Advanced Transmission Electron Microscopy

Jian Min Zuo • John C.H. Spence

Advanced TransmissionElectron MicroscopyImaging and Diffraction in Nanoscience

123

Jian Min ZuoFrederick-Seitz Materials ResearchLaboratory, Department of MaterialsScience and Engineering

University of Illinois, Urbana-ChampaignUrbana, ILUSA

John C.H. SpenceDepartment of PhysicsArizona State UniversityTempe, AZUSA

ISBN 978-1-4939-6605-9 ISBN 978-1-4939-6607-3 (eBook)DOI 10.1007/978-1-4939-6607-3

Library of Congress Control Number: 2016947937

© Springer Science+Business Media New York 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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To my family, Xiurong, Yuan and Ling,and in memory of my parents,Yijun and Qiaozhen

—Jian Min Zuo

To my family, in gratitude

—John C.H. Spence

Preface

This book is written and organized around three topics: electron diffraction, electronoptics, and electron crystallography. It is intended as an advanced undergraduate-or graduate-level text in support of course materials in materials science, physics, orchemistry departments. High-resolution transmission electron microscope imagingand scanning transmission electron microscopy are treated as major applications ofelectron optics, as well as powerful electron crystallographic techniques for struc-ture determination. The emphasis here is on the fundamentals and applications ofelectron diffraction and imaging in materials research, especially in the study ofnanoscience. For this purpose, we have included theory for electron wave propa-gation, electron diffraction and imaging, and a detailed treatment of electron optics,aberration correction, and instrument techniques, on a level that can be followed bya materials science or physics graduate student. For crystallography, we haveemphasized the fundamentals of symmetry, structure and bonding, diffuse scatter-ing, imaging of defects, strain measurement, and determination of nanostructures.Structure determination of large crystals, including polymeric and biological sam-ples, is not discussed specifically in this book, although the electron diffraction andimaging theories presented here and instrumental techniques apply equally to thesetopics.

Transmission electron microscopy (TEM) traditionally refers to electrondiffraction and imaging techniques that are enabled by a transmission electronmicroscope (with the same TEM acronym). Scanning transmission electronmicroscopy (STEM) embodies a separate set of techniques. The development ofmodern TEMs that function as both TEM and STEM has brought them together, ascomplementary techniques, often abbreviated as S/TEM. For this reason, we havesimply used TEM in the book’s title. STEM, more than TEM, is associated withpowerful analytical techniques, such as electron energy loss spectroscopy andenergy-dispersive X-ray spectroscopy. This aspect of TEM is not covered here, andreaders are referred to the excellent books on these subjects by Egerton (2011),Hawkes and Spence (2007), and Pennycook and Nellist (2011).

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The materials included here come from multiple sources. Firstly, we haveupdated our previous book on “Electron Microdiffraction” (Plenum, New York,1992, by J.C.H. Spence and J.M. Zuo). The previous Chaps. 2–4, 7, and 9 are nowparts of Chaps. 3, 5, 12, 13, and 10, respectively. The new Chap. 10 on instrumentaltechniques also incorporates the previous Chap. 6. The previous Chap. 8 is nowseparated into Chap. 14, which discusses atomic-resolution imaging and Chap. 15 onthe characterization of defects. Secondly, we have incorporated much new teachingmaterial throughout the book, such as waves and wave properties (Chap. 2), kine-matical theory of electron diffraction (Chap. 4), electron optics (Chaps. 6 and 7),diffuse scattering (Chap. 13), and electron imaging (Chaps. 14 and 15). This materialis based on the lectures given to graduate students at University of Illinois,Urbana-Champaign in two courses: diffraction physics and advanced electronmicroscopy. The writing of Chaps. 6 and 7 has benefitted from the special invitedlectures given by Prof. Harald Rose in 2011 to the advanced electron microscopyclass.

In writing this book, we have also relied on the original research work by manygraduate students, post-docs, and our collaborators. To them, we owe specialthanks, especially to Profs. Michael O’Keeffe, Ragnvald Hoier (1938–2009),Miyoung Kim, Randi Holmestad, Jerome Pacuad, Jean-Paul Morniroli, SyoMatsumura, Yoshitsugu Tomokiyo and Drs. Bin Jiang, Weijie Huang, Jing Tao,Jiong Zhang, Min Gao, Celik Ayten, Shankar Sivaramakrishnan, Amish Shah,Ke Ran, Wenpei Gao, and Honggyu Kim. The work at University of Illinois wasfunded by the Department of Energy, Basics Science and Division of MaterialsResearch, National Science Foundation. Especially, JMZ wishes to thank Dr. JaneZhu at the Department of Energy for the support of the electron nanocrystallog-raphy project.

On reading the literature, one is struck by the enormous variety of applications ofTEM/STEM. These include studies of various defects, grain boundaries andinterfaces in a broad range of materials, analyses of the symmetry changes whichaccompany phase transitions, polarization and charge ordering includingcharge-density waves in layered structures, accurate mapping of the distribution ofvalence electrons in crystals, phase identification and strain measurement arounddefects, precipitates and interfaces in alloys or semiconductors, in addition to thecharacterization of all sorts of nanostructures. To review all this work, published ina vast number of papers, and draw out its implications for materials physics wouldbe a Herculean task. Our aim has been a limited one, to explain the principles ofTEM, to provide the theory in a consistent format and to convey enough under-standing to students and researchers to let them get started with modern TEM formaterials characterization. Thus, to experts in the field, some examples in this bookmay seem somewhat oversimplified. Also, we have cited references that are directlyrelated to our discussions. We offer our apologies to many of our colleagues whoseworks were not covered or cited here. With regret, for reasons of space, we have notbeen able to include the topics of structure determination (see Zou et al. 2011),electron tomography, or coherent diffractive imaging.

viii Preface

Several chapters were written during the sabbatical stay of JMZ at CEA,Grenoble, France, in the fall of 2014. He is therefore grateful to Drs. Jean-LucRouviere and Alain Fontaine for their hospitality and also to the NanoscienceFoundation, Grenoble, for the Chair of Excellence position which made his visitpossible.

The study of electron diffraction and imaging can be significantly helped bycomputer simulations. For this purpose, we have made available of computerprograms listed in the “Electron Microdiffraction” book on the website http://cbed.matse.illinois.edu/, as well as links to other online resources.

Urbana, USA Jian Min ZuoTempe, USA John C.H. Spence

ForMemRS

References

Egerton, R.F.: Electron Energy-Loss Spectroscopy in the Electron Microscope, 2nd edn. Springer,New York (2011)

Pennycook, S., Nellist, P. (eds.): Scanning Transmission Electron Microscopy, Imaging andAnalysis. Springer, New York (2011)

Rose, H.: Electron Optics. University of Illinois, Urbana-Champaign (2011) http://cbed.matse.illinois.edu/download/Rose_optics_of_magnetic_lenses.pdf

Hawkes, P.W. and Spence, J.C.H. (eds) Science of Microscopy. Springer, New York (2007) , andSpringer Handbook of Microscopy, (2017) to follow.

Spence, J.C.H., Zuo, J.M.: Electron Microdiffraction. Plenum, New York (1992)Zou, X., Hovmöller, S., Oleynikov, P.: Electron crystallography, electron microscopy and electron

diffraction. Oxford University Press (2011)

Preface ix

Preface to “Electron Microdiffraction,”Plenum, New York, 1992

Much of this book was written during a sabbatical visit by J.C.H.S. to the MaxPlanck Institute in Stuttgart during 1991. We are therefore grateful to Profs.M. Ruhle and A. Seeger for acting as hosts during this time and to the Alexandervon Humboldt Foundation for the Senior Scientist Award which made this visitpossible. The Ph.D. work of one of us (J.M.Z.) has also provided much of thebackground to the book, together with our recent papers with various collaborators.Of these, perhaps the most important stimulus to our work on convergent beamelectron diffraction resulted from a visit to the National Science Foundation’sElectron Microscopy Facility at Arizona State University by Prof. R. Hoier in 1988and from a return visit to Trondheim by J.C.H.S. in 1990. We are therefore par-ticularly grateful to Prof. R. Hoier and his students and coworkers for theirencouragement and collaboration. At ASU, we owe a particular debt of gratitude toProf. M. O’Keeffe for his encouragement. The depth of his understanding of crystalstructures and his role as passionate skeptic have frequently been invaluable.Professor John Cowley has also been an invaluable sounding board for ideas andwas responsible for much of the experimental and theoretical work on coherentnanodiffraction. The sections on this topic derive mainly from collaborations byJ.C.H.S. with him in the seventies. Apart from that, we have tried to review theliterature as impartially as possibly and at the same time bring out the underlyingconcepts in a clear and unified manner, so that the book will be useful for graduatestudents. We are particularly grateful to Dr. J.A. Eades for his critical review ofChap. 7. We apologize to those authors whose work may have been overlookedamong the many hundreds of papers. In order to make the book more practicallyuseful, we have included some FORTRAN source listings, together withPOSTSCRIPT code which allows the direct printing of Kikuchi and HOLZ line

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patterns on modern laser printers from the programs. Support from NSF awardDMR-9015867 (“Electron Crystallography”) and the facilities of the NSF-ASUNational Center for High Resolution Electron Microscopy is gratefullyacknowledged.

Tempe, USA John C.H. SpenceJian Min Zuo

xii Preface to “Electron Microdiffraction,” Plenum, New York, 1992

Contents

1 Introduction and Historical Background . . . . . . . . . . . . . . . . . . . . . . 11.1 Electrons and the Electron Wavelength. . . . . . . . . . . . . . . . . . . 11.2 Electron and Sample Interaction . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Transmission Electron Microscope . . . . . . . . . . . . . . . . . . . . . . 51.4 Electron Microdiffraction and STEM . . . . . . . . . . . . . . . . . . . . 71.5 Analytical TEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 A Brief History of Electron Microdiffraction . . . . . . . . . . . . . . 121.7 A Note to Students and Lecturers . . . . . . . . . . . . . . . . . . . . . . . 16References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Electron Waves and Wave Propagation . . . . . . . . . . . . . . . . . . . . . . 192.1 Wave Functions and the Wave Equation . . . . . . . . . . . . . . . . . 192.2 Quantum Mechanical Wave of Electrons and Schrődinger

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 The Principle of Wave Superposition . . . . . . . . . . . . . . . . . . . . 242.4 Amplitude and Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Coherence and Interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Wave Packets and the Uncertainty Principle . . . . . . . . . . . . . . . 282.7 The Gaussian Wave Packet and Its Propagation . . . . . . . . . . . . 312.8 Temporal Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.9 Spatial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.10 Electron Refraction and the Refractive Index . . . . . . . . . . . . . . 382.11 Wave Propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.11.1 Huygens–Fresnel Principle. . . . . . . . . . . . . . . . . . . . . . 392.11.2 Propagation of Plane Wave and Fresnel Zones . . . . . . 412.11.3 Fresnel Diffraction—The Near-Field Small-Angle

Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.11.4 Fraunhofer Diffraction—Far-Field Forward

Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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3 The Geometry of Electron Diffraction Patterns. . . . . . . . . . . . . . . . . 493.1 Bragg’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Laue Diffraction Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Lattice d-Spacing and Crystal, Real,

and Reciprocal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 Transmission Electron Diffraction Patterns . . . . . . . . . . . . . . . . 543.5 Excitation Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6 Kikuchi Lines and Their Geometry (Kinematic) . . . . . . . . . . . . 593.7 Diffraction Pattern Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.8 One-Dimensional (Systematics) CBED . . . . . . . . . . . . . . . . . . . 653.9 Two-Dimensional CBED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.10 High-Order Laue Zone (HOLZ) Lines . . . . . . . . . . . . . . . . . . . 71References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4 Kinematical Theory of Electron Diffraction . . . . . . . . . . . . . . . . . . . 774.1 First-Order Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . 784.2 Weak-Phase-Object Approximation. . . . . . . . . . . . . . . . . . . . . . 804.3 Electron Atomic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4 Kinematical Electron Scattering from a Monoatomic Small

Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 Electron Crystal Structure Factors and the Diffracted Intensity

from a Small Crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.6 Integrated Diffraction Intensity of a Rotating Crystal . . . . . . . . 904.7 Atomic Thermal Vibrations and Effect on Electron

Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.8 Electron Structure Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.9 Electron-Optical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Dynamical Theory of Electron Diffraction for Perfect Crystals . . . . 1015.1 Many-Beam Theory, Wave-Mechanical Approach . . . . . . . . . . 1025.2 Howie–Whelan Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3 Two-Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.4 The Concept of the Dispersion Surface. . . . . . . . . . . . . . . . . . . 1125.5 Absorption and Its Effects in a First-Order Approximation . . . . 1175.6 Many-Beam Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.6.1 Three-Beam Theory and Particular Solutionsfor Centrosymmetric Crystals . . . . . . . . . . . . . . . . . . . 122

5.6.2 Two-Beam Theory with Weak-Beam Effects . . . . . . . . 1255.6.3 Three-Beam Theory—Noncentrosymmetric Crystals

and the Phase Problem . . . . . . . . . . . . . . . . . . . . . . . . 1265.6.4 Dynamic HOLZ Intensities and Positions.

Dispersion Surfaces for HOLZ Lines.How the Bragg Law Dependson Local Composition . . . . . . . . . . . . . . . . . . . . . . . . . 133

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

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6 Electron Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.1 Magnetic Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446.2 Fundamental Rays and Conjugate Planes . . . . . . . . . . . . . . . . . 1496.3 Thin Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.4 Thick Lenses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.4.1 Glaser’s Bell-Shaped Model . . . . . . . . . . . . . . . . . . . . 1546.4.2 Cardinal Points and Planes . . . . . . . . . . . . . . . . . . . . . 1566.4.3 Lens Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586.4.4 Determination of Cardinal Points

from the Electron Path. . . . . . . . . . . . . . . . . . . . . . . . . 1596.5 The Objective Lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.6 The Objective Prefield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

7 Lens Aberrations and Aberration Correction . . . . . . . . . . . . . . . . . . 1657.1 Lens Aberrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1657.2 Aberration Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.3 Multipole Fields and Quadrupole Focal Properties . . . . . . . . . . 1787.4 Aberrations of Hexapole Fields. . . . . . . . . . . . . . . . . . . . . . . . . 1807.5 Cs Correctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

8 Electron Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.1 Source Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938.2 Thermionic Emission Source . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.3 Schottky Emission Source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1988.4 Cold-Field Emission Source . . . . . . . . . . . . . . . . . . . . . . . . . . . 202References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9 Electron Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2079.1 Scintillator–Photomultiplier Detectors . . . . . . . . . . . . . . . . . . . . 2079.2 Characteristics of Point Detectors . . . . . . . . . . . . . . . . . . . . . . . 2109.3 Characteristics of ADF Detectors . . . . . . . . . . . . . . . . . . . . . . . 2119.4 CCD Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2159.5 Detector Characteristics of CCD Cameras. . . . . . . . . . . . . . . . . 2189.6 Direct Detection Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2239.7 Film and Image Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

10 Instrumentation and Experimental Techniques. . . . . . . . . . . . . . . . . 23110.1 Electron Beam Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

10.1.1 Illumination Using Two Condenser Lenses . . . . . . . . . 23210.1.2 The Use of Condenser Minilens . . . . . . . . . . . . . . . . . 23410.1.3 A Third Condenser Lens and Kohler Illumination . . . . 23510.1.4 Beam Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23610.1.5 Coherence and Coherent Current . . . . . . . . . . . . . . . . . 237

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10.2 Probe Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24110.3 Beam Deflectors and Scanning . . . . . . . . . . . . . . . . . . . . . . . . . 24610.4 Electron Diffraction Techniques . . . . . . . . . . . . . . . . . . . . . . . . 250

10.4.1 Selected Area Electron Diffraction (SAED) . . . . . . . . . 25010.4.2 Nanoarea Electron Diffraction (NAED)

and Nanobeam Diffraction (NBD) . . . . . . . . . . . . . . . . 25110.4.3 Convergent-Beam Electron Diffraction (CBED). . . . . . 25210.4.4 Large-Angle Methods . . . . . . . . . . . . . . . . . . . . . . . . . 25510.4.5 Precession Electron Diffraction . . . . . . . . . . . . . . . . . . 25910.4.6 Selected Area Diffraction in STEM . . . . . . . . . . . . . . . 26110.4.7 Scanning Electron Nanodiffraction. . . . . . . . . . . . . . . . 263

10.5 Specimen Holders and Rotation . . . . . . . . . . . . . . . . . . . . . . . . 26610.6 Energy Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

10.6.1 First-Order Focusing by Magnetic Sectors. . . . . . . . . . 27210.6.2 Energy Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27610.6.3 Vertical Focusing Using Fringing Fields . . . . . . . . . . . 27710.6.4 Sector Fields, Paraxial Equations,

and Second-Order Aberrations . . . . . . . . . . . . . . . . . . . 27910.6.5 In-Column Energy Filters . . . . . . . . . . . . . . . . . . . . . . 28210.6.6 Post-Column Imaging Filters . . . . . . . . . . . . . . . . . . . . 28310.6.7 Isochromaticity, Filter Acceptance, and Distortion. . . . 285

10.7 Radiation Effects and Low-Dose Techniques . . . . . . . . . . . . . . 288References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

11 Crystal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29711.1 Symmetry Operations and Symmetry Groups . . . . . . . . . . . . . . 29711.2 Point Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29911.3 Lattice and Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30311.4 Symmetry Operation in Real and Reciprocal Spaces . . . . . . . . 31111.5 Symmetry Determination Using Kinematic Diffraction

Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31211.6 Symmetry Determination by CBED . . . . . . . . . . . . . . . . . . . . . 315

11.6.1 Point Symmetry in Dynamic Diffraction . . . . . . . . . . . 31711.6.2 Point Group Determination by CBED . . . . . . . . . . . . . 323

11.7 Bravais Lattice Determination. . . . . . . . . . . . . . . . . . . . . . . . . . 32811.8 Space Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32911.9 Quantification of CBED Pattern Symmetry and Symmetry

Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33511.10 Symmetry and Polarization in Ferroelectric Crystals . . . . . . . . . 339References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

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12 Crystal Structure and Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34712.1 Description of Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . 34712.2 Common Structure Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35012.3 Chemical Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

12.3.1 Bonding of a Diatomic Molecule. . . . . . . . . . . . . . . . . 35812.3.2 Atomic Sizes and Electronegativity . . . . . . . . . . . . . . . 36012.3.3 Bonding in Infinite Crystals. . . . . . . . . . . . . . . . . . . . . 36112.3.4 Types of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36212.3.5 Characteristics of Bonds . . . . . . . . . . . . . . . . . . . . . . . 36312.3.6 Charge Density as the Ground-State Property

in Density Functional Theory . . . . . . . . . . . . . . . . . . . 36512.4 Experimental Measurement of Charge Density . . . . . . . . . . . . . 366

12.4.1 X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36612.4.2 Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 36812.4.3 Combined Electron and X-Ray Analysis . . . . . . . . . . . 37412.4.4 Multipole Expansion of Electron Density . . . . . . . . . . 374

12.5 Crystal Electron Density and Bonding . . . . . . . . . . . . . . . . . . . 37612.5.1 Covalent Bonding in Diamond Structure . . . . . . . . . . . 37612.5.2 Ionic Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38112.5.3 Metallic Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38812.5.4 Transition Metal Oxides . . . . . . . . . . . . . . . . . . . . . . . 391

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

13 Diffuse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40313.1 Electron Diffuse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40413.2 Thermal Diffuse Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40613.3 Diffuse Scattering from Small Lattice Defects . . . . . . . . . . . . . 41113.4 Scattering by Solid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 41813.5 Modulated Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42513.6 Multiple Scattering Effects in Diffuse Scattering . . . . . . . . . . . . 430References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

14 Atomic Resolution Electron Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 44114.1 Introduction and a Brief History . . . . . . . . . . . . . . . . . . . . . . . . 44114.2 Abbe’s Theory of Coherent Imaging. . . . . . . . . . . . . . . . . . . . . 44314.3 Coherent Imaging in an Ideal Lens . . . . . . . . . . . . . . . . . . . . . . 44514.4 Coherent Imaging in a Real Lens . . . . . . . . . . . . . . . . . . . . . . . 44814.5 Linear Imaging Theory and Contrast Transfer Function

(CTF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44914.6 The Effects of Electron Energy Spread

and Partial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45514.7 Electron Probes for High-Resolution STEM and Analysis . . . . 45914.8 Probe Size and Resolution in Bright-Field STEM. . . . . . . . . . . 46114.9 Ronchigrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46414.10 Coherence in STEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

Contents xvii

14.11 HAADF-STEM (Z-Contrast) Imaging . . . . . . . . . . . . . . . . . . . . 47614.12 Aberration-Corrected STEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 47814.13 Three-Dimensional Imaging in STEM . . . . . . . . . . . . . . . . . . . 48114.14 Channeling, Bound States, and Atomic Strings. . . . . . . . . . . . . 48514.15 Image Simulation Using the Multislice Method . . . . . . . . . . . . 489References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496

15 Imaging and Characterization of Crystal Defects . . . . . . . . . . . . . . . 50115.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50115.2 Atomic Displacements, Strain, and Stress . . . . . . . . . . . . . . . . . 50515.3 Diffraction Contrast Imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . 510

15.3.1 Column Approximation . . . . . . . . . . . . . . . . . . . . . . . . 51215.3.2 Thickness Fringes and Bend Contours. . . . . . . . . . . . . 51315.3.3 Diffraction Contrast from Lattice Defects . . . . . . . . . . 51515.3.4 Weak-Beam Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 525

15.4 Howie-Basinski Equations and the Dynamical Theoryof Electron Diffraction from Crystal Defects. . . . . . . . . . . . . . . 529

15.5 Defect Analysis Using LACBED, Defocused CBED,and CBIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

15.6 Atomic Structure Determination of Defectsfrom High-Resolution Electron Images . . . . . . . . . . . . . . . . . . . 53815.6.1 Atomic Structure of Dislocation Cores . . . . . . . . . . . . 53915.6.2 Grain Boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550

16 Strain Measurements and Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 55316.1 Local Lattice Parameters and Strain . . . . . . . . . . . . . . . . . . . . . 55316.2 Electron Beam-Based Strain Measurement Techniques . . . . . . . 55516.3 Limitations of Electron Beam Techniques . . . . . . . . . . . . . . . . 55916.4 Electron Diffraction-Based Strain Measurement Techniques

and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56016.4.1 Nanobeam Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 56016.4.2 Diffraction Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 56016.4.3 Strain Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56216.4.4 Convergent Beam Electron Diffraction (CBED) . . . . . 56516.4.5 3D Strain and Deformation Gradient Matrix . . . . . . . . 56716.4.6 HOLZ Line Splitting from 3D Strain. . . . . . . . . . . . . . 568

16.5 Electron Imaging-Based Strain Measurement Techniquesand Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57016.5.1 Strain Mapping Using GPA. . . . . . . . . . . . . . . . . . . . . 57016.5.2 STEM and Its Application for Strain

Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57216.6 Off-Axis Electron Holography . . . . . . . . . . . . . . . . . . . . . . . . . 574References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577

xviii Contents

17 Structure of Nanocrystals, Nanoparticles, and Nanotubes . . . . . . . . 58117.1 Nanostructures and Nanoscale Phenomena . . . . . . . . . . . . . . . . 58117.2 Structure of Nanocrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

17.2.1 Nanocrystal Equilibrium and Kinetic Shapes . . . . . . . . 58317.2.2 Nanocrystal Facet Determination . . . . . . . . . . . . . . . . . 58517.2.3 Identification of Planar Faults Using Coherent

CBED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59117.2.4 Nanocrystal Surface Reconstruction. . . . . . . . . . . . . . . 59317.2.5 Surface Atoms of a Twinned Nanocrystal . . . . . . . . . . 60017.2.6 The Equilibrium Shape of Supported Nanocrystals . . . 60217.2.7 Triple Junctions and Line Tension. . . . . . . . . . . . . . . . 60617.2.8 Interaction with Surface Steps . . . . . . . . . . . . . . . . . . . 608

17.3 Structure of Nanoclusters and Nanoparticles . . . . . . . . . . . . . . . 61117.3.1 Diffraction by Free Clusters. . . . . . . . . . . . . . . . . . . . . 61117.3.2 Structure and Energetics of Metallic Nanoparticles . . . 613

17.4 Carbon Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61817.4.1 Carbon Allotropes and Bond Lengths . . . . . . . . . . . . . 61817.4.2 Electron Diffraction of Carbon Nanotubes . . . . . . . . . . 62217.4.3 Chirality and Diameters of Single-Walled Carbon

Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62717.4.4 Structure of Multiwalled Carbon Nanotubes . . . . . . . . 63117.4.5 Defects in Graphene and Carbon Nanotubes . . . . . . . . 63317.4.6 Van der Waals Forces and Molecular Interactions . . . . 637

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

Appendix A: Useful Relationships in Electron Diffraction . . . . . . . . . . . . 653

Appendix B: Electron Wavelengths, Physical Constants,and Atomic Scattering Factors. . . . . . . . . . . . . . . . . . . . . . . 655

Appendix C: Crystallographic Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667

Appendix D: Indexed Diffraction Patterns with HOLZ . . . . . . . . . . . . . . 675

Appendix E: Fourier Transforms, d-Function, and Convolution . . . . . . . 685

Appendix F: Crystal Structure Data, Mean Inelastic Free Path,and Mean Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

Contents xix

Symbols and Abbreviations

Symbolsa, b, c, a, b, c Lattice constants and angles~a;~b;~c Unit cell vectors

~a�;~b�;~c� Reciprocal space unit cell vectorsa~k and~e Vibrational amplitude and directionA Amplitude, structure matrix, scalar magnetic potential,

lattice matrixAi AstigmatismAo Cross-sectional area, lattice matrix

A ~k� �

Objective aperture function

~b Burger’s vectorB Magnetic field, Debye–Waller factorBi ComaBg Normalized eigenvector coefficientBj x; yð Þ Electron lateral eigen functioncijkl Stiffness tensor, fourth-rankC Eigenvector matrix, speed of light, centered~C Chiral vectorCg EigenvectorC1 DefocusCc Chromatic aberration constant. PositiveCs Spherical aberration constant. Positive for round magnetic

lensesCIJ The 6 � 6 elastic stiffness matrixd d-spacing, distance, diameterdhkl d-spacing of hkl lattice planedlm� Real spherical harmonic functionD Diameter, size, aberration coefficientD ~Kt� �

STEM detector response function

xxi

e Electron chargeE Electron energy, Young’s modulusEc Df ;Kt� �

, EK as;Ktð Þ Envelope functions in contrast transfer functionEI Bending stiffnessf Atomic scattering factor, focal lengthf x X-ray atomic scattering factor

f ~k;~ko� �

Scattering amplitude as function of the scattered andincident wave vectors

f 0 sð Þ Atomic absorptive coefficient

fr ¼ 1þ eU=mec2

1þ 2eU=mec2Relativistic factor

F Fano noise, deformation gradient matrix, face-centered,functional

~F ForceFg, F(hkl), Fhkl Structure factorFBg Electron structure factor

Fxg X-ray structure factor

~g or g Reciprocal lattice vector, reflection or reflected beamG 2pg, gain, group, G matrix, Gibbs energyG ~r;~r0ð Þ;G ~r �~r0ð Þ Green’s functionh Planck’s constanth zð Þ Multipole characteristic functionh ~r;Eð Þ Lens resolution function~h or h Reciprocal lattice vector, reflection or reflected beam�h h/2p, reduced Planck's constantH Hamiltonian, Eikonal, principle plane, a subset of a group,

parameterHv 2 � 2 matrixH ~Kt;E� �

Fourier transform of h ~r;Eð Þh, k, l Miller indicesI Intensity, body-centered, unit diagonal matrixJ Current densityjn Spherical Bessel function of nth orderJ Joule, a unit of energyJn Bessel function of nth orderk Kilo, one thousand, wave numberkB Boltzmann constant~k Electron wave vector, phonon wave vectorK Wave number inside the crystal~K Scattered wave vector~Kt Tangential component of wave vector~Ko Incident wave vector inside the crystalL Length, camera length

xxii Symbols and Abbreviations

Lc Coherence lengthLT Transverse coherence lengthL ~rð Þ Lattice function

L ~S� �

Reciprocal lattice function

me and m Electron mass and relativistic massm Mirror symmetry, mixing factorM Magnification, number, moleculeM ¼ B sin h=kð Þ2 Thermal damping factorMA Atomic massn Normal, integer number, n-fold rotation axisN Integer number, nodal point~n Surface normal vector~n Roto-inversion axiso Zero order, originp momentum, pointP Perturbation function, primitive, populationPz x; yð Þ, P ~r �~r0ð Þ Fresnel propagatorPgð~rÞ Phase mapq 2pk, charge, quantum efficiencyq ~rð Þ Object transmission functionQ ~Kt� �

Fourier transform of q ~rð Þ~Q 2p~gr Radius~r Real space vector~r� Reciprocal space vector~R Lattice vectors s ¼ sin h

k , path lengthS Symmetry, symmetry operation, surface area, pole-piece

gap size, signalSi Aberration coefficientSIJ The 6 � 6 elastic compliance matrix~S Scattering vectorS ~qð Þ Diffuse scatteringSg Excitation error of reflection gS2 ~r1;~r2ð Þ Ursell function

S ¼ C e2picit

n oC�1 Scattering matrix

t ThicknessT Temperature, translation, transformation matrix, time,

transmittance~T Translation vectorT x; yð Þ Objective lens resolution functionT(z) Paraxial lens function

Symbols and Abbreviations xxiii

u Electron ray pathu, ~u Atomic displacement amplitude and vectorU Electron interaction potential, object distanceU′ Imaginary electron interaction potential or absorption

potentialUg Fourier coefficient of electron interaction potential, or

electron structure factorUo Mean electron interaction potentialUC ~rð Þ Real crystal potential�U ~R� �

Projected electron interaction potential

Ujk ¼ bjk2p2a�j a

�k

Averaged, squared, thermal displacements

v Electron velocityV Electrostatic potential, image distance, volume, electron

velocityVo Mean electrostatic potential, electron speed~V or Vx, Vy, Vz Electron velocityVg Fourier coefficient of electron potential, or electron

structure factor�V ~R� �

Projected electron potential

Wadh Work of adhesionx, y, z Cartesian coordinates, variableX ElectronegativityX, Y, Z Cartesian coordinatesZ Atomic number

a Angle, phaseb Angle, variablebs Source brightnessbjk ¼ 2p2 ujuk

� �Averaged, squared, thermal displacements

v Phase shift due to lens aberrations, parameter,wave function

v2 Least-square functiond d function, small number, phasee Small number, coefficient matrix, dielectric constant,

energyeij Strain, i, j = x, y, z/ Wave functionc Relativistic constant, fundamental ray, correlation,

eigenvalue, dispersion, surface energyη Parameteru ~rð Þ Bloch wave functionj Curvaturek Wavelength, Lamè constantl Lamè constant

xxiv Symbols and Abbreviations

lo Vacuum permeabilitym Poisson’s ratioh AnglehB Bragg's angleq Charge density, radius in the cylindrical coordinates,

densityr Electron interaction constant, scattering cross section,

standard deviation, line charge densityrij Stressrjk ¼ ujuk

� �Averaged, squared, thermal displacements

s Time delay, eigenvaluet Visibility- Complex coordinate (x − iy)x Frequency, complex coordinate (x + iy), parameter

x ~k� �

Phonon frequency

ng Extinction distancew Wave functionD Interval, distanceDf DefocusU Electron acceleration voltageUg Wave functionC Correlation, dispersion rateK Lattice shape functionKmin ¼ 1

KtmaxInformation transfer limit

HD Debye temperatureR Density of coincidence sites, in its reciprocal form� Eigenvalue matrixX Solid angleW Phase invariant, multipole potential

AbbreviationsADF Annular dark fieldBCC Body-centered cubicCA Condenser apertureCBED Convergent beam electron diffractionCBIM Convergent beam imagingCCD Charge-coupled deviceCL CathodoluminescenceCTF Contrast transfer functionDQE Detector quantum efficiencyDR Dynamic rangeDWBA Distorted wave Born approximation

Symbols and Abbreviations xxv

EBSD Electron backscattered diffractionfcc or FCC Face-centered cubicFFT Fast Fourier transformFT Fourier transformG–M Gjønnes–MoodieHAADF High-angle annular dark fieldHCP Hexagonal close packedHIO Hybrid input and outputHREM High-resolution electron microscopyICSD Inorganic Crystal Structure DatabaseIP Imaging platesLACBED Large-angle convergent beam electron diffractionLEED Low-energy electron diffractionMPB Morphotropic phase boundaryMTF Modulated transfer functionNAED Nanoarea electron diffractionNBD Nanobeam diffractionNBED Nanobeam electron diffractionPED Precession electron diffractionPMN-xPT (1 − x)Pb(Mg1/3Nb2/3)O3−xPbTiO3

PMT Photomultiplier tubePSF Point spread functionRHEED Reflection high-energy electron diffractionSAED Selected area electron diffractionSEM Scanning electron microscopy or scanning electron

microscopeSEND Scanning electron nanodiffractionSNR Signal-to-noise ratioSTEM Scanning transmission electron microscopyTEM Transmission electron microscopy or transmission elec-

tron microscopeWPO Weak phase objectXAFS X-ray absorption fine structure

xxvi Symbols and Abbreviations