Table Of ContentElastic and Inelastic
Scattering in Electron
Dff"actionandlmagmg
Transmission electron diffraction pattern recorded at 100 kV from a [100] SrTi0
3
crystal, showing thermal diffuse scattering streaks due to phase correlation between
atomic vibrations. Related contents are described in Chapters 7 and 12.
High-angle annular-dark-field STEM (or "Z-contrast") image of [110] GaAs
Ino.2sAlo.7sAs multi-quantum-well structures showing the direct correlation
between image contrast with local chemical composition and crystal structure
(Courtesy ofLiu and Cowley, 1991). Related constants are described in Chapters
11 and 13.
Elastic and Inelastic
Scattering in Electron
Diffraction and Imaging
Zhong Lin Wang
School of Materials Science and Engineering
Georgia Institute of Technology
Atlanta, Georgia
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
On file
ISBN 978-1-4899-1581-8 ISBN 978-1-4899-1579-5 (eBook)
DOI 10.1007/978-1-4899-1579-5
© Springer Science+Business Media New York 1995
Originally published by Plenum Press, New York in 1995
Softcover reprint of the hardcover 1st edition 1995
10987654321
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in
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or otherwise, without written permission from the Publisher
Preface
Elastic and inelastic scattering in transmission electron microscopy (TEM) are
important research subjects. For a long time, I have wished to systematically
summarize various dynamic theories associated with quantitative electron micros
copy and their applications in simulations of electron diffraction patterns and
images. This wish now becomes reality. The aim of this book is to explore the
physics in electron diffraction and imaging and related applications for materials
characterizations. Particular emphasis is placed on diffraction and imaging of
inelastically scattered electrons, which, I believe, have not been discussed exten
sively in existing books. This book assumes that readers have some preknowledge
of electron microscopy, electron diffraction, and quantum mechanics. I anticipate
that this book will be a guide to approaching phenomena observed in electron
microscopy from the prospects of diffraction physics.
The SI units are employed throughout the book except for angstrom (A), which
is used occasionally for convenience. To reduce the number of symbols used, the
Fourier transform of a real-space function P'(r), for example, is denoted by the same
symbol P'(u) in reciprocal space except that r is replaced by u. Upper and lower limits
of an integral in the book are (-co, co) unless otherwise specified. The (-co, co) integral
limits are usually omitted in a mathematical expression for simplification.
I very much appreciate opportunity of working with Drs. J. M. Cowley and J.
C. H. Spence (Arizona State University), J. Bentley (Oak Ridge National Labora
tory), R. F. Egerton (University of Alberta), C. Colliex (University of Paris-Sud),
and A. Howie (University of Cambridge), who gave me advice and encouragement
on many topics. Particular thanks go to Dr. Nea S. Wheeler for her careful and
critical review ofthe manuscript. Thanks to Drs. J. Bentley, S. Dudarev, C. Fanidis,
A. V. Latyshev, J. Liu, R. F. Loane, Ping Lu, L. Reimer, C. J. Rossouw, A. E. Smith,
D. J. Smith, J. C. H. Spence, B. Viguier, M. H. Yao, and J. M. Zuo, who have kindly
allowed me to use their illustrations and results in the text.
v
vi Preface
This book was written in my spare time after working hours. I am heartily
grateful to my wife, Cui Xia Yuan, for her encouragement and understanding. This
book would have been impossible without her support.
Zhong Lin Wang
Introduction
Diffraction and imaging of transmitted high-energy electrons are important experi
mental techniques for determining crystal structures. The steady improvement of
TEM, scanning transmission electron microscopy (STEM), and their associated
analytic techniques has made it possible to perform structural determination at
atomic resolution. The recently commercialized charge-coupled device (CCD)
camera and the electron energy filter open a new era in quantitative electron
microscopy. With these attachments, images and diffraction patterns formed by
electrons with distinct energy losses can be separated and digitally recorded. In
practice, quantitative data analysis depends entirely on theoretical simulations. This
book is about the theories and techniques of elastic and inelastic electron diffraction
and imaging and their applications in quantitative electron microscopy. In particu
lar, attempts were been made to summarize and develop various dynamic theories
developed over years from the studies of elastic and inelastic electron multiple
scattering and to apply them to explore new experimental techniques.
There are many excellent books dedicated to the various techniques associated
with TEM. The book by Hirsch et al. (1977) is a fundamental text book for
conventional TEM, particular diffraction contrast imaging. The book by Reimer
(1984) is more associated with the physics of TEM. The book by Cowley (1981)
is the only one that gives a systematic description of the unified theory of electron,
neutron, and X-ray diffractions. A recent book edited by Buseck et al. (1988) gives
a full coverage of high-resolution TEM and associated applications in various
fields. The book by Spence and Zuo (1992) and those edited by Cowley (1992,
1993) are dedicated to electron diffraction techniques. Books on electron energy
loss spectroscopy (Egerton, 1986) and energy-dispersive X-ray microanalysis are
more related to analytical applications ofTEM. Subjects associated with TEM are
rather diverse, so each book could describe only a small portion of these topics.
Diffraction and imaging theories of transmitted electrons are an important part
ofTEM. In most of the existing books, elastic scattering theory is usually described,
and in some cases, the kinematic scattering approach is adopted. Bethe's elastic
vii
viii Introduction
scattering theory was comprehensively reviewed by Humphreys (1979). Elastic
scattering multislice theory was extensively summarized by Van Dyck (1985).
Recently an excellent review was given by Amelincks and Van Dyck (1993) of the
theory of diffraction contrast imaging. Inelastic scattering, however, is usually
treated as an absorption effect and included in the theory by introducing an
imaginary potential. The book by Ohtsuki (1983) is more devoted to inelastic
excitations of charged particles, but diffraction and imaging of inelastically scat
tered electrons are not extensively covered. Therefore there is a need for a compre
hensive summary of the following subjects that are important in quantitative data
analysis:
• Dynamic elastic scattering theories: Elastic electron diffraction is usually
approached from the convenient theoretical scheme for the specific subject of the
book, such as Bloch wave theory for convergent beam electron diffraction and
multislice theory for high-resolution TEM image simulation. There is lack of a
systematic description for all existing elastic-scattering theories.
• Reflection high-energy electron diffraction (RHEED): This has been an
important technique for observing in situ surface structure evolution in thin film
growth, but the quantitative analysis of RHEED patterns is far from satisfactory. It
is necessary to summarize dynamic approaches and compare them for quantitative
RHEED data analysis.
• Absorption effect: Inelastic scattering is usually considered an absorption
effect when considering elastically scattered electrons, so it is phenomenologically
represented by an imaginary potential. However no systematic description of the
theoretical basis and related calculations of the absorption potential due to various
inelastic-scattering processes are given. These calculations are important in quan
titative electron diffraction and imaging.
• Diffraction of inelastically scattered electrons and resulting Kikuchi pat
terns: The theory of inelastic electron diffraction is complex and difficult because
of incoherent scattering characteristics of inelastic electrons; thus it is rarely
discussed in the existing books. However inelastic electron scattering has been
studied for more than three decades, and it is necessary to have a book that
summarizes all existing theoretical achievements. Although the contribution of
electrons with an energy loss greater than a few e V can be removed from the
diffraction pattern by an energy filter, phonon-scattered electrons remain because
of small energy loss « 0.1 e V).
• New imaging techniques using inelastically scattered electrons: In recent
years, great interest has developed in structural determination using inelastically
scattered electrons. High-angle annular dark-field imaging in STEM, for example,
is based on the signal of high-angle phonon-scattered electrons. This imaging
technique may provide atomic-number-sensitive structural information with a
resolution superior to conventional bright-field imaging.
Introduction ix
The ultimate goal of electron microscopy is to determine the structure of
materials quantitatively. This process relies on using the best possible theory and
energy-filtered high-resolution diffraction patterns and images. It is anticipated that
electron diffraction and imaging will become as indispensable as X -ray and neutron
diffractions for materials research, a status they have not yet achieved. This book
focuses on dynamic diffraction and imaging ofe lastically and inelastically scattered
electrons and their applications for structural determinations, with particular em
phasis on preceding topics.
The book covers the following principles. First since elastic-scattering theory
is described in various books, diffraction and imaging of inelastically scattered
electrons are the main emphasis of this book. Second all theoretical derivations are
initiated from first principles wave mechanics for high-energy electrons (Eo> 20
keV). Elastic scattering is initiated from the SchrOdinger equation, and inelastic
scattering is determined by the solution of Yoshioka's coupled equations. Third
since the interaction between incident electrons and crystals is very strong, all
approaches presented are dynamic theories. Fourth all theories are linked to one
another and presented in the form best suited for numerical calculations. All
mathematical operations are given in detail, so that the book is self-contained. The
entire text is linked together, so it forms a unity based on first principles. Finally
all physical quantities are defined in SI units except where stated; for example,
angstrom (A) is used occasionally for convenience.
This book consists of two parts: Part I covers the diffraction and imaging of
elastically scattered electrons. Chapter 1 introduces fundamental concepts related
to electron diffraction based on the kinematic scattering approach. Some basic
quantities are defined and illustrated. Chapters 2-3 give comprehensive quantum
mechanical description of elastically transmitted electron scattering. Chapter 2
describes the Bloch wave theory (or Bethe theory) and its applications in convergent
beam electron diffraction, diffraction contrast imaging, and weak-beam imaging.
Chapter 3 gives the wave mechanics basis of the Cowley-Moodie (1957) multislice
theory and its applications for simulating high-resolution electron microscopy
images. Substantial discussion is given to explore the methods of introducing
high-order Laue zone reflections in the multislice calculations. Real-space
multislice theory is also introduced. Existing theories are compared to exhibit
their uniqueness and disadvantages for treating particular problems. As an
important part of high-energy electron diffraction, Chapter 5 is devoted to
dynamic theories of RHEED from bulk crystal surfaces. Various theoretical
approaches are derived and compared. This is the only comprehensive review
of RHEED theories so far.
Part II of this book is devoted to inelastically scattered electrons. This is the
first book to discuss fully inelastic scattering in electron diffraction. The main topics
include the following:
