in the 60–80 kHz range. Lower frequency of 10kHz and the second flexural mode of the probes (400–500 kHz) were most often chosen for welec. In addition to the conducting probes, the KFM and EFM measurements can be performed with regular Si probes, because many of them [84] have a conductivity

Fig. 9.12 Microphotographs of conducting probes made by Olympus (a, b), MikroMasch (c). The micrograph in (b) was obtained with SEM and others with TEM

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level sufficient for these operations. If the use of sharp conducting probes is beneficial for high-resolution EFM and KFM measurements, the same probes might not be optimal for PFM measurements, which are conducted in the contact mode. In this case, the conducting probes with larger tip apex are less subjected to wear and such probes (e.g., NSC14Ti–Pt) are most suitable for reproducible PFM studies. These probes have stiffness around 3–5 N/m and might be too destructing for soft materials. This problem might be solved by using soft probes, e.g., CSC17Ti–Pt (MikroMasch) with stiffness of ~0.1N/m). Besides the batch-processed Si probes with the rectangular shape and symmetric placement of the tip, there is also a room for more specialized probes that might complement the regular probes in studies of electric properties. Two types of the specialized probes are worth mentioning here, Fig. 9.13a–c. In some instances it might be desirable to use the probes with much longer tips [61]. The dimensions of microfabricated tips of Si are confined by its crystal lattice that limits manufacturing of the probes with longer tips because of the substantial increase of their mass. In such case the novel probes made of sapphire cantilevers with metal-bonded single diamond tips might be a possible alternative. Such tips have very small opening angles (~9°) and can be >50 mm, Fig. 9.13b, c. The probes with sharp and long tips allow high-resolution profiling of highly corrugated surfaces. A use of conducting diamond, which can be naturally or specially doped for this purpose, offers unique capabilities for such probes [84]. Despite the fact that resistivity of the naturally doped diamond probes was high (R=5kW versus R~3–5 Ohm for conducting Olympus probes) they have been successfully applied for the lift-based EFM studies of polymer samples loaded with carbon black. In vicinity of the sample surface, the probes with long tips subjected to less damping and keep their Q-factor very high. Therefore, compared to the conducting Si probes they are more sensitive (~3× if judged by phase changes) to the electrostatic forces in the lift mode [84]. The probes with asymmetrically positioned tips [17] (Fig. 9.13a) provide strong signal at torsional resonance (at frequencies close to 1MHz) that can be used for detection of electrostatic forces. The amplitude-versus-frequency spectra of Olympus conducting probe and the asymmetric probe are shown in Fig. 9.14a, b. The spectra, which were detected for vertical and lateral signals, show the pronounced response

Fig. 9.13 Micrographs of AFM probes made by MikroMasch (a) and MicroStar Technologies (b, c). The micrograph in (a) was made with SEM and others with TEM

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Fig. 9.14 The vibrational spectra of the Olympus (a) and the asymmetric probe (b) recorded for vertical and lateral signals

of first and second Eigen modes of the Olympus probe, and strong torsional resonance in the lateral signal for the asymmetric probe. Summarizing the description of conducting probes applied for AFM-based electric measurements we would like to admit that they can be further improved. The use of fully conducting probes with long tips will be most advantageous for EFM and KFM measurements with high sensitivity. In respect of the tip apex dimensions, one needs to find a compromise between spatial resolution, which is higher for small apexes, and sensitivity, which is better for tips with larger apex.

Practical Studies with AFM-Based Electric Modes This part deals with practical electrostatic and electromechanical measurements in the AFM. At the beginning, we will discuss the general issues regarding the experiments, sample preparation, and the results of evaluation of EFM and KFM approaches. This will be accompanied by the description of studies of the AFM substrates and polymers, semifluorinated alkanes, metals and semiconductors. Finally, we present the results obtained on molecular self-assemblies and will discuss what useful information can be extracted from these data.

General Comments Most of the experimental results described in this chapter were obtained with the 5500 Agilent scanning probe microscope and a large scanner (scans up to 100mm). The performance of this microscope with the improved electronic controller was checked in the atomic-scale imaging of the bc-plane of polydiacetylene crystal,

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Fig. 9.15 (a)–(c) Topography images of the bc-crystallographic plane of PDA crystal, whose molecular arrangement is shown in (d). Scan size is 16 × 12.5 nm in (a) and 18 × 14 nm in (b) and (c)

which was performed in AM mode. The high-resolution images of these samples (Fig. 9.15a–c) were obtained regularly and their periodical patterns are consistent with the overall crystallographic structure of t his surface (Fig. 9.15d). However, the images exhibit definite variations. The computer simulation of these images [25, 28] revealed that tip dimensions and level of the tip–sample force interactions are the main reasons of these changes. The 5500 microscope is equipped with the environmental chamber that allows imaging of samples in different gas and vapor media. For studies at different humidity the chamber was either purged with dry argon or with the same gas blowing through the water. The humidity is controlled with the sensor installed inside this chamber (Fig. 9.16). Some of the KFM data, which were obtained at different humidity, will be discussed below. The purging can be also done with vapor of organic solvents (alcohols, toluene, benzene, xylene, etc.) that allows diversification of imaging capabilities and monitoring of sample changes caused by different agents. In AFM studies it is important to realize that the electrostatic tip–sample forces can modify the probe behavior that should be taken into account in the experiment. The simple way to detect the electrostatic forces is to perform measurements with a conducting probe and apply DC voltage to it. The effects of the applied voltage are seen from the images of semifluorinated alkane adsorbate on Si substrate (Fig. 9.17a, b) and thermoplastic vulcanizate (TPV) (Fig. 9.18a–c).

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Fig. 9.16 A photo of the environmental chamber of Agilent 5500 scanning probe microscope

Fig. 9.17 Topography images of F14H20 adsorbate on Si obtained with the probe voltage of 0 V (a) and varying from -3 to +3 V (b). Scan size is 1 mm. The contrast covers the topography variations in the 0–7.5 nm range in (a) and in the 0–25 nm range in (b)

Drastic variations of the topography contrast, which are caused by the voltage applied to the probe, are observed in Fig. 9.17b. This illustrates the fact that the researchers should be very careful in the assignment of the topography image contrast to real surface corrugations in cases when the samples have sources of electrostatic forces. The effect is less pronounced when the regular Si probes with lower conductivity are applied. This issue has been already discussed in [85] and the

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Fig. 9.18 (a, b) Topography images of TPV sample, which were obtained with the probe voltage of 0 and -3 V. (c) The image obtained by subtraction of those in (a) and (b). Scan size 5 mm. The contrast variations cover the topography corrugations in the 0–90 nm range (a, b) and in the 0–50 nm range in (c)

compensation of electrostatic forces in KFM is the direct way of getting most reliable topographic contrast. In the other example, the electrostatic tip–sample force interactions enhanced the contrast of a large number of surface locations of TPV, Fig. 9.18a, b. These sites are most pronounced in the subtraction image in Fig. 9.18c. The examined material consists of a matrix of isotactic polypropylene and rubber (EPDM) filled with carbon black (CB) particles, which form a percolation network to provide electric conductivity of the sample. The locations with the CB particles are enhanced in the topography image when the negative probe voltage is applied. Therefore, imaging with the biased probe is a most straight way of compositional analysis of similar samples. However, thermal drift of the AFM microscope complicates the complete separation of the topography and electrostatic effects. A simple solution to this problem was found with EFM in the lift mode, which is usually applied for studies of similar samples [40, 86]. The experiment in scanning probe microscopy has a number of common and specific features for different techniques. As regarding sample preparation for EFM and KFM measurements, it has the same requirements as for other AFM modes. Particularly, an examined surface should be relatively flat to get best data, which are not complicated by tip shape and its interactions on steep corrugations. When studies of bulk material are of prime interest then polishing and ultramicrotomy (or cryoultramicrotomy) can be applied for preparation of flat surfaces. As we mentioned the electric field in the tip–sample junction is usually made by applying the voltage to the conducting probe and grounding the sample. For this purpose the conducting glue or paste are usually applied to connect the sample with a metallic puck and a grounded sample stage or plate. For AFM studies of thin layers and individual macromolecules the material is deposited on flat substrates, such as Si wafer, mica, graphite, and Au (111). Therefore, in EFM and KFM applications to different thin and ultrathin layers and deposits on such substrates one would like to know the possible background signals of the substrates. The conducting substrate is usually grounded yet it might be not necessary. We have checked two Si wafers with different doping levels. In case of

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Si substrate with higher resistivity (0.1O hm cm) there was no stable response related to the electrostatic forces. The Si sample with lower resistivity (~0.02– 0.05 Ohm cm) exhibited a stable level of the electrostatic forces and surface potential was around –0.1 to –0.2V. More curious observations were made with a sheet of mica, which due to its atomically flat surface is often used in AFM. It did not show any stable electrostatic response when it is placed on thick non-conducting support. The situation is different when mica is glued to a metal puck, which is fixed on the sample plate with a small magnet imbedded into the plate. In this case one can detect a stable electrostatic force and surface potential around +0.85 V. It is worth noting that this surface potential is measured versus the Pt-coated probe. As humidity increased from 20 to 90% the surface potential became 0.45 V. In the EFM and KFM experiments it is important to realize in what imaging regime the electrostatic forces between the conducting probe and the samples are measured. Most of such measurements were performed with the probe staying in the non-contact regime where electrostatic forces dominate over van-der-Waals forces. This might undermine the spatial resolution which will be improved as the tip comes into intermittent contact with the sample. At this situation the cross-talk between topography and the electrostatic force might become evident and this effect was of special concern in our applications. We conducted EFM and KFM studies in the AM mode with the Asp (e.g., A0=10–15nm, Asp=0.6A0) chosen at the level common for the intermittent contact regime. This is clearly seen in Fig. 9.19. At distances far away from the surface there is a gradual drop of the AvZ curves. One can get the signature

Fig. 9.19 Amplitude-versus-distance (top) and phase-versus-distance (bottom) curves recorded with a conducting probe and the sample exhibiting electrostatic forces

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of the electrostatic response on Asp chosen along this part of the curve where the tip is not tracking the sample topography. With the tip approach to the sample the AvZ curve changes its character and became steep. When topography image is recorded at Asp chosen along the steep part of the AvZ curve it provides a high-resolution visualization of surface features and the probe senses electrostatic force with better sensitivity and resolution. The use of different frequencies for detection of topography and electrostatic interactions allows their simultaneous and independent measurements thus avoiding the lifting the probe needed for the operation at single frequency. In the intermittent contact mode the tip–sample mechanical interactions can be varied by changing Asp. Weak tip–sample mechanical interactions can be judged by minimal variations of the phase signal at the wmech. This might be the best operation regime for electrostatic measurements. The phase changes are enhanced at lower Asp due to increased tip–sample forces. A substantial decrease of Asp in EFM and KFM experiments is not recommended especially when imaging is performed on conducting samples. Such operation as well as the application of high voltage to the probe during tuning process, which is useful for optimization of the KFM servo operation, might cause an unwanted tip–sample discharge and related damage of the sample location and/or the probe (Fig. 9.20a). The analysis of the tip force interactions with surface charge was described in the first experiments and the authors pointed out the AFM capability of detection even <100 electrons [32]. Actually the tip-induced deposition of surface charges might be very useful for lithographic or other purposes. In the evaluation of such capabilities we conducted several experiments similar to those described in [32, 33, 87]. In these trials, surface charges were deposited by a tip–sample voltage discharge on surface of PMMA and normal alkane C60H122 layers on Si and graphite, respectively. The charges were deposited in 2ms pulses above the voltage threshold, which is around 5–10 V (depending on a layer thickness and annealing state). The imaging of the deposited charge in KFM is straightforward yet the cross-talk might imprint the charges’ signature to the topography image as shown in Fig. 9.20b, c. The surface potential image in Fig. 9.20c shows the charge pattern depositedon PMMA

Fig. 9.20 (a) Topography image of the fluoroalkane deposit on graphite showing a location disturbed by the unwanted tip–sample voltage pulse during KFM tuning procedure. Scan size is 3 mm in (a) and 8 mm in (b) and (c)

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Fig. 9.21 Topography (a, d), phase (b, e) and surface potential (c, f) images of the PMMA location with surface charge deposited by the tip voltage pulse. In all images the scan size is 1 mm. The images in (a)–(c) were obtained immediately after the pulse, and the images in (d)–(f) were recorded 5 h later

by the voltage pulses of different polarity. The positive charges have induced the “elevated” locations in the topography image in Fig. 9.20b. The observation of the single charge deposited on PMMA surface by the voltage pulse of 15V (Fig. 9.21a–f) shows that it slowly dissipated in air and the cross-talk signature in topography image disappear much faster than the charge. The chargerelated pattern in the phase images exhibit intermediate behavior showing that this signal is only partially influenced by the charge. The quantitative estimates of the described effects can be made from the crosssection profiles taken across the charge location (Fig. 9.22a–d). After the 5-hr period the cross-talk effect in topography images was drastically reduced yet the charge dissipation was most obvious at the periphery of the charge location. The observed charge dissipation proceeds differently on other materials and long-term charge storage was observed on fluorinated surfaces. The charged patterns similar to that shown in Fig. 9.20c can be used for further deposition of other electroactive species for catalytic or lithographic purposes [87]. We have also examined the effect of the charge deposition in thin layers of normal alkanes on graphite. It is well established that ultrathin adsorbates of normal alkanes on graphite form lamellar domains in which the chains are aligned parallel to the surface [88]. This lamellar order is preserved even at temperatures much higher (30–50°)

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Fig. 9.22 The cross-section topography and surface potential profiles taken across the charge location shown in Fig. 9.21a–f. The profiles in (a) and (c) were measured in images in Fig. 9.21a, c, respectively. The profiles in (b) and (d) were taken in the image in Fig. 9.21d, f, respectively

than melting temperature of the same alkane in bulk crystals [89]. The voltage pulse applied to the C60H122 layer on graphite induced a circular damage pattern in the adsorbate and even in graphite substrate visible as a hole in the center (Fig. 9.23a). The phase image in Fig. 9.23b emphasizes only the edges of the surface structures. In contrast, the surface potential (Fig. 9.23c) shows a bright–contrast circular pattern, which can be assigned to the surface potential of the substrate. In addition, dark patterns surrounding the disk-like region represent the generated negative charges on the elevated alkane domains. The negative charges on the alkane domains were dissipating for several days. Further high-resolution AFM images (not shown here) demonstrated that the lamellar order of the domains was destroyed and the material of the domains displays a granular morphology. Most likely, the discharge caused a variety of different chemical processes. Therefore, this approach can be applied not only for lithography but also for local initiation and monitoring of chemical reactions [90].

Evaluation of Different Approaches in EFM and KFM Recent developments of scanning probe microscopy are related with use of frequency modulation and multifrequency studies of local mechanical and electric properties. In the multifrequency experiments the optimization of measurements of the particular surface features or properties is the outstanding challenge. In relation to AFM-based electrostatic methods this implies finding the most accurate and sensitive approach for detection of surface potential and dC/dZ as well as the mapping of their variations with

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Fig. 9.23 (a)–(c) Topography, phase and surface potential images of a location of the ultrathin adsorbate of C60H122 after the 15 V pulse for 2 ms. (d) A cross-section profile taken in (c) along the direction marked with a white line. The contrast in (a) covers the height corrugations in the 0–8 nm range. The phase changes in (a) are in the 0–20° range. Scan size 7 mm

high spatial resolution. These questions are addressed in the comparison of EFM and KFM results obtained in the AM–AM and AM–FM modes on several materials. First of all, we applied these modes in studies of semifluorinated adsorbates and TPV – the materials that exhibit strong electrostatic effects in the images obtained at different tip voltage (Figs. 9.17 and 9.18). The experiments, which were conducted on the adsorbates of semifluorinated alkanes on different substrates, have been useful for comparison of AM–AM and AM–FM approaches in EFM and KFM. The images of F12H20 on Si, which are shown in Fig. 9.24a–e, were obtained in AM–AM and AM–FM operations. A domain of self-assembled structures in the center exhibits the pronounced contrast in the EFM images (Fig. 9.24b, d) obtained in both approaches. The multiple circular structures surrounding this domain do not show any signature in the EFM image. Also the contrast of the domain’s pattern in Fig. 9.24d is higher than in Fig. 9.24b. Remarkably, the image of amplitude at 2welec in the AM–FM mode (Fig. 9.24e) clearly shows the same pattern (with different contrast variations) but the similar image obtained in the AM–AM mode (Fig. 9.24c) is practically featureless. As expected, the KFM images of the same location (Fig. 9.25a–f), which were obtained in the AM–AM and AM–FM modes, exhibit the similar features seen in the

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Fig. 9.24 The EFM images of F12H20 adsorbate on Si substrate: (a) – topography image; the images of the same location using the X-vector component at 10 kHz (b) and amplitude at 20 kHz (c); the images of the same location using the X-vector component at 5 kHz (d) and amplitude at 10 kHz (e). Scan size is 1.2 mm in all images

EFM images. In this case, however, the images provide quantitative data about the surface potential of the F12H20 self-assembled domain. Actually this surface potential detected in the AM–FM mode is higher (–0.75 V on the main domain surface and –1.25V in the center) than that measured in the AM–AM mode (–0.5 V on the main domain surface and –0.75V in the center). The amplitude image of this area, which was obtained in the AM–FM mode, presents spatial variations of dC/dz that can be generally assigned to changes of the dielectric constant of this adsorbate. This fact might explain that the center part of the domains, where the adsorbate is thicker, exhibits darker contrast compared to the rest of the domain, which is darker than the areas outside of the domain. The higher values of the surface potential measured in the AM–FM mode have been also found in the studies of KBr layers on Au (111) [62]. In KFM examination of F14H20 adsorbates on graphite we also detected the higher surface potential values when the AM–FM mode was applied. In this case, the topography image shows different patches of the adsorbate, and only few of self-assemblies are exhibiting a pronounced surface potential contrast (Fig. 9.26a–f). As usual, the imaging was performed in the intermittent contact and the tip–sample force interactions did not cause strong variations of phase contrast (Fig. 9.26b). The surface potential images recorded in the AM–AM and AM–FM modes (Fig. 9.26c, d) show similar patterns, which are very different from topography that excludes the cross-talk between these

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Fig. 9.25 The KFM images and profiles of F12H20 adsorbate on Si substrate: (a) topography image; (b) the surface potential image at welec = 10 kHz (AM–AM); (c) the surface potential cross-section profile taken across the center of the image in (b); (d) the surface potential image at welec = 5 kHz (AM–FM mode); (e) the image of the amplitude at 2welec = 10 kHz; (f) the surface potential cross-section profile taken across the center of the image in (d). Scan size 1.2 mm

signals. Furthermore, only few self-assembled structures exhibit a well-defined dark contrast in the surface potential images, whereas most of the adsorbate (for the exception of several other bright patches) has a relatively homogeneous contrast. As we will see below, only F14H20 self-assemblies with molecular dipoles oriented perpendicular to the sample surface provided strong negative potential. The comparison of the crosssection profiles taken across the surface potential images revealed higher potential values in the image recorded in the AM–FM mode. The similar finding regarding the comparison of the surface potential measurements in the AM–AM and AM–FM modes was made in the KFM studies of CdTe nanostructures, which are formed on mica substrate during drying of water suspension of the CdTe nanoscale spheres [91]. This structural transition was accompanied by emergence of positive surface potential. The topography and surface potential images of these molecular nanostructures are presented in Fig. 9.27a–e. The comparison of the topography and surface potential patterns of CdTe nanostructures shows obvious differences. Linear structures of different size (“nanowires”) are clearly distinguished in the topography image. In surface potential image these structures exhibit practically identical contrast whereas the individual spheres and their aggregates do not show the noticeable contrast. The cross-section profiles, which were taken along the top parts of the images in (b) and (c), reveal the potential variations in the 0.3V range in the AM–FM study and the potential changes in the 0.2V range for the AM–AM approach. This observation

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Fig. 9.26 The images obtained in KFM study of F14H20 adsorbate on graphite. (a, b) Topography and phase images, which are practically identical for the AM–FM and AM–AM modes; (c, d) Surface potential images of the same location obtained respectively in the AM–FM and AM–AM modes. (e, f) The surface potential cross-section profiles taken in the images in (c) and (d) along the directions indicates with white dashed lines. Scan size is 3 × 5 mm

suggests that the formation of stronger molecular dipoles or their vertical realignment accompanies the self-assembly of CdTe spheres into nanowires. Most likely, this is due to a creation of structural defects (i.e., truncation) in CdTe structures with cubic symmetry when the nanoparticles are transforming into nanowires [92]. The conclusion that the surface potential values obtained in the AM–FM mode are higher than those obtained in the AM–FM is consisted with the predictions of [61] and KFM study of KCl islands on Au in UHV [62]. The latter study also indicated that the FM detection gives surface potential values identical to that detected with ultraviolet photoelectron spectroscopy.

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Fig. 9.27 (a)–(c) Topography and surface potential images of CdTe nanostructures on mica. The surface potential images in (b, c) were taken in the AM–FM and AM–AM modes. Scan size 5 mm. (d)–(e) The cross-section profiles taken along the top part of the images in (b) and (c)

The better sensitivity of the AM–FM detection also leads to higher spatial resolution achieved in mapping of surface potential [93, 94]. The sub-10nm features observed in the topography and surface potential images of the F14H20 adsorbate on graphite illustrate this point (Fig. 9.28a–c). The surface area shown in these images is covered by two types of self-assembled structures: toroids and ribbons (Fig. 9.28a). The surface potential of the ribbons is only slightly different from that of the toroids, and there are few locations with pronounced bright contrast (Fig. 9.28b). These areas present the voids in the packing of toroids and ribbons at which the probe detects the underlying substrate. The 2-nm wide strip is clearly distinguished in the 200-nm image of surface potential of the area (Fig. 9.28c). This feature can be used as a measure of spatial resolution of KFM in the AM–FM operation in the intermittent contact mode. Therefore, the demonstrated spatial resolution of this method is close to the 3-nm resolution recently reported for the similar imaging of KBr/InSb system in UHV [64]. Several examples of sensitive KFM imaging in the AM–FM mode were obtained in studies of Au (111) and highly oriented pyrolitic graphite, which are commonly used for AFM studies. The crystalline domains with lateral dimensions reaching 1mm are seen in the topography images of the hydrogen-flame annealed Au film (Fig. 9.29a). The edges of the domains as well as crystalline steps are seen in the

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Fig. 9.28 (a, b) Topography and surface potential images of F14H20 adsorbate on graphite obtained in the AM–FM mode. (c) The surface potential image taken in the 200-nm area marked with a white dashed square in (b). The insert in the top of (c) shows a cross-section profiles taken along the white strip indicated with the arrow. Scans are 1 mm in (a, b) and 200 nm in (c)

Fig. 9.29 (a)–(c) Topography, phase and surface potential images of Au (111) obtained in the AM–FM mode. The scan size is 3 mm. The contrast covers the 0–100 nm corrugations in (a), the 0–55° variations in (b) and the 0.25 to -0.05 V changes in (c). (d) Topography image of Au (111) domain showing the contamination traces at the domains edges. Scan size is 800 nm

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phase image (Fig. 9.29b). The same edges exhibit bright contrast in the surface potential image in Fig. 9.29c. The most likely reason of the contrast is an air-borne contamination, which is clearly seen at the domain edges in the topography image with higher magnification (Fig. 9.29d). These images were recorded after only short-time exposure of the Au (111) sample to ambient air. Therefore, a special care should be taken to avoid such contamination. Similar conclusion was derived from KFM study of freshly cleaved sample of graphite. The long-term KFM imaging of this sample is documented in Fig. 9.30a–f. The topography and surface potential images taken 2hr after the cleavage show a surface region with several steps of graphite planes and few dark patches with different potential, Fig. 9.30a, b. Such patches were not seen at shorter times after the cleavage.

Fig. 9.30 (a, b) the topography and surface potential images of graphite 2 h after cleavage. (c, d) the surface potential images of graphite 2.25 and 2.5 h after the cleavage. Scan size is 20 mm

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Therefore, they can be assigned to the air-borne contamination of the surface. As time progresses, these patches have increased in size, and new patches originated. This process is visible in surface potential images in Fig. 9.30c, d, which were recorded in 15-min intervals after the ones in Fig. 9.30a, b. The difference in the potentials of the fresh and contaminated areas was ~60mV, and it did not change as the contamination grew. At longer times the contamination has covered the entire area and the surface potential image became homogeneous again. Amazingly, phase images (not shown here) are less sensitive to the growing contamination than the surface potential images.

Metals and Semiconductors Work function is the essential characteristic of metals, which defines their important properties as electron emission, corrosion, photosensitivity, surface potential, etc. Practical measurements of work function are subjected to a number of variables such as a level of contamination, type of crystalline morphology, the examination method, etc. The rigorous work function data for elements have been collected from UHV experiments performed on well-defined samples and the work function of metals falls in the 4.2–5.7 V range (Ag – 4.26 V, Fe – 4.5 V, Cu – 4.65 V, Au – 5.1V, Pd – 5.12 V, Ni – 5.15 V, Ir – 5.25 V, and Pt – 5.65 V) [95]. The surface potential, which is determined in the KFM experiment, is a difference between work functions of the sample and tip material (usually Pt). The most reliable quantitative surface potential data can be obtained in UHV. When such measurements are performed at ambient conditions, one should be aware about the tip and sample variations and possible influence of contaminations. These factors have definitely influenced our KFM measurements performed on a test structure [96], which was prepared by deposition of 150-nm thick strips of Pt and Au on Si substrate. The images of this structure are shown in Fig. 9.31a–c. The surface potential variations between Pt and Au lines are minimal whereas the surface potential of the Si substrate is ~0.3 V higher. Although the recorded potential changes are generally consistent with the expectations of the work function table [95], absolute surface potential values are influenced by poorly controlled ambient conditions. For example, the surface potentials of the AFM probe and Pt strip should be close to each other but a 0.5-V difference was measured instead. Therefore, the comparison of local surface potential data made in the particular image is more reliable than the absolute potential values. The other KFM application to metals has been attained in study of soldering material – an alloy of Bi and Sn. The sample of this alloy was prepared by a hot pressing a melted piece of the material between two mica sheets. The pressed “sandwich” was cooled to RT and the metal sample was released from the mica sheets. The KFM data obtained on the surface of Bi/Sn sample are shown in Fig. 9.31d–f. In addition to a domain morphology, which is seen in topography image, the surface potential image revealed that the domains exhibit different contrast.

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Fig. 9.31 (a, b) Topography and surface potential images of the SiAuPt structure. The Pt and Au lines are 150 nm in height. Scan size is 30 × 40 mm. (c) The cross-section profile taken along the direction indicated with white dashed line in (b). (d, e) Topography and surface potential images of Bi/Sn soldering material. The contrast in (a) covers height variations in the 0–50 nm scale. Scan size 8 mm. (f) The cross-section profile taken along the direction indicated with white line in (e)

This finding suggests that the material is actually a partial solid solution. The profile in Fig. 9.31f shows the potential alternation of 0.2V that corresponds to difference of work function of the metals. It is worth noting that these results were obtained on freshly prepared sample. After several hours of storage of the sample at ambient conditions the potential variations between the individual domains became very small most likely due to oxidation of Sn. The KFM inspection of semiconductors and related structures provide useful technological information about local impurities and defects [97], thickness of gate oxide [98] and its leakage, 2D doping profiles [99] and silicon pn junctions [100]. In many instances, the reliability of such measurements needs further improvement. We strongly believe that these applications will benefit from higher-resolution AM–FM studies in the intermittent contact work. The evaluation of this approach to a number of semiconductor structures is presented below. The initial goal was to check the possible cross-talk between topography tracking and recording of surface potential. The images presented in Fig. 9.32a–d were taken from KFM studies of SDRAM and SiGe structures. The topography and surface potential patterns of the same sample regions show dissimilar patterns. Surface potential images revealed the surface locations of different doping types and levels and a presence of small defects. The darker areas with lower surface potential can be tentatively assigned to p-doped regions. The comparison of the topography and surface potential images of SRAM sample in Fig. 9.33a, b confirms that the cross-talk between these patterns is

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Fig. 9.32 (a, b)Topography and surface potential images of SDRAM. Scan size is 60 mm. (c, d) Topography and surface potential images of SiGe structure. Scan size is 25 mm

Fig. 9.33 (a, b) Topography and surface potential images of SRAM. Scan size 25 mm. (c) Top and bottom – the cross-section profiles along the directions indicated with white lines in (a) and (b), respectively

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p ractically absent. This is enforced by the cross-section profiles in Fig. 9.33c taken in the images along the directions marked with white arrows. Furthermore, the signal-to-noise ratio of the cross-sectional profile is high and it allows distinguishing the potential variations as small as 10 mV. In the control experiments we obtained surface potential images of SRAM in the AM–AM mode with different combinations of wmech and welec (first flexural resonance/10 KHz, first flexural resonance/ second flexural resonance; second flexural resonance/first flexural). The comparison of the quantitative values in the surface potential profiles showed that the variations did not exceed 10%. In addition, we found that the AM–AM shows quite similar if not better quality surface potential images compared to the AM–FM technique. This might be the consequence of large-scale structures of the examined semiconductor samples. The situation has drastically changed when we decided to detect the electrostatic force responses at 2welec that is related to dC/dz. In this case, the AM–FM technique has definite advantage. The comprehensive characterization of local electric properties goes well beyond the measurements of surface potential. The response of the electrostatic force at 2welec is related to the capacitance between the probe and the sample. The local capacitance variations will be reflected in the amplitude response at this frequency, particularly, when the tip–sample distance will be maintained constant. The images of SiGe structures in Fig. 9.34a–c illustrate the simultaneous measurements of topography, surface potential and amplitude at 2welec. All three patterns are quite different therefore they provide simultaneous and independent information. It is worth noting that these particular images were recorded with NSC14Ti–Pt probe, which is not as sharp as the conducting Olympus probes. This might be behind a low-contrast surface potential image. On the other hand, with the applied probe the contrast of the amplitude (dC/dz) image was more pronounced. Therefore, there is definite room for further optimization of the KFM and EFM images by a proper selection of the conducting probe. The crucial question about the possible use of surface potential to quantitative analysis of doping density is still open. Doping profiles can be deduced from KFM

Fig. 9.34 (a, b) Topography and surface potential images of SiGe structure. (c) Amplitude image at 2welec. Scan size is 15 mm in all images

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Fig. 9.35 (a, b) Topography and surface potential images of a cross-section of the staircase sample with layers of different doping density. Scan size is 20 × 24 mm. The contrast in (a) covers the height variations in the 0–200 nm range. (c) The profiles of concentration and resistivity of the staircase sample. (d) The surface potential profile taken across the layers in (b)

measurements to the extent that variations of work function and capacitance are related to the dopant type and concentration at or near the sample surface [99, 101]. The correlation between the surface potential and doping concentration if often examined on a test samples in which surface layers have different dopant concentration. The KFM images of such staircase sample, which was prepared by crosssectioning in a direction perpendicular to the layers, are shown in Fig. 9.35a, b. The surface topography on the sample cross-section is smooth whereas the contrast variations in the surface potential image revealed the layers with different dopant concentration. This is evident from the comparison of the concentration, resistivity and surface potential profiles in Fig. 9.35c, d. As the dopant concentration varied from 1014 to 1019 the potential has changed ~250mV. The results are consistent with findings reported in [101]. High sensitivity to doping profiles was also demonstrated by scanning microwave microscopy [102]. At present, the important question is which of these

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approaches is most reliable for quantitative 2D dopant mapping. There are a number of hurdles, which are general (such as laser-induced photovoltaic effects and surface contamination) or more specific to a particular technique (probe geometry and conductivity level). Hopefully, the problems can be eliminated by the proper calibration of the probe, the use of reliable test structures, appropriate sample preparation and thoroughly performed experiments.

Examination of Molecular Self-Assemblies Semifluorinated compounds, which exhibit a strong polar nature due to the dipole of the –CH2–CF2-bond, have been examined with KFM [51, 53]. We applied KFM to molecular self-assemblies of the perfluoroalkyl-alkanes or semifluorinated alkanes, F(CF2)n–(CH2)2mH or shortly FnHm. The sketch of the molecules in Fig. 9.36 illustrates that the molecules consist of hydrocarbon and fluorocarbon parts. The F atoms are larger than C atoms. Therefore the planar zigzag conformation of -CH2-sequence evolves into a bulky helical conformation of -CF2-chain. The dimensions of these parts for F14H20 and F12H8 molecules are also indicated. These materials exhibit the peculiar phase behavior due to incompatibility of the fluorocarbon and hydrocarbon segments and their volume differences. In bulk, these molecules form a large number of stable smectic phases depending on temperature and length of the blocks [103, 104]. In confined geometry of thin and ultrathin layers semifluorinated alkanes self-assemble into the nanoscale ribbons, spirals or toroids and their different intermediates [105, 106]. These structures exist in a delicate energetic balance, and they undergo transformation into each other when solvent, temperature or humidity changing [106]. Similar self-assemblies of semifluorinated long-chain acids were examined with AFM and FTIR [107, 108]. Particularly, it was found that the height of these structures is smaller than extended length of the molecules.

Fig. 9.36 Sketch illustrating the chemical structure of semifluorinated alkanes and several geometrical parameters of two representatives of these compounds

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That is consistent with a slightly tilted vertical arrangement of the molecules with fluorinated parts closer to the air–layer interface. In case of the semifluorinated alkanes, the additional structural information was obtained from X-ray reflectivity studies of F8H18 and F14H20 layers [105, 106]. The best modeling of height dependence of electron density obtained from the X-ray reflectivity data of such layers was achieved using a disk-like structure with quarter circles at the borders. The extended molecules with fluorocarbon groups point to air and form the disk. At the edges the bending of the molecules towards the surface compensates the mismatch of electron density in the hydrocarbon and fluorocarbon parts. A rational modeling of molecular arrangement in the F14H20 ribbons implied that the air-pointing fluorinated segments are arranged perpendicular to the surface and the hydrocarbon chains are tilted 122° with respect to the fluorocarbon segments. The described structural model differs from a smectic bilayer of FH2F-type arrangement for F8H18 layer [109], which was proposed in the analysis of surface pressure versus molecular area isotherms and X-ray reflectivity. Furthermore, the diagrams of surface pressure and surface potential versus molecular area isotherm were interpreted to explain the build-up of the bilayer from a single layer [110]. Particularly, a macroscopic Kelvin probe, which was suspended above the spread perfluoroalkyl-alkane layer, was used for measurements of surface potential. On compression of the single layer the surface potential gradually changed from zero to –0.8V. These observations were explained by the transformation of the single layer with FH (fluorinated segments – to air, hydrocarbon segments – to substrate) and HF (hydrocarbon segments to air, fluorinated segments to substrate) to the bilayer of FH2F-type. By applying KFM to self-assemblies of semifluorinated alkanes on different substrates we extended macroscopic measurements of surface potential to the sub-micron scale. This improves the characterization of these structures and helps understanding of the nanoscale architecture. We have examined self-assemblies of a number of semifluorinated alkanes (F14H20, F12H20, F12H12, F12H10, and F12H8) on Si, mica and graphite using KFM in the AM–FM mode. These samples are prepared by spin-casting of dilute solutions of the semifluorinated alkanes in different solvents (decalin, perfluorodecalin, octane, xylene, tetrahydrofuran, etc.). Spirals, ribbons, toroids and their intermediates are the typical self-assemblies of these compounds on Si and mica. These structures are surrounded by thinner and self-organized layers of the semifluorinated alkanes. One of the F12H20 domains formed by the spiral structures is shown in Fig. 9.24a. The high-magnification topography and surface potential images of similar F12H20 domain on Si are presented in Fig. 9.37a, b. The height of the structures is in the 3.0–3.5nm range, and their surface potential is 0.75–0.8V more negative than that of the surrounding surface. Topography and surface potential images of F12H8 self-assemblies on the same substrate are presented in Fig. 9.37c, d. The F12H8 ribbons are less curved and more extended. Only few toroids are formed in this adsorbate. The ribbons are ~2nm in height and surface potential is ~0.65V more negative compared to the rest of the sample. The self-assembly of the semifluorinated alkanes is governed by a delicate balance of intermolecular interactions therefore the changes of their shape, dimensions and

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Fig. 9.37 (a, b) Topography and surface potential images of the F12H20 domain on Si substrate. Scan size is 400 nm. (c, d) Topography and surface potential images of the F12H8 self-assembled structures on Si substrate. Scan size is 500 nm

molecular architecture can be induced by vapors of different solvents [106]. We have monitored similar alternations during the long-term studies of F14H20 adsorbate on Si substrate in humid atmosphere (RH>90%). A spreading of spirals, which have formed an initially compact aggregate, was noticed in the first 24h (Fig. 9.38a, b). The reference points marked with the white arrows help to identify these changes. Several factors such as a lower adhesion of the assemblies in humid air, charge interactions of the spirals and a tip-assisted material transfer, might contribute to this phenomenon. At longer time the spirals converted to toroids which are more thermodynamically favorable structures (Fig. 9.38c). In humid air, these toroids, which have a diameter in the 45–55nm range, exhibit a negative surface potential that is 0.5V, i.e., slightly smaller than in

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Fig. 9.38 (a, b) Topography images of the same location of F14H20 adsorbate on Si in dry air and in humid air after 1 day exposure. Scan size 2.5 × 1.25 mm. (c, d) Topography images of F14H20 adsorbate on Si in humid air after 1 and 2 day exposure. Scan size 1 × 0.5 mm. (e, f) Topography and surface potential images of one location of area in (d). Scan size 400 nm

dry air (Fig. 9.38e, f). This drop of the surface potential is less substantial compared to the partial shielding of surface charges observed in the non-contact KFM studies of semiconductor samples [111]. This difference might be related to the fact that our studies were performed in the intermittent contact regime. The self-assemblies of F14H20, F12H20, and F12H8 on mica are presented in Figs. 9.39 and 9.40. The domains of curved ribbons are the main features of the

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Fig. 9.39 (a, b) Topography and surface potential images of F14H20 adsorbate on mica. The height of the domains is ~3 nm and surface potential image of the domains is ~1.4 V more negative than the surrounding. Scan size is 3 mm. (c, d) Topography and surface potential images of F12H20 adsorbate on mica. The height of the domains is ~2.1 nm and the surface potential of the domains is ~1.1 V more negative than the surrounding. Scan size is 1.5 mm

F14H20 adsorbate (Fig 9.39a, b). Their height is comparable with that of the F14H20 spirals and toroids on Si yet the surface potential of these domains is substantially more negative (–1.4V) than surface potential of the spirals and toroids (~–0.8V) on Si. The F12H20 adsorbate on mica is characterized by the domains formed of more straight ribbons. The topography image in Fig. 9.39c shows that there are preferable orientations of the ribbons, which might reflect the epitaxial order of the semifluorinated alkane chains on the surface of mica. Both, height and surface potential of the F12H20

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Fig. 9.40 (a) Topography image of F12H8 layer on mica immediately after the spin-casting. (b, c) Topography and surface potential images of the same location 1 h later. White arrows in (b) indicate the first voids, which were formed due to sublimation. (d, e) Topography and surface potential images of the same location 2 h later. (f) The cross-section profiles along the directions indicated with white dashed lines in (d) and (e). Scan size 800 nm

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s elf-assemblies on mica are different from those of the F14H20 structures. The drop of height from 3 to 2.1nm is more substantial than a lowering of negative surface potential from 1.4 to 1.1V. The possible reasons of these deviations will be discussed below. In the examination of adsorbates of shorter semifluorinated alkanes we followed a sublimation of the material from the substrates. This effect is demonstrated in the images shown in Fig. 9.40a, b. The topography image (Fig. 9.40a), which was taken immediately after the deposition of F12H8 on mica, revealed the multiple nanoscale domains separated by curved boundaries. Within the next hour this pattern transformed into another one with numerous toroids, which are connected to each other by short links (Fig. 9.40b). The surface potential image of this morphology showed dark patterns of the toroids and brighter boundaries (Fig. 9.40c). In the top part of this image there are two brightest spots, which correspond to newly appeared voids marked with white arrows in Fig. 9.40b. These are the locations where the probe started to feel the higher surface potential of the substrate. As scanning of this area has continued, the void formation intensified, and the images in Fig. 9.40c, d revealed an increasing number of voids, which were also growing in size. The cross-section profiles in Fig. 9.40f showed that the toroid-like features are ~1nm in height and their surface potential is 0.8V lower than that of the substrate. The continued monitoring of the surface demonstrated that after 5–6hr the F12H8 adsorbate completely sublimated from mica. The morphology of layers of semifluorinated alkanes on graphite is more diverse than on other substrates, particularly, due to strong epitaxial order of these compounds in the immediate vicinity of the graphite surface. The images of F14H20 adsorbate on graphite in Fig. 9.41a, f pointed out a presence of toroids (two of them are indicated with white arrows in Fig. 9.41a) as well as a disordered material and linear structures oriented in one of the preferable directions consistent with the threefold symmetry of the substrate. Therefore, similar to normal alkanes their semifluorinated analogs have tendency to align themselves along the basal plane of this substrate. The self-assemblies of different types are most likely responsible for the contrast of the surface potential image in Fig. 9.41b. The areas with less coating and locations covered by the featureless material exhibit the brightest contrast and the toroids and their aggregates are seen as darkest spots with the potential difference of 0.8V. Many linear structures are seen as locations with intermediate surface potential, and one of such regions is magnified in Fig. 9.41c, d. The linear structures of two types, which are marked as “1” and “2”, are distinguished in the topography image in Fig. 9.41c and in the cross-section profiles taken along the related stacks (Fig. 9.41e). The profiles show that the stacks are characterized by (~5nm) and (~9nm) spacing. These dimensions are close to the single and double length of the extended F14H20 molecule. Therefore, the type “1” structures are tentatively assigned to lamellar structures formed of single molecules that partially interdigitate with each other (Fig. 9.41f). The type “2” structures are most likely formed of the bilayer entities with slightly curved shape that reflects the deviations of the molecular chain orientation from the flat one. As we will discuss below the variations of the surface potential of these structures can be related to this circumstance.

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Fig. 9.41 (a, b) Topography and surface potential images of F14H20 adsorbate on graphite. Scan size 800 nm. The height of toroids (two of them are indicated with white arrows) is ~3.5 nm and their surface potential is ~0.7 V more negative than that of the substrate. (c, d) High-magnification topography and surface potential images of the F14H20 adsorbate on graphite. (e) The cross-section profiles taken along the directions marked with dashed white lines in the lamellar regions “1” and “2” in the image in (c). (f) Sketches of suggested molecular order in the lamellar regions “1” and “2”

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Although the main structural features of the F14H20 adsorbates on graphite were also found in studies of F12H20 layers on the same substrate, there are some specific findings. The images in Fig. 9.42a, b and the cross-section profiles in Fig. 9.42c show a sheet of single molecules and the aggregate formed of ribbons and one distorted toroid. The aggregate is seen as the darkest patch in the surface potential image yet the toroid is slightly brighter than the ribbons. The molecular sheet exhibits slightly darker contrast and the surrounding shows only faint features that can be assigned to the boundaries of molecular domains spread on the substrate. In another example, an aggregate of F12H20 ribbons with linear substructures is positioned on top of the lamellar layer (Fig. 9.42d–f). The spacing of the lamellar layers, which are most pronounced in amplitude image in Fig. 9.42e, is around 7.5nm. The white arrow indicates the toroid with 3.5nm in height. Their surface potential is ~0.5V below that of the background. The ribbons exhibit more negative potential than that seen on the lamellae in (Fig. 9.41f). The potential variations across the lamellar layers are of few tens of millivolts. The imaging of F12H8 layers on graphite revealed the morphology changes accompanying their slow sublimation and, therefore, the structural hierarchy of the layers can be observed from the top to the bottom, down to the substrate. The structural characteristics of the same location immediately after the material deposition and 8hr later are seen in the images in Fig. 9.43a–d. The parquet-like morphology of the top layer gradually transformed to the threefold symmetry “skeleton” made of the F12H8 ribbons lying directly on the graphite surface. The surface potential differences became pronounced with the variations of 0.75V between the bright substrate location and the dark ribbons (Fig. 9.43d). This implies that the transition from lamellae to ribbon includes the reorientation of the fluoroalkanes molecular chains from horizontal to vertical direction with respect of the substrate. One of the intermediate morphologies is presented by the images in Fig. 9.44a–c. At this sublimation stage, the areas between the straight ribbons are filled by lamellar domains with the molecular-scale spacing. The surface potential contrast varies for the domains that have different spacing and also inside the individual domains. The analysis of the imaging of the semifluorinated adsorbates on different substrates shows the KFM capability of compositional mapping of the samples with heterogeneities of local electric properties. In our case, the heterogeneities are most likely related to different molecular self-assemblies, which were governed by intrinsic properties of the semifluorinated molecules and their interactions with the substrates. The method is also useful for monitoring molecular transformations caused by change of humidity and for observations of the molecular organization at various depths in the layers of low-molecular weight semifluorinated alkanes. The significance of these findings will be enhanced when the surface potential data can be reliable interpreted in terms of molecular properties of the samples. We will analyze the topography and KFM data of the semifluorinated alkanes, which are collected in Tables 9.1 and 9.2. Table 9.1 contains the contour length of different parts of the fluoroalkanes molecules and the height of their self-assembled structures on the mica and Si substrates. The height of the F14H20 and F12H20 self-assemblies on all three substrates is larger than the contour length of the fluorocarbon and hydrocarbon parts but it is smaller

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Fig. 9.42 (a, b) Topography and surface potential images of F12H20 adsorbate on graphite. Scan size 700 nm. (c) The cross-section profiles taken along the directions marked with the white dashed lines in (a). (d)–(f) Topography, amplitude and surface potential of another location of F12H20 adsorbate on graphite. Scan size 400 nm. White arrows in (d) and (f) indicate the same toroid

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Fig. 9.43 Topography (a, c) and surface potential (b, d) images of F12H8 adsorbate on HOPG. Scan size is 1.4 mm in (a, b) and 1 mm in (c, d). The identical areas in (a) and (c) are marked with the red dotted parallelograms. The images in (a, b) were measured immediately after the spincasting and the images in (c, d) 8 h later. The 15-nm wide and 1-nm high ribbons in (c, d) have surface potential ~0.75 V lower than of the bright patches of the substrate

than total contour length of the molecule. This is consistent with the bending of the hydrocarbon parts suggested in [106]. The height of the F12H8 structures on mica is comparable to the length of the fluorinated part whereas the self-assemblies on Si and graphite are much lower. Therefore, the inclination of the fluorocarbon segment of F12H8 with respect to the surface normal can be expected in these cases. Table 9.2 presents the surface potential data obtained in the AM–FM studies as well as the experimental and theoretical data from the study [110], in which the F8H18 Langmuir layer was examined with macroscopic Kelvin probe and negative surface potential of -0.8V was detected. This value was not far from the estimate of the surface

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Fig. 9.44 Topography (a), phase (b) and surface potential (c) images of the location inside the area of F12H8 adsorbate on graphite shown in Fig. 9.43. These images were taken 6.5 h after the spin casting. Scan size is 500 nm Table 9.1 Geometrical parameters of semifluorinated alkanes and the experimental height values for their self-assemblies on different substrates Self-assemblies Self-assemblies Contour SelfContour on HOPG, length,assemblies on on mica, Fluoroalkane, length,height, nm (CF2)nF, nm (CH2)mH, nm Si, height, nm height, nm FnHm 1.67 0.98 1.0 1.5 1.0 (ribbons) F12H8 F12H20 1.67 2.48 3.0 2.2 2.0 (ribbons) 1.93 2.48 3.1 3.2 3.5 (toroids) F14H20 Table 9.2 Surface potentials (SP) of self-assemblies of semifluorinated alkanes on different substrates obtained in the KFM studies, Kelvin probe measurements and theoretical estimates Theoretical SelfSelfSelfSelfFluoroalkane, assemblies on assemblies on assemblies on assemblies on estimate, SP, V water, SP, V HOPG, SP, V mica, SP, V Si, SP, V FnHm F12H8 0.69 0.87 0.7 (ribbons) n/a 1.39 F12H20 0.80 0.93 0.69 (ribbons) n/a 1.39 F14H20 0.75 1.40 0.70 (toroids) n/a 1.39 F8H18 n/a n/a n/a 0.8 [F8] 0.55 [F8]

potential (–0.5V), which was made for the FH2F bilayer [110]. In our analysis, we calculated the surface potential for the layer of vertically oriented fluoroalkanes taking into consideration only the fluorinated part of the molecules. For this purpose we applied the known formula, which relates surface potential V with electric potentials of the sample – jsample and the tip – jtip, charge of electron – e, the dipole moment of a molecule – m, molecular area – A, dielectric constant – e, and permittivity of vacuum – e0:

V=−

(j

sample

e

− j tip

)+

m A × e × e0

The contact potential difference (CPD) is defined as

(9.15)

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CPD = −

(j

sample

− j tip

e

)

293

(9.16)

If we neglect the dipole of the end –CH3 group then m of the fluorinated part will be as sum of the dipoles of –CF3 and –CF2–CH2-groups, which are oriented in the same direction:

m = m − CF3 + m − CF2 − CH2 −

(9.17)

The m was estimated as 3.1D [110]. If we use molecular area A = 3×E-19m2, dielectric constant e = 2.8 and permittivity of free space e0 = 8.85×E-12C2/Nm2 then the surface potential of the fluorinated part V will be as follows:

V = CPD + 1.39V

(9.18)

Therefore, in the idealized case of self-assemblies with vertically oriented fluorocarbon groups we can expect the surface potential of ~1.39V. This is close to the surface potential of F14H20 spirals on mica. The surface potential of the self-assemblies of shorter molecules on mica becomes smaller; particularly for F12H8 structures whose height is less than the contour length of the fluorocarbon part. This implies a deviation of the m − CF3 and m − CF2 − CH2 − dipoles’ orientation from the vertical direction. The upwards orientation of the dipole moment of the end –CF3 group that is responsible for a half of m = 3.1D value [111] might explain the surface potential of F12H8 self-assemblies on mica. However, this is not very realistic due to high mobility of the end groups. The F14H20, F12H20 and F12H8 self-assemblies on Si exhibit the lower surface potentials than their counterparts on mica but they are still close to those expected from the dipole moments of the fluorocarbon parts. The mentioned differences might be assigned to specifics of the molecular packing inside the self- assemblies that requires further study. The height and surface potential data are in favor of the structural models that suggest the preferably vertical orientation of the fluorinated parts with hydrocarbon blocks spread on the substrate [105, 106]. The semifluorinated alkanes form a larger variety of self-assemblies on graphite due to specific adsorbate-graphite interactions. In addition to toroids and ribbons the self-assemblies include also various lamellar structures that developed in the immediate vicinity of the substrate. The lamellar structures, which are oriented in registry with the substrate symmetry, also influence the orientation patterns of thicker ribbons and even toroids. As judged by the height (3.5nm) and surface potential (0.7V) data, the molecular orientation inside the F14H20 toroids on graphite is not much different from that in the toroids formed on Si. Similar surface potentials of F12H20 and F12H8 ribbons on graphite, which are only ~1-nm high structures, also imply the dominant contributions of the fluorocarbons that presumably are stronger tilted with respect of the vertical direction. The surface potential of the lamellar structures (Figs. 9.40–9.42) is in the 0.1–0.4V range that most likely reflects different and more planar orientation of the fluorocarbons’ and hydrocarbons’ dipoles compared to that

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in the toroids. This suggestion is in line with the geometrical models of the lamellar domains, which are marked as “1” and “2” in the images in Fig. 9.41c–f. When single molecules of semifluorinated alkanes form the lamellar layers of ~0.5nm in thickness their dipoles are oriented practically along the substrate surface. As soon as bilayer-type structures appear (Fig. 9.41f, bottom), the orientation of the dipoles of the top molecules start to deviate from the planar one and the surface potential changes towards more negative values as seen in the image in Fig. 9.41d. Therefore, the analysis of the KFM results obtained on self-assemblies of semifluorinated alkanes on different substrate pointed out that surface potential is very sensitive to the orientation and strength of molecular dipoles.

Conclusions and Further Outlook The recent developments in the AFM-based electric techniques related with use of multifrequency measurements in the contact and intermittent contact regimes are outlined in this chapter. The novel results obtained on different materials clearly demonstrated the advanced capabilities of the AM–FM approach in detection of the electrostatic tip–sample interactions and, particularly, the improved sensitivity and higher spatial resolution compared to more traditional AM–AM measurements. These findings enhance the value of the electric measurements in characterization of materials at the nanometer scale. This statement is strongly supported by the measurements of self-assemblies formed by semifluorinated alkanes on different substrates. The quantitative studies of surface potential and its spatial distribution on the samples as well as the interpretation of surface potential images in terms of strength and orientation of dipoles of individual molecular groups allow better understanding of the structural organization of these self-assemblies and monitoring their transformations caused by various stimuli. The demonstrated advances in the characterization of materials with EFM, KFM, and PFM are related to the multifrequency studies which are becoming the essential part of the contemporary AFM. The detection of the mechanical and electrical tip–sample force interactions at different frequencies in the intermittent contact regime is essential for independent and simultaneous tracking of the sample topography and measurements of its local electric properties. EFM and KFM imaging in intermittent contact mode is essential for high-resolution mapping of electric heterogeneities of a broad range of samples. This helps overcoming the resolution limitations of the probing of electrostatic force, which is commonly performed in the lift and non-contact operation with the probe staying away from the sample surface. It will be fair to claim that the multifrequency studies are bringing a variety of experimental approaches that not only broaden the AFM applications but also complicate them. The latter is related to the need of finding the most rational approach for detection of the particular sample property. The example of such optimization is given above by evaluation of AM–AM and AM–FM techniques. The superior character of AM–FM approach has been proven in studies of many samples. The capabilities of simultaneous high-resolution studies of

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surface potential and dC/dz variations were also demonstrated exclusively in the AM–FM approach. Nevertheless it might be premature to use solely AM–FM mode in EFM and KFM applications because the AM–AM technique might have undiscovered advantages in studies of other samples or when different probes are employed. In case of PFM, one can also perform a variety of experiments using AC voltages at different frequencies either to enhance the mechanical sample response to the electric stimuli or to learn about electromechanical behavior at different time scales. The analysis of the described experimental data shows several challenges that are needed to be addressed in further developments. Despite the fact that surface potential studies lead to quantitative data their quality is far from being perfect due to a number of reasons. The main complication is the sudden changes of the surface potential, dC/dz values, or piezoresponse that happen due to sample and tip contamination and related instabilities of imaging. These effects undermine the accuracy of absolute measurements. Help might come from a cleaner environment and improved sample preparation. The role of the probe in these effects and in optimization of imaging in different AFM-based electric modes has not yet been properly explored. This is related not only to the geometrical parameters of the tip and cantilever but also to quality and type of the conducting coating when commercial Si probes are used. The fully conducting probes similar to the hand-made solid W probes used in first EFM and KFM experiments [19] might be useful as well. No doubt that the improvements of the overall performance of AFM instruments, which are related to lower electronic noise, better laser stability, quality of scanners and reduced thermal drift, will be extremely beneficial for local electric measurements. There are several applications that require novel developments of AFM-based electric techniques. It is well known that electric studies of semiconductors and materials employed in photovoltaic devices might be influenced by the laser beam inherent for optical deflection schemes employed in AFM instruments for monitoring the probe response. Some relieve might come when IR lasers are applied. The better solution can be achieved with the use of self-sensing probes (piezoelectric levers, tuning forks) that do not require the laser detection. The use of surface potential detection of material changes caused by adsorption of light was suggested earlier for high-resolution optical adsorption studies [112]. The practical implementation of such function might revolutionize the AFM by bringing chemical sensitivity to the nanometer-scale. Another expansion of AFM that can be realized in the immediate future is simultaneous measurements of topography, mechanical and electric properties by performing multifrequency measurements in the repulsive force regime as presented in [113]. These applications, which might be facilitated by using the special probes [17, 18], do need further efforts in finding optimal conditions of minimal cross-talk between various individual signals. Acknowledgements The samples examined in the studies, which are described in this Chapter, were kindly provided by our colleagues: Prof. M. Moeller (RWTU Aachen, Germany), Prof. N. Kotov (University of Michigan, Ann Arbor, USA), Prof. A. Gruverman (University of Nebraska, Lincoln, USA), Prof. M. Cota (University of Campinas, Campinas, Brazil) – to whom we are very thankful.

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Chapter 10

Quantitative Piezoresponse Force Microscopy: Calibrated Experiments, Analytical Theory and Finite Element Modeling Lili Tian, Vasudeva Rao Aravind, and Venkatraman Gopalan

We present quantitative experiments, analytical theory and finite element modeling (FEM) of vertical and lateral piezoresponse force microscopy (PFM) across a single antiparallel (180°) ferroelectric domain wall. There are three important aspects in making quantitative measurements. (1) Calibration and background subtraction of PFM displacements; (2) characterization of the tip shape and contact area; and (3) analytical theory and numerical simulations that incorporate all the relevant property tensors (dielectric, piezoelectric, and ferroelectric), tip shape, contact geometry, and the relevant physics of the feature being studied, such as the width of the wall. By calibrating the displacement of the tip, and using a reference sample, one can measure nanoscale piezoelectric coefficients, which are shown to be independent of tip size for a uniform sample. The shape of the contact area of a tip with the sample is characterized by field emission scanning electron microscopy (FE-SEM) to be disk-like. Only a true-contact with zero dielectric gap between the tip and the sample can explain the experimental PFM wall width versus tip radius measurements. Finally, in the limit of the tip disk-radius approaching zero, one can estimate the ferroelectric wall width from the vertical PFM profiles across the wall. The most complete analytical theory and finite element modeling to date are presented that can realistically simulate the PFM profile across a single wall. While vertical PFM signal agrees well with theory and simulations, the lateral PFM signal shows excellent qualitative agreement only. The experimental width of the lateral PFM signal across a wall is significantly wider than that predicted by FEM, suggesting elements of surface physics that are not captured in the current electromechanical theory of PFM.

V. Gopalan (*) Materials Science and Engineering, Pennsylvania State University, University Park, PA 16803, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_10, © Springer Science+Business Media, LLC 2010

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Ferroelectric Wall Width as a PFM Challenge Piezoelectric effect leads to an electrical polarization in a material under stress, called the direct effect, and a strain in the material under an electric field (converse effect). The latter is the basis for piezoelectric force microscopy, where applied alternating current (AC) voltages to a material using an atomic force microscopy (AFM) tip leads to distortions of the materials that pushes the tip up or down (vertical PFM signal) or twists it laterally (lateral PFM signal). Piezoresponse Force Microscopy (PFM) detects the surface displacements, Ui, related to piezoelectric strain e ij = dkijEk induced by applying an oscillating electric field Ek to the tip in contact with the sample surface [1, 2]. Clearly, the materials that can be probed this way should be piezoelectric, which from symmetry arguments [3] of Neumann’s law, require that the material be non-centrosymmetric (lack inversion symmetry). All ferroelectric materials possess a built-in spontaneous electrical polarization in their crystal structure in the absence of an external field; hence, they lack inversion symmetry and are all piezoelectric. PFM has thus become an extremely important and popular technique for probing ferroelectric domain structure. Since piezoelectric effect is a third rank polar tensor property, dijk, the spatial orientation of the electric polarization can be effectively probed by probing different tensor coefficients of the domains under the tip, which depends on the direction of the electric fields, and the type of displacement measured in relation to the unit cell coordinates of the material. The piezoelectric tensors for different point groups symmetries in nature are given in Ref. [3]. A single ferroelectric 180° wall is predominantly Ising-like (Fig. 10.1) and is a particularly ideal challenge for the PFM technique. The extremely small width of ferroelectric domain walls, typically of the order of 1–2 lattice units (~0.5–1nm) [4–6], has applications in data storage [7] and tunable optics [8]. It can also effectively test the resolution limit of the PFM technique. Conversely, piezoelectric tensor is a property that is directly related to ferroelectric polarization in the material. Since piezoelectric coefficients are related to the order parameter components as dijk = gijklPl, where Pl = Ps is the spontaneous polarization, and gijkl is the electrostric

Fig. 10.1 Schematic of Ising type of ferroelectric domain wall

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tive coefficient of the material, measurement of the piezoresponse across a wall is expected to provide direct information on the primary order parameter, Ps across a wall. (The fourth-order electrostrictive tensor, gijkl is not expected to change across the wall, since it is a property of the prototype paraelectric phase and is symmetric with respect to inversion symmetry across the wall.) Hence, PFM can be used as an effective probe to measure polarization profiles across a wall, or study its switching dynamics under external fields applied to the tip. Domain wall width also directly influences the dynamics of wall motion [9, 10]. The significance of probing domain wall width directly is illustrated by Fig. 10.2, which depicts the phase-field modeling calculations of the threshold field required for the motion of a domain wall by one unit cell as a function of the wall width in LiNbO3 [11–13]. One can see that even a wall broadening from 0.5 to 2 nm can lead to over two orders of magnitude decrease in the threshold field for wall motion. Thus determining the nanoscale structure of a wall is of fundamental interest to the field of ferroelectrics. To date, the primary means of investigating wall widths on unit cell level has been transmission electron microscopy (TEM)[14–18]. The original TEM studies of a ferroelectric 180° domain wall in a related material, lithium tantalate, concludes that wall width cannot be resolved down to their resolution limit of 0.28 nm [10]. Recently, an improved TEM technique has demonstrated that charged 180° walls in lead zirconate titanate thin films can be up to 4–5 nm [19]. In parallel, direct imaging of strain at these walls using synchrotron X-ray [20, 21], index contrast using nearfield scanning optical microscopy [22], and excitation emission spectroscopy

Fig. 10.2 Phase-field simulation of the threshold coercive field Eh for the motion of a single domain wall in LiNbO3 (LN) and LiTaO3 (LT) versus domain wall width 2wo. The thickness of the film in phase-field simulation was t = 96a, and the simulation size was 128a × 2a × 128a for 2wo < 2 nm and 512a × 2a × 128a for 2wo > 2 nm, where a = 0.515 nm. Inset shows the phase-field simulation of the polarization profile at the junction between the wall and one of the surfaces of the film. The bulk phase-field simulation was infinite in all dimensions. Analytical theory based on (10.1) in [11]. With permission from the AIP is also plotted for LiNbO3

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[23, 24] reveal property changes on length scales of 1–30 mm. So far the scale of 1nm–1 mm linking atomic structure of the wall and macroscopic properties of the ferroelectric has been less explored. Atomic force microscopy measurements of surface topography at twinned 90° walls have been used to derive wall widths of ~1.5 nm and determine defect effect on wall broadening [25, 26]. However, there is no intrinsic topography associated with 180° domain walls, necessitating detection of primary order parameter, Ps. Recent work using scanning nonlinear dielectric microscopy (SNDM) has shown domain wall widths of up to 200 nm [11, 27]. While SNDM has excellent resolution of better than 1nm based on published evidence [28, 29], a quantitative theory for this technique is not fully developed. Here we review primarily the experimental and finite element simulation work of Tian [11, 30, 31], Vasudevarao [32] and Scrymgeour [33, 34], and the analytical theory work by Morozovska et al [35], which to date, constitute a rigorous and complete quantitative theory (both analytical and FEM) of PFM imaging. PFM technique has been reviewed in many places, and has been extensively used to study ferroelectric domains [34, 36–39]. However, to date there has been very little experimental and theoretical investigation of the resolution limits of PFM [40, 41] in order to understand widely varying wall width studies [11, 42, 43]. Thus, the exact limits of domain wall width are still highly debated. The combination of experiments and theory can enable extraction of the polarization profile across a single ferroelectric domain wall. Since the goal is to extract information regarding the intrinsic width of an antiparallel domain wall, the first step is to quantitatively understand the PFM imaging technique, and its resolution limits. As mentioned in the abstract, there are three key aspects to achieving this goal. Calibration and background subtraction of PFM displacements (Section “Calibration, Background Subtraction, and Frequency Dispersion of PFM Displacements”), characterization of the tip shape and contact area (Section “Tip Shape and Contact Geometry”), and numerical simulations (Sections “Finite Element (FEM) Simulation: General Approach,” “FEM Simulation of the Potential and Electric Fields Under the PFM Tip,” and “FEM Simulation of the PFM Surface Displacements”) and analytical theory (Section “Analytical Theory of the PFM Response Across a Diffuse Domain Wall”) incorporate all the relevant property tensors (dielectric, piezoelectric, and ferroelectric), as well as tip shape, contact geometry, and the relevant physics of the materials being studied. By varying the effective radius of the probe, carefully characterizing the details of the tip shape and tip-sample contact region (Section “Calibration, Background Subtraction, and Frequency Dispersion of PFM Displacements,” “Tip Shape and Contact Geometry” and “Tip Size Dependence of PFM Amplitude and Profile”), and combining it with 3-dimensional Finite Element Modeling (Sections “Finite Element (FEM) Simulation: General Approach,” “FEM Simulation of the Potential and Electric Fields Under the PFM Tip” and “FEM Simulation of the PFM Surface Displacements”), and analytical theory (Section “Analytical Theory of the PFM Response Across a Diffuse Domain Wall”), the vertical PFM profiles can be quantitatively understood. Intrinsic or extrinsic wall broadening can be extracted by a careful comparison of

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experiments and theory (Section “Comparison Between Vertical PFM Experiments, Simulation, and Theory”). Finally lateral PFM theory and experiments are compared in Section “Lateral PFM: Experiments, Simulation, and Theory.”

Calibration, Background Subtraction, and Frequency Dispersion of PFM Displacements It is important to calibrate the displacement of the cantilever in picometers rather than arbitrary units. The displacement detection sensitivity of ~pm is enabled through the use homodyne detection using lock-in amplifier. This can be performed as follows: The vertical response of the AFM system was calibrated by using three standard gratings and then with a piezoelectric ceramic with known piezoelectric properties. The z-piezoresponse of the AFM system for these calibrated height should be in linear relationship with the z height. The slope of the calibrated curve gives the z-piezo response of the system in pm/V. The PFM amplitude of the piezoelectric cemamic used for calibration here was a poled PZT ceramic sample with uniform electrodes on both sides, whose piezoelectric coefficient in pm/V was also independently measured using a piezometer, which applies a stress and measures the open loop potential generated across the material. The surface displacement of this PZT sample in response to an applied voltage from a PFM tip is then used to calibrate the PFM signal. Next, it is important to eliminate the background, so that the pure PFM electromechanical signal is studied. The PFM signal is typically a complex displacement, U = UR+iUI = Uoeiq. The phase, q, refers to the relative phase of the tip displacement with respect to the phase of the alternating voltage applied to the PFM tip. The pure electromechanical signal U can be clearly distinguished in the complex plane from any background signal as described in literature and depicted in Fig. 10.3 [44].This background subtraction described in Ref. [44] is critical for quantitative analysis of the PFM profiles. Figure 10.4a, b depicts example images of the as collected signal of UR and UI, respectively, measured as x-channel and y-channel signals from a lockin amplifier, as a function of the wall normal coordinate x. The amplitude |Uz| and phase, q are shown in Fig. 10.4d, while the real and imaginary signals are plotted in Fig. 10.4c. Note that with appropriate background subtraction, the pure electromechanical displacement is entirely along the real part UR =Uocosq, while UI = Uosinq is zero (Fig. 10.4e). Since the displacement component normal to the surface (Uz) is measured in vertical PFM, UR~±Uo = ±Uz, and the phase q is 0 or p (Fig. 10.4f). Such experimental wall profiles should be measured for a series of tip radii, which will be discussed further in relation to theory and simulation in Section “Comparison Between Vertical PFM Experiments, Simulation, and Theory.” The frequency dispersion of the PFM signal is important to characterize. Figure 10.5 shows the typical the frequency dependence of PFM signal. In this case, the PFM signal does not show any frequency dispersion between 20 and 100KHz, which is the range where predictable measurements can be made[30].

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Fig. 10.3 The schematic diagram showing the measured PFM signal on both +c and –c surface of the ferroelectric material away from the resonance frequency of the probe. The measured signal includes the pure piezoelectric response signal, which is denoted as “electromechanical” signal, and a background signal, which is due to electrostatic response and non-local nonlinear response acting on the cantilever. PR+ and PR- represent the measured PFM signals on +Ps and –Ps domains, respectively

Tip Shape and Contact Geometry Since PFM is a contact mode technique, abrasion of the tip easily occurs during imaging [45, 46] which changes the tip shape and the field distribution under the tip. The PFM literature typically approximates the tip to be an ideal sphere or disc with a radius r. The contact of the tip to the sample is either considered to be an ideal point contact, or more commonly, a dielectric gap of the order of 0.1–1nm is assumed between the tip and the sample. Exact analytical expressions for the field distribution around such tip shapes are well-known [47, 48]. However, Tian et al. have recently shown [31] that these assumptions of tip shape and tip-sample contact region are limited. They imaged the end of each tip using Field Emission Scanning Electron Microscope (FESEM), after scanning a few PFM line scans across a domain wall. As seen in Fig. 10.6a, tips look more disk-like, in that its end is flat with a circular contact area of radius r. Even tips that look spherical (Fig. 10.6b) have a finite contact region as described by Tian et al. [31]. This is an important insight in quantitative interpretation of the experimental data as shown further below.

Tip Size Dependence of PFM Amplitude and Profile We focus the discussion here on antiparallel domain walls in congruent lithium niobate. The point group symmetry of LiNbO3 is 3m and the polarization is along +z direction (+Pz) or –z direction (–Pz). The walls are typically parallel to the crystallographic y-z mirror planes. Hence the wall coordinates are defined

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Fig. 10.4 An example of complex PFM displacement, ŨR=Uo cos ~ UZ as a function of wall normal coordinate, x across a 180° domain wall in lithium niobate. (a) UR=Uo cos (q) ~ UZ; (b) UI=Uo Sin(q); (c) line scan of real part, UR (x-channel of the lockin amplifier, Uo is labeled as A), and imaginary part, UI (y-channel of the lock-in amplifier, Uo is labeled as A) signal of the as-measured piezoresponse across the domain wall. (d) line scan of the amplitude |UZ|, and phase, of the piezoresponse across the domain wall after background subtraction; (e) the real part, UR, of the pure electromechanical response across the wall after the “background” subtraction; (f) the phase, of the pure electromechanical response across the wall after the “background” subtraction with permission from the american institute of physics [31]

as x perpendicular to the wall, y along the wall, and z- along the polarization direction. The vertical PFM (bending mode) detects the displacement of sample surface perpendicular to the sample surface. For lithium niobate (point group 3m), the piezoelectric tensor for lithium niobate has four independent nonzero coefficients (d31, d33, d22, and d15). Experimentally, there are two important experimental parameters that are extracted from the PFM p rofiles across a wall (see Fig. 10.4e): amplitude, |Uz|, and width, wPFM. The calibrated amplitude of

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Fig. 10.5 The piezoresponse across the signal 180° wall in lithium niobate single crystal. (a) The measured x and y channel lockin amplifier signal at selective frequencies plotted in a complex XY plane. Solid line was the fitting, whereas the scattered dots are experiment data; (b) The maximum amplitude of the electromechanical contribution to the PFM signal at different frequencies after background subtraction

Fig. 10.6 (a) FESEM image of a typical used PFM tip with a circular disk-like end with a radius of r; (b) FESEM image of a used sphere-like PFM tip. The radius r of the contact circle for a weak indentation (h~1–2 nm or 1 unit cell depth) is used to characterize the radius r of the tip as shown (h is not to scale in the figure) with permission from the american institute of physics [31]

the PFM response away from the wall in units of deff (pm/V) is plotted as a function of the experimentally determined tip radius r, in Fig. 10.7. An important conclusion seen from Fig. 10.7 is that the deff is independent of the tip radius used. Remarkably, thus calibrated magnitude of the d eff in pm/V as well as its invariance with respect to tip radius also agrees with both the analytical theory [41, 49] and numerical simulations described next in Section “Comparison between vertical PFM experiments, simulation, and theory”. Figure 10.8 shows the width 2wPFM of the PFM response across a single 180° domain wall as a function of the experimentally determined tip radius r. The wPFM refers to the half width where the PFM response reaches ±0.76 of the saturation

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Fig. 10.7 The measured d33, eff along with the FEM prediction and analytical theory results as function of tip radius, r, is shown.. Also shown overlapped is the maximum amplitude, Uz, away from the domain wall in lithium niobate as a function of tip radius, r is shown with permission from the american institute of physics [31]

Fig. 10.8 PFM wall width as a function of tip radius for sphere-like (tip set 1) and disk-like (tip set 2) PFM tips. Also shown are analytical theory predictions generated from (10.4) and (10.5) with permission from the american institute of physics [31]

PFM value away from the center of the wall. Note the tanh(1) ~0.76, which is motivated from the fact that phenomenological Ginzburg–Landau–Devonshire theory of polarization variation across the wall follows the expression P/Ps tanh(x/w0 ). This saturation value was taken by Tian et al. as the PFM value at ~±1.8 mm from the center of the wall. The PFM width decreases linearly with the tip radius, except at the smallest tip radii, where deviations from linearity are observed. The relation-

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ship between these deviations and the intrinsic wall width is discussed in Section “Comparison Between Vertical PFM Experiments, Simulation, and Theory.” Two modeling approaches can be employed to analyze this data, namely, analytical theory and Finite Element Modeling (FEM) as described below.

Finite Element Simulation: General Approach Finite Element modeling (FEM) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to vibration systems [53]. FEM consists of a computer model of a material or design that is stressed and analyzed for specific results. There are generally two types of analysis that are used: 2-D modeling and 3-D modeling. In the model, FEM uses a complex system of points called nodes, which makes a grid called a mesh, which is then programmed to contain the material and structural properties that define how the structure will react to certain loading conditions. In order to understand the PFM response quantitatively, Tian et al. performed Finite Element Method (FEM) modeling of the imaging process using the commercial ANSYS program. The FEM approach described in Ref. [30] and Ref. [31] in detail, includes a complete description of the geometry of the tip, the sample, and the contact region, numerical calculation of the electric field distribution with a constant potential applied to the tip, and using the computed potential distribution on the sample surface as boundary condition to calculate the piezoelectric deformation of the surface. Input to the FEM includes the complete dielectric tensor, elastic tensor and piezoelectric tensor of the sample, thus providing a rigorous 3-D approach. A single domain wall, parallel to one of the three degenerate y-z physical crystal planes was defined in the simulation by flipping the crystal physics axes, y and z across the wall. To find out the surface deformation induced by the PFM probe due to pure converse piezoelectric effect, one needs to solve the coupled equations,

s ij = Cijkl e kl + dijk Ek

(10.1)

Dl = d lij e ij + klk Ek

(10.2)

where E and D are electric field and electrical displacement, respectively, C and d are elastic stiffness tensor and piezoelectric stress tensor, respectively, and s and e are stress and strain tensors, respectively. As suggested by these equations, both the piezoelectric stress and the dielectric displacement depend on both strain and field. Scrymgeour and Gopalan [34] and Tian and Gopalan [31] have solved these equations in a decoupled way. First, the electric field distribution inside the ferroelectric sample was calculated purely by treating the material as a dielectric; then

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the deformation of the sample was calculated under the field calculated in the first step. This method is justified for the rigid dielectric system [50, 54]. Each of these steps is described below.

FEM Simulation of the Potential and Electric Fields Under the PFM Tip Two different models for the tip-sample surface interaction were explored by Tian et al. [31]: (a) the sphere-plane model, and (b) the disc-plane model, where the sphere or the disc refers to the tip shape and the plane refers to the sample surface. In both models, there are three electrostatic boundary conditions to satisfy: (1) an equipotential AFM tip surface equals the electrical potential, V, applied to the tip; (2) the tangential component of electric field E is continuous across the interface between the dielectric medium (air) and the dielectric specimen; and (3) the normal component of electric displacement D is continuous across the interface between the dielectric medium and dielectric specimen. The conical part the AFM tip contribution in a spherical tip can be modeled using the line charge model developed by Huang et al. [50]. The conical part of AFM tip contribution in a disc tip was modeled using ANSYS. Sphere-plane model: The field distributions inside the dielectric (for instance, radial dielectric constant, kr = 85.2 and depth dielectric constant, kz = 28.7 for congruent lithium niobate) under a spherical tip with radius of 50 nm in contact (d = 0 nm) are plotted in Fig. 10.9, in which the tip is located at (0, 0, 0) on top of the sample surface. The contribution of the conical part of a spherical tip can be modeled using the line charge model developed by Hao et al. [59]. As can be seen, the maximum potential and maximum electric fields inside the dielectric are

Fig. 10.9 Simplified sphere model of an AFM tip over an anisotropic material in PFM system

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located at the very point that is right under the tip apex with coordinates (0, 0, 0). Therefore, the resultant electrical fields inside the dielectric sample are extremely high (Er = 7.53 × 108 V/m and Ez = 4.35 × 109 V/m under a tip with a radius of 50 nm and 5V on). These field values are much higher than the coercive fields of lithium niobate. However, no domain switching in a ferroelectric that is in contact with the AFM tip with such a low applied voltage has ever been reported. The reason for not observing domain reversal with such low voltages may be that the volume of the sample over which the field is above the coercive field (~20 nm in radius) may be smaller than the critical volume for a stable nucleation of a domain under the tip, which was theoretically predicted to be around ~20 nm according to [51]. The second assumption in the sphere–plane model is that there is no extra charge other than the induced charge on the dielectric surface. However, for lithium niobate, lithium tantalate and other ferroelectric materials, there is always an argument about under compensated or over compensated ferroelectric surfaces [57, 58], which may modify the field distribution and lower the effective fields under the tip. Disc-plane model: In reality, the AFM coated tip can be blunted soon. Normally, after a few lines of scan, the very end of the AFM tip can be blunted into a flat plane (Fig. 10.6). In this case, the tip is in contact with the ferroelectric surface over a finite area. Also, in the case that the tip indents into the ferroelectric surface, the tip is also in contact with the ferroelectric over a finite area. In both these cases, the tip can be simplified as a metal disc with a radius R (Fig. 10.10a), which is in contact with the sample. In this model, the potential distribution inside the specimen can only be solved numerically by taking care of the three boundary conditions mentioned earlier in this section. For a 50-nm radius flat tip with applied 5V voltage on, the potential field distribution for lithium niobate is plotted in Fig. 10.10b. The conical part of AFM tip contribution in a disc tip was modeled using ANSYS. The conical part was simplified by Tian et al. [31] as a 20-µm long metallic coated silicon with a full cone angle of 30°.

Fig. 10.10 (a) Schematic of the simplified disc model; (b) The potential distribution on the z = 0 nm plane inside z-cut lithium niobate under a flat 50-nm radius AFM tip with 5 V applied voltage on

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FEM Simulation of the PFM Surface Displacements By applying the potentials for spherical or disk tips, as described above, as the boundary condition for the piezoelectric coupling simulation using ANSYS, the deformation of the ferroelectric resulting from the AFM tip can be simulated. Piezoelectric response of a single domain lithium niobate in true contact with the tip was simulated with finer meshing of the sample surface below the tip and coarser meshing away from the tip. Figure 10.11 shows the surface deformation of lithium niobate under a 50-nm radius spherical tip and a 50-nm radius disc tip with 5V applied. The domain wall is a y-wall (mirror plane in lithium niobate structure) located at x=0. The AFM tip is located at (0, 0, 0), right on the domain wall. As we move the tip location across the domain wall, the piezoelectric response across the domain wall can be obtained. One can study this response as a function of the distance between the tip and the specimen surface, as well as the tip radius. Figure 10.12a, b show the examples of the piezoelectric response predicted by FEM across the single 180° domain wall in lithium niobate for a 50-nm radius spherical tip and a 50-nm radius disc-type tip with 5V applied, respectively. The distance between the tip and the specimen surface is 0 nm (no gap). The surface in left region (x < 0) under the tip is +z surface and the surface in right region (x > 0) under the tip is –z surface. The domain wall is located at x = 0. As can be seen, for both the sphere and disc-type of tip, the displacement Uz changes from being negative on +z surface (left region) to being positive on –z surface (right region), which means that +z surface contracts and –z surface expands under the 5V applied. Hence, the +z surface is outof-phase (180°) to the driving voltage in PFM and –z surface is in-phase (0°) to the driving voltage in PFM system. The amplitude of Uz reaches minimum while the tip is sitting right on top of the domain wall. The amplitude of the displacement Ux and Uy are symmetrical across the domain wall, however, the sign of Uy changes across the domain wall. It is worthy pointed out that even though the tip shapes are different for sphere and disc tips, the shapes of the induced surface deformation in the ferroelectric, and thus, the displacements Ux, Uy,and Uz, are similar. The amplitudes of the Uz are the same for both sphere tip and disc tip. Due to the surface deformation, in vertical PFM, the tip follows the out-of-plane surface deformation component (displacement Uz). Therefore, the amplitude of displacement Uz across the domain wall gives the vertical PFM signal (Fig. 10.12c, d). The 3D displacements (Ux, Uy, and Uz) also cause the tip not only experience the out-of-plane displacement, but also the in-plane displacement, which gives the lateral PFM signal. While the tip scans, the AFM tip experiences the slope of the surface normal to tip beam. The deformed surface oscillates at the same frequency as that of the driving voltage applied on the tip. By processing the FEM, the slope of the deformation surface normal to tip beam arm can be obtained, which is proportional to the 0°-lateral PFM signal (Fig. 10.12e, f ). The slope of the deformed surface depends on both tip geometry and size.

Fig. 10.11 The FEM simulation of the deformation Ux, Uy, and Uz on z = 0 nm plane of a uniform lithium niobate under a 50-nm radius AFM tip with 5 V applied voltage on. (a–c) for a sphere tip; (d–f ) for a disc tip

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Fig. 10.12 The FEM results of the piezoelectric vertical and lateral response across the domain wall (located at 0) in lithium niobate for a tip with radius of 50 nm, which is in contact (d = 0 nm) with lithium niobate. The driving voltage on the tip is 5 V. The spontaneous polarization in the in the negative tip location coordinates points up, which means +z surface, whereas the spontaneous polarization in the positive tip location coordinate points down and the surface under the tip is –z surface. (a) displacement with a sphere tip; (b) displacement with a disc tip

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Fig. 10.13 (a) FEM simulated surface potential on LiNbO3 surface under a 50-nm radius disk tip in contact with sample with 5 V applied. (b) FEM simulated piezoelectric displacements, Uz of a LiNbO3 z-surface. Displacements for three different tip locations are shown: tip located on the wall (location 2) and away from the wall on either side (locations 1 and 3) with permission from the american institute of physics [31]

Figure 10.13 depicts the surface potential (a), and the corresponding surface deformations (b) simulated by FEM for three different locations of the tip across a single step-like 180° domain wall. (Three separate simulations for the three tip positions have been merged in this plot.) Using a series of positions of tip across the wall, continuous FEM line profiles of Uz were generated. All simulations with FEM were performed only with step-like wall, and did not include any intrinsic broadening or diffuseness, since it was not numerically feasible in the software used.

Analytical Theory of the PFM Response Across a Diffuse Domain Wall Mozorovska and Eliseev [52] have developed an analytical theory for the displacement and piezoelectric response across a single domain wall. This theory has been applied to LiNbO3. The surface displacement vector is given by the convolution of piezoelectric tensor coefficients d klj (x ) with the resolution function components Wijkl (x, y ) as proposed in Ref. [41], where x is the wall normal. For most inorganic ferroelectrics, the elastic properties are weakly dependent on orientation and hence the material was approximated as elastically isotropic. Using decoupling approximation, and resolution function approach [40] for transversally isotropic media [41], the vertical piezoelectric response of isolated 180°-domain wall in an inhomogeneous electric field of the probe tip is derived [53]. The surface displacement of a step-like infinitely thin domain wall is derived in Ref. [41]. Below we list the final close-form expression:   x (1 + n ) f313 x f333  x f333 d33 x f351 d15  U zstep (x ) = V  d31  − + + ,  x + C333 zo  x + C333 zo x + C351 zo    x + C313 zo

(10.3)

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 (1 + ν) zo C133 f113 zo C133 f133  z C f d zC f d − d 31 + o 133 133 33 + o 151 151 15 . (10.4)   x + zo C113 x + zo C133  x + zo C133 x + zo C151

U xstep (x ) = V 

Here dlm ≡ dlkjbulk = dljkbulk in Voigt notation, V is the applied voltage to the tip, and n eff is the Poisson ratio. Note that U zstep (x ) = V d 33 (x ) , U ystep (x ) ≡ 0 , and U xstep (x ) = V d35eff (x ) at the distance x normal to a flat domain wall. Characteristic distance zo is determined by the parameters of the tip. In the effective point charge model it is the charge–surface separation. If we approximate the tip by the metallic disk of radius r in contact with surface, then Z0 = 2r/p. The expressions for material anisotropy constants fijk and Cijk are given in [31]. Using (10.4), one can derive a simple approximation for the effective width, wPFM of infinitely thin domain wall (measured as distance between the points where the response is equal to ± (1 − η) fraction of saturation polarization:

w PFM ≈ 2 zo

(

) )

1 − η (1 + ν) f313 C313 − f333 C333 d 31 + f333 C333 d 33 + f351C351 d15 η

(

(1 + ν) f313 − f333 d 31 + f333 d 33 + f351 d15

(10.5)

The authors neglected the contribution of d 22 and related terms, since their contribution far from the wall is exactly zero in the framework of the decoupling approximation model [54]. To estimate the cone effects in PFM imaging, the conical part was modeled by a line charge [50], and the contact area by a disk touching the sample surface, as proposed elsewhere [50]. Analytical theory predictions of the vertical PFM response near the single domain wall in LiNbO3 are shown in Fig. 10.14. The influence of the tip radius itself on the wall profile is shown in Fig. 10.14a for a step-like wall (diffuseness wo = 0), clearly indicating the PFM wall width increase with radius r. The influence of domain wall

Fig. 10.14 (a) Normalized theoretical PFM response profile near a step-like 180° domain wall (wo = 0) in LiNbO3 for disk radius r = 17, 25, 36, 50, 70, 88, 110, 150, and 200 nm (curves 1–9, respectively), a cone with length of L = 20 mm and half cone angle of q = 15° for the PFM tip and an intrinsic half width wo = 20 nm. (b)Theoretical vertical PFM response near the single domain wall in LiNbO3 as a function of the distance from the diffused wall with different half width values wo = 3, 30, 60, and 90 nm (curves 1, 2, 3, and 4, respectively); r = 30 nm, L = 20 mm, and q = 15°. bulk The intrinsic wall diffuseness d lkj ( x ) = d lkj tanh( x / ωo ) is schematically shown in the inset with permission from the american institute of physics [31]

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diffuseness is shown in Fig. 10.14b, where the piezoelectric coefficient profile is dlkj ( x) = dlkjbulk tanh (x / xω 0 − ω 0 )(i.e. the intrinsic profile b ( x) = tanh( x / w 0 )). This is chosen to mimic the polarization variation, P3 across a 180°-domain wall, given by P3 ( x ) = P3bulk tanh (x / ω 0 ), since the piezoelectric coefficients and the polarization are linearly related by the electrostriction tensor. As expected, domain wall diffuseness broadens the PFM wall profile for a give tip radius r. The combination of both a change in tip radius, r and a change in the wall diffuseness wo was previously shown in a series of theory plots in Fig. 10.8, along with experimental data points. The PFM wall width increases linearly with the tip radius r for a step-like wall. Domain wall diffuseness, wo adds significant nonlinearity to these curves for approximately tip radii r < wo. The general quantitative agreement between the theory and experiments is excellent. It also suggests that there may be significant domain wall diffuseness in LiNbO3 crystals as discussed next.

Comparison Between Vertical PFM Experiments, Simulation, and Theory Figure 10.7 (shown earlier) plots the maximum effective piezoelectric coefficient, deff = Uz/V measured and simulated away from the wall, where V is the maximum voltage on the sample surface. The experiments, analytical theory and FEM simulations show excellent quantitative agreement with each other. Within error bars, both experimental and simulated values of the maximum displacement Uz and the eff corresponding d33 are relatively insensitive to the tip radius, and the presence of the cone. Figure 10.15 shows the quantitative comparison of the vertical PFM profiles in LiNbO3 and LiTaO3 with FEM simulations. Note that the FEM analysis predicts a ratio of PFM amplitude of congruent LiTaO3 to LiNbO3 to be 0.67, which matches very well with experiments as well. Figure 10.8 shows a comparison of the wPFM as a function of tip radius r for various tip models. It is clear from the comparisons that spherical tips with a point contact do not agree quantitatively with experimental results. The predicted PFM widths in this case are much smaller than those experimentally observed. Introducing an imaginary dielectric layer gap (air) of 2 nm (which is large) between the spherical tip and the surface increases the predicted wPFM, but still is considerably less than the experimentally measured widths. Only the disk-type tip model, including the conical section shows the best agreement with the experiments. These results rule out the spherical tip models and the possibility of dielectric gaps. True contact with a disk type tip and no dielectric gap therefore reflects the true nature of the PFM imaging. Finally, we address the question of the resolution of the PFM technique. Finite Element Method and analytical theory show that for a point contact of the tip, and in the absence of the conical part of the tip, the PFM wall width tends to zero. In other words, theoretically infinite resolution is possible. However, this is not practical due to the presence of the cone. With the cone part of the tip (20-mm long metallic coated

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Fig. 10.15 Comparison of the the piezoresponse in lithium tantalate between the FEM modeling and PFM measurement. The tip radius size is ~50 nm. Congruent periodically poled lithium niobate was used as the reference. In FEM, the ratio between the vertical piezoresponse across the single wall in congruent lithium tantalate and congruent lithium niobate is ~0.73 ± 0.07, whereas in the experiment, it was determined to be ~0.67 ± 0.02

silicon with a full cone angle of 30°), the linear extrapolation of the FEM predicted PFM width to zero tip contact radius is ~11±10 nm, which is statistically zero width. However, practical consideration of a finite tip contact radius, and abrasion of the tip while in contact with the sample results in experimentally measured PFM widths on the order of ~100 nm. In contact mode PFM geometry, the experimental tip radii below ~15–20 nm realistically is not possible. Nonetheless, one can extrapolate the PFM width versus tip disk-radius measurements down to zero tip radius. Experimental results in Fig. 10.8 show that such an extrapolation of tip radius to zero still would reflect a significant scatter in PFM widths up to 100 nm. Does this reflect information regarding the intrinsic ferroelectric wall width at the surface? The domain wall tilts cannot account for full range of scatter of PFM widths (at zero tip contact radius) over 10 nm. One can incorporate an intrinsic wall diffuseness, 2wo, in the analytical theory in terms of piezoelectric tensor distribution dlkj ( x) = dlkjbulk tanh (x / ω 0 ). Figure 10.8 shows the theoretical PFM width predictions for different values of intrinsic domain wall diffuseness. The PFM wall width versus tip contact radius becomes nonlinear for small tip radius, and reaches a saturation value equal to the intrinsic wall diffuseness in the limit the tip radius equals zero. The scatter of experimental data points fall below the wo ~100 nm theory curve, suggesting a range of intrinsic wall diffuseness at the surface of congruent LiNbO3. Figure 10.16a shows that for large tip radius (r = 200 nm), the match between experiments, FEM and analytical theory is excellent, and wo has little influence on the wall profile. However, as the tip

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Fig. 10.16 Experimental PFM profile, Uz (open circle) as a function of wall normal coordinate, x across a 180° domain wall in lithium niobate. Measurements were made with a Ti/Pt coated Si tip with a disk shaped tip actual tip of (a). r = 36 nm and (b) r = 200 nm, determined after PFM scan by SEM imaging. An oscillating voltage of 5 Vrms, at 42.35 kHz was applied to the tip. Theoretical PFM profiles using FEM (solid squares, wo = 0 nm), and analytical theory (solid lines) are also shown with permission from the american institute of physics [31]

radius r approaches wo, the simulation of a sharp (step-like) domain wall does not faithfully match the experimental line profile. Figure 10.16b shows that an excellent fit is obtained only when 2w = 60 nm for that particular line profile. This analysis, when repeated for all the data points in Fig. 10.8, results in 2wo varying from 20 to 200 nm. Independent measurements of domain wall width in congruent LiNbO3 using scanning nonlinear dielectric microscopy (SNDM) technique also show such widths [27, 55]. Cho et al. have recently demonstrated imaging of the Si(111)7×7 surface [28, 29] atomic structure using similar second and third order capacitance

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Fig. 10.17 SNDM images of a circular domain in a 40-nm thick z-cut single crystal lithium niobate [(a) and (b)] and a 31-nm thick z-cut single crystal lithium tantalate [(c) and (d)] at first harmonic, wp = 6 kHz [(a) and (c)] and second harmonic, 2wp = 12 kHz [(b) and (d)] modulation frequencies. Figures 10.1d and 10.1e show typical line profiles from these images as labeled, and pairs of arrows indicate the wall region. Images (a)–(d) are plotted in three-dimensional orthographic view with a ~21° rotation about the horizontal axes of the images. The length scales directly correspond to the horizontal axes of the images. The same domain region is imaged in (a) and (b), and similarly in (c) and (d) Need copyright permission; will get it now [11]

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terms in SNDM showing <0.5 nm resolution of SNDM. Surface domain wall widths of 20–150 nm have been measured in LiNbO3 and LiTaO3 crystals by this technique [27, 56]. Figure 10.17 [11] reveals that the larger wall widths arise when large polar and dielectric defects exist adjacent to the wall. Thus, it is quite likely that dielectric/polar defects and surfaces can broaden antiparallel ferroelectric domain walls, and this broadening, on the scale of tens of nanometers is being detected in these PFM studies as broadening of the polarization profile. A caveat in the above study is the influence of moisture on the tip radius. It has been shown that high level of humidity can form a meniscus around the tip and effectively broaden the tip [56]. Thus, the above studies should ideally be performed in vacuum and/or at higher temperatures, to confirm the wall width assertions unequivocally. Nonetheless, the above studies highlight the ultimate capabilities of vertical mode PFM in terms of imaging, quantitative measurement and analysis of piezoelectricity on the nanoscale, and limits of resolution.

Lateral PFM: Experiments, Simulation, and Theory The tip rotation (a twist) around an axis parallel to the cantilever axis or a rotation about an axis perpendicular to the cantilever axis and parallel to the sample surface (buckling) in PFM gives the lateral PFM signal. This mode, though a powerful imaging modality, is not fully evaluated quantitatively. Below, we compare experiments and FEM simulations to highlight what can be understood and what not. The “buckling” of the tip cannot be clearly distinguished from vertical PFM signal, hence we focus on only the “twist” mode of the lateral PFM signal. There are two “twist” type scans one can make across a wall: either twisting in a plane perpendicular to the wall (called 0° scan) or a twist in the planes parallel to the wall (90° scan) (Fig. 10.18). The 90° scan signal is very weak in LiNbO3 and hard to

Fig. 10.18 The importance of symmetry in lateral images in LiNbO3. (a) The domain structure relative to the x-y crystallographic axes. The circled area is expanded in (b–e). Cantilever parallel to domain wall is shown in top view (b) and side view (d). Scanning is in the horizontal direction shown by arrows. Cantilever perpendicular to domain wall is shown in top view (c) and side view (e) scanning in vertical direction shown by arrows. Loops indicate torsion on cantilever

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probe. Nonetheless, Scrymgeour has imaged in this mode and modeled the response [34]. Below we focus on the 0° lateral scans. Figure 10.19 shows lateral PFM scans on both congruent lithium tantalate and lithium niobate. To compare the signals on lithium niobate and lithium tantalate, same tip, same applied voltage on the tip, same frequency, and same time constant as well as same sensitivity of the lock-in amplifier has been employed. As can be seen, the lateral signal reaches its maximum right at the domain wall and almost zero when it is far away from the domain wall region in both congruent lithium niobate and congruent lithium tantalate. The ratio (Rlateral(CLT/CLN)) between lateral response of congruent lithium niobate and that of congruent lithium tantalate is around ~0.68±0.01, which is very close to the FEM prediction of ~0.70. The lateral PFM comes from the deformation slope of the sample surface. Unlike the vertical PFM signal where the amplitude of the signal does not depend on the tip size and tip geometry, in the lateral PFM signal, the slope of the surface does depend on the tip size, tip geometry as well as the contact condition. Figure 10.20a, b show the relationship between the slope of the deformation in congruent lithium niobate under different tip radii and geometry. It shows that if the

Fig. 10.19 0° lateral PFM measurement across the single domain wall in (a). congruent lithium niobate (CLN); (b). congruent lithium tantalate (CLT); (c) line scan of normalized 0° lateral amplitude signals of both CLN in (a) and CLT in (b); (d). 0° lateral amplitude signals of both CLN and CLT by FEM simulation. The ratio between congruent lithium tantalate (CLT) and congruent lithium niobate (CLN) in experiment is ~0.68 ± 0.01; whereas it is ~0.70 ± 0.05 in FEM

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Fig. 10.20 FEM result of the amplitude (slope) of the surface deformation normal to the domain wall vs. the tip size and tip geometry. (a) the disc tip without the cone, disc tip with cone and sphere tip with cone; (b) disc tip with cone and disc tip without cone. Solid line are the fitting using second exponential decay function y = y0 + A1exp(– x/t1) + A2exp(– x/t2). (c) FEM results of the slope of surface for both congruent lithium niobate (CLN) and congruent lithium tantalate (CLT) for disk shape tip in contact. Rslope(CLT/CLN) is the ratio between slopes of FEM surfaces in CLT and CLN, which is ~0.70 ± 0.05 and independent of the tip size

contact area is larger, for instance in a disc type tip, the amplitude of the slope is much lower as compared to a spherical tip with the same radius. The cone part of the tip does not change the amplitude significantly. The main contribution arises from the tip-sample contact area. As the tip radius exceeds 50nm, the slope of the surface deformation quickly decreases and become insensitive to both spherical tip and a disk-type tip. However, the ratio (Rslope(CLT/CLN)) between the surface slope in congruent lithium niobate and the slope in congruent lithium tantalate is independent of the tip size and tip geometry (Fig. 10.20c), and found to be ~0.70±0.05, which is in excellent agreement with the measured result (~0.67±0.02) (Fig. 10.19). Figure 10.21 shows the comparison between the domain interaction width in FEM simulation and the experimental measurement. The qualitative agreement between the measured and simulated curves is excellent. The relative magnitudes of the maximum slope of lateral PFM signal for LiNbO3 and LiTaO3 also agrees

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Fig. 10.21 (a) 0° lateral amplitude signal along with FEM simulation result. The ratio between congruent lithium tantalate (CLT) and congruent lithium niobate (CLN) in experiment is ~0.68 ± 0.01; whereas it is ~0.70 ± 0.05 in FEM (b). (b) Comparison of slope of FEM surface in CLN and CLT. The tip radius is ~55 nm, which was determined by FESEM image

well with the FEM simulations. However, it can be seen that the full domain wall width (FW ~95±5nm) in FEM is ~10 times smaller than the experimental measurement (FW~800±10nm). Unlike the vertical PFM signal, where such quantitative agreement was excellent, the discrepancy in the width of the lateral PFM signal between experiments and FEM simulations suggests that there are additional broadening mechanisms that may be related to surface physics and surface relaxation effects. Jungk et al. [57] suggest that this is due to surface charges. However, no measurable surface potential was observed on these samples by Scrymgeour et al. [34]. Despite these excellent qualitative agreements in lateral signal presented here, that lead to a surface tilt on the order of ~0.2-0.5 degrees (Figure 10.20a,b), J. Guyonnet et. al. [59, 60] however argue that the lateral force predicted by such a tilt would be small as compared with experimental measurements, and hence they rule this mechanism out as the reason for lateral PFM signal. Instead they conclude that a lateral shear motion of the surface under an electric field is responsible for the lateral PFM signal. If so, one possibility is that the PFM signal arising from lateral shear might have the same qualitative features as those arising from the surface tilts explored here. This issue is thus still open.

Conclusions In conclusion, we have shown that piezoresponse force microscopy (PFM) can be a quantitative tool for probing piezoelectric materials, and particularly ferroelectric domains and domain walls. A PFM tip with finite contact area and in true contact with the sample surface gives the best agreement between PFM experiments, analytical theory, and finite element modeling of the PFM response across a domain wall. The PFM amplitude is independent of the tip radius. The PFM width across a sharp domain wall is linear with tip contact radius, and is predicted by theory to

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provide infinite resolution for a point contact. However, practically, the presence of conical part of the tip, any slight tilts of the wall with respect to surface normal, and importantly, the abrasion of the tip on contact with the sample leads to finite resolution on the order of ~10–20nm. Theory predicts that walls with finite intrinsic diffuseness will lead to a nonlinear relationship between PFM width and tip contact radius, particularly for small tip radii on the scale of wall diffuseness. Using a combination of PFM experimental line profiles across 180° domain walls, analytical theory, and FEM simulations, we conclude that real domain walls on the z-surface of lithium niobate show broadening on the scale of 2wo~ 20–200nm, with considerable scatter from location to location. The scatter arises from the surface influence, as well as the presence of dielectric and polar defects adjacent to walls, which have been imaged by SNDM technique and reported in literature in these materials [56, 58]. The PFM results show that these defect-domain wall interactions lead to an effective broadening of the walls in terms of polarization. The lateral PFM signal, interpreted as the slope of the surface, agrees qualitatively between experiments and FEM simulations. However, the experimental width of the lateral PFM signal across a domain wall is 10 times that simulated by FEM, suggesting additional surface physics not captured by the electromechanical problem. Acknowledgments The authors wish to gratefully acknowledge financial support from the National Science Foundation grant numbers DMR-0820404, 0602986, 0908718, and 0512165. Research was also sponsored in part by the Center for Nanophase Materials Sciences, Office of Basic Energy Sciences, U.S. Department of Energy with Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC.

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Chapter 11

High-Speed Piezo Force Microscopy: Novel Observations of Ferroelectric Domain Poling, Nucleation, and Growth Bryan D. Huey and Ramesh Nath

Introduction The application of ferroelectric materials into oxide-based electronic devices has led to tremendous interest in understanding local ferroelectric properties at the micron and submicron scale. A wide spectrum of noncontact, semicontact, and contact mode scanning probe techniques have been successfully implemented in the imaging, characterizing, and manipulation of nanoscale ferroelectric structures [1–3]. Among the most commonly used are electrostatic force microscopy (EFM), scanning surface potential microscopy (SSPM), and piezoresponse force microscopy (PFM). The imaging mechanism in noncontact EFM and SSPM is related to the total charge distribution on the ferroelectric surface, including both polarization and screening charges. If measurements are carried out in ambient conditions, the emanating electric field can be minimized or screened by surface electronic states or adsorption from the atmosphere [4–6]. This creates challenges for quantitatively imaging as surface charge screening is a function of humidity, temperature, exposure time, etc. In contrast, the PFM image mechanism stems from the electromechanical response from the sample surface induced by an external field. The images obtained can be directly correlated to the polarization through the linear coupling between the piezoelectric and ferroelectric parameters. Similarly, PFM is also sensitive to the surface charge via capacitive interactions between the biased tip/cantilever and sample [7]. However, with careful calibration measurements, the capacitive interactions can be systematically removed from the PFM signal, therefore providing reliable quantitative imaging [8, 9]. Most importantly, PFM has the versatility to be operated in several modes to probe and modify domain structures. The static imaging mode is used to map polarization magnitudes and orientations in two- or threedimensions [10–12]. A spectroscopy mode obtains local hysteresis measurements to B.D. Huey (*) Materials Science and Engineering Program and Institute of Materials Science, University of Connecticut, Storrs, CT 06269, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_11, © Springer Science+Business Media, LLC 2010

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map the variation in properties due to inhomogeneities, defects, or grain boundaries [13, 14]. A lithographic mode is often used to selectively pole specific regions of the ferroelectric surface using the tip as a positionable top electrode [15, 16]. The advantages this flexibility offers over EFM and SSPM have thus made PFM emerge as the most widely used method to study ferroelectric properties at the nanoscale. The PFM-based polarization dynamics studies have also been implemented to observe the mechanisms and kinetics of the domain switching process [17–21]. Previous studies [18, 20, 22] have clearly shown that the switching phenomena occur through nucleation and growth of new domains, whereby interfacial defects such as dislocations present in the film, grain boundaries, and/or electrode geometries can either promote or hinder the switching process. However, because of low conventional AFM imaging speeds (typically 4min/frame), PFM-based studies of polarization domain evolution phenomena are either done in static or pseudodynamic modes where the evolution of the switching process is captured on the order of hours, days, or weeks, often as a function of domain relaxation time [18], poling bias [20], or pulse width [17, 19, 21]. Of course this can be disadvantageous in terms of both maintaining optimal temporal resolution while also investigating long enough durations to observe the entire domain switching phenomena, including the nascent stages of nucleation and growth. Although temporal resolution has been improved by applying shorter pulses to ferroelectric capacitor structures using pump-to-probe schemes, the spatial resolution, which is a bilinear function of the ferroelectric film and top electrode thicknesses, sets the applicability limits for dynamic studies conducted in this manner [23]. This book chapter outlines a novel extension to the PFM technique that has been developed to substantially increase AFM operating speeds by more than two orders of magnitude. This new high-speed AFM-based method allows complete polarization domain images to be obtained in fractions of a second. Furthermore, it uses common commercial equipment and therefore can be easily implemented on any AFM system. The nanoscale resolution expected of traditional PFM [24, 25] is nevertheless maintained, and 10-ns resolution per image pixel has been achieved. Because images are acquired in seconds instead of minutes, a reasonable half-day microscope session can thus yield literally 10,000 images. In addition to improving temporal studies, this provides a powerful platform for rapidly investigating multiple experimental conditions, where equivalent measurements previously required days or weeks. The resulting nanoscale resolution “movies” based on consecutively acquired images uniquely provides significant statistical evidence of dynamic events such as switching mechanisms and kinetics along with their spatial densities and distributions. Challenges associated with such high-speed work, including data analysis, are discussed. Finally, the chapter concludes with an investigation into domains written with tip speeds up to 1cm/s that are availed by the HSPFM platform.

High-Speed Piezo Force Microscopy In general, PFM consists of a conducting AFM probe brought into contact with a sample to locally bias the piezoelectric material, causing detectable surface displacements due to the converse piezoelectric effect. In a typical experiment setup,

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Fig. 11.1, the sample is mounted or grown on a conducting substrate that is grounded during the PFM measurement. A bias is applied to the tip with a function generator, including a periodic AC signal (Vac oscillating at frequency w) and possibly also a DC bias (Vo) for domain poling or hysteresis measurements. Any resulting surface displacements beneath the tip are then detected as deflections of the integrated AFM cantilever. This transduction is achieved in most commercial AFM systems by reflecting a laser or LED beam off of the lever onto a position sensitive photodiode, the output of which is monitored by a lock-in amplifier for enhanced signal-to-noise resolution. A computer controller is often employed as well for automated poling, patterned domain writing, and repetitive measurements. At its simplest, the sample vibration normal to the surface at the excitation frequency (w) follows (11.1), and is proportional to the piezoelectric coefficient dzz, the AC bias amplitude, and the orientation of the ferroelectric domain beneath the AFM tip ( ϕ ). The domain orientation is usually defined in terms of the angle away from the surface normal, such that for domains aligned out of the sample surface ϕ = 0° and cos ϕ = 1, for domains pointing into the surface ϕ = 180° and cos ϕ = -1, for domains parallel to the surface ϕ = 90° and cos ϕ = 0, etc. Therefore, the amplitude signal maps piezoelectric coefficients within the sample, while the phase signal indicates local domain orientations. For adjacent but oppositely oriented domains, the phase exhibits a 180° phase shift, and the amplitude should vreveal little if any contrast. A minima is reached in the amplitude at the domain wall, such that amplitude images essentially appear as outlines of oppositely oriented domains.

zω = dzzVac cos (ϕ ) = R cos (θ )

(11.1)

Phase

For the results presented here, resonant-frequency piezo force microscopy was employed. In this case, an AC frequency is selected near the cantilever resonance to enhance the signal-to-noise ratio. The contact resonance frequency is selected instead of the free resonance, however, shifting the measurement into the low MHz

Amplitude

position

position Vac+Vdc Domain orientation

PZT

Fig. 11.1 Sketch of HSPFM setup

SrRuO3 SrTiO3

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range instead of hundreds of kHz. This has two important advantages over traditional PFM imaging. First, using such higher frequency contact resonances provides more AC periods of piezoactuation per imaging pixel for a given scan resolution and scan rate, yielding higher signal-to-noise imaging. For example, for an image of 256 by 256 pixels scanned at a typical rate of 1line/s (both directions), the tip resides at each pixel for approximately 2ms while traversing the sample in each direction (trace and retrace). The number of AC periods sampled at each pixel for 10kHz and 1MHz PFM measurements is therefore 20 or 2,000 periods, respectively, a clear advantage when employing the higher frequency contact resonance. Of course this approach can be problematic for heterogeneous samples due to contrast contributions from local variations in mechanical properties (instead of ferroelectric effects) [7], but for the homogeneous films characterized in this chapter such challenges are avoided. Alternatively, this challenge can be easily overcome by resonant frequency tracking [26]. Second, and even more significantly, for a given signal-to-noise with traditional PFM, the possibility obviously exists to scan significantly faster for contact resonance PFM without sacrificing image quality. Continuing the 10kHz and 1MHz imaging examples, scan rates of 100lines/s are therefore uniquely possible. Of course the feedback in most AFM systems is not designed for such high-speed operation, but high rate imaging is certainly feasible for flat samples such as those considered here where the role of the AFM feedback is primarily to compensate for instrumental/thermal drift. In this manner, line rates of 100Hz, image frame rates of 2.56s, and tip speeds of 1mm/s have been achieved with high-speed PFM imaging [24]. In the following, full resolution imaging at 128Hz line rates is demonstrated; the high frame rates of HSPFM are employed to characterize nucleation and growth of 180° domains in epitaxial Pb0.8Zr0.2TiO3 (PZT) thin films on SrTiO3 (STO) substrates, and domain poling at tip speeds as fast as 1cm/s is analyzed. Extensions to the technique for a variety of related measurements are also introduced, including mapping electronic, magnetic, mechanical, optical, and/or coupled properties.

High-Speed Imaging The contact resonance used for these HSPFM measurements was first determined by applying a constant AC voltage to a stationary conducting tip and sweeping the AC frequency until the contact resonance was identified (typically in the range of 900–1,600kHz depending on the probe). The amplitude and phase of the corresponding cantilever deflection, and hence sample piezoactuation, are monitored by a lock-in amplifier (200MHz SRS844) and/or high-speed data acquisition card (20MHz, Measurement Computing DAS4020). As an example, for conducting diamond coated probes with a spring constant of approximately 40nN/nm and free resonance frequency of roughly 250kHz, the contact resonance used was around 1,450kHz (Veeco DESP), Fig. 11.2.

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Fig. 11.2 HSPFM contact resonance spectra

Fig. 11.3 Amplitude (top) and phase (base) images acquired with HSPFM operating at 10 Hz (left) and 128 Hz (right, slightly zoomed in)

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To achieve purely topographic high-speed AFM imaging, several approaches have been employed including resonant tip and sample scanners [27], multiple probes operating in parallel [28, 29], and specially designed actuators [30–33]. For property mapping at high speeds, specialized high frequency levers [34, 35] and/or sample and hold techniques were necessarily implemented [36]. Here, commercial hardware is used along with high-speed data acquisition for mapping the amplitude and phase of the piezoresponse pixel by pixel. For example, Fig. 11.3 presents HSPFM images of a ferroelectric domain pattern in a PZT thin film [18] acquired at line rates of 10Hz (left) and 128Hz (right). The distinct domains are readily apparent in the phase images (base), while the domain walls appear in the amplitude contrast as anticipated. There is clearly some blurring in the faster image, to be expected given this increase in data acquisition speed of at least 100 times over standard PFM imaging (usually achieved at 1Hz line rates or less). In any case, higher speed and resolution data acquisition hardware now available will improve this resolution still further.

Domain Wall Motion One of the most obvious applications of HSPFM is to study the dynamics of domain polarization. Focusing first on domain wall motion, a region of a 120-nmthick polydomain PZT film was prepoled shown in the phase and amplitude image at t = 0s in Fig. 11.4. The darker area in the phase image poled tip with + 10V with respect to the bottom electrode and slowly scanning the surface (1-s line rates) represents DOWN polarization direction (i.e., pointing towards the bottom electrode), while the brighter areas the UP direction which is biased with a the tip voltage of -10V. The poled region of 1.0mm × 2.5mm area was then used to study domain switching phenomena. The tip was driven near the contact resonance frequency as in Fig. 11.2 with a 5V peak-to-peak AC component for imaging, revealing the prepoled pattern as in the upper left frames of Fig. 11.4 (labeled “0s” for phase and then amplitude images). The HSPFM images were acquired consecutively at scanning rates of 40lines/s; amounting to approximately 6s per image frame but clearly still maintaining standard PFM resolution. A DC offset of + 0.5V was simultaneously applied causing gradual switching of the field of view as the tip repeatedly scanned the surface, because the maximum tip bias (AC + DC) is just at the coercive potential. This is clear in the remainder of the frames in Fig. 11.4, showing every tenth amplitude image acquired during this investigation which overall comprised of > 150 images acquired in just 15min. The frames are labeled as a function of the cumulative poling time per pixel, a function of the time the tip spends at each pixel location while scanning. It is important to note that the growth direction progresses both parallel, and perpendicular, to the fast scanning direction (horizontally), such that the gradual switching is enabled by the poling probe, but is not influenced by the fast scanning direction.

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Fig. 11.4 Frames of 40-Hz HSPFM imaging during in situ domain poling

Considering Fig. 11.4 in more detail, at t = 0s, the amplitudes for the antiparallel 180° domains (poled up or down) are nearly equal, while they exhibit a 180° phase difference and are therefore exceedingly simple to detect and identify. A subtle grid of dark lines is also apparent, corresponding to 90° tetragonal domains (‘a’ domains oriented within the film) as observed elsewhere [18]. The separation between the 90° domains is approximately 150nm and roughly periodic, with the resolved domain wall width of roughly 20nm. As predicted, the detected piezo-amplitude is stronger for the out-of-plane actuation with 180° domains than for the in-plane 90° domains. Indeed, the only reason any amplitude contrast is observed when the tip is positioned above 90° domains is that they are narrow with respect to the film thickness, and they are inclined at 45° within the film [37, 38]. Therefore, some out-of-plane material above and below the tilted slivers of in-plane domains are always actuated by the AFM tip. An outline of the initial domain wall boundary is drawn in each frame of Fig. 11.4, particularly highlighting the uneven progression of the domain boundary as the area switches from one polarity to the opposite. Similar effects have been studied elsewhere [18], though without the advantages of high temporal resolution provided by HSPFM. The influence of the 90° tetragonal domains on 180° ferroelectric domain switching is very clear, effectively pinning domain wall motion at certain sections of the advancing domain front. The instantaneous domain wall speed therefore depends greatly on the local environment, including both microstructural defects directly beneath the probe as well as the energy of the curved domain wall.

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Domain Nucleation and Growth Depending on the sample and experimental conditions, nucleation or growth dominated switching can occur. As an example, Fig. 11.5 presents 48 consecutive HSPFM frames from a movie of domain switching, taken from Nath et al. [24]. The full movie of phase and amplitude frames is available online [39]. Initially, a reverse L shape of oppositely oriented 180° domains was prepoled, and then both an AC and DC bias were applied to simultaneously switch, and monitor, the ferroelectric film as in Fig. 11.4. This 30-nm PZT film is strained instead of relaxed (as compared to the polydomain sample), and clearly exhibits extensive new domain nucleation in the initial white regions. Some growth is also apparent, both for these newly nucleated domains as well as in the gradual expansion of the elbow shape. Analyzing such large datasets presents several challenges, as the number of images to consider is enormous considering the point-and-click routines common in most commercial SPM software packages. Conveniently, though, the optics community has already dealt extensively with similar quality images as well as even higher acquisition rates, so automated routines for assembly, visualization, and quantification of multiple image frames are widespread. Three software packages are primarily used in this research, the freely available, NIH-developed ImageJ, and the commercial programs Volocity by Improvision, as well as the more computational Matlab by The MathWorks. In each case, a series of images as in Fig. 11.5 is first converted to a more common image format from the AFM system (or from externally assembled files as with the highest speed images discussed here), usually as TIF files. These are then rendered as a time series, as above, or into a volumetric element that is known in the optical community as a “stack.” This is depicted in three-dimensional in Fig. 11.6, prepared by stacking all of the images from Fig. 11.5 such that the Z-axis (labeled) represents time during switching while the XY plane is the same as in the original AFM images (though mirrored by the XZ plane for better visualization). Similar stacks have also been assembled where each new plane in the volume is acquired with a distinct bias, temperature, or other variable. Such volumetric visualization is valuable for several reasons. First, it is an efficient way to assemble the large datasets created during HSPFM. Second, a subtle tilt in the XZ direction for all of the features evidences some drift during imaging, which is easily correctable if necessary using automated routines that are widely available and implemented in the optics community. Planar cross sections to visualize any plane can also be simply displayed (e.g., the XZ and YZ planes through the origin are already obvious in Fig. 11.6), as are cross sections along any line to present the HSPFM contrast as a function of position by “drilling” along any userspecified direction. Simple visual inspection makes the onset of nucleation clear in terms of poling time (height) and location (note that in this case, new domains are black and near the base of the stack, while unswitched domains at the end of the experiment are still white near the top). The relative rate of growth is also obvious by observing domain expansion while traversing from base to top. The variability in these parameters, and the distribution of sites, is readily apparent as well.

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Fig. 11.5 2 µm × 2 µm frames from a HSPFM movie evidencing domain nucleation and growth [24]

Of course quantitative two-dimensional and full three-dimensional volumetric analyses of features within such images and stacks are possible. Individual domain sites, nucleation times, and growth rates are of particular interest here, and thus feature tracking was employed. To identify distinct domains, a simple contrast threshold was implemented, whereby switching is assumed if the color, i.e., phase of the 16-bit data, falls to within the bottom 25% of contrast levels (dark gray to black). Their area was also required to be greater than six pixels squared. Within the resulting set of 539 individual features in all 48 images in the stack, 82 were next found to exist within close enough proximity from one frame to the next that they can reasonably be assumed to be a real feature instead of random fluctuations. This was visually confirmed. Out of these “tracked” domains, the 46 that were imaged consecutively for the longest duration (up to 20 frames)

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Fig. 11.6 Volumetric view (area in xy dimensions and time in z direction) for a stack assembled from the consecutive 2 µm × 2 µm images of Fig. 11.5, depicting domain nucleation, growth, and coalescence

are depicted in Fig. 11.7, labeled by color, an identification number, and outlines for each frame in which they exist in the stack of Fig. 11.6 (most recent frames are darkest in color). Domains are only tracked when completely isolated, between the time of nucleation (when they first appear) until they either impinge on another domain, touch the image edge (as their size and shape cannot be known beyond the field of view), or the experiment ceases. The size, area, perimeter, and centroid location for each of these tracked domains in every single frame of the stack were then analyzed. The perimeters from this set are plotted in Fig. 11.8 as a function of poling time per pixel, with only the 12 longest duration domains shown for clarity. Generally, the domain perimeters grow linearly, and individual domains nucleate and grow at distinct times and rates. Assuming a circular domain shape, the domain radii can be calculated by dividing the perimeter by 2p. The resulting slopes of the various tracked domain radii with time then provide a reasonable estimate for the directly measured growth velocity of individual domains. Figure 11.9 presents a histogram of the velocities determined in this manner from the domains in Fig. 11.8, as well as 29 other domains from this dataset that were discrete for at least five frames (to improve the statistical significance of the calculated slopes). The average velocity is 32mm/s, with a standard deviation of 11.53mm/s. The variation is not a function of experimental error, but rather absolutely distinct growth rates for different domains. Such distributions of local properties are important to consider for analyses of ultimate switching speeds in capacitive elements, domain switching mechanisms, etc., which previously had to assume uniform domain wall velocities. Similarly, Fig. 11.10 displays a histogram of calculated nucleation times for the same 41 domains as considered in Fig. 11.9. The average nucleation time is 1.96ms, with a standard deviation of 1.45ms, but again, the results are not single valued.

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Fig. 11.7 Forty-six isolated domains individually tracked from the 2 µm × 2 µm images of Fig. 11.5, labeled by color and an identification number

Fig. 11.8 Domain perimeter tracked as a function of time for ten of the distinct domains in Fig. 11.7

It is important to note that biases just at the coercive potential are employed in this work in order to maximize the number of image frames for each growing domain.

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Fig. 11.9 Histogram of the domain wall velocity for 41 domains from Fig. 11.7

Fig. 11.10 Histogram of the nucleation time for all 44 domains considered from Fig. 11.7

With higher voltages, and/or narrow pulses, nucleation times in the nanosecond and domain wall velocities of meters per second have been directly observed. Since the locations of each of the tracked domains are known, maps of the domain properties are also possible. Even activation energies can be mapped, the subject of ongoing work performing similar measurements as a function of poling duration and potential HSPFM thereby provides a unique method to quantify switching dynamics for ferroelectric thin films.

High-Speed Domain Writing Finally, leveraging the high tip speeds achievable with HSPFM, domain poling was assessed as a function of tip speed. For square voltage pulses of 20ms, domains were switched from tip velocities greater than 1cm/s down to static conditions. Accordingly, Fig. 11.11 presents PFM phase images of domains written with eight distinct tip speeds, ranging from 0.244 to 10.2mm/s. At these rates and pulse widths, the tip traversed distances ranging from 4.8 to 200.4nm during poling.

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0.2 mm/sec

1.3 mm/sec

2.0 mm/sec 2 mm

10.4 mm/sec

Fig. 11.11 Domains poled in PZT with varying tip speeds as shown (scale bar appropriate for all images)

Figure 11.12 plots the corresponding measured domain radius as a function of tip poling velocity (“measure”). This is determined by PFM imaging at 0.5mm/s (regardless of the tip speed during domain writing). A dotted vertical line in the plot indicates this measurement speed, itself roughly ten times faster than standard PFM imaging as this represents scanning a 50-um image at 5Hz line rates (both directions). Generally, the domain size can be seen to increase with tip speed, but it also exhibits a decrease at low speeds. Figure 11.12 also includes several other features. For points labeled “traverse,” the domain size is predicted assuming that they will be directly proportional to the distance traversed by the moving tip during poling (d), which increases linearly with tip speed (v) and pulse duration (t) according to d = vt. This explains the general slope in the measured data, but not the apparent offset of nearly 40nm nor the initial decline in domain size with speed. The offset is easy to understand based on the real tip/sample interaction size, though. Simply, for this tip radius, contact force, and sample, there is a minimum domain size that can be achieved. This is estimated at 38nm (labeled “contact” on the plot) by averaging the difference between the measured radii and those predicted by the simple traverse model for the highest four tip speeds only. The initial smooth decrease in Fig. 11.12 is also straightforward to understand, as it is essentially equivalent to the well-studied decrease in domain size with

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domain radius (nm)

150

100

50

0

0

2

4

6

8

10

12

write speed (mm/sec)

Fig. 11.12 Measured domain radii following domain writing at a range of tip speeds to beyond 1 cm/s, with contributions to this radius as described in the text (the vertical dotted line indicates the speed at which the domain radii were measured, itself ten times faster than standard PFM imaging)

decreasing pulsing times for a given location. As the tip speed increases, any given region sees the bias for shorter and shorter pulses, which has been shown in pulsing experiments with static tips to exponentially decrease the resulting stable domain size. Consequently, for high tip speeds (i.e., fast pulses) the domain size should decrease exponentially and approach a theoretical limit as observed in Fig. 11.12 (labeled “remaining”). This overall behavior is similar to writing with a fountain pen: the lines will increase in radius as the pen is moved slower and slower, even growing with time at a static location, whereas for faster strokes their radius might decrease to a minimum but the overall area written in a given period of time will still increase via elongation.

Other Applications The HSPFM provides several other advantages over standard SPM imaging. The high speeds are ideal for efficient large area scanning, rapid assessment of properties on multiple samples, and combinatorial film gradients. They are also amenable to pump-toprobe measurements akin to those achieved by Gruverman et al. [19]. Finally, by mapping the same region repeatedly at different offset voltages, maps of hysteresis loops similar to those acquired in a point-by-point fashion [40] are feasible, only they are acquired frame by frame instead. Ongoing measurements with different durations and offsets are especially enlightening, as they allow mapping of nucleation and growth activation energies as a function of film processing, nano/micro-scale structural defects, and/or fatigue. More generally, high speed scanning property mapping (HSSPM) is similarly applicable for high-speed contrast of mechanical properties, magnetic, electric fields, optics, and coupled systems [7]. Such efficient and accurate measurements are increasingly necessary for characterizing the varied behavior of complex systems such as multiferroics, optoelectronics, etc.

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Conclusion Clearly, the HSPFM method presented here significantly extends the range of available characterization techniques for nanoscale ferroelectric and piezoelectric materials. Images are shown with line rates of 128Hz, and still higher speeds are feasible and underway. By repeatedly imaging a given sample area while applying sufficient voltage to the scanning tip to enable switching, domain dynamics can also be quantified and mapped. This includes directly measuring nucleation times and growth rates. The resulting statistical distributions of materials properties are crucial for next generation ferroelectric applications, especially as device dimensions shrink towards the feature densities observed in this research. The influence of microstructural effects is similarly important, and resolvable with this technique. Finally, tip speeds beyond 1cm/s enabled by the HSPFM approach were employed to characterize and analyze high-speed poling. Given the growing importance of ferroelectric applications, including coupled systems such as multiferroics, efficient and accurate measurement capabilities such as HSPFM are crucial.

References 1 M. Alexe and A. Gruverman, Nanoscale Characterisation of Ferroelectric Materials – Scanning Probe Microscopy Approach (Springer, Berlin Heidlberg, 2004). 2 A. Gruverman, O. Auciello, and H. Tokumoto, Annu. Rev. Mater. Sci. 28, 101–123 (1998). 3 S. Hong, Nanoscale Phenomena in Ferroelectric Thin Films (Kluwer Academic Publishers, Dordrecht, 2004). 4 S. V. Kalinin and D. A. Bonnell, Phys. Rev. B 63, 125411/1–13 (2001). 5 J. W. Hong, K. H. Noh, S. Park, S. I. Kwun, and Z. G. Khim, Phys. Rev. B 58, 5078–5084 (1998). 6 H. O. Jacobs, H. F. Knapp, S. Muller, and A. Stemmer, Ultramicroscopy 69, 39–49 (1997). 7 B. D. Huey, Annu. Rev. Mater. Res. 37, 351–385 (2007). 8 S. V. Kalinin and D. A. Bonnell, Phys. Rev. B 65, 125408 (2002). 9 S. Hong, J. Woo, H. Shin, J. U. Jeon, Y. E. Pak, L. C. Enrico, S. Nava, K. Eunah, and N. Kwangsoo, J. Appl. Phys. 89, 1377–1386 (2001). 10 O. Kolosov, A. Gruverman, J. Hatano, K. Takahashi, and H. Tokumoto, Phys. Rev. Lett. 74, 4309–4312 (1995). 11 C. S. Ganpule, V. Nagarajan, B. K. Hill, A. L. Roytburd, E. D. Williams, R. Ramesh, S. P. Alpay, A. Roelofs, R. Waser, and L. M. Eng, J. Appl. Phys. 91, 1477–1481 (2002). 12 L. M. Eng, H. J. Guntherodt, G. A. Schneider, U. Kopke, and J. M. Saldana, Appl. Phys. Lett. 74, 233–235 (1999). 13 R. Nath, R. E. Garcia, J. E. Blendell, and B. D. Huey, JOM 59, 17–21 (2007). 14 S. Jesse, B. J. Rodriguez, S. Choudhury, A. P. Baddorf, I. Vrejoiu, D. Hesse, M. Alexe, E. A. Eliseev, A. N. Morozovska, J. Zhang, L. Q. Chen, and S. V. Kalinin, Nat. Mater. 7, 209–215 (2008). 15 C. H. Ahn, T. Tybell, L. Antognazza, K. Char, R. H. Hammond, M. R. Beasley, O. Fischer, and J. M. Triscone, Science 276, 1100–1103 (1997). 16 S. V. Kalinin, D. A. Bonnell, T. Alvarez, X. Lei, Z. Hu, R. Shao, and J. H. Ferris, Adv. Mater. 16, 795–799 (2004).

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17 T. Tybell, P. Paruch, T. Giamarchi, and J. M. Triscone, Phys. Rev. Lett. 89, 097601 (2002). 18 C. S. Ganpule, A. L. Roytburd, V. Nagarajan, B. K. Hill, S. B. Ogale, E. D. Williams, R. Ramesh, and J. F. Scott, Phys. Rev. B 65, 01401/7 (2002). 19 A. Gruverman, D. Wu, and J. F. Scott, Phys. Rev. Lett. 100, 097601/4 (2008). 20 S. Hong, E. L. Colla, E. Kim, D. V. Taylor, A. K. Tagantsev, P. Muralt, K. No, and N. Setter, J. Appl. Phys. 86, 607–613 (1999). 21 Y. W. So, D. J. Kim, T. W. Noh, J. G. Yoon, and T. K. Song, Appl. Phys. Lett. 86, (9), 092905/3 (2005). 22 A. Roelofs, U. Bottger, R. Waser, F. Schlaphof, S. Trogisch, and L. M. Eng, Appl. Phys. Lett. 77, 3444–3446 (2000). 23 S. V. Kalinin, B. J. Rodriguez, K. Seung-Hyun, S. K. Hong, A. Gruverman, and E. A. Eliseev, Appl. Phys. Lett. 152906-1-3 (2008). 24 R. Nath, Y. H. Chu, N. A. Polomoff, R. Ramesh, and B. D. Huey, Appl. Phys. Lett. 93, (7), 072905/4 (2008). 25 R. Nath, Ph.D. Thesis, University of Connecticut, 2008. 26 B. J. Rodriguez, C. Callahan, S. V. Kalinin, and R. Proksch, Nanotechnology 18, (47), 475504/6 (2007). 27 A. D. L. Humphris, M. J. Miles, and J. K. Hobbs, Appl. Phys. Lett. 86, (3), 034106/3 (2005). 28 P. Vettiger, M. Despont, U. Drechsler, U. Durig, W. Haberle, M. I. Lutwyche, H. E. Rothuizen, R. Stutz, R. Widmer, and G. K. Binnig, IBM J. Res. Dev. 44, 323–340 (2000). 29 S. R. Manalis, S. C. Minne, and C. F. Quate, Appl. Phys. Lett. 68, 871–873 (1996). 30 L. M. Picco, L. Bozec, A. Ulcinas, D. J. Engledew, M. Antognozzi, M. A. Horton, and M. J. Miles, Nanotechnology 18, (4), 044030/4 (2007). 31 P. K. Hansma, G. Schitter, G. E. Fantner, and C. Prater, Science 314, 601–602 (2006). 32 T. Ando, N. Kodera, E. Takai, D. Maruyama, K. Saito, and A. Toda, Proc. Natl. Acad. Sci. USA 98, 12468–12472 (2001). 33 Y. G. Cui, Y. Arai, T. Asai, B. G. Ju, and W. Gao, Int. J. Precis. Eng. Manuf. 9, 27–32 (2008). 34 H. Kawakatsu, S. Kawai, D. Saya, M. Nagashio, D. Kobayashi, H. Toshiyoshi, and H. Fujita, Rev. Sci. Instrum. 73, 2317–2320 (2002). 35 S. Hosaka, K. Etoh, A. Kikukawa, and H. Koyanagi, J. Vac. Sci. Technol. B 18, 94–99 (2000). 36 T. Takahashi and S. Ono, Ultramicroscopy 105, 42–50 (2005). 37 I. B. Misirlioglu, A. L. Vasiliev, S. P. Alpay, M. Aindow, and R. Ramesh, J. Mater. Sci. 41, 697–707 (2006). 38 C. S. Ganpule, V. Nagarajan, H. Li, A. S. Ogale, D. E. Steinhauer, S. Aggarwal, E. Williams, R. Ramesh, and P. De Wolf, Appl. Phys. Lett. 77, 292–294 (2000). 39 R. Nath and B. D. Huey, http://ftp.aip.org/epaps/appl_phys_lett/E-APPLAB-93-044832/ elbowall.mov (2008). 40 B. J. Rodriguez, S. Jesse, M. Alexe, and S. V. Kalinin, Adv. Mater. 20, 109–114 (2008).

Chapter 12

Polar Structures in Relaxors by Piezoresponse Force Microscopy V.V. Shvartsman, W. Kleemann, D.A. Kiselev, I.K. Bdikin, and A.L. Kholkin

Introduction Relaxor ferroelectrics (or relaxors) comprise a special group of polar oxides widely used in capacitor and actuator applications. A generic feature of relaxors is a broad maximum in temperature dependence of the dielectric permittivity, whose position, Tm, is shifted to lower temperatures as the frequency of the probing field decreases [1]. In contrast to conventional ferroelectrics this maximum does not correspond to a phase transition from a paraelectric into a long-range ordered ferroelectric state with macroscopic polarization [2, 3]. Instead, in relaxors the polarization is correlated on a local scale resulting in the appearance of polar nanometer-size regions (PNRs). The attracting feature of these systems: the large dielectric permittivity in a broad temperature range and the extremely high piezoelectric effect (piezoelectric coefficient up to 2,500pm/V) are associated with contributions from PNRs. In spite of intense studies of relaxors, the nature of the PNR and their evolution under variation of external conditions (temperature, electric field, pressure) are still controversial. Most of the information about the size and dynamics of the PNR was obtained by indirect methods, mainly from X-ray and neutron scattering experiments. The direct observation of PNRs, e.g., by optical methods, which are commonly used for domain visualization in normal ferroelectrics, is restricted due to limited resolution. However, recent developments of scanning probe microscopy techniques, namely piezoresponse force microscopy (PFM), provided a powerful tool for studyingferroelectrics at the nanoscale [4–6]. Extremely high spatial resolution of the PFM (down to few nanometers) and high sensitivity to the local polarization makes this method attractive also for investigating relaxors. Here we present a detailed review of our recent results on the domain structures and local piezoelectric

A.L. Kholkin (*) Center for Research in Ceramic and Composite Materias (CICECO) & DECV, University of Aveiro, 3810-193, Aveiro, Portugal e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_12, © Springer Science+Business Media, LLC 2010

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properties of several important families of relaxors [SrxBa1–xNb2O6 (SBN), (1–x) PbZn1/3Nb2/3O3–xPbTiO3 (PZN–PT), (1–x)PbMg1/3Nb2/3O3–xPbTiO3 (PMN–PT), and (Pb1–xLax)(Zr1–yTiy)O3 (PLZT)] by means of PFM.

Polar Nanoregions: Experimental Evidences, Structure, Mechanisms Macroscopically the inversion symmetry of relaxors is not broken below Tm. However, there are many evidences that a break of the symmetry of paraelectric phase occurs locally on a nanoscale level, which results in the appearance of PNRs at temperatures much higher than Tm [3, 7]. The PNRs manifest themselves in the deviation of certain macroscopic characteristics of relaxors from the behavior predicted for the paraelectric state. In the absence of an electrical field there exist randomly distributed locally polarized regions, so that ∑ Pl = 0, i.e., there is no measurable remanent polarization. However, since ∑ Pl 2 ≠ 0 one can expect that the existence of PNRs becomes apparent in properties depending on P2, e.g., in the quadratic electro-optic effect, which is reflected in the refractive index or birefringence [7]. This was shown by Burns and Dacol for PbMg1/3Nb2/3O3 (PMN) single crystals by measuring the temperature dependences of their refractive index, n(T) [8]. For normal perovskite ferroelectrics, starting in the high-temperature PE phase, n decreases linearly with decreasing T down to the Curie temperature, below which n(T) deviates from linearity [9]. This deviation is proportional to the square of the polarization. Burns and Dacol found that in the case of PMN this deviation starts already at TD=620K (so-called Burns temperature), which is about 350K above Tm (at f=10kHz). Similarly, a deviation from the linear behavior above Tm has been found for the thermal expansion, due to the electrostrictive contribution from PNRs [2, 7]. The existence of PNRs was later confirmed by elastic diffuse neutron scattering experiments [10, 11]. So, in PMN single crystals a significant diffuse scattering appears below TD with the intensity increasing with decreasing temperature [11]. This diffuse intensity has been associated with PNRs. Various neutron and X-ray scattering experiments have been carried out in order to investigate how PNRs are formed, their size and polarization at different temperatures [12]. The correlation length of the atomic displacements contributing to diffuse scattering, which is a direct measure of the PNR’s size is inversely proportional to the width of the diffuse peak. According to high-resolution neutron diffuse scattering performed on PMN single crystals, the size of PNRs is about 1.5nm at TD and is almost temperature independent [13]. Only below 300K the correlation length starts to increase on cooling and reaches ~10nm at 10K. The most significant change of the correlation length was observed around so-called freezing temperature Tf~210K. This temperature is obtained from the analysis of the dynamic dielectric response of relaxors and corresponds to the slowing down of the relaxation related to the reorientation of PNRs (see below). The number of PNRs may be estimated from the integrated intensity of scattering. It shows a monotonic increase on cooling starting from TD, and then an abrupt decrease at about Tf due to merging of smaller PNRs into larger

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ones [13]. Qualitatively, a similar behavior was observed for another relaxor, PbZn1/3Nb2/3O3 (PZN), but with larger PNRs (they grow from ~7nm at TD to 18nm at 300K) [14, 15]. The shape of PNRs and their polarization are still a matter of discussion. From dynamical structural analysis of the diffuse neutron scattering in PMN crystals, it was found that the B-site cations and O2- anions are displaced with respect to the Pb cations along the <111> directions of the cubic unit cell, resulting in a local rhombohedral distortion [16]. The rhombohedral symmetry of PNRs in PMN was derived from the analysis of ion-pair displacement correlations obtained from diffuse X-ray scattering [17]. Nuclear magnetic resonance measurements on PMN also confirmed the <111> polar directions of PNRs [18]. It was proposed that PNRs have an ellipsoidal shape [17]. However, recent results of three-dimensional mapping of diffuse X-ray scattering intensities performed for PZN–PT single crystals indicate that PNRs have a pancake-like shape resulting from <110> planar correlations of 1 10 polarizations [19]. The diameter of these “pancakes” is about 5–10 times greater than their thickness. Similar planar type correlation of local polarization have been observed in other relaxor compounds such as Na1/2Bi1/2TiO3 (NBT) [20] and K1–xLixTaO3 (KLT) [21]. Xu et al. suggested that <111> type of polarization established in PZN–PT below the Curie temperature can be a result of global averaging of <110>-type polarized “pancake” entities [19], i.e., <111> polarized PNRs could be a result of overlapping of three different <110> -type pancakes [19]. There are different approaches to explain the formation of PNRs. The early model by Smolenskii considered that disorder in the distribution of different ions in the same crystallographic position (e.g., Mg2+ and Nb5+ for PMN) leads to a local fluctuation of their concentration [22]. In turn, it will result in spatial fluctuations of local Curie temperatures. It was suggested that upon cooling a local ferroelectric phase transition occurs first in regions with higher TC, whereas in other locations, the sample remains in the paraelectric phase. In this view, PNRs correspond to regions with elevated Curie temperature. This model was later developed by Bokov, who considered the appearance of PNRs as a result of the local condensation of the soft phonon mode [23]. Indeed, a low-energy transversal optic mode was found in PMN far above TD [24]. On cooling down to TD this mode softens in the same way as in displacive ferroelectrics following the Cochran law. Below TD it becomes overdamped due to interaction between acoustic and optic modes [24–26]. But below the freezing temperature it is recovered again, with a temperature evolution consistent with the typical behavior of a ferroelectric soft mode below the Curie temperature [27]. To explain the results of diffuse and softmode scattering experiments, Hirota et al. proposed a model of TO phonon condensation with a uniform phase shift [28]. According to this model, the total displacement of atoms inside a PNR can be divided into two components. The first component is created by the condensation of the soft mode and gives rise to spontaneous polarization inside the PNR. The second component results from a uniform displacement of all atoms inside PNR parallel to the polarization vector. Such uniform shift of PNRs relative to the surrounding nonpolar matrix may create an energy barrier, which prevents the macroscopic ferroelectric transition.

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Later the validity of the uniform phase-shift model has been established for the PZN–PT relaxors [29]. The structural disorder, which is inherent to relaxors, results in randomness of microscopic forces responsible for an onset of the spontaneous polarization. Therefore, Bokov and Ye suggested that each PNR might consist of unit cells polarized in different directions [30]. This model of “soft nanoregions” implies that due to thermally activated reorientation not only the direction of individual PNRs, but also the magnitude of the dipole moment of individual PNRs can strongly change with time. But the size of PNRs remains unchanged. A random field model to explain relaxor behavior was proposed by Kleemann et al. [31–33]. In this case, the lattice and corresponding charge disorder of relaxors are considered to be the origin of quenched electric random fields (RFs). At temperatures below TD, RFs promote nucleation of PNRs with polarity controlled by the fluctuations of RFs (at higher temperature due to large thermal fluctuations there are no well-defined dipole entities) [32]. On the other hand, according to the argument by Imry and Ma the quenched RFs conjugate to the order parameter will destroy the transition into long-range ordered state in the systems with continuous symmetry of the order parameter [34]. In the case of relaxors with the perovskite structure (“cubic” relaxors), like PMN, the order parameter, i.e., the local polarization, has eight allowed directions (taking into account rhombohedral symmetry) and, hence, can be considered as quasi-continuous. Therefore an equilibrium phase transition into long-range ordered ferroelectric state is excluded. Indeed, the lowtemperature state in PMN or in PZN is a short-ranged ordered glassy like state. On the other hand, for the uniaxial relaxors with tungsten-bronze-type structure, like SBN, the order parameter has only two directions. Such systems belong to the three-dimensional Ising model universality class. In this case, theory predicts the existence of a phase transition into a state with a long-range order, even in presence of quenched RFs [34]. Such conclusion has been experimentally proved for SrxBa1– Nb2O6 (SBN) single crystals [35]. x

Temperature Evolution of Polar Structures in Relaxors As follows from the aforementioned, the appearance of PNRs at the Burns temperature drastically affects the properties of relaxors. Therefore, it was proposed to consider the relaxor state below TD as a new phase different from paraelectric one [3]. The dipole moments of PNRs are weakly correlated and free to reorient, so after some excitation the system returns to the state with the lowest free energy. It is always the same state regardless of the initial conditions. Therefore, such a phase is often called an ergodic relaxor (ER) phase [3]. In the ER phase, the PNRs are very small and may be considered as dynamical entities with their dipole moments thermally fluctuating between equivalent directions [2, 36]. The thermally activated reorientation of dipole moments of PNRs yields the major contribution to the dielectric permittivity of relaxors [36, 37].

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Another contribution associated with PNRs is the sideway motion of their boundaries, without a change of the orientation of their polarization [38, 39]. This motion looks like breathing of PNRs. Glazounov and Tagantsev considered the vibration of PNRs boundaries in terms of randomly pinned interface, where the pinning centers are associated with quenched RFs and other defects [38]. On cooling, the number of PNRs increases and the interaction between them become substantial, which results in slowing down of PNRs dynamics. A broad distribution of PNRs sizes and randomness of interactions between them result in a broad distribution of the relaxation times. This gives rise to a broad peak the dielectric permittivity vs. temperature. Finally, the divergence of the longest relaxation time at a finite temperature results in freezing of PNRs and in the transition into a glassy-like state in systems like PMN [40, 41]. The freezing temperature, Tf, may be derived from fitting the frequency dependence of the temperature of the maximum of the dielectric permittivity, Tm, to the phenomenological Vogel–Fulcher law   Ea (12.1) , f = f0 exp  k (Tm − Tf )   where Ea and f0 are parameters and k is the Boltzmann constant [36]. In this state relaxors display various characteristics of the non-ergodic behavior typical for dipolar glasses, namely an anomalously wide relaxation time spectrum [42], aging [43], and dependence of the thermodynamic state on the thermal and field history of the sample [31, 41, 44, 45]. Therefore, this state is also called the non-ergodic relaxor state (NR). In relaxors with perovskite structure, like PMN or PZN, the average symmetry of NR phase remains cubic [46–48]. In spite of some similarity between relaxors and dipolar glasses, there are some features inherent only for relaxors. For example, a special feature of relaxors in the NR phase is their possibility to undergo an irreversible transition into the ferroelectric phase by applying an electric field larger than a critical value (for PMN, Ecr~1.7kV/ cm at 210K [49]). In the case of PMN the field-induced phase has rhombohedral symmetry, i.e., the cubic symmetry is macroscopically broken [49, 50]. Nevertheless, the structure remains locally inhomogeneous, e.g., traces of cubic phase were found at low temperatures by X-ray diffraction on poled PMN single crystals [50]. The NMR studies of a PMN crystal poled along the [111] direction revealed that only a 50% of Pb ions are displaced along the poling direction, while the other 50% exhibit a spherical layer-type displacement characteristic of paraelectric phase [18]. Upon heating the poled samples exhibit a first-order phase transition from the induced ferroelectric phase into the ER state at a well-defined temperature (~210K in PMN) [51]. This transition is accompanied by a step-like drop of the spontaneous polarization and a sharp peak of the dielectric permittivity. To describe the transition from the ergodic to the NR state, Pirc and Blinc proposed the spherical random bond random field (SRBRF) model [52, 53]. They introduced pseudospins, which  2 are proportional to the dipole moments of PNRs and satisfy the relation ∑ Si = 3N (N is the number of PNRs). Because in cubic i

( )

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

relaxors, like PMN, the number of allowed orientations of Si is rather high, the authors considered Si as continuously varying vectors −∞ < Siµ < +∞ (m=x, y, z). Then the Hamiltonian of the system can be written as H=−

   1 J ij Si S j − ∑ hi Si − ∑ ESi . ∑ 2 ij i

(12.2)

Here Jij describes the random interaction (bonds) between PNRs, which are assumed to be infinitely ranged. The second and third terms  introduce the interactions of h pseudospins with quenched random electric fields, i , and an external electric field  E , respectively. Both random bonds and RFs obey Gaussian probability distributions with rms variances of J / N and D, respectively. The mean value for the random bonds is J0/N, and zero for RFs. In the absence of RFs (D=0) the theory predicts a transition either into an inhomogeneous ferroelectric state with a nonzero spontaneous polarization (if J>J) or into a spherical glass state without long-range order (if J0<J). In the latter case, the N order parameter is Edwards–Anderson-type order parameter, qµEA = 1 ∑ Si µ 2 , N

i =1

which decreases linearly from 1 at T=0 to zero at T=J. In the presence of RFs (D¹0) the phase transition is destroyed such that qEA remains nonzero at T=J and approaches zero only at higher temperatures. The local polarization distribution function predicted by the SRBRF model for PMN and confirmed experimentally from the NMR line shape is quite different from those observed in dipolar and quadrupolar glasses [52]. Therefore, it was supposed that the NR phase in PMN is a new type of glass, which was called “spherical cluster glass” [52]. The predictions of the model are in a good agreement with temperature dependence of qEA obtained from NMR experiments on PMN single crystal [52]. The SRBRF model was successfully applied to describe the dispersion of the linear and nonlinear dielectric susceptibilities [54]. The drawback of the SRBRF model is that it considers PNRs as already existing entities. Kleemann proposed that fluctuations of the quenched RFs give rise to the formation of PNRs at high temperatures [32]. Upon further cooling of cubic relaxors, a transition into the spherical cluster glass state then occurs due to random interactions between PNRs [55]. While in the case of PMN the low-temperature state is the glass state, in other relaxor systems a spontaneous (i.e., in the absence of an external electric field) transition from ER into the ferroelectric-like state may occur. Well-known examples are SrxBa1–xNb2O6, PMN–PT, and PZN–PT with a large titanium concentration, etc. Typically the transition takes place at a temperature TC, which is up to several tens of degrees lower than Tm. At TC the macroscopic symmetry of the system is changed from cubic to rhombohedral or tetragonal. The mechanism of the transition from the ER state with dynamic PNRs into a long-range ordered state is still a matter of discussion. It has been proposed that the growth of PNRs leads to the situation where they become large enough that percolation occurs over the entire sample resulting in a static cooperative phase transition into a ferroelectric state [29, 56]. This mechanism found a confirmation from neutron diffuse scattering measurements by Xu et al. in PMN [13]. Another mechanism presumes that such a

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transformation occurs via thermally activated formation of critical ferroelectric nuclei from some PNRs at the Curie point and their isothermal growth [57]. This process is supposed to be analogous to the formation of a new phase at a “normal” first-order phase transition. The low-temperature long-range ordered state in relaxors exhibits typical ferroelectric properties, such as ferroelectric hysteresis, pyro- and piezoelectric as well as linear electrooptic effects. Nevertheless, there are some differences between this state and a “normal” ferroelectric state. In particular, a coexistence of rhombohedral and cubic phases was revealed by X-ray diffraction in PZN single crystals below TC [58]. In PMN–PT with 35% of Ti the central peak of the Brillouin scattering, which is related to relaxation of PNRs, was observed not only above, but also below TC. This may indicate that PNRs persist in ferroelectric phase [59]. Another particular feature of some cubic relaxors is their specific macroscopic phase inhomogeneity. Namely, diffraction experiments performed on PZN single crystal with X-ray of different energies (hence with different penetration depths) revealed the structural phase transition into the rhombohedral FE phase to occur only in an outer layer (thickness several tens of microns), while the interior part remains cubic [60]. The same feature was also observed in PZN–PT [61] and PMN–PT [62] crystals with small x. It was suggested that this cubic phase (named X-phase) is similar to the NR phase of pure PMN [62]. The nature of this phenomenon is still not clear.

Piezoresponse Force Microscopy Investigations of Uniaxial Relaxors SrxBa1–xNb2O6 SrxBa1–xNb2O6: Structural Considerations Strontium barium niobate, SrxBa1–xNb2O6 (SBN), has a tetragonal tungsten-bronze-type structure described by the general formula AB2O6. This structure consists of a framework of BO6 (NbO6 in case of SBN) octahedra sharing corners in such a way that three types of interstitial channels result: square A1, pentagonal A2, and triangular C ones [63, 64]. For SBN the A1 positions are occupied only by Sr, the A2 sites are filled with both by Sr and Ba, while the C channels remain empty (Fig. 12.1). Since there are only five Sr and Ba atoms for six A1 and A2 positions, one of A-sites remains unoccupied. On cooling SBN undergoes a phase transition from a paraelectric 4/mmm state to a polar 4mm one [65]. The spontaneous polarization is oriented parallel to the c-axis. Because no optic soft mode was found for SBN [66], its properties can be described by an order–disorder pseudospin model with a one-component order parameter (Ising model) [35]. On the other hand, randomly distributed A-site vacancies are sources of frozen (quenched) random electric fields. Therefore, it was supposed that SBN exemplifies the three-dimensional random-field Ising model (3D-RFIM) universality class [35]. In this case, contrary to conventional relaxors like PMN, a transition into a longrange ordered ground state is predicted below the Curie temperature, TC. In reality, however, due to extreme critical slowing down the dynamic PNRs are expected to merely transform into a RF controlled metastable domain state.

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Fig. 12.1 Projection of the SBN unit cell along the c axis: the tetragonal channels A1 are occupied by Sr, the pentagonal channels A2 by Sr and Ba, while the trigonal channels C remain empty. Five Sr and Ba atoms are distributed over six A1 and A2 sites (adapted from [67])

Fig. 12.2 (a) Temperature dependences of the dielectric susceptibility, c ¢(T), of SBN40 and SBN75 measured on cooling at decade stepped frequencies within the range 1 £ f £ 105 Hz. (b) Concentration dependences of Tm measured at 1 and 105 Hz, and TC evaluated from best fits of f(Tm) to (12.3) (adapted from [67])

Macroscopically, SBN shows a cross-over from the ferroelectric-like to relaxorlike properties with increasing Sr/Ba ratio [64]. Figure 12.2a shows the temperature dependences of the dielectric susceptibility, c¢(T), of the SBN single crystals with the compositions referring to ferroelectric and extreme relaxor behavior, respectively [67]. For SBN40 (x=0.4) a relatively sharp peak is observed, whose position, Tm»412K, does not shift with the probing frequency. Such behavior is typical for ferroelectrics, and Tm coincides with the Curie temperature, TC. Contrastingly, for the SBN75 single crystals the maximum of the dielectric susceptibility is much broader and the peak position shifts to lower temperatures when decreasing the

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frequency of the probing field as is typical of relaxors [2]. The frequency dependence of Tm is well described by an activated dynamic scaling law predicted for RFIM systems [68]

nq

f = f0 exp  −Ta / (Tm − TC )  ,

(12.3)

where f0 and Ta are parameters, and n and q are critical exponents [69]. A best fit yields the transition temperature TC»299K for SBN75 [69]. Intermediate SBN compositions show a progressive change from ferroelectric to relaxor behavior on increasing the Sr content as illustrated by the increasing Tm vs. TC splitting as x increases in Fig. 12.2b. It is considered that the cross-over to the relaxor behavior in SBN with larger x is related to an enhancement of structural and related charge disorder [64]. This disorder is primarily due to randomly distributed vacancies in A-sites. The missing charges on these vacancies are most intense sources of RFs. On the other hand, occupation of A2 sites by different cations introduces disorder of the oxygen ion positions due to different Ba–O and Sr–O bonding lengths, whereas the presence of vacancies in both A1 and A2 sites enhances this disorder [63]. Accommodation misfits of the different oxygen octahedra give rise to local buckling and tilting deformation creating localized electric multipole moments, which also are sources of random electric fields. Thus, more structural disorder results in stronger RFs. By statistical consideration the most ordered structure is expected in SBN20, where all A2 sites are occupied solely by Ba2+ cations, while the Sr2+ ions and vacancies are randomly distributed on the A1 sites. In other compositions, the entropy of the cations distribution, which is a measure of the structural disorder, increases with increasing Sr/Ba ratio and reaches a maximum level at x~0.65 [67, 70]. It has also been argued that vacancies at A2 sites have a stronger impact on polar properties of SBN, than those at A1 sites. Since the vacancy tends to increasingly occupy the A2 sublattice as x increases [67, 71, 72], the corresponding “active” charge disorder will be enhanced at large Sr2+ content. Obviously, enhancement of structural and charge disorder has to be reflected in changes of polar structures of SBN.

SrxBa1–xNb2O6: Polar Structures Below TC Figure 12.3 shows the PFM images acquired on c-cut single crystals with different Sr/Ba ratios at room temperature [73]. All samples have been aged over about 1 year at RT before being scanned by PFM. This temperature lies below the transition temperature for all of the specimens. They can, hence, be considered to be in quasistable final states, which nevertheless reflect different initial subdivisions into differently sized domains by RFs of different strengths. The images are presented in a trimodal false color code. The blue and red colors correspond to domains with the spontaneous polarization oriented up and down relative to the figure plane. The yellow contrast corresponds to regions with negligible piezoresponse. One can see that the

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Fig. 12.3 The PFM images observed on c-cut SBN single crystals with various compositions, SBN40 (a), SBN50 (b), SBN61 (c), and SBN75 (d) adapted from [73]

domains become smaller and their boundaries increasingly jagged in samples with higher Sr content. At the same time “yellow” regions take up a larger total area. We can attribute these piezo-inactive areas to very fine polar structures with size below 10nm being unresolved under our experimental conditions. Figure 12.4a shows the domain size distributions, NS(S) vs. S, for the different SBN compositions. Here NsdS is the number of nanodomains within the range of areas S … S + dS. One can see that even in the ferroelectric SBN40 the sizes of the domains (curve 1) are distributed in a broad range. On increasing the Sr/Ba molar ratio the number of smaller domains increases, while large domains become rare. As a result, the mean domain size

∞

∞

0

0

< R >= 2 p −1 ∫ N S SdS / ∫ N S dS

(12.4)

estimated from the NS(S) distributions decreases with increasing x (see inset to Fig. 12.4a). The domain size distributions follow power laws with exponential cutoff [74],

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Fig. 12.4 (a) Domain-size distributions measured at room temperature on single crystals of SBN40 (1), SBN61 (2), and SBN75 (3). The broken lines are best fits to (12.3). The inset shows the dependence of the mean domain size on the Sr concentration x. (b) Domain perimeter (L) vs. area (S) for the determination of the Hausdorff dimension (DH) of the domain boundaries for the PFM images taken on SBN40 (1) and SBN75 (2) single crystals. The inset shows the concentration dependence, DH vs. x (from [73])

N S ∼ S − δ exp (− S / S0 ),

(12.5)

where S0 corresponds to a cutoff size, above which domains become rare. It was found that on increasing of Sr/Ba ratio d increases continuously from the value 1.2 in SBN40 to d=1.6 in SBN75, while S0 drops down from 1.1 to 0.05mm2, respectively. Similar behavior of NS(S) was also reported for a Ce-doped SBN61 single crystal [74]. It is worth comparing the obtained experimental data with theoretical predictions for the RFIM systems. In particular, it was shown that the domain-size distributions are described by a power law with exponential cutoff in the case of two-dimensional RFIM systems [75, 76]. While in our case the domain structure is in reality a threedimensional system, PFM probes a projection of this three-dimensional domain pattern on a sample surface. This projection is actually a two-dimensional object and may be described by two-dimensional RFIM statistics. In the two-dimensional model both parameters, S0 and d, depend on the strength of RFs [75, 76]. Namely, S0 decreases with increasing RFs, while the exponent d approaches the value 1.55 in the limit of strong RFs. A similar tendency is observed for the experimental domain size distributions with increasing Sr/Ba ratio. It points out that the effect of RFs becomes stronger in the composition with higher Sr content, i.e., in compositions showing more pronounced relaxor behavior. It is worth mentioning that the exponent d has approximately the same value for SBN75 [73] and for SBN61 doped with only 1.1% of Ce [74]. This conforms to a substantially stronger effect of heterovalent substitution on relaxor properties, which is probably due to the different RF activities of hetero- and isovalent doping. The shape of the domains reminds of fractal-like objects similar to those found in Monte Carlo simulations of the RFIM systems [77]. Quantitatively, the shape of the domain walls may be analyzed by determining their fractal dimension. The PFM images show cross-sections of two-dimensional domain walls within the sample surface.

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These cross-sections, which we call domain boundaries, are topologically one-dimensional objects. The area of an individual domain, S, is related to its perimeter, L, as

L1/ DH S −1/ 2 = const ,

(12.6)

where DH is the Hausdorff (or fractal) dimension of domain boundaries [78]. In the case of an “irregular” domain structure, DH deviates significantly from the topological dimension D=1. Figure 12.4b shows the experimental dependences, L vs. S, which follows power laws according to (12.6) quite satisfactorily. The x dependence of the best-fitted values of DH is shown in the inset of Fig. 12.4b. One can see that the fractal dimension of domain walls increases with increasing Sr content. Theoretically, previously determined fractal dimensions for the two-dimensional RFIM have been reported to be DH=1.18 using Monte Carlo simulations [77] and DH=1.96 from the exact solution for the ground state [76]. The large difference between these two theoretical values is partly due to the fact that Monte Carlo simulations are far from thermal equilibrium. In addition, their small DH value refers to rather weak RFs and to finite temperatures [77], both parameters favoring weak domain wall roughness, since local pinning can easily be overcome by thermal fluctuations. The experimental values obtained from analysis of the PFM images DH(0.4)=1.23, DH(0.5)=1.29, DH(0.61)=1.48, and DH(0.75)=1.60 lie between the two above theoretical values. The increase of DH indicates that “irregularity” of domain structures caused by polar disorder, becomes increasingly pronounced with increasing Sr content. A substantial deviation of the Hausdorff dimension from the topological one is observed even in ferroelectric SBN40, where 180° domains have an “irregular” shape in comparison to those typical of conventional uniaxial ferroelectrics, e.g., “ovals” or “lenses” in triglycine sulphate [79, 80]. It is important to note that the total fractal dimension of the domain surfaces in SBN crystals finally comes out to be Dtot=D⊥+D||=2.23–2.6, if we accept an additional longitudinal dimension D||=1 along the polar c direction. Indeed, a strong anisotropy of 180° domains is typical of SBN single crystal [73, 81] giving rise, e.g., to a huge anisotropy of the second harmonic light scattering as reported recently for SBN61 by Betzler et al. [82]. It can be illustrated by comparison of vertical (VPFM) image for the c-cut and lateral PFM (LPFM) image for the a-cut (with polarization in-plane) of the SBN40 single crystals (Fig. 12.5) [73]. In the plane perpendicular to the polar axis a complex maze-type domain pattern is seen, as it was discussed above. Contrastingly, on the a-cut sample stripe domains extended along the polar axis were observed. The size of domains along the polar axis is several tens of microns, which is about two orders of magnitude larger than that perpendicular to the polar axis. A simple explanation of this shape anisotropy can be found under the assumption of a corresponding anisotropy of the inter- and intralayer interactions, Jc and Ja, acting between the ferroelectric O–Nb–O “spin” chains along the c- and a(b)-axes, respectively. Within the framework of an anisotropic Ising model, the excess of the free energy for RF-induced domains [33] can be considered in a first approximation as [73]

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Fig. 12.5 The PFM images of a SBN40 single crystal observed on a c-cut sample in the vertical mode (a) and on an ab cut in the lateral mode (b). The blue and red colors correspond to domains with the spontaneous polarization oriented up and down relative to the figure plane (vertical PFM) and left and right in the figure plane (lateral PFM), respectively. Regions with negligible piezoresponse are shown in yellow adapted from [73]

W = 2 L2a J c + 4 La Lc J a − hLa Lc .

(12.7)

Here a square column-shaped domain has length Lc (in lattice units) and cross-section La2, h is the average RF acting on the statistical excess N = L2a Lc of up (or down) “spins.” Minimization of the free energy with respect to both La and Lc yields the relationship La / Lc = J a / J c . Hence, La << Lc as observed experimentally seems to indicate the relationship J a << J c . In a more complete consideration of the domain stability also the effects of depolarization fields have to be taken into account. To this end one should take care of the condition div P ≈ 0 to be fulfilled at the domain boundaries for minimizing the domain wall energy [9]. For uniaxial ferroelectrics this means that domain walls perpendicular to the polar axis are energetically unfavorable and the domains will have needle-like rather than square column shapes. This is an additional factor promoting the observed anisotropy of domain shape in SBN. The results of our PFM investigations point out that the SBN compositions even with small Sr/Ba ratio differ from conventional ferroelectrics and may be related to a RFIM system with weak RFs. The RFs pin and distort domain walls to form a metastable pattern even after an aging time of 1 year at room temperature. The statistics of the domain sizes and the fractality of their wall contours corroborate theoretical predictions of RFIM systems. The polar disorder is enhanced in compositions with larger x indicating a strengthening of RFs with increasing Sr content. At a certain Sr/Ba substitution level, x³0.6, the effect of RFs becomes dominant and a crossover to typical relaxor behavior takes place.

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SrxBa1–xNb2O6: Temperature Evolution of the Polar Structures The evolution of the polar structures in the vicinity of the transition temperature was investigated in the SBN with x >0.6 [73, 83], and in Ce-doped SBN61 [84, 85] single crystals. For all these compositions relatively large quasi-static regions of correlated piezoresponse were found on cooling already above the nominal Curie temperature (Fig. 12.6a, b). The term “quasi-static” means here that the shape and the location of these regions do not change substantially during several consecutive scans, i.e., they are static on the timescale of the PFM experiment, t»102–103s. Due to the small value of the signal, the existence of regions of correlated piezoresponse (i.e., correlated polarization) was verified by an autocorrelation function analysis. Autocorrelation images (Fig. 12.6c, d) were obtained from the original PFM images by the following transformation:

C (r1 , r2 ) = ∑ D( x, y ) D( x + r1 , y + r2 ),

(12.8)

x, y

Fig. 12.6 The PFM and corresponding autocorrelation images of a SBN61 single crystal (TC = 346 K) at 295 K (a, c), and 354 K (b, d) adapted from [73])

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Fig. 12.7 (a) Spatial dependence of the autocorrelation function, , averaged over all in-plane directions for SBN61 at 295 K (1) and 354 K (2). (b) Temperature dependence of the average correlation radius <x > of SBN61 (from [73])

where D(x, y) is the value of the piezoresponse signal. Positive or negative values of the autocorrelation function correspond to probabilities to find a region with parallel or antiparallel direction of the polarization after a shift by (r1, r2) from an arbitrary point in the original PFM image. Typically, a peak arises in the center of the autocorrelation images. For the paraelectric state, in the absence of piezoactive regions (PRs), the width of the “central” peak is determined by the resolution of the PFM image as well as by scan parameters (scanning velocity, probing frequency, and time constant of the lock-in amplifier). The condition that the width of the central peak in the autocorrelation image becomes larger than the minimal size related to noise was used as a criterion of appearance of a region of correlated polarization. Both in SBN61 and SBN75 quasi-static PRs were resolved approximately 15K above the corresponding transition temperature (Fig. 12.7a, curve 2). The width of the central peak is a measure of the size of PRs. To describe correlations of the piezoresponse signal (polarization) inside individual PNRs, the autocorrelation function can be presented as

2h < C (r ) >= σ 2 exp  − (r / < x > )  ,  

(12.9)

where r is the distance from the central maximum [86]. A similar expression has been used for the analysis of rough surfaces [87]. Here is the autocorrelation function averaged over all in-plane directions, <x> is the average correlation radius and the exponent h (0 < h < 1) describes the “roughness” of a polarization interface [87]. Figure 12.7b shows the temperature dependence of <x> for SBN61. It appears above the resolution limit at T»TC+15K, and increases on cooling when approaching TC. Below TC the PRs continue to grow, but less abruptly. Such regions of correlated polarization observed above TC were attributed to relatively large “mesoscale” PNRs. The existence of PNRs in SBN61 at temperatures well above TC was evidenced by the measurements of the optic index of refraction [88], birefringence [88, 89], second harmonic generation [90], dynamic

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[91], and inelastic Brillouin [92] light scattering. The estimated TD is about 750K [89, 92]. On cooling the PNRs grow and in the vicinity of the transition temperature some of them becomes “frozen” (immobilized) on the experimental timescale due to their large size. Their polarity is controlled by fluctuations of RFs. These “frozen” PNRs are surrounded by non-PRs, which consist of still dynamic PNRs and probably on smaller static PNRs not resolved in the PFM experiment. The occurrence of quasi-static PNRs above the transition temperature provides an additional evidence of the transition from the high-temperature ER state into the low-temperature one (long-range ferroelectric state in the case of SBN). It points out that the transition does not takes place at a fixed temperature, but there is a certain temperature range above TC where the system contains both small dynamic and large quasi-static PNRs. The latter ones are precursors of domains in the lowtemperature ferroelectric state. The intermediate state has a non-ergodic character, which was verified for the studied single crystal by observation of aging (as indicated by an isothermal decay of the dielectric permittivity) in the same temperature range above TC, where frozen PNRs were visualized [83, 84]. This aging is related to isothermal growth of quastistatic PNRs at the expense of the dynamic ones. Another macroscopic manifestation of the existence of quasi-static PNRs is the appearance of a particular relaxation mode in the dielectric spectra of SBN single crystal, which was related to a “breathing” motion of the boundaries of “frozen” PNRs [85]. It has to be mentioned that the appearance of frozen PNRs in the close vicinity above TC may explain a long time disputed controversy between the experimentally observed critical behavior in 3D-RFIM systems and theoretical predictions [84, 93, 94]. Namely, the values of critical exponents found in experiments are close to those theoretically predicted for the two-dimensional Ising model, but deviate strongly from those of the 3D-RFIM systems. Kleemann et al. suggested that the appearance of quasi-stable PNRs excludes true equilibrium criticality when approaching TC [84]. While these large PNRs are frozen on a finite timescale, the unfrozen interfaces can be considered as regions with very short-ranged correlation of RFs, which are not able to select sizable polar clusters via field energy gain. In these essentially two-dimensional regions, forming a percolating network through the sample, a phase transition may take place under the constraint of a weakly disordered quasi-staggered field. At the same time, two-dimensional Ising model criticality of interface system is preserved [84]. Even below TC the domains in SBN are metastable and the system needs an extremely large time to reach its equilibrium state. Figure 12.8 shows PFM images acquired on a Ce-doped SBN61 single crystal, Sr0.61–xCexBa0.39Nb2O6 (x=0.0113), at different times after cooling from the paraelectric state to room temperature [95]. For this composition, TC is about 320K. The domains observed immediately after the cooling procedure (Fig. 12.8a) are relatively small and their borders are strongly jagged. Between the domains large areas show gray contrast corresponding to vanishing piezoresponse. In the course of time significant changes of the images take place: the domains are coarsening and become more compact with smoother borders. The growth of the domains occurs at the expense of both

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Fig. 12.8 Piezoresponse images of a SBN61:Ce single crystal acquired at room temperature after cooling down from the paraelectric state: (a) after 1 h, (b) after 72 h. (c) Dependence of the mean domain size, R, in a SBN61:Ce single crystal cooled down from the paraelectric state to room temperature as a function of the waiting time. The logarithmic growth law, (12.10), is best fitted (broken line) to the data points

embedded small “patches” of opposite polarity and gray areas. After 72h (Fig. 12.8b) the “white” and “black” domains occupy the major part of the scanned area, the mean domain size is nearly two times larger than in the initial situation. Nevertheless, the domains are still very irregular. Their size varies in a wide range from tens of nanometers to several microns. Some of the large nanodomains still contain small “patches” of opposite polarity. Continuous growth of domains is observed even 1 month (720h) after the sample cooling. Figure 12.8c shows the temporal dependence of the mean domain size estimated from the domain size distribution, NS(S), according to (12.4). The temporal evolution of domains clearly indicates the metastability of the domain state in SBN:Ce, which is in accordance with theoretical predictions [96, 97] for the RFIM systems. If the system is cooled down from the paraelectric state, the domains emerging in the vicinity of TC are closely related to the fluctuations of RFs, where the smallest domain size corresponds to the largest attainable correlation length of the ferroelectric order parameter at T>TC. The size of the domains is a function of temperature (it increases on cooling) and depends on the relation between the strength of RFs (i.e., their average magnitude) and domain wall energy. The domain growth occurs via absorption of smaller domains of opposite sign both in the interior part and near to the borders. These smaller domains are stabilized by RFs of appropriate orientation, therefore a potential barrier has to be overcome to absorb them. The waiting time being necessary to overcome this barrier is related to its height by the Arrhenius law, t=texp(U/kT), where t is a microscopic attempt time. On the other hand, the height of the potential barrier, U, is proportional to the size of absorbing domain [96]. Thus, whereas the small domains may be absorbed relatively fast, the larger ones are long lived providing that transformation into equilibrium long-range ordered state needs time, which – in principle – may largely exceed the experimental scale. Villain [96] has shown that the typical radius of the domains, R, is related to the waiting time, t, as

R = AT ln(t / t ),

(12.10)

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where A is a constant, which depends on the strength of RF, domain wall energy, and value of the order parameter. The experimental dependence of the mean domain size on the waiting time (Fig. 12.8c) is in good agreement with this theoretical prediction.

Piezoresponse Force Microscopy Studies of Cubic Relaxors Pb(Mg1/3Nb2/3)O3–PbTiO3 Single Crystals To the largest group of the relaxors belong “cubic” relaxors with the perovskite structure ABO3. In this case the relaxor behavior is closely related to charge disorder, which is caused by cations of different valency randomly occupying equivalent crystallographic positions, i.e., Pb2+ and La3+ on A sites of (Pb1–xLax)(Zr1–yTiy)O3 (PLZT) or Mg2+ and Nb5+ on B-sites of PMN. Pure PMN exhibits a transition into a cluster glass state with random orientation of the local polarization [52]. The introduction of ferroelectrically active Ti4+ ions in (1–x)PMN–xPT solid solutions enhances the interactions between PNRs leading to a breaking of the macroscopic cubic symmetry at x > 0.27 [98]. In compositions with 0.05 < x < 0.25 a transition into a state with macroscopic rhombohedral symmetry was found in a surface layer, while the interior regions of the samples remain macroscopically cubic [62]. The thickness of the outer layer is several tens of microns. In compositions with x> 0.27 the structural phase transition is observed in the entire sample. The low-temperature state has rhombohedral symmetry at 0.27 <x <0.3 [98], a monoclinic one at 0.3< x <0.35 [99], and a tetragonal one at x>0.35 [100]. The structural transition brings about an evolution of the polar structure from PNRs to ferroelectric domains. Indeed, large micron-sized domains were observed by polarized optical microscopy at x > 0.25 [101, 102]. However, this method becomes less applicable for compositions with lower titanium content due to the limited resolution of this technique. The PFM due to its high spatial resolution is ideally suited for studying them. Recently, PFM has been successfully applied to investigate polar structures in “cubic” relaxors such as PMN–PT [86, 103–106] and PZN–PT [107, 108] single crystals, PLZT [109, 110] and BaTiO3-based [111, 112] relaxor ceramics. Figure 12.9a shows PFM images taken on the (001)-oriented Pb(Mg1/3Nb2/3)0.9 Ti0.1O3 (PMN–PT10) single crystal at room temperature [86]. A clear piezoresponse contrast is seen, in spite of the fact that for this composition the macroscopic TC is about 280K [113], at least for the subsurface layer probed by PFM. The size of PRs varies in the range from tens to hundreds of nanometers. The boundaries between PRs of opposite polarity are wavy and diffused. Their distribution looks irregular at first sight. Nevertheless, in the corresponding autocorrelation images (Fig. 12.9c) a pronounced interchange of bright and dark contrast along the crystallographic [110] direction is seen. Such behavior of the autocorrelation function indicates some regularity of the PRs structure with a characteristic well-defined period in this

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Fig. 12.9 The PFM and corresponding autocorrelation images of (001)-oriented PMN–PT10 (TC = 280 K) (a, c) and PMN–PT20 (TC = 360 K) single crystals (b, d) taken at room temperature. On the PFM images bright and dark contrasts correspond to regions with the polarization oriented up and down (relative to the figure plane), respectively. Intermediate contrast corresponds to regions with negligible piezoresponse

direction. In order to describe it, the autocorrelation function can be presented as a sum of two contributions:

  r  2h   r  πr  C ( r ) = σ exp  −    + (1 − σ 2 ) exp  −  cos   ,  a   x    rc  2

(12.11)

where r is the distance from the central maximum. The second term corresponds to long-range correlations related to the regularity of the observed pattern. Correspondingly, rc is the long-range correlation length and a is the period of the structure. The best fit of the curve shown in Fig. 12.10 with (12.11) yields values of a=(180 ± 40)nm and rc=(800±300)nm. The first term, which is analogous to (12.9), describes short-distance correlations of the polarization (piezoresponse signal) inside an individual PR. Here x has a meaning of a short-range correlation radius. In order to estimate the mean size of the PRs we averaged the autocorrelation image over all in-plane direction and then approximate it just by the first term

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Fig. 12.10 Cross-sections of the autocorrelation images along the [110] direction for the PMN–PT10 and PMN–PT20 single crystals at 295 K. Dashed lines show the best fit of the experimental curves to (12.11) (from [86])

of (12.11). Such a procedure obviously erases all information on the long-range correlations. The best fit gives the value of the average correlation radius, <x>»70nm and the roughness exponent h of about 0.75. A relatively large value of h signifies that the “polarization interface” is smooth at room temperature, i.e., the polarization distribution inside an individual nanodomain is quite homogeneous. It is worth mentioning that the exponent h is related to the fractal dimension of an interface, d: d=3–h [87]. While on heating the piezoresponse becomes weaker, the long-range correlations in <110> directions are still observed up to 360K (Fig. 12.11). Individual regions of correlated polarization exist even at higher temperatures (»385K), whereas their average size becomes smaller. The average correlation radius estimated from decreases from 70 to 20nm with increasing temperature (Fig. 12.12). At the same time, the value of the roughness exponent decreases from 0.75 to approximately 0.3 (Fig. 12.12). This indicates that the response inside the individual nanodomains becomes less homogeneous and, at the same time, their boundaries become more jagged. In other words, the correlation between individual dipole units within the nanodomains becomes weaker. In the composition with Ti 20% (PMN–PT20) a quasi-regular domain pattern with domain walls oriented preferentially along a [110] direction has been observed below the macroscopic TC=360K (Fig. 12.9b) [103]. These domains have a complex structure, i.e., a large number of small “nanodomains” of opposite orientation are embedded in both “white” and “black” large “macrodomains.” The long-range correlation length, rc, is about 2–2.5mm at room temperature, which is approximately three times larger than that in PMN–PT10 (Fig. 12.10). This signifies more regular domain patterns in compositions with higher Ti content. Similarly to PMN–PT10, regions with nonzero piezoresponse were found above the nominal TC (Fig. 12.11d). They are not arbitrarily distributed, but form structures resembling

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Fig. 12.11 The PFM images of a PMN–PT10 single crystal (TC = 280 K) at 295 K (a) and 335 K (b) and of a PMN–PT20 single crystal (TC = 360 K) at 295 K (c) and 405 K (d)

Fig. 12.12 Temperature dependences of the average correlation radius <x> (squares) and the fractal dimension of the “polarization interface” d = 3–h (circles). The lines are drawn to guide the eye (from [86])

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the domains observed at lower temperatures [103]. The size of individual PRs at high temperatures is larger than in PMN–PT10, <x>»110nm at TC+45K. As in the case of the uniaxial SBN system quasi-static mesoscale PRs were found in cubic relaxors PMN–PT above TC. By analogy, we can attribute them to large PNRs “frozen” on the experimental timescale in the vicinity of the transition temperature. Nevertheless, it is worth noting that they are observed in a much wider temperature range above TC in PMN–PT than in SBN. Another particular feature is that these PRs are not fully random, but exhibit some regularity, in particular, a preferential orientation of the PR boundaries along <110> directions. This feature becomes more pronounced in compositions with higher titanium content. It was suggested that it is related to an effect of mechanical strains resulting from the rhombohedral distortion of unit cell in cubic relaxors [103, 105]. There are three main factors influencing the polar structure in relaxors: (a) minimization of the depolarization field energy, which controls formation of 180° domains; (b) the influence of RFs which break domains into smaller entities; (c) minimization of mechanical stress, which is achieved by formation of ferroelastic domains (twins). Obviously, in uniaxial SBN only the first and the second factors are acting. On the other hand, the stress accommodation factor is important for PMN–PT resulting in a preferential orientation of domain walls parallel to {110} habit planes, which are invariant for systems with rhombohedral symmetry. It is supposed that large domains in compositions with high Ti4+ content are self-assembled agglomerates of smaller nanosized domains [105]. The driving force for such self-organization may be due to stress relief, i.e., minimization of the free energy related to mechanical deformation. The RFs may essentially modify the energy balance resulting in diffused domain patterns in composition with less Ti4+. Nevertheless, self-assembly of the nanodomains still persists in PMN–PT10 on a small scale. Furthermore, one can assume that this tendency to self-organization may stimulate the formation of relatively large agglomerates of PNRs already at high enough temperatures above the nominal TC. The observed regions of correlated piezoresponse [86, 103] probably correspond to such agglomerates. It should also be taken into account that PMN–PT relaxors are spatially inhomogeneous materials. As it was mentioned in Sect.114 3 ferroelectric transition occurs in these compounds not homogeneously, but only in the outer layer, several tens of microns thick. The nature of this phenomenon is unclear. In particular, a large compressive strain was found in the near-surface layer in PMN single crystals [14]. Such strain may stabilize the ferroelectric state shifting the phase transition to higher temperatures. The effect may be similar to one observed in ferroelectric thin films, where the Curie temperature depends on a misfit strain, um, between the film and the substrate [115, 116]:

TC = TC0 + 2Ce 0

2Q12 um , s11 + s12

(12.12)

where TC0 is the Curie temperature for the free-standing film, Q12 the electrostriction coefficient, s11, s12 the components of the elastic compliance tensor, C the Curie–Weiss constant, and e0 the dielectric permittivity of vacuum. Moreover, it

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was recently found that the misfit strain may induce a ferroelectric state in nominally relaxor thin films of PLZT [117]. Suggesting that in PMN–PT a gradient of the strain across the outer layer might exist, then the shift of the Curie temperature will be also nonuniform and the transition point in a very thin subsurface layer, which gives the main contribution to the PFM signal, may be considerably higher than the average (nominal) Curie temperature estimated from X-ray measurements. Similar arguments may be applied, if we assume that the existence of the outer layer in PMN–PT single crystals is due to a gradient of the composition, e.g., the Ti content, in the vicinity of the sample surface.

PbZn1/3Nb2/3O3–PbTiO3 Single Crystals The domain structure and electromechanical properties of relaxor single crystals of solid solutions (1–x)PbZn1/3Nb2/3O3–xPbTiO3 (PZN–PT) have been intensively investigated during the last few years. The attracting feature of this system is the extremely high piezoelectric effect (piezoelectric coefficients up to 2,500pm/V) and electrically induced strain (up to 1.7%) measured for (001)-cut faces of rhombohedral crystals of the composition close to the morphotropic phase boundary (MPB) with tetragonal phase (x~0.08–0.10) [118]. It was also observed that the corresponding coupling coefficients are very high (~95%) making these crystals a promising material for piezoelectric transducers for medical imaging, active vibration damping, and underwater sensing [119, 120]. It was clearly shown [121] that the unusually high strain in these crystals is somehow related to the phase transition from rhombohedral to tetragonal phase under high electric field applied along the <001> pseudocubic directions. For compositions close to the MPB a new monoclinic phase (intermediate between rhombohedral and tetragonal phases) has been observed by both X-ray diffraction and direct domain study [122, 123]. This phase provides the possibility of domain rotation between rhombohedral and tetragonal phases and was suggested as a reason for the extraordinarily large electromechanical response of PZN–PT under an applied electric field [124]. Several domain studies have been performed on PZN–PT crystals that revealed the existence of ferroelastic twins that exist even in poled specimens [107, 125, 126]. In another study, a peculiar dendritic domain pattern formed under pulsed poling conditions has been documented making domain structure very complicated [127]. The disadvantage of these investigations is that they have been performed mainly using optical techniques with a resolution limited to the wavelength of the light. However, submicron-size domains can exist in such materials and contribute to their final electromechanical performance. Indeed, using PFM technique it was possible to visualize irregular micron-size domains in PZN–PT single crystals with compositions both at the MPB [107] and far from MPB [108]. Figure 12.13 shows domain images of the PZN–4.5% PT crystals at four different scales: 100, 15, 5, and 2mm taken on (001) and (111) faces. The contrast is

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Fig. 12.13 Piezoresponse images of Pb(Zn1/3Nb2/3)O3–4.5% PbTiO3 single crystal: (a, c, e, g) – (100)-oriented sample, (b, d, f, h) – (111)-oriented sample. The inset to Fig. 12.13b shows the central part of self-correlation image. The inset to Fig. 12.13f is the variation of piezoresponse signal along the marked cross-section (1, 3 – signal from domains with polarization tilted to crystal surface, 2, 4 – signal from domains with polarization normal to crystal surface)

determined by the out-of-plane component of polarization (P⊥), where dark and bright areas correspond to the opposite directions of P⊥. Inset to Fig. 12.13b demonstrates the image of the self-correlation function calculated according to (12.8) from the corresponding domain picture. It illustrates that, along certain directions, there exist spatial correlations of seemingly irregular domains. These directions should coincide with the intersections of allowed domain walls with the crystal surface. It is clearly seen that the self-correlation image of the (111) surface reveals the symmetry of the rhombohedral phase (3m), where domain walls are preferably aligned parallel to ;110= and ;100= planes. In contrast, no clear correlation for the domain wall directions has been observed for the (001)-oriented surface. The rhombohedral ferroelectric phase allows for the existence of eight equivalent domain states and the contrast for a (111)-oriented crystal should be made up of four different signal levels corresponding to all possible values of the polarization projections on the [111] axis. The cross-section in Fig. 12.13f demonstrates these levels corresponding to different angles between polarization direction and [111] direction. The domain boundaries often appear L-shaped. (Fig. 12.13d). It has previously been shown that such boundary should comprise both charged and uncharged domain walls [125]. It can be noticed that domains in (001)-oriented crystals are much finer than in (111) samples. In a larger magnification (Fig. 12.13e, g) it is clearly seen that the micron-size domains on (001)-oriented surface actually consist of much finer irregular structures of the size down to 20nm (limited by PFM resolution). Seemingly “dark” areas are actually split into numerous “bright” inclusions and vice versa. The statistical

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Fig. 12.14 Nanodomains size distribution for (001)-oriented Pb(Zn1/3Nb2/3)O3–4.5% PbTiO3 single crystal

distribution of the domain area inside a micron-size domain is shown in Fig. 12.14. The average size of fine domains is ~ 65nm, but it is the upper limit because of the limited resolution. No nanodomains have been observed in (111)-oriented crystals. The complex domain structures seen on a (001)-cut are similar to those observed in PMN–PT single crystals with Ti content 0.20 £ x £ 0.35 [103, 105]. All these compositions exhibit an “intermediate” behavior between canonical relaxor and ferroelectric ones. While the PbTiO3 addition favors the long-range ferroelectric order, relaxor features still remain in zero-field-cooled crystals. So, the existence of short-range order manifested in the formation of nanoscale polar region (polar clusters) is the principal feature of the relaxor state [128]. For example, frozen PNRs with an average size of about 10nm were found in PZN–4.5% PT at room temperature by neutron scattering [129]. Appearing far above TC the PNRs grow in size when cooled in zero field. At the same time the presence of the highly polarizable Ti4+ ions promotes interaction between PNRs, which thus can reorient and merge into micron-size domains upon cooling. The nanoscale inclusions embedded into these domains may represent regions where RFs were especially strong to prevent reorientation and growth of PNRs. As was shown by Viehland and Li [130], such locations can serve as heterogeneous nucleation centers for polarization switching in “soft” ferroelectrics. In this context, it is worth noting that the macroscopic coercive field is much smaller for the (001) cut than for the (111) one [131]. This may be a consequence of the peculiar domain structure shown in Fig. 12.13. The average size of nanodomains observed in the PFM experiment is significantly larger than estimated by neutron scattering experiments (65nm vs. ~10nm). This can result from the natural difference in the two techniques and from the fact that the PFM method is only sensitive to the surface of the crystal (see the arguments for explaining the

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Fig. 12.15 Local piezoresponse hysteresis loops of PZN–4.5% PT crystals: (a) (100)-oriented crystal. (b) (111)-oriented crystal

PMN–PT results). The growth and coalescence of PNRs may apparently differ near the surface and in the crystal’s bulk. The natural question arises why no nanodomains are observed on a (111) surface. Results of diffuse neutron scattering on PZN–5% PT [129] indicate that, even though the polarization in the nanodomains is oriented along [111] direction, their easy growth direction is along the <001> axes. This results in a larger correlation length and, as a consequence, in a higher piezoelectric signal measured by PFM on the (001) cut. If the piezoelectric signal is lower than the vertical resolution of PFM it may not be observed on a (111) cut. An attempt was made to measure nanoscale hysteresis loops in both crystallographic directions in order to verify the ability of the PFM tip to induce a phase transition in the tetragonal phase and, therefore, creating a high piezoelectric signal in the vicinity of the PFM tip. The results are shown in Fig. 12.15. The sequence of the measurements was as follows: first the crystal was prepoled by pulses of variable height and then the piezoelectric response was measured at the same location in the absence of an electric field. Since the electric field is highly nonuniformly distributed under the surface [4], even small voltages applied by the PFM tip are expected to induce the phase transition. However, no evidence of such transition has been observed for the (001)-oriented crystals (see Fig. 12.15). The loops are very similar for both interfaces. In macroscopic measurements, the difference in respective piezoelectric coefficients for (001) and (111) cuts is more than an order of magnitude [118]. Thus, our observation contradicts the macroscopic data and underlines the importance of nanoscale measurements performed on the scale comparable with the domain size.

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Fig. 12.16 The PFM images of a PMN–PT14 ceramics (TC = 330 K) at 360 K (a), 315 K (b), 295 K (c). Grain contours are indicated by solid lines

Fig. 12.17 Temperature dependence of the remanent polarization of PMN–PT14 ceramics

Polycrystalline Materials (Ceramics) Temperature Evolution of Domains in the PMN–PT Ceramics The investigation of polar structures of polycrystalline relaxors is of special interest. In this case the microstructure of the samples, namely grain size effect and grain boundaries, may play an important role thus giving PFM unique possibilities to study these effects via the domain structure. Figure 12.16 shows the PFM images taken on the PMN–PT ceramics with 14% of titanium (PMN–PT14) on cooling from high temperatures (420K). This temperature is well above the Curie temperature for this sample (TC~330K), which may be estimated from the temperature dependence of the remanent polarization, Pr (Fig. 12.17). Like in the case of single crystals, quasi-static PRs of several hundred nanometers in size appear already above the nominal TC. It is important that they are concentrated near grain boundaries, especially near “triple points,” where several

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Fig. 12.18 Temperature evolution of the poled area in PMN–PT14 ceramics. (a) initial state; (b–e) after the area 3 × 3 mm2 was scanned under -30 V: at 295 K (b), 330 K (c), 350 K (d), and 360 K (e). (f) – temperature dependence of the piezoactive (polar) area relative to initially poled one

grains are joining. Obviously, in these places mechanical stresses are accumulated, which promote coarsening of PNRs and stimulate their agglomeration into bigger entities, as it was discussed above. Both the growth of already existing domains and the appearance of new ones take place on cooling. Even below TC the domain pattern remains metastable and an isothermal growth of domains with time was observed. The transition from the ferroelectric into the ER state is often associated with a macroscopic depolarization of the sample, for example, in a field cooling-zero field heating cycle [3]. To simulate such conditions in a PFM experiment, a square area 3×3mm2 was scanned under bias voltage Udc=-30V at RT. As a result this area was almost uniformly polarized in one direction (Fig. 12.18b). The created polarization is stable, no decay of the polarized area was observed during 24h. Then the sample was heated, and piezoresponse images of the poled area were acquired at different temperatures (Fig. 12.18c–e). From the analysis of the distributions of the piezoresponse signal inside the poled square, a relative amount of polar (piezoactive) phase (polarized area) was estimated as a function of the temperature (Fig. 12.18f). It was found that the artificial domain disappears abruptly in the narrow temperature range 330–340K, which agrees well with the temperature dependence of the remanent polarization (Fig. 12.17). Thus, the Curie temperature indeed corresponds to the decay of polar ordering on a macroscopic (micron-size) scale. It is worth noting that during the decay of the artificial domain not only areas of zero-piezoresponse, but also regions with opposite polarity appear. Such polar regions of both polarities are still present at higher temperatures on a mesoscale (100nm
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Grain Size Effect in Pb0.9125La0.0975(Zr0.65Ti0.3)0.976O3 Ceramics It has long been known that the dielectric properties of ferroelectric ceramics are dependent on their average grain size. Most of the studies were focused on BaTiO3based compositions, because they have been used as a capacitor material since the 1940s. A strong increase in the dielectric constant in fine-grain BaTiO3 ceramics [132, 133] has motivated a large number of research groups all over the world to study this effect in detail. In a seminal paper, Arlt et al. [134] have reported that the dielectric constant in BaTiO3 ceramics first increases with decreasing grain size d and then passes a pronounced peak at d»1µm followed by a sharp decline in a fine-grain material. This effect imposed a serious limitation for the application of these materials in multilayer capacitors, where submicrometer grain size is required to decrease the separation between inner metallic electrodes [135]. Ferroelectric behavior was also found to be strongly influenced by the grain/particle size, i.e., the ferroelectric properties disappear at some critical size accompanied by a strong decrease of the Curie temperature (see, e.g. a recent review [136] and references therein). The origin of such a strong deterioration of the properties with decreasing grain/particle size is not well understood today, as there are many extrinsic and intrinsic factors influencing this behavior [136]. For example, the dielectric constant effect has been attributed either to the increase of residual internal stress in submicron grains [137] or to enhanced domain wall contributions to the dielectric response [134]. Also, severe damage during the prolonged milling or sintering processing step [138] and the formation of a surface layer with reduced properties (the so-called dead layer) [139] were found to be an apparent source of size-induced phenomena. As such, nanoscale studies of the electrical properties of ferroelectric materials are extremely important as they allow establishing a relationship between the grain size, local properties, and macroscopic response paving the way for improvement in the ceramics performance. The situation is much more complicated in relaxor ceramics, where the compositional disorder causes the appearance of PNRs [32, 33]. Since the dimensions of PNRs are extremely small, relaxors should not be susceptible to size effects even in fine-grain ceramics. However, the apparent grain size dependence may appear via indirect mechanisms, e.g., due to mechanical stress appearing during ceramics sintering and consequent cooling to room temperature or due to inhomogeneous distribution of defects and their segregation at the grain boundaries. In relaxor thin films (solid solutions of PMN with PbTiO3), a clear grain size effect has been observed recently by Shvartsman et al. [140] using the PFM technique (see below). Figure 12.19a shows an example of the polar structure observed by PFM in PLZT ceramics with the concentration of La x=9.75% [109, 110]. The PRs form a complex labyrinth structure, which reflects spatial inhomogeneities determined by both RFs and random strains introduced with La3+ doping at the Pb2+ sites. The random polarization pattern corresponds to a 180°-domain structure. The value of the PFM contrast depends on the angle between the crystallographic orientation of the grain and the polarization direction lying along the pseudocubic [111]

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Fig. 12.19 Representative piezoresponse image of an individual PLZT 9.75/65/35 grain (a) and variation of the correlation radius x and average piezoresponse across the grain (b). The values of x are taken from scans shown in square boxes in (a)

direction. Applying the autocorrelation function method (see above) it is possible to determine the piezoresponse (polarization) correlation radius, x, within an individual grain. The strict relationship between the “surface” correlation radius estimated from the PFM experiment and its “true” bulk value (that determines the peculiar dielectric properties of relaxors) is not clear today, but it is believed that x is directly related to the average size of PNR and thus can serve as a measure of the polarization disorder on the surface of polycrystalline relaxor ferroelectrics. Relatively large grain size of the studied PLZT ceramics (d~10– 15µm, at least one order of magnitude greater than the correlation radius [109]) gives a possibility to determine x as a function of the coordinate inside the grain. This is illustrated in Fig. 12.19 where the value of x was measured in six scans consecutively distant from the grain boundary. It was found that the correlation radius varies with the position being greater in the center of the grain while gradually decreasing upon approaching the grain boundary. Such behavior was noted in several tens of grains investigated by PFM and thus indicates that the polarization disorder notably increases close to the grain boundary (most significant variation at ~1–1.5µm). There are several reasons that can qualitatively explain this observation. First, it can tentatively be attributed to the compressive mechanical stress arising at the grain boundaries upon cooling from the sintering temperature. The origin of the stress is due to directional differences in thermal expansion coefficients, plastic deformation, or elastic properties. In ferroelectric materials, the stress may also appear due to structural distortion arising upon the ferroelectric phase transition. As was shown by Samara [56, 141] the application of compressive hydrostatic pressure to relaxors leads to an apparent decrease in the correlation radius, i.e., to the reduction of the dimensions of PNRs. This results in a shift and broadening of the dielectric maximum in several relaxor systems including PLZT [141]. For example, the correlation radius is decreased by a factor of 2 when the pressure changes from atmospheric to 20kbar in PLZT 12/40/60. At much higher pressures, the correlation radius may increase again [142], but this pressure cannot be achieved in ceramics. The measurements of the correlation radius reported in [110] are consistent with the mechanical pressure

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effect [141] and thus may be responsible for the grain size dependence in PLZT ceramics reported in the past (see [143] and references therein). Indeed, finegrain PLZT ceramics exhibit much more diffuse phase transition as compared to coarse-grained samples [143]. An alternative explanation for the observed spatial dependence of the correlation radius is due to composition gradients across the grain. This is reminiscent of the first model of relaxors [1, 144] where the peculiar dielectric response is explained by the local fluctuations of the Curie points due to compositional disorder. Indeed, Lin and Chang [145] reported segregation of defects and second phases at the PLZT grain boundaries during hot pressing under PbO excess. This resulted in a gradient of the La concentration upon approaching the grain boundary. This behavior could, in turn, be translated into a variation of the correlation radius, as it is known that La substitution disrupts long-range ferroelectric order and leads to a pronounced diffuseness of the phase transition. Another important feature of the observed grain size effect is that the average value of the piezoelectric response (proportional to the average value of the polarization) also decreases upon approaching the grain boundary closely following the correlation radius dependence (see Fig. 12.19b). It means that the mechanical stress or composition gradients may be responsible for the broad variation of the dielectric and piezoelectric properties in polycrystalline relaxors [143]. Since the correlation radius is sensitive to the position inside the grain, it is natural to expect that its value taken on a larger scale (several micrometers) will depend on the grain size, too. This is illustrated in Fig. 12.20, where the autocorrelation function is shown for two neighboring grains. The grains are contrasting with their different average piezoelectric signals thus allowing the correct determination of the grain area. It is clearly seen that the correlation radius (Fig. 12.20b) is greater for the larger grain 1, confirming the above conclusion. The correlation radius as a function of the grain size is shown in Fig. 12.21. At least five grains of the same size but of different

Fig. 12.20 (a) Piezoresponse images of two neighboring grains with areas of 12.8 (#1) and 5.2 µm2 (#2), (b) the corresponding autocorrelation functions showing a difference in the correlation radius in these grains. The inset presents a zoom of the selected area (dashed rectangle) near the grain boundary

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Fig. 12.21 Average correlation radius in PLZT ceramics taken over the entire grain as a function of its effective diameter

contrasts were chosen in order to average the possible effect of the grain orientation on the observed dependence. The correlation radius is about 80nm for grains of 1-µm size and rapidly increases to about 100nm for the grains with a size of 2–3µm. These data are consistent with those shown in Fig. 12.19, which demonstrate the spatial dependence of the correlation radius. It is obvious that the increase in x occurs mainly at 1µm
Thin Films In polycrystalline relaxor films it was also found that the transition into ferroelectric state is influenced by the properties of the individual grains (due to their different orientation, chemical composition, mechanical stress, etc.). For example,

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Fig. 12.22 Topographic (a) and PFM (b) images of the 0.9PMN–0.1PT thin film. Black regions on piezoresponse image correspond to grains with self-polarization. Typical local piezoresponse hysteresis loops (c) for initially nonpiezoactive (1) and self-polarized (2) grains. The remanent dzz value as a function of the grain size (d)

Fig. 12.22 shows typical topography and piezoelectric images of polycrystalline PMN–0.1PT thin films [140]. The majority of the grains exhibit a weak piezoelectric activity (an intermediate contrast on the PFM image), while others have relatively high d33. It means that they are strongly polarized without any external bias, i.e., self-polarized. Two conclusions were made: (a) self-polarization can be found at the nanoscale in the relaxor state, while it is hardly found in macroscopic measurements, and (b) since piezoelectricity in relaxors is possible only under a bias field, the distribution of self-polarization represents also the distribution of internal field in these materials. Thus relaxor thin films on the microscopic level behave like a nanocomposite, the macroscopic response of which can be derived by the averaging of local responses of the individual grains. The induced piezoresponse inside individual grains was found to depend on the grain size: the bigger the grain, the higher remanent dzz was observed (Fig. 12.22d). There are several possible explanations of the observed phenomena. First, the mechanical clamping of a grain caused by surrounding grains is apparently dependent on its size. It is likely that the

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mechanical interaction between the grains leads to a high level of the compressive pressure inside small grains, while a mean stress value (or the stress in the center of the grain) remains relatively low for large grains. The growth and reorientation of PNRs under an external bias may be considerably restricted in smaller grains via the stronger coupling between local strains and polarization inherent for cubic relaxors. Therefore, the piezoelectric response is substantially reduced in those grains. In very small grains, PNRs may be completely frozen-in due to high mechanical clamping and the response is due to paraelectric matrix. Another reason for the observed size effect can be related to the possible grain size dependence of the mean size of PNRs inside a single grain, possibly through the mechanical clamping effect again. Indeed, studies on PMN ceramics [146] have revealed a pronounced decrease in the size and corresponding weakening of the dielectric response of an average polar cluster with decreasing grain size. The chemical inhomogeneity and local variations of the compositions (PbTiO3 content) could also be partly responsible for the change of the physical properties from grain to grain. A transition into the ferroelectric state can be induced in relaxors after application of a strong enough external electric field [3]. In PFM experiments, the corresponding value of the electric field may be easily achieved locally. In particular, field-induced effects were investigated by Shvartsman et al. in epitaxial (001) oriented PMN thin films [147]. At room temperature, the films exhibit a negligible piezoresponse as is expected for this material, where the transition from the ergodic into the NR state occurs at Tf=220K. Nevertheless, a strong PFM signal, which indicates the onset of a polar phase, could be induced after applying a dc voltage exceeding a threshold value of 1.5–2V. The induced piezoresponse is unstable and relaxes within tens of minutes after switching off the dc bias (Fig. 12.23). The time dependence of local dzz exhibits an initial fast decay followed by a slow decrease within a longer time. The slow stage of the dzz decay can well be approximated by 35

local dzz [arb.units]

30

2

25 20 1

15 10 5

1

10

100

1000

t [s]

Fig. 12.23 Time dependences of the induced piezoresponse in the epitaxial (001)-oriented PMN thin film measured after applying voltage pulses V = +3 V (1) and +9 V (2) during 0.5 s. The solid lines show the best fits of the slow stage of the relaxation process with a Kohlrausch–Williams– Watt-type dependence

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a Kohlrausch–Williams–Watt-type dependence: dzz(t)~exp[–(t/t)b], where t is the time, and t and b are the fitting parameters. Both the KWW type of the slow relaxation stage and the increase of the parameter b observed with increasing field indicate a narrowing of the relaxation time spectrum of PMN under higher dc bias. It should be mentioned that the characteristic time t is in the range of 20–30min as compared to seconds reported for the macroscopic experiments [148]. To explain the obtained results the following model was proposed. Applying short voltage pulses of moderate magnitude results in reorientation of the polar clusters and their subsequent coarsening in bigger entities, nanodomains. The slow relaxation kinetics reflects thermally activated breakup of these nanodomains. Longer relaxation times as compared to macroscopic experiments are attributed to the stabilization of the polar regions near the surface as expected under the experimental conditions of PFM. If the voltage is higher than the critical value, a macroscopic single domain ferroelectric state can be created. In this case, the relaxation is governed by depolarization process as in normal ferroelectrics [149]. The decay is much faster because of the existence of strong macroscopic depolarizing fields.

Outlook As was shown in the preceding sections, relaxor ferroelectrics have been intensively studied by PFM during the past several years. Significant progress has been achieved in understanding the appearance of the piezoelectric contrast on the surface of nominally cubic relaxors and those having intermediate properties between relaxors and “normal” ferroelectrics. It has been shown that nanodomains (actually agglomerates or surface-stabilized “polar nanoregions”) may exist far below the transition point and dielectric maximum. The mean size of the nanodomains is 50–150nm at room temperature and was found to decrease upon heating. The results can be interpreted as an evidence for a gradual transformation between the low-temperature ferroelectric state with static nanodomains and the high-temperature relaxor state with dynamic PNRs. Grain size effect in relaxor ceramics and films demonstrates that the degree of disorder in these materials varies as a function of the local position inside the grain. The surface polarization correlation radius can be directly extracted from these measurements and depends on the grain size. The variation of the correlation radius inside ceramics is still an open question and requires additional experiments. Undoubtedly, future theoretical models and experiments will allow better understanding of the nature of relaxor phases in ferroelectrics and determine their practical applications that are still to come. Acknowledgments Intense cooperation on SBN-type relaxor materials with Prof. J. Dec, University of Silesia, Katowice, Poland, is gratefully acknowledged. The work of V.V. Shvartsman has partly been supported by the EU STREP “MULTICERAL.” D.A. Kiselev is grateful to the Foundation for Science and Technology of Portugal for the financial support (SFRH/BD/22391/2005). The work of the Portuguese co-authors is partly supported by the project PTDC/FIS/81442/2006.

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Chapter 13

Symmetries in Piezoresponse Force Microscopy Andreas Ruediger

Conventional Interpretation of PFM Images The tremendous success of piezoresponse force microscopy (PFM) in various fields of material research has been driven by its promise to locally reconstruct the piezoelectric tensor and thus the polarization state by value and orientation. In order to achieve such a reconstruction, PFM generally allows for the collection of five different signals at once: topography, vertical amplitude, vertical phase, lateral amplitude, and lateral phase. In the most common configuration, the metalized AFM tip will be placed directly onto the sample surface without a top electrode. This configuration without top electrode will therefore be considered from now on; the difference to a system with a top electrode will become clear once the complexity of the common case is fully elaborated. Without restrictions to the generality of what follows, we will also consider the technologically relevant c-axis-orientated, tetragonal thin film configuration for all considerations. All other scenarios (with substantially lower crystal symmetry) follow from simple coordinate transformations. The c-axis is therefore perpendicular to the substrate plane and to the surface of the film. For the sake of simplicity, we chose the laboratory coordinate system x, y, z parallel to the crystallographic, orthogonal axes a, b, c. The complexity of the matter requires to carefully designate the constituents of a PFM setup. What is eventually labeled as “vertical piezoelectric amplitude” on the screen is in fact the local Lock-In signal of the four-quadrant photodiode (upper–lower). PFM relies on the assumption that this signal is a linear function of the vertical sample deformation as much as the “lateral piezoresponse amplitude” on the screen is a linear function of the lateral deformation. In that sense, conventional PFM represents a very special and rare case under realistic experimental conditions, as will be explained in what follows. A. Ruediger (*) Laboratory of Ferroelectric Nanoelectronics, Institut National de la Recherche Scientifique, Université du Québec, Québec, 1650, Blvd. Lionel-Boulet, Varennes, Canada, J3X 1S2 e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_13, © Springer Science+Business Media, LLC 2010

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The aim of this chapter is thus to identify and highlight the signal contributions from sample and device that contribute to ambiguities of the classical signal interpretation. Whenever possible, experiments to discriminate those contributions against the conventional PFM interpretation are suggested. The understanding of these contributions opens a way for various types of new experiments that will also be briefly mentioned.

Orientation of the Polarization and the Piezoelectric Tensor In a tetragonally distorted perovskite ferroelectric, the polarization will point along the long c-axis. The piezoelectric tensor is also pinned to the crystallographic axes,1 which makes piezoresponse the method of choice to identify a polarization reversal as an inversion of the piezoelectric tensor. While in many perovskites [1] the piezoelectric tensor element d33 is proportional to the polarization, the relative orientation of the polarization can be extracted from the phase shift of the piezoelectric response. Antiparallel domains are ideally 180° out of phase. With respect to the quantitative interpretation of PFM data, the point contact between tip and sample plays a crucial role. Under ambient conditions, an adsorbate layer covers the surface and acts as a voltage divider [2, 3], therefore, PFM data are typically overestimating the actually applied voltage and consequently underestimating the piezoelectric response by up to an order of magnitude. The work around to use stiffer cantilevers provides a better point contact but is extremely abrasive and modifies the electromechanical boundary conditions at the point of interaction. Under a stiff cantilever, the sample often suffers from nanoindentations that void the assumption of PFM as a method of constant stress. To conclude this section, conventional PFM will interpret a vertical deflection of the laser beam as a consequence of the vertical deformation due to d33. The existence of a lateral signal is necessarily associated with the existence of an in-plane polarization as illustrated in Fig. 13.1. Figure 13.1a, b shows an important detail: the ferroelectric is covered by a thick top electrode that assures a uniform electric field distribution underneath the tip, an idealized situation that does not apply for most PFM experiments. The next section will describe in some detail to what extent PFM can be considered a local probe.

Sometimes, especially in engineering literature on piezoelectric ceramics, e.g., d33 is indexed according to the laboratory coordinate system. This might sometimes be useful to facilitate the phrasing but the piezoelectric tensor is defined in the crystallographic coordinate system and can be transferred by a respective coordinate transformation into the laboratory system [4]. Often, especially in polycrystalline materials, only an effective piezoelectric tensor element out of plane can be determined that is not d33.

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Fig. 13.1 (a) Conventional explanation of the origin of a vertical PFM signal. The structure expands or contracts as a function of the (here: homogeneous) electric field between tip and bottom electrode due to d33. The lateral deformation due to d31 is not detectable. (b) PFM configuration that evokes a lateral signal. Such a scenario corresponds, e.g., to 90° domains in a ferroelectric thin film. For a polarization in a different direction, the two scenarios depicted in (a) and (b) are to be superimposed

Nonlocal Piezoelectric Response The mechanical contact between the tip and the sample surface has been subject to a sequence of publications and is determined by their mechanical properties and the applied forces. The resulting electrical contact is necessarily a function of the mechanical contact and cannot be considered as a point. For most experimental conditions, the exact contact conditions cannot be identified with satisfactory precision but as a rule of thumb, an area of pr2, where r denotes the tip curvature, is a reasonable approximation [3]. As a particularity of PFM, the tip simultaneously acts as an electrical stimulus and a mechanical sensor, coupling an external electric excitation field to the piezoelectric response.2 This external electrical field first needs to be translated into an electric field distribution inside the sample, a process that is mediated by the anisotropic dielectric permittivity tensor. A typical potential distribution is sketched in Fig. 13.2. The anisotropy can be considerable, e.g., 20 in BaTiO3, which means that the electric field distribution underneath the tip is far from being spherical [5, 6]. Wherever there is an electric field, a piezoelectric response will follow from the respective volume. The AFM tip as a stimulus is thus generating an electric field distribution in a volume underneath the tip while the same AFM tip as a mechanical sensor monitors the sum of the piezoelectric response integrated over the entire volume of the electric field distribution at the very point of contact. It is essential for the application and the understanding of PFM that it monitors the integral of the piezoelectric response at the point of contact. As a first consequence we see that in an ideal c-axis-oriented ferroelectric any lateral signal is prohibited by symmetry, which does not mean that there is no lateral piezo-

“simultaneous” is not meant in the sense that the piezoelectric response to an electrical signal is instantaneous, there might be both electrical and mechanical phase shifters in the tip–sample interface system.

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Fig. 13.2 Sketch of the radial potential distribution underneath the tip. For symmetries lower than tetragonal, the radial distribution can be elliptical. Depending on the dielectric anisotropy, the ellipsoid of the equal potential is either oblate or prolate

electric response in the volume. Directly underneath the tip, the electric field is along z while the electric field lines in general will be both perpendicular on the tip surface (semisphere) as well as on the bottom electrode. The resulting field distribution has radial symmetry around the point of contact. All vertical field components will result in a vertical movement which will add to the vertical deflection of the cantilever. For the lateral response, the situation is dictated by symmetry. We consider the point of contact as the origin of our laboratory coordinate system. For any lateral field vector at (x, y) there will be an oppositely directed electric field vector at (-x, -y). These two field vectors cause equal piezoelectric responses (lateral expansions or contractions due to d31) on opposite sides of the contact point the response of which will cancel out as depicted in Fig. 13.3. Despite the fact that the entire neighborhood of the contact is in lateral motion, the sum of all these movements at the point of contact is zero, a lateral signal is thus prohibited by symmetry in a c-axis-oriented ferroelectric. We will address this point again in the discussion of experimental findings of lateral piezoresponse in those materials.

Device Asymmetries This section addresses consequences of the detection scheme for PFM data. The optical lever method provides a high sensitivity for small sample deformations. Its realization by a mechanical cantilever and a laser that is reflected from the backside imposes a specific geometry that is not only inherent to PFM but to various other

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Fig. 13.3 Illustration of the radial symmetry around the contact point of a metalized AFM tip on a sample surface. The two areas indicated by black squares experience the same electric field and will therefore contribute equal piezoelectric response. Their effect on the tip–sample contact point therefore cancels out

types of AFM. The following considerations are related to the effect of the cantilever orientation with respect to structures on the sample surface.

Cantilever Axes: Optical Amplification During a PFM scan the cantilever is displaced line by line with a fast scanning direction (x) and a slow scanning direction (y). In addition to the choice of the fast and the slow axis, there is the degree of freedom for the long cantilever axis with respect to the sample surface. As we will see, the orientation of the long cantilever axis plays an important role for many aspects of PFM. Let us consider two scenarios: in the first one, the sample surface is moving one nanometer upwards while in the second one it moves one nanometer sidewise. What will be the resulting PFM signals? They will be different in their magnitude. A closer look at the cantilever geometry and the laser deflection method (Fig. 13.4) shows that the optical lever method is not monitoring the displacement of the tip itself but the corresponding angles.3 It is worthwhile mentioning at this point that the tip–sample contact point has three linearly independent directions of displacement. One vertical and two lateral ones. The photodiode can only monitor two different directions. This has dramatic consequences for the interpretation of the vertical PFM channel as will be discussed in the next section.

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Fig. 13.4 Deflection angles of the laser beam for equally large displacements that lead to vertical bending (a) and lateral torsion (b). The respective levers are determined by the length of the bending beam (a) and the cantilever height (b) which yields different optical amplifications for the vertical and the lateral channel

Peter et al. [7] calculated the different displacements of the laser beam on the photodiode for a constant displacement of the tip–sample contact point. The optical amplifications are determined by the cantilever beam geometry, in particular the height h and the length l. For a given cantilever the optical amplification ratio R between lateral and vertical PFM is determined by the following equation:

2L 3h where L denotes the cantilever length and h is the cantilever height between the contact point with the sample and the cantilever reflective coating, i.e., height of the tip plus the thickness of the cantilever. The factor of 2/3 accounts for the fact that the beam is bending (instead of staying stiff). For typical beam deflection cantilever dimensions in contact mode R is about 15–20. As a direct consequence, lateral piezoresponse scans are generally less noisy with a signal-to-noise ratio of approximately R1/2. In conclusion, vertical and lateral response must not be directly compared quantitatively as they differ by the optical amplification factor R. Ideally, both channels are calibrated by a piezoelectric standard. This procedure has to be repeated after any cantilever replacement and is quite tedious; experiments of this kind are therefore rare. R=

Deconvolution of Cantilever Deflection Modes:Mechanical Crosstalk As mentioned above, the tip–sample contact point has three linearly independent directions of displacement (x, y, z), one vertical and two lateral, that are translated into a vertical and a lateral PFM channel. What happens to the second lateral displacement? Figure 13.5 illustrates the three basic deformation modes of a cantilever. While the vertical displacement of the tip apex translates into a vertical bending and thus a vertical deflection of the laser beam onto the photodiode, a lateral displacement of

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Fig. 13.5 Three different deformation modes of the beam deflection cantilever. The respective elastic constants are considerably different which does not play a role unless the tip starts to slip. The two upper modes vertical bending (top) and vertical torsion (middle) both result in a deflection of the laser beam in a vertical direction. The lateral torsion causes a beam deflection in a lateral direction. This signal is therefore unambiguously transferred from the tip to the photodiode

the tip in the direction perpendicular to the long cantilever axis (along x) results in a lateral torsion and a lateral displacement of the laser beam on the photodiode. A displacement of the tip parallel to the long cantilever axis causes a vertical torque and a “buckling” of the cantilever. This buckling also causes a deflection of the laser beam in the vertical channel of the photodiode. The vertical channel of PFM thus contains both the information on the vertical piezoelectric response and the lateral piezoresponse along the long cantilever axis. These three modes of cantilever deformation are associated with three different spring constants, the torque constants are at least one order of magnitude larger than for the bending. However, as the tip is in contact with the sample surface and typically slightly deforming it, the tip will follow the movement of the sample surface at the contact point and slip does not occur. If slip occurred, it would be detectible in the amplitude and the phase signature as well. In conclusion, the vertical piezoelectric detection channel contains the sum of the vertical tip deflection and the lateral deflection along the long cantilever axis. Only the lateral channel contains a pure, lateral signal from the lateral torsion of the cantilever. Is it possible to achieve an unambiguous result from the vertical piezoelectric response? One way to deconvolute the signals in the vertical channel is to determine either the vertical torsion or the vertical bending contribution independently. After a simple rotation of the sample by p/2 the tip movement that caused a vertical torsion is now causing lateral torsion and vice versa. What is simple on a globally homogeneous sample, where the rotation axis can be anywhere, turns out to be experimentally demanding on a particular nanostructure of interest. It is necessary to place the rotation axis of the sample directly underneath the AFM tip with a precision better than the scan range which is typically restricted to 10mm. Those

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experiments have been conducted by different groups and turn out to be extremely time consuming. The sample has to be placed on an xy stage that is placed on a rotational stage that rests on the sample scanner. This structure tends to introduce mechanical resonances that can render PFM results entirely useless [8]. Even if a specific structure can be identified after the rotation, there is still an open question related to the subtraction of the signals in the vertical PFM channel that should now be possible. The lateral piezoelectric response after rotation has been monitored with a different optical amplification than the sum of this signal and the vertical piezoelectric response in the vertical channel prior to rotation. For the three possible forces Fx, Fy, and Fz acting on the tip the rotation matrix around an angle of p/2 in the x, y plane toggles Fx and Fy: π  cos 2   sin π  2  0  

π 2 π cos 2 0

− sin

 0   Fx   Fy    0   Fy  =  Fx  .     F  1   Fz   z  

So in principle this concept is suitable for a decoupling of the two contributions to the vertical channel; however, the noise in PFM data and the limited control of the tip position leave it a challenging task for the future to experimentally achieve two scans at the same position after a rotation and to subtract them with the respective optical amplification factor.

Alignment of the Photodiode: Optical Crosstalk The signal chain in a PFM experiment continues after the laser had been reflected from the backside of the cantilever to the photodiode. The reflective plane of the cantilever can be tilted around the aforementioned two angles. So for a tilt around the small bending angle j, irrespective of the origin of this tilt, the laser will be moving up and down on the four-quadrant photodiode while a tilt around the large torsion angle a causes a left–right movement of the laser on the photodiode. Taking into consideration the optical amplification ratio, the center of the laser beam will thus move on a horizontally elongated surface. Let us consider a perpendicular incidence of the laser beam onto the photodiode for the sake of simplicity. This leaves one degree of freedom for the rotation of the photodiode around the k-vector of the incident beam, i.e., the surface normal of the photodiode. For any given angle g of this rotation that is unequal zero, some part of the vertical deflection of the laser will be detected in the lateral channel and vice versa according to the following equation:

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Sv = Sv0 + Sl0 ·sin γ , where Sv denotes the effective vertical signal as detected in Fig. 13.6b that is composed of the initial vertical signal Sv0 (assumed to be only moderately changing in absolute values) and Sl0 denotes the lateral signal that is mixed to the vertical channel. Given the different optical amplifications for lateral and vertical displacements of the cantilever tip, the coupling from the lateral into the vertical channel will be much more pronounced than for the other direction. For an optical amplification ratio of 20, already a misalignment angle g of only about 3° will couple a portion of the lateral signal into the vertical channel that corresponds to the average vertical signal, larger misalignment angles will see the lateral signal dominate both channels, a rather frequent observation in PFM. In fact, in various cases, both vertical and lateral PFM channel look quite the same, which requires the aforementioned level of understanding of the signal origin. This optical crosstalk is by no means restricted to PFM and has regularly been rediscovered throughout AFM literature. Soergel et al. [9] published an interesting electronic approach to compensate for the crosstalk. Knowing that the first harmonic vertical oscillation at resonance is free of lateral contributions, this signal can be used to tune the vertical response to a maximum and the lateral signal to a minimum. While a mechanical rotation is generally unfeasible with the existing microscopes, the AC signals can be mutually subtracted in a small electronic circuit described in [9]. It may turn out that a vertical excitation requires some knowledge of the microscope as the standard non-contact holders that allow for a piezoelectric stimulation of the cantilever oscillation are typically not suitable for a subsequent PFM measurement. In conclusion, identical vertical and lateral PFM signals are often an indicator of a misaligned photodiode. This can be electronically compensated, a procedure that should be applied after any modification to the PFM setup, including cantilever changes.

Fig. 13.6 Effect of a tilted photodiode with respect to the laboratory coordinate system by an angle g. The red area illustrates the illuminated area of the photodiode (exaggerated) reflecting the difference in optical amplification between lateral and vertical piezoresponse

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Beam Asymmetry The linearity of PFM detection depends on the small angle approximation in the optical lever method as well as the beam profile of the laser beam on the photodiode. For small variations close to the maximum of the beam intensity profile, the Gaussian beam profile or that of a solid state laser diode is almost constant. So for small deflections of the laser beam the difference of intensity for the respective channels (up–down or left–right) is proportional to the displacement of the laser beam. For larger displacements of the order of the beam width, the difference in intensities is no longer linear in the displacement of the beam. More frequent than large displacements is a poorly focused laser or scattering from the cantilever. If the diameter of a distorted beam is larger than the photodiode the difference in intensities will not only be determined by the difference of intensities close to the center of the photodiode but also by the intensity that enters at the perimeter of the photodiode. A regular optical inspection of the beam quality is therefore vital for the interpretation of PFM results.

Sample Asymmetries While the previous section highlighted frequent technical subtleties of the PFM signal detection that can become misleading in the interpretation of images, this paragraph will now address sample properties that contribute to a signal in the lateral PFM channel. The lateral channel is prone to pick up all different types of asymmetries as we will see in this section. In previous section we have derived from symmetry considerations that a lateral response on a c-axis-oriented tetragonal ferroelectric thin film is prohibited. This consideration relied on a global symmetry that the tetragonality is preserved anywhere on the sample surface. Following Neuman’s principle, a material tensor has at least the symmetry of the crystal structure. This important theorem of solid-state physics implies that the dielectric and the piezoelectric tensor will also have tetragonal symmetry. The only deviations from a globally homogeneous and symmetric sample that we have considered so far are ferroelectric domains. However, there is a lot more to a sample and its surface than just that and several of these contributions will subsequently be investigated with respect to their impact on the movement of the point of contact.

Topography Uneven surfaces create the simplest asymmetry underneath the tip as they partially remove material on one side of the tip. So the material response from one side is no longer completely counterbalanced by the opposite side. For the case of a lateral expansion

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Fig. 13.7 Sketch of the strongest electric field vector in the proximity of the surface depending on the position on a sloped surface. A detailed description can be found in [6]. The field vector has to be perpendicular on the metal tip and on the bottom electrode, therefore the lateral component of the electric field increases with the derivative of the topography

from the side of the material excess, the tip is pushed towards the other side. For the case of a contraction, the tip often loses contact which creates additional noise. The scenario of the piezoelectric response on a slope of a c-axis-oriented ferroelectric was numerically simulated by a finite element approach by Peter et al. [6]. The situation is typically too complex for an analytical expression, especially as the electric field distribution in-plane now has to be quantitatively considered (Fig. 13.7). As the most important consequence of topography on PFM, we notice the occurrence of a signal in the lateral PFM channel that only originates from the uneven surface (Fig. 13.8). In conventional PFM such a signal would necessarily be interpreted as an in-plane component of the polarization vector. However, in reality, this signal is well present even in perfectly c-axis-oriented films. Figure 13.8 also illustrates the relation of the lateral piezoelectric response perpendicular to the long cantilever axis and the derivative of the topography in the direction perpendicular to the long cantilever axis. Figure 13.9 sketches the two different tip positions with respect to the nanoisland. Again, the scan direction is of no importance to these findings, the orientation of the long cantilever axis determines the detection scheme. Induced Topography On materials with 180° domains such as LiNbO3, the nonlocal piezoelectric response in the proximity of a domain wall causes an expansion of one domain and the contraction of the adjacent one. So a sample that had initially been flat will now experience a tilt at the very domain wall. Following the same line of argument as in the previous paragraph, this tilt represents an uneven surface and therefore causes a signal in the lateral channel. Scrymgeour et al. [11] reported on this effect

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Fig. 13.8 Topography (a), vertical PFM amplitude (b) and lateral PFM amplitude (c) on a BaTiO3 nanoisland grown on (100) SrRuO3 on (100) SrTiO3. The scan range is 250 by 250 nm. The polarization vector is out of plane and is expected to be constant. However both vertical and lateral response show a pronounced perimeter enhancement. The key to its understanding is the reduction of radial symmetry at the respective slopes of the grain. The vertical response in particular stems from mechanical crosstalk from vertical torsion into the vertical channel. The factor of 8 was quantitatively described in [10] within 5% experimental error and is illustrated in Fig. 13.10

Fig. 13.9 Different channels of detection of a lateral response on a ferroelectric nanoisland. The upper image illustrates the lateral torsion as we look along the long cantilever axis while the lower sketch indicates the case of vertical torsion. The details of the lower case are described in the next figure. The upper scenario describes the origin of the perimeter enhancement in the lateral channel while the lower one describes the crosstalk into the vertical channel

in periodically poled LiNbO3. It should be mentioned that Jungk et al. [12] s uggested an alternative model for this tilt that they attribute to induction charges on the PFM cantilever. If the argument of effective surface charges under AC

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c onditions of a PFM experiment holds true it requires further investigations that are currently on the way. Topography Crosstalk The symmetry reduction due to slopes on an uneven sample surface works in both principle axes of the cantilever symmetry, i.e., perpendicular to the long cantilever axis this signal causes a lateral signal that is prohibited by symmetry and that can therefore clearly be identified as an artifact. Parallel to the long cantilever axes, the lateral deformation of the cantilever will appear in the vertical channel. Peter et al. [10] observed an optical amplification ratio of the vertical signal that did not correspond to the expected values for the cantilever geometry. A direct comparison to the topography that provided a geometry factor of tan v/l as illustrated in Fig. 13.10 revealed that the tip slips on the slope. Details can be found in [10].

Local Heterogeneities Even on flat surfaces the lateral symmetry of the sample structure can be broken by, e.g., misfit or threading dislocations. These local heterogeneities act on the permittivity tensor as well as on the piezoelectric tensor and the exact description of the physical properties is still in its infancy. While the coupling of homogeneous strain to both tensors is well understood, only very recent phase field simulations started to bridge the gap between the dislocation core the properties of which are governed by quantum mechanics and the bulk ferroelectric. But regardless of the possible complexity of the tensor properties in the proximity of dislocations, their strain

Fig. 13.10 Difference between mechanical crosstalk for a tip that sticks on the surface (left) and a tip that slips on the surface (right). Both scenarios are possible; however, their optical amplifications should be different due to the different optical levers. In the left case, the ratio between lateral perimeter enhancement and vertical perimeter enhancement should be unity. The right case corresponds almost to vertical bending with an expected R of 1:18 for the particular cantilever geometry. The experimentally determined value of 8 (cf. Fig. 13.8c) is quantitatively explained if the slope of the nanoislands is taken into consideration. In the case of this BaTiO3 nanoislands the slope was independently determined and confirmed the ratio of the optical amplifications within 5% error margins

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Fig. 13.11 Cross-section of a flat ferroelectric thin film with a c-axis-oriented ferroelectric (top) that has a local variation of the dielectric permittivity. For the case of these numerical simulations, the permittivity was enhanced by only 10% relatively to the bulk value. The lower figure plots the lateral (or in-plane) piezoelectric response as a function of the tip position. The coordinates of both figures coincide

fields reach about as far as the electric field from the tip reaches into the sample, typically a few tens of nanometers. As the tip approaches a local heterogeneity from any given direction, the field distribution underneath the tip will not overlap with the volume into which the local heterogeneity reaches out. With decreasing distance between the tip and the heterogeneity, the field will penetrate into this volume and interact with different permittivity and piezoelectricity in the bulk as illustrated in Fig. 13.11. Typically, the piezoelectric response will be reduced as compared to the bulk. This locally modified piezoelectric response breaks the radial symmetry around the tip and unbalances the forces on the tip–sample contact point; as a result, the cantilever senses a lateral torsion. This torque on the cantilever increases while the tip approaches the heterogeneity as simulated by Peter et al. [13] by finite elements for only a small local variation of the permittivity. As the tip is directly above the heterogeneity (assuming that it only extends along z), the symmetry is restored at this very point and the lateral response drops to zero. On the opposite side of the heterogeneity, the lateral signal

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Fig. 13.12 Topography (left) and amplitude of the lateral piezoelectric response (right) of heteroepitaxial PZT nanoislands on (100) SrTiO3:Nb. The islands are flat and c-axis oriented. Nonetheless, the lateral response shows a wealth of information on the local structure that cannot be attributed to a lateral polarization and probably stems from the superposition of different types of dislocations and substrate steps. The scan range is 500 by 500 nm

jumps to a high value again and slowly decays with the distance between the tip and the heterogeneity. Note that the phase of the lateral signal is 180° out of phase for the opposite sides of the heterogeneity. Structures with this kind of signature have been reported for conventional piezoelectric response force microscopy scans (Fig. 13.12) as well as for scans at double frequency that are sensitive to the electrostriction of a sample.

Substrate Signatures While the previous section discussed the heterogeneities with respect to intrinsic sample defects such as dislocations, there is another important contribution to strain fields in (ferroelectric) thin films. Heteroeptitaxial thin films with the polarization perpendicular to the substrate plane are grown on (100) faces of conductive perovskite lattices such as SrRuO3 or SrTiO3:Nb. These substrate surfaces are never ideally flat but exhibit terraces that can be etched to one unit cell height with a uniform termination and that are typically several dozen nanometers wide depending on the miscut angle. In very good samples, the terrace width can be as large as several hundred nanometers. However, a terrace step imposes irregular electromechanical boundary conditions to the thin film. Even for an ideal match of in-plane lattice constants and a fully commensurate lattice, the out-of-plane lattice constants do not match at terrace steps, the ferroelectric that grows on such a terrace step is vertically compressed which leads to strain fields. These fields have been measured on PZT nanoislands grown on Niobium-doped SrTiO3 substrates [14] and are clearly visible on flat surfaces where a lateral response is otherwise prohibited.

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Conductivity Most of the ferroelectrics that are under investigation by PFM are wide bandgap semiconductors with only a very small conduction. Under illumination, this conduction can be enhanced, a frequent observation in thin films that are exposed to ambient light sources. This conduction is not necessarily due to band–band excitation as already dopant concentrations in the order of a few parts-per-million are known to create a noticeable light-induced charge transfer when pumped at the appropriate wavelength. In addition to this, the high absorption coefficients for band–band excitation create a photoconduction in a thin layer of only a few tens of nanometers. What is generally negligible in bulk becomes dominant in films with an overall thickness of the same order. The interaction of free carriers with PFM stems from the migration of carriers along the electric field towards or away from the contact point. If the response of these free carriers to the electric field was instantaneous, the screening of the electric field would be symmetric around the tip leaving the local radial symmetry unaltered. However, free carriers will only respond within the Maxwell time constant, determined by the conductivity and the permittivity. The space charge distribution that follows the electric field underneath the tip is therefore retarded and follows the tip with a finite delay. As a result, the screening is now asymmetric and should induce a torque on the tip [15]. This effect has not yet been observed beyond doubt but it should provide a lateral signal that linearly increases with the scan speed along the fast axis.

Contact Asymmetries The aforementioned sections were devoted to the reduction of radial symmetry around the tip as they result in a lateral signature that should not be present within the conventional interpretation of PFM. This section now addresses the properties of the point contact and two different effects that are known to or might contribute to unexpected results in PFM.

Tip Curvature No AFM tip, commercially available or homemade, meets the assumption of a hemisphere and often the tip radius is a function of the radial angle in the surface plane (xy). Two examples of commercially available AFM tips are given in Fig. 13.13. Different tip curvatures do imply different air gaps (generally filled with surface water due to capillary forces) on the respective sides of the tip which again causes a radial asymmetry. Different from the scenarios discussed in the previous paragraphs, this asymmetry is constant as long as the tip does not change its shape and therefore only causes

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Fig. 13.13 SEM images of typical metal-coated AFM tips prior to PFM scans. Even the tips in the virgin state are asymmetric and prone to a pronounced reduction of the radial symmetry of the electric potential around the tip. This effect causes a constant torque on the tip independent of the position on the sample and can therefore be subtracted as a constant background

a constant background signal in the lateral channel. There is no way to compensate for it, it is an omnipresent source of error that provides a nonzero lateral signal.

Bow Waves As the AFM tip is scanning the surface, it is dragging through a viscous medium of surface water, typically few nanometers thick. This might give rise to a kind of bow wave in front of the PFM tip which again causes a radial asymmetry that depends on the scan speed just as for a conductivity of the sample. To date, there are no systematic studies on the effect of scan speed on the lateral PFM channel. Both the effect of viscous surface water as well as for sample conductivity should become detectable for large variations of the scan speed, which in itself might introduce additional sources of error.

Summary PFM implies an important symmetry as the point of tip–sample contact is very often the center of symmetry for the electric field distribution and therefore the piezoelectric response. For the undisturbed translational symmetry of an ideal, flat

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crystal, the piezoelectric response should therefore create zero (for tetragonal symmetry) or a constant (lower than tetragonal symmetry), but always locally invariant lateral piezoelectric response. Additional image contrast in the lateral channel is created by any kind of symmetry reduction in the proximity of the tip–sample contact. The most important physical reasons that cause such an image contrast are an uneven topography, domain walls, dislocations, and substrate steps. They all have a specific fingerprint in PFM and can therefore be identified and discriminated against each other while this lateral signal must not be interpreted as the presence of an inplane polarization. In other words, a lateral PFM signature is a necessary, but not a sufficient criterion for an in-plane polarization. Acknowledgement The author would like to thank Mischa Nicklaus for his assistance with the figures.

References 1. D. Damjanovic, J. Am. Ceram. Soc. 88, 2663 (2005) 2. F. Peter, K. Szot, R. Waser, B. Reichenberg, S. Tiedke, J. Szade, Appl. Phys. Lett. 85, 2896 (2004) 3. S.V. Kalinin, D.A. Bonnell, Phys. Rev. B, 65, 125408 (2002) 4. I.P. Kaminov, Introduction to electro-optic devices (Academic Press, NY, 1974) 5. T. Otto, S. Grafstroem, L.M. Eng, Ferroelectrics 303, 149 (2004) 6. F. Peter, A. Rüdiger, R. Dittmann, R. Waser, K. Szot, B. Reichenberg, K. Prume, Appl. Phys. Lett. 87, 082901 (2005) 7. F. Peter, A. Rüdiger, R. Waser, K. Szot, B. Reichenberg, Rev. Sci. Instr. 76, 046101 (2005) 8. T. Jungk, A. Hoffmann, E. Soergel, Appl. Phys. Lett. 90, 163507 (2006) 9. A. Hoffmann, T. Jungk, E. Soergel, Rev. Sci. Instr. 78, 016101 (2007) 10. F. Peter, A. Rüdiger, R. Waser, Rev. Sci. Instr. 77, 036103 (2006) 11. D.A. Scrymgeour, V. Gopalan, Phys. Rev. B 72, 024103 (2005) 12. T. Jungk, A. Hoffmann, E. Soergel, Appl. Phys. Lett. 89, 042901 (2006) 13. F. Peter, A. Rüdiger, R. Waser, K. Szot, B. Reichenberg, Rev. Sci. Instr. 76, 106108 (2005) 14. M.W. Chu, I. Szafraniak, R. Scholz, C. Harnagea, D. Hesse, M. Alexe, U. Goesele, Nat. Mater. 3, 87 (2004) 15. F. Peter, B. Reichenberg, A. Rüdiger, R. Waser, K. Szot, IEEE Trans. Ultrason. Ferroelectrics Freq. Contr. 53, 2253 (2006)

Part V

Novel SPM Concepts

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Chapter 14

New Capabilities at the Interface of X-Rays and Scanning Tunneling Microscopy* Volker Rose, John W. Freeland, and Stephen K. Streiffer

Introduction Nanoscale structures are at the forefront of fundamental research as well as the keystone for whole new classes of potential applications. Beyond doubt, the ability to manipulate and organize matter at small length scales together with the utilization of functional structures has enormous potential to change society. It is anticipated that global research efforts will lead to new medical treatments, more efficient energy production, lighter and stronger materials, advances in agriculture, and powerful nanoelectronics [1], and these are just a few of the more significant ways in which people are discussing the use of nanotechnology. Certainly, nanoscale science and engineering will be an essential component in gaining a better understanding and control of nature in the next decades. The fascination of nanoscience and nanotechnology is driven by the fact that nanoscale structures often exhibit novel physical, chemical, and biological properties, substantially different from those displayed by bulk materials. The study of these new, emerging phenomena will facilitate an exciting new voyage of discovery. However, proper understanding of nanoscale systems though requires tools with both the ability to resolve nanometer structure and to provide detailed information about chemical, electronic, and magnetic state. Scanning probe microscopies achieve the requisite high spatial resolution; however, direct elemental determination is not easily accomplished with scanning tunneling microscopy (STM) or other scanning probe variants. Only in very specific cases do various * The submitted manuscript has been created by the University of Chicago as Operator of Argonne National Laboratory (“Argonne”) under Contract No. DE-AC02-06CH11357 with the US Department of Energy. The US Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. V. Rose (*) Advanced Photon Source Argonne National Laboratory, Argonne, IL 60439, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_14, © Springer Science+Business Media, LLC 2010

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effects lead to chemical contrast [2, 3, 4]. X-ray microscopies, on the other hand, provide elemental selectivity, but currently have spatial resolution of typically only tens of nanometers [5, 6]. Here we describe a radically different concept for high-resolution microscopy that utilizes the detection of local X-ray interactions by a scanned probe, in which the scanning probe provides spatial resolution and X-ray absorption directly yields chemical, electronic, and magnetic sensitivity. The strength of X-rays is the ability to excite core electrons of a specific level by tuning the incident photon energy to the binding energy. Hence, specific excitations allow discrimination between different chemical species. The combination of the spatial resolution of STM with the energy selectivity afforded by X-ray absorption spectroscopy (XAS) provides a powerful analytical tool. In Section “Basic Interactions of X-Rays with Matter” of this review, we will first outline the basic interactions of X-rays with matter. The fundamental concepts of electron tunneling under X-ray illumination are discussed in Section “The Physics of X-Ray-Enhanced Scanning Tunneling Microscopy.” Several groups worldwide are working on the integration of STM and synchrotron-based radiation. An historical outline and important achievements are highlighted in Section “The Development of Synchrotron Radiation-Enhanced Scanning Tunneling Microscopy.” The utilization of STM in combination with synchrotron-based radiation necessitates the development of specialized insulator-coated tips, which is the subject of Section “Fabrication of Insulator-Coated Smart Tips.” Generally, we distinguish between experiments in which the tip/sample separation lies within the tunneling regime (near field), and separations that exceed the tunneling distance for STM (far field). Photoelectron detection in the far field is discussed in Section “Photoelectron detection using a scanning tunneling microscope”, and considerations about nearfield studies are presented in Section “X-ray assisted scanning tunneling microscopy”. Concluding remarks about the new capabilities at the interface of X-rays and STM are recapitulated in Section “Concluding Remarks.”

Basic Interactions of X-Rays with Matter There are five basic interactions of X-rays with matter: elastic (coherent) scattering, Compton scattering, the photoelectric effect, pair production, and photodisintegration. The latter two are only possible at very high photon energies (>1.02 and > 8MeV, respectively), and thus, are not further considered here. In an elastic scattering event (Fig. 14.1a), also called Rayleigh scattering, an X-ray photon is elastically scattered, i.e., only the direction of the photon changes. Ionization does not occur. An event in which the incident X-ray photon loses energy (increases its wavelength) and is deflected from its original path by an interaction with an electron is called Compton scattering (Fig. 14.1b). The photon continues its travel on an altered path, while simultaneously the electron containing the remaining energy is ejected. The photoelectric effect is a quantum electronic phenomenon in which electrons are

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Fig. 14.1 (a) An incident X-ray photon is elastically scattered. Ionization does not occur. (b) In Compton scattering, an incident X-ray photon is deflected from its original path by an interaction with an electron. An electron is ejected from its orbital position and the X-ray photon loses energy

Fig. 14.2 Schematic representations of the photoelectric effect. (a) An incident photon ejects a photoelectron. The photon must have an energy greater than the binding energy of the electron. Subsequently, there are two competing paths for energy dissipation. (a) The vacancy is filled by a second atomic electron from a higher shell, and a third electron, an Auger electron, escapes carrying the excess energy. (b) The vacancy is filled by an electron from a higher shell and an X-ray photon removes the energy

emitted from matter after the absorption of photons (Fig. 14.2a). The probability of the photoelectric effect occurring depends on a number of factors. Obviously, the incident photon must have energy greater than the binding energy of the electron that gets ejected. The X-ray absorption coefficient for photoabsorption decreases smoothly with increasing photon energy. However, when the photon energy reaches one of the deep inner-shell ionization energies of an atom, a sharp jump (absorption edge) marks the opening of an additional photoabsorption channel. The removal of an inner shell electron produces a vacancy, which either results in an Auger cascade (Fig. 14.2b) or an X-ray emission (Fig. 14.2c) by the following mechanisms.

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In an Auger process, a second electron from a higher shell fills the inner-shell vacancy. The energy difference between the two shells must be simultaneously released. Consequently, a third electron, the Auger electron is ejected, carrying the excess energy in a radiationless process; hereby, the excited ion decays into a doubly charged ion. This is the most probable process for low-atomic number elements during which a K-level electron is ejected by the energy absorbed from the X-ray photon, an L-level electron drops into the vacancy, and another L-level electron is ejected. Highatomic number elements have LMM and MNN transitions that are more probable. In contrast to an Auger process, a characteristic X-ray photon can remove the energy when the inner shell vacancy is filled by a higher-level atomic electron (Fig. 14.2c). The probability of a core hole in the K or L shells being filled via a radiative process, also referred to as fluorescence yield, increases with atomic number [7]. In addition to the primary photoelectrons (from the photoabsorption of the incident X-rays) and primary Auger electrons (from the de-excitation after photoionization), inelastic interactions with valence electrons of atoms can lead to the emission of secondary electrons. Such secondary photoelectrons are caused by photoabsorption of fluorescent radiation in the sample, and secondary Auger electrons from the relaxation of secondary excited atoms. Electron emission is then dominated by low-energy (a few electron volts) secondary electrons [8, 9]. Utilizing the polarization of X-rays allows for probing of magnetic properties of matter. Generally, in X-ray absorption an atom absorbs a photon, giving rise to

Fig. 14.3 This simplified model illustrates the origin of X-ray circular dichroism, which is caused by the excitation of spin-orbit split 2p3/2 and 2p1/2 levels to empty d valence states. Right circularly polarized and left circularly polarized photons transfer an opposite angular momentum to the excited photoelectrons, and electrons with opposite spins are created in the two cases

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the transition between a core level and an empty state above the Fermi level. X-ray magnetic circular dichroism (XMCD) measures the dependence of X-ray absorption by a magnetic material on the helicity of the X-rays [10, 11]. It is the X-ray equivalent of the Faraday or Kerr effects, which occur in the visible range. The underlying physical mechanism of XMCD can be illustrated in 3d metal systems by a two-step model (Fig. 14.3) proposed by Stohr and Wu [12, 13]. Magnetic properties of 3d transition metals are determined primarily by their d valence electrons [14]. The core level is split into p3/2 and p1/2 states, where spin and orbit are coupled parallel and antiparallel, respectively. In the first step, the emission with the light helicity vector parallel (antiparallel) to the 2p orbital moment results in photoelectrons of preferred spin up (down) direction. In the second step, the spinsplit valence band acts as a detector for the spin of the excited electrons. Theoretical sum rules relate the integrated difference signal in absorption between left- and right circularly polarized (LCP, RCP) X-rays at the 2p absorption edges to the ground-state magnetic moment of the 3d transition metals [15, 16]. Hence, due to the dipole selection rules, final states with different symmetry can be probed by choosing the initial state.

The Physics of X-Ray-Enhanced Scanning Tunneling Microscopy The development of the family of scanning probe microscopes was initiated by the original invention of the STM by Gerd Binnig, Heinrich Rohrer, and co-workers from the IBM Research Laboratory at Rüschlikon more than a quarter century ago [17, 18]. The STM is based on a quantum mechanical effect. When a sharp conducting tip is brought close to a conductive sample, a bias voltage can allow electrons to tunnel through the vacuum gap between them. As the tip is scanned across the sample surface, while a feedback loop keeps the tunneling current constant by adjusting the tip-sample separation. In this fashion, the STM tracks the topography of the sample, assuming that the local density of states related to the tunneling process is independent of topography. The complication is therefore obvious – more precisely, the STM measures a mixture of both topography and the local STM density of electronic states. The tunnel current I tunnel depends exponentially on both vacuum gap distance and the local barrier height EF. Hence, the STM delivers realspace information on the morphology and electronic structure of surfaces with atomic resolution [19]. The applied bias voltage between the tip and the sample determines the direction of the electron tunneling. When the sample is negatively biased (Fig. 14.4a), electrons tunnel from the occupied states of the sample into the unoccupied states of the tip. In the case of a positively biased sample (Fig. 14.4b), electrons tunnel from the occupied states of the tip to the unoccupied states of the sample. Only electrons close to the Fermi energy EF contribute to the tunneling current. Unfortunately, these electrons do not necessarily carry any direct chemical information about their originating atoms.

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Fig. 14.4 Bias dependence for conventional STM. (a) When the tip is positively biased, electrons tunnel from the occupied states of the sample to the unoccupied states of the tip. (b) If the tip is negatively biased, electrons tunnel from the occupied states of tip to the unoccupied states of the sample. Only electrons close to the Fermi energy EF contribute to the tunneling. The tunnel current I STM depends exponentially on both vacuum gap distance and the local barrier height EF tunnel

The combination of STM with synchrotron-based X-ray radiation has the potential to enable element-specific microscopy that goes beyond the well-established topographic contrast. If during tunneling a sample is illuminated with monochromatic X-rays (Fig. 14.5), characteristic absorption will arise. When the excitation energy is tuned to a core level energy EC, a jump-wise increase in the number of excited electrons occurs, and core electrons are excited into unoccupied states above the Fermi level EF. Depending on the sign of the applied bias between tip and sample, those excited electrons might increase or decrease the conventional tunnel STM STM current I tunnel . When the sample is negatively biased (Fig. 14.5a), I tunnel may be X-ray enhanced by an additional tunneling current I tunnel that originates from the tunneling of excited electrons through the vacuum barrier into the tip. Because the energy of the incident X-rays is well known, the increase in the total tunnel current could potentially directly reveal chemical information of the sample surface. Tunneling is also well localized; thus, high spatial resolution is obtained. When the sample is positively biased (Fig. 14.5b), the excitation of core-level electrons in the sample STM manifests itself as a decrease of I tunnel . In this case, electrons tunnel from occupied states of the tip to unoccupied states of the sample. Yet excited electrons are lifted into states close to EF, diminishing the number of available empty states for tunnelSTM ing. Hence, independent of the applied bias the modulation of I tunnel caused by variations in the energy of the incident X-rays allows a chemical mapping of the surface. Due to the nature of tunneling, atomic resolution may be expected.

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Fig. 14.5 Schematic representation of the local density of states for X-ray-enhanced electron tunneling. (a) Incident X-rays excite core electrons with energy EC to unoccupied states close to X-ray STM the Fermi energy E F. The conventional tunnel current I tunnel is then enhanced by I tunnel . (b) When the sample is positively biased, electrons that are excited to states close to the EF reduce the number of available empty states for tunneling from the tip

Although the concept that we outline above takes advantage of the full potential of STM in terms of spatial resolution, so far only excitation of electrons to energies below the work function EF are considered. But if an electron absorbs the energy of a photon and has more energy than EF , it is ejected. In addition, inelastic interactions in the sample can cause the emission of secondary electrons. Photoejected electrons will therefore always provide an additional channel for chemical mapping. Figure 14.6 shows a schematic representation for the origin of different contributions to the photocurrent. Because the separation between tip and sample is only a few nm under tunneling condition, the tip always gets illuminated in an actual experiment. Therefore, photoejected electrons are generated at the sample as well as at the tip. The photoelectrons emerge with all velocities from zero up to a more or less sharply defined maximum velocity [20]. Electrons that escape from sample the sample and are detected at the tip cause a current I pass , while the remaining sample electrons that escape without detection carry a current I loss away. Simultaneously, tip tip currents I pass and I loss are generated, which describe electrons that leave the tip and reach the sample, or electrons that escape into the continuum without detection at the sample, respectively. Generally, the number of photoejected electrons exhibits a sharp jump when the photon energy reaches one of the inner-shell ionization energies. This accounts for the chemical sensitivity of the photocurrent channel. It is clear that the number of electrons in each channel depicted in Fig. 14.6 strongly depends on the bias between tip and sample. In particular, secondary electron emission is dominated by low-energy electrons, which are extremely sensitive to an electric field and, hence, can in principle be varied via application of low voltages.

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Fig. 14.6 Electrons with energy greater than EF can escape from the sample and generate a phosample sample tocurrent, which consists of I pass (electrons that pass to the tip) and I loss (electrons that escape into the continuum). Because the tip always gets illuminated in an actual experiment, tip tip additional photoelectrons I pass and I pass are generated at the tip

To sum up, when a sample is illuminated with X-rays, the conventional tunnel current X-ray STM I tunnel experiences a modulation due to an absorption-induced tunnel current I tunnel , and contributions caused by photocurrents. The resulting signal for synchrotron X-rayenhanced scanning tunneling microscopy (SXSTM) at a negatively biased sample is tip tip sample STM X-ray I SXSTM = I pass + I loss − I pass − I tinnel − I tunnel .

(14.1)

In this convention, currents that arrive at the tip are counted as negative, while currents that leave the tip have a positive sign. When the sample is positively biased, the SXSTM signal is tip tip sample STM X-ray I SXSTM = I pass + I loss − I pass + I tinnel − I excited ,

(14.2)

where I excited denotes the reduction of tunnel current caused by the excitation of electrons into unoccupied sample states close to EF (cf., hatched area in Fig. 14.5b). After electrons are ejected from either the sample or the tip they are accelerated in the electric field generated by the bias voltage between tip and sample. If the sample is positively biased (Fig. 14.7a) photoelectrons are prone to be recollected by the sample, while the tip due to the relatively negative potential repels those photoelectrons. The situation is reversed when the sample is negatively biased (Fig. 14.7b). Thus, the sign and the size of the bias voltage allow for controlling the X-ray

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Fig. 14.7 The bias between tip and sample determines the number of photoejected electrons that sample tip arrive at the tip ( I pass ) or at the sample ( I pass ). (a) A negatively biased tip repels photoelectrons from the sample, while (b) a positively biased tip attracts those electrons. Electrons that are ejected sample tip from the tip or sample without detection carry a current I loss and I loss away, respectively

photoelectron channel. Particularly, the detection of low-energy secondary electron emission can be tuned to meet the experimental needs.

The Development of Synchrotron Radiation-Enhanced Scanning Tunneling Microscopy The concept of achieving elemental information with ultra-high spatial resolution by combining X-ray absorption and STM is based on two different experimental realizations. In the spectroscopy mode (Fig. 14.8), an STM tip is positioned over a well-defined surface area. Then, the energy E of the incident X-rays is scanned and the tip current I is recorded while the tip remains at the same height over the surface. Equivalently, the change of the tip height z can be recorded when I is kept constant. The spectra obtained on different surface positions directly yield local chemical information. In the imaging mode (Fig. 14.9), a region of interest is first scanned in the conventional constant-current mode. Then, the same region is scanned again under monochromatic X-ray illumination. If the X-ray energy E is tuned to an absorption edge of an elemental species that is present at the surface, the SXSTM current will be modified at those locations as described in Section “The Physics Of X-Ray-Enhanced Scanning Tunneling Microscopy.” In order to keep the current constant, the tip height z is adjusted by a feedback loop. By subtracting the previous obtained topographic scan from the one under X-ray illumination, an elementsensitive image is obtained. In the case of samples with low corrugation, an element-sensitive image can also be achieved by scanning in the constant-height mode. Here, instead of analyzing the tip height z, the tip current I is studied. A few years after the invention of STM, the idea of combining this technique with photon excitations was proposed. Walle et al. [21] presented the first photoassisted STM measurements utilizing a HeNe laser and a halogen lamp in 1987. The team demonstrated the feasibility of using photoexcitation on semi-insulating

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Fig. 14.8 Spectroscopy mode of SXSTM. The tip is positioned over the sample surface and the energy E of the incident X-rays is varied. X-ray absorption spectra are obtained by recording the tip current I at constant tip height (or tip height z at constant current) as a function of X-ray energy

Fig. 14.9 Schematic representation of the SXSTM imaging mode. A region of interest is first scanned in the conventional STM constant current mode. In a second step, the same area is scanned under X-ray illumination with energy E. The subtraction of both scans yields an elementselective image

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material for the application in STM. In their accompanying discussions they already envisioned the possibility of investigating infrared-excited vibrations of adsorbed molecules and electronic states with STM. This optically-assisted spectroscopy would distinguish between different chemical species on a surface and convey information impossible to obtain solely with STM. This idea was widely adapted in the following years and launched various research activities in the emerging field of photoassisted STM [22, 23]. We would like to mention that ongoing efforts utilizing the combination of a related technique, atomic force microscopy (AFM), and synchrotron-based radiation have likewise let to considerable advances in nanoscale characterizations [24, 25]. In 1995, exactly 100 years after Röntgen discovered what he later called X-rays, Tsuji et al. presented the first measurement of X-ray-excited current using STM equipment [26]. In this experiment, a tip was separated from a sample surface by about 600nm, where tunneling current could not be detected (far field). The sample was then directly irradiated with polychromatic X-rays from an X-ray tube, which caused a tip current that was amplified in air. From that, they concluded that the origin of this current was the electrons emitted from the sample due to the photoelectric effect. The amplification of the tip current was explained by the ionization of air molecules by emitted electrons. The influence of the presence of molecular gases was studied in more detail later on [27, 28]. It has also been shown that it is in principle possible to obtain a regular STM image under X-ray irradiation, when the tip/sample separation lies within the tunneling regime (near field) [29, 30, 31]. However, because the STM tip cannot energy analyze electrons that arrive from the sample, it was first necessary to change the energy of the incident X-rays in order to obtain unambiguous chemical information. Ten years ago, the first experiments with monochromatic hard X-ray radiation were carried out at the Photon Factory in Tsukuba, Japan [32]. By changing the photon energy from 7.5 to 12keV, the research team was able to obtain extended X-ray absorption fine structure (EXAFS)-like and X-ray absorption near-edge structure (XANES)-like spectra of patterned Au–Ni thin-film samples with an STM tip in the far field. The experiment was performed under normal air pressure. The tip current exhibited a jump-like increase, when the X-ray energy approached the Ni K-edge or the Au L III-edge. The group also identified the important role of the tip properties in the enhancement of spatial resolution. Generally, electrons emitted from the sample surface are detected not only at the tip apex but also at the side of the tip. This is the main reason why the surface area that contributes to the measured tip current was about 2mm2 for an uncoated STM tip [27]. Therefore, in order to make the analyzing region small, the tip was next coated with an insulating polymer film that covered the sidewalls of the tip. The measurements suggested that the tip now collected electrons from a surface area of about only 1mm in diameter. In the following years, the development of “smart” tips, i.e., tips with different kinds of insulating coverage, was recognized as a key task for the success of X-ray-enhanced scanning probes. In 2004, a group from the University of Tokyo reported on the development and trial measurement of synchrotron-radiation-light-illuminated STM [33].

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By measuring photoexcited electron current together with the conventional STM current, Si 2p soft XAS were obtained from a Si(111) surface in the near field. The spatial resolution of the XAS measurements by an uncoated tip has been estimated to be of the order of several 10s of microns. To improve the spatial resolution, they repeated the measurements with a polymer-coated tip [34]. A tungsten tip was covered except for an area of around 100 mm from the tip apex. The detected photocurrent was reduced to about 1–10% of the current detected by a bare tip. However, measurements on microsize dots suggest that the spatial resolution was above 5 mm. Yet another option to improve the spatial resolution is to use smaller X-ray spots. Saito et al. attempted elemental analysis by STM using a sample of Ge nanoislands on a Si(111) surface, based on the core-level interaction between highly brilliant X-rays and surface atoms [35, 36]. An uncovered STM tip made from electrochemically etched tungsten wire was used. It appears that the ideal ambient experimental condition is ultra-high vacuum (UHV), rather than gases or air, to prevent the unnecessary excitation of many electrons, which cause undesired noise. A modification of the tunnel current at the Ge absorption edge was detected with a spatial resolution on the order of 10nm utilizing a hard X-ray microbeam. It is nevertheless surprising that the average tip current over the Ge island was smaller than over the Si substrate for an X-ray energy that was slightly above the Ge Ka absorption edge. Because hard X-rays penetrate through air and beryllium windows, the analysis chamber can be disconnected from the hard X-ray flight path, and thus, the mechanical vibrations from the beamline can be eliminated. Experiments with soft X-rays are more challenging because, due to the short free mean path of the X-rays, the STM chamber has to be directly connected to the beamline. Further, the cross section of photoionization is higher for soft X-rays. Hence, employing smart tips in order to eliminate the undesired background component in the photoinduced current caused by electrons impinging at the side of the tip is essential. Utilizing a tungsten tip covered with a glass layer except for an area of less than 5 mm from the tip apex [37], a spatial resolution of about 14nm was demonstrated with soft X-rays [38]. An array of mm-size Ni dots fabricated by electron-beam lithography was used as a sample. An increase of the tip current of a few pA was observed, when the tip was located over a Ni dot. The sample bias was set to -10V in order to maximize the signal detection; however, this widens the reduced-barrier area and thus deteriorates the spatial resolution. The authors concluded that the bias voltage could be adjusted to optimize both signal detection and spatial resolution when advanced synchrotrons with higher brightness are used. A spatial resolution of around 10nm was observed using soft X-rays on a patterned Ni, Fe, and Au sample [39]. Okuda et al. attributed the high spatial resolution to a local reduction of the surface potential barrier caused by the close proximity of tip and sample. Consequently, secondary electrons that could not overcome the surface barrier when the tip was in the far field might be detected under tunneling condition. Furthermore, excited electrons with lower kinetic energies than the reduced surface barrier might tunnel through the barrier.

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Fabrication of Insulator-Coated Smart Tips In order to achieve high-resolution imaging in STM, the properties of the tip are of particular importance. From the very first experiments with STM it has been known that resolution depends on the sharpness of the tip [17, 40]. A STM tip is usually fabricated with platinum–iridium (Pt–Ir) or tungsten (W) wires. Well-established electrochemical methods result in sharp tips [41, 42]. On the other hand, sharp Pt–Ir tips are also obtained by remarkably simple and unsophisticated means, such as cutting Pt–Ir wire with pliers. The overall shape of the cut tips may not be well defined, but the apex can be very sharp. The Pt/Ir tips do not oxidize as quickly as W, and are therefore superior in ex situ studies. While the sharpness of the tip is a basic requirement even for conventional STM, the combination with X-ray absorption necessitates some additional processing. Generally, a conducting tip can detect electrons that are ejected from the sample surface not only at the tip apex but also at the sidewalls. Thus, in order to spatially limit the tip active area, a specific nanofabricated tip structure is required. An insulating tip coating with conducting apex can limit the probe collection area. This requirement is similar to that for electrochemical STM work, where coated tips with a small exposed area are routinely employed [43, 44]. For studies that combine STM with X-rays, several coatings have been utilized such as nail polish [27], polymers [34], glass [37], or SiO2 [45]. The fabrication of coated tips can be extremely sophisticated and sometimes consists of multiple steps, including focused ion-beam etching in order to remove insulating material from the tip apex [37, 44]. In contrast, a fairly easy dip-coating technique for the fabrication of boron–nitride (BN)-covered Pt–Ir tips [46] is depicted in Fig. 14.10.Boron nitride is isoelectronic with carbon and exhibits excellent dielectric and thermal properties. A tip is prepared by first cutting a Pt90Ir10 wire with a diameter of 250 mm and subsequent mechanical polishing. The tip is then immersed into a solution of BN with ethanol and slowly extracted. After drying (i.e., removal of the volatile alcohol component) the tip exhibits an insulating coverage except at the very end of the tip apex. The dry composition is 97% BN, 2% SiO2, and 1% MgO. Most likely, the surface tension at the sharp tip

Fig. 14.10 Fabrication of “smart” tips: A Pt–Ir tip is dipped into a boron nitride solution and afterwards dried. Eventually, a thin insulating boron nitride film covers the tip except at the very end of the tip apex

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apex is responsible for the uncovered cone. In Fig. 14.11, we present scanning electron micrographs of a mechanically polished bare Pt90Ir10 tip, and a different tip after dip-coating in a BN solution. To confirm the formation of a BN coating on the tip surface, energy dispersive X-ray (EDX) microanalysis was performed. The spectra at the bottom of Fig. 14.11 were taken over a BN-coated area and the free uncoated tip apex as schematically indicated in the micrograph. The characteristic peaks of the Pt90Ir10 tip get screened over the coated area. In contrast, the uncoated region at the tip apex does not present peaks that are related to the coating. The size of the uncoated area at the tip apex is estimated to be below 1 mm. An exact quantitative assessment of this area is nevertheless relatively difficult. The spatial resolution of EDX is determined by the probe size, beam broadening within the specimen, and the effect of backscattered electrons on the specimen around the point of analysis. This inherently encumbers the quantitative investigation. To facilitate nanoscale analysis utilizing X-ray-enhanced STM, the passivation of the tip except the apex is indispensable for adequate signal and noise levels. The size reduction of the tip active area down to the nanometer scale is a prerequisite

Fig. 14.11 Scanning electron microscopy images of (a) a mechanically polished Pt–Ir tip and (b) a different Pt–Ir tip after coating with a boron nitride film. The EDX spectra at the bottom were taken over a coated area and over the free uncoated tip apex

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for improving measurement accuracy and spatial resolution. Therefore, it is expected that the advancing development of smart tips constitutes an active research field with vital importance for the utilization of synchrotron-based scanning probe microscopy.

Photoelectron Detection Using a Scanning Tunneling Microscope As discussed in Section “The Physics of X-Ray-Enhanced Scanning Tunneling Microscopy,” both tunneling as well as photoejected electrons contribute to X-rayenhanced STM measurements. This inherent convolution of the signals makes the understanding and quantification of the involved processes difficult. Therefore, we focus in this section exclusively on the detection of photoejected electrons in the far field, where quantum mechanical tunneling does not take place [45]. Consequently, the tunneling associated with terms in (14.1) and (14.2) vanish, and a tip current

tip tip sample I TIP = I pass + I loss − I pass

(14.3)

is obtained. Likewise, total electron yield (TEY) can be measured directly at the sample via the sample-current method. The electrically conductive sample has to be grounded and the sample drain current

sample sample tip I TEY = I pass + I loss − I pass ,

(14.4)

i.e., the charge that is needed to compensate for the incurred loss by emitted electrons is measured. A schematic view of the experimental configuration is shown in Fig. 14.12.The STM tip can be placed at a variable distance perpendicular to the sample surface using a calibrated one-axis linear piezomotor. Then the sample is irradiated at an angle of 10° with respect to the plane of the sample, while the tip current ITIP and ITEY are measured simultaneously. In order to manipulate the ratio of passed and lost free electrons, a bias Ubias can be applied to the sample. The sign of ITIP is positive for currents leaving the tip. Likewise, the positive ITEY direction is defined for currents leaving the sample. Due to the close proximity between sample and tip, we have to consider that both are always illuminated in an actual measurement. All experiments reported in the following were performed at beamline 4-ID-C of the Advanced Photon Source at Argonne National Laboratory using a focused beam size of around 100 × 100mm2 [47]. The circularly polarized undulator delivers a flux of 1 × 1012 photons/s at 1,000eV. Emitted electrons were detected at a BN-coated tip of a modified Omicron cryogenic scanning probe microscope head [48]. A custommade support with fluoroelastomer damping was utilized in order to reduce the mechanical vibrations of the beamline from the microscope head. The studied sample consisted of a patterned Cu (3-nm)/NiFe (5-nm) multilayer film, which was fabricated on a Si(001) wafer by means of electron beam evaporation and lift-off.

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Fig. 14.12 Origin of the photocurrent at the tip (ITIP) and sample (ITEY) under X-ray illumination. The positive direction of the measured current ⊕ is indicated. The bias Ubias is applied at the sample while the tip is always grounded

The TEY of the sample for X-ray energies close to the Ni L absorption edges is shown in Fig. 14.13a.The BN-coated tip was placed 1,600nm away from the surface, and spectra were recorded with different biases Ubias between -5 and +5V applied at the sample. In all spectra, peaks due to the metallic Ni L absorption edge are clearly visible at 852.4 and 869.6eV. The multiplet structure of the L edges indicates a slight oxidation of the sample. The intensity ratio of the Ni L2 to Ni L3 peak amounts to 0.19 ± 0.01, independent of the applied bias. This value is in good agreement with calculations of the expected Ni L peak ratio [49]. The peaks are superimposed on a monotonically increasing background caused by the continuum of photoejected electrons. In order to remove this background, the derivative dI/dE was formed for each spectrum (Fig. 14.13b), and peak-to-peak intensities were obtained for further analysis. In Fig. 14.13c, we show the normalized peak-to-peak intensities of the Ni L3 and Ni L2 peaks as a function of applied bias. The peak intensities decrease from around 1.4 to 0.4 for increasing bias from -5 to +5V. In addition to the variation of the peak intensity, the TEY spectra exhibit a characteristic offset for each spectrum as a function of the applied bias. In Fig. 14.13d, we present the bias dependence of the nonresonant mean TEY, normalized to the spectrum obtained at 0V. The values are obtained from the mean current for the nonresonant energy range of 860–865eV in each spectrum. The curve progression reflects the trend already found for the peak-to-peak intensities. The mean TEY drops with increasing positive bias. Obviously, the net amount of departing phototip sample sample ejected electrons I pass and arriving charge I pass gets smaller when the + I loss sample bias is more positive. Nevertheless, ITEY remains positive, indicating that the number of arriving electrons at the sample is always smaller than the number of ejected electrons. The simultaneous measured tip current spectra are presented in Fig. 14.14.X-ray energies close to an absorption edge lead to negative peaks in the spectra. It is a

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Fig. 14.13 (a) Total electron yield of the Cu/NiFe sample with X-ray excitation between 840 and 880 eV as function of applied sample bias Ubias. (b) The derivative of TEY spectra yields peak-topeak heights for subsequent analysis. (c) Normalized peak-to-peak intensities of the Ni L2 and Ni L3 peaks. (d) The normalized nonresonant mean TEY describes the vertical offset of the spectra as a function of Ubias

straightforward proposition to understand the appearance of negative peaks considering the sign of ITIP. Electrons that leave the tip give rise to a positive current, while electrons that arrive from the sample yield a negative current. The peaks are superimposed on a monotonically decreasing background caused by an increasing number of photoejected electrons arriving from the sample. Unlike the TEY spectra, the tip current can change its sign, as presented in Fig. 14.14b. If no bias is applied between tip and sample, ITIP amounts to around 2pA at 840eV. At this photon sample tip tip energy, I pass is greater than I pass . But at around 850eV the tip current ITIP + I loss sample changes its sign due to the emergent pre-edge of the Ni L3 peak. Now I pass tip tip becomes greater than I pass + I loss . Finally, at 880eV, ITIP yields a background of around -2pA. As in the case of the TEY spectra, the intensities of the Ni L3 and L2 peaks decrease with increasing Ubias (Fig. 14.14c). However, the bias dependence is much stronger in the case of the tip. With respect to the unbiased situation, the peak-to-peak intensities double for Ubias of -5V. On the other hand, only extremely small peaks are detectable for a bias of 5V. In addition to the peak intensities, the

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Fig. 14.14 (a) Tip current simultaneous recorded with the TEY spectra shown in Fig. 14.13. The tip/sample separation is 1,600 nm and the bias is applied to the sample. (b) The spectrum obtained at 0 V clearly shows that the tip current changes its sign. For photon energies smaller than around 850 eV the tip current is positive, indicating that more electrons leave the tip than arrive. If the amount of arriving electrons exceeds the number of leaving electrons, the net current is negative. (c) Normalized peak-to-peak intensities of the Ni L2 and L3 peaks. (d) The nonresonant tip current increases as a function of the applied bias

mean tip current varies as a function of the applied bias, which causes a DC offset of the spectra. The nonresonant mean tip current increases with more positive biases (Fig. 14.14d), which is contrary to the behavior found at the sample (Fig. 14.13d). The nonresonant tip-current background varies between 23pA with a sample bias of 5V, and -28pA at -5V. At the sample, the background TEY is maximal for the -5V bias and amounts to 250pA. If no bias is applied, the background TEY decreases to 190pA and is only 80pA at 5V. Figure 14.15 shows the tip-to-sample peak-to-peak ratios of the Ni L2 and Ni L3 absorption edges. The ratio describes how many of the resonant photoejected electrons contribute to the signal at the tip. The sensitivity is independent of the sample/ tip separation examined here. Data that are obtained in the far field with separations of 400, 800, and 1,600nm can be fitted by one linear function. Generally, the tip/ sample ratio decreases with increasing voltage. With Ubias of -5V, around 20% of

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Fig. 14.15 Relative peak-to-peak intensity ratio of the Ni L3 and Ni L2 peaks at the tip and sample as a function of applied sample bias. Data are obtained for tip/sample separations of 400, 800, and 1,600 nm

the ejected electrons are detected at the tip. The ratio decreases to around 12% at 0V, and only 3% at +5V. Thus, a small bias between tip and sample allows an accurate and effective control of the background currents as well as the desired signal obtained at absorption edges. Measuring X-ray absorption spectra yields local chemical and electronic properties. But in addition to the elemental selectivity, exploiting the polarization of the X-rays allows for elucidation of magnetic properties. The underlying effect is XMCD, which measures the dependence of X-ray absorption on the helicity of the X-ray beam by a magnetic material (cf., Fig. 14.3). The experimental geometry used in the experiment described here is illustrated in Fig. 14.16a.Left and right circularly polarized X-rays probe the averaged magnetization M component of a NiFe film parallel to the direction of the incoming beam. The incidence angle qi of the X-ray beam amounts to around 10°. At each energy step, the tip current is meaRCP LCP sured for LCP and RCP radiation ( I TIP and I TIP , respectively). Figure 14.16b shows a XMCD difference spectrum for a BN-coated tip that is placed around 200nm away from the surface. A bias Ubias=-2V is applied to the sample. The specRCP LCP trum is obtained from the difference of I TIP and I TIP , and exhibits a net negative 2p3/2 and positive 2p1/2 peak. The different number of spin-up and spin-down holes available in the 3d electron states causes this. Because the 3d states are the origin of the magnetic properties of this material, the spectrum contains information on spin and orbital magnetic moments. Therefore, the distinctive and unique potential of the combination of X-rays and STM lies not only in the ability to localize spins, but also in the capability to measure their size. It is obvious that a tip placed in the far field only provides a limited spatial resolution. Recently, Chiu et al. [50] have simulated electron trajectories for photoelectrons

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Fig. 14.16 (a) The magnetization M of a NiFe film is derived from tip current (ITIP) spectra, one taken with LCP, and one taken with RCP light. The incidence angle of the X-rays is qi. (b) X-ray magnetic circular dichroism spectrum measured with the tip

escaping from the surface. Assuming a constant photon flux homogenously delivered to a sample area Aph on a metallic surface, they obtained an effective probing diameter Dprob of the tip that is defined as the diameter of a circular region on the surface within which emitted electrons contribute 90% to ITIP. According to this definition, the absolute value of ITIP is estimated from the sample current Isample delivered by the incident X-rays: 2  10 Dprob  I sample (14.5) I TIP =   9 2  Aph Generally, decreasing the probing area requires a small tip/sample separation, a smart tip with minimized area of detection, and a strong acceleration field resulting in a large negative Ubias. The best resolving power is achieved when the tip/sample

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separation lies in the tunneling regime, i.e., in the near field. Chiu et al. were able to show that in this case, the signal that is typically detected in SXSTM measurements is too large to be only caused by photoejected electrons. Thus, if the tip/ sample separation lies within the conventional tunneling regime, both channels photoejected as well as X-ray-excited tunneling electrons must be considered. The illustration of this regime will be the subject of the following section.

X-Ray-Assisted Scanning Tunneling Microscopy In order to achieve the ultimate spatial resolution in SXSTM, the separation between the tip and sample obviously has to be minimized. Therefore, unlike in the studies of photoelectron detection in the far field, a separation within the quantum mechanical tunneling regime has to be employed. Consequently, we have to expect two qualitatively different current contributions in the SXSTM signal ISXSTM, that is to say, the current caused by photoelectrons as well as the tunneling current. The signal ISXSTM represents a convolution of those signal channels. While the tunneling current is highly localized and, in principle, is capable of delivering atomic resolution, the photocurrent contribution can be considered as an undesired noise that degrades resolution. Nevertheless, spatial resolutions of 10nm for hard X-rays [35], and 14nm for soft X-rays [38] have already been demonstrated and it is expected that further developments, e.g., improvements of high-performance tips, may significantly improve the spatial resolution. STM In conventional STM the feedback loop keeps the tunneling current I tunnel constant, so that the consequential adjustment of the tip height (i.e., the z-piezo voltage) in principle yields a topography image of the sample surface. But as shown in Section “The Physics of X-Ray-Enhanced Scanning Tunneling Microscopy,” the illumination with X-rays during scanning produces several additional current contributions. Although some of those additional contributions provide the chemical sensitivity of SXSTM, large currents can overstrain the dynamical range of the feedback loop. The situation is illustrated in Fig. 14.17.The tip was positioned over a NiFe film and stabilized under X-ray illumination at 850eV, a tunneling current of 0.2nA, Ubias=-3.5V, and z-piezo voltage of 5V. Then the X-ray energy was scanned from 850 to 856eV. An X-ray absorption spectrum was simultaneously obtained using an energy standard (dotted line), which exhibits the Ni L3 edge. The z-piezo voltage remains constant up to an X-ray energy of 851.7eV. Then the z-piezo voltage decreases rapidly and reaches -137V at 852.4eV. Here the z-piezo is fully retracted. Generally, the z-piezo voltage ranges between +137V (maximal approach) and -137V (maximal retracted tip). For energies larger than 855.7eV the z-piezo voltage returns to its initial value, i.e., the tip approaches the sample surface again. Although the z-piezo starts to retract the tip for X-ray energies larger than 851.7eV, the SXSTM current ISXSTM continues to decrease (gets more negative, i.e., the net amount of charge arriving at the tip still increases). Apparently, the feedback loop is not able to maintain a constant ISXSTM because the current caused by the interaction with X-rays is too large. In the case of the fully retracted tip, ISXSTM appears to mirror the

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Fig. 14.17 Response of the SXSTM signal and the z-piezo voltage for an energy scan of a NiFe film. The dotted line represents the spectrum simultaneously obtained from a NiFe standard. Between 852.4 and 855.0 eV the z-piezo is fully retracted at -137 V. (Ubias = -3.5 V; It = 0.2 nA)

progression of the intensity of the energy standard. Not until ISXSTM falls below the threshold of -0.11nA does the tip approach the sample again. Although such strong responses still allow the spectroscopy mode of SXSTM, it is undesirable for the imaging mode. In order to maintain sound tunneling conditions during the experiment, it is crucial to maintain a somewhat small current that lies within the dynamic range of the STM feedback loop. An easy way to adjust the conditions of an SXSTM experiment is the variation of Ubias. In Fig. 14.18, we present a point spectroscopy measurement on the same NiFe film with a bias of only -3V. Here, the tip was stabilized at 845eV with a z-piezo voltage of 11V. The appearance of the Ni L3 peak causes a notable decrease of the z-piezo voltage to -61V. However, tunneling conditions are preserved during the energy scan. The Ni L2 peak does not produce a response of the z-piezo voltage at all, because the z-piezo voltage reacts only to a normalized intensity of the energy standard larger than 0.6. The Ni L2 peak exhibits an intensity of only 0.5. The feedback loop keeps ISXSTM more or less constant over the whole energy scan, with only small peaks related to the up- and downward slope of the z-piezo voltage. From this spectroscopy experiment we can expect that appropriate tunneling conditions can enable an imaging mode of SXSTM. The feasibility of imaging with an STM under synchrotron-based X-ray illumination is demonstrated in Fig. 14.19.At first a sputtered Cu film on a Si substrate was scanned with conventional STM in constant-current mode. The image obtained from the reaction of the z-piezo (Fig. 14.19, left) shows the rough topography of the Cu film featuring elongated elevations. The tip current (Fig. 14.19, top) reflects these structural features, but is nevertheless relatively flat. For comparison, the same surface area was scanned under X-ray illumination. The energy was tuned to the Cu L3

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Fig. 14.18 Response of the z-piezo and the SXSTM current as a function of X-ray energy for a point spectrum over a NiFe film. The z-piezo shows a strong response when the energy approaches the Ni L3 absorption peak, but does not reach the saturation voltage

Fig. 14.19 Topography image of a Cu film (left) and the associated conventional tunneling current (top). The bottom image shows the current image for a sequence of illumination at the Cu L3 absorption edge (bright stripes) and without illumination (dark stripes). The image was obtained with an uncovered Pt-Ir tip (Ut = -2 V; It = 1 nA; scan speed was 2,000 nm/s)

absorption edge at 931.2eV. During the scan, the X-ray shutter was repetitively opened and closed at every 25th scan line. While the topography image provides basically the same information as the one shown on the left side of Fig. 14.19, the tip current image exhibits a pronounced stripe structure. The image shows dark

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stripes when the X-ray shutter is closed. By contrast, if the sample is illuminated with X-rays, bright stripes are recorded. Although this is a basic demonstration of the achievability of imaging under X-ray illumination, care has to be taken in the analysis of chemical species in such images. As shown in Section “Photoelectron Detection Using a Scanning Tunneling Microscope,” illumination with X-rays can cause a DC offset of current spectra, which may also lead to a similar response. So far, we have seen that ISXSTM as well as the z-piezo voltage can directly provide elemental selectivity, while Ubias largely controls the sensitivity of SXSTM. Further, the feasibility of imaging under X-ray illumination was demonstrated. In Fig. 14.20, we present finally the experimental justification of the SXSTM mechanism proposed in Section “The Physics of X-Ray-Enhanced Scanning Tunneling Microscopy.” Point energy scans were carried out on a patterned NiFe (20 nm) ring structure over position S as shown in Fig. 14.20a. The sample was fabricated on a Si(001) wafer by electron beam evaporation and lift-off. An X-ray absorption spectrum, which is simultaneously obtained from a NiFe energy standard, is shown in Fig. 14.20b. The peak is caused

Fig. 14.20 (a) The topography scan of patterned NiFe rings. Current spectra were obtained at the indicated sample position S. The feedback loop was switched off for the energy scans. (b) The X-ray absorption spectra of a NiFe standard shows the Ni L3 peak. (c) The SXSTM signal exhibits a decrease when the sample is positively biased at 1.5 V. (d) In the case of tunneling from the sample into the tip (Ubias = -1.5 V), the modulus of the SXSTM signal increases, i.e., the current gets more negative when the photon energy reaches the Ni L3 transition

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by the Ni L3 absorption edge. Before each scan, the tip was stabilized under X-ray illumination at 845eV, Ubias=±1.5V, and ISXSTM of ±0.2nA. Hereby, a positve Ubias causes tunneling from the tip into the sample, and therewith a positive ISXSTM. Likewise, a negative ISXSTM results from a negative bias. The X-ray energy was scanned from 845 to 858eV, while the feedback loop was switched off. This assures a constant tip/sample separation during the scan. In the case of the postive Ubias, the SXSTM current drops from around 0.20 to 0.07nA when the energy reaches the Ni L3 peak (Fig. 14.20c). This can be explained by electrons that are excited to states close to EF, which reduce the number of available states for tunneling from the tip into sample the sample (cf., Fig. 14.5b). Furthermore, the current I pass that is caused by photoejected electrons is opposite to the initial tunnel current (cf., Fig. 14.6). The scenario is reversed when the sample is negatively biased. Fig. 14.20d shows the response of ISXSTM for a bias of -1.5V. The appearance of the Ni L3 transition causes an increase of the absolute value of ISXSTM by about 0.34nA. We would like to remind the reader that the sign of ISXSTM is only indicative of the current direction, which means that the more negative ISXSTM gets, the more charge arrives at the tip. Incident X-rays excite core electrons to unoccupied states close to EF. Consequently, the conventional tunnel X-ray sample current is enhanced by I tunnel as well as by I pass (cf., Figs. 14.5a and 14.6).

Concluding Remarks Depending on the scientific community asked, the benefits and prospects of SXSTM could be viewed from two different perspectives. On the one hand, the synchrotron community experiences a fundamentally different and unusual approach to high-resolution X-ray microscopy. Currently, all efforts to improve spatial resolution for the X-ray sciences try to modify the beam properties by means of sophisticated optics [6, 51]. In contrast, SXSTM has the potential to achieve those resolutions even with relatively large beams, because the close proximity of the tip, and not the small footprint of an incoming X-ray beam, achieve the high resolution. On the other hand, in the STM community chemical sensitivity has been a dream since the early days of this amazing technique. Additionally, SXSTM allows for direct detecting and quantifying of magnetic properties. Therewith, SXSTM is not just a marginal extension of the established STM. It has the potential to revolutionize the way in which we are able to study nanostructures. However, several remaining challenges have to be overcome before SXSTM can be put into widespread use. The technique is still in the early stages and a huge effort has to be undertaken to further develop and better understand this novel tool. One example is the development of dedicated electronics. The interaction of X-rays with the sample and the tip introduces currents that are not present in conventional STM experiments. Hence, one must emphasize that care must be taken in the analysis of signals obtained with conventional STM electronics. It is further desirable that electronics take advantage of the fast time resolution of third-generation synchrotrons. A detailed discussion of dedicated electronics for SXSTM would far exceed

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the subject of this chapter, and therefore is omitted here. A second example is the fabrication of “smart” tips. The size reduction of the tip active area is key to successful nanoscale imaging. So far, several more or less sophisticated approaches have been implemented, but there is still room for further optimization, which would improve the spatial resolution of SXSTM. Today, the emerging fields of nanoscience and nanotechnology are leading to unprecedented understanding and control over the fundamental building blocks of nature. The combination of STM and synchrotron radiation with nanoscale resolution has the potential to provide a major impact on nanoscale research by enabling fundamentally new methods of characterization. Further vigorous and diversified development of the technique is critical to fully realizing its potential for nanotechnology. Acknowledgements The authors would like to express their gratitude to several people who contributed to this project. Special thanks go to Kenneth Gray for the generous allocation of experimental equipment, which made this work possible in the first place. We thank Vitali Metlushko for the growth and patterning of the studied samples. Curt Preissner is acknowledged for his engineering support and Matthias Bode for several fruitful discussions. This work has been supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

References 1. M.C. Roco, W.S. Bainbridge (eds.), Nanotechnology: Social Implications – Maximizing Benefits for Humanity (National Science Foundation Report, Arlington, VA, 2001) 2. R. Wiesendanger, M. Bode, R. Pascal, W. Allers, U.D. Schwarz, J. Vac. Sci. Technol. A 14, 1161 (1996) 3. T.A. Jung, F.J. Himpsel, R.R. Schlittler, J.K. Gimzewski, Chemical information from scanning probe microscopy and spectroscopy, in: Scanning Probe Microscopy: Analytical Methods, Chap. 2, R. Wiesendanger (ed.) (Springer, Berlin, 1998), p. 11 4.. J. Viernow, D.Y. Petrovykh, A. Kirakosian, J.-L. Lin, F.K. Men, M. Henzler, F.J. Himpsel, Phys. Rev. B 59, 10356 (1999) 5. B. Kaulich, M. Kiskinova, in Synchrotron Radiation X-ray Microscopy Based on Zone Plate Optics, Lecture Notes in Physics, vol. 588 (Springer, Berlin, 2002) 6. H.C. Kang, H. Yan, R.P. Winarski, M.V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A.T. Macrander, G.B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008) 7. M.O. Krause, J. Phys. Chem. Ref. Data 8, 307 (1979) 8. B.L. Henke, J.A. Smith, D.T. Atwood, J. Appl. Phys. 48, 1852 (1977) 9. B.L. Henke, J. Liesegang, S.D. Smith, Phys. Rev. B 19, 3004 (1979) 10. J.L. Erskine, E.A. Stern, Phys. Rev. B 12, 5016 (1975) 11. G. Schütz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, G. Materlik, Phys. Rev. Lett. 58, 737 (1987) 12. J. Stohr, Y. Wu, in New Directions in Research with Third-Generation Soft X-ray Synchrotron Radiation Sources, ed. by A.S. Schlachter, F.J. Wuilleumier (Kluwer Academic Publishers, Netherlands, 1994), p. 211 13. J. Stohr, J. Electron Spectrosc. Rel. Phenom. 75, 253 (1995) 14. O. Eriksson, B. Johansson, R.C. Albers, A.M. Boring, M.S.S. Brooks, Phys. Rev. B 42, 2707 (1990) 15. B.T. Thole, P. Carra, F. Sette, G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992) 16. P. Carra, B.T. Thole, M. Altarelli, X. Wang, Phys. Rev. Lett. 70, 694 (1993)

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Chapter 15

Scanning Ion Conductance Microscopy Johannes Rheinlaender and Tilman E. Schäffer

Introduction In 1981, the age of the scanning probe microscopes (SPMs) began when Binnig, Rohrer, and cowokers developed the first scanning tunneling microscope (STM) [1]. Their setup was based on measuring an electrical tunneling current between a sharp metal tip and a conducting sample. For the first time, a sample surface could be imaged with true atomic resolution in real space. The STM launched the development of several other types of SPMs. In general, these microscopes consist of a small, submicrometer probe, which senses a certain physical interaction with the sample and which is scanned over the sample to generate an image. For example, Pohl et al. invented the scanning near-field optical microscope (SNOM) in 1984 [2], which uses an evanescent electromagnetic field in the subwavelength range to image the sample. In 1986, Binnig and co-workers developed the atomic force microscope (AFM), which is based on measuring the mechanical forces between a sharp tip and the sample [3]. The AFM is not limited to conducting or transparent samples and has become one of the most important tools in nanoscale science. The AFM also works in aqueous environments, such as buffer solutions and so is well suited for biological samples [4]. Since then, several related SPMs have been developed, such as the magnetic force microscope [5, 6], the electrical force microscope [7], and the scanning electrochemical force microscope (SECM) [8]. In this chapter we focus on the scanning ion conductance microscope (SICM), which was introduced by (Hansma 1989) [10]. The central component of a SICM is a small, electrolyte-filled aperture, which is usually formed by a nanopipette pulled from a glass capillary. An ion current through the pipette is established between two electrodes, one inside and one outside the pipette (“pipette electrode” and “bath electrode,” respectively). By scanning the pipette over the sample images of the sample

T.E. Schäffer (*) Institute of Applied Physics, University of Erlangen-Nuremberg, Staudtstr. 7, Bldg. A3, 91058 Erlangen, Germany e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_15, © Springer Science+Business Media, LLC 2010

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topography can be obtained. Despite its broad application range, the SICM is not yet widely spread and only a few setups are described in the literature [9–17]. Although some SICM experiments have been done with microfabricated probes [9], most studies use pulled glass capillaries with a tip opening diameter of typically 50–100nm. This parameter determines the lateral resolution of the microscope, which is on the order of the diameter of the pipette opening [18]. When the pipette tip approaches the sample surface, access to the opening is partly blocked, resulting in a decrease of the ion current. Thus, the ion current can be used as input to a feedback loop, which keeps the distance between pipette and sample constant while the pipette is scanned. Topography images of soft samples can thereby be obtained in this “noncontact” configuration [19, 20]. In order to improve the distance control of the feedback loop, the pipette can be modulated vertically above the sample. In this case, the amplitude of the modulated ion current signal is used as input to the feedback loop [14, 17]. In a standard SICM setup, the ion current is used to measure sample topography. With the detection of additional interaction parameters between tip and sample, further sample properties can be imaged. For example, with a distance control based on such an additional parameter, the ionic conductivity of the sample can be mapped in addition to its topography. This was achieved by using a complementary AFM distance control [15] or a complementary shear-force distance control [12, 21]. On the other hand, complementary sample properties can be imaged while using the ion current for distance control, for example, with a combined SNOM–SICM setup [22–24]. This chapter is structured in the following way: first, we delineate the fundamental principle of the SICM and the basics of ion conduction. In Section “Ionic Currents in SICM,” we present theoretical descriptions of ion currents in SICM nanopipettes. Section “Basic Imaging Modes” describes basic imaging modes and their improvements. Finally, in Section “Advanced Imaging Modes” some advanced applications showing the huge potential of the SICM are presented, including high-resolution SICM imaging, combinations with other scanning techniques and elasticity measurements.

Fundamental Principles Basic Experimental Setup A schematic of a standard SICM setup is shown in Fig. 15.1. An electrolyte-filled nanopipette with an opening diameter in the submicrometer range is positioned above a structured sample in an electrolyte bath. A voltage, typically in the range between 100 and 1,000mV, is applied between the pipette electrode and the bath electrode. The resulting ion current flows from the pipette electrode through the pipette, the pipette opening and the bath volume to the bath electrode and is measured with a current amplifier. Usually, a commercial patch-clamp current amplifier is used. The magnitude of the current strongly depends on the distance from the pipette to the surface.

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Fig. 15.1 Schematic of a basic SICM setup. A nanopipette with an opening in the submicrometer range is filled with an electrolyte and placed close to a sample in an electrolyte bath. A voltage, U, is applied between the electrodes and results in an ion current, I, in the nanoampere range. The current is measured with a current amplifier, which is connected to a computer-based controller. The controller drives the x-y-z scanner. The inset shows a scanning electron image of a typical borosilicate glass pipette

The closer the pipette is to the surface the smaller the gap between the pipette walls and the sample is. Therefore, the presence of the sample can be detected although there is no actual mechanical contact between the tip and the sample. The current is monitored by a computer-based controller, which also actuates a three-dimensional scanner. The scanner is usually piezo-electric and allows nanometer-precise positioning of the pipette relative to the sample in all three axes. Possible are pipettescanning, sample-scanning, and dual-scanning setups. By using a feedback loop implemented in the controller with the ion current as input, the distance between the pipette and the surface can be kept constant. While scanning the pipette over the sample a topography image can thereby be generated. Due to the absence of mechanical interaction, the SICM is especially well suited for imaging soft samples like living cells, as described in more detail later in this chapter.

Nanopipette Probes One important part of the setup is the nanopipette, which has similar functions and requirements as pipettes for intracellular recording techniques: providing an electrical connection from the macroscopic to the microscopic scale with reproducible geometrical parameters. Nanopipettes are optimized for small opening diameters in order to improve the lateral resolution of the SICM. The nanopipettes are usually drawn

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from glass capillaries (e.g., borosilicate or quartz glass) with an outer diameter of 1–2mm. Some pipette pullers use heating coils, others are based on heating the glass capillary in an infrared laser beam and drawing apart the ends by a sequence of force pulls. The shape and diameter of the pipette tip can be modified by adjusting the intensity of the laser beam and the number and magnitude of the force pulls. Therefore, probe fabrication is straightforward compared with that for other SPMs. The inset in Fig. 15.1 shows a scanning electron microscope (SEM) image of a SICM pipette tip. Typical pipette opening diameters are on the order of 50–100nm, but experiments with diameters on the order of 10nm were also reported [25]. A small opening diameter results in a relatively large pipette resistance. The maximum electrode voltage U is on the order of 1V and the electrolyte composition is usually set by the requirements of the sample. Physiological electrolyte solutions have a conductivity of about 1S/m, resulting in typical pipette resistances on the order of 100MW and in ion currents in the range of 1–10nA.

Half-Cell Electrodes In SICM, the electrodes usually consist of silver/silver chloride, used as electrochemical half cells. They consist of a solid silver core or wire (Ag) covered with solid silver chloride (AgCl) and are immersed in an electrolyte containing chloride ions (e.g., a solution partially composed of NaCl or KCl). In aqueous environment the silver chloride dissolves:

AgCl(solid) ↔ Ag + (aq) + Cl− (aq ).

(15.1) The reaction’s equilibrium constant is K=1.76×10-10 (at T=25°C). In equilibrium, the concentration of ions in solution is therefore very small. The redox reaction at the Ag/AgCl electrode, in which the chloride atom receives an electron and dilutes as a chloride ion, leaving metallic silver, and vice versa, is described by

AgCl(solid) + e − ↔ Ag(solid) + Cl− (aq ).

(15.2) This reaction occurs close to the solid interface (<1nm distance). The electrochemical reduction potential in thermodynamic equilibrium is described by the Nernst equation, which relates the potential to the activity of chloride ions. For a low ion concentration, the activity can be replaced by the concentration [26]:

E = E0 +

RT F

ln

1 Cl − 





≅ 0.222V − 0.0591V × ln Cl − 

(15.3)

for T =298K

Here E0 is the standard potential of the Ag/AgCl electrode (relative to the standard potential of the hydrogen electrode at T=25°C), R=8.315J/(mol K) is the molar gas constant, T is the absolute temperature, and F=96,485C/mol is the Faraday constant. The square brackets mark the ion concentration in SI units. Generally, in

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SICM experiments two identical electrodes are used as anode and cathode, resulting in a simpler setup than that used in voltametric experiments, where three different electrodes are used (working electrode, auxiliary electrode, and reference electrode). In SICM, just an external voltage between the electrodes is applied, which drives a Faraday ionic current. This current is typically measured with a current-to-voltage converter with a sufficient large amplification factor for current measurements in the nanoampere range. We note that usually the electrochemical conditions are identical at both electrodes. But differences, for example in the ion concentration, can cause potential offsets, which are added to the applied voltage U, resulting in a different effective applied voltage. The chloride ion mass transport through the electrolyte can be described by

m =

dm dQ M = × = I × 3.674 × 10−4 g / C, dt dt F

(15.4)

where Q is the transported charge and M=35.45g/mol is the molar mass of chloride. So for an ion current of I=1nA=10-9C/s the mass transport from the cathode to the anode is 3.674×10-13g/s of chloride or 6.241×109ions/s.

Ionic Conductivity The conductivity of an ionic solution can be estimated from the electrophoresis of charged particles in water. For that purpose, we consider the ions as spherical particles with radius a and charge Z×e, pulled by an electrical field of magnitude E through a fluid with viscosity h. In stationary movement, the electrical force Fel=ZeE and the viscous friction force Ffric=-6phavion compensate each other:

Fel + Ffric = 0.

(15.5)

Therefore, the stationary velocity is proportional to the electrical field

vion = µion E ,

(15.6)

where

mion =

Ze 6pha

(15.7)

is the ionic mobility. This simple model works as a surprisingly good estimation. The value for Z=1, h=1mPa s in water and a»0.2nm is µion»4×10-8m/s (V/m)-1, which is accurate for most small ions in aqueous solution. The radius used here is not the bare ion radius (typically <0.1nm), but the larger so-called hydrated radius, considering some layers of hydrating water molecules. Furthermore, the ionic mobility is directly connected to the ionic conductivity sion of the electrolyte. The ionic current density is defined by

jion ≡ σion E

(15.8)

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and here given by

jion = Zecion vion .

(15.9)

Comparison with (15.6) gives

(Ze )

2

σion = Zecion µion =

cion

6πηa

.

(15.10)

For example, an ionic concentration of cion=150mM=150mol/m3 results in an ionic conductivity of sion»0.6S/m. Although the structure of fluids is quantized on the length scale of intermolecular interactions (typically <1nm), the electrolyte appears continuous on the length scale relevant for the SICM. So it is usually sufficient to deal with macroscopic material properties, such as conductivity and viscosity. In ionic solutions, also surface charges and their screening are important. The range, in which surface charges are compensated by accommodation of ions, is given by the Debye length [27]

λ Debye =

εkBT 2 (Ze ) cion 2

.

(15.11)

For example, for water with a dielectric constant of e =78 e0 with vacuum permittivity e0=8.854×10-12As/Vm an ion concentration of cion=100mM results in lDebye=0.95nm at room temperature (kBT=4.043×10-21J). Therefore, surface charges are screened within 1nm for the high ion concentrations typically used in SICM experiments, and the solution can be treated as electrically neutral on a macroscopic scale.

Ionic Currents in SICM Geometry of the Pipette–Sample System The electrolyte can be described as a homogeneous medium with a certain conductivity sion. The lower part of the pipette below the electrode can be approximated as a cone (Fig. 15.2) with a half cone angle a, where r −r r tan α = u i ≈ u . (15.12)

L

L

The inner opening radius ri is on the order of 100nm and much smaller than the upper pipette radius ru and the cone length L, which are in the range of millimeters. Typical values for a are 1–10°. The ratio between the pipette’s outer radius and inner radius, typically 1.5–2.0, can be taken as constant over the lower part of the pipette [28]. The pipette is placed close to the sample, here taken as an ideally flat surface, at a pipette–sample distance of z0.

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Fig. 15.2 Schematic of the pipette–sample system. The tip region of the pipette is modeled as a hollow cone with half cone angle, a, upper opening radius, ru, and length, L. The pipette opening is defined by the inner opening radius, ri, and the outer opening radius, ro. The equivalent circuit in the case of a nonconducting surface consists of three electrical resistances in series: The pipette resistance, Rp, the resistance of the tip region, Rt, and the resistance of the electrolyte bath, Rb

Analytical Model The electrical circuit for the ion current from the pipette electrode through the pipette to the bath electrode can be described by a series of three resistances

Rtot = Rp + Rt + Rb .

(15.13)

1. The pipette resistance Rp can be calculated analytically for the small cone angles used here by [12, 28–30]

Rp =

1 L 1 1 ≈ . σion πru ri σion πri tan α

(15.14)

The pipette resistance is usually on the order of several 100MW using sion»1S/m, a =1–10° and ri=50–100nm. 2. The resistance of the tip region, Rt, is strongly dependent on the pipette–sample distance, z0, and diverges for z0→0, when the pipette is completely closed. Nitz et al. [12] developed a simple analytical model, which estimates the resistance of the tip region as

Rt =

r 1 3 ln o . σion 2πz0 ri

(15.15)

3. The remaining resistance is given by that of the electrolyte bath, Rb, whose upper limit can be estimated by the access resistance of the pipette opening. This can be derived by assuming the pipette opening being a pore, facing an infinite halfspace, resulting in [31]

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Rb ≈

1 1 . σion 4ri

(15.16)

Due to the typically small pipette cone angles, Rp is at least one order of magnitude larger than Rb, which can therefore be neglected in this rudimentary model. The ion current is then given by

I ( z0 ) ≈

I0 r 3 U = with I 0 = σion πri tan αU and p = tan α ln o . (15.17) 2 Rtot 1 + p (ri / z0 ) ri

The ion current decreases to zero for z0®0 and approaches the saturation current, I0, asymptotically for z0® ∞. It can be derived that the ion current drops at very small pipette–sample distances, for example to I=I0/2 at z0=pri, which is typically on the order of 0.01–0.05ri, i.e., in the range of nanometers.

Finite Element Model The method of finite element modeling (FEM) [32] can be used to calculate the distance behavior of the ion current through the nanopipette numerically. In FEM, partial differential equations (PDEs) are solved numerically by dividing the domain into elements of finite size and solving the PDE on this discrete mesh. This discretization can be done, for example, in the spatial or in the time domain. Here, we use FEM to solve the Poisson equation for the electrical potential  ∇2j (r ) = 0

(15.18)

inside the conducting electrolyte. Ñ denotes the Nabla operator. The two boundary conditions used here are “constant potential” on an electrode surface A1,

j

A1

= constant

(15.19)

and “electrical insulation” on glass and sample surfaces A2,

  n·∇j (r )

A2

=

∂j ∂n

= 0.

(15.20)

A2



∂/∂n is the derivative in the direction of n , which is the unit vector perpendicular to the regarding boundary. The electrolyte can be described as a homogeneous Ohmic resistor with conductivity sion. So the ion current density, j , is given by the electric field, E , with where

    j (r ) = σion E (r ),

(15.21)

15 Scanning Ion Conductance Microscopy

    E (r ) = −∇j (r ).

441

(15.22)

The current, I, through an electrode surface A1 is obtained by the integration of the perpendicular component of the current density, jn, over the boundary area

∂j    I = ∫ n· j (r )dA = ∫ jn (r )dA = −σ ion ∫ dA. A A1 A1 ∂n

(15.23)

The numerical solution of the Poisson equation on a complex mesh leads to a PDE with several thousand degrees of freedom. It is useful to decrease the size of the domain in order to reduce the computation time (or the computation error, respectively). For this purpose, the electrical resistance of the electrolyte can be divided into three parts, equivalent to the analytical model (Fig. 15.2). Then FEM is used to calculate the resistance of the electrolyte in the vicinity of the pipette tip, Rt. Figure 15.3 shows the model of the tip region used here. Rt is given by the potential difference between the upper boundaries inside and outside the pipette wall (red and blue areas in Fig. 15.3) divided by the induced current. The total current through a pipette of full length is given by

I=

U U = , Rtot Rp + Rt

(15.24)

where U is the voltage applied between the macroscopic electrodes. The pipette– sample distance, z0, was changed stepwise and the finite element model was solved for each z0, which takes typically about 10s on a current computer workstation. Figure 15.4 shows numerical data for I(z0) for different values of the half cone angle and the pipette wall thickness. It can be seen that, upon approaching the sample, the ion current decreases the earlier, the wider the pipette is and the thicker the

Fig. 15.3 Schematic of the finite element model (FEM). (a) Three-dimensional view and (b) cross-section of the tip region that is modeled. The resistance of the upper part of the pipette, Rp, is added in series (reprinted with permission from Rheinlaender and Schäffer [18]. Copyright 2009 American Institute of Physics)

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Fig. 15.4 Calculated ion current, I, as a function of the pipette–sample distance, z0, calculated with FEM for different pipette geometries. (a) Smaller pipette cone angles (a) and (b) smaller pipette outer radii, ro, result in a stronger dependence of the ion current on the pipette–sample distance (reprinted with permission from Rheinlaender and Schäffer [18]. Copyright 2009 American Institute of Physics)

pipette wall is. This is because the influence of the tip region is more significant for a smaller pipette resistance, Rp. The effect of the wall thickness is minor compared to the influence of the pipette cone angle. FEM has also been applied to the investigation of SICM image formation and to the determination of the lateral resolution [18]. By placing an object on the sample plane in the finite element model (Fig. 15.3), the influence of topographic features on the ion current can be determined. Moreover, when the lateral and vertical position of the pipette tip relative to the sample is varied, the imaging process of the SICM can be simulated. For example, by calculating topography images of two small cylindrical particles it was shown that they can be resolved from each other in the image if their distance, d, is larger than three times the inner radius of the pipette (Fig. 15.5). This distance (d=3ri) was defined as the lateral resolution of the SICM [18]. Furthermore, it was found that the image formation is strongly dependent on the height of the particles and on the pipette–sample distance. This results in the particles being imaged as bell- or ring-shaped distributions, indicative of bell- or ring-shaped special point spread functions. This also results in the height of small objects being imaged with a height that is only a fraction of its actual height [18]. This has significant consequences for the quantitative interpretation of SICM images. To demonstrate different imaging effects, FEM was used to calculate topography images of more complex samples. Figure 15.6 shows images of a “sicm”-shaped object, which is convoluted with bell- or ring-shaped special point spread functions.

Experimental Ion Current vs. Distance Curves Current–distance curves provide useful information about the pipette–sample interaction. In an experiment, the pipette is approached towards the sample surface by using a piezo actuator. Simultaneously, the ion current is recorded. A typical experimental curve is shown in Fig. 15.7a. When the pipette is far away from the sample surface, the current is approximately constant and independent of the pipette–sample distance (right-hand side of the graph). When the pipette–sample distance is reduced to about the pipette opening radius, the electrical resistance of the tip region increases

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Fig. 15.5 Topography images of two cylindrical particles (radius r = ri/2) at a distance d on a planar surface calculated with FEM. (a–d) Outline of the actual topography. (e–h) Calculated topography images for two high particles (h0 = ri) and a large pipette–sample distance (z0 = 1.6ri). (i–l) Images for two low particles (h0 = ri/2) and a small pipette–sample distance (z0 = 0.6ri). (e) At a large particle distance (d = 6ri), each high particle is imaged as a single bell shape (black trace in cross-section e¢). (f, f¢) For a decreasing distance (d = 4ri), the bell shapes start to overlap and are not resolvable from each other any more for (g, g¢) d = 2ri and (h, h¢) d = ri. (i) At a large particle distance (d = 6ri), each low particle is imaged as a ring (black trace in cross-section i¢). (j, j¢) The rings start to overlap for d = 4ri. (k, k¢) For a smaller particle distance, the rings overlap in the center between the particles. (l, l¢) For an even smaller distance, the largest overlap is above and below the x-axis. In this case, the image appears as if two particles rotated by 90° were present. The color bars indicate a height range of 0.0–0.3h0. The pipette tip opening is outlined dashed in (e, i) (reprinted with permission from Rheinlaender and Schäffer [18]. Copyright 2009 American Institute of Physics)

and therefore the current starts to drop. In the case of moderately soft samples, the gap between pipette and sample surface can be closed completely by sample deformation and the ion current can decrease to zero. For example, this is the case for polycarbonate. In contrast, on stiff samples like glass the current does not reach zero (Fig. 15.7b), because a gap with a finite size always remains, due to a slightly tilted or rough pipette opening. Sometimes, the current increases again if the pipette is further lowered. This effect has been interpreted as a bending of the lower part of the pipette, which causes the pipette opening to uncover again [12].

Basic Imaging Modes Imaging with Simple Ion Current Feedback The strong distance dependence of the measured ion current can be used to control the pipette–sample distance by using a feedback system. This allows scanning the pipette laterally over a sample with the spacing between pipette and sample

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Fig. 15.6 Topography images of an object shaped as the letters “sicm” calculated with FEM. (a) Actual topography. (b) Topography image of a high object (h0 = ri) imaged with a large pipette at a large pipette–sample distance (z0 = 1.6ri). (c) Topography image of a low object (h0 = ri/2), imaged with a large pipette at a small pipette–sample distance (z0 = 0.6ri). (d) Topography image of the same object, imaged with a pipette half the size, at a corresponding large pipette–sample distance (z0 = 1.6ri). The letters are now clearly resolvable. The color bar indicates a height range of 0.0–0.8h0. The cross- sections of the respective pipette tip openings are outlined dashed in the images (reprinted with permission from Rheinlaender and Schäffer [18]. Copyright 2009 American Institute of Physics)

c onstant. In general, the movement is driven by piezo actuators, which allows relative pipette-to-sample positioning with subnanometer accuracy. Optional position sensors can help to remove positional hysteresis and nonlinearity of the piezo actuators. By recording the z-actuator position while the lateral x- and y-actuators are raster scanned, a topographic image of the sample surface is generated.

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One characteristic of the ion current feedback is that topographic data are obtained without mechanical contact between pipette and sample. Therefore, damage- and even deformation-free imaging of extremely soft samples such as living cells is possible. For example, Korchev et al. reported topographic imaging and cell volume measurements of living cells at a resolution of 2.5×10-20L [33] and localization of single ion channels on the cell surface and detection of their dynamics [34].

Limitations of Simple Ion Current Feedback Although simple ion current feedback has successfully been applied to topography imaging of very delicate samples such as living cells, some inherent technical problems make the scanning process quite unstable. The main drawback of a DC-based mode is its strong dependence on the long-time stability of the electrochemical conditions of the whole setup. For example, temperature drifts or changes in the ion concentration as they usually occur during hour-long scanning cause significant DC offsets. Due to the strong distance dependence of the ion current, these DC offsets have a strong effect on the current-based feedback system, ultimately making it impossible to track the sample topography for a long time. If the SICM is 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. Another inherent disadvantage of a current-based feedback system results from the low directional sensitivity of the ion current. The pipette is practically insensitive to protrusions on the sample surface if they are approached from the side during the lateral scanning. As a consequence, the pipette runs into these protrusions and bends due to the lateral movement of the scanner. Such a bending will result in a small tilt of the opening with respect to the sample surface, leading to an increase of the ion current (Fig. 15.7b). The feedback loop will then force the pipette even further toward the sample, ultimately causing a pipette crash, damaging both sample and pipette.

Fig. 15.7 Measured ion current vs. pipette–sample distance. (a) On a soft sample (polycarbonate), the ion current decreases rapidly from the saturation current, I0 » 1.5 nA, to approximately zero after touching the surface. (b) On a stiff sample (glass), the ion current usually does not reach zero, because the pipette bends when lowered further, thereby uncovering the pipette opening again

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For the same geometric reasons, the pipette 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 SICM system. As a first solution, the average distance between pipette and the sample could be kept rather large by simply setting the current setpoint of the feedback loop close to the saturation current. Such a workaround, however, would degrade the sensitivity and the resolution of the SICM. Therefore, routine applications require overcoming the inherent technical limitations of a simple current-based feedback system. A number of different approaches have been implemented to achieve this goal. They can be subdivided into two classes: SICM systems of the first class still rely on using the ion current as a feedback signal for distance control, but apply special modulation techniques to minimize drift, sensitivity and noise effects mentioned above. They will be described in the next section. In the second class of 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 pipette–sample distance. The local ion current is no longer used for distance control but recorded simultaneously with the sample topography as it is measured by the separate feedback system. Typical implementations are based on concepts well known from other SPMs such as dynamic mode or shear force concepts, which are discussed in detail in Section “Combination with Other Scanning Techniques.”

Distance-Modulation Methods Different methods for improving the performance of a SICM while retaining the original mode of using the local ion current as feedback signal for controlling the pipette–sample distance have been introduced. In particular, modulation techniques have proved to be quite effective minimizing the basic problems of a DC-currentbased feedback loop. They share the common idea that the risk of hitting a topographical sample feature during lateral scanning can be effectively reduced by modulating the pipette–sample distance. Gitter et al. [35], Mann et al. [36], and Happel et al. [37] employed a type of point spectroscopy technique for SICM imaging as it is known from other scanning probe methods such as STM (“hopping mode”) [38], AFM (“force mapping mode”) [39], and “picking mode” in SECM [40]. In SICM, the pipette is approached to and then retracted from the sample surface by typically several micrometers for each x-y position on the scanning area. Thus, the lateral movement of the pipette can be performed while it is far above the sample. Consequently, the risk of the pipette laterally colliding with a feature of the sample or getting trapped in a pit is minimized. However, this also results in a slow scanning speed due to the time-consuming large vertical pipette movements while scanning. For each x-y position, the pipette is approached toward the sample until a predefined trigger condition is reached, i.e., until a drop of the ion current below a set threshold value signals the pipette being in close proximity to the sample. The z-position of this trigger point is then stored and the pipette is lifted again before moved to the next x-y position. Novak et al. [41] applied the hopping mode to

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high-resolution cellular imaging. They also implemented an adaptive scan algorithm, which first identifies feature-rich regions in the scan area by performing a low-resolution prescan. Then, the final image is obtained with higher pixel density in the featurerich regions, e.g., where parts of the imaged cell are located. This results in a decreased acquisition time, but it is still large compared to conventional scanning. To address problems caused by time-varying DC offsets in the ion current when scanning in the simple ion current feedback additional measures have to be taken. One option is applying voltage or current pulses [36, 37] instead of applying a constant voltage between the pipette and the bath electrode. 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 system has been proposed by the groups of Shao [14, 23] and Korchev [17]. They keep the voltage between the two electrodes constant, but the z-position of either the sample or the pipette is modulated with amplitude, Dz, of a few tens of nanometers. This can be done with the same piezo actuator as used for the vertical scanning or with an additional one with little technical effort. A schematic drawing of the extended setup is shown in Fig. 15.8a. For the pipette far away from the sample, the ion current is constant at the saturation value. When the pipette is approached to the sample surface, the ion current drops as soon as the pipette comes into the vicinity of the sample. A small distance modulation thus leads to an AC-modulated component in the ion current, which can be detected, for example, with a lock-in amplifier. The amplitude of the AC ion current increases with decreasing average vertical pipette position, z0 (Fig. 15.8b), so this amplitude can then be used as feedback signal to control the average pipette–sample distance, keeping a constant distance during the scanning process. Figure 15.9 shows the dependence of the average ion current and the ion current amplitude on the pipette–sample distance. This AC imaging mode has some advantages compared with the conventional DC-current-based approach. First, the system is less sensitive to drifts in the electrochemical circuit allowing very small setpoint values. This allows scanning at a large pipette–sample distance. Second, the pipette approaches the sample surface mainly in vertical direction, because the lateral pipette movement is small within one modulation period due to modulation frequencies in the range of some 100Hz. Third, the bandwidth of the current measurement is centered at the modulation frequency, where usually less electronic noise is present. All together these advantages minimize the risk of pipette crashes and improve the performance of the SICM, allowing the imaging of highly structured samples with long-term stability. Figure 15.10 shows a typical SICM topography image of a living cell.

Applications The improvements by the modulation techniques, as they were described in the previous section, allow scanning delicate biological samples such as living cells. For example, 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

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Fig. 15.8 Principle of distance-modulated SICM. (a) The basic SICM setup is expanded by a piezo actuator for modulating the vertical pipette position. The amplitude of the resulting ion current is measured. (b) Simulation of a modulated vertical pipette position at three different pipette– surface distances, z0 (upper graphs, here with a modulation amplitude Dz = ri), and the resulting ion current as calculated with (15.17) (lower graphs, p = 0.02)

Fig. 15.9 Measured ion current and ion current amplitude vs. pipette–sample distance in distance-modulated SICM. When approaching the sample, the ion current decreases and the measured ion current amplitude increases (modulation amplitude Dz = 150 nm)

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Fig. 15.10 The SICM topography image of a living mouse fibroblast cell growing on a polycarbonate surface in culture medium recorded with distance-modulated SICM. Image size: 80 µm x 80 µm. Vertical range: 4 µm

Fig. 15.11 Topography image of a monolayer of living A6 epithelial cells. Numerous microvilli can be seen (reprinted with permission from Gorelik et al. [42])

as depicted in Fig. 15.11 [42]. Because the distance-modulation mode reduces the risk of uncontrolled contact between the pipette and the sample and also improves the longterm stability, it becomes possible to continuously image specific areas for hours. For example, microvilli dynamics [43, 44] and the formation and rupturing of freely suspended, artificial phospholipid membranes [19] have been studied. Another application is to employ the SICM for performing patch-clamp experiments at specific surface sites [42, 44]. In such experiments, the SICM is first used to image the sample topography. The resulting high-resolution images of the cell membrane topography allow identifying regions or structures of interests. The pipette is then positioned over a selected spot. Finally, the pipette is approached 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 spots with lateral position control in the range of tens of nanometers. Such high position accuracy cannot be obtained

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Fig. 15.12 (a) The SICM topography image of a living A6 cell surface and (b) ion channel recording on top of a single microvillus. The pipette’s position is marked by an arrow in the line profile on the top. The channel was identified as a K+ channel by the reversal potential of the current–voltage curve (reprinted with permission from Gorelik et al. [42])

with light microscopy alone, which is conventionally used for patch-clamp pipette positioning. With the SICM it becomes possible to investigate the activity of single ion channels in the membrane of living cells. For instance, the SICM pipette can be deliberately positioned on top of a single microvillus on an epithelial cell and used for subsequent patch-clamp recording (Fig. 15.12). Another application is the combination of SICM imaging and nanolithography, as demonstrated by the selective destruction of individual, freely suspended phospholipid membrane patches [19].

Advanced Imaging Modes High-Resolution Imaging In 2006, Shevchuk et al. imaged single proteins and their dynamics on the surface of living cells with SICM [25]. The resolution of the SICM, which is limited by the diameter of the pipette, can be increased by using glass capillaries made of, for example, quartz glass. While standard pipettes have diameters in the range of 50–100nm, diameters on the order of 10nm have been reported [25]. Imaging with these small pipettes is more difficult than with standard pipettes. First, the magnitude of the ion current is in the range of 100pA, about ten times smaller than in standard SICM experiments. Second, the pipette–sample distance during scanning is smaller by a factor of 10. So the mechanical stability of the setup has to be significantly improved, because mechanical or acoustical noise can easily cause the pipette to crash into the sample. As shown in Fig. 15.13, Shevchuk et al. recorded high-resolution topography images of a living boar spermatozoon. Additionally, in consecutive images of the same area spontaneous morphological changes of the cell

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Fig. 15.13 The SICM topography image of a living boar spermatozoon undergoing a spontaneous acrosome reaction. (a) Low-resolution overview image and (b) higher-resolution image of the equatorial segment (EqS) of the sperm marked in (a). (c and d) Two consecutive images of the area indicated by the white box in (b) recorded 10 min apart with both stable regions (examples highlighted with dashed lines) and spontaneous morphological changes. The proteins are imaged with widths of typically 10–20 nm and heights on the order of 1–5 nm (Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission from Shevchuk et al. [25])

surface were recorded. The proteins were imaged with widths of typically below 20nm and heights on the order of 1–5nm. This shows that imaging of molecular features on living samples is possible, which opens a wide field of possible applications of imaging with molecular resolution, which has thus far been dominated by other SPMs, especially the AFM.

Combination with Other Scanning Techniques The ability to image biological samples in vitro makes the SICM an interesting candidate for combination with other scanning microscopy techniques. In doing so,

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different strategies can be pursued. The first strategy 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 (see Section “Combination with Optical Microscopy”). The second 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. Local variations in the ion current are then recorded, which can reflect variations in the sample’s local ionic conductivity or permeability (see Sections “Combination with Atomic Force Microscopy” and “Combination with Shear Force Microscopy”). Combination with Optical Microscopy The groups of Korchev, Klenerman, 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 scanning near-field optical microscopy (SNOM) [22–24]. This was 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 is transparent, the transmitted light can then be collected through an objective and detected by a photomultiplier located underneath the SICM head. Living cells were successfully imaged with such a combined SICM–SNOM setup [22–24]. A method to utilize the SICM probe as a localized light source for SNOM was suggested by Bruckbauer et al. [45, 46]. The method is based on fluorescence, which occurs when a calcium indicator, with which the pipette is filled, binds with calcium in the sample solution while being illuminated with laser light. The mixing zone, where the fluorescent complex forms, serves as the localized light source. The SICM has also been successfully combined with scanning confocal microscopy (SCM) [47]. The setup comprises an inverted light microscope fully configured for SCM, with the SICM head placed above. During lateral scanning, the vertical position of the sample is controlled by a standard SICM, i.e., by a distancemodulated ion-current-based feedback loop. As a consequence, the optical confocal volume – which is positioned just below the pipette’s opening – follows the topography of the sample. This allows capturing fluorescence images of a sample volume simultaneously with topographic data. For example, the interaction of fluorescent virus-like particles with the surface of fixed or living cells can be studied [47]. Combination with Atomic Force Microscopy If the nanopipette is bent [48, 49], it can be used as a cantilever for AFM [15]. With such a bent nanopipette, the sample topography can be imaged using a standard AFM setup. Simultaneously, the ion current through the pipette can be monitored

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and recorded as a complementary measurement signal. The nanopipette thus acts as both force sensor and ion conductivity probe. While the force on the pipette tip is used to generate the topography signal, the ion current is recorded to construct a complementary image of the sample’s ion conductivity. 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 conductivity) is generally obtained in tapping mode [50, 51] where the probe is modulated vertically and lateral forces are minimized. Combination with Shear Force Microscopy Another option for a simultaneous measurement of ion current and sample topography lies in the use of the nanopipette as a shear force sensor. Shear force measurements are well known from SNOM setups, where an optical fiber is kept at a constant distance to the sample while scanning and optical signal recording [52, 53]. In a shear force SICM configuration, a small piezo actuator is attached to the nanopipette exiting transverse mechanical vibrations in the tapered end of the nanopipette (Fig. 15.14). The vibration amplitude strongly depends on the pipette–sample distance due to shear forces. The shear forces increase sharply at small pipette– sample distances, thereby reducing the vibration amplitude. The technical challenge is detecting this vibration amplitude on the nanometer scale. Several methods

Fig. 15.14 Schematic of a combined SICM and shear force microscope. The basic SICM setup is extended with a dither piezo driving the pipette tip into lateral oscillations. The oscillations are recorded by detecting the deflection of a laser beam focused on the tapered pipette end via a segmented photodiode and a lock-in amplifier. The controller-based feedback loop keeps the pipette’s oscillation amplitude and therefore the pipette–sample distance constant using shear force interaction between pipette and sample. Simultaneous ion current recording allows topography and ion conductivity imaging in parallel [21]

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have been established so far, including optical detection [54] and detection using a piezo-electric tuning fork device [55]. Piezo-electric detection methods face some principal problems when they are used for imaging in liquids, because electrical shortcuts occur easily in electrolyte media. Special shear force setups solving this problem have been implemented [56–58]. For the optical detection method [21], a laser beam is focused onto the vibrating section of the nanopipette near its tip (Fig. 15.14). 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 and generates a signal that is proportional to the vibration amplitude of the nanopipette. This signal is then used as the feedback signal. Simultaneously, the ion current image is recorded. Because the electrical resistance through the pipette straight to the bath electrode is indirectly held constant by the shear-force-based feedback loop, changes in the ion current signal are induced by changes in the sample’s ion conductivity. Figure 15.15 shows an image of a polycarbonate filter membrane, where the sample’s topography and ion conductivity are imaged independently of each other [21]. This shows, for example, which pores in the filter membrane are permeable for ions and which are not. The possibility of imaging the sample’s local ion conductivity, for both stiff and soft samples, opens perspectives for further applications in fields as diverse as biochemistry/pharmacology or material science.

Fig. 15.15 Topography and ion current of a polycarbonate membrane with pores of 1 µm d iameter. The topography (3D-relief) is overlaid with the ion current (color; brighter colors correspond to larger currents). The pores are visible in both the topography and the ion current. Arrow: this pore displays no significant conductivity increase, showing that the pore is blocked. Height scale: 2.5 µm (reprinted with permission from Böcker et al. [21])

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Elasticity Measurements with SICM A novel application of the SICM was demonstrated by Sánchez et al. [59] by using the pipette as a contact-free elasticity probe. While the main SICM setup is left unmodified, a fluid flow through the pipette is induced by a macroscopic pressure on the order of kilopascals applied at the upper pipette opening. Based on the Hagen–Poiseuille law, the fluid flow through a conical pipette with a free opening, Q0, can be described by [59]:

Q0 =

3p tana 3 ri p0 , 8h

(15.25)

where a is the pipette half cone angle, ri is the pipette inner radius, h is the fluid viscosity, and p0 is the pressure applied to the upper pipette opening relative to the pressure in the bath. Due to the small value of a and ri being in the micrometer range for p0 on the order of kilopascals the flow is typically Q0»10-12m3/s=1nL/s. If the pipette is now placed in the vicinity of a sample, the hydrodynamic becomes quite complicated. Therefore, it is useful to consult numerical methods. Figure 15.16 shows the results of finite element calculations, solving the Navier–Stokes

Fig. 15.16 The FEM of a nanopipette with a pressure p0 applied to the upper pipette opening. (a) Distribution of flow velocity, v, in units of vmax » 1 cm/s• p0/kPa. (b) Corresponding pressure distribution. (c) Pressure profile at the sample surface for a pipette–sample distance z0 = 0.5ri. The maximum pressure at the surface is only about 50% of the applied pressure, due to a significant pressure drop along the pipette. (d) Resulting surface force, F, in units of F0 = pr 2i • p0 and fluid flow, Q, in units of saturation flow, Q0, as a function of pipette–sample distance, z0. The pipette parameters were ri = 1 µm, ro = 1.5 µm, a = 3° [59]

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equation in the tip region for a planar sample. The flow velocity in the pipette inner room (Fig. 15.16a) has a Hagen–Poiseuille-like parabolic profile. The magnitude of the velocity decreases to zero when passing the gap between pipette wall and sample. It can be seen in Fig. 15.16b that the hydrodynamic pressure decreases along the pipette so that only approximately 50% of the applied pressure, p0, is present at the sample surface directly below the pipette. Due to the high hydraulic flow resistance in this gap, the remaining pressure drops over the gap. This can also be seen in the pressure distribution at the sample surface (Fig. 15.16c). By integrating the pressure distribution over the sample surface, the total surface force (F) is obtained (Fig. 15.16d). F is strongly dependent on the pipette–sample distance, z0. For the pipette far away from the sample (right-hand side of Fig. 15.16d), the fluid leaves the pipette almost freely with little additional flow resistance in the tip region, resulting in a maximum fluid flow, Q»Q0. Therefore, most of the pressure drops along the pipette and does not “reach” the sample. For a decreasing pipette– sample distance, the flow resistance in the tip region increases, so the fluid flow decreases. According to (15.25), this results in less pressure drop along the pipette and an increasing pressure at the sample surface. Hence, the surface force increases (left-hand side of Fig. 15.16d). Note that the maximum surface force for z0→0 is larger than the pipette opening area times the applied pressure, F0=pr 2i • p0. This is due to the finite thickness of the pipette walls, resulting in a larger effective area on where the pressure is exerted. To sum up, the pipette is used as a pressure channel from the macroscopic to the microscopic domain which allows localized force application without mechanical contact between pipette and sample [59]. The pressure-induced fluid flow through the SICM pipette can be used to measure mechanical properties of microscopic samples [59]. Figure 15.17 shows the determination of the Young’s modulus of a red blood cell. The pipette was placed above the cell surface with the ion current feedback keeping the distance constant

Fig. 15.17 The SICM pressure experiment on a red blood cell. (a) Schematic of the pipette positioned above the cell with the current feedback keeping a constant pipette–sample distance. (b) Plot of applied pressure and resulting cell deformation as a function of time. The cell deformation is linear with applied pressure. The inset shows an optical phase contrast image of the cell with the pipette visible as a cone-shaped shadow. (c) Plot of pressure vs. cell deformation from the data in (b), resulting in a Young’s modulus of E = 4.2 kPa (reprinted with permission from Sánchez et al. [59])

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and no pressure applied. Then the pressure was slowly ramped up (Fig. 15.17a) causing the cell to deform (Fig. 15.17b). The relation between the applied pressure and the resulting cell deformation is linear (Fig. 15.17c) and can be used to determine the local Young’s modulus of the sample. For quantifications of the local Young’s modulus, a calibration method is needed which can be either numerical (Fig. 15.16) or experimental in the form of a calibration measurement on a sample with known elastic properties. The SICM elasticity measurements were performed on oil droplets and on various kinds of fixed and living cells, allowing, for example, to study the effect of fixation on the elasticity of epithelial cells or to study the cytoskeleton of cardiomyocytes and neurons [59].

Outlook We have shown that the SICM has a broad range of applications in imaging science especially in the field of bioscience with a resolution down to the nanometer range. Next to topographic information, it can also extract material properties, e.g., the local conductivity [21] or elasticity of the sample [59]. Due to its noncontact character the SICM is highly suitable for imaging soft and delicate samples. It has been applied to the imaging of suspended phospholipid membranes [19], which may be used as a model system for cell membranes. Another challenge is using the pipette as a source for localized material delivery. The material flux from the pipette to the sample substrate may be driven by an electric field, capillary force, hydraulic pressure, or ultrasonic excitation. Recently, pipette-based nanolithography became possible by the deposition of metals [60–63], chemicals [49, 64–66], or biological material [29, 45, 67–70]. Another exciting prospect is the construction of pipette-based biosensors [71, 72].

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Chapter 16

Combined Voltage-Clamp and Atomic Force Microscope for the Study of Membrane Electromechanics Arthur Beyder and Frederick Sachs

Introduction Tools for Membrane Electromechanics at Molecular Scale Most mammalian ion channels have conductances of 5–50pS that produce single-channel currents of 0.5–5pA under physiologic conditions [1]. For example, Shaker’s unitary conductance is ~11pS. Voltage clamp of cell membranes with pipettes, also known as “patch-clamping,” is routine and allows an electrical resolution of individual molecules at thermal noise levels <0.1pA [2]. For the study of molecular mechanics, requires resolution at the level of kBT=4.11×10-21J=4.11pNnm. Atomic force microscope (AFM) should resolve pico-Newton forces and nanometer displacements. The response time of both the AFM and the voltage clamp are both <1ms and well matched to be able to study the correlated relaxation rates. Exploring the influence of membrane potential on the mechanical motion of cell membranes is difficult for two primary reasons: (1) the allowable voltage range is limited by membrane breakdown [3] and (2) the membrane is soft and the surface is not mechanically well defined [4–9]. Critical to understanding the mechanics is that the membrane is much more complex than a bilayer and includes many cytoskeletal proteins capable of supporting stress in three dimensions [6]. There are two basic methods of measuring membrane mechanics: measurements over large areas to increase sensitivity or local measurements with more scatter to increase speed. Large area measurements include optical measurements by contact angle tensio meters [10, 11], differential confocal microscopy [12], and single-particle tracking [13]. The local area measurement techniques include fluorescence- and lanthanideresonance transfer [14], stroboscopic interferometry [15], optical [16] and magnetic tweezers [17], surface force apparatus [18], biomembrane force probe [19], and AFM [20, 21].

A. Beyder (*) Department of Medicine, Mayo Clinic, 200 First Street SW, Rochester, MN 55905, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_16, © Springer Science+Business Media, LLC 2010

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A few groups have succeeded in combining patch clamp electrical recording with mechanical measurements, using AFM [20, 22–26], optical tweezers [16], and direct cell stimulation with a micropipette [27]. These setups need two “arms” – voltage and mechanical – with the ability for precise positioning and stability. Technological improvements are continually driven by “lab-on-a-chip” technology. To this end, a study recently published utilized planar patch clamp in combination with AFM reducing the need for two arms to one [25]. Our long-term goal is to directly measure the displacements of the moving parts of single ion channels. Below we detail our VC-AFM setup and recent experiments of MEM of Shaker-transfected cells. Our motivation arose from a body of knowledge about the coupling transmembrane electric fields to membrane mechanics. Some notable examples include swelling of neurons during action potentials [28, 29], voltage-induced effects on mechanosensitive channels (MSCs) [30], mechanical effects on voltage-activated channels [31, 32], voltage-induced membrane movement of outer hair cells (OHCs) [33] as well as voltage-induced displacements in nonspecialized cells [21], and finally, voltage-induced effects on lateral diffusion in the membrane [34]. While some of the above findings have physiological links, especially those in OHCs, where the electrically driven motion is used to tune the resonance of the basilar membrane, the others have not yet been associated with a distinct physiologic phenomena, but the linkage of mechanics to function is a rapidly expanding area [35–38]. We selected voltage-gated ion channels for our experimental target because these channels are nature’s prototypical EM coupling machines. They are the transistors of the excitable tissues, having voltage sensitive gates that are physically displaced in response to shifts in voltage, and these characteristics place ion channels squarely into the category of EM transducers [39–42]. The delayed-rectifier Kv1.1 is a prototypical voltage-gated channel and was one of the first eukaryotic channels to be cloned [43]. This channel is affectionately named Shaker channel, because of its host ,Drosophila melanogaster, shakes when this channel is rendered nonfunctional. Structurally, Shaker is a homo-tetramer, with each of the four subunits made up of six transmembrane segments (TMs), and each subunit has three functional domains: a K+-selective pore, voltage sensors, and a gate. The S5/P-region/S6 constitutes the K+-selective pore [44], while S1–S4 has been implicated as the voltage sensor domain (Fig. 16.1). Each S4 harbors 4–8 positively charged residues [45]; the electric field does work on these charged amino acids of the S4 segment [46]. Shaker’s voltage sensors have been shown to move in a changing electric field [47, 48], but the precise sequence and displacement of the voltage sensors during activation remains hotly contested. Activation of the four voltage sensors (which move independently) is coupled to a concerted step of opening of the channel gate. This gate consists of the four intracellular portions of the S6 segments which line the pore. Gating involves bending at the “elbow,” more formally known to be the PVP motif, near the middle of the bilayer [49].

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Fig. 16.1 Shaker channel structure. (a) Profile view of a single subunit of a Shaker channel. Each identical subunit of the channel consists of six transmembrane (TM) segments named S1–S6. There is a voltage sensor domain, consisting of S1–S4 on each subunit. The pore is composed of four S5 and S6 segments, with four signature sequences “VGYG” that make up the selectivity filter. (b) Top view of a Shaker channel as a homo-tetramer. The ion channel pore is shown as an open circle in the middle, surrounded by S5–S6 TMs and flanked by the voltage sensor domains

Ion channels are embedded in lipid membranes and their function is affected by the composition and stress in the membrane [36]. In the best studied case of reconstituted prokaryotic MSCs, the bilayer tension is responsible for transmitting force to the channels [50]. There is significant recent evidence demonstrating mechanical effects of lipid bilayers on voltage-sensitive ion channels [36, 51–56] including voltage-gated potassium channels [31, 36, 52, 56–60]. This evidence supports the notion that as a resident of a lipid bilayer, an ion channel’s function must be studied as a system including the lipids and not just a protein [36].

Background on Voltage-Clamp AFM on Cells Mosbacher et al. voltage clamped wtHEK cells, and then recorded the displacement of an AFM cantilever, preloaded with a small force (0.5–3nN) against the cell, with a 10-mV peak–peak AC carrier stimulus [20]. They found that wild-type human embryonic kidney (HEK) cell membranes moved the cantilever 0.5–1.5nm normal to the plane of the membrane. The movement was outward with depolarization, and surprisingly, the holding potential (HP) had a negligible effect on the amplitude of the movement. The movements were also present in HEK cells that were transfected with noninactivating mutant Shaker K+ channels. Unlike the movements of the nontransfected membranes, these movements were sensitive to the HP, decreasingbetween -80 and 0mV. Still, the maximal movements of transfected cells were slightly larger than untransfected cells. The authors suggested that the inverse flexoelectric effect [61] may be responsible for this movement. Further analysis showed that for typical values, the peak–peak force exerted on the AFM tip could range from 0.7 to 70pN and account for the observed nanometer movements.

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In recent work from our lab, we characterized MEM in wtHEK cells in more detail using step potentials [21]. The results were modeled by a Lippman tension (s) [62] that depends on the concentration of mobile charges in the double layer (Q in C/m2). That charge density is a function of the ionic strength of the solution bathing the interface, the fixed charge density, and the transmembrane potential (E),

 ds  − = Q.  dE  m , T , p This solution requires two assumptions: (1) the double layer is a parallel plate capacitor and (2) the membrane capacitance (C) is independent of the transmembrane potential, C = Q/E. The assumption of independence of capacitance from potential seems valid because for small potential excursions (±100mV) the capacitance of black lipid membranes is essentially constant [11]. The above equation may then be integrated, resulting in,

s = −0.5CE 2 + T0 , where T0 is the voltage-independent tension [21]. In the physiological voltage range, Zhang et al. showed that the voltage-induced movement of wtHEK membranes was linear with voltage, and also dependent on the initial tension, or indentation (in the experimental sense). The results were similar to those of Mosbacher et al: depolarization produced an outward displacement and an increase in tension at normal ionic strength [20]. With hyperpolarization the membranes slacked, allowing the cantilever to settle toward the cell interior. As predicted, the Lippman tension also depended on the surface charge of the membrane which was varied with ionic strength and addition of salicylate [21]. In a normal bath solution, the voltage-induced displacement was linear with voltage (1nm per 100mV). In low ionic strength solutions, however, the relationship of potential to movement reversed as the outer surface potential became equal to the inner surface potential at about –20mV. These studies demonstrated the experimental power of the VC-AFM approach and allowed resolution of the physical phenomena previously inaccessible. The energies uncovered by these studies are small (Lippman tension ~1% of lytic), they are relevant for operation of ion channels. Yet, in order to resolve the Angstrom sized movements and energies at the kBT level, Zhang et al. used 50× signal averaging, which limited temporal resolution [21]. Further improvements in the VC-AFM setup allowed us to extend the studies on membrane electromechanics (MEM) in live cell membranes with Shaker K+ channels. We found that the opening of these channels resulted in transient changes in the mechanical properties of the membrane. We detail our approach and the results below. We also discuss the obstacles encountered during our exploration, our solutions, and the inherent limitations.

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Methods Channel cDNA In order to focus on primary gating of Shaker, we used a noninactivating ShakerH4 (ShIR) [63] with five mutations: deletion of residues 6–46 to remove N-type inactivation; T449V to inhibit C-type inactivation, and for future studies using cysteine modification C301S, C308S, and A359C (Dr. Richard Horn, Thomas Jefferson University, Philadelphia, PA) [64]; “IL” mutant of ShakerH4 with additional V369I and I372L mutations (Dr. Gary Yellen, Harvard University, Cambridge, MA); Acetylcholine receptor (AChR) a, b, d, e subunits used in 2:1:1:1 ratio, respectively (Dr. Anthony Auerbach, SUNY at Buffalo, Buffalo, NY).

Cell Culture and Transfection All MEM experiments were performed on tsA201 cells (HEK-293, ATCC CRL1573, stably transfected with SV40 large T antigen). Cells were grown in Dulbecco’s Modified Eagle Media (DMEM), supplemented with 10% fetal bovine serum (FBS) and 1% penicillin–streptomycin, at 37°C in air/5% CO2 incubator. Cells were cultured in T25 flasks and upon confluence, cells were lifted using trypsin, resuspended in fresh “low growth” media (DMEM with 1% FBS) and plated at 10–20% confluency onto poly-lysine coated glass cover slips in 35mm Petri dishes. After 12–24h, cells were transfected with cDNA for the channel and GFP using Fugene-6 (Roche). Transfections followed the manufacturer’s protocol [65]. Specifically, for four culture dishes we added the transfection media: 390µL serum-free DMEM, 12µL Fugene, 2µg channel DNA, 2µm GFP DNA. Cells were transfected for 12–24h.

Recording Solutions Immediately prior to the experiments, cells were removed from the growth media and placed in the “physiologic” bath solution that contained (mM) 137 NaCl, 5.4 KCl, 0.5 MgCl2, 1.8 CaCl2, 10 HEPES, 5 d-Glucose. “NMDG bath” solution contained 142.4 mM NMDG replacing NaCl and KCl. “Physiologic” intracellular (pipette) solution contained 145 KCl, 5 NaCl, 0.5 MgCl2, 10 EGTA, 10 HEPES, 5 d-Glucose, pH = 7.4 with KOH and 300 mOsm. “NMDG pipette” contained 149 mM NMDG and 1mM KCl instead of 145 mM KCl and 5 mM NaCl. “Symmetric K+” solutions were “physiologic pipette” and an extracellular solution with all NaCl replaced with KCl (142.4 mM) All solutions were adjusted to pH = 7.4 and 300mOsm with mannitol.

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Voltage Clamp Standard whole-cell recording methods were used to clamp the voltage and record current [2]. Electrode resistances were 1–3MW when filled with the pipette solution. HEK cells were whole-cell clamped and voltage was manipulated using an Axopatch 1-B amplifier (Molecular Devices, Sunnyvale, CA). Seal resistances were >1GW. Capacitance and series resistance compensation were routinely used during whole cell current recordings. Data was sampled at 10 kHz and filtered at 5 kHz. Between runs the cell was held at a HP of –90mV. A typical stimulus was a voltage ladder with 20ms long steps from –120mV in 20 mV increments.

Atomic Force Microscopy We used a modified [22, 24] Nomad AFM (Quesant Instrument Corp., Agoura Hills, CA). The AFM was operated using modified Quesant software and data was collected using custom VIs in Labview [22]. We calibrated the photodetector (PDT) sensitivity with both the Quesant and Labview software. In the Quesant we used the automated routine that followed cantilever engagement on glass to measure PDT sensitivity using a 0.4V piezo-transducer (PZT) displacement that had been previously calibrated. In Labview, we performed a force–distance (FZ) curve on glass and adjusted the gain on the PDT using the previously calibrated PZT ( DzPZT/DzPDT =1).

AFM Cantilevers For these experiments we used Micro-Lever B 0.02nN/nm rectangular cantilevers (Veeco, Instruments, Inc., Plainview, NY). The cantilever chip was glued to the nosepiece of the AFM and calibrated in air and liquid using Sader’s method [66]. Liquid calibration was more difficult, because soft levers in liquid have a low Q (for the Micro-lever B Qliquid =1.6±0.22, n =11) making difficult the identification of the resonance peak. When compared to other methods, Sader’s calibration method proved reliable and we used it to calibrate levers quickly in air or liquid [67]. The Micro-Lever B spring constants were higher than the manufacturer’s quoted values, kair= 0.026±0.0035N/m (n =21), and had slightly higher resonant frequencies in air than specified by Veeco, f0,air =15.7±0.6kHz (n =21). Once the cantilevers were submerged in liquid, the resonant frequency decreased to f0,liquid =3.45±0.4kHz (n =11). In our experience, kliquid was estimated to be higher than kair by about 20%.

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AFM Measurement of Cell Stiffness The HEK cells were analyzed in various experimental conditions to determine their relative stiffness. After the cells were plated, Microlevers A and B (k = .01 and 0.02 N/m, respectively) were used in a FZ mode to softly push on the cell surface. We used calibrated deflections of the PZT (zPDT) and PDT (zPZT) and calibrated cantilever spring constant (k) to estimate cell stiffness (kcell),

kcell =

zPDT * k zPZT

We then used kcell and Hooke’s law to calculate membrane displacement in response to a point force on the AFM cantilever.

VC-AFM Setup Individual components of the Voltage Clamp AFM (VC-AFM) setup were tested and tuned as above. The combined setup is represented in a cartoon form in Fig. 16.2a. Multiple components were necessary, requiring careful organization for spatial and noise concerns. The AFM and patch-clamp head-stage were positioned atop an Olympus IX70 inverted microscope. The entire setup, including the microscope was mechanically isolated in three steps (Fig. 16.2b). First, the microscope was mounted on a plywood stage that was suspended on four bungee cords to provide mechanical isolation of less than 10Hz as previously described [68]. Stiffness of the bungee cords required careful adjustment for optimal damping. The setup was then rested on an air table to provide further mechanical isolation above 10Hz. Finally, the setup was surrounded by an acoustic isolation chamber to minimize the effects of air currents and other acoustic interference. Meticulous care was required for mechanical and electrical isolation. The AFM and patch-clamp head-stages needed to be electrically grounded to reduce interference. All wires connecting the headstages were also suspended for mechanical isolation. Electrical control units were kept several feet away from the setup.

VC-AFM Data Acquisition Transfected cells were identified by GFP fluorescence and small, rounded cells were selected. In a typical experiment, once the cell was voltage clamped in wholecell configuration. The AFM cantilever was positioned about 50µm above the cell surface. The cantilever was then stepped down in 2µm steps (range of the PZT), and an FZ routine performed at the end of each step to detect the cell surface. Once in

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Fig. 16.2 VC-AFM setup. (a) Cartoon form of the setup highlights the personal computer (PC) control of two dedicated amplifiers (AFM Amp and VC Amp). (b) Components of the setup are outlined in gray. Bottom left, a photograph of the instruments (inverted microscope, AFM, and patch-clamp head-stages) is suspended on a bungee cord supported stage, resting on an air table and surrounded by an acoustic isolation chamber. Recording was done with the acoustic chamber door closed. Electronic control interfaces and microscope light source (not pictured) were kept outside the acoustic isolation chamber. Bottom right in blue is a photograph of an upright AFM on a micromanipulator stage and patch-clamp head-stage. Above in yellow is shown the AFM head modified to improve performance in fluid [22]; the patch pipette is brought in under the sloped edge of the AFM head. Above, a 40× optical image of the AFM cantilever and patch pipette engaged on a HEK cell

the vicinity of the surface (2 µm), we engaged the cantilever at the desired force (0.02–0.5 nN). We then stimulated the cell with the voltage protocol and recorded the bending of the cantilever. Cell current and voltage and cantilever displacement were digitized by a data acquisition board (AT-MIO-16E2, National Instruments) controlled by custom software in Labview [69]. The displacement of the cantilever was recorded as a voltage output from the bottom–top output (B–T) of the PDT. The signal was acquired from the breakout on Quesant Instruments’ electronic interface unit and was typically ~3 mV/nm. We filtered (typical bandwidth 0.1–500 Hz) and amplified (100×) the signal using an active eight-pole Bessel filter (Krohn-Hite Model 3341). We then averaged 5–50 (mean =12) traces to reduce random noise.

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Data Selection and Analysis The data with cells that were sealed (Rseal) >1GW and had a series resistance (Rs) <5 MW were selected for analysis. For each dataset, the mean cantilever displacement was calculated at multiple time locations during the pulse – early, late, and off. “Early” signified a displacement in response to a voltage step once steady-state current was achieved. “Late” occurred 20–50 ms later, just before repolarization. Finally, “off” was the displacement at the new baseline, 40–70 ms after the depolarization step. Displacement values for each data point in the baseline were smoothed by an average of ±2.5 ms and by ±1ms for each location during the pulse.

Results In this section we present the details of the VC-AFM experiments with live cells, provide experimental particulars, and then present our data on MEM of HEK and Shaker-transfected HEK cells. Majority of our experimental difficulties concerned AFM operation in fluid and experiments that involved force-clamping (FC) of very soft, metabolically active living cells, which are soft substrates with ill-defined surfaces.

Force Clamp Quantitative measurements with AFM have several potential sources of error. Instrument quantification errors in AFM studies are due to uncertainties in calibration and noise of each AFM element, including the cantilever [67], the PZT, and the PDT [70]. We will circumvent a detailed discussion of the measurement errors by assuming that the measurements are no better than the worst of errors. This dubious distinction generally belongs to the cantilevers and may be as much as 30% [71]. Cantilever-induced errors increase in significance when the AFM is operated in fluid where viscous drag and mass coupling are substantial. Limitations of a Cantilever as a Force Sensor The non-equilibrium environmental noise may be minimized by vibration and acoustical uncoupling from the environment [22, 68, 72] and by off-line data processing [73]. However, the absolute noise minimum is the thermal noise energy, kBT. There are only two approaches to minimize this noise, either lower the temperature or limit the recording bandwidth [74]. The vertical mechanical noise performance of an AFM probe is related to its transfer function. The fluctuation–dissipation theorem predicts that the linear response of a system to external perturbations is the same as

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the fluctuation properties about thermal equilibrium. Thus, in a dissipative environment, the transfer function can be estimated from an order of magnitude calculation of the diffusion constant (D),

D=

kBT g

where g is the friction of the probe [75]. Probes with high diffusion constants move further and faster in response to thermal excitation. Consequently, a practical aim of probe design is to reduce the friction, which may be accomplished by a reduction of solution viscosity and/or reduction in probe size. Commercially available cantilevers (Fig. 16.3a) were designed to work in vacuum and air [76] which are low-drag media when compared to liquid. For biological samples in their native environment, smaller cantilevers were designed and fabricated [77–79]. These probes demonstrated a substantial improvement in force sensitivity [77], but they retained other problems inherent to the cantilever design, such as drift [80, 81], optical clearance [82], and nonlinear cantilever bending [83]. We developed a micro-fabrication process suitable for mass production of levers with moving area only as large as necessitated by the target size for the optical lever (Fig. 16.3b) [84]. Because the noise level of the mechanical sensors depends directly on the size of the moving area, reduction of this area improves noise performance and increases bandwidth. Moving area of this lever is roughly ten to a hundred times smaller than the equally soft commercially available cantilevers. Our preliminary experiments with these probes exhibited radically improved baseline noise and drift resulting in decrease of the minimal resolvable force (Fig. 16.3c). In addition to lower noise, these torsion levers have significant advances in other areas: composite design, easily customizable spring constants as low as 0.001N/m, no warping due to changes in temperature or solution, reduced drag, increased frequency response, and calibrated orthogonal sensitivity [84–86]. Cell is a Mechanically Imperfect Substrate Large errors are introduced into AFM measurements when cantilevers are used to examine soft substrates such as living cells [4, 87–89]. The AFM is routinely used to perform FZ experiments to measure the stiffness of homogeneous substrates and other surface properties [4, 89–93]. This involves positioning a cantilever above the cell and then stepping it downward until a force, arbitrarily chosen to define the substrate, is encountered (Fig. 16.4a). In solution, the cantilever encounters no appreciable force when advancing and therefore no deflection (red portion of Fig. 16.4a). However, when the tip encounters the substrate, the front end of the cantilever arches according to Hooke law, F= kleverDz. Since the cantilever is then coupled to the substrate, one needs to consider the spring constant of the system (ksys), which is,

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Fig. 16.3 Torsion Cantilever. (a) Scanning electron microscope (SEM) image at 150× magnification of a set of commercially available Microlevers (Veeco). These cantilevers are manufactured from micrometer thick silicon nitride and have nominal spring constants between 0.01 and 0.5 N/m. Rectangular Microlever B, at the right of the image, has the lowest surface area and thus it was used for the MEM experiments. (b) SEM image at 150× showing soft torsion probes (~0.01 N/m). Torsion probes use moving parts that are 10–100-fold smaller in surface area than traditional cantilevers (small pad on SEM). Given very small size of the torsion probe, stiff extension forks (100-um long, k ~ 10 N/m) are used to reduce optical interference. (c) Baseline noise for Microlever A (gray, k = 0.01 N/m, f0,liquid ~ 1 kHz, = 7 pN) compared to a prototypical torsion probe (black, k = 0.002 N/m, f0,liquid ~ 1 kHz, = 0.4 pN)

ksys

 1 = ∑   n kn 

−1

where kn are spring constants of individual components in series. For stiff substrates (such as glass and mica), system stiffness is well approximated by the cantilever stiffness alone, and therefore contact point is well defined (Fig. 16.4a). On the other hand, cells are generally considerably softer than cantilever (~100×) [91, 94] so ksys is much closer to km (membrane) than kc (cantilever). Therefore, very small forces (F) (small cantilever deflections) result in large deflections (z) of the cell membrane, z = – F/ ksys [91]. In addition, one must realize that the cell membrane is not a sharp planar structure, but a furry and topologically heterogeneous surface so there is no absolute standard for contact. Experimentally, the relative softness and ill-defined shape of cell surface are evident by a slow and nonlinear rise in detected force after the tip makes contact with the cell membrane (Fig. 16.4b, Cantilever).

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Fig. 16.4 AFM Force-Distance (FZ) approach to a cell. (a) Cartoon of a typical AFM FZ approach. An AFM cantilever stepped downward through solution toward the substrate (red) does not encounter a sustained resistance at the tip, and thus does not show deflection. When the cantilever tip touches the substrate surface (blue), it reports this interaction force as an upward cantilever deflection. (b) The upper trace shows an FZ approach using a typical soft cantilever (Veeco Microlever A, k = 0.02 N/m, = 7 pN in 0.2-kHz bandwidth). Lower trace shows FZ approach curve using a typical torsion cantilever (k = 0.01 N/m, = 2 pN in 1-kHz bandwidth)

Because of this, routine identification of the cell surface generally requires a cell deflection of several hundred nanometers. When using torsion levers in FZ experiments, the lower baseline noise allows for earlier spatial recognition of the cell membrane. In our FZ experiments on HEK cells, the cell membrane can be readily identified online within about 50–100nm of contact (Fig. 16.4b, Torsion Lever). The FZ relationship of the cell surface depends upon the cell stiffness – the stiffer the cell the more force and the faster the response it will generate for the

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AFM. In our preliminary experiments we aimed to find a suitable expression system for our ion channels. Thus, we explored relative stiffness of various cell types using the FZ experiments (Fig. 16.5). All studied cells were softer than the cantilever (k ~ 0.01N/m = 10 mN/m); astrocytes and cardiac myocytes were about 40-fold softer than the softest commercially available cantilever (Veeco Microlever A), while wtHEK cells were almost 200-fold softer, and transfected HEK (ShHEK) were a whopping 1,000 times softer than the cantilevers! For FZ experiments, this implies that in order to achieve a 20-pN FC, cell surface of astrocytes and cardiac myocytes will deflect ~75 nm, while the wtHEK surface will deflect nearly 400nm and the ShHEK will deflect ~1,800 nm (Fig. 16.5, data in red). In the literature, the cell membrane is often stiffened by Gd3+ [95], while the supporting cytoskeletal network can be cross-linked and surprisingly only slightly stiffened by formaldehyde, suggesting that it was naturally cross-linked prior to treatment [87]. While we found that these approaches do stiffen the cell, they only have moderate effects and often they jeopardize cell’s viability and reduce the physiologic relevance (Fig. 16.5). We took advantage of the access of the patch clamp to the inside of the cell to increase the intracellular pressure and stiffen the

Fig. 16.5 Cells are soft. Force-distance (FZ) experiments used to estimate the linear stiffness of astrocytes, cardiac myocytes, and wtHEK cells in various conditions (Gd3+ 100 µM, formaldehyde 1% v/v, and positive pressure through the patch pipette of 10 and 20 mmHg). Cell stiffness shown in red, (kcell) is estimated from the initial deflection (400–660 nm) of the AFM cantilever upon cell contact kcell = DzPDT/DzPZT ·klever. In green is the approximate cell deflection (Dzcell) needed to reach 20 pN (Dzcell = 20 pN/kcell)

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cell by inflation. Application of 10–20 mmHg positive pressure through the pipette effectively stiffened the cell membrane, reducing noise and thus simplifying membrane detection and FC. At +20 mmHg, the cell is about ten-fold stiffer than a resting HEK cell, requiring only ~30 nm of indentation to reach a 20-pN force on the cantilever. Using Laplace’s law, we estimate that these pressures produced about 0.013–0.026 N/m of tension in a typical cell with a radius of ~10 µm. In the experiments on voltage-dependent motion performed by Zhang et al., the indentation was a gentle 0.46 nN, lower than the earlier experiments by Mosbacher et al. that had a resting force of 0.5–3 nN [20, 21]. In this study, we were able to further lower the FC set point. We could engage the cantilever at the cell surface for about 1min with forces as low as 20 pN in a 1-kHz bandwidth (for comparison, a 0.02 N/m cantilever has a thermal noise =9 pN).

Electrical Behavior of Wild-Type and Shaker-Transfected HEK Cells The HEK cells are often used for patch-clamp experiments because they are easy to culture, grow quickly, and efficiently express exogenous ion channels [64]. The background voltage-sensitive currents are small, but finite [96, 97]. Several studies have found in wtHEK cells endogenous expression of a-subunits of up to ten different delayed rectifier Kv channels, transient outward Kv channels as well as Kvb2 subunit [96–98]. These channels influence the outwardly rectifying background current. For all MEM experiments, we used various voltage ladder protocols (Fig. 16.6a). A typical set of membrane currents in response to the above voltage protocol is graphed in Fig. 16.6b. As expected, the background wtHEK current typically showed some outward rectification, but that typically amounted to ~200pA/cell at +30mV [97, 98]. This background current was always much smaller than the currents for the Shaker-transfected HEK cells (ShHEK). Typical Shaker current was several nanoamperes and highly voltage dependent (Fig. 16.6c). If the conductance of the membrane is plotted against voltage shortly after channel activation, we observe a highly nonlinear increase due to opening of the channels. The normalized conductance vs. voltage can be fit to a Boltzmann function (Fig. 16.6d). In our experiments, mean activation kinetics (V1/2 = – 41mV, slope=9.6mV) were similar to the published data (V1/2 = – 40mV, slope = 6.8mV) [99].

Membrane Electro-Mechanics (MEM) Wild-Type HEK MEM The wtHEK cells were voltage clamped in whole-cell mode with the AFM in FC. Depolarizing voltage steps induced outward membrane movement, i.e., upward motion of the cantilever (Fig. 16.7a). In agreement with earlier data [21], displacement

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Fig. 16.6 Whole-cell electrophysiology of wild-type HEK cell (wtHEK) and Shaker-transfected HEK cell (ShHEK). (a) Voltage-ladder protocol is applied across the membrane. (b) wtHEK cell membranes pass a small current, typically accounting for a few hundred pA. (c) ShHEK cell membranes, generally pass several nanoamperes of current with the same stimulation. (d) Conductance of ShHEK may be fit with the Boltzmann function, with V1/2 at –41 mV and slope factor of 9.6 mV

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Fig. 16.7 Wild-type HEK (wtHEK) MEM. (a) In voltage-clamp mode, a voltage ladder protocol (top) produced a typical membrane displacement of several Å (bottom). Positive displacement is an upward membrane movement, while a negative displacement is downward. (b) The MEM was linear at all voltage steps at multiple (100, 200, 400, 500 pN) force clamp settings tested over 200 mV from –120 to +80 mV. (c) The MEM scaled linearly with force. A 100-mV depolarization step resulted in MEM of 2.82, 3.35, 5.66, and 6.75 pm/mV at 100, 200, 400, and 500 pN force-clamp (FC)

from –120 to +60 mV MEM was linear at all set points of force from 100 to 500 pN (n = 8) (Fig. 16.7b, c). The amplitude of the displacements, »1nm/100 mV, was comparable to that of previous studies [21, 25]. At 500pN, a 100-mV depolarization caused a peak outward displacement of 6.75Å, corresponding to 13.5pN for a 0.02 N/m cantilever, similar to the previously published value of ~10pN/100 mV [21]. The MEM of wtHEK cells was linear with respect to voltage at all examined FC set points (Fig. 16.7b). Data also scaled linearly with FC; MEM for a 100-mV depolarization step creates forces that are about 5.6, 6.7, 11.3, and 13.5 pN for 100, 200, 400, and 500 pN FCs, respectively (Fig. 16.7c). We expected that the relation of FC vs. displacement should be nonlinear as suggested by Hertzian indentation mechanics [100]. This, however, was not the case, implying, not surprisingly, that the Hertzian assumptions of uniformity and isotropy are not correct [6]. Our optimized VC-AFM allowed for low noise recording with minimal averaging and time dependence in the experimental protocols. The extremely low noise level at high bandwidth makes the practical limits of biological AFM <1Å in 100ms if there is a repeatable stimulus. A typical voltage ladder protocol depolarized the

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membrane for 20 ms, and while membrane current was at steady state within a few milliseconds, we consistently noted a gradual downward drift of lever position during the length of the pulse and even after the membrane was stepped back to the HP (Fig. 16.7a, especially red and green traces). But the membrane position did return to baseline prior to the start of the next pulse. The long timescale of this process and the apparent voltage independence imply that these effects are likely due to mechanical relaxation of the cortical cytoskeleton. Acetylcholine Receptor Transfected HEK MEM Our objective was to elucidate MEM for Shaker-transfected HEK cells as a probe for voltage driven channel movements. To rule out transfection-induced artifacts we sampled the MEM of HEK cells transfected with nicotinic acetylcholine receptors (AchRs). AchRs can be expressed at a density similar to Shaker (~1channel/µm2) [101]. However, these channels are not voltage sensitive, and the IV curves from the AchR-transfected HEK cells are roughly linear and similar in current density to wtHEK cells (Fig. 16.8a). Also, similar to the wild-type HEK cells, MEM for AchR transfected cells is linear (Fig. 16.8b). Shaker-Transfected HEK MEM Typical experimental data from ShHEK MEM experiments is pictured in Fig. 16.9a. A voltage ladder protocol induced an outward membrane current of nanoampere scale, that achieved steady state within about 3ms. The basic AFM response to a jump in potential is an instantaneous (early) jump in probe position (Fig. 16.9a) followed by relaxation to a potential-dependent steady-state position (late) and after the end of the pulse, a slow relaxation back to rest (off). For the early steps, over the nonactivating voltage range of -120 to -40mV where Shaker remains closed, MEM was similar to wtHEK (Fig. 16.9a, b, top). For the nonactivating steps (–120 to – 40 mV), the early steps’ raw displacements (pm/mV) were larger for ShHEK than they were for wtHEK at all resting force set points. The difference MEMShHEK – MEMwtHEK increased from 0.89 to 4.25 pm/ mV for FC of 100 and 500 pN, respectively. In the Lippman tension model, this difference would suggest that channel expression produced a larger surface charge. However, ShHEK MEM was notably different than the MEM of wtHEK and AchR HEKs. The difference can be seen as a marked nonlinearity around the voltage of channel activation. The Shaker-activating voltage steps (depolarization to Vm greater than – 40 mV) produced MEM that saturated at the early steps (Fig. 16.9b, top). Such nonlinearity was observed in 83% (19/23) of the experiments, and only the experiments lacking nonlinear behavior were performed at the lowest FC where mechanical noise often dominated the displacement recording.

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Fig. 16.8 Acetylcholine receptor (AchR) MEM. (a) Current–voltage (IV) relationship for AchR (red ) was within the range of IVs for wtHEK (grey). (b) The MEM scaled linearly with voltage at 100, 300, and 500 pN force-clamp settings

The MEM linearity returned late in pulse (20 ms) as shown in Fig. 16.9b (middle panel). This was the case in 56% (10/18) of the experiments. For these displacements, there was a large deviation from average, which we attributed to the decay in the FC during the voltage step. Straightening of the MEM depended neither on the current density nor on FC. At maximum depolarization, displacement late in the pulse was similar to wtHEK and ShHEK with an average difference of only 2±1Å. The ShHEK MEM continued to be nonlinear even after the membrane voltage was stepped back to baseline (off). For nonactivating voltage steps, ShHEK MEM off displacement at constant voltage showed a downward drift similar to that seen with wtHEK (Fig. 16.9b, bottom panel). Our version of the Shaker channel lacks both N- and C-type inactivation, so this effect is unlikely to be due to channel inactivation. Instead, as described above for wtHEK MEM, the timescale of this process implies a relaxation of the cytoskeleton. For the voltage steps that activated the channels there was no typical downward drift of the lever position upon return to HP.

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Fig. 16.9 Shaker-transfected HEK cell (ShHEK) MEM. (a) In response to a voltage ladder (top), ShHEK membrane conducts typical delayed rectifying current (middle) and also produces upward membrane movement (MEM, bottom panel). ShHEK MEM behavior is similar to wtHEK MEM but some significant differences were noted; ShHEK and wtHEK MEM were compared at early, late,and off time points, measured as displacements from baseline. (b) Average fractional lever displacement (z/|z|max, n = 5) for ShHEK MEM (black traces) compared to the range of wtHEK MEM (gray shaded areas) at early (top), late (middle) and off (bottom) time points. In response to depolarizing voltage steps, wtHEK MEM increases linearly at early and late time points (top and middle). ShHEK MEM is distinctly nonlinear with respect to voltage for the early steps, but straightens late in the pulse. Upon return to baseline voltage (holding potential), the off drift of wtHEK MEM is inversely proportional to size of the depolarizing step. This results in an AFM tip position negative compared to prestimulus baseline. ShHEK MEM shows similar drift in position for preceding voltage stimuli that did not result in channel opening, but for stimuli that resulted in channel opening, the positive cantilever displacement persists

Instead, we saw an outward driving force, again likely due to cytoskeletal restructuring connected with channel opening (detailed below). NMDG + Ion Replacement One explanation for the saturation of ShHEK MEM is that background MEM is overwhelmed and negated by a current-related process, secondary to channel opening. In order to assess current dependence of the early movement nonlinearity, we

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e xamined MEM in the absence of potassium voltage-dependent current. Shaker channels have superb selectivity for potassium ions and the removal of potassium from the inside of the cell greatly reduced ionic current [1, 102]. However, we retained a small amount of K+ because there is evidence suggesting that Shaker K channels irreversibly inactivate in solutions with zero K+ [103]. To examine the coupling of current to MEM, we replaced all of the extracellular monovalent cations with N-methyl-d-glucamine (NMDG), large impermeant monovalent cation. The NMDG was also substituted inside, but we left 1-mM KCl in the intracellular solution as a tracer of channel activity. This ion replacement resulted in a drastic reduction in the voltage-dependent K+ current (sub-nA) (Fig. 16.10a), but the voltage and time dependent properties of the channels remained the same.

Fig. 16.10 ShHEK MEM with Na+ → NMDG+ replacement. (a) ShHEK IV with NMDG replacing K+ internally shows dramatic reduction in current. (b) At all examined force-clamps, examined at the early portion of the pulse, ShHEK MEM with intracellular NMDG (black) nearly overlapped the ShHEK MEM with intracellular K+ (gray)

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The MEM of NMDG ShHEK is similar to the MEM of ShHEK (Fig. 16.10b). Again, for the early steps, the MEM is large and roughly linear with voltage of up to -40mV. Moreover, the amplitudes of the early MEM for NMDG ShHEK agree well with those for ShHEK with intracellular K+; the differences are £1pm/mV. Yet, for the channel-activating voltage steps we again noted the saturation of early MEM. Thus, the saturation is not likely due to the effects of an ion flux through the open channels. The MEM is voltage dependent and not current dependent.

Symmetric K+ To further rule out the flux hypothesis, we reversed the K gradients. For HEK cells in physiologic solutions, high Na+ outside and high K+ inside, VK = – 84mV. Thus, for values of Vm that are more depolarized than VK, there is an outward K+ driving force. In the following experiments, our aim was to shift VK in order to uncouple the amplitude and direction of the ionic flux from the opening of Shaker channels. To this end, we replaced all Na+ in the extracellular solution with K+, making [K]o= [K]i. According to the Nernst equation, this substitution shifts VK to 0mV, meaning that channel opening initially results in an inward K+ flow and the reversal in K+ current direction (to the outside) occurs at 0mV. The resulting IV curve is plotted in Fig. 16.11a. The differences between the IV relationships for ShHEK in physiologic solutions (grey) and that in the symmetric K+ solutions are that upon channel opening, current in symmetric K+ solutions is inward (negative) up to VK and becomes positive after 0mV. We hypothesized that if the inflection in the MEM curves of ShHEK is related to current flow then the rightward shift in ShHEK outward K+ current should in principle affect the inflection point by rightward shifting it. The results show a good correlation between for ShHEK in physiologic solutions with that recorded in symmetric K+ solutions (Fig. 16.11b). The implication of these experiments is that the MEM nonlinearity is due to channel activation but is independent of the flux. Shaker IL Mutant MEM In the IL mutant of the Shaker channel, two noncharged residues on the voltage sensor are mutated conservatively (V369I and I372L), but produce large effects on channel activation. The voltage sensing apparatus of the ILT mutant, which is a close relative of the IL mutant but poorly expressing channel, is intact and behaves similarly to Shaker (G. Yellen, personal communication). Our data confirms that the activation of the IL channel is shifted to depolarized potentials, and in our hands, ShIL V1/2 = +9.4mV (DV=12mV), which is slightly right shifted from the published results (V1/2 = –1.9mV, DV = 8.6mV) (Fig. 16.12a) [99].

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Fig. 16.11 ShHEK MEM with symmetric K+. (a) IV for symmetric K+ solutions. (b) ShHEK MEM with symmetric K (black) overlapped ShHEK MEM with extracellular NaCl (gray)

Figure 16.12b shows that MEM for the early steps is again linear at the nonactivating depolarizations. The ShIL MEM curves show curvature upon channel activation, which occurs at a more depolarized voltage than for wildtype Shaker channels. As before, we observed a straightening of the MEM for the late step. For the off step, some traces showed nonlinearity, but on average we did not observe a nonlinear behavior. The main finding in these experiments is that MEM saturation for ShIL is at a more depolarized potential than that of ShHEK MEM. The finding that the ShIL MEM also becomes nonlinear upon channel activation suggests that the observed nonlinearity is not due to the conformational movements associated with voltage sensing, but rather the movements associated with channels opening.

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Fig. 16.12 Shaker IL transfected HEK MEM. (a) Boltzmann fit of the conductance–voltage relationship (GV) for Shaker IL mutant (ShIL) is shifted +50 mV (black) when compared to Shaker IR (ShIR) GV curve (gray). The slopes of the GV curves for ShIL and ShIR were not significantly different. (b) When compared to ShIR, MEM of the ShIL (black) shows a voltagedependent shift of the nonlinearity toward channel opening. The gray trace shows the average MEM of ShHEK

Discussion We began this work as an extension of previous studies in our laboratory and others [20, 21, 24]. We confirmed the earlier findings of membrane electromechanics (MEM) in HEK cells. For wtHEK cells, MEM was linear with voltage and with FC set points at all examined time steps, early, late, off. We also observed a time-dependent relaxation of the membrane at the onset of the depolarizing step. In this work, we show nonlinear MEM of Shaker-transfected HEK cell membranes. The findings presented here are also supportive of an earlier study by

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Mosbacher et al., which showed a decrease in outward MEM in Shaker-transfected HEK cells at depolarized HPs in response to a 10-mV AC coupled stimulus. The MEMs reached 0nm displacement at 0mV. The small amplitude of the stimulus ensured that the channel state was unperturbed during the experiment, so their finding of no MEM at 0mV is consistent with ours of no additional MEM at 0mV. Another reported finding paralleled by our results was that at negative potentials, ShHEK MEM amplitudes were 25 –125% larger than those for nontransfected cells [20]. Similarly, our results showed an increase in MEM slope for nonactivating steps (–120 to – 40mV) of ShHEK MEMs of 31–70% over wtHEK MEM. Yet, the biggest surprise was finding the saturation of MEM with depolarizations that opened the channels.

Channel Density Our results show that the nonlinearity in MEM at early times disappears at late time points, where MEM is again linear. At the onset of this work, we expected that Shaker would produce only local changes in membrane properties so that the background motion would be superimposed on any channel-induced motion. Thus, complete saturation was entirely unexpected. The return of linear MEM at long times suggested that the saturation was probably a kinetic effect, but the simplest explanations required a high channel density. We can estimate the number of Shaker channels in our transfected cells using the single-channel conductance of ShakerH4 ~10pS [104], and assuming current densities common to our experiments (several nA/100mV), we had a few thousand channels per cell [1]. We had picked small rounded cells for our experiments with membrane capacitance (Cm) around 10pF. Assuming a specific capacitance of 0.01pF/ µm2 for cell membranes [1], these cells had membrane area ~1,000 µm2 and a channel density ~1 per µm2. At this density, and assuming a uniform distribution, they are separated by distances much larger than the Debye length. If the channels separated by distances larger than the Debye length, they could not readily alter the mean surface charge of the membrane. According to the Lippman equation [21, 61], saturation of MEM implies that the inner and outer monolayers had the same surface charge. The surface charge of HEK cells is about – 20 mV [21], and to for the gating current that is associated with a transit from activated to open (1–2e-) to neutralize that amount of charge would have required a much higher expression density. Furthermore, a sudden change in charge density would cause a jump in MEM rather than saturation, since a change in charge density at constant potential will produce motion.

Flux Effects Perhaps a cell volume change due to water and ion flux reduces the background MEM and produces the nonlinearity in MEM of Shaker-transfected HEK? The currents

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in the transfected cells were significant and we spent considerable effort to show that the movement was independent of current. A primary piece of evidence against flux dependent effects is that the MEM remains independent of potential over a wide range of voltage where the current is increasing linearly. Furthermore, when the reversal potential is changed so that the currents are inward instead of outward, MEM effects persist with the same sign. Finally, when internal K+ concentration is reduced to 1mM or replaced with NMDG+, the nonlinear MEM persists, suggesting that MEM is truly a voltage and not current dependent process. The problem with a flux explanation is that charge and water are supplied from the patch pipette at the same rate as it is dissipated across the membrane. Thus, there should be no continuous change in volume, although there will be transients as the new equilibrium is established in the double layer. This conclusion is supported by previous work showing that transmembrane water diffusion coefficients were not significantly affected during electrical stimulation of neural tissues [105]. However, it is worth examining the expected change in volume from transient current flow. Our experimental results on Shaker-transfected HEK MEM were recently published [106]. Here we will just summarize the models discussed in that paper to explain the saturation of MEM associated with channel opening (Fig. 16.13). We refer the reader to the original paper for more detail. The surface charge models utilize the prediction of the Lippman equation that a membrane with symmetric surface charges will have minimal motion. However, even if we utilized all the gating charges in Shaker, the low density of active channels would not change the surface charge significantly. There are several geometric models of MEM where the channels change in-plane area or push normal to the membrane. If the channels were directly pushing on the cantilever and changed z dimension upon opening, that would produce a step of distance superimposed upon the background MEM, and we did not observe that. The change of in-plane area affecting tension also suffers from the low density of channels. However, the gating of Shaker is proposed to involve a splay of the inner transmembrane helices from a hinge point in the middle of the bilayer [49, 107]. If this torque generated buckled ripples in the bilayer when the channel opened, the AFM could touch the upper folds and the voltage would not drop across the fold since it would be small compared to the length constant for current spread. Thus, the fold motion would not be highly voltage dependent and hence would appear as a saturation of EM (Fig. 16.13b). The relaxation of saturation with time would fit with a viscoelastic response of the membrane and cytoskeleton to the stress induced by channel opening (Fig. 16.13c, d). As required by any reasonable model, the steady-state MEM following a step of potential should be similar to the background because the channels are dispersed in the membrane and the response should be the series response of both. These experiments on MEM using combined AFM and patch clamp show the potential resolution of the system in time and space. Despite using large changes in potential, there is little artifactual crosstalk between the AFM and the patch clamp with proper shielding. We envision a variety of new experiments using the system. For example, these experiments were done as background for experiments where the cantilever was covalently linked to a mutated site in S4 to directly

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Fig. 16.13 Shaker-membrane-cytoskeleton model of ShHEK MEM. (a) At baseline, AFM tip rests on a lipid membrane (gray) containing a closed Shaker channel (red), which is further supported by the membrane’s attachment to the underlying cytoskeleton (blue). (b) Depolarizing voltage pulse produces channel opening which results in early local buckling of the bilayer (top), resulting in apparent softening of the bilayer, reducing early MEM and transient sinking of the tip toward cell interior (bottom). (c) Cytoskeletal restructuring decreases membrane restraint and allows late catch-up of MEM. (d) Long-term cytoskeletal restructuring and stiffening results in prolonged outward position of the AFM tip and slow return to baseline

easure S4 motility. Another possibility is scanning of local channel topography m while driving the cell potential to make a spatial map or relative motion. The MEM would provide the modulation contrast for imaging. The general technique can be extended to other channels, pumps, and membrane-bound enzymes because they all undergo voltage-dependent changes in conformation. The patch clamp and the AFM with proper cantilevers have similar frequency responses and promise to provide new insights into biological structure and dynamics.

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Chapter 17

Dynamic and Spectroscopic Modes and Multivariate Data Analysis in Piezoresponse Force Microscopy B.J. Rodriguez, S. Jesse, K. Seal, N. Balke, S.V. Kalinin, and Roger Proksch

Introduction Electromechanics on the Nanoscale Coupling between electrical and mechanical phenomena is one of the fundamental physical mechanisms manifested in materials ranging from polar inorganic materials to ferroelectrics and piezoelectric semiconductors, and to electroactive polymers and biological systems [1]. Electromechanics refers to a broad class of phenomena in which mechanical deformation is induced by an external electric field, or, conversely, a redistribution of electric charge is generated by the application of an external force. Examples of electromechanical coupling include piezoelectricity in inorganic piezo- and ferroelectrics, flexoelectricity in nanomaterials, two-dimensional (2D) crystals, cellular membranes and broad range of bias-induced strain phenomena associated with phase transitions, diffusion, and electrochemical reactions [2]. Additionally, electromechanical coupling is a key component of virtually all electrochemical transformations, in which changes in oxidation state are associated with changes in molecular shape and bond geometry. Electromechanical activity is directly related to the structure and static and dynamic functionalities of the material. In polar compounds such as III–V nitride semiconductors, spontaneous polarization can play both advantageous and detrimental roles in device design. In ferroelectric materials, electromechanical coupling is directly related to a switchable spontaneous polarization. Thus, electromechanical behavior in ferroelectrics can be used to probe large and small field hysteresis for information technology and piezoelectric device applications, respectively, and to study a wide range of phenomena including polarization reversal mechanisms, domain wall motion and pinning, cross-coupled phenomena in multiferroic materials, and complex time and field dynamics in ferroelectric relaxors. B.J. Rodriguez (*) University College Dublin, Belfield, Dublin 4, Ireland e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_17, © Springer Science+Business Media, LLC 2010

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The wide variety of processes related to electromechanical coupling in ferroelectrics requires a tool capable of studying the variability of ferroelectric and electromechanical responses in space, time, and voltage domains. Below, we discuss piezoresponse force microscopy (PFM) as a method to study electromechanics in space, and complex PFM-based time and voltage spectroscopies to probe dynamic processes. Combined with the necessity to understand the dynamic behavior of tip– surface junction and first-order reversal curve measurements, this can give rise to very complex multidimensional scanning probe microscopy (SPM) methods requiring advances in data acquisition and analysis. The key factor that hindered broad studies of electromechanical phenomena in inorganic, biological, and molecular systems is the smallness of corresponding coupling coefficients, as illustrated in Fig. 17.1. In typical piezoelectric crystals with d ~ 2 – 100 pm/ V, the piezoelectric displacements induced by moderate (1–100V) biases are generally in the low nanometer range, necessitating the use of precise strain and interferometric sensors. The properties of thin films have become accessible only in the last two decades with the advent of single- and double-beam interferometry techniques [3–7] that allow ~pm displacements to be probed at the low (0.1–10V) biases required to obviate film damage. Finally, 1D and 0D systems

Fig. 17.1 Electromechanical coupling in functional materials. The solid blue and red lines c orrespond to estimated limits of nanoindentation and SPM, respectively (note that applicability can be limited by elasticity). The systems below the spatial resolution limit (albeit not sensitivity) can be studied using model systems with large defect separations or isolated molecules. Reproduced from [8]. Copyright 2008, Elsevier

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such as nanowires, nanoparticles, polar molecules, as well as spatially resolved probing of biological systems have until recently remained elusive due to the lack of enabling instrumentation. The breakthrough in electromechanical imaging and spectroscopy of polar and ferroelectric materials has been enabled in the last decade by the emergence of PFM and piezoresponse force spectroscopy (PFS). PFM(S) refers to a broad class of techniques in which a conductive SPM probe is used to concentrate electric bias in a small volume of a material (tip as electrode), and bias-induced local strain is measured through detection of bias-induced surface displacements. Alternatively, a top electrode can be used thus establishing a uniform field within material, and local strain is detected by the SPM probe. The spectroscopies in PFM typically refer to measurements at a single location as a function of bias (hysteresis loops), excitation frequency or amplitude, or temperature. Finally, spectroscopic mapping methods utilize spectra measured over a grid of locations on sample surface, giving rise to three- or higher dimensional data sets. Below, we provide a brief overview of PFM and related spectroscopic methods as applied for ferroelectric and biological mate rials. A special emphasis is made on the methods for rapid analysis and interpretation of complex high-dimensional data in terms of relevant material behaviors.

Piezoresponse Force Microscopy In single-frequency PFM, an electrically conductive cantilevered tip traces surface topography using standard deflection-based feedback (contact mode imaging). A sinusoidal electrical bias, Vtip = Vdc + Vac cos (w t ) , is applied to the tip during scanning, and the electromechanical response of the surface is detected as the first harmonic component of the bias-induced tip deflection, d1w cos (w t + j ), or tip torsion, as shown in Fig. 17.2a, b. The response amplitude, d1ω , provides a measure of the local electromechanical activity of the surface. The phase of the piezoresponse, j , yields information on the polarization direction below the tip. For c– domains (polarization vector pointing downward), the application of a positive tip bias results in the expansion of the sample and surface oscillations are in phase with the tip voltage, j = 0. For c+ domains, j =180°. Traditionally, the PFM signal is plotted either as a pair of amplitude phase, A = d1w / Vac , j , images, or a mixed signal representation, PR = Acosj , is used [9–16]. Both flexural and torsional components of tip displacement can be probed, giving rise to vertical and lateral PFM (VPFM and LPFM, respectively). The operation of the PFM is complementary to that of conventional SPMs as shown in Fig. 17.2c. For techniques such as scanning tunneling microscopy (STM), an electrical bias is applied to a metal tip, and the tunneling current is measured. In the case of atomic force microscopy (AFM), a force is applied, and the resulting tip deflection is measured. In PFM, an electrical bias is applied to a tip, and the tip deflection resulting from the deformation of the sample is measured. In AFM, this results in an image formation mechanism that is sensitive to

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Fig. 17.2 Diagrams showing PFM imaging principles for (a) vertical and (b) lateral surface d isplacements. (c) Diagram showing how PFM compares to several other SPM techniques. Adapted with permission from [1]. Panels a and b copyright 2007, Annual Reviews

surface topography and can be enhanced by resonance techniques [17]. However, the contact stiffness of the tip–surface junction scales proportionally with the contact radius, resulting in the topographic cross-talk inherent to many force-based SPM imaging modes (e.g., phase imaging). In contrast, the electromechanical response in PFM depends only weakly on the tip-contact area, thus minimizing cross-talk for low-frequency imaging. Due to its quantitative nature, PFM has emerged as the primary tool for studying polarization dynamics in ferroelectric materials, bringing about advances such as sub-10nm resolution imaging of domain structures, domain patterning for nanostructure fabrication, and local spectroscopic studies of bias-induced phase transitions. The development of the high-sensitivity PFM has allowed probing of piezoelectric phenomena in weakly piezoelectric compounds such as piezoelectric semiconductors [18–24] and biopolymers [25–34] (~1–5pm/V), providing spectacular sub-10nm resolution images of material structures based on the difference in the local piezoelectric response.

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This approach for electromechanical response measurement using an AFM-based technique was first demonstrated in 1992 by Günther and Dransfeld on thin copolymer films of polyvinylidene fluoride and trifluoroethylene [35]. Examples of domain imaging of the out-of-plane (VPFM) and the in-plane (LPFM) [36] polarization of a lead zirconate titanate (PZT) ceramic are shown in Fig. 17.3a–d. In the VPFM and LPFM amplitude images (Fig. 17.3a, c, respectively), regions of bright contrast, indicating a significant out-of-plane or in-plane component of polarization, respectively, are separated by boundaries marking either grain boundaries or in some cases, parallel domain walls within grains. The V(L)PFM phase images (Fig. 17.3b, d) provide information on the orientation of the polarization within the grain or of the domain, allowing components of polarization pointing away from the sample surface to be distinguished from those pointing into the sample, and those pointing to the right side of the figure to be distinguished from those pointing to the left side. The PFM data can be represented in a vector form and thus combining VPFM data with (ideally two orthogonal) LPFM data has enabled an approach for determining the 3D direction of polarization in thin films [37], crystals [38], and ferroelectric capacitors [39]. This “vector” PFM [40] imaging is illustrated in Fig. 17.3f for a PZT ceramic sample (Fig. 17.3d), where color indicates orientation and intensity indicates the magnitude of the response.

Fig. 17.3 Vertical (a, b) and lateral (c, d) PFM amplitude and phase images, respectively, of a PZT ceramic sample. (e) Topography and (f) vector representation of 2D PFM data. The orientation angle is coded by the color as reflected in the “color wheel” legend, while the intensity provides the magnitude of the response (dark for zero response, bright for strong response). Imaging voltage is 1 Vpp. Sample is courtesy of C. Watson, Sandia National Labs

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Fig. 17.4 Schematic relating the contact stiffness to surface topography via variations in contact area and local mechanical properties, resulting in topographic cross-talk in resonant PFM. Reprinted with permission from [47]. Copyright 2006, American Institute of Physics

Typical driving frequencies in PFM are chosen to be above the bandwidth of the topographic feedback loop (> 1–3kHz) so that the feedback loop does not compensate for the voltage-induced surface deformations. In low-frequency PFM, imaging is performed well below the first contact resonance frequency of the probe, providing for quantitative data. At the same time, operation at high frequencies allows the signal-to-noise ratio to be increased by moving away of 1/f noise corner (~10 kHz for conventional electronics), and effectively utilize cantilever tip–surface resonances to enhance weak electromechanical responses. The dynamic behavior of cantilever has been extensively studied in the context of atomic force acoustic microscopy (AFAM) [41–43]. Similarly, in high-frequency and resonance-enhanced PFM, the operation is performed at or above the first or higher resonance frequency. However, the use of single-frequency excitation in this frequency range is limited, because variations in the tip–sample contact stiffness can cause strong coupling between the topography and the apparent PFM signal, as illustrated in Fig. 17.4. Unlike AFAM, the use of standard phase-locked loop (PLL)-based circuitry to maintain cantilever at the resonance by adjusting excitation frequency is impossible in PFM, necessitating development of dual frequency [44] and band excitation (BE) [45] modes, as well as dual-cantilever systems [46]. Imaging is one of several aspects of PFM. The PFS is also widely used to probe local bias-induced phase transformations in PFM that can be detected through changes in amplitude, phase, or resonance frequency of the cantilever as a function of bias offset, excitation amplitude, or other external control parameters at a single location. The voltage and frequency spectroscopic measurements in PFM are discussed below.

Switching Spectroscopy-PFM PFM Spectroscopy In piezoresponse force voltage spectroscopy (local hysteresis loop measurements), a bias is applied to the tip in contact with a free surface or top electrode, and the

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electromechanical response is measured during and/or after the application of the pulse. In this manner, local piezoelectric hysteresis loops can be acquired. The detected electromechanical hysteresis loop is directly linked to tip-induced formation of switched domain at the tip–surface junction. The domain growth process in classical ferroelectric material (not ferroelectric relaxor) consists of a domain nucleation stage followed by subsequent forward and lateral domain growth. On reverse bias, both (a) shrinking of the formed domain and (b) nucleation of the domain of opposite polarity are possible as shown in Fig. 17.5. Note that in the tipelectrode experiments, the domain formation occurs only below the probe, and this measured hysteresis loop contains information on deterministic mechanism of polarization switching, in drastic comparison to macroscopic hysteresis loops that contain information on multiple domain nucleation, growth, and interaction controlled both by external field and frozen and thermal disorder. The in-field hysteresis loop measurements were first reported by Birk et al. [49] using an STM tip and Hidaka et al. [11] using an AFM tip. In this method, the response is measured simultaneously with the application of the dc electric field, resulting in an electrostatic contribution to the signal. To avoid this problem, a technique to measure remanent loops was reported by Guo et al. [50]. In this case, the response is determined after the dc bias is removed, minimizing the electrostatic

Fig. 17.5 Domain evolution with bias depending on different pinning strength of the material. (a) Time dependence of voltage, (b) schematics of hysteresis loop, and (c) schematics of the domain growth process. In the purely thermodynamics case (dashed arrows), the domain shrinks with decreasing voltage (path 3–4). To account for realistic loop, the domain size does not change on (3–4) and the domain of opposite polarity nucleates on path (4–6). At point 6, antiparallel domain walls annihilate. Reprinted with permission from [48]. Copyright 2007, American Institute of Physics

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contribution to the signal. However, after the bias is turned off, domain relaxation is possible. In a parallel development, Roelofs et al. [37] demonstrated the acquisition of both vertical and lateral hysteresis loops. This approach was later used by several groups to probe crystallographic orientation and microstructure effects on switching behavior [38, 51–55]. The progress in experimental methods has stimulated a parallel development of theoretical models to relate PFM hysteresis loop parameters and materials properties. A number of such models are based on the interpretation of phenomenological characteristics of PFS hysteresis loops similar to macroscopic polarization–electric field (P–E) loops, such as slope, imprint bias, and vertical shift. In particular, the slope of the saturated part of the loop was originally interpreted as electrostriction; later studies have demonstrated the linear electrostatic contribution to the signal plays the dominant role. Several groups analyzed the effect of nonuniform material properties, including the presence of regions with nonswitchable polarization on parameters such as imprint and vertical shift. In thin films, the vertical shift of the PFM hysteresis loops was interpreted in terms of a nonswitchable layer by Saya et al. [56] Alexe et al. [57] analyzed the hysteresis loop shape in ferroelectric nanocapacitors with top electrodes, obtaining an estimate for the switchable volume of a nanocapacitor. Similar analysis was applied to ferroelectric nanoparticles developed by the self-patterning method [58] by Ma [59]. In all cases, the results were interpreted in terms of ~10nm of nonswitchable layers, presumably at ferroelectric–electrode interface. A number of authors attempted to relate local PFM hysteresis loops and macroscopic P–E measurements, often demonstrating good agreement between the two [60]. This suggests that despite the fundamentally different mechanism in local and macroscopic switching, there may be deep similarities between tipinduced and macroscopic switching processes. A framework for analysis of PFM and macroscopic loops was developed by Ricinschi et al. [61–63], demonstrating an approach to extract local switching characteristics from hysteresis loop shape using first-order reversal curve diagrams. In parallel with tip-induced switching studies, a number of groups combined local detection by PFM with a uniform switching field imposed through the thin top electrode to study polarization switching in ferroelectric capacitor structures. Spatial variability in switching behavior was discovered by Gruverman et al. and attributed to strain [64] and flexoelectric [65] effects. In subsequent work, domain nucleation during repetitive switching cycles was shown to be initiated at the same defect regions, indicative of the frozen disorder in ferroelectric structures [66, 67]. Finally, in a few cases, “abnormal” hysteresis loops having shapes much different than that in Fig. 17.5b have been reported. Abplanalp et al. have attributed the inversion of electromechanical response to the onset of ferroelectroelastic switching [68]. Harnagea has attributed the abnormal contrast to the in-plane switching in ferroelectric nanoparticles [60, 69]. Finally, a variety of unusual hysteresis loop shapes including possible Barkhausen jumps and fine structures associated with topographic and structural defects have been observed by Jesse et al. [70].

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The rapidly growing number of experimental observations and recent developments in PFS instrumentation, data acquisition, and analysis methods requires understanding not only phenomenological, but also quantitative parameters of hysteresis loops, such as numerical value of the coercive bias, the nucleation threshold, etc. Kalinin et al. [52] have extended the 1D model by Ganpule et al. [71] to describe PFM loop shape in the thermodynamic limit. Kholkin has postulated the existence of nucleation bias from PFM loop observations, in agreement with theoretical studies by Abplanalp [72], Kalinin et al. [73], Emelyanov [74], Morozovska and Eliseev [75]. Finally, Jesse et al. [76] have analyzed hysteresis loop shape in kinetic and thermodynamic limits for domain formation. The full 3D models for hysteresis loop formation based on rigid dielectric approximation and time-dependent Ginzburg–Landau theory are becoming available.

Switching Spectroscopy-Piezoresponse Force Microscopy of Films Understanding polarization dynamics on the nanometer scale has spurred the development of PFM spectroscopy, wherein the dc bias offset on the probe follows a triangular (~1 Hz) waveform envelope. In ferroelectric materials, the dc bias induces polarization switching, and the size of the ferroelectric domain formed below the tip is related to a change in the electromechanical response detected at a probing frequency of ~10 kHz – 2 MHz. Electromechanical hysteresis loops obtained in this manner are illustrated in Fig. 17.6. The key considerations for high-veracity high-resolution imaging of polarization dynamics is (a) the tradeoff between data acquisition speed resolution and (b) the reproducibility of single-point measurements. To establish the optimal data acquisition parameters, the effect of the waveform on imaging has been analyzed, where t1 is the duration of the reading state, t2 is the duration of the writing state (duration of dc bias), t3 is the delay from the end of the writing state to the beginning of the measurement window having duration t4 , where t1 ³ t3 + t4. The time per pixel is then ts = aM (t1 + t2 ) , where M is the number of bias steps per loop and a is the number of hysteresis loops measured per point. Typically, a = 3.25 with data collected over three cycles. Examples of several hysteresis loops collected for {ts , t1 , t2 , t3 , t4 } = {21.6,78,26,30,30} (waveform 1), {10.8,39,13,15,20} (waveform 2), and {3.5,13,4,6,7} (waveform 3), where the units for ts are [s] and ti are [ms], are shown in Fig. 17.6a. The total image acquisition time is then T = N 2 (ts + tm ), where N 2 is the number of pixels in an N × N image, ts is the spectrum acquisition time, and tm is the time required for tip motion between positions. The latter can be arbitrary small and depends on the AFM system (e.g., < 1ms for standard AFMs). The hysteresis loops for waveforms 1 and 2 coincide, while waveform 3 exhibits significant broadening. Of these waveforms, the optimal time per single voltage point was found to be 52ms, corresponding to 3.3s per loop for 64 voltage points, and 10.7s per pixel if a = 3.25. The repetition time per pixel, a, can

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Fig. 17.6 (a) PFM hysteresis loops measured on a BiFeO3 (BFO) film surface at 5 V, 310 kHz with three different waveforms with pulse durations of 26, 13, and 4 ms and measurement times of 30, 20, and 7 ms for waveforms 1, 2, and 3, respectively. The delay between the end of the pulse and the beginning of the measurement is roughly equal to the pulse duration in each case. (b) A set of PFM hysteresis loops as a function of dc bias measured using waveform 2 at 0.75 V, 1 MHz from a different location than (a). The loops in (a) are averages of 12 loops from the same location with a tip with a spring constant of 40 N/m. The loops in (b) were measured with a cantilever of spring constant of 4.5 N/m and are averages of 12 loops each. Sample courtesy of R. Ramesh. Panel b reproduced partially, with permission, from [77]. Copyright (2007) National Academy of Sciences, USA

be reduced if the signal is sufficiently strong. Acquisition times as small as 2s/loop with 128points/loop in the waveform {4.5,10.9,4.7,1.5,9} have been demonstrated, corresponding to a pixel time of 4.5 s. In this case, increasing the number of voltage points allows the fine structure of the loop to be observed, as discussed later. The optimal conditions for hysteresis loop measurements must be established for each sample independently. Notably, large bias windows can significantly affect the loop shape as shown in Fig. 17.6b. When the bias becomes sufficiently large, an anomalous loop is observed, however, upon reduction of the bias, the loop returns to the original shape. As demonstrated in Fig. 17.6b, loops typically do not saturate until more than 10-V bias is applied. Attempts to bring the loops to saturation occasionally resulted in dielectric breakdown in the film. From a general viewpoint, the hysteresis loops in PFM are analogous to forcedistance spectroscopy in AFM. The unfolding spectroscopy allows probing the kinetics and thermodynamics of macromolecular reactions using force applied to a molecule as an external control variable. The PFS allows probing bias-induced phase transition in the solid containing a number of defects that serve as nucleation centers and pinning cites for domain wall. However, in PFS the location of the defects on the surface is generally unknown, necessitating measurements at multiple location and subsequent exploration of spatial variability of measured hysteresis loops.

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These considerations have lead to the development of the switching spectroscopy PFM (SS-PFM), in which hysteresis loops are acquired over a spatially resolved 2D grid on a sample surface, yielding a 3D data array, as shown schematically in Fig. 17.7a. The typical acquisition time for a 64 × 64 point image is ~2–4 h [70]. However, recent progress in fast data acquisition methods and ultrafast cantilevers holds the promise of reducing the data acquisition times to that of conventional scanning (10–20min/image). The 3D data can be further analyzed to yield 2D maps of relevant hysteresis loop characteristics, including the local work of switching (the area within a loop), coercive and nucleation biases, remanent response, and other parameters that define the operation of ferroelectric devices. Alternatively, hysteresis loops can be extracted and analyzed individually. The spatial variability of the switching behavior in BiFeO3 (BFO) films is shown in Fig. 17.7. The topography of an area containing a topographic defect is shown in Fig. 17.7b. The SS-PFM maps of PFM and the work of switching are shown in Fig. 17.7c, d, respectively. Several loops extracted from the full data set along the line marked in Fig. 17.7b are shown in Fig. 17.7e. The inspection of the hysteresis loops acquired on an evenly spaced mesh demonstrates that topographic defects are often associated with “abnormal” hysteresis loops as shown in Fig. 17.7e. The spectra with characteristic discontinuities are concentrated in the areas adjacent to the topographic defect. We attribute this behavior to the interaction between the growing domain and the strain field of the defect, which results in rapid transition between two domain configurations, similar to Barkhausen jumps in macroscopic measurements. Hence, in addition to analysis of early stages of switching processes, PFS allows insight into the domain–defect interactions. In certain cases, highly reproducible fine structure in the hysteresis loop shape has been observed, as shown in Fig. 17.7e. The 14 × 14 SS-PFM map (Fig. 17.7f) with 1-nm pixel size contains the equivalent of 392 loops from effectively the same location, each having a pronounced shoulder. The fine structure can be attributed to domain–defect interactions, as analyzed elsewhere [78]. An SS-PFM map of the work of switching in multiferroic heterostructures is shown in Fig. 17.8a. The work of switching is reduced in the vicinity of the interfaces; however, this is a purely geometric effect due to the reduction of the domain volume. Small (5–10%) variations of switching behavior within the BFO are observed and can be attributed to long-range strain fields. There is a clear spatial dependence on the loop shapes (Fig. 17.8b) and on the switching properties as indicated by line profiles (Fig. 17.8c, d). Note that the coercive biases and the imprint vary weakly with position, while the work, remanent piezoresponses, and the switchable piezoresponse all follow the same trend across the line profile. Recently, the statistical studies of polarization switching in thin BFO films combined with the Ginzburg–Landau theory have revealed that polarization switching on a defect-free surface [77] is intrinsic, as can be rationalized by extremely small volume (104nm3) of probing volume is PFM, as compared to 1011nm3 for capacitor. Following the development of imaging capability, the single defect resolution [78] and mesoscopic disorder potential component mapping in epitaxial ferroelectric thin films [79] have been demonstrated. The SS-PFM has

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Fig. 17.7 (a) Schematic of tip motion during SS-PFM, (b) topography image of a BFO thin film, 2D SS-PFM maps of (c) PFM and (d) effective work of switching, and (e) hysteresis loops along the line in (b). (f) Representative loops and (g) SS-PFM map of switchable piezoresponse from a 14 nm region of a BFO film. Panels (c) and (d) are SS-PFM maps of the 1 × 1 mm2 area shown in panel b. Panel a reprinted with permission from [76]. Copyright 2006, American Institute of Physics. Sample courtesy of R. Ramesh

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also been employed to study nanoparticles [80] and nanostructures such as multiferroic heterostructures (see Fig. 17.8) [81]. Finally, mesoscopic polarization switching mechanisms can be deciphered using atomically engineered defect structures such as 24° grain boundary in a (100) rhombohedral ferroelectric [82, 83], twin walls in tetragonal (100) ferroelectrics [84], 180° domain walls in uniaxial ferroelectrics [85], and ferroelastic 71° walls in (100) rhombohedral ferroelectric [86]. As an example, SS-PFM studies of switching across a grain boundary fabricated using a bicrystal substrate revealed differences in the propensity for ferroelectric versus ferroelastic switching in the vicinity of the grain boundary (Fig. 17.9). The characteristic asymmetric shape of switching profile is related to the domain-induced stabilization of the unusual ferroelectric–ferroelastic twin domain, as shown at locations B, C in Fig. 17.9e. While the full potential of SS-PFM is yet to be revealed, the results to date suggest tremendous potential of this method to explore mesoscopic mechanisms of

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bias-induced phase transitions on a single defect level, ultimately allowing the defect functionality to be linked directly to atomic and electronic structure.

SS-PFM of Capacitors Local ferroelectric hysteresis loops can also be measured by the PFM tip through the top electrode in ferroelectric capacitor structures, allowing comparison between local measurements of polarization reversal with macroscopic measurements of polarization current. Typically, the tip is kept at the same potential as the top electrode. Using the tip in contact with the top electrode allows the application of the switching field to be decoupled from the measurement of the local electromechanical response. The field within material in this case is uniform (whereas in the free-surface case, when the tip is used as the top electrode, the applied field is highly inhomogeneous), simplifying the interpretation of the PFM imaging data. Furthermore, the measurements are less susceptible to a ubiquitous problem in PFS measurements – the erosion and wear of the conductive coating from the tip apex.

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However, the switching measurements on the capacitors are generally more difficult to interpret, because the nucleation can proceed at the material-specific defect site and not necessarily below the tip. Due to the broad applicability of ferroelectric capacitor structures in nonvolatile memory technologies, the mapping of internal polarization dynamics in these was actively explored in the last decade. A significant progress was achieved by the groups of Gruverman [87, 88], Noh [89], and Setter [90–94]. In many of these studies, the domain structure in capacitor was visualized through the top electrode at the different stages of switching, thus yielding the information on the domain nucleation sites and wall motion kinetics. Recent studies by Gruverman have demonstrated the role of boundary and top electrode effects on the domain dynamics in small capacitors that often lead to the formation of ting-like structures and back switching. Recently [95], PFM imaging of capacitors allowed the mapping of domain profiles at different biases and the comparison of macroscopic P–V loops directly with local measurements (Fig. 17.10). The information obtained in these

Fig. 17.10 Domain structure evolution delineated by means of bias-dependent PFM in a ferroelectric capacitor. (a) Positive switching domain profiles for dc biases of 0.40, 0.55, 0.60, and 0.70 V and (b) negative domain switching profiles for dc biases of �0.40, �0.45, �0.50, and �0.60 V in a 0.5 × 0.5 mm2 capacitor (0.65 × 0.65 mm2 scan size). (c, d) Domain structure evolution delineated by means of bias-dependent PFM in a 2.0 × 2.0 mm2 capacitor for (c) 0.4, 0.5, 0.55, and 0.7 V and (d) �0.45, �0.50, �0.55, �0.60, and �0.65 V. Reprinted with permission from [95]. Copyright 2009, American Institute of Physics

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Fig. 17.11 Spatial maps of SS-PFM ferroelectric switching parameters from an electroded s urface. (a) Initial piezoresponse, (b) imprint, (c) work of switching, (d) switchable polarization, (e) positive nucleation bias, and (f) absolute value of the negative nucleation bias. Reprinted with permission from [97]. Copyright 2009, American Institute of Physics

measurements is similar in nature to SS-PFM measurements in that both methods probe the space- and voltage-dependent evolution of domain structures. However, in the bias-dependent PFM measurements, the space domain is sampled at each voltage step, while in SS-PFM, voltage dynamics at each location are probed sequentially. The information will be equivalent for a system with microscopic return point memory in the quasi-stationary state. The applications of SS-PFM for capacitor structures have been pioneered by Seal [96] and Bintacchit [97]. These measurements yield a 3D data set formed by the hysteresis loop at each spatial location. The loop at each location can be analyzed to yield local ferroelectric switching parameters. The spatial distribution of the remanent piezoelectric response, imprint, work of switching, switchable response, and nucleation biases are shown in Fig. 17.11a–f. The remanent response image is similar to the standard PFM image and remains almost identical after multiple switching cycles, suggesting that at the end of the switching cycle each pixel returns to the same polarization state, i.e., the system possesses microscopic return point memory. Interestingly, all images show clear details within the scan area, with regions of high contrast with typical feature sizes of the order of 200–500nm, which is several times the grain size. That is, the SS-PFM images taken using large top electrodes show regions that are substantially larger than the grain size that display similar contrast, indicating correlated switching behavior of these regions. In many cases, these regions are characterized by a comparatively low nucleation bias. This, in turn, suggests that within the volume probed, a region of material switches, and that cross-granular coupling

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drives switching in adjacent grains. These studies then provide the first measure of the volume of correlated switching in PZT films, which is of the order of 1–2mm. The distribution functions for each of the loop parameters are Gaussian in nature. The characteristic aspect of hysteretic behavior is its strong dependence on the field history. In general, knowledge of the hysteretic behavior for one field history (e.g., major hysteresis loop) does not allow to predict hysteretic response for a different field history (e.g., minor hysteresis loops). This consideration has stimulated the development of Preisach-type models of hysteresis [98, 99] that describe the system as a superposition of bistable hysteretic units having a defined 2D distribution function in the space of positive and negative coercive biases. Experimentally, getting insight into these behaviors necessitates the measurements of first-order reversal curves or families of minor hysteresis loops.

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In SS-PFM, the evolution of the switchable polarization as a function of bias window in hysteresis measurements is illustrated in Fig. 17.12. The remanent response images are virtually independent of bias window, indicative of intrinsic inhomogeneity of material and in agreement with wipe-out property often observed in hysteretic systems. The switchable response images show a gradual evolution with increase in the excitation window. Note that there is considerable spatial variability in this local preferred polarization and the excitation window dependence of local hysteresis loops differs significantly from macroscopic (e.g., local loops tend to open vertically asymmetrically, Fig. 17.12g, h). At the same time, when all of the loops are averaged, no preferred polarization is apparent and the resulting behavior matches closely with the Preisach calculations for the same field excursions. For minor loops, the switchable response is small, as would be expected for excitation at field levels well below the coercive field. For higher excitation windows (e.g., as the sample is driven through larger and larger minor loops), the average switchable response value increases, indicating progressive opening of the hysteresis loop. The corresponding image develops a clearly visible contrast which is different from the piezoelectric response images (thus excluding possible crosstalks in signal acquisition or hysteresis loop analysis). The statistical analysis of this data has recently allowed the characteristics length scales corresponding to the transition from Rayleigh to Preisach behaviors to be determined [96]. The future development in the field will undoubtedly see the capabilities for probing Preisach densities on the nanoscale and correlating them with microstructural behavior.

High-Frequency PFM and Topographic Cross-Talk The second major example of spectroscopic measurements in PFM has emerged in the context of resonance-enhanced PFM. High-resolution imaging and spectroscopy of ferroelectrics and imaging of weakly piezoelectric materials including nitrides [100] and biopolymers greatly benefits from resonant enhancement of the piezoresponse signal, with anticipated improvement in signal-to-noise ratio by 10–100 for resonant PFM. The contact resonance frequency of the cantilever in PFM, similarly to AFAM [41–43] is strongly dependent on surface topography, resulting in large (up to ~10–100 kHz) position dependent frequency shifts (see Fig. 17.4) [47]. Yet standard frequency tracking techniques (e.g., based on a PLL) cannot be applied to PFM directly due to the strong dependence of response phase on local polarization [44]. These considerations traditionally limited PFM to the frequency regime well below first cantilever resonance and relatively large driving amplitudes, limiting both smallamplitude spectroscopy and topographic cross-talk free high-frequency imaging. Here, we discuss several approaches to obtain resonant enhancement that minimize the inherent problems with cross-talk, diminishing the need for high

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biases which can damage thin films and biological materials, and limit the energy resolution in PFS. We also discuss advances in PFS to counteract the limitations arising from thermal drift of the tip, and further explore the marriage of the two (spectroscopy and novel excitations for resonance enhancement) which allow for dynamic, resonance enhanced, acquisition of switching data at every point in an image. This combination introduces novel challenges in data analysis and interpretation of multidimensional data, which are also discussed.

Historical Notes For the first several years of PFM (~1995–2001), the groups active in the field developed in-house setups based on the combination of a commercial AFM system with a function generator – lock-in amplifier assembly. The use of standard lockins limited the frequency range to ~100kHz, well below the first contact resonance of the cantilever. Hence, the imaging was performed in a low-frequency regime where the cantilever transfer function is almost flat. The instrumental corrections, lock-in and low-frequency cantilever holder resonances, have been extensively studied in the series of work by the group of Soergel [101–103], as well as by Agronin et al. [104]. While the sensitivity of modern AFM instrumentation is very impressive, and deflections of order of ~10pm can be reliably detected, the use of the low-frequency PFM is still subject to strong limitations. The natural limiting factor in PFM detection is the thermomechanical noise of the cantilever [105] that limits the measured deflection signal. Practically, environmental vibrations and laser shot noise limit the detection limit to hc~10pm in the 1-kHz measurement bandwidth. In ferroelectric or piezoelectric material, the surface displacement is determined as h = d eff Vac , where d eff is effective piezocoefficient and Vac is driving voltage. Detectability of PFM signal requires h = deff Vac >> hc . For example, for PZT material or thin film with d eff ~100 pm/V imaging can be performed with biases as small as 100mV and good-quality images can be obtained for Vac =1V. In comparison, in material such as lithium niobate with d eff ~10pm/V the driving voltages should be of the order of several volts. Finally, for biological systems with d eff ~1–100 pm/V, driving voltages of the order of 10–100 V are required. While for many systems such as insulating single crystals and biological tissues the use of high voltage amplitudes is possible, this is not universally the case. For example, thin ferroelectric films often have reduced electromechanical responses and at the same time cannot support high biases due to the onset of polarization switching and dielectric breakdown. Similarly, flexoelectric biological membranes and conductive materials limit the bias amplitudes prior to the onset of electrochemical processes. The special requirements is imposed by the PFM spectroscopy, where voltage amplitude is the factor limiting the energy

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resolution of the method. For ferroelectric capacitor with ~3–5 V coercive bias the energy resolution of ~0.05–0.1V is required, limiting the measured signal to ~3–5 pm, well below the reliable detection limit. Even for macroscopic samples, the use of high voltage amplitudes may result in stray electrochemical processes, charge injection, redistribution of oxygen vacancies, and many other processes that complicate the interpretation of PFM data or lead to the irreversible changes in the surface state. The classical approach to enhance weak response signals adopted in virtually all force-based SPM techniques is the use of the resonance enhancement. By operating close to the resonance, the cantilever acts as a mechanical amplifier that magnifies the weak responses prior to the addition of laser and electronic noises. Furthermore, the high-frequency operation also adds the advantage of the separation from 1/f noise corner (~1–10 kHz for most commercial systems). Hence, the use of the resonance enhancement in PFM offers a clear pathway to improved signal-to-noise ratios, imaging weakly piezoelectric materials, and high-resolution voltage spectroscopy. The high-frequency cantilever dynamics in the 0–1 MHz has been explored experimentally by Harnagea [17, 69] and in the 0–10 MHz range by Seal et al. [106]. Harnagea has experimentally shown that resonance frequency of the cantilever strongly depends on position on the surface due to variation of contact resonance frequency. The relevant theoretical framework has been extensively studied in the context of AFAM [42, 107, 108] and a number of analyses relating resonance frequency with cantilever dynamics and contact stiffness [109] and dissipation [110] at the tip–surface junction are available. The theory was extended to PFM by Jesse et al. by including the effects of distributed and localized electrostatic forces [111]. The approximate expressions for contact resonance frequencies were obtained by Mirman [47, 112]. Recently, detailed studies of resonance behavior in PFM have been reported by Salehi-Kohjin et al. [113], suggesting an approach for the identification of system resonances and oscillation modes. The fundamental conclusion of these studies is that for all practically important surfaces the contact resonance frequency changes very significantly between locations (typically by ~10 s of kHz), well above the typical width of the contact resonance peak (~3–5 kHz). Consequently, for PFM imaging at a single frequency close to the cantilever resonance, the contrast due to the variations of contact resonance frequency will greatly exceed that due to electromechanical response, resulting in a significant topographic cross-talk. The latter can be identified from the strong correlation between PFM signal and topographic images, strong contrast variations in phase images (rather than sharp grouping of the data in two peaks separated by 180°), abnormally sharp domain walls at topographic features, etc. Multiple examples are summarized in recently published PFM support note by Asylum Research (Figs. 17.13 and 17.14) [114]. A number of approaches to address this problem based on ingenuous cantilever designs that decouple the driving and resonances [115] and improved control methods have been suggested. We discuss two such approaches below.

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Fig. 17.13 (a) PFM signal at the contact resonance for a smooth sample. (b) PFM signal at the contact resonance for a rough surface, demonstrating how changes in the contact resonance resulting from the surface roughness can make interpretation of the domain structure very difficult. Reproduced with permission from [114]

Fig. 17.14 PFM phase image of a polished PZT sample measured near the contact resonance demonstrating cross-talk between the sample topography and the PFM signal. Red arrows indicate “roughness” where the contact stiffness modulates the phase. The yellow arrows indicate a sudden tip change caused a change in the contact resonance. Scale: 4 µm scan. Reproduced with permission from [114]

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Dual AC Resonance Tracking in PFM The alternative solution to resonance-enhanced PFM is the use of resonance frequencytracking methods, in which the frequency of the driving voltage applied to the tip is adjusted at each point on the sample surface to maintain resonance. Classically, the frequency tracking is implemented using analog or digital PLL circuitry. PLL maintains the resonance by keeping the phase of the response at 90° compared to excitation signal, in this case the driving applied to the probe. However, the direct use of PLL in PFM is impossible, because the local response phase depends on domain orientation. For upward domains, the resonance corresponds to 90° phase shift, and for downward to �90° phase shift, thus precluding direct application of PLL to PFM. An approach for resonance frequency tracking in PFM based on the use of two closely spaced excitation frequencies has been employed by Rodriguez et al. [44]. In the dual AC resonance tracking (DART) method, the cantilever is driven with a linear combination of sinusoidal voltages, one above ( f 2 ) and one below ( f1) the resonant frequency ( f 0 ) as shown in Fig. 17.15. This signal is used to modulate

Fig. 17.15 In dual AC resonance tracking, the cantilever is both driven and measured at a f requency below and a frequency above resonance. The total drive voltage is the sum of voltages at frequencies f1 (blue) and f2 (red). The cantilever deflection then contains information at both of those frequencies (purple). The amplitude and phase at the two frequencies are then separated again by the two lock-ins and passed on to the controller. Resonance tracking uses the difference of the two amplitude measurements as the error signal input for a frequency feedback loop. Reproduced with permission from [44]. Copyright 2007, Institute of Physics and IOP Publishing Limited

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the tip–sample interaction. This modulation could be at the base of the cantilever, the potential across the legs of the cantilever, the current through the legs (temperature), or the sample height. The resulting motion of the cantilever is measured with the standard position-sensitive detector. This signal in turn is used as the input for two lock-in amplifiers, where the sinusoidal drive at f1 is used as a reference for one lock-in and f 2 is used as a reference for the other. The output of the lock-in amplifiers consists of the two amplitude and phase pairs measured at each frequency A1 , j1 A2 , and j 2 . To implement resonance tracking, the difference between the two amplitudes, A1 - A2 , is used as the error signal for frequency feedback. The DART-PFM has been used to image samples with uneven topography (Fig. 17.16) and the effect of switching in ferroelectrics (Fig. 17.17). Shown in Fig. 17.16 are DART-PFM images of a BFO fibril (see [116] for sample details). Roughness and micro- to nanoscale structure of materials can lead to topographic cross-talk in PFM; however, the DART-PFM images showing these detrimental effects can be avoided. The DART-PFM images of switched domains in lithium niobate (Fig. 17.17) reveal a resonance shift for the nucleated domain. The quasi-stable domain may not extend through the thickness of the sample and may be pinned by defects.

Fig. 17.16 (a) Resonant frequency, (b) PFM amplitude, (c) AFM height, and (d) PFM phase images of a BFO fibril. The PFM data (b, d) is overlaid onto the height image. Sample courtesy of Jiangyu Li, University of Washington

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Fig. 17.17 (a, d, g) Resonance frequency, (b, e, h) DART-PFM amplitude and (c, f, i) phase images of a periodically poled lithium niobate surface. Shown are images of the (a–c) engineered domain structure, (d–f) an intrinsic domain, and (g–i) domains switched by ± 176 V [locations marked in (e)]. Reproduced with permission from [44]. Copyright 2007, Institute of Physics and IOP Publishing Limited

The basic DART method allows the resonant PFM signal and the contact resonance frequency to be measured. The phase of the responses can be further analyzed to yield the energy dissipation signal [117]. The DART is concerned primarily with dynamically tracking the resonance frequency and ensuring the measurement takes place at the resonant frequency to take advantage of the increased signal to noise, without the limitations of increased topographic cross-talk. Another approach aims to measure the response over a defined frequency window using nonsinusoidal excitation.

Band Excitation PFM The Need for Band Excitation The fundamental limitations of single-frequency detection methods can be readily illustrated from simple physical considerations. The resonant system in the simplest case can be described as a linear harmonic oscillator. The system response is determined

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by three independent components – response at the resonance, resonance frequency, and Q-factor (resonance/peak width at half maximum). In the context of AFM, response at the resonance provides the measure of the force exciting the system, resonant frequency is determined by the stiffness of the cantilever and tip–surface force gradient, and Q-factor is the measure of dissipation in the system. In the general case, the three parameters are independent. The situation is further complicated in the presence of nonlinear interactions. For example, weak cubic nonlinearity will result in cross-over to Duffing type dynamics, in which system is characterized by four independent parameters. For more general nonlinearity, the number of independent parameters describing system response can increase further. At the same time, classical lock-in and PLL detection schemes that form the mainstay of modern single-frequency SPM techniques allow detection of only two parameters defining system response – amplitude and phase at a single frequency for lock-in and resonant frequency and amplitude for PLL. Because the number of measured parameters is smaller than the number of unknowns, the system is underdetermined. In SPM techniques with acoustic excitation, the driving force is set by the voltage set to the piezo and is constant, providing additional constraint (and hence enabling classical SPM). However, even in this case the nonconstant cantilever transfer function may result in large systematic errors in, e.g., dissipation measurements [118]. In techniques with electrical excitation such as PFM and Kelvin probe force microscopy, this additional constraint is absent, precluding the use of frequency tracking, resonance amplification, and dissipation probing.

Principles of Band Excitation An approach to circumvent these problems is offered by the BE method [45, 119]. In BE, the system is excited by a digitally synthesized signal having a finite spectral density in a band centered on the resonance (Fig. 17.18). This signal substitutes a single sine wave used in classical SPM (that corresponds to a delta function in Fourier space). Note that BE is not limited to PFM, and excitation through acoustic, magnetic, or any other of the multiple SPM schemes is possible. The response is detected in a usual manner, e.g., using photodetector, and is Fourier transformed. The ratio of the fast Fourier transforms of response and excitation signals yields the segment of the transfer function of the system. This routine is repeated at each sample location in the point-by-point or line scanning modes, yielding the response curve at each spatial location. The BE data can be acquired at the same rate as standard SPM (10–20min for 256 × 256 pixel image), thus suggesting the broad potential for its implementation on existing platforms. The resulting frequency dependence of the response at each pixel can be analyzed to yield, e.g., the parameters of simple harmonic oscillator, including the amplitude, resonant frequency, and Q-factor, or more complex functional fits (e.g., by Duffing equation). Alternatively, the multivariate statistical data analysis methods can be used to determine the structure of the data, dimensionality of the parameter space, and spatial correlations that constitute the 2D image, as discussed below.

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Fig. 17.18 Operational principle of BE in an SPM. The excitation signal is digitally synthesized to have a predefined amplitude and phase in the given frequency window. The cantilever response is detected and Fourier transformed at each pixel in an image. The ratio of the fast Fourier transforms of response and excitation signals yields the cantilever response (transfer function). Fitting the response to the simple harmonic oscillator yields amplitude, resonance frequency, and Q-factor, plotted as 2D images or used as feedback signals. Reproduced with permission from [45]. Copyright 2007, Institute of Physics and IOP Publishing Limited

Data Analysis in Band Excitation The development of the BE method necessitates the interpretation of the 3D {A,q }(x, y,w ) data arrays, where {A,q }is the response amplitude and phase and w is frequency acquired at each spatial location (x, y ) . Ideally, the interpretation of the data includes the (a) dimensionality reduction and (b) extraction of parameters. Dimensionality reduction includes the transformation of the three- or higher dimensional data set to 2D spatially resolved maps that can be readily interpreted by a human observer. These transformations can be based on “physical meaning” of the data or based on statistical projection techniques. The extraction of parameters assumes the presence of a welldefined physical model describing the image formation mechanisms, and the model parameters are then extracted to yield 2D parameter maps. Below, we briefly summarize two examples of the parameter extraction and statistical data analysis.

Fitting Models In PFM, for linear contact resonances with small damping, the amplitude-frequency response of the cantilever in contact with the surface can be described using the harmonic oscillator (SHO) model [47]

Ai (w ) =

(w

A imaxw i20 / Qi 2 i0

-w

) + (ww

2 2

/ Qi )

2

i0

(17.1)

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Fig. 17.19 (a) Experimental setup for PFM measurements with resonance enhancement. (b) Map max of the resonant PFM amplitude signal Ai across a PZT surface. (c) Response spectra and fits using (17.1) at selected locations within the domains and at a domain wall. Reprinted with permission from [47]. Copyright 2006, American Institute of Physics

where Aimax is the signal at the frequency of the ith resonance, w i 0 , and Qi is the Q-factor that describes the energy losses in the system. Consequently, the amplitude– frequency curve can be fitted to (17.1) at each point, providing maps of the corresponding coefficient. An example of this imaging mode in the fitted data is shown in Fig.17.19 for a PZT ceramic sample. Note the 40-kHz shift in resonance frequency during a PFM scan. Using this approach, the 3D data can be fit to the SHO model to extract and plot 2D maps of SHO parameters: response amplitude, resonance frequency, and Q-factor as shown in Fig. 17.20. The topographic image in Fig. 17.20a shows topographic changes at the grain boundary and features associated with preferential domain etching. The PFM amplitude and phase images at 2.0MHz, i.e., far from the resonance, detail the local domain structure (Fig. 17.20b, c, respectively). Note the strong enhancement of the PFM signal at the grain boundary (region I in Fig. 17.20b), the presence of 180° domain walls within the grains (region II), and the region with zero response corresponding to an in-plane domain or an embedded, nonferroelectric particle (region III). Figure 17.20d shows a convergence map delineating points with successful (white) and unsuccessful (black) fits to (17.1). Unsuccessful convergence occurs where the response signal is too weak (e.g., region II in Fig. 17.20b) and no resonance peak is detectable. The maximal electromechanical activity map in Fig. 17.20e illustrates variations in the piezoelectric response between the domains. Note that there is no

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Fig. 17.20 (a) Surface topography, (b) PFM amplitude, and (c) PFM phase images of the PZT surface. (d) Map showing (white) successful SHO fitting locations. 2D maps of SHO fitting parammax eters (e) electromechanical response Ai at (f) the local resonant frequency, and (g) the quality factor. Reprinted with permission from [47]. Copyright 2006, American Institute of Physics

significant enhancement of the response at the grain boundary, while there is a significant 150-kHz shift in the local resonant frequency (Fig. 17.20f). Note also that there are significant variations between the images, thus signifying that they provide complementary data about the local material properties. Finally, a map of the Q-factor in Fig. 17.20g illustrates that losses are almost uniform throughout the sample. Principle Component Analysis In the SHO fitting approach, the knowledge of underlying physical model is required for successful data deconvolution. An alternative approach based on the purely statistical variations in the data is offered by the statistical projection techniques such as principal component analysis (PCA), independent component analysis, and factor analysis [120, 121]. These methods explore the hidden statistical similarities within the data set, allowing the high-dimensional data (spectra at each spatial point) to be projected on to low-dimensional space (several 2D images) while preserving the relevant and significant details. Below, we consider one of the simplest examples of multivariate statistical analysis, namely PCA. In PCA, the spectroscopic image of N × M pixels formed by spectra containing P points is represented as a superposition of the eigenvectors w j,

Ai (w j ) = aik wk (w j )

(17.2)

where aik º ak ( x, y) are position-dependent expansion coefficients (PCA loadings), Ai (w j ) º A( x, y,w j ) is the 3D spectroscopic image at a selected frequency, and w j are the discrete frequencies at which response is measured. The eigenvectors wk (w )

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and the corresponding eigenvalues l k are found from the covariance matrix, C = AA T, where A is the matrix of all experimental data points A ij , i.e., the rows of A correspond to individual grid points (i = 1,..., N · M ), and columns correspond to frequency points, j = 1,..., P. The eigenvectors wk (w j ) are orthogonal and are chosen such that corresponding eigenvalues are placed in descending order, l1 > l 2 >...l In other words, the first eigenvector w1 (w j ) contains the most information within the spectral-image data set, the second contains the most common response after the subtraction of the first one, and so on. In this manner, the first p maps, ak (x, y ), contain the majority of information within the 3D data set, while the remaining P-p sets are dominated by noise. Mathematically, the eigenvalues and corresponding eigenvectors can be determined by direct matrix diagonalization, or, more efficiently, through singular value decomposition of the A matrix [122]. The number of significant elements, p, can be chosen based on the overall shape of l k (k ) dependence, where the linearly decaying part corresponds to the significant elements and the constant part corresponds to noise-dominated components [123–125]. An example of PCA decomposition is shown for a BE-PFM image of lead– zirconate–titanate ceramic in Fig. 17.21. In this case, the large changes in the contact resonance frequency due to surface topography variations result in a large dispersion of the peak positions, (w i - w av ) / 0.1w av. PCA analysis illustrates that images ai (x, y ) contain visible structural elements up to p = 20, as further illustrated in Fig. 17.21g. As expected, the width of w1 w j component is significantly higher than the characteristic peak width, consistent with strong variability of resonant frequency within the data set. However, in this case the NMP element data array (P = 45) can be reduced only to an NMp element data array (p = 20), i.e., reduced by ~2 without significant information loss. It is likely that this compression ratio as well as PCA characterization of the data set can be improved with preconditioning of the data set that would, for instance, center and/or normalize the resonance peaks. Radial correlation analysis on each individual PCA component for BE-PFM data set, ai (x, y ) , is shown in Fig. 17.21k [126]. As expected, correlations virtually disappear for j > 3, indicative of the lack of spatial correlations in noise-dominated images ak (x, y ) for k > p. Unexpectedly however, strong correlation peaks reappear for a few high order ( j = 16,.., 25 ) components, which are extremely weak and cannot be identified on the ln (l i ) vs. i plot. Inspection of corresponding eigenvectors reveals the presence of nonlinear interaction features (peak on the PCA eigenvectors). The corresponding maps are essentially featureless and contain linearly varying background within the image. While the origins of this behavior are unclear, this analysis illustrates the sensitivity of PCA-correlation function analysis to weak signal variations within a 3D data set, as well as a powerful approach for background subtraction. Hence, the behavior of corresponding correlation function C (ak (x, y )) provides additional test for the presence of the image data, e.g., C (a ) (where a is the pixel size) is a measure of short range order, and decay length of C (ak (x, y )) is a measure of spatial correlations in each PCA component (and hence different length scales in the image can be differentiated).

( )

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Fig. 17.21 BE-PFM of PZT. (a–i) PCA images for the first nine eigenvalues. (j) The slow drop-off of the magnitudes of eigenvalues illustrate that the first 30 eigenvalues contain significant information. (k) The first four eigenvectors. Reproduced with permission from [126]. Copyright 2009, Institute of Physics and IOP Publishing Limited

Band Excitation Piezoresponse Spectroscopy The synergy of BE and PFM spectroscopy allows the efficient use of resonance enhancement in PFM spectroscopy, and hence offers a pathway to high energy resolution mapping of polarization dynamics on free surfaces and in ferroelectric structures. The combination of BE and SS-PFM waveform can give rise to very large excitation waveforms and complex 4D data sets representing 2D voltage–frequency spectra

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acquired at each location on the sample surface. However, the analysis of this data allows not only to utilize resonance enhancement in PFM, but also to trace the evolution of contact resonance frequency, dissipation, and associated nonlinearities with bias at single location and between dissimilar locations on sample surface. The implementation and several examples of BE SS-PFM (BEPS) imaging are discussed below.

BEPS of Free Surfaces Conventional PFS utilizes a pulse train of increasing magnitude to switch the probed volume of the ferroelectric, while a single-frequency AC signal is used to measure the piezoresponse of the switched volume between the pulses [50]. In BEPS [127], the AC signal with chirp pulse centered at the resonance frequency of the cantilever following the recently developed BE method [45]. The BE waveform exciting the tip during the bias-off step allows for the rapid (~3 ms) sampling of the full response-frequency curve in the vicinity of the resonance. The BE excitation is superimposed on the standard voltage pulse train in PFS, in which bias-on step is used to create a polarization pattern in the material and read-out is performed during bias-off state. The amplitude, A (w ), and phase, q (w ) -frequency response curves are acquired at each point of the image, (x, y ), as a function of the dc pulse magnitude, Vdc , yielding the 4D data set of {A,q }(x, y,w ,Vdc ) (Fig. 17.22). After the acquisition, the resulting 4D data set is analyzed to yield voltage dependence of materials properties. Briefly, the frequency response curve, {A,q }(w ) , at each spatial and voltage point, (x, y,Vdc ), is fitted using simple harmonic oscillator model to yield local electromechanical response, Q-factor (i.e., dissipation), and resonant frequency. The bias dependence of the electromechanical activity, defined as either response at maximum, or integrated peak intensity, yields local PFM hysteresis loop. The hysteresis loop data can be analyzed to provide the local nucleation and coercive biases, remanent response, and work of switching that can be plotted as 2D maps [70]. Provided that resonant frequency and Q-factor are bias independent, these can be plotted as local maps of elastic and dissipative properties. Alternatively, the bias-related changes in resonant frequency and Q-factor within a cycle (reversible) and between the cycles (irreversible) provide insight into previously inaccessible polarization- and voltage-related changes in local contact mechanics and dissipation. An example of a single point BEPS sweep obtained on a BFO surface in ultrahigh vacuum is shown in Fig. 17.23. The amplitude and phase spectra are calculated as an inverse Fourier transform of the cantilever response to the BE pulse after each dc-recording pulse. The points where the ferroelectric switching occurs along the voltage axis are seen as vertical lines in the amplitude image of Fig. 17.23a and each point is associated with a change of the phase by 180° in Fig. 17.23b. The direct examination of the amplitude-frequency-bias response diagram illustrates that PFS at resonance conditions necessitate the use of BE or other frequencytracking method, because the resonance frequency shifts are large enough compared

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Fig. 17.22 Data acquisition and processing in BEPS. (a) Ferroelectric switching is induced by a pulse train of increasing DC bias while the changes in the piezoresponse, contact stiffness, and dissipation are measured by exciting the cantilever with a narrow frequency band around its contact resonance (b, c). The ferroelectric hysteresis is measured across a grid of points (d) resulting in a spatially resolved 4D data set, where each point represents the cantilever’s resonant response along the local hysteresis loop (e, f). The resonant response (amplitude and phase) is then fit by a simple harmonic oscillator model to yield the resonance amplitude, resonance frequency, and Q-factor (g) as a function of the dc bias. Reprinted with permission from [127]. Copyright 2008, American Institute of Physics

to resonant peak widths. Therefore, signal at a constant frequency is dominated by changes in resonant frequency, rather than piezoresponse amplitude. The cantilever response at every voltage point was fitted with the SHO model, yielding the evolution of the contact resonance frequency of the cantilever and the Q-factor along the ferroelectric hysteresis loop, as shown in Fig. 17.23d, e. The biggest change of the resonance frequency (by 1–2kHz) happens in the vicinity of the ferroelectric switching events. In particular, for both polarities the resonance frequency increases at the onset of domain nucleation under the cantilever and gradually decreases after the switching event. Contact dissipation follows almost the same trend as the resonance frequency, and is almost linearly correlated with it. From the basic properties of the SHO model dw r /dQ = g / (mw r ), where w r is the resonance frequency of the cantilever, is the energy dissipation, and is the mass of the cantilever. The linear correlation between the dissipation and the resonance frequency suggests that the resonance frequency shifts are due to the changes in the contact dissipation associated with domain formation. Therefore, observation of resonance frequency shifts provides insight into previously inaccessible dissipative micromechanics of ferroelectric domain formation. Note that in this case the process is reversible and the hysteresis loops reproduce between the cycles. The resulting 3D data array formed by hysteresis loops at each point can be further processed to yield standard SS-PFM maps of ferroelectric parameters, as shown in Fig. 17.24a–c.

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0.56

0.52

0.52

0.48

0.48

0.44

0.44

0.40

0.40

0.36

0.36

−8 −6 −4 −2

b

0

2

4

6

8

4

6

8

DC bias, V

Piezoresponse, a.u.

e

c

0.56

dissipation, a.u.

d

ωr

a

523

0.8 0.6 0.4 0.2 0.0 −0.2 −0.4 −0.6 −8 −6 −4 −2

0

2

DC bias, V

Fig. 17.23 BEPS experiment at a single point of the BFO (100) surface in ultrahigh vacuum. (a) Amplitude and (b) phase of the resonant response of the cantilever as a function of the (c) varying dc bias that is used to obtain the ferroelectric hysteresis loop. Switching events are clearly seen as straight vertical lines where the amplitude of the response goes effectively to the noise value while the phase changes by p. (d) Evolution of the resonance frequency (black) and dissipation (red) along the (e) ferroelectric hysteresis loop. The resulting array of hysteresis loops can be processed to yield maps of (f) imprint, (g) switchable polarization, and (h) effective work of switching (as illustration, shown are maps for PZT thin film in ambient). Reprinted with permission from [127]. Copyright 2008, American Institute of Physics

Fig. 17.24 The array of hysteresis loops can be processed to yield maps of (a) imprint, (b) switchable polarization, and (c) effective work of switching (shown are maps for a PZT thin film in ambient, 1.8 mm × 1.8 mm scan size). Reprinted with permission from [127]. Copyright 2008, American Institute of Physics

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Summary and Outlook Piezoresponse force microscopy and spectroscopy are rapidly emerging as universal techniques for probing local structure and functionality in materials with strong biasstrain coupling. Historically, these systems included inorganic piezo- and ferroelectrics in the form of crystals, polycrystalline ceramics, thin films, or device structures. The need to understand the local bias- and time-induced polarization dynamics in these materials has stimulated emergence of a large number of spectroscopic techniques. In parallel, the use of resonance enhancement in PFM has stimulated the development of band excitation and DART detection methods that allow efficient sampling of the frequency response of the cantilever-tip system, and hence efficient use of resonance enhancement in PFM. The synergy of these methods is bringing life to new generations of high-dimensional spectroscopic methods providing 3, 4, and even 5D data sets. This progress in turn stimulates the development of multivariate data analysis methods based on data mining and artificial intelligence algorithms required to find intrinsic correlations in higher-dimensional data, link them to local microstructure, and extract underpinning physical behaviors. Finally, PFM and associated spectroscopies are not limited to the realm of inorganic piezo- and ferroelectrics. The recent progress in imaging piezoelectric biopolymers is but a harbinger of future progress. In particular, dynamic phenomena in biological systems from tissue to cellular to subcellular and molecular levels are typically associated with electromechanical transformations [128]. Similarly, functionality of many electrochemical systems, most notably energy storage and conversion materials, is strongly linked to bias-induced strains. Undoubtedly, the future will see the broad range of applications of PFM to these systems. Acknowledgments This research was conducted at the Center for Nanophase Materials Sciences, which is sponsored at Oak Ridge National Laboratory by the Scientific User Facilities Division, U.S. Department of Energy. BJR also acknowledges the support of UCD Research. SVK gratefully acknowledges the collaborations with R. Ramesh (UC Berkeley), S. Trolier-McKinstry, V. Gopalan, and L.Q. Chen (Penn State), and A. Morozovska and E. Eliseev (National Academy of Science, Ukraine). The authors also thank A. Gruverman for permission to reproduce data in Fig. 17.10 and Jiangyu Li for the sample imaged in Fig. 17.16. PFM, SS-PFM, BE SPM, and other SPM modes are available as a part of the user program at the Center for Nanophase Materials Sciences (CNMS), www.cnms.ornl.gov.

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Chapter 18

Polarization Behavior in Thin Film Ferroelectric Capacitors at the Nanoscale A. Gruverman

Introduction A physical principle of most of ferroelectric-based devices is electrically induced polarization reversal, which on a microscopic level occurs via the nucleation and growth of a large number of domains. The dynamic characteristics of domain growth as well as static properties of domain structure to a large extent determine the ferroelectric device performance. Recent advances in the synthesis and fabrication of micro- and nanoscale ferroelectric structures [1–4] make it imperative to understand the domain switching behavior at this scale. A major limitation in acquiring this crucial information is the lack of experimental methods to characterize the domain kinetics with the nanometer length and nanosecond time resolution. The most effective approach to visualization of domain kinetics is based on linear coupling between ferroelectric and piezoelectric parameters, which on the experimental level can be detected either by X-ray scattering or by scanning force microscopy. High-resolution studies using time-resolved X-ray microdiffraction imaging [5–7] have demonstrated reproducible switching behavior of polarization from cycle to cycle and allowed direct measurements of domain wall velocity at high electric fields. Over the last several years majority of the polarization imaging studies in ferroelectric materials are done by piezoresponse force microscopy (PFM). Application of PFM provided a breakthrough in understanding the static and dynamic behavior in ferroelectric films [8, 9]. One of the special features of PFM is its capability to detect the polarization state through the top electrode (TEL) [10–12], which provides a unique possibility of nanoscale studies of the statics and dynamics of domain structure in ferroelectric capacitors under the uniform field conditions. This paper reviews recent advances in this area of ferroelectric research.

A. Gruverman (*) Department of Physics and Astronomy, University of Nebraska, Lincoln, NE 68588-0111, USA e-mail: [email protected] S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8_18, © Springer Science+Business Media, LLC 2010

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Experimental Approach In conventional PFM studies of ferroelectrics, a periodic bias, Vtip = V0 cos ωt , with an amplitude V0 well below the coercive voltage is applied to a conductive probing tip in contact with a bare surface of the sample, i.e., without a deposited TEL. Due to the converse piezoelectric effect, application of the ac bias results in a local surface displacement, d = d0 cos(ωt + φ) . Domain imaging is performed by detecting the piezoelectric strain using the same tip. A nanoscale dimension of the tip-sample contact determines a high spatial resolution of domain features [13]. Due to a highly localized electric field of the tip, application of the bias above the switching threshold results in local polarization reversal and formation of a single nanoscale domain (Fig. 18.1a). In PFM imaging of the ferroelectric capacitors, the probing tip is in contact with the deposited TEL (Fig. 18.1b). Although in this case the whole volume underneath the electrode is electrically excited, the electromechanical response is still probed locally. Scanning the electrode surface while measuring the local strain provides spatially resolved information on domain structure underneath the electrode. The lateral resolution, determined as the domain wall image profile, linearly scales with the thickness of the ferroelectric layer H and TEL L. For typical material parameters and thin TEL (L<
Fig. 18.1 Sketch of domain switching in (a) standard PFM geometry and (b) ferroelectric capacitor structure

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the relative contribution of nucleation and domain wall motion into polarization reversal process, field-dependent motion of domain walls and capacitor size effect on its switching behavior. Generally, application of PFM to investigation of dynamic processes is limited by its low time resolution determined by an acquisition time for a single frame (of the order of several minutes). A high-speed version of PFM (HSPFM) has been developed by Huey’s group to allow complete image acquisition in several seconds thus increasing time resolution by two orders of magnitude over the conventional PFM imaging [15, 16]. This approach, which involves high-speed scanning of a bare ferroelectric surface with a tip under a superposition of a switching and imaging bias, allows effective studies of the dynamics of domain nucleation and growth but requires relatively smooth surfaces. Recently, time resolution of PFM imaging has been improved even further (into the 100 ns range) by using an interrupted switching PFM (IS-PFM) method based on visualization of domain configurations developing in ferroelectric capacitors during step-by-step polarization reversal [17]. Switching characteristics such as nucleation rate and domain wall velocity can be calculated from a set of PFM snapshots taken at different time intervals by measuring the time dependence of the number and size of growing domains. The time resolution of the IS-PFM method is determined by the rise time and duration of the switching pulses and, depending on the capacitor size and time constant of the external circuitry, can be in the order of 10 ns. Two main modifications of the IS-PFM method have been suggested (Fig. 18.2). In one approach, the domain switching behavior is visualized by applying a series of short input pulses with fixed amplitude and incrementally increasing duration ( τ1 < τ 2 <  < τ n < t s τ < t s , where ts is a switching time for a given voltage). PFM imaging of the resulting domain pattern is performed after each pulse (Fig. 18.2a)

Fig. 18.2 Schematic of pulse sequences for IS-PFM imaging of domain switching kinetics: (a) step-by-step switching approach and (b) successive switching approach. Reprinted with permission from [20]

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[18]. At the beginning of each switching cycle, the capacitor is reset into the initial polarization state. Applicability of this approach depends upon the reproducibility of domain switching kinetics from cycle to cycle. Indeed, it has been shown, that in each switching cycle, domain nucleation occurs in the predetermined sites most likely corresponding to the local defects at the film-electrode interface. Figure 18.3 shows a two-dimensional map of on nucleation probability obtained by Kim et al. in epitaxial Pb (Zr, Ti)O3 (PZT) capacitors [19]. This map is a direct illustration of the fact that ferroelectric switching is a result of heterogeneous nucleation. Most sites in epitaxial capacitors have nucleation probability above 90% while in polycrystalline capacitors probability is close to 100%. In another approach, proposed to reduce the detrimental effect of stochastic nucleation events in epitaxial structures, after a capacitor is being set, the switching pulses of the same duration are applied to the capacitor ( τ1 = τ 2 =  = τ n < t s ) with PFM imaging between the pulses (Fig. 18.2b) [20]. In this case, it is assumed that the PFM image obtained after the nth pulse is the same as that after a single pulse with duration of t = τ1 + τ 2 +  + τ n . Then, all the PFM images taken before the (n + 1)th pulse reveal the successive domain wall evolution during the time period of t. It should be mentioned that both variations of the IS-PFM method rely on stability of instantaneous domain patterns between pulse applications. Whether polarization relaxation takes place or not can be checked by comparing the PFM switching with the transient current measurements. Little or no discrepancy between the two sets of data obtained in most reports is a solid proof of the reliability of the IS-PFM approach. A small dc offset during PFM imaging can be used to account for an internal bias (imprint) in the capacitors and stabilize the domain patterns [21].

Fig. 18.3 A two-dimensional map of nucleation probability acquired in epitaxial PZT capacitor. The scan area is 6 × 6 µm2. Reprinted with permission from [19]

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Capacitor Scaling Effect on Domain Switching Kinetics Application of the IS-PFM approach to fast switching processes (in the 100-ns range) allowed identification of the effect of capacitor size on the rate-limiting mechanism. Figure 18.4 shows PFM images of instantaneous domain configurations developing in 1 × 1.5 and 5 × 5 µm2 polycrystalline (111) PZT capacitors at different stages of polarization reversal process [18]. IS-PFM switching data in Fig. 18.4 indicate the difference in switching mechanisms in larger and smaller capacitors. Generally, in larger capacitors, two distinct stages of polarization reversal can be observed: a fast switching stage dominated by domain nucleation (usually up to 60–70% of the total volume) and a slower stage where switching occurs mainly via lateral domain wall motion. On the other hand, in the small capacitors the switching proceeds mostly via lateral growth of just few domains, i.e., contribution of domain nucleation to polarization reversal is significantly reduced. Due to the qualitatively different domain dynamics, the relative switching speed of capacitors of smaller dimensions is field dependent: they switch faster than the larger capacitors in the high field range (in agreement with earlier reports by macroscopic measurements [22]), but slower in the low fields (Fig. 18.5a) [18]. Quantitative insight into the capacitor scaling effect on switching behavior has been obtained by measuring the field dependency of nucleation rate R and wall velocity v (Fig. 18.5b) obtained from the time dependence of domain number and domain radius, respectively [17]. It is found that both parameters are exponential functions of the applied field: R ∼ exp( −a n / E ) and v ∼ exp( −a w / E ) , where an=7.1 × 107 V/m and aw=8.3 × 107 V/m. A difference between activation field values an and aw in combination with the capacitor size-dependent polarization switching mechanism explains a faster switching in 1 × 1.5µm2 capacitors in comparison to 5 × 5 µm2 capacitors in the high field range by virtue of transition from walllimited to nucleation-limited switching (NLS). As contribution of nucleation to the polarization switching in smaller capacitors is insignificant (they switch mainly via the domain wall motion), in the high field range, where nucleation is a rate-limiting

Fig. 18.4 PFM phase images of domain configurations developing in polycrystalline (111) PZT capacitors, (a) 1 × 1.5 µm2 and (b) 5 × 5 µm2, at different stages of step-by-step polarization reversal in the low-field range (E = 100 kV/cm). The scanning area is 1.6 × 1.6 µm2 in (a) and 5.5 × 5.5 µm2 in (b). Reprinted with permission from [18]

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Fig. 18.5 (a) Switched capacitor volume as a function of time for 1 × 1.5 and 5 × 5 µm2 (111) PZT capacitors in the low-field range (E = 100 kV/cm) showing faster switching in larger capacitors. Smaller capacitors start to switch faster in the fields above 200 kV/cm. Dashed line, fitting of the PFM switching data by the NLS model using Lorentzian distribution for log t0. (b) Nucleation rate and domain wall velocity vs. electrical field for 5 × 5 µm2 capacitors. Reprinted with permission from [18]

mechanism, the smaller capacitors will switch faster than the larger capacitors. In the low-field range, on the contrary, domain wall speed will limit the switching rate. For this reason, larger capacitors characterized by considerable contribution of nucleation mechanism to polarization reversal will be switching faster in low fields. Theoretical estimations show that the nucleation time becomes a rate-limiting parameter for polycrystalline PZT capacitors smaller than 0.6 µm2 in the fields above 200 kV/cm [18].

Effect of Film Microstructure on Domain Switching Kinetics Direct imaging of domain evolution during switching provides a possibility to understand the effect of film microstructure on polarization reversal behavior. A traditional approach to model the switching kinetics – the KAI model – is based on the classical statistical theory of nucleation and unrestricted wall motion proposed by Kolmogorov and Avrami, and extended to ferroelectrics by Ishibashi [23]. In the KAI model, the volume fraction of the polarization switched by time t is described by p(t ) = 1 − exp[ −(t / t 0 )n ] , where t0 is the characteristic switching time and n can be treated as domain dimensionality reflecting the mechanism of polarization reversal. The KAI model has been successfully applied to describe the switching in epitaxial thin films and single crystals. However, switching in polycrystalline films proceeds much slower than predicted by the KAI model [24] and continuous domain wall motion is presumably interrupted due to pinning by structural defects. Validity of this assumption has been confirmed by IS-PFM observation of spatial variations in domain wall velocity by almost two orders of magnitude [7]. It has been also shown that the KAI model cannot adequately fit the domain switching kinetics. A NLS model featuring independent switching kinetics in individual grains and an

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exponentially wide variation of local nucleation times has been proposed to explain switching in polycrystalline capacitors [25, 26]. The time-dependent behavior of switched polarization is expressed by p(t ) = 1 − ∑ F (log t0 ) exp( −(t / t0 )n ) , where F(log t 0) is a distribution function for local switching times. To account for variations in the local electrical field related to the randomly distributed dipole defects Jo et al. [27] suggested to use the Lorentzian distribution of logt 0. This treatment allows description of the switching current data within a wide range of electric fields and temperatures. IS-PFM measurements show that this approach provides the best fit for experimentally observed domain kinetics in relatively large polycrystalline PZT capacitors (Fig. 18.5a). In contrast, the switching kinetics in smaller capacitors is close to the logarithmic time dependence, which can be a result of a much narrower distribution of log t 0. Snapshots of domain structure evolution during switching in epitaxial (001) PZT capacitors are shown in Fig. 18.6. Note that the switching as a whole occurs via dynamics of 180° domains and no 90° wall formation has been observed in agreement with earlier conclusion based on transient current measurements [28]. For E=700 kV/cm, the nucleation density was estimated to be 7.1 × 108 cm -2 (Fig. 18.6a), which is well below the nucleation density of 3.2 × 1012 m-2 measured in polycrystalline PZT capacitors [18]. Given a higher concentration of point defects associated with grain boundaries and relatively rough interfaces in polycrystalline films this difference is reasonable. The time-dependent evolution of domain structure reveals that the wall velocity is isotropic and independent of domain size in the range at least up to 400nm in diameter. Considerable decrease

Fig. 18.6 PFM phase images of domain configurations developing in epitaxial (001) PZT capacitors (a) E = 700 kV/cm and (b) E = 500 kV/cm. The scan size is 2.5 × 2.5 µm2

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in the wall velocity has been observed for the domains in the close proximity (<100 nm) to another growing domains. Significant anisotropy of the wall velocity has been detected during switching in the lower-field range (<500 kV/cm). This anisotropy is manifested by the formation of characteristically elongated domains (Fig. 18.6b). Measurements of domain size as a function of time show that in the y-axis direction the wall velocity vy is almost three times higher than velocity vx in the orthogonal x-axis direction (0.39 and 0.12 m/s, respectively). Overall, the switching kinetics in the low fields is characterized by a long-range (up to 1 µm) lateral growth of just a few nucleated domains in stark contrast to the switching in the high fields. For epitaxial capacitors, the KAI model provides excellent description of the timedependent switching behavior in epitaxial capacitors, contrary to the case of polycrystalline PZT capacitors. It has been found that n values are different for low and high fields and very close to integer values: 1.12 for 500 kV/cm and 1.96 for 700 kV/cm, indicating a change in the switching mechanism. In the KAI model, n = 2 corresponds to a growth of cylindrical domains (2D growth) and n = 1 indicates a lateral expansion of lamellar domains (1D growth). An increase in n with an applied field, also reported by So et al. [29], is consistent with the change in domain growth dimensionality revealed by IS-PFM studies – transition from one-dimensional anisotropic growth under 500 kV/cm to two-dimensional isotropic growth under 700 kV/cm. A general approach adopted by Noh et al. [30] treats domain wall motion as a nonlinear dynamic process resulting from competition between elastic and pinning forces. Investigations of domain growth in a wide range of temperature and electric field showed that the wall dynamics could be classified into the creep, depinning and flow regimes. Wall velocity exhibits high-temperature dependence in low fields, while in high fields a crossover to the flow regime makes it temperature independent. In the creep regime at low fields, the wall dynamics can be described as thermally activated propagation between pinning sites: v ∼ exp[ −(U / kBT )( Ec / E )µ ] , where U is an energy barrier and µ is a dynamic exponent, reflecting the nature of the pinning potential. It was found that for epitaxial PZT capacitors µ = 0.9 ± 0.1 over a wide temperature range suggesting a long-range pinning potential due to defect structure.

Mechanical Stress Effect on Static Polarization Behavior The static polarization state in ferroelectric capacitors, determined by the thermodynamic minimum of free energy, is a function of electrical and mechanical boundary conditions. Reduction of the free energy via formation of ferroelectric domain structures gives rise to a profound mechanical stress effect on the physical properties of ferroelectrics. Accommodation of the misfit strain between a substrate and a thin film in epitaxial heterostructures can result in the appearance of new phases forbidden in bulk samples and significant size dependence on the dielectric and piezoelectric properties [31]. Numerous attempts have been made to develop a thermodynamic

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theory that would account for the effect of the mechanical boundary conditions on structural transformations in ferroelectric films [32–34]. However, up to now almost all of these studies took into account the mechanical boundary conditions only at the film/substrate interface while effect of the mechanical stress imposed by the TEL has largely been ignored. Application of PFM reveals that boundary conditions at the top interface drastically affect the polarization stability in polycrystalline PZT capacitors. Figure 18.7 shows surface topography, PFM amplitude, and phase images of the poled 1´1.5 µm2 capacitors with a 50-nm thick TEL [35]. Capacitors in the upper row have been poled by -5 V, 1s voltage pulses applied to the TELs, while the bottom row capacitors have been poled into opposite direction by the +5 V, 1s voltage pulses. Uniform amplitude and phase contrast indicate complete and uniform switching of the capacitors into a stable polarization state. On the other hand, PFM imaging of the poled capacitors with the 250-nm thick TEL (Fig. 18.8) reveals unusual domain patterns: after application of poling voltage the central regions of the capacitors exhibit polarization opposite to the polarity of

Fig. 18.7 (a) Topographic, (b) PFM phase, and (c) PFM amplitude of the poled (111) PZT capacitors with 50-nm thick top electrodes. Upper row capacitors poled by negative voltage pulses (-5 V, 1 s), bottom row capacitors poled by positive pulses (+5 V, 1 s). The scanning size is 6 × 6 µm2. Reprinted with permission from [35]

Fig. 18.8 (a) Topographic, (b) PFM phase, and (c) PFM amplitude of the poled (111) PZT capacitors with 250-nm thick top electrodes. Upper row capacitors poled by negative voltage pulses (-5 V, 1 s), bottom row capacitors poled by positive pulses (+5 V, 1 s). The scanning size is 6 × 6 µm2. Reprinted with permission from [35]

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the applied voltage. The observed effect is symmetric with respect to the voltage polarity, i.e., the central parts of the capacitors always exhibit polarization opposite to the polarity of the applied bias. It should be mentioned that imaging the same capacitors in the PFM mode with an additional dc bias superimposed on the ac imaging voltage resulted in complete switching of polarization in the whole capacitor indicated by a uniform PFM signal across the TEL. However, after the dc bias is turned off, the inverse domain in the center region appears again. This behavior is indicative of spontaneous backswitching occurring in the center of the capacitors after application of the poling voltage. These results also suggest that the backswitching effect in the PZT capacitors is not affected by the substrate but is the result of the presence of a thick (250 nm) TEL. Free energy calculations using a fully coupled ferroelectric finite element phase (FFEP) field model [36] show mechanical constraints due to the thick TEL create residual shear stresses during ferroelectric switching in (111) PZT capacitors leading to complex, twinned domain structure (Fig. 18.9). The residual elastic energy results in spontaneous polarization backswitching in the central regions of capacitors. Residual elastic energy is larger in the center away from the stress-free edges, which explains why backswitching mainly occurs in the central parts of the capacitors. Also, recently residual stress has been suggested to cause faceting in poled epitaxial (001) PZT capacitors [37]. It has been shown that within a period of time much longer than the switching time metastable circular domains develop facets along (100) directions (Fig. 18.10). Faceting can be attributed to anisotropic contribution to the wall velocity along the crystallographic axes [38]. The observed effect is interpreted as mechanical relaxation resulting from highly inhomogeneous stress distribution in the circular capacitors with high-stress conditions in the center due to electrode-film lattice mismatch and presumably stress-free edges.

Fig. 18.9 Finite element phase field map of the out-of-plane (111) polarization component (P2) in two short-circuited ferroelectric capacitors with a lateral size of 1 µm under equilibrium conditions: (a) 250-nm thick TEL and (b) 50-nm thick TEL. Reprinted with permission from [35]

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Fig. 18.10 PFM phase images of the poled circular epitaxial PZT capacitor (a) right after poling and (b) 24 h after poling. Reprinted with permission from [37]

Summary Rapid development of ferroelectric-based devices with reduced dimensions generated a strong need for extensive investigation of the size effects in ferroelectric materials. Application of PFM provides a unique opportunity to study physical mechanisms underlying the static and dynamic properties of ferroelectric structures at the nanoscale level. Inhomogeneous domain nucleation results in domain kinetics high reproducibility of switching kinetics and allows application of the interrupted switching approach to study fast polarization reversal processes. Direct measurements of the field-dependent wall velocity and nucleation rate provide information on mechanism of domain growth in conjunction with microstructural and scaling effects. Acknowledgments This work was supported by the National Science Foundation (Grant No. MRSEC DMR-0820521) and the Nebraska Center for Materials and Nanoscience at University of Nebraska-Lincoln. The author would like to thank Prof. T.W. Noh for his kind permission to use his data.

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7. A. Grigoriev, R. Sichel, H.-N. Lee, E.C. Landahl, B. Adams, E.M. Dufresne, P.G. Evans, Phys. Rev. Lett. 100, 027604 (2008) 8. M. Alexe, A. Gruverman (eds.), Nanoscale Characterization of Ferroelectric Materials: Scanning Probe Microscopy Approach, Springer, (2004) 9. S. Jesse, A.P. Baddorf, S.V. Kalinin, Appl. Phys. Lett. 88, 062908 (2006) 10 A. Gruverman, B.J. Rodriguez, R.J. Nemanich, A.I. Kingon, J.S. Cross, M. Tsukada, Appl. Phys. Lett. 82, 3071 (2003) 11. P. Bintachitt, S. Trolier-McKinstry, K. Seal, S. Jesse, S.V. Kalinin, Appl. Phys. Lett. 94, 042906 (2009) 12. S. Hong, E.L. Colla, E. Kim, D.V. Taylor, A.K. Tagantsev, P. Muralt, K. No, N. Setter, J. Appl. Phys. 86, 607 (1999) 13. L. Tian, A. Vasudevarao, A.N. Morozovska, E.A. Eliseev, S.V. Kalinin, V. Gopalan, J. Appl. Phys. 104, 074110 (2008) 14. S.V. Kalinin, B.J. Rodriguez, S.-H. Kim, S.-K. Hong, A. Gruverman, E.A. Eliseev, Appl. Phys. Lett. 92, 152906 (2008) 15. R. Nath, Y.-H. Chu, N.A. Polomoff, R. Ramesh, B.D. Huey, Appl. Phys. Lett. 93, 072905 (2008) 16. N.A. Polomoff, R. Nath, J.L. Bosse, B.D. Huey, J. Vac. Sci. Technol. B 27, 1011 (2009) 17. C. Dehoff, B.J. Rodriguez, A.I. Kingon, R.J. Nemanich, A. Gruverman, J.S. Cross, Rev. Sci. Instrum. 76, 023708 (2005) 18. A. Gruverman, D. Wu, J.F. Scott, Phys. Rev. Lett. 100, 097601 (2008) 19. D.J. Kim, J.Y. Jo, T.H. Kim, S.M. Yang, B. Chen, Y.S. Kim, T.W. Noh, Appl. Phys. Lett. 91, 132903 (2007) 20. S.M. Yang, J.Y. Jo, D.J. Kim, H. Sung, T.W. Noh, H.N. Lee, J.-G. Yoon, T.K. Song, Appl. Phys. Lett. 92, 252901 (2008) 21. A. Gruverman, B.J. Rodriguez, C. Dehoff, J.D. Waldrep, A.I. Kingon, R.J. Nemanich, J.S. Cross, Appl. Phys. Lett. 87 082902 (2005) 22. T. Hase, T. Shiosaki, Jpn. J. Appl. Phys. 30, 2159 (1991) 23. Y. Ishibashi, Y. Takagi, J. Phys. Soc. Jap. 31, 506 (1971) 24. O. Lohse et al., J. Appl. Phys. 89, 2332 (2001) 25. X.F. Du, I.W. Chen, Appl. Phys. Lett. 72, 1923 (1998) 26. A. Tagantsev et al., Phys. Rev. B 66, 214109 (2002) 27. J.Y. Jo, H.S. Han, J.-G. Yoon, T.K. Song, S.-H. Kim, T.W. Noh, Phys. Rev. Lett. 99, 267602 (2007) 28. W. Li, M. Alexe, Appl. Phys. Lett. 91, 262903 (2007) 29. Y.W. So, D.J. Kim, T.W. Noh, J.-G. Yoon, T.K. Song, Appl. Phys. Lett. 86, 092905 (2005) 30. J.Y. Jo, S.M. Yang, T.H. Kim, H.N. Lee, J.-G. Yoon, S. Park, Y. Jo, M.H. Jung, T.W. Noh, Phys. Rev. Lett. 102, 045701 (2009) 31. N.A. Pertsev, A.G. Zembilgotov, A.K. Tagantsev, Phys. Rev. Lett. 80, 1988 (1998) 32. N.A. Pertsev, G. Arlt, A.G. Zembilgotov, Microelectron. Eng. 29, 135 (1995) 33. J.S. Speck, A. Seifert, W. Pompe, R. Ramesh, J. Appl. Phys. 76, 477 (1994) 34. W. Pompe, X. Gong, Z. Suo, J.S. Speck, J. Appl. Phys. 74, 6012 (1993) 35. A. Gruverman, J.S. Cross, W.S. Oates, Appl. Phys. Lett. 93, 242902 (2008) 36. Y. Su, C. Landis, J. Mech. Phys. Solids, 55, 280 (2007) 37. J.F. Scott, A. Gruverman, D. Wu, I. Vrejoiu, M. Alexe, J. Phys.: Condens. Matter 20, 425222 (2008) 38. W. Kleemann, J. Dec, S.A. Prosandeev, T. Braun, P.A. Thomas, Ferroelectrics 334, 3 (2006)

Index

A Abplanalp, M., 498, 499 AC240 cantilever, 136 AFAM. See Atomic force acoustic microscopy AFM. See Atomic force microscopy Agronin, A., 509 Akhremitchev, B.B., 202 Albrecht, T.R., 72, 142 Alexander, J., 233 Alexe, M., 498 Amplitude modulation (AM) mode, 72, 73. See also Tip–sample force Aono, M., 28 AP-TD. See Atmospheric pressure thermal desorption Aravind, V.R., 301e Arlt, G., 373 Arrhenius law, 361 Arthrobacter oxydans bacteria–mineral interactions, 64–66 gram-positive and chromium (VI)-resistant bacterium, 40 ASAP. See Atmospheric-pressure solids analysis probe Atmospheric pressure mass spectrometry (AP-MS) AP-TD, 184–186 laser desorption ionization mass spectrometry AP-MALDI (see Atmospheric pressure matrix assisted laser desorption ionization) LA-ICP technique, 187–188 LD-APCI, 188–189 LD-ESI, 190–191 ruby laser, 186 Atmospheric pressure matrix assisted laser desorption ionization (AP-MALDI) Abbe criterion, 193

diffraction limit, 195 IR laser, 192 laser irradiation, 191 NSOM, 193 SNOM system, 193, 194 spatial resolution, 192 UV nitrogen laser, 192 Zenobi’s technique, 195 Atmospheric-pressure solids analysis probe (ASAP), 185, 186 Atmospheric pressure thermal desorption (AP-TD) ambient surface sampling technique, 184 ASAP, 185, 186 DAPCI, 184–185 DART, 185 GC/MS, 185 Atomic force acoustic microscopy (AFAM), 96, 213, 496 Atomic force microscopy (AFM), 96 AM–AM approach EFM, 250–252 electrostatic response detection, 244 AM–FM approach, 253–254 amplitude and frequency modulation, 236, 237 amplitude vs. distance, 265–266 amplitude vs. separation (AvZ) curve, 236 atomic-resolution imaging, 234 Bacillus atrophaeus, 44–46 carbon black particles, 264 CdTe nanostructure, 271–273 C60H122 layer, 268, 269 cross-section topography, 267, 268 electrostatic response detection conducting probe, 238 feedback mechanism, 239–240 flexural mode, 239 lift mode/two-pass technique, 240

S.V. Kalinin and A. Gruverman (eds.), Scanning Probe Microscopy of Functional Materials: Nanoscale Imaging and Spectroscopy, DOI 10.1007/978-1-4419-7167-8, © Springer Science+Business Media, LLC 2010

541

542 Atomic force microscopy (AFM), (cont.) probe–sample separation, 238–239 quantitative analysis, 239 sensitivity and resolution reduction, 240 electrostatic tip–sample force, 235 environmental chamber, 262, 263 F14H20 adsorbate, 273, 274 F12H20 adsorbate, EFM image, 269, 270 fluoroalkane deposit, 266–267 FM–FM method, 244 force gradient detection, 244 grating-type probe, 235 KBM averaging method, 236–237 KBr island, 245 KFM (see Kelvin force microscopy) “lattice-plus-defect,” 238 LIA, 250 MACIII accessory, 250 mechanical properties, 234 metals and semiconductors concentration, resistivity and surface potential profile, 280 doping density, 279–281 SDRAM structure, 277, 278 SiAuPt structure, 276–277 SiGe structure, 279 SRAM, 277–279 work function, 276 MicroStar Technologies, 260 molecular self-assemblies adsorbate-graphite interaction, 293 contact potential difference, 292–293 F12H8 adsorbate, graphite, 289, 292 F12H20 adsorbate, graphite, 289, 290 F14H20 adsorbate, graphite, 287–289 F12H8 adsorbate, HOPG, 289, 291 F12H20 adsorbate, mica, 285, 287 F14H20 adsorbate, mica, 284–285 F14H20 adsorbate, Si, 283–284 F12H20 domain, 282, 283 FH2F-type, 282 F12H8 layer, mica, 286, 287 geometrical parameters, 289, 291, 292 hydrocarbon and fluorocarbon, 281 lamellar structure, 293–294 semifluorinated alkanes, 282 surface potentials, 291–292 X-ray reflectivity, 282 oscillating-plate Kelvin probe, 244–245 PFM (see Piezoresponse force microscopy) phase vs. distance, 265 PMMA, 267 polydiacetylene crystal, 261–262 potential resolution, 245

Index probe–sample force detection, 233 Pt-coated Si probe, 258–259 quasistatic/contact mode, 236 SEM and TEM, 259 semifluorinated alkane adsorbate, 262–264 Si wafer, 264–265 spatial resolution, 243 surface structure visualization, 233 surface topography, 236 tip–sample force interaction, 236 tip–sample junction, 264 tip–sample mechanical interaction, 266 TPV, 262, 264 ultraviolet photoelectron spectroscopy, 272 van-der-Waals force, 265 vibrational spectra, 260–261 B Bacillus and Clostridium spores AFM amorphous layer, 54 CotEgerE spores, 54 DNA and dipicolinic acid, 40 environmental resistance and dormancy, 40 environment change, 44–46 high-resolution structure and assembly exosporium and crystalline layer, 47 genomic and proteomic analysis, 46 G medium, 51–52 hollow cores, 49 honeycomb structure, 48, 49 hydrophobins, 48 multi-domain rodlet structure, 48 nutrient broth, 51 protection and germination, 46 protein crystals, 50 self-assembly crystallization process, 50 spore coat structure, 51 surface architecture and structure– function, 47 immunolabeling, 52–53 morphology, 41–43 peptidoglycan, 40 size distributions, 43–44 causes, 40 genome sequence, 39 germination mechanism amyloid fibrils, 58 crystalline honeycomb structure, 57 cytoskeleton network, 59 GerP proteins, 55 gram-positive streptomycetes, 58

Index hexagonal coat layer, 55 higher-order rodlet structure, 55–56 hydrolysis propagation, 57 spore-forming bacteria, 55 vegetative cells (see Vegetative cells) Balke, N., 491 Band excitation-NanoTA and Z-therm method AFAM, 213 amyloid fibrils, 212, 213 characteristics, 215, 216 contact mechanics, 210 DART technique, 210 glass transition temperature, 214 Oliver–Pharr method, 213 PET sample, 210–212 PVDF, 210, 211 quality factor, 214 SAN/PMMA phase polymer, 214 SThEM and TA-AFAM, 209–210 tip–sample contact, 210 Young’s modulus, 214–215 Band excitation piezoresponse force microscopy data analysis fitting models, 516–518 PCA, 518–520 Duffing type dynamics, 515 operational principle, 515–516 Q-factor, 515 single-frequency detection methods, 514 Band excitation piezoresponse spectroscopy (BEPS), 520–523 Bdikin, I.K., 345 Becker, J.S., 183, 187, 188 Betzler, K., 356 Beyder, A., 461 Bimodal magnetic force microscopy, 140, 141 Binnig, G., 210, 409, 433 Bintacchit, P., 506 Birk, H., 497 Blinc, R., 349 Bokov, A.A., 347, 348 Boltzmann function, 474, 475 Burns, G., 346 C Ceramics grain size effect autocorrelation function, 375 BaTiO3 ceramics, 373 correlation radius, 375–376 dielectric constant, 373 diffuse phase transformations, 376 extrinsic and intrinsic factors, 373

543 piezoelectric response, 375 PLZT, 373–375 temperature evolution, PMN-PT, 371–372 Chang, Y.H., 375 Chiu, C-Y, 423, 425 Cho, Y., 320 Cleveland, J.P., 79, 83, 142 Clostridium spores. See Bacillus and Clostridium spores Cochran law, 347 Compton scattering, 406, 407 Constant-amplitude (CA) driving scheme, 73 Constant-excitation (CE) driving scheme, 73 Contact resonance force microscopy methods (CR-FM) AFAM, 96 AFM, 96 buried interface film/substrate adhesion, 117–118 Hertzian contact, 117 hybrid wrinkling–buckling mechanism, 120 impedance-radiation theory, 118 stress-induced buckling mechanics, 119–120 UFM method, 117 disadvantages, 96 elastic modulus, single-point measurements data analysis models (see Data analysis models) dynamic motion and contact mechanics, 98 indentation modulus, 103–104 LIA, 101 multiple data sets, 102–103 noncontact/intermittent contact probes, 97 piezoelectric actuator, 101 single-crystal silicon, 97 vibrational resonant modes, 97–98 force–displacement techniques, 96 force modulation microscopy, 96 nanomechanics, definition, 95 qualitative stiffness imaging, 111–112 quantitative imaging, modulus mapping applications, 113 elastic stiffness, 111, 112 fiber–matrix interface, 115–116 frequency-tracking electronics, 113–115 lyocell fiber-polypropylene matrix composite, 116–117 nanomechanical mapping, 113 point-measurement techniques, 113

544 Contact resonance force microscopy methods (CR-FM) (cont.) spatial resolution, 115 wood-polymer composites, 115 shear elastic property measurement AFM photodiode detector, 107 cantilever flexural modes, 104 flexural and torsional experiment results, 107 fused silica, 106 isotropic materials, 106 tangential contact stiffness, 105 thin-film behavior, 104 tip–sample contact mechanics, 106 vertical contact stiffness, 104–105 small-scale mechanical properties, 95 viscoelastic property measurement contact mechanics model, 109 contact resonance spectra, 110 elastic spring stiffness, 108 harmonic excitation model, 110 loss modulus, 110–111 normalized contact stiffness, 108 oscillatory nanoindentation technique, 109 PMMA, 109 polymer fabrication, 107–108 Cook, R.G., 184 Cook, S., 134 Courant, R., 310 CR-FM. See Contact resonance force microscopy methods Crick, S.L., 202 Cubic relaxors Pb(Mg1/3Nb2/3)O3–PbTiO3 single crystals autocorrelation images, 362–363 Curie temperature, 366–367 macroscopic rhombohedral symmetry, 362 nonzero piezoresponse, PMN-PT10, 364–366 polarization interface, 363 polar structure factors, 366 quasi-regular domain pattern, 363, 364 stress accommodation factor, 366 PbZn1/3Nb2/3O3–PbTiO3 single crystals diffuse neutron scattering, 370 macroscopic coercive field, 369 monoclinic phase, 367 nanoscale hysteresis loop measurement, 370 piezoelectric effect, 367 piezoresponse images, PbTiO3, 367–368 polar clusters, 369 statistical distribution, 368–369

Index D Dacol, F.H., 346 Damped harmonic oscillator, 75 DAPCI. See Desorption atmospheric pressure chemical ionization DART. See Dual AC resonance tracking Data analysis models cantilever dynamics, 98–99 contact mechanics model, 99–100 finite-element analysis methods, 98 Hertzian contact, 100–101 normalized contact stiffness, 99 tip–sample interaction force, 98 Derjaguin, B.V., 201 Derjaguin–Landau–Verway–Overbeek theory, 201 Derjaguin–Muller–Toporov (DMT) model, 77, 157, 201 Desorption atmospheric pressure chemical ionization (DAPCI), 184–185 Desorption atmospheric pressure photoionization (DAPPI), 185 Dipalmitoylphosphatidylcholine (DPPC) monolayer, 242–243 Domke, J., 91 Doroshenko, V.M., 191, 192 Dransfeld, K., 495 Dual AC resonance tracking (DART), 210 dual frequency signal, 143 model parameters, 143–144 multiple frequency measurement, 143 tip–surface junction, 143 topography, contact resonance and Q factor, 144–146 Dynamic force microscopy and spectroscopy AM mode, 72, 73 CA driving scheme, 73 cantilever dynamics, 71, 74 CE driving scheme, 73 driven and self-driven cantilevers, 75–77 feedback loop, 71 FM mode, 72, 73 function generator, 72 lock-in amplifier, 72 microstructured cantilever, 71 NC-AFM, 75 quality factor, 74, 75 self-driven oscillator, 72 spring-mass-model, 74 “tapping mode,” 75 time and phase shift, 74 tip–sample force (see Tip–sample force) tip–sample interaction force (see Tip–sample interaction force)

Index Dynamic nanomechanical characterization harmonic force, mechanical properties damped harmonic oscillator, 156 DMT model, 157 elastic modulus, 157, 158 Fourier transform, 159 frequency domain characteristics, 158 higher harmonic vibrations, 159 spatial characteristics, 157, 158 temporal characteristics, 158 high-speed and high spatial resolution, 153 polymer blends (see Polymer blends) sharper probes and lower interaction force, 155 stiffness, adhesion and dissipation (see Single tapping-mode scan) sub-molecular resolution mechanical mapping, 172–174 torsional harmonics, mechanical measurements, 166–169 Hookian spring, 155 measurement speed, 154 multiple force sensor continuum mechanics, 159 force–distance relationship, 164, 165 high-and low-density polyethylene sample, 165–166 multiple vibration modes, 159 periodic flexural and torsional vibration signal, 163–164 position-sensitive photo detector, 160 pyrolytic graphite, 163 time-resolved tip–sample force measurement, 164–165 torsional harmonic cantilever (see Torsional harmonic cantilever) nanoimaging method, 154 non-destructive measurements, 154 sensitivity and dynamic range, 153 tapping cantilever (see Tapping cantilever) tapping-mode AFM (see Tapping-mode atomic force microscopy) E Electric force microscopy (EFM) AM–AM approach, 250–252 AM–FM approach, 253–254 Electron impact ionizer (EI) source, 182 Eliseev, E.A., 316, 499 Emelyanov, A.Yu., 499 Ethylene-propylene-diene monomer elastomer (EPDM), 169–171 Euler–Bernoulli equation, 236

545 F F12H20 adsorbate, 282, 283 EFM image, 269, 270 graphite, 289, 290 mica, 285, 287 F14H20 adsorbate, 273, 274 graphite, 287–289 mica, 284–285 Si, 283–284 F12H8 layer graphite, 289, 292 HOPG, 289, 291 mica, 286, 287 Field emission scanning electron microscopy (FE-SEM), 301 Finite element method (FEM) ANSYS program, 310 dielectric, elastic and piezoelectric tensor, 310 disc-plane model, 312 electric field distribution, 310–311 electrostatic boundary conditions, 311 lithium niobate piezoelectric coupling simulation, 313 piezoelectric vertical and lateral response, 313, 315 surface deformation, 313, 314 surface potential, 316 mesh, definition, 310 partial differential equation, 440 pipette–sample distance, 441–442 Poisson equation, 440–441 Ritz method, 310 sphere-plane model, 311–312 topography images, 442–444 Force clamp force sensor, 469–471 instrument quantification errors, 469 mechanically imperfect substrate astrocytes and cardiac myocytes, 473 cell membrane, 471 cytoskeletal network, 473 Hooke law, 470 Laplace law, 474 spring constant, 470–471 torsion lever, 472–473 Freeland, J.W., 405 Frequency-modulation (FM) mode, 72, 73. See also Tip–sample force G Ganpule, C.S., 499 Garcia, R., 126

546 García, R., 79 Gas chromatogram mass spectrometer (GC-MS), 183 Gerber, C., 210 Ginzburg–Landau theory, 499 Gitter, A.H., 446 Glazounov, A.E., 349 Gleiche, M., 89 Goeringer, D.E., 182 Gopalan, V., 301, 310 Gorelik, J., 447, 450 Gray, A.L., 187 Gruverman, A., 256, 342, 498, 505, 529 Günther, P., 495 Guo, H.Y., 497 Guo, S.L., 202 Guyonnet, J., 325 H Hagen–Poiseuille law, 455 Hahn, Junhee, 145 Half-cell electrodes, 436–437 Hammiche, A., 207, 208, 218 Hao, 311 Happel, P., 446 Harnagea, C., 498, 510 Harrison, W.W., 189 Henningsen, N., 27 Hertz, 201 Hertz-plus-offset model, 77 Hidaka, T., 497 Highly oriented pyrolitic graphite (HOPG), 133 High-speed piezo force microscopy (HSPFM) AFM feedback role, 332 applications, 342 contact resonance frequency, 331–332 disadvantages, 330 domain nucleation and growth domain perimeter, 338, 339 ferroelectric thin film, 340 NIH-developed ImageJ, 336 nucleation time calculation, 338–340 phase and amplitude frame, 336, 337 PZT film, 336 stack optical community, 336–338 “tracked” domains, 337–339 wall velocity determination, 338, 340 domain orientation, definition, 331 domain poling/hysteresis measurement, 331 domain switching process, 330 domain wall motion, 334–335 ferroelectric materials, 329 high-speed domain writing, 340–342

Index high-speed imaging amplitude and phase, 333, 334 contact resonance spectra, 332, 333 lock-in amplifier, 332 laser/LED beam, 331 lithographic mode, 330 piezoelectric material, 330 static imaging mode, 329 surface charge screening, 329 Hirota, K., 347 Hohenberg–Kohn–Sham formalism, 8 Hölscher, H., 71 Hooke law, 470 Horkay, F., 201 HSPFM. See High-speed piezo force microscopy Huang, H., 311 Huey, B.D., 329, 531 Human embryonic kidney (HEK) cell membranes, 463 Hurley, D.C., 95 I Imaging modes, SICM applications, 447, 449–450 distance-modulation methods, 446–449 high-resolution imaging, 450–451 simple ion current feedback limitations, 445–446 pipette–sample distance, 443–444 position sensor, 444 topographic image, 444–445 Imry, I., 348 Inelastic electron tunneling spectroscopy (IETS) action spectroscopy, 9–12 localized chemical reactions, 5 Interrupted switching piezoresponse force microscopy (IS-PFM) method capacitor scaling effect, 533–534 film microstructure effect, 534–536 polarization relaxation, 532 time resolution, 531 J Jaasma, M.J., 202 Jesse, S., 214, 215, 491, 498, 510 Johnson–Kendall–Roberts (JKR) model, 201 Johnson, K.L., 201 Jo, J.Y, 535 Jungk, T., 325

Index K Kalinin, S.V., 214, 491 Kalinin, V., 499 KBM averaging method. See Krylov– Bogoliubov–Mitropolsky averaging method Keldysh Green’s function method, 10 Kelvin force microscopy (KFM) alkanethiols, 243 AM-AM approach, 250–252 AM-FM approach, 253–254 aromatic thiols, 243 chlorobenzene-and toluene-cast film, 241 conducting polymer, 241 DPPC monolayer, 242–243 F14H20 adsorbate, graphite, 270–272 F12H20 adsorbate, Si substrate, 269–271 hydrogen-flame annealed Au film, 273–275 hydrophobicity, 240–241 Langmuir–Blodgett layer, 242 organic thin-film transistor, 241 pn-structure, 240 polyfluorene-based photodiode, 241 quantum dots, 241 surface dipoles, 242 topography and surface potential, 275–276 Kendall, K., 201 Kholkin, A.L., 345 Kim, D.J., 532 King, W.P., 204, 219, 223 Kiselev, D.A., 345 Kleemann, W., 345, 348, 350, 365 Klenerman, D., 452 Korchev, Y.E., 445, 447, 452 Krylov–Bogoliubov–Mitropolsky (KBM) averaging method, 236–237 Kuhn’s segment, 200 L Lab-on-a-chip technology, 462 l-a-dipalmitoyl-phosphatidycholine, 89–90 Lambda-digest deoxyribonucleic acid, 134 Langmuir–Blodgett technique, 89 Laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS), 183 Laser ablation inductively coupled plasma (LA-ICP) technique, 187–188 Laser deflection method, 389, 390 Laser desorption atmospheric pressure chemical ionization (LD-APCI), 188–189

547 Laser desorption electrospray ionization (LD-ESI), 190–191 Laser/LED beam, 331 Lee, J., 223 LIA. See Lock-in amplifiers Li, J-F., 369 Lin, D.C., 201, 202 Lin, W.K., 375 Lithium niobate (LiNbO3) diffuse domain wall, 316–318 disc-plane model, 312 domain interaction width, 324–325 FEM modeling, 318, 319 frequency dispersion, 305, 308 vs. LiTaO3, 323 phase-field simulation, 303 piezoelectric coupling simulation, 313 piezoelectric vertical and lateral response, 313, 315 point group symmetry, 306–307 SNDM, 320–322 surface deformation, 313, 314 surface potential, 316 “twist” mode, 322 Lithium tantalate (LiTaO3) domain interaction width, 324–325 FEM modeling, 318, 319 vs. LiNbO3, 323 phase-field simulation, 303 SNDM, 320–322 Lock-in amplifiers (LIA), 175, 250, 255 M Magonov, S., 233 Majumdar, A., 206–208, 220 Maksymovych, P., 3, 23, 28 Malkin, A.J., 39 Mann, S.A., 446 Ma, S.K., 348 Matrix-assisted laser desorption ionization (MALDI), 181–182 Maugis approximation, 77 Maugis, D., 201 Ma, W., 498 Mechanical crosstalk buckling, 391 optical amplification factor, 392 piezoelectric response, 392 vertical and lateral displacement, 390–391 Membrane electromechanics (MEM) acetylcholine receptor transfected HEK, 477, 478 AFM cantilevers, 466

548 Membrane electromechanics (MEM) (cont.) Axopatch 1-B amplifier, 466 Boltzmann function, 474, 475 cell culture and transfection, 465 cell stiffness, AFM measurement, 467 channel cDNA, 465 channel density, 484 data selection and analysis, 469 flux effects, 484–486 force clamp (see Force clamp) ion channels, 463 lab-on-a-chip technology, 462 local area measurement techniques, 461 modified Quesant software, 466 patch-clamping, 461 PZT, 466 recording solutions, 465 ShHEK (see Shaker-transfected human embryonic kidney) symmetric K+ flux hypothesis, 481 IL mutant, 481–483 Nernst equation, 481 physiologic solutions, 481, 482 VC-ATM (see Voltage-clamp atomic force microscope) voltage-gated ion channels, 462 voltage ladder protocols, 474, 475 wild-type HEK, 474, 476–477 Mesoscopic polarization switching mechanisms, 503 Microbial and cellular systems. See Bacillus and Clostridium spores Mii, T., 10 Mirman, B., 223, 510 Morozovska, A.N., 221, 304, 316, 499 Mosbacher, J., 463, 464, 484 Muller, V.M., 201 Multi-frequency atomic force microscopy adhesive and elastic force, 125 attractive and repulsive mode, 131–132 band excitation technique, 126 bimodal magnetic force microscopy, 140, 141 cantilever resonant modes and boundary conditions acoustic and ultrasonic force microscopy, 129 contact resonance, 128, 129 feedback system, 127 “fundamental” mode, 127 Hertz model, 130 mechanical properties, 129 phase and amplitude curve, 130–131 Si and SiN cantilever, 128, 129

Index contact stiffness, 127, 128 DART dual frequency signal, 143 model parameters, 143–144 multiple frequency measurement, 143 tip–surface junction, 143 topography, contact resonance and Q factor, 144–146 energy analysis feedback loop, 136 power dissipation, 135 rat tail tendon, collagen fiber, 136–137 second mode phase, 137 tip–surface interaction, 135 two-dimensional histograms, 137, 138 feedback, 132–133 frequency tracking, 142–143 higher mode resonance, 131 high resolution, low force bimodal imaging, 138–139 intermittent-and non-contact bimodal results color-coded force curve, 134 HOPG, 133 lambda-digest deoxyribonucleic acid, 134 Olympus Bio-Lever, 134–135 second resonance mode phase, 134 surface topography, 133–134 Z-feedback loop, 133 intermodulation AFM, 146–147 multi-frequency nano-scale measurement, 147 nonlinear tip–surface interactions, 127 piezo coefficient, 142 SHO model, 127 spring constant, 142 steady-state amplitude, 140 STM, 125 tip–sample interaction, 126 transfer function, 142 van der Waals interactions, 126–127 N Nakamura, Y., 22 Nanoscale chemical imaging AFM/MS technique, 182, 183 AP-MS (see Atmospheric pressure mass spectrometry) chemical analysis, 181 EI source, 182 GC-MS, 183 LA-ICP-MS, 183 laser desorption/ablation, 182

Index MALDI, 181–182 NF-ICP-MS, 183 NSOM fiber, 182 quadruple mass spectrometer, 182 surface sampling, 181 TOF mass spectrometer, 182 Nath, R., 329, 336 Near-field inductively coupled plasma mass spectrometry (NF-ICP-MS), 183 Near-field laser ablation mass spectrometry, 182 Near field scanning optical microscopy (NSOM), 193 Nelson, B.A., 204, 219 Nikiforov, M.P., 199, 214 Noh, T.W., 505, 535 Noncontact atomic force microscopy (NC-AFM), 75 Novak, P., 446 O Ohara, M., 27 Okawa, Y., 28 Okuda, T., 416 Oliver–Pharr method, 213 Oliver, W.C., 202 Optical lever method, 388 Oulevey, F., 208, 209 Ovchinnikova, O.S., 181 P Palmer, R.E., 26 PBS buffer, 136 PCA. See Principle component analysis Persson, B., 20 Persson, B.N.J., 11 Peter, F., 390, 395, 398 PET surface. See Polyethylene-terephthalate surface Pharr, G.M., 202 Piezoelectric tensor, 386–387 Piezoresponse force microscopy (PFM) advantage, 530 amplitude vs. frequency response, LiNbO3, 255 band excitation (see Band excitation piezoresponse force microscopy) BEPS, 520–523 bow waves, 401 calibration, background subtraction and frequency dispersion displacement detection sensitivity, 305 frequency dispersion, 305, 308

549 lithium niobate, 305, 307 pure electromechanical signal, 305, 306 tip shape and contact geometry, 306, 308 cantilever displacements, 247 capacitor scaling effect, 533–534 conductive probe and cantilever, 246 contact mode imaging, 493 data storage system, 248 device asymmetries beam asymmetry, 394 mechanical crosstalk (see Mechanical crosstalk) optical amplification, 389–390 optical crosstalk, photodiode, 392–393 optical lever method, 388 diffuse domain wall decoupling approximation, 316 effective width, 317 LiNbO3, 317–318 piezoelectric tensor coefficients, 316 resolution function approach, 316 surface displacement, 316–317 dual AC resonance tracking, 512–514 dynamic behavior, cantilever, 496 electromechanical response measurement, 495 electrostatic contribution, 249 FEM (see Finite element method) ferroelectric capacitor imaging, 530 ferroelectric materials, 245, 529 ferroelectric wall width antiparallel domain wall, 304–305 direct effect, 302 Ising type wall, 302–303 Neumann’s law, 302 phase-field simulation, 303 piezoelectric effect, 302 SNDM, 304 TEM, 303 FE-SEM, 301 film microstructure effect, 534–536 hysteresis loop, 258 in-plane shear deformation, 247 lateral PFM congruent LiTaO3 and LiNbO3, 323 domain interaction width, 324–325 surface deformation vs. tip size and tip geometry, 323–324 “twist” mode, 322–323 lead zirconate titanate ceramic, 495 LIAs, 255 longitudinal piezoelectric constant, 247 matrix equation, 246 multivariate data analysis methods, 524

550 Piezoresponse force microscopy (PFM) (cont.) nanoscale electromechanics, 491–493 nanoscale piezoelectric coefficient, 301 nonlocal piezoelectric response, 387–389 orthogonal coordinate system, 246 PbTiO3 nanoparticles, 248 piezoelectric coefficient, 249 piezoelectric materials, 245 polarization and piezoelectric tensor orientation, 386–387 polarization dynamics, 494 polarization imaging, 529 PZT capacitors, 537, 538 quantitative measurement, 301 quantitative piezoelectric constant, 249–250 sample asymmetries conductivity, 400 electric field vector, 395 ferroelectric nanoisland, 395, 396 induced topography, 395–397 local heterogeneities, 397–399 Neuman’s principle, 394 polarization vector, 395, 396 substrate signatures, 399 topography crosstalk, 397 signal contributions, 386 vs. SPM, 493, 494 SrBiTaO, 256, 257 SS-PFM (see Switching spectroscopypiezoresponse force microscopy) stochastic nucleation events, 532 tetragonal thin film, 385 time resolution, 531 tip curvature, 400–401 tip size dependence calibrated magnitude, 308, 309 LiNbO3, 306–307 wall width, 308–310 topography image, 257, 258 transverse piezoelectric constant, 247 vector PFM, 248 vertical experiments, simulation and theory effective piezoelectric coefficient, 309, 318 LiNbO3, 319–320 LiTaO3, 318, 319 SNDM, 320–322 wall width, 309, 318–319 X-Y-vector components, 255 Piezo-transducer (PZT), 466 Pirc, R., 349 Plomp, M., 39 PMMA. See Poly(methyl methacrylate)

Index PMMA-PIB-PMMA. See Polymethylmethacrylatepolyisobutylenepolymethylmethacrylate Pohl, D.W., 433 Polar nanometer-size region (PNR) dipole moments, 348 ferroelectric phase, 351 “frozen” PNR, 360, 366 piezoelectric effect, 345 piezoresponse signal, 359 polar nanoregions, 346–349 quadratic electro-optic effect, 345 reorientation and growth, prevention, 369 SBN61, 359 SRBRF model, 350 Pollock, H.M., 218 Poly(methyl methacrylate) (PMMA), 109, 174–178 Polyethylene-terephthalate (PET) surface, 204–205 Polymer blends effective reduced elastic modulus, 175, 177 glass transition temperature, 174–175, 178 LIA, 175 qualitative stiffness map, 175–178 time-varying force measurement, 175, 177 Polymethylmethacrylate-polyisobutylenepolymethylmethacrylate (PMMAPIB-PMMA), 169–171 Polypropylene matrix, 169–171 Polystyrene (PS), 174–178 Principle component analysis (PCA), 518–520 Proksch, R., 125, 199, 491 Q Qiu, X.H., 15 Quate, C.F., 210 R Radmacher, M., 91 Rao, B.V., 15 Rayleigh scattering, 306, 307 Reading, M., 185 Relaxor ferroelectrics capacitor and actuator applications, 345 PFM cubic relaxors (see Cubic relaxors) uniaxial relaxors (see Uniaxial relaxors, SrxBa1−xNb2O6) polar nanoregions Burns temperature, 346

Index high-resolution neutron diffuse scattering, 346 low-energy transversal optic mode, 347 macroscopic ferroelectric transition, 347 paraelectric state, 346 TO phonon condensation model, 347 PZN-PT single crystals, 347 random field model, 348 uniform phase-shift model, 348 polycrystalline materials (see Ceramics) temperature evolution Brillouin scattering, 351 dielectric permittivity, 349 ergodic relaxor (ER) phase, 348 linear and nonlinear dielectric susceptibilities, 350 non-ergodic relaxor state, 349 spherical cluster glass, 350 SRBRF model, 349 Vogel-Fulcher law, 349 thin films Kohlrausch-Williams-Watt-type dependence, 378–379 mechanical clamping effect, 377–378 polycrystalline relaxor film, 376 topography and piezoelectric images, 377 Rheinlaender, J., 433 Ricinschi, D., 498 Roberts, A.D., 201 Rodriguez, B.J., 140, 491, 512 Rodriguez, T.R., 79, 126 Roelofs, A., 498 Rohrer, H., 409, 433 Rose, V., 405 Ruediger, A., 385 S Sachs, F., 461 Sader method, 115 Sahin, O., 153 Sainoo, Y., 10 Saito, A., 416 Salehi-Khojin, A., 510 Samara, G.A., 374 Saya, Y., 498 Scanning electrochemical force microscope (SECM), 433 Scanning ion conductance microscopy (SICM) AFM, 433, 452–453 complementary AFM distance control, 434 elasticity measurements, 455–457 electrolyte-filled nanopipette, 434 feedback loop, 434

551 half-cell electrodes, 436–437 imaging modes (see Imaging modes, SICM) ionic conductivity, 437–438 ionic currents analytical model, 439–440 vs. distance curves, 442–443, 445 FEM (see Finite element method) pipette-sample system, 438–439 nanopipette probes, 435–436 optical microscopy, 452 patch-clamp current amplifier, 434 piezo-electric scanner, 435 shear force microscopy, 453–454 Scanning near-field optical microscope (SNOM), 433 Scanning nonlinear dielectric microscopy (SNDM), 320–322 Scanning thermal expansion microscopy (SThEM), 206–208 Scanning tunneling microscopy (STM) atomic-scale resolution, 3 CO molecules, 3, 4 excitation rate parasitic capacitance, 6 photochemical process, 7 Poisson distribution, 7 statistical analysis, 8 vibrational relaxation time, 6 excited electronic and anionic state reactions azobenzene derivative isomerization, 12 dipole moment, 14 ground electronic state, 12 image charge stabilization, 13 inelastic excitation mechanisms, 12–13 kinetic measurements, 12 NiAl surface, 14–15 Kondo resonance, 3, 4 molecular aggregate, collective reactivity diacetylene, self-assembled monolayer, 28–30 electron–hole pair excitation, 28 hydrogen bond exchange, 31–32 LUMO, 30 self-assembled diacetylene compound, 28 molecular junction, delocalized excitation attenuation function, CH3SSCH3, 25 chlorine atom diffusion, 22 C60 polymers, 22, 23 electron-induced dissociation, 23–25 radial probability, 25–26 silicon surface, 23 statistical analysis, 22

552 Scanning tunneling microscopy (STM) (cont.) surface-resonance scattering techniques, 26 total reaction rate, 25 single-and multiple-electron process anharmonic coupling, 16, 17 electron-vibration coupling, 19 excitation/coupling mechanism, 20 Hamiltonian operator, 19 high-frequency mode decay, 20, 21 inelastic tunneling current, 16 kinetic analysis, 20–21 LUMO and HOMO orbitals, 19 molecular desorption, 16, 18 Pauli master equations, 17, 20 RC deexcitation and excitation, 19 reaction rate, 20, 21 single-quanta population, 22 thermal fluctuations, 17 total relaxation time and reaction time, 20 single-molecule reaction measurement, 5–6 rate constant, 14–15 tip effects and field-induced manipulation, 26–27 tunneling electrons, 3 vibrationally mediated reactions and action spectroscopy acetylene molecules, 8, 9 adsorbate-induced resonance, 11 anharmonic coupling, 11 average excitation time, 8 density functional methods, 8 IETS, 9–10 inelastic losses, 12 Keldysh Green’s function method, 10 non-equilibrium Green’s function, 9 tunneling bias function, 10–11 Schäffer, T.E., 433 Schmutz, J-E., 71 Schwarz, U.D., 71 Scrymgeour, D.A., 304, 310, 323, 325, 395 Seal, K., 491, 506 Secondary ionization mass spectrometry (SIMS). See Atmospheric pressure thermal desorption Setter, N., 505 Shaker channel, 462, 463 Shaker-transfected human embryonic kidney (ShHEK) Lippman tension model, 477 NMDG and ion replacement, 479–481 nonactivating voltage steps, 478

Index Shao, Z., 447, 452 Shen, S., 220 Shevchuk, A.I., 450 Shiea, J., 190 Shvartsman, V.V., 345, 373, 378 SICM. See Scanning ion conductance microscopy Silicon heater, 219 Simple harmonic oscillator (SHO) model, 127 Single tapping-mode scan peak attractive force, 171 tapping-mode operation, 169 tip–sample energy dissipation, 171–172 TPV, 169–171 triblock copolymer, 169–171 Sloan, P.A., 26 Smolenskii, G.A., 347 SNDM. See Scanning nonlinear dielectric microscopy Soergel, E., 393, 509 So, Y.W, 535 Spherical random bond random field (SRBRF) model, 349 Stephan–Boltzmann law, 220 SThEM. See Scanning thermal expansion microscopy Stipe, B., 8, 9, 23 STM. See Scanning tunneling microscopy Stohr, J., 409 Streiffer, S.K., 405 Sun, Y.J., 202 Switching spectroscopy-piezoresponse force microscopy (SS-PFM) Barkhausen jumps, 497 capacitors applications, 506 domain structure evolution, 505 hysteretic behavior, 507 local ferroelectric hysteresis loops, 504 nonvolatile memory technologies, 505 Preisach-type models, 507 switchable polarization, 507, 508 ferroelectric nanocapacitors, 498 films BFO films, 501, 502 dielectric breakdown, 500 force-distance spectroscopy, 500 image acquisition time, 499 mesoscopic polarization switching mechanisms, 503 multiferroic heterostructures, 501, 503 surface topography and nucleation bias map, 503–504 free surface/top electrode, 496–497

Index Ginzburg–Landau theory, 499 in-field hysteresis loop measurements, 497 phenomenological characteristics, 498 piezoelectric hysteresis loop measurement, 496–497 Synchrotron X-ray-enhanced scanning tunneling microscopy (SXSTM) absorption-induced tunnel current, 412 element-sensitive image, 413 Ge absorption edge, 416 imaging mode, 413, 414 Ni dots, 416 NiFe rings, topography scan, 428–429 quantum mechanical tunneling regime, 425 sound tunneling conditions, 426 spectroscopy mode, 413, 414 X-ray absorption near-edge structure, 415 X-ray-enhanced scanning probe, 415 X-ray illumination, 426–428 z-piezo voltage, 425 Synge, Edward, 195 T TA-AFAM method. See Thermally assisted atomic force acoustic microscopy method Tabor, 201 Tagantsev, A.K., 349 Tapping cantilever elastic modulus, 169 frequency response, torsional mode, 163 nanoimaging method, 154 sinusoidal trajectory, 155–156 spring constant and quality factor, 157 vibration amplitude and phase response, 160 Tapping-mode atomic force microscopy damped harmonic oscillator, 156 displacement response, 159 quadrant photo detector, 160 sinusoidal trajectory, 155–156 tip–sample force, 157–159 TEM. See Transmission electron microscopy Thermally assisted atomic force acoustic microscopy (TA-AFAM) method, 208–209 Thermo-mechanical properties AFM BE-nanoTA and Z-therm (see Band Excitation-NanoTA and Z-therm method) force–distance curve, 202 SThEM, 206–208

553 TA-AFAM method, 208–209 TTM (see Transition temperature microscopy) cantilever probe, 217 constant temperature mode, 218 contact mechanics model elastic media, 221–222 tip–surface contact resonance, 222–223 diffusive heat transfer, 220 high spatial resolution, nanoindentation data analysis, 200 DMT theory, 201 elastic and plastic deformation, 200 Hertz model, 201 JKR model, 201 Mooney–Rivlin strain energy function, 201 NI and AFM data, 202 tip–surface interactions, 200–201 Young’s modulus, 202 localized mechanical analysis, 199–200 radiative heat transfer, 220 resistive probe, 217 silicon heater, 219 solid–solid contact, 220 surface phonon polaritons, 220 technique development prospects and limitations calorimetric measurements, 225 displacement measurements, 224 local thermal analysis, 223–224 phase transition, polymers, 225 polymer linear expansion coefficient, 225 tip–surface heat transfer, 220 variable temperature mode, 218 Thermoplastic vulcanizate (TPV) AFM, 262, 264 single tapping-mode scan, 169–172 Thin film ferroelectric capacitors domain switching kinetics capacitor scaling effect, 533–534 film microstructure effect, 534–536 IS-PFM method, 531 mechanical stress effect, 536–539 PFM imaging, 530 polarization reversal dynamics, 530 PZT capacitors, 532 scanning force microscopy, 529 switching characteristic, 531 Tian, L., 301, 304, 306, 310–312 Tikhodeev, S.G., 17, 31 Time-of-flight (TOF) mass spectrometer, 182

554 Tip–sample force AM mode theory amplitude vs. frequency curve, 80–82 Fourier series, 78–79 oscillation amplitude, 80–81 resonance frequency, 82 steady-state solutions, 78 FM mode theory energy dissipation, 87 Fourier series, 86 frequency shift, 86 “large amplitude approximation,” 86 normalized frequency shift, 87 sinusoidal cantilever oscillation, 85 Tip–sample interaction force in air, 77–78 amplitude and phase vs. distance curve, 84, 85 AM vs. FM mode adhesion force, 89 contact stiffness, 89 silicon cantilevers, 87 spectroscopy measurements, 88 biological sample mapping, 89–91 cantilever oscillation, 82 energy dissipation, 83, 85 fourth-order Runge–Kutta method, 84 “large amplitude approximation,” 83 tip–sample distance, 82 Topographic cross-talk, 508–511 Toporov, Y.P., 201 Torsional harmonic cantilever damped harmonic oscillator, 162–163 flexural response, 161 Fourier transform, 163 frequency response, torsional mode, 163 operation, 160, 161 vibration spectra, 161–162 TPV. See Thermoplastic vulcanizate Transition temperature microscopy (TTM) glass transition temperature, 203 localized thermal expansion, 203 PET surface, 204–205 probe deflection vs. temperature curve, 203 thermal noise, 206 tip–sample junction, thermal coupling, 204 Transmission electron microscopy (TEM), 172–174, 303 Tsuji, K., 415 U Ueba, H., 11, 17, 19–21, 31 Ultrasonic force microscopy (UFM) method, 96, 117

Index Uniaxial relaxors, SrxBa1−xNb2O6 polar structures domain size distributions, 354–355 maze-type domain pattern, 356 Monte Carlo simulations, RFIM system, 355, 356 SBN40 single crystals, 356–357 trimodal false color code, 353, 354 structural considerations electric multipole moments, 353 order-disorder pseudospin model, 351 SBN75 single crystals, 352–353 tetragonal tungsten-bronze-type structure, 351 three-dimensional random-field Ising model, 351 temperature evolution, SBN61 Arrhenius law, 361 autocorrelation function analysis, 358–359 Ce-doped single crystal, 360–361 domain radius, 361–362 quasi-static PNR, 360 quasi-static regions, 358 3D-RFIM systems, 360 spatial and temperature dependence, 359 V Van der Waals force, 77 Varesi, J., 206–208 Vastola, F.J., 186 Vasudevarao, A.N., 304 Vegetative cells dormant spore populations, 63 etch pits, 59 lytic enzymes, 62 parallel/orthogonal model, 63 peptidoglycan structure, 61 pore structures, 64 surface structure, 60, 61 Vertes, A., 190–192 Viehland, D., 369 Villain, J., 361 Vogel–Fulcher law, 349 Voltage-clamp atomic force microscope (VC-ATM) data acquisition, 467–468 HEK cell membranes, 463 inverse flexoelectric effect, 463 Lippman tension, 464 mechanical isolation, 467 Olympus IX70 inverted microscope, 467 Voltage ladder protocols, 474, 475

Index W Walle, H., 413 Wickramasinghe, H.K., 217 Wollaston probe, 217–218 Wu, S., 233 Wu, Y., 409 X X-ray magnetic circular dichroism (XMCD), 409 X-rays and scanning tunneling microscopy chemical sensitivity, photocurrent, 411, 412 Cu film, topography image, 426, 427 electron tunneling, 409, 410 element-specific microscopy, 410 insulator-coated smart tips, 417–419 monochromatic X-rays, 410–411 morphology and electronic structure, 409 nanoscale structures, 405 photoejected electrons, 412–413 photoelectron detection Cu/NiFe, total electron yield, 420, 421 fluoroelastomer damping, 419 NiFe film magnetization, 423, 424 photocurrent, 419, 420

555 probing diameter, definition, 424 sample-current method, 419 tip and sample peak ratio, 422, 423 tip current spectra, 420–422 tip/sample separation, 425 spatial resolution, 406 SXSTM (see Synchrotron X-ray-enhanced scanning tunneling microscopy) X-ray interactions Auger cascade, 407–408 compton scattering, 406, 407 elastic scattering, 406 photoelectric effect, 406–407 X-ray circular dichroism, 408, 409 Xu, G., 347, 350 Xu, J.B., 220 Y Ye, Z-G., 348 Yin, F.C.P., 202 Z Zenobi, R., 182, 193, 195 Zhang, P.C., 464, 474