Transmission Electron Energy Loss Spectrometry

TRANSMISSION ELECTRON ENERGY LOSS SPECTROMETRY IN MATERIALS SCIENCE AND THE EELS ATLAS Second Edition Edited by C. C. ...

Author: Channing C. Ahn

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TRANSMISSION ELECTRON ENERGY LOSS SPECTROMETRY IN MATERIALS SCIENCE AND THE EELS ATLAS Second Edition

Edited by

C. C. Ahn

Editor Channing C. Ahn California Institute of Technology Materials Science Faculty 1200 California Boulevard Pasadena, CA 91125 USA

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are avised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: Applied for British Library Cataloguing-in-Publication Data: A catalogue record for this book is available from the British library Bibliographie information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie: detailed bibliographie data is available in the Internet at: http:dnb.ddb.de

c 2004 WILEY-VCH Verlag GmbH & Co. KGaA Weinheim Printed in the Federal Republic of Germany Printed on acid-free paper All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form ± by photoprinting, microfilm, or any other means ± nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printing: Betz-Druck GmbH, Darmstadt Bookbinding: Grossbuchbinderei J. SchaÈffer GmbH & Co KG, GruÈnstadt ISBN 3-527-40565-8 ISBN 0-471-32407-8

Contents

Preface to the Second Edition

ix

Preface to the First Edition

xi

Contributors 1 Introduction Brent Fultz 1.1 1.2 1.3 1.4 1.5

Overview of Concepts Electron Excitations in Materials Instrumentation Core Edges Analyses

2 Experimental Techniques and Instrumentation Ray F. Egerton 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Introduction Electron Spectrometers and Imaging Filters Influence of the Microscope Optics Recording Systems for Spectra and Images Comparison of Serial and Parallel Recording Comparison of EFTEM and Energy Filtered STEM Imaging Experimental Conditions for Spectroscopy and Mapping Future instrumentation References

3 EELS Quantitative Analysis Richard Leapman 3.1 3.2 3.3 3.4

Introduction Basic Formulas for Quantification Inner-Shell Cross-Sections Background Subtraction and Signal Estimation

xiii 1 1 4 12 15 20 21 21 22 25 31 36 37 38 42 45 49 50 50 54 62 v

vi

CONTENTS

3.5 3.6 3.7 3.8

Factors Influencing Accuracy of Quantification Detection Limits and Comparison with EDXS Quantitative Elemental Mapping Summary References

4 Energy Loss Fine Structure Peter Rez 4.1 4.2 4.3 4.4 4.5 4.6

Introduction The Low Loss Region Inner-Shells Extended Fine Structure Near-Edge Structure Conclusions References

5 Energy Filtered Diffraction Ludwig Reimer 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Introduction Instrumentation Survey of inelastic scattering processes Diffraction of Amorphous and Polycrystalline specimens Diffraction at Single-Crystalline Specimens Convergent-Beam Diffraction Methods Imaging of the Dispersion of Plasmon and Compton Scattering References

6 Elemental Mapping Using Energy Filtered Imaging Ferdinand Hofer and Peter Warbichler 6.1 6.2 6.3 6.4 6.5 6.6

Introduction EFTEM Instrumentation Energy Filtered Imaging Element distribution images Examples of Applications Conclusions and Outlook References

7 Probing Materials Chemistry Using ELNES Richard Morgan Drummond-Brydson, Hermann Sauer and Wilfried Engel 7.1 7.2 7.3 7.4

Introduction Experimental Considerations Determination of Nearest-Neighbor Coordinations Determination of Valencies

76 79 82 88 88 97 97 98 101 102 104 118 118 127 127 128 129 131 136 140 150 153 159 159 160 164 165 181 201 202 223

224 224 227 240

CONTENTS

7.5 7.6 7.7

Further Applications Application to Interfaces and Defects Conclusions References

8 Application to Ceramics, Catalysts and Transition Metal Oxides James Bentley and Jason Graetz 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Introduction General Experimental Points Composition Determination Low-Loss/ELNES Studies EXELFS Analysis Other Topics Conclusions References

9 EELS of the Electronic Structure and Microstructure of Metals J. K.Okamoto, D. H. Pearson, A. Hightower, C. C. Ahn, and B. Fultz 9.1 9.2 9.3 9.4 9.5 9.6

Introduction Plasmons Chemical Shifts in Threshold Energy of Metallic Alloys Near-edge Structure in the Study of Band Structure Extended Energy Loss Fine Structure Conclusions References

10 Electron Energy Loss Studies in Semiconductors Philip E. Batson 10.1 Introduction 10.2 General Considerations 10.3 Interband Excitations 10.4 Soft X-ray Core Excitations: Fingerprints 10.5 Core Excitations: JDOS Interpretation 10.6 Core Edge Intra-Gap Scattering 10.7 Si L-Edge Excitonic Analysis 10.8 Band Structure Interpretation of Si ELNES 10.9 Specialized Equipment 10.10 Conclusions References 11 Electron Energy Loss Spectroscopy of Magnetic Materials Jennifer A. Dooley 11.1 Introduction 11.2 Chemical and Microstructural Information

vii

249 252 256 256 271 272 272 275 280 299 303 307 307 317 317 318 321 323 330 345 346 353 353 355 357 361 363 367 368 371 378 380 381 385 385 386

viii

CONTENTS

11.3 11.4 11.5 11.6 11.7

Determination of DOS White Line Probe of d-band Occupancy Magnetic Dichroism Effect in EELS Energy Filtered Lorentz Microscopy Conclusion References

12 Electron Energy Loss Spectroscopy of Polymers Matthew R. Libera and Mark M. Disko 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Index

Introduction Elements of polymer morphology Elastic Scattering and Staining Methods Inelastic Scattering and Spatially Resolved Spectroscopy Core-loss spectra, profiles, and images Low-loss spectra, profiles, and images Constraints imposed by radiation damage Conclusion and Outlook References

391 392 395 398 414 415 419 419 421 426 430 432 439 442 446 448 455

Preface to the Second Edition

The first edition of this book, Transmission Electron Energy Loss Spectrometry in Materials Science, published eleven years ago, was written to show the broad range of capabilities of EELS for studies on materials. In the intervening years, advances in instrumentation, techniques, and applications have motivated us to prepare this second edition. While the original book concentrated on the spectroscopic capabilities of EELS, advances in instrumentation have made energy-filtered imaging a practical method for chemical analysis of materials on the nanometer scale. Consequently we have added several chapters to cover important aspects of energy-filtered imaging in the transmission electron microscope. The spectroscopic capabilities of electron energy loss analysis remain important for transmission electron microscopy of materials, especially when used in conjunction with diffraction, imaging tools, and other spectroscopic techniques. We chose to keep the spirit of our first edition by organizing this second edition along the same lines as the first. The book is divided approximately into halves. Chapters 1 to 6 develop the principles of electron energy loss methods that are common to all materials studies. The remainder of the book, Chapters 7 to 12, focus on applications of EELS to particular classes of materials—minerals, ceramics, catalysts, metals, magnetic materials, semiconductors, and polymers. The scope of this book thus spans the topics of inorganic materials science and includes synthetic organic polymers. Biological materials are not covered in great detail but Chapter 3 and Chapter 12 cover important aspects of biological analysis and imaging. This second edition includes on CD-ROM a digital reference library of EELS spectra of the elements and simple compounds–the EELS Atlas. The Atlas data were compiled over a period of several months in 1982 using a Gatan 607 serial EELS spectrometer, and great care was taken to obtain data of high statistical quality. More recent instrumentation provides higher rates of data acquisition, but the good energy resolution and statistical quality of the Atlas data have made them a popular reference. These data should be even more useful in digital form. The development of the EELS Atlas was originally supported by the Center for Solid State Science at Arizona State University and by Gatan, Inc. The CD-ROM also contains a number of important spectroscopic images, kindly provided by Ferdinand Hofer. Mark Disko and Brent Fultz co-edited the first edition and deserve thanks for the ix

x

PREFACE TO THE SECOND EDITION

time and effort they expended in getting this second edition launched. Greg Franklin arranged for publication of this edition through John Wiley. The staff at Wiley; Brendan Codey, Rachel Witmer, and Johanna van Gendt, were of great help through the production of this edition. John Hunt and Mike Kundmann of Gatan kindly provided the EELS Atlas data in digital form. During the preparation of the second edition, Professor Ludwig Reimer, a pioneer in the field of electron microscopy passed away. This edition is dedicated to his memory. CHANNING AHN

Preface to the First Edition

Transmission electron microscopy is a workhorse for characterizing the microstructure of materials that range from polymers to superconductors. Electron energy loss spectrometry is a relatively new addition to the group of diffraction, imaging and spectroscopic techniques available for the study of materials with the transmission electron microscope. Many of the methods that are required to convert from an understanding of electron scattering physics to a new understanding of a material can be daunting. Transmission Electron Energy Loss Spectrometry in Materials Science is the result of a growing demand from materials scientists to combine analytical tools and problem-solving approaches. We hope that this emphasis on applications can help to shorten the learning curve necessary to achieve productive EELS experiments. Rapid changes in instrumentation and the associated analytical methods require an update for even the seasoned veterans of energy-loss studies. These changes include high quantum efficiency parallel detection systems, powerful analysis algorithms, and advanced spectroscopic imaging techniques. Our approach to meet these requirements begins with Chapters 1 to 5 which are dedicated to energy loss methods that are general to all areas of materials studies. The remainder of the book, Chapters 6 to 10 focus on applications of EELS to particular areas of materials science. An emphasis on studies that involve fine structure measurements is common to the applications chapters. This is important since the review literature in this area is sparse and the methods of fine structure analysis are challenging. Finally, we hope you share our enthusiasm for a technique that combines ultimate limits of spatial resolution with information on bonding, electronic properties, and composition. Applications of transmission EELS will certainly take the lead in the development of materials with custom microstructures into the twenty-first century. MARK DISKO, CHANNING AHN, BRENT FULTZ January, 1992

xi

Contributors

CHANNING AHN, California Institute of Technology, Pasadena, California PHILIP BATSON, IBM Thomas J. Watson Research Center, Yorktown Heights, New York JAMES BENTLEY, Oak Ridge National Laboratory, Oak Ridge, Tennessee MARK DISKO, ExxonMobil Research and Engineering Company, Annandale, New Jersey JENNIFER A. DOOLEY, Jet Propulsion Laboratory, Pasadena, California RICHARD MORGAN DRUMMOND-BRYDSON, University of Leeds, Leeds, United Kingdom RAY EGERTON, University of Alberta, Edmonton, Canada WILFRIED ENGEL, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany BRENT FULTZ, California Institute of Technology, Pasadena, California JASON GRAETZ, California Institute of Technology, Pasadena, California ADRIAN HIGHTOWER, California Institute of Technology, Pasadena, California FERDINAND HOFER, Graz University of Technology, Graz, Austria RICHARD LEAPMAN, National Institutes of Health, Bethesda, Maryland MATTHEW LIBERA, Stevens Institute of Technology, Hoboken, New Jersey JAMES OKAMOTO, California Institute of Technology, Pasadena, California DOUGLAS PEARSON, California Institute of Technology, Pasadena, California LUDWIG REIMER HERMANN SAUER, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany PETER WARBICHLER, Graz University of Technology, Graz, Austria

xiii

1

Introduction Brent Fultz Division of Engineering and Applied Science California Institute of Technology Pasadena, CA 91125

1.1

OVERVIEW OF CONCEPTS

Electron energy loss spectrometry (EELS) measures electronic excitations in materials and condensed matter. This book presents the theory and practice of EELS measurements in a transmission electron microscope (TEM). These measurements are now widely possible with commercial EELS spectrometers on TEMs intended for “analytical work” (i.e., microcompositional spectroscopy). Since analytical electron microscopy (AEM) provides compositional and chemical information to complement the structural information obtained with TEM imaging, AEM is an indespensible tool for studying the microstructures of materials. The EELS spectrometer is usually an instrument of lower priority on an analytical TEM in relation to the X-ray ﬂuorescence spectrometer in the form of an energy dispersive X-ray (EDX) detector. This prioritization can be understood from the standpoint of compositional analysis, for which EDX data can usually be interpreted simply within the thin ﬁlm approximation. EELS data generally offers much more physical insight, however, in our understanding of the materials or phases under study and does so within a context that can be readily assimilated from our understanding of solid-state physics and chemistry. While the EDX spectrum also originates with electronic excitations in materials. The fundamental difference is that EELS is a probe of the initial excitation of an atom in a material to an excited or ionized state, whereas EDX measures the X-rays emitted from the decay of this excited state. For comparison, the energy range of EEL spectra is typically from 0 to 3 keV, whereas X-ray spectra are typically acquired over a range of 1–40 keV. The detection of light elements, which have low energy X-rays and low X-ray ﬂuorescence yields, is therefore a typical reason for using EEL spectrometry. The energy resolution of an EELS spectrometer is typically a factor of a hundred better than that of an energy dispersive X-ray spectrometer, however, so EELS offers advantages as a spectroscopic measurement. The ability to resolve energy shifts of 0.1 eV allows 1

2

INTRODUCTION

studies of chemical bonding and local atomic conﬁgurations.1 More recently, the technique of energy ﬁltered imaging has propelled EELS into a unique position of being a practical experimental technique for compositional mapping of specimens with atomic-scale spatial resolution. All these methods will be covered in detail in this book, with applications to all classes of materials. In order to obviate many of the suspicions and to provide a sense of the utility of EELS as both a microanalytical technique, and as a means of providing insights into the atomic physics of solids, this introductory chapter presents an introduction to the materials scientist in the form of a review of the basic concepts of atomic and solid-state physics that underlie the EELS method, and discusses some experimental features of EELS. Subsequent chapters will elaborate on the details of the technique and analysis. In this chapter, we will present descriptions of the various ionization processes in increasing levels of elaboration, starting with simple physical descriptions, and then continuing with more detailed mathematical derivations of these ionization processes. We start with a most basic description of EELS. A nearly monochromatic beam of high energy electrons passes through a material, perhaps as a tightly focused probe beam. Coulombic interactions with the electrons in the material cause inelastic scatterings of the high energy electrons, and the energy spectrum of the high energy electrons is measured after they pass through the material. Energy is conserved, so the energy lost from the high energy electron is gained by the electrons in the material. The excitation spectrum of the material therefore can be deduced from an EELS spectrum. The ways in which the incident beam gives up energy to the material occur over speciﬁc energy ranges that are dependent on the nature of the material, but that can be categorized into distinct processes and features. Generally, the most intense feature is the zero loss peak, which consists of those electrons that have given up no energy to the sample (although we note that phonons may contribute to broadening of the zero loss peak but these are probed most effectively in an inelastic neutron scattering experiment). Zero loss electrons are the elastic unscattered or Bragg scattered electrons, whose intensity is dependent on the sample thickness and orientation with respect to the incident beam. The inelastic electronic excitations of the material that the EELS technique effectively probes, include low energy (7–30 eV) plasmon excitations, which dominate the inelastic scattering in an EEL spectrum. Core loss events involve the excitation of inner electrons out of the atom, and the approximate energies and intensities of core loss features in an EELS spectrum can be understood with some concepts of atomic physics. There is, in addition, ﬁne structure to these core loss features because solid-state chemistry alters the energies and shapes of the core edge excitation onset. Finally, an extended ﬁne structure above an absorption edge originates with the backscattering of the outgoing electron by nearby atoms. 1 Measuring sample thicknesses from plasmon intensities, and measuring radial distribution functions are other capabilities of EELS.

OVERVIEW OF CONCEPTS

1.1.1

3

Overview of an Experimental Spectrum

A typical EEL spectrum is presented in Fig. 1.1, which is a graph of the number of detected electrons in a narrow energy interval (1–2 eV) versus the energy loss, E. The enormous peak at “zero energy loss” is from those electrons of 200,000 eV that traveled through the microscope without any energy transfer to the specimen. The sharpness of this peak indicates that the energy resolution of the spectrometer is ∼ 2 eV. The next feature at an energy loss of 25 eV (from electrons having energies of 199,975 eV) is the “ﬁrst plasmon peak”, caused by the excitation of one plasmon in the sea of conduction electrons. With thicker specimens there may also be peaks at multiples of 25 eV from electrons that excited two or three plasmons in the specimen. This is not the origin of the small bump in the data at 68 eV, however. This small feature is caused by a core loss, speciﬁcally a Ni M2,3 absorption edge caused by the excitation of 3p electrons out of the Ni atoms. The background in the EELS spectrum falls rapidly with energy and the spectrum is rescaled at an energy loss of ∼ 375 eV to better show the weaker features at higher energy losses.

Fig. 1.1 EELS spectrum from sample of fcc nickel metal showing (from left to right) zero-loss peak, bulk plasmon, Ni M -edge, detector gain change, and Ni L-edges.

The next feature of the Ni spectrum in Fig. 1.1 is a core loss edge at 855 eV. The jump in intensity at 855 eV is caused by the excitation of a 2p3/2 electron out of the nickel atom, and is called the “L3 -edge.” The L2 -edge at 872 eV is caused by the excitation of a 2p1/2 electron. Right at the edge are some sharp prominent peaks, which are known as “white lines”. The white lines originate from the excitation of 2p electrons into unoccupied 3d states at a nickel atom. Such “electron near-edge ﬁne structure” (ELNES) features are typical of transition metals and their alloys, and are indicative of the ﬁlling of the 3d band. At energy losses over a range of

4

INTRODUCTION

1000 eV beyond the edge threshold, gradual oscillations in the intensity, termed “extended electron energy loss ﬁne structure”, are expected, but the experimental data of Fig. 1.1 are not of sufﬁcient quality or range to show these features. 1.2 ELECTRON EXCITATIONS IN MATERIALS 1.2.1

Plasmons

The simplest approximation of electrons in a material, especially a metal, is that the electrons behave as a gas of free electrons. The positive ion cores are neglected, except insofar as they provide a uniform potential to conﬁne the electron gas. This approximation, crude as it may be, is used effectively in solid-state physics to understand the origin of the electrical and thermal conductivity of metals. Here we use it to understand the origin of plasmon excitations in the energy range of 7–30 eV. Plasmons can be understood as collective oscillations of the free electron gas, which result from the passage of our incident electron through the free electron gas. The free electron gas electrons are initially repelled by the passing electron, within the limits imposed by the solid. After the incident electron has passed through the solid, the free electron gas electrons which were repelled, attempt to restore themselves to their original positions but usually “overshoot”. The resulting oscillation from many such processes has a characteristic frequency. To ﬁnd the characteristic frequency (and hence the energy) of the oscillation, we can freeze this simple picture of this oscillation at a moment when the repulsed electrons are displaced to their maximum positions away from their equilibrium positions. We can view this by considering a rigid translation of a thick slab of electron density by a small amount, x, as in Fig. 1.2. We can regard this translation causally as having resulted from the passage of the incident electron.

Fig. 1.2 Displacement of a slab of electric charge, leading to doubling of the charge density at the top of the slab over thickness x, and depletion of charge at the bottom of the slab.

Changes in electron density caused by this translation will set up an electric ﬁeld, E, that provides a restoring force to push the slab back to its original position. The restoring force per unit volume of slab, of density ρ [electrons/cm3 ] is F = −eρE

(1.1)

ELECTRON EXCITATIONS IN MATERIALS

5

The electric ﬁeld, E, that provides the restoring force is just the ﬁeld of a parallel-plate capacitor E = 4πσ

(1.2)

where σ is the surface charge density. From Fig. 1.2 we see that at the bottom face all the electrons are removed, but at the top face the electron density is doubled. The surface charge density, σ, is therefore equal to the product of electron charge, e, the electron density, ρ, and the displacement, x σ = eρx

(1.3)

Substituting Eq. 1.3 into Eq. 1.2, and then into Eq. 1.1 provides F = −e2 ρ2 4πx

(1.4)

The Newtonian equation of motion per unit volume of electron slab is F = ρme

d2 x dt2

(1.5)

Substituting Eq. 1.4 into Eq. 1.5 4πe2 ρ d2 x x =− dt2 me

(1.6)

Equation 1.6 is the equation of an undamped harmonic oscillator with the characteristic frequency 4πe2 ρ √ ωp = = 5.64 × 104 ρ (1.7) me where ρ is in [electrons/cm3 ], and ωp is in [Hz]. With analogy to a mechanical oscillator, the electron density provides the stiffness. The higher the electron density, the higher the plasmon frequency. For metals, assuming an approximate free electron density of ρ = 1023 electrons cm−3 , ωp ∼ 2 × 1016 Hz. The energy of such an oscillation, E = h ¯ ωp = (6.6 × 10−16 ev s)(2 × 1016 s−1 ) = 13 eV. Intense plasmon peaks are typical in EELS spectra at energy losses of 10–20 eV. Plasmons are not long-lived, however, because they promote excitations of electrons near the Fermi energy. Plasmon peaks therefore tend to be broadened in energy inversely with the plasmon lifetime.2 Free electron metals such as aluminum have sharper plasmon peaks than do alloys of transition metals, which have a high density of states at the Fermi energy.

2 “Lifetime broadening” is understandable from the uncertainty principle: ∆E∆τ ∼ h = ¯ ; a short lifetime ∆τ requires a large uncertainly in energy, ∆E.

6

INTRODUCTION

1.2.1.1 Thickness The characteristic length for a 100-keV electron to excite a plasmon is ∼ 100 nm in metals and semiconductors. This is an average length, so in a TEM specimen of even 50 nm, some electrons will excite one, two or even more plasmons. The probability for the excitation of n plasmons is determined by the statistics of Poisson processes n 1 t −t Pn = (1.8) exp n! λ λ Here the ratio of the sample thickness to the characteristic plasmon scattering length is t/λ. If the plasmon energy in a material is 7 eV, in thick specimens a series of plasmon peaks will be observed at multiples of 7 eV with intensities set by Eq. 1.8. We can see this in the example of a Li ﬁlm as shown in Fig. 1.3.

Fig. 1.3 Low loss spectrum from thick Li metal ﬁlm showing multiple plasmon peaks.

The intensities of plasmon peaks can sometimes provide a practical way of determining the thickness of a specimen, even when the specimen is so thin that only the elastic peak (n = 0 in Eq. 1.8) and the ﬁrst plasmon peak (n = 1 in Eq. 1.8) are observed. Deviations from the intensities of Eq. 1.8 are of course expected when the electron beam passes through regions of non-uniform thicknesses. 1.2.1.2 Dielectric Constant Analysis of Plasmons It is usually unwise to attempt any quantitative analysis of plasmon spectra with the free electron gas model and Eq. 1.7. For example, it is arguable that insulating oxides have no free electrons, yet they exhibit clear plasmon spectra. A more rigorous analysis of the plasmon spectrum uses a wavevector- and frequency-dependent dielectric constant of the material, ε(k, ω). The electron oscillations in the plasmon provide an oscillating electric ﬁeld, which propagates the electron oscillation. In other words, the electrons serve as a medium for propagation of a wave with the functional form u(x, t): u(x, t) = exp[i(kx − ωt)]

(1.9)

ELECTRON EXCITATIONS IN MATERIALS

7

where the wavevector, k, for a nonmagnetic material is √ ω (1.10) ε c All information about the conduction electrons is contained in the dielectric constant, ε(k, ω), which in general is complex k=

ε(k, ω) = ε + iε

(1.11)

We therefore expect the wavevector, k, to be complex k 2 Substituting Eq. 1.12 into Eq. 1.9, we ﬁnd dissipation of the wave −k x u(x, t) = exp[i(k x − ωt)] exp 2 k = k + i

(1.12)

(1.13)

Equations 1.12 and Eqs. 1.10 and 1.11 provide ω 2 k 2 + ik k = (ε + iε ) 4 c so we equate the imaginary and real parts of the dielectric constant k 2 = k 2 +

ik k = iε

ω 2

k 2 + k 2 = ε

c ω 2

c In the case where k >> k , we divide Eq. 1.15 by Eq. 1.16:

(1.14)

(1.15) (1.16)

ε k = (1.17) k ε From Eq. 1.13, we note that the intensity of the wave, u∗ u, decreases with distance with a characteristic length k . The ratio k /k = k λ/2π of Eq. 1.17 is the fractional attenuation in intensity of the electron disturbance over the distance λ/2π. By energy conservation, this attenuation of the wave intensity is the excitation spectrum of the electron gas, σ(k, ω). This plasmon spectrum is a function of k and ω through the dielectric constant ε(k, ω). This is emphasized in the standard expression for σ(k, E) through the fractional attenuation k /k and Eq. 1.17 σ(k, ω) ∝ Since ε << ε

ε ε = ε ε ε2

(1.18)

8

INTRODUCTION

σ(k, ω) ∝

ε ε = Im 2 ε + ε2

−ε + iε (ε + iε )(ε − iε )

ε = Im

−1 ε(k, ω)

ε

(1.19) Plasmons are quantized with energy E = h ¯ ω, we convert from ω to E, which is more appropriate for EELS spectra −1 (1.20) σ(k, E) ∝ Im ε(k, E) Equation 1.20 is useful for comparing optical spectra with EELS data, and provides a formalism to calculate the EEL spectrum itself. 1.2.2

Core Edge Energies

Core electron excitations yield the most important features of the energy loss spectrum. The two quantities of immediate interest are the binding or ionization energy, and the ionization cross-section. In the same way that we use the X-ray energy in the EDS spectrum to indentify the element of interest, the binding energy in the EEL spectrum identiﬁes the element of interest as these energies are normally well-separated and unique in Z. In the same way that we use the X-ray intensity to determine the relative quantities of elemental constituents, the EELS ionization cross-section also quantiﬁes the amount so that relative elemental quantities can be determined. In this section, we will consider the value of the ionization energy for different electrons within an atom and consider the cross-section (which we will need for quantitative analysis) later in this chapter. To understand the origin of core electron excitation energies, it is necessary to go beyond the free electron gas model and to consider the actual atom and it’s electronic shells. There is a long history in quantum mechanics devoted to understanding the ionization energy required to remove an electron from an atom. The formal basis for understanding that these energies occur at discrete values and the starting point for a quantitative understanding of the potential energy associated with the motion of an electron with respect to the nucleus, uses the time independent Schr¨odinger equation. We can begin, however, to develop a good appreciation of the details by considering the treatment by Bohr, who used a naive but effective model, which proved highly successful in explaining spectroscopic data using hydrogen as a basis for understanding all atoms. We can start by assuming that the atoms are isolated and are brought together without chemical bonding. The spectroscopic evidence revealing the nature of the hydrogen atom was known well over a century ago. The frequencies of spectral lines are given by Ei − Ek h Rydberg and Ritz originally noted that the reciprocal wavelength of spectral lines had a form νik =

ELECTRON EXCITATIONS IN MATERIALS

9

1 = τi − τk λik where τn = −R/n2 for a positive integer n. R is the Rydberg constant 109,678 cm−1 . The ionization energy of hydrogen is then just hcR −hcτ1 = = 13.6eV e e and the hydrogen atom states can be expressed as V0 =

En = hcτn =

−hRc 13.6 = − 2 eV 2 n n

He+ , Li2+ and Be3+ are isoelectronic with H and the energy levels can be scaled proportionally to Z 2 hcR 13.6 = −Z 2 2 eV 2 n n In the parlance of quantum mechanics, this is known as an accidental degeneracy as this model only works for one electron atoms. A complete mathematical treatment should consist of the appropriate wave equations that satisfy the spectroscopic observations noted above. The appropriate formulation is the time independent Schr¨odinger equation for a one-electron atom En = −Z 2

Ze2 −¯ h2 2 ∇ Ψ(r, θ, φ) − Ψ(r, θ, φ) = EΨ(r, θ, φ) (1.21) 2m r To simplify the problem, we seek solutions that are spherically symmetric, so that the derivatives of the electron wave function, Ψ(r, θ, φ), are zero with respect to the angles θ and φ of our spherical coordinate system. With spherical symmetry, the Laplacian in the Schr¨odinger equation takes a relatively simple form Ze2 −h2 1 ∂ 2 ∂ r Ψ(r) − Ψ(r) = EΨ(r) (1.22) 2 2m r ∂r ∂r r Since E is a constant, acceptable expressions for Ψ(r) must provide an E that is independent of r. Two such solutions, which can be veriﬁed by direct substitution, are −Zr Ψ1s (r) = exp (1.23) a0 Zr Zr exp (1.24) Ψ2s (r) = 2 − a0 2a0 and with these solutions for n = 1 and n = 2 the values of energy are

10

INTRODUCTION

En = −

1 2 Z n2

me4 2¯ h2

=−

1 2 Z ER n2

(1.25)

which is the same as that derived above. In Eq. 1.25 we have deﬁned the energy unit, ER , the Rydberg, which is +13.6 eV. The integer, n, in Eq. 1.25 is sometimes called the “principal quantum number”. The integer, n, is 1 for Ψ1s , 2 for Ψ2s , and so on. It is well known that there are other solutions for Ψ that are not spherically symmetric, for example: Ψ2p , Ψ3p , and Ψ3d . Perhaps surprisingly, for ions having a single electron, Eq. 1.25 provides the correct energies for these other electron wave functions, where n = 2, 3, and 3 for these three examples. This is sometimes called an “accidental degeneracy” for the one electron Schr¨odinger equation, but it does not hold true when the atom has more than one electron. The presence of additional electrons around atoms can be dealt with in what is called the independent-electron approximation. While it is normally important in determining the potential energy to consider each electron of an atom, making it necessary to know its position at a given time, we treat all of the electrons as composing a probability distribution about the nucleus. The attractive force of the electron to the nucleus is then roughly Ze2 /r2 when the electron is close to the nucleus. When the electron is farther than the Z − 1 other electron of the atom, the potential is e2 /r2 . Core loss features in EELS spectra occur at the binding energies of the atomic electrons. This differs from the more familiar energies of characteristic X-ray emission, which depend on the difference in atomic energy levels. It is useful to review these X-ray energies for atoms with one electron and more than one electron. For a one-electron atom, the energy difference between energy levels, ∆E, depends only on differences in the principal quantum number n. Our sign convention is such that the electron energy is more negative when the electron is bound more strongly to the nucleus, and the energy of an X-ray is positive, of course. Consider an X-ray emitted from a one-electron atom or ion when an electron from a 2p state falls into an empty 1s state 1 1 3 Z 2 E R = Z 2 ER ∆E = E2 − E1 = − − (1.26) 22 12 4 The characteristic X-ray energy depends on the state of the higher energy electron (assumed to be in the atom’s L shell, where n = 2) as well as the deep core electron state (assumed to be in the atom’s K shell, where n = 1). The electronic transition L → K emits a particular X-ray called a Kα X-ray. A Kβ X-ray originates with an M → K transition (the M shell has n = 3). Equation 1.26 works well for photon emission from atoms or ions with one electron, but electron–electron interactions complicate the calculation of energy levels of heavier atoms. The essential behavior was determined experimentally by Moseley. Moseley’s laws provide the energies of K series and L series X-rays:

ELECTRON EXCITATIONS IN MATERIALS

1 1 EK = (Z − 1) ER − 2 = 10.204(Z − 1)2 12 2 1 1 EL = (Z − 7.4)2 ER − = 1.890(Z − 7.4)2 22 32

11

2

(1.27) (1.28)

Equations 1.27 and 1.28 are good to ∼ 1% accuracy for X-rays with energies from 3 to 10 keV (not bad for work published in 1914). Moseley correctly interpreted the offsets for Z (1 and 7.4 in Eqs. 1.27 and 1.28) as originating from shielding of the nucleus by other electrons. For the 1s electron of the atom, the other electron in the K shell provides partial shielding of the nuclear charge, so the effective charge seen by one K electron is ∼ Z − 0.5. Electrons in the L shell will be shielded by the two K electrons (1s), and partly shielded from the nucleus by L electrons (2s and 2p), of which there are seven others. For atoms with full L and K shells, Moseley’s empirical law for EK , Eq. 1.27, could perhaps be “improved” by using an expression (Z − 0.5)2 for the K energy and ER (Z − 4.2)2 /4 for the L energy. such as ER . These constants would Doing so, however, would require new constants ER and ER be different from the Rydberg energy because electron–electron interactions affect strongly the electron energy levels of multielectron atoms, and not just because of screening of the nuclear charge. The value of 7.4 in Eq. 1.28 for L series X-rays should be regarded as an empirical parameter. This discussion of the historical Moseley’s laws for X-ray energies shows that it is possible to obtain information about core edges in EELS by using analytic expressions from the hydrogen atom for electron energies or wave functions. For example, the K and L electron wave functions can be assumed to be hydrogenic wave functions, which are well known (Eqs. 1.23 and 1.24 are the correct forms for 1s and 2s electrons, e.g.). The required modiﬁcation is a change of the nuclear charge. The rule that works reasonably well for K electrons is to use Z − 0.5 rather than Z, and for L electrons the rule is to use Z − 4.2. It is sometimes even possible to use hydrogenic wave functions for M electrons, but here the choice of effective Z depends on both Z and the angular momentum. The need for wave functions will be obvious in the discussion of transition matrix elements in Section 1.4 below. The shape of an atomic wave function depends on its orbital angular momentum quantum number, l (which must be < n). In a multielectron atom, the energy of the atom will depend on l. The general trend is for electrons with larger l to have less favorable binding energies, since their larger angular momentum will keep them further from the nucleus. Finally, the third quantum number, m, which describes the orientation of l (and −l ≤ m ≤ l) causes some smaller effects on the electronbinding energy. Outer electrons of an atom are affected by m through chemical bonding to neighboring atoms. The energies of core electrons are affected by m through their interactions with the electron spin, s (which is ± 1/2) and the ﬁne structure of the spin–orbit interaction. The spin can be either parallel or antiparallel to the orbital angular momentum, and the spin–orbit energy depends on the sum l + s, a quantity termed “j”. This spin–orbit splitting is possible to see in Fig. 1.2 for the L edge of Ni. The L3 edge, which is from excitation of the 2p electrons with

12

INTRODUCTION

j = 3/2 (2p3/2 ), is at a slightly lower energy (855 eV) than the L2 edge (872 eV), which is from the excitation of the 2p electrons with j = 1/2 (2p1/2 ). Note that the jump over background (and the white line) at the L3 edge is approximately twice as large as the jump at the L2 edge, since twice as many of the six 2p electrons have j = 3/2 (−3/2, −1/2, +1/2, +3/2) than j = 1/2 (−1/2, +1/2). The ordering of the electron energy states is presented in Table 1.1, together with the nomenclature. n 5 5 5 5 5

l 1 1 0 3 3

j 3/2 1/2 1/2 7/2 5/2

shell O3 O2 O1 N7 N6

4 4 4 4 4

2 2 1 1 0

5/2 3/2 3/2 1/2 1/2

N5 N4 N3 N2 N1

3 3 3 3 3

2 2 1 1 0

5/2 3/2 3/2 1/2 1/2

M5 M4 M3 M2 M1

2 2 2

1 1 0

3/2 1/2 1/2

L3 L2 L1

1

0

1/2

K

Table 1.1 Notation and relative energies of atomic states for electrons. More strongly bound states are shown lower in the diagram; vacuum level is toward the top.

In summary, the energies of the core edges depend on the quantum numbers n, l, and j. These energies are deﬁned precisely for isolated atoms. In a solid, however, effects of local chemistry and coordination can alter these energies by up to several eV. Sometimes these “solid-state effects”, or “electron energy loss near edge structure (ELNES)”, can be interpreted fairly directly. The white lines of transition metals, for example, reﬂect the number of unoccupied 3d states at the atom. More typically, however, the interpretation of ELNES is a complicated multielectron problem. The difﬁculty is that measurements are made on energy levels of innermost electrons, whereas the chemical bonding involves the outer electrons of the atom. Chemical

INSTRUMENTATION

13

bonding affects core electrons because changes in the outermost electrons perturb the intermediate electrons, and these perturbed intermediate electron states in turn perturb the core electrons. Calculations of these inner shell polarization phenomena require some expertise with atomic structure computations. Some trends with valence can be understood easily. For example, the K edge of a metal atom in oxide form typically lies at a larger energy than in the metallic state. The loss of outer electrons in the oxide gives the atom a greater ratio of nuclear charge to electron charge, so the energy levels of the electrons are lowered a little bit, typically a couple of eV (so the K edge is seen at larger energies in an energy loss spectrum). A number of such cases can be found in spectra in the EELS Atlas. 1.3 INSTRUMENTATION To measure the spectrum of electron energy losses, an EELS spectrometer can be mounted after the projector lenses of the TEM. The heart of most EELS spectrometers is a magnetic sector, which provides the energy dispersion of the electrons. In the homogeneous magnetic ﬁeld of the sector, Lorentz forces will bend electrons of equal energies into arcs of equal curvature. Some such electron trajectories are shown in Fig. 1.4. The spectrometer must accept an angular spread of the electrons entering the magnetic sector, both for reasons of intensity and for measuring how the choice of scattering angle, φ, affects the features of the spectrum (φ is proportional to ∆k in Eq. 1.44). A well–designed magnetic sector provides good focusing action. Focusing in the plane of the paper of Fig. 1.4 (the equatorial plane) is provided by the magnetic sector because the path lengths of the outer trajectories are longer than the path lengths of the inner trajectories. It is less obvious, but also true, that the fringing ﬁelds at the entrance and exit boundaries of the sector provide an axial focusing action. With proper electron optical design, the magnetic sector can be made “double focusing” so that the equatorial and axial divergences are focused to the same point on the right of Fig. 1.4. In a “serial spectrometer”, a pair of slits is placed at the focal plane of the magnetic sector, and a scintillation counter is mounted after the slits. Intensity is recorded only from those electrons bent through the correct angle to pass through the slits. The energy loss, E, is scanned by ramping the current in the coils of the magnet; with a higher magnetic ﬁeld, intensity is measured from electrons of higher energy. (The higher energy electrons have undergone smaller energy losses, so are of smaller E.) Since the energy losses are small in comparison to the incident energy of the electrons, the energy dispersion at the focal plane of typical magnetic sectors is only a few microns per eV. With narrow slits and a stable magnet, an energy resolution of ∼ 2 eV is possible with this arrangement. A “parallel spectrometer” does not use a pair of slits, but instead uses a scintillator and a position–sensitive photon detector such as a linear photodiode array. It is typical to use a set of postﬁeld lenses to magnify the energy dispersion before the electrons reach the scintillator. A parallel spectrometer has an enormous advantage over a serial spectrometer in its rate of data acquisition.

14

INTRODUCTION

Fig. 1.4 Some electron trajectories through a magnetic sector with uniform magnetic ﬁeld, B. The light curves are trajectories for lower energy electrons (those with larger energy loss, E), and heavier curves are for higher energy electrons.

The optical coupling of a magnetic sector spectrometer to the microscope is usually done by arranging the object plane of the spectrometer to be at the back focal plane of the ﬁnal projector lens. This back focal plane contains the diffraction pattern of the sample when the microscope is in image mode. When the microscope is operated in image mode, the spectrometer is therefore said to be “diffraction coupled” to the microscope. With diffraction coupling, the collection angle, β, of the spectrometer is controlled by the objective aperture of the microscope. Alternatively, the microscope can be operated in diffraction mode. In diffraction mode, the back focal plane of the projector lens contains an image, and the spectrometer is said to be “image coupled” to the microscope. With image coupling, the collection angle, β, is controlled by an aperture at the entrance to the spectrometer. 1.4 1.4.1

CORE EDGES Core Electron Matrix Elements

Here we calculate the strength of an inelastic scattering process involving the excitation of a core electron. A high energy electron excites a second electron from a bound atomic state into a state of higher energy. Since two electrons are involved, for conciseness we employ the Dirac bra and ket notation. The high energy electron (with coordinates r1 ) is initially in a plane wave state |k0 , and after scattering it is

CORE EDGES

15

in the state |k. The atomic electron (with coordinates r2 ) is initially in the bound state |α. After scattering, electron 2 is in the state |β, which may be either a bound state that is initially unoccupied, or a plane wave state if electron 2 is ejected from the atom. For inelastic scattering, |k| = |k0 |, and α = β. The Schr¨odinger equation with the initial state is written as H0 |k0 , α = E|k0 , α

(1.29)

So long as our two electrons are far apart and therefore non-interacting, our twoelectron system obeys the unperturbed Hamiltonian H0 = −

¯2 2 h ¯2 2 h ∇1 − ∇ + V (r2 ) 2me 2me 2

(1.30)

With different coordinates, each Laplacian in Eq. 1.30 acts on only one of the two electrons, and the potential energy term involves only electron 2. In such problems we can express the initial state as a product of one-electron wave functions: |k0 , α = |k0 |α, and the ﬁnal state as |k, β = |k|β As an example of using a product wave function in Eq. 1.30, the factor for electron 2, |β, is a constant under the operations of ∇21 , and |k is a constant under the operations of ∇22 and V (r2 ). A “constant factor” does not affect the solution of the Schr¨odinger equation for the other wave function of the product.3 The Hamiltonian of Eq. 1.30 is therefore equivalent to two independent hamiltonians for two independent electrons. This is as expected when the two electrons have no interaction. As the high energy electron approaches the atom, we must consider two perturbations of our two-electron system. One perturbation is the Coulombic interaction of electron 2 with the electron 1, which is e2 /|r1 − r2 | . The second perturbation is the interaction of electron 1 with the potential from the rest of the atom, V (r1 ). The perturbation Hamiltonian is H =

e2 + V (r1 ) |r1 − r2 |

(1.31)

This perturbation H couples the initial and ﬁnal states of the system. The stronger the coupling, the more probable will be the transition from the initial state |k0 |α to the ﬁnal state |k|β. It is a result from time-dependent perturbation theory that the wave function of the scattered electron 1 will be an outgoing spherical wave times f (k, k0 ), where −me β| k|H |k0 |α (1.32) 2π¯ h2 Substitution of Eq. 1.31 into 1.32 gives Eq. 1.33 below. In the second term of Eq. 1.33, the coordinates of electron 2 appear only in the atomic wave functions |α and |β, f (k, k0 ) =

3 The

wave functions for electron 1 commute with the terms for electron 2, and vice versa.

16

INTRODUCTION

so these wave functions are moved out of the integral involving the coordinates of electron 1

f (k, k0 ) =

−me 2π¯ h2

e2 β| k|

1 |k0 |α + β|α k|V (r1 )|k0 |r1 − r2 |

(1.33)

For inelastic scattering we have α = β, so the second term is zero by the orthogonality of the atomic wave functions. To be explicit in notation, we denote the inelastic contribution to f (k, k0 ) as fin (k, k0 ). To calculate fin (k, k0 ), we use spatial coordinate representations for our wave functions. We change variables: R = r1 − r2 and ∆k = k − k0 , and separate the integrations in the nonzero ﬁrst term of Eq. 1.33, which becomes fin (k, k0 )

=

−me 2 e 2π¯ h2

∞ exp(−i∆k · R) ∞

∞

1 3 d R |R|

exp(−i∆k · r2 )Ψβ ∗ (r2 )Ψα (r2 )d3 r2 (1.34)

∞

We see that the only dependence of fin on k and k0 is through their difference, ∆k. The integral over R is 4π/∆k 2 −2me e2 fin (k, k0 ) = 2 2 h ∆k ¯

∞ exp(−i∆k · r2 )Ψβ ∗ (r2 )Ψα (r2 )d3 r2

(1.35)

∞

We can now obtain the total cross section for inelastic scattering, dσin /dΩ, as ¯ 2 /me e2 fin ∗ fin . With the deﬁnition of the Bohr radius, a0 = h +∞ 2 4 dσin (∆k) 3 ∗ = 2 4 exp(−i∆k · r2 )Ψβ (r2 )Ψα (r2 )d r2 dΩ a0 ∆k

(1.36)

−∞

The electron–electron interaction potential of e2 /|r1 − r2 | is the perturbation that caused the transfer of energy between the two electrons, as required for the simultaneous transition of the two-electron state from |k0 |α to |k|β. Energy is conserved in the overall process, so a spectrum of electron energy losses of the high energy electron will show enhanced intensity at energy losses, E, greater than the energy difference of the states of the atomic electron, Eαβ ≡ Eβ − Eα . If we had accurate wave functions for Ψα and Ψβ , we could use Eq. 1.36 to calculate the strength of this inelastic scattering, and the intensity of the electron energy loss spectrum at energies > Eαβ . To do so, however, we must ﬁrst relate the experimental

CORE EDGES

17

conditions to the idealized form of Eq. 1.36. Speciﬁcally, we need to know how a typical experimental conﬁguration selects ∆k at different energy losses. This is the topic of the next section 1.4.2. Finally, we note a subtle deﬁciency of Eqs. 1.35 and 1.36. The excitation of a core electron changes the electronic structure of the atom. It is not necessarily true that free atom wave functions are appropriate for Ψα or Ψβ when a core hole is present, so the second term in Eq. 1.33 may not be strictly zero by orthogonality. 1.4.2

Experimental Kinematics and Cross-Sections

Inelastic scattering involves a change in both the direction and magnitude of the wavevector of the high energy electron. Equation 1.36 includes explicitly the change in momentum, ∆k, and the states |α and |β differ in energy, of course. The two changes are interrelated, and we need to know how an experimental measurement will sample a range of momentum transfer and energy loss. Total momentum is conserved, and before scattering the total momentum is with the incident electron, p0 = me v0 = h ¯ k0 . After scattering, the momentum transfer to the atomic electron must be ¯ h∆k = h ¯ (k−k0 ). From Fig. 1.5 we see that for a given energy loss (denoted E, and determined from the change in length of the wavevector of the electron), the ˆ 0 . Larger values of ∆k are minimum change in wavevector is ∆kmin = (|k| − |k0 |)k possible for this same E, but momentum conservation requires that the heads of the wavevector ∆k lie along the circle of radius k. In EELS, we measure the spectrum of energy losses from electrons in some range of ∆k, set by the angle, β, of a collection aperture (see Fig. 1.4). To understand the intensity of core loss spectra, we need to know how the inelastic scattering depends on both the scattering angle φ and E. We start with the φ-dependence for ﬁxed E. For small ∆k we can approximate, as shown on the right of Fig. 1.5 2 ∆k 2 = k 2 φ2 + ∆kmin

q

k0kq0q kq ~ kq k

k0 6k min

(1.37)

6kmin

6k

6k

Fig. 1.5. Kinematics of inelastic electron scattering. Left: deﬁnitions. Right: Enlargement (valid for small φ, or equivalently for small ∆k).

The increment in solid angle covered by an increment in φ (making a ring centered about k0 ) is dΩ = 2π sin φ dφ By differentiating Eq. 1.37 (for ﬁxed E, ∆kmin is a constant):

(1.38)

18

INTRODUCTION

φ dφ =

∆k d∆k k2

(1.39)

so that ∆k d∆k k2 Substituting Eq. 1.40 into Eq. 1.36 provides dΩ = 2π

(1.40)

+∞ 2 3 ∗ exp(−i∆k · r2 )Ψβ (r2 )Ψα (r2 )d r2 −∞ (1.41) EELS spectra are measurements of the intensity of inelastic scattering as a function of energy. When Ψβ is a bound state of the atom, Eq. 1.41 can be used directly to obtain an EELS intensity at the energy corresponding to the transition α → β. On the other hand, in the more typical case when Ψβ is in a continuum of states (such as an energy band), we need to scale the result of Eq. 1.41 by the number of states in the energy interval of the continuum, which is ρ(E)dE. Here, ρ(E) is the “density of unoccupied states” available to the atomic electron when it is excited. These unoccupied states can be free electron-like (meaning that the atomic electron leaves the atom), or they can be atomic states (if there are unoccupied atomic electron states at the atom). Accounting for the density of states of Ψβ gives the double differential cross section: dσin dΩ 8π dσin (∆k) = = 2 2 3 d∆k dΩ d∆k a0 k ∆k

+∞ 2 8π d σin (∆k, E) exp(−i∆k · r2 )Ψβ ∗ (r2 )Ψα (r2 )d3 r2 = 2 2 3 ρ(E) d∆kdE a0 k ∆k −∞ (1.42) The “generalized oscillator strength”, GOS, or Gαβ (∆k, E) is a property of the atom deﬁned as: 2

2m Gαβ (∆k, E) = Eαβ 2 2 h ∆k ¯

+∞ 2 3 ∗ exp(−i∆k · r2 )Ψβ (r2 )Ψα (r2 )d r2 (1.43) −∞

Here Eαβ is the difference between the energies of the states Ψα and Ψβ . The GOS is the probability of the transition α → β, normalized by a factor related to the energy and momentum transfer. Using this deﬁnition for Gαβ (∆k, E) of Eq. 1.43 in Eq. 1.42 2π¯ h4 d2 σin (∆k, E) 1 = 2 2 ρ(E)Gαβ (∆k, E) d∆kdE a0 m Eαβ T ∆k

(1.44)

ANALYSES

19

where we have deﬁned the kinetic energy, T , as: T =

h2 k 2 ¯ p2 = 2me 2me

(1.45)

It is interesting to compare inelastic electron scattering to inelastic X-ray scattering. For electric dipole transitions induced by X-rays, the GOS differs from Eq. 1.43 in that the exponential, exp(−i∆k · r2 ), is replaced by a dipole operator, er2 . For small values of ∆k, the integral in Eq. 1.43 is identical for both electron and X-ray inelastic scattering, and X-ray and electron absorption edges look very similar.4 Nevertheless, the ∆k dependence of EELS spectra is signiﬁcantly different from that for inelastic scattering spectra of photons. Speciﬁcally, the double differential cross-section of Eq. 1.42 decreases strongly with ∆k as ∆k −4 . Reviewing Eq. 1.35, we see that this originates from the nature of electron scattering by a Coulomb potential. Evaluation of the GOS of Eq. 1.43 requires explicit wave functions for atomic electrons. We can, however, understand a general feature of the GOS that occurs at large energy losses, well above the energy transfer needed to excite the core electron from the atom. In this case, the scattering is equivalent to the scattering of the high energy electron by an unbound, free electron. For a√particular energy loss E , the function Gαβ (∆k, E ) shows a peak when ∆k = 2mE /¯h. This peak in ∆k corresponds to the momentum transfer in the classical problem of elastic scattering of two particles of equal mass. This peak is less well deﬁned at energy transfers closer to the energy of the atomic transition, but is still visible. In a two-dimensional plot of Gαβ (∆k, E ) versus ∆k and E , this peak is called the “Bethe ridge”. 1.5 ANALYSES Subsequent chapters of this book will describe the speciﬁc details related to the concepts discussed here. These range from details of instrumentation, low loss, core loss, near and extended ﬁne structure, covered in the ﬁrst half of the book. The latter half of this book describes how this information has been applied to speciﬁc material systems. All of the chapters provide reviews and references that will be of use for the study of speciﬁc systems and that can be applied universally.

4 For small ∆k, exp(−i∆k · r ) ≈ 1 − i∆k · r . After substituting this expression into Eq. 1.43, we 2 2 ﬁnd the term β|1|α = 0 by orthogonality, and the term β|i∆k · bf r 2 |α is the dipole term.

2

Experimental Techniques and Instrumentation Ray F. Egerton Department of Physics University of Alberta Edmonton, Canada T6G 2J1

Abstract To ensure reliable and/or quantitative results from electron energy loss spectroscopy or energy ﬁltered imaging, it is desirable to have some basic understanding of the operation of an electron spectrometer (including the electron optics and performance of the electron detector) and an appreciation of certain aspects of electron scattering. In this chapter, we discuss the commercially available instrumentation and how it is used, with some emphasis on elemental microanalysis. 2.1 INTRODUCTION The most notable progress in EELS during the last decade has been in its application to materials problems, a subject that is dealt with in the succeeding chapters of this book. In this chapter, we review the commercially available instrumentation and the basic techniques involved in its operation. Most EELS and EFTEM systems now use a diode-array detector to record the energy loss spectrum or an energy-ﬁltered image, and employ a personal computer for data acquisition and data analysis. Another recent innovation is the availability of reliable ﬁeld-emission electron sources for several microscopes, which offer the possibility of improved spatial and energy resolution. Specimen preparation remains a key ingredient in most applications of EELS, and still demands careful technique and patience [2.1–2.3]. However, the recent trend towards higher accelerating voltages (200 kV or more) for materials-science TEMs somewhat alleviates the specimen-thickness requirements. Moreover, specimens which are initially “clean” now remain so within the beam, thanks to the improved vacuum conditions of modern microscopes.

21

22

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

2.2

ELECTRON SPECTROMETERS AND IMAGING FILTERS

To separate the different energy losses that fast electrons (kinetic energy E0 of the order of 105 eV) incur while passing through a thin specimen, we need a spectrometer that is capable of a resolution of the order of 1 part in 105 . This requirement rules out electrostatic designs, which sufﬁce for electron energies below 1000 eV, in favor of magnetic ones. 2.2.1

The Magnetic Prism

The basic component employed in all EELS and energy-ﬁltering systems is the magnetic prism: essentially an electromagnet whose polepieces produce a magnetic ﬁeld B perpendicular to the incoming electron beam. As illustrated in Fig. 2.1, this ﬁeld has several effects on the electron trajectories. O

x (a)

z u R

Bn Bx

Ix B

vx

(b) O

Bz

B

y x

z

Iy

B

z

y

y vy

Fig. 2.1 Focussing and dispersive properties of a magnetic prism, represented in terms of curvilinear coordinates, the x and y axes being always perpendicular to the central trajectory (the z-axis). (a) In the x–z plane, the electrons are deﬂected by an amount which is larger at lower kinetic energy (dashed lines), while being focused to an image Ix . (b) In the perpendicular direction, electron trajectories are bent at the magnetic ﬁeld boundaries to give an image Iy ; this image is coplanar with Ix in the case of a double-focusing spectrometer.

First, the magnetic force Bev on each electron (speed = v) bends the entire beam through an appreciable angle, often chosen as 90◦ for mechanical convenience.

ELECTRON SPECTROMETERS AND IMAGING FILTERS

23

Second, because the force depends on v, the electrons emerging from the magnetic ﬁeld are spatially separated (dispersed) according to their kinetic energy. Third, and less obvious, the spectrometer has a focusing action: electrons that stray from the optic axis return to the axis at the spectrometer image plane where those of a given energy loss are brought to a focus. Focusing within the bend (x–z) plane takes place because electrons that deviate from the optic axis traverse different path lengths within the ﬁeld (Fig. 2.1a). Focusing in the perpendicular (y) direction arises from the fact that electrons traveling off-axis experience a restoring force at the entrance and exit faces (Fig. 2.1b) and this force increases with deviation from the axis. By appropriate choice of the inclinations of the entrance and exit edges of the prism, the focusing powers in the x and y directions can be made equal; a point source at a particular location O then gives rise to a single point image and the prism is described as fully stigmatic or double-focusing. In practice, the focusing is not perfect: Electrons originating from a point source in the object plane do not pass through precisely the same point in the image plane. However, such aperture aberration can be largely corrected by curving the entrance and exit boundaries of the magnet to calculated radii, resulting in a reduced disk of confusion in the image plane and consequently better energy resolution in the energy loss spectrum. Alternatively, the angular spread of the entrance beam can be larger for a given energy resolution, giving the aberration-corrected spectrometer a greater collection efﬁciency. The single magnetic prism can also be used as an energy ﬁlter, by placing an narrow energy-selecting slit in the spectrum plane. An electron detector placed behind the slit will record only electrons of a well-deﬁned energy loss (depending on the slit location and width). In the case of a scanning-transmission electron microscope (STEM), which forms its image by scanning a focused electron beam across the specimen, the time-dependent output of the detector is then an energy-ﬁltered image. Alternatively, we can obtain energy-ﬁltered images using the ﬁxed-beam TEM mode. A magnetic prism mounted beneath a TEM column normally employs the projector-lens crossover as its object point O. But the microscope also produces a magniﬁed image of the specimen (or its diffraction pattern) at the level of the viewing screen, closer to the spectrometer entrance. This screen-level intensity distribution is imaged by the magnetic prism at a plane further along the z-axis, behind the energy loss spectrum. If a slit is inserted in the spectrum plane, this subsequent image will be an energy-ﬁltered image or diffraction pattern. However, the spectrometer is doublefocusing and aberration-corrected only for a single image plane, corresponding to the spectrum formed at I. The ﬁltered image will therefore be astigmatic and will suffer from distortion and various types of aberration. Fortunately these effects can be corrected by subsequent electron optics; the Gatan GIF ﬁlter uses a series of quadrupole and sextupole lenses to project either the energy loss spectrum or a distortion-free energy-ﬁltered image onto a two-dimensional electronic detector (a CCD camera), as shown in Fig. 2.2

24

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

Fig. 2.2 Gatan GIF 200 imaging ﬁlter, based on a 10-cm radius magnetic prism followed by quadrupole and sextupole lenses [2.4].

2.2.2

Prism-Mirror Spectrometer

We have said that a point object, placed at an appropriate distance from a magnetic prism, gives rise to a point image after the electrons have passed once through the prism. If an electrostatic mirror (consisting of one or more electrodes biassed at approximately the electron source potential) is placed at this image plane, the electrons will be reﬂected back into the prism and suffer a further deﬂection; see Fig. 2.3a. For an original deﬂection angle of 90◦ , the electrons will arrive back onaxis, allowing the prism-mirror device to be inserted within the column of a TEM, below the objective lens.

Fig. 2.3 Optics of the prism-mirror energy ﬁlter, showing the real-image points R1 -R3 and virtual-image points V1 -V3 . Bias voltages Va and Vc adjust the apex position and curvature of the electrostatic mirror.

INFLUENCE OF THE MICROSCOPE OPTICS

25

Furthermore, as demonstrated by Castaing and Henry [2.5] at the University of Paris, an object placed at a second point V1 can give rise to achromatic images at V2 and V3 . Although V2 and V3 are virtual images, as indicated by the dotted-line extrapolations in Fig. 2.4b, a postobjective lens can be employed to place a virtual image of the specimen at V1 and its diffraction pattern at R1 . The remaining TEM lenses can be used to project the specimen image V3 (as a real magniﬁed image of the specimen) onto the viewing screen, and this image will be energy-ﬁltered if a slit is inserted at the energy-dispersed diffraction plane R3 . Alternatively, the postobjective lens may produce a diffraction pattern at V1 and an image of the specimen at R1 . An energy-selecting slit inserted at the spectrum plane R3 will then result in a ﬁnal image which is an energy-selected diffraction pattern of the specimen. In either case, the energy loss spectrum can be displayed by removing the energy-selecting slit and projecting R3 onto the TEM screen, or recorded by lifting the screen to expose a CCD array. 2.2.3

Omega-Filter Spectrometer

The omega ﬁlter is an alternative energy-ﬁltering device that achieves similar results to the prism-mirror system but employs only magnetic prisms. Because it requires no connection to the microscope high-voltage supply, it is the favored option for operating voltages > 100 kV. As illustrated in Fig. 2.4, the ﬁlter consists of four separate magnetic prisms which bend the electron beam in the shape of a Greek letter Ω, such that the exit beam emerges in-line with the entrance beam. It is inserted below a post-objective lens, as in the case of the prism/mirror ﬁlter. The dispersions within each magnetic prism are additive; an energy loss spectrum is formed at position D2 , which is conjugate to the real image D1 . To produce an energy-ﬁltered image of the specimen, D1 is a diffraction spot and planes O1 , O2 , and O3 contain images of the specimen; an energy-selecting slit is inserted at D2 and the remaining microscope lenses project and magnify the O3 image onto the TEM screen. For energy ﬁltering of a diffraction pattern, the microscope optics are readjusted to form a specimen image at D1 and a diffraction pattern at O1 . In either case, the energy loss spectrum is displayed by refocusing the TEM lenses so as to project D2 onto the TEM screen or recording plane. Because of the symmetry of the omega ﬁlter, second-order aperture aberrations are completely compensated provided the system is well aligned. In some designs [2.6], the second and third prisms (whose magnetic ﬁelds point in the same direction) are joined, although one advantage of the four-prism design is that a sextupole lens can be inserted at the midplane to compensate for other second-order aberrations [2.7]. 2.3

INFLUENCE OF THE MICROSCOPE OPTICS

The performance of an electron spectrometer or imaging ﬁlter system depends on how all the lenses in the electron-optical column are utilised. Because the microscope

26

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

Fig. 2.4 Optics of the omega ﬁlter: achromatic images of the specimen are formed at O1 , O2 , and O3 , while D2 , contains an energy-dispersed diffraction pattern. Second-order aberrations can be controlled by careful choice of the design parameters or by the use of sextupole lenses S.

operator has control of these lenses, their settings will affect the quality and reliability of the data obtained. 2.3.1

Dedicated-STEM System

A dedicated scanning transmission electron microscope (STEM) uses a ﬁeld-emission electron source and relies on scanning of the incident beam to produce its images. In most cases, the electron spectrometer directly follows the specimen, which acts as the electron-optical object for the spectrometer [2.8]. An aperture mounted on the optic axis (near the entrance to the spectrometer) selects the range β of scattering angles being analysed. In the case of higher resolution STEM machines, this simple situation is complicated by the fact that the specimen is immersed within the magnetic ﬁeld of the objective lens and an appreciable post ﬁeld focuses scattered electrons toward the optic axis. The angular range of electrons entering the spectrometer is thereby reduced, resulting in less spectrometer aberration and higher energy resolution [2.9].

INFLUENCE OF THE MICROSCOPE OPTICS

2.3.2

27

TEM Optics for EELS

An electron spectrometer added to a conventional (as opposed to scanning) transmission electron microscope (TEM) is mounted beneath the camera chamber, so electrons that emerge from the specimen must pass through all of the microscope imaging lenses before reaching the spectrometer. Although these lenses can actually improve the spectrometer performance, any improvement is dependent on the way in which the lenses are used. In normal TEM operation, the projector lens forms a crossover SO located just below the lens bore, a distance h (typically 30 – 40 cm) above the TEM viewing screen; see Fig. 2.5. In commercial systems, this crossover is used as the object point for the spectrometer. With suitable excitation of the intermediate lenses (located between the objective and projector), the angular divergence of the electrons emerging from the crossover can be kept small enough to ensure good energy resolution, as in the STEM case discussed above. However, the TEM situation is more complicated because of the greater ﬂexibility of the optics and the presence of several apertures in the lens column. IMAGE MODE S ß

DIFFRACTION MODE S O OA SAD I

P SO

SO

h VS SEA

SEA

Fig. 2.5 TEM optics for microscope image mode (on the left) and for diffraction mode (on right). S represents the specimen; O, I, and P represent the objective lens, intermediate-lens system and the ﬁnal projector lens; SO and VS are the spectrometer object point and viewing screen; OA, SAD and SEA are the objective, selectedarea-diffraction and spectrometer entrance apertures.

If the microscope is operated in image mode (an image of the specimen of magniﬁcation Mvs being present on the viewing screen), the projector-lens crossover contains a diffraction pattern of very small camera length L = h/Mvs (typically < 1 mm) and the spectrometer is said to be diffraction coupled. In this mode of operation, the collection angle β (the maximum scattering angle contributing to the recorded data) can be controlled by an objective aperture, while the area of analysis is normally selected by a spectrometer entrance aperture (SEA) located just below the viewing screen, as shown in Fig. 2.5. If the TEM is operating in diffraction mode, the projector crossover contains an image of the illuminated area of the specimen, of magniﬁcation M = h/Lvs (typically of the order of unity) and the spectrometer is image coupled. Usually, the collection angle β is determined by the SEA and the region of analysis by the incident beam diameter, or possibly by a selected-area-diffraction (SAD) aperture.

28

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

For both types of spectrometer coupling, there exist values of L and M that optimize the energy resolution of the system, by reducing the angular divergence of electrons entering the spectrometer without too much increase in spectrometer object size. In practice, the energy resolution is better than 3 eV provided Mvs > 5000 (in imaging mode) or Lvs > 10 cm (for analyzed areas < 1-µm diameter) in diffraction mode [2.9]. 2.3.3

TEM-Lens Aberrations

Like the spectrometer itself, each TEM imaging lens suffers from aperture aberration and these aberrations have several unwanted effects on energy loss spectroscopy. 1. At the level of the spectrometer object plane, aberrations broaden the crossover and thereby degrade the energy resolution; see Fig. 2.6. This effect is usually small [2.9] but might be important for ﬁne-structure studies.

SO CTEM screen SEA SI

Fig. 2.6 Aberration at the spectrometer entrance plane SO gives rise to a broadening in the spectrometer image SI (indicated by vertical arrows) and a corresponding loss in energy resolution. Aberration at the level of the spectrometer entrance aperture may cause a loss in spectrometer collection efﬁciency if resulting electron trajectories (dashed) are intercepted by the SEA.

2. At the level of the SEA (just below the TEM screen), aberrations broaden a single point in the image or diffraction pattern into a disk of confusion. If the diameter of this disk exceeds the SEA diameter (Fig. 2.6), the collection efﬁciency of the spectrometer will be reduced. Chromatic aberration is especially serious, since the corresponding reduction factor increases with energy loss and will vary between different ionization edges, thereby altering the measured atomic ratios. 3. The chromatic and spherical broadening at any image plane containing an area-determining aperture (the SEA in image mode, selected-area aperture in diffraction mode) can be referred back to the specimen plane (by dividing by image-plane magniﬁcation) to give an equivalent reduction in spatial resolution. The following is a semiquantitative discussion of effects (2) and (3), to show their effect in the two modes used in energy loss spectroscopy.

29

INFLUENCE OF THE MICROSCOPE OPTICS

2.3.4

Avoiding Lens-Aberration Artifacts in Image-Mode EELS

Because image-plane angular divergence decreases down the lens column, the main contribution to image blurring comes from aberrations of the objective lens. Spherical aberration sets a resolution limit (referred to the specimen plane) of bs = Cs β 3 or 3 (whichever is smaller), where θE ∼ E/2E0 , E = energy loss, E0 = incident Cs θE beam energy. For core loss spectroscopy, a typical value of β or θE is 5 mrad, giving bs ≈ 2 nm. Chromatic aberration leads to a resolution limit bc which is the smaller of Cc βE/E0 and Cc θE E/E0 . Both expressions give bc = 100 nm if E = 1000 eV, E0 = 100 kV, Cc = 2 mm and b = θE = 5 mrad. Clearly, chromatic aberration dominates over spherical aberration in terms of its effect on the spatial resolution of core loss spectroscopy in imaging mode, so we neglect spherical aberration in the remaining discussion. There are two possibilities for avoiding the E-dependent reduction of collection efﬁciency in image mode. The ﬁrst is to ensure that the total width of the illumination at the entrance-aperture plane, including chromatic aberration, is less than the SEA diameter; see Fig. 2.7a. In other words Mvs (r + bc ) < R

(2.1)

where r is the radius of the incident beam at the specimen and R is the SEA radius. Taking R = 1 mm and bc = 100 nm, this condition can be fulﬁlled (with r > 0) only if Mvs < 104 . In this case, the area of analysis is being selected (at the specimen plane) by the incident probe, the small value of r required being obtained by operating the TEM in a small-probe mode.

Fig. 2.7 Proﬁles of electron intensity at the plane of the spectrometer entrance aperture, without (solid lines) and with (dashed lines) chromatic aberration of postspecimen lenses. The shaded areas in cases (b) and (e) represent electrons which are rejected by the SEA, as a result of chromatic aberration..

30

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

The second possibility is to use sufﬁciently broad illumination at the specimen (large r), so that the chromatic broadening of the illumination occurs completely outside the spectrometer entrance aperture (see Fig. 2.7c) and does not inﬂuence the measured core loss intensity. The necessary condition is Mvs (r − bc) > R

(2.2)

For R = 1 mm, bc = 100 nm and Mvs = 104 , Eq. 2.2 requires r > 200 nm, a value easily achieved. Provided the specimen is uniform within a radius r, there is no net reduction in the number of core loss electrons passing through the aperture, a condition that has been previously referred to as compensation [2.10]. This is a useful option for specimens whose thickness and composition are slowly varying, but is not applicable to some microanalysis applications. 2.3.5

Avoiding Lens-Aberration Artifacts in Diffraction-Mode EELS

It can be shown that the main chromatic effect arises from the intermediate lenses [2.11], which cause the central spot of the diffraction pattern (and the Bragg spots, for a crystalline specimen) to be broadened into a disk of radius Bc = Ci

(2.3)

where Lvs is the camera length of the pattern, d is the diameter of the area selected for analysis, f is the focal length of the objective lens, E is the energy loss being analyzed and E0 is the electron incident energy. Ci is a property of the intermediatelens system; for a Philips CM12, it has been measured to be 30 cm [2.11]. To avoid E-dependent loss of collection efﬁciency (and consequent errors in elemental ratios), all the electrons in the central diffraction disk should pass through the spectrometer entrance aperture, of radius R. This occurs (Fig. 2.8d) if (αLvs + Bc ) < R

(2.4)

where α is the angular width of the illumination incident on the specimen, giving a radius αLvs for the central disk of the diffraction pattern. Under most TEM conditions, the ﬁrst term in Eq. 2.4 can be neglected. If so, Eqs. 2.3 and 2.4, together with the relation β = R/Lvs (assuming that the spectrometer entrance aperture determines the collection semiangle), are equivalent to: 2 f E0 d< β (2.5) Ci E Taking β = 5 mrad, f = 2 mm, Ci = 30 cm, E0 /E = 100, Eq. 2.5 gives d < 6.7 µm. This condition is readily achieved by using a reasonably small SAD aperture or a small probe. In practice, aberration effects may arise as a result of misalignment of the intermediate lenses, unless the ‘voltage center’ is adjusted in TEM diffraction mode [2.11].

RECORDING SYSTEMS FOR SPECTRA AND IMAGES

2.3.6

31

Electron Optics of Energy Filtered (EFTEM) Imaging

The aberrations of the TEM imaging lenses also affect the spatial resolution in an energy-selected image or an elemental map formed using an in-column or postcolumn imaging ﬁlter. In this case, however, chromatic aberration can be reduced by raising the microscope high voltage by an amount equal to the desired energy loss, leaving the ﬁlter excitation constant. If the specimen image has been focused with the zero-loss peak passing through the center of the energy-selecting slit, chromatic broadening remains zero for electrons that pass through the center of the slit. Averaging over all electrons that pass though a slit of energy width ∆, the loss of resolution (if deﬁned according to the Rayleigh criterion) is [2.10] ri ≈

Cc 4

∆ E0

2 (2.6)

and is typically quite small (ri = 0.02 nm for Cc = 2 mm, ∆ = 20 eV and E0 = 100 kV). 2.4 RECORDING SYSTEMS FOR ENERGY LOSS SPECTRA AND ENERGY FILTERED IMAGES Nowadays, electronic methods of recording spectra and images are generally preferred to the use of photographic recording. In the case of spectral recording there are two alternative procedures, serial and parallel, depending on whether the different energy losses are recorded in sequence or simultaneously. 2.4.1

Serial EELS

In a serial-recording EELS system, a narrow slit is located at the spectrometer image plane, in front of a single-channel electron detector. Electrons passing through this slit can be detected in various ways but the most common system uses a scintillator (to convert the electrons to visible light) followed by a photomultiplier tube (PMT) to produce an electrical signal; see Fig. 2.8. Current from the PMT can be digitized, using an analog-to-digital or a voltageto-frequency converter, to allow electronic storage in some form of multichannel analyzer (MCA), usually a microcomputer. Alternatively, at low intensity levels each energy loss electron gives rise to an individual pulse at the PMT anode, enabling the spectral intensity to be recorded by counting these pulses. A low-level discriminator can be set to reject spurious pulses arising from PMT noise. The MCA supplies a ramp signal that is applied to the spectrometer; by changing the magnetic ﬁeld slightly, the spectrum is scanned across the detector slit and a time-dependent signal (representing the energy loss spectrum) is stored in successive memory locations (channels) of the MCA, a technique known as multichannel scaling. If an extended range of energy loss is of interest (e.g., from the zero-loss peak to inner-shell edges), the electron intensity will vary over many orders of magnitude.

32

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

Fig. 2.8 A serial-recording EELS system. The value of Rs determines the energy scale of the spectrum (eV/channel).

To avoid storage problems, the spectrum can be broken up into two or more ranges, separated by a gain change (typically a factor of 100–1000) achieved by varying the sensitivity of the photomultiplier, either by changing the photocathode voltage or by changing from analogue detection to electron counting. The transition can be made automatically at a chosen MCA channel by sending a signal from the multichannel analyzer to the PMT power supply (as in Fig. 2.9) or to the signal-processing circuitry. The electron slit and its surroundings must be carefully designed so as to minimize the spectrometer background which arises (mainly) from high-energy electrons which are backscattered from the slit blades and which, through multiple scattering, eventually reach the scintillator. If the spectrum extends to several kilovolt loss, it may be necessary to subtract this E-dependent background from the acquired data [2.9, 2.12, 2.13]. It is convenient to make the slit width variable. A narrow slit provides the best energy resolution (limited by spectrometer aberrations and by the energy width of the electron source, typically 1–2 eV) but gives the lowest amount of signal, so is more suitable for recording low energy losses. To record inner-shell edges, a wider slit may be necessary to obtain data which are not dominated by shot noise. A plastic scintillator allows pulse counting up to several MHz but suffers from radiation damage and must be changed or mechanically shifted from time to time. It may be necessary to correct for response nonlinearity for count rates > 1 MHz [2.14]. 2.4.2

Parallel EELS

The main disadvantage of the serial method of spectrum recording is that it imposes a low collection efﬁciency on the whole recording system, since at any instant a

RECORDING SYSTEMS FOR SPECTRA AND IMAGES

33

large fraction of the energy loss range (typically 99% or more) is intercepted by the energy-selecting slit. Greater overall efﬁciency is possible by replacing the detector slit and singlechannel electron detector by a position-sensitive detector of electrons. Such detectors are available in the form of photodiode arrays (PDA) and charge-coupled diode (CCD) arrays. The Gatan model 666 PEELS instrument (Fig. 2.9) uses a 1024-element linear photodiode array, whereas spectra produced by an imaging ﬁlter are recorded by the CCD area array that also handles image recording [2.15, 2.16]. While these arrays respond directly to electrons, there is some risk of radiation damage; therefore commercial systems employ indirect exposure, where a thin transmission screen of a scintillator (single-crystal yttrium aluminum garnet: YAG) converts the electron distribution into one of visible photons which are then imaged by lenses or ﬁbre optics onto the diode array. In the Gatan 666 PEELS system, a single count in the multichannel analyzer is the result of ∼ 30 primary electrons, which create 30,000 photons at the scintillator, resulting in 3000 photodiode electrons [2.17].

Fig. 2.9 Gatan model 666 parallel-recording spectrometer. The components labelled Q1–Q4 are quadrupole electron lenses.

The spacing between photodiode elements is typically 25 µm and the spatial resolution of the system is somewhat worse than this, as a result of electron-beam spreading in the scintillator. The dispersion of a compact spectrometer is typically 2 µm/eV, so to obtain an energy dispersion of the order of 1 eV (limited by the energy spread of the electron source) some form of electron-optical magniﬁcation between the spectrometer and detector is essential. In the Gatan PEELS system, this magniﬁcation is achieved through the use of quadrupole lenses, as shown in Fig. 2.9.

34

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

However, multiple reﬂection of the light emitted within the YAG generates extended tails on the zero loss peak and similarly degrades the energy resolution throughout the spectrum [2.14]. These tails can be minimized by using the highest possible quadrupole magniﬁcation (low eV/channel setting), although this results in a limited energy range of the recorded data. Spectrum deconvolution procedures can also be used to reduce the effect of these tails, as described in the Gatan EL/P software manual. Diode-array detectors operate as follows. Immediately prior to spectrum recording, all array elements are charged to the same potential (typically 5 V). During electron exposure, each diode is discharged by an amount which is proportional to the time-integrated electron ﬂux at the corresponding energy loss. At the end of this integration period, the diodes are interrogated and the spectrum read out serially as a chain of pulses, the height of each pulse representing electron intensity. The signal is fed into a multichannel analyzer via multichannel scaling (MCS) circuitry, as in the case of serial recording. The diode capacitors lose charge not only through irradiation but also as a result of their thermal leakage current, which is slightly different for each element of the array. In order to obtain data values which are proportional to spectral intensity, a leakage or bias spectrum must be subtracted. This bias spectrum is recorded while electrons are excluded from the array (e.g., TEM screen lowered) and will remain the same provided the integration time and array temperature do not vary. To minimize the noise content of recorded data and allow longer integration times (without total discharge by thermal leakage), the photodiode array is cooled to −20◦ C by a thermoelectric device. Note that bias subtraction removes only the detector background and not the spectrometer background which arises from backscattering of the zero-loss beam from a beam-trap aperture inserted in front of the detector (see Fig. 2.9). The latter is most noticeable at high energy loss (> 1 kV) and with very thin specimens. To correct for it, the spectrometer background can be recorded with no specimen in the beam, and a scaled version of this ‘spectrometer spectrum’ subtracted from the experimental data, analogous to the case of serial recording [2.9,2.12–2.14]. Besides having different leakage currents, the individual diodes have slightly different sensitivities to radiation. Such variations are supposedly no more than a few percent for a good-quality photodiode array, but other effects, such as imperfections in the scintillator and in its optical coupling to the array, contribute further to the ‘gain variations’ between channels. In addition, YAG is an electrical insulator and tends to accumulate internal charge in strongly irradiated regions, which has the effect of locally (and temporarily) increasing its conversion efﬁciency. It has been found possible to reduce this artifact by ‘annealing’ the YAG scintillator by prolonged illumination with a broad (undispersed) beam of electrons. The same form of broad illumination can be used to record a gain spectrum of the array and in an area-array CCD detector it is possible to eliminate the effect of the gain variations by dividing any recorded spectrum by this gain spectrum. But in the case of a linear diode array, the broad beam illuminates diode elements

RECORDING SYSTEMS FOR SPECTRA AND IMAGES

35

over their entire horizontal width, rather than the (smaller) width of the energy loss spectrum, and this procedure is not reliable. Scanning a spectrum across the array (using the same spectrometer entrance aperture, same dispersion, etc.) provides an alternative gain spectrum, but correction is still not exact because the width of the actual spectrum varies with energy loss. Another way of dealing with the gain variations is to record two or more spectra, shifted slightly (by a few eV), and subtract these electronically to give a difference spectrum (see, e.g., Sect. 3.4.3). Although gain variations still modulate the difference spectrum, their effect is reduced; in the case of core loss spectra, forming a difference signal largely eliminates the slowly varying background which is the major component of the intensity and (therefore) of the gain modulation. 2.4.3

Dynamic Range of Parallel-Recorded Spectral Data

When the bias (dark-current) spectrum is subtracted, its associated statistical noise remains. This noise level, together with shot noise of the primary electrons, determines the minimum signal which can be recorded with the array detector. The maximum recordable signal is determined by saturation of the output, corresponding to complete discharge of a diode during the integration period. The dynamic range of the detector is the ratio of these two intensity levels, and is of the order of 104 for a 1024-element photodiode array. In contrast, the range of electron intensities typically exceeds 108 , for energy losses extending from 0 to 2000 eV, so a large energy range must be recorded in several segments. One way of doing this is to change the electron intensity entering the spectrometer (by changing the emission current, illumination focus, etc.). It may be desirable to shift the zero-loss peak off the array when recording higher energy losses, to avoid ‘burning in’ an artifact in the gain response, as described above. The gain (or sensitivity) factors between the different acquisitions can be found by least-squares ﬁtting or measurement of equivalent areas within the regions of overlap. Another method of extending the dynamic range is to acquire two or more spectra with different integration times. The gain factors should again be determined by comparison of intensities in the regions of overlap; experience shows that the recorded intensity is not necessarily proportional to the acquisition time entered at the computer keyboard [2.14]. The Gatan 666 spectrometer is ﬁtted with an “electron attenuator”, which uses scanning coils to deﬂect the spectrum in a horizontal plane, perpendicular to the direction of dispersion. Between acquisitions, the spectrum is deﬂected to one side, off the array. When acquisition starts, the spectrum is scanned quickly across the array, giving an effective exposure time of the order of 1 ms, which is useful for recording the low-loss region. In practice, the detector sensitivity appears to vary with total acquisition time, so the safest way of measuring the appropriate gain factor is again by examining the regions of overlap.

36

2.4.4

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

Multiple Spectral Readouts

A further technique for increasing the dynamic range of a diode-array detector is to accumulate multiple readouts in computer memory. If the total acquisition time T is divided into m separate acquisitions, the maximum intensity that can be recorded without saturation is increased by a factor of m. The minimum recordable intensity is also increased, but by a factor < m1/2 [2.18], so the possible dynamic range of the accumulated data is increased relative to its value in a single-acquisition spectrum [2.9, 2.18]. If m is made too large, however, the sensitivity of the detector falls so much that the energy loss signal is eventually swamped by readout noise; in other words, the detective quantum efﬁciency (DQE) of the detector is impaired. To avoid loss of DQE, the number of accumulated readouts should not exceed 10 and the peak signal should not be too far below the saturation limit in each readout. 2.5 COMPARISON OF SERIAL AND PARALLEL RECORDING The major advantage of parallel recording is its higher collection efﬁciency (typically a factor of 100–1000), which enables spectra of adequate signal/noise ratio to be obtained in a shorter period of time and with a lower electron dose to the specimen. The decreased irradiation dose minimizes radiation damage and/or hydrocarbon contamination of the specimen. The shorter time makes it easier to attain good spatial resolution, since there will be less specimen drift. Short recording time also results in less drift of the high voltage and spectrometer power supplies, so one might expect better energy resolution from parallel recording. But in single-acquisition serial recording, such drift results in a distortion of the energy scale or the introduction of spectral artifacts and not in a loss of resolution. Moreover, the energy resolution in parallel-recorded data is degraded by light spreading in the scintillator, unless a large spectral dispersion is used (giving data with a restricted energy range). However, parallel recording may offer better resolution at high energy loss, where serial recording with an acceptable noise level would require a wide detector slit. In exchange for this increase in performance, there is an increase in the complexity of the instrumentation and its operation. If quantitative results are required, a recently acquired bias spectrum must be subtracted from each acquired spectrum and the result divided by a recently-recorded gain spectrum. Moreeover, the intensity scale of photodiode-recorded data is sometimes found to be slightly nonlinear and may require a correction procedure [2.14]. Because of the limited dynamic range and sensitivity of current parallel-recording systems, an extended range of energy loss must be recorded in several segments and (for quantitative work) care taken to measure the relevant gain factors. This requirement reduces the advantages of parallel recording in terms of specimen dose and recording time.

COMPARISON OF EFTEM AND ENERGY FILTERED STEM IMAGING

37

In the case of a serial-recording spectrometer, it is straightforward to obtain STEMmode energy-selected images corresponding to losses before and after a particular ionization edge, and by computer processing obtain an elemental map [2.19]. With parallel-recording spectrometers, the whole spectrum can be read out at each pixel, giving rise to a large data set known as a spectrum-image [2.20], which can be subsequently processed off-line to extract the maximum amount of information from the specimen [2.21]. The dose advantage of parallel recording is then a factor N , where N is the number of energy-selected images that would be required to yield the same information in serial-recording STEM mode. 2.6

COMPARISON OF EFTEM AND ENERGY FILTERED STEM IMAGING

As noted earlier, energy-ﬁltered images can be acquired either from a ﬁxed-beam TEM equipped with an imaging energy ﬁlter (EFTEM imaging) or by using a microscope operating in scanning-transmission mode and ﬁtted with a simple spectrometer, such as the single-prism design. Some instruments (e.g., FEGTEM ﬁtted with scanning attachment) can operate in both modes. We now discuss the relative advantages of these two modes for elemental-mapping applications in terms of the irradiation damage to the specimen, which (in the absence of instrumental limitations) provides the ultimate limit to elemental detectivity. We will assume that the TEM and STEM spatial resolutions are similar, that the spectrometer performance (its collection efﬁciency and energy resolution) is the same and that an electron detector with similar noise performance (DQE) is used in each mode. Under these conditions, the timeintegrated inelastic signal produced from each resolution element would be the same in each case, provided the incident-electron dose to the specimen were identical. Initially let us assume that the STEM uses a serial-recording spectrometer so that a single energy loss is recorded in each image, just as for EFTEM imaging. Fitting and removing the preedge background requires three images in both modes so the specimen irradiation remains the same, for the same information content; the only difference is that the electron dose is delivered continuously in EFTEM mode but for only a small fraction of the frame time in the STEM case. For many specimens this same total exposure will result in the same amount of beam-induced damage (or no damage at all) but this will not be the case for materials whose damage is dependent on dose rate. In the case of a ﬁeld-emission STEM, the total beam current might be a factor of 103 lower than in the TEM case (requiring longer image-recording times) but the current density in the focused probe is typically a factor of 103 higher. This larger current density may cause a greater temperature rise in the specimen, although the factor involved depends on the geometry of heat ﬂow. If so, STEM mode is more likely to cause thermal degradation of organic specimens. Likewise, some inorganic compounds damage radiolytically but only above a threshold current density [2.22] and the STEM procedure would again be more damaging.

38

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

However, if time-dependent diffusion processes limit the degradation process, the STEM may create less observable damage because of the comparatively short irradiation time per pixel. Diffusion is more likely to be rate limiting if the specimen is cooled to reduce beam damage. As judged from the disappearance of EELS ﬁne structure, bond breakage in cooled (97 K) poly(ethylene terephthalate) samples occurred at doses of the order of 0.1 C/cm2 for a 100-nm diameter incident beam of 200 kV electrons but was reported to require 1000 C/cm2 for a 1-nm diameter probe [2.23]. Perhaps for similar reasons, scanning X-ray microscopy caused less mass loss from hydrated chromosome specimens when these were imaged in a single scan, rather than multiple scans [2.24]. If the STEM system uses a parallel-recording spectrometer, the three images required to analyze each element are recorded simultaneously, reducing the recording time and specimen dose by a factor of three compared to the EFTEM case. This factor becomes 3n if n elements are being analyzed simultaneously. In the case of spectrum-imaging, where an N -channel energy loss spectrum is acquired at each pixel, the dose will be lower by a factor of N compared to that required to record the same information in EFTEM mode (by the sequential acquisition of N images). For N = 1000, the recording time in both modes should actually be comparable, the smaller beam current in the STEM mode being compensated by the greater collection efﬁciency of an arrangement which does not reject a large part of the energy loss spectrum at an energy-selecting slit. 2.7

EXPERIMENTAL CONDITIONS FOR ENERGY LOSS SPECTROSCOPY AND ELEMENTAL MAPPING

Most EELS applications, described in later chapters in this volume, require the recording of ionization edges. In discussing data acquisition, we will therefore concentrate on core level spectroscopy; similar considerations apply to elemental mapping. Three variables (each under control of the operator) affect the quality and usefulness of the data, namely, the collection angle, incident energy and specimen thickness. 2.7.1

Choice of Specimen Thickness

Specimen thickness is important because it determines the degree of plural scattering of the transmitted electrons. Poisson statistics describes the probability Pn that an electron is inelastically scattered n times in a specimen of thickness t n 1 −t t Pn = exp (2.7) n! λ λ where λ is a mean free path (the average distance between all inelastic-scattering events or those corresponding to a particular energy loss process) and is inversely

EXPERIMENTAL CONDITIONS FOR SPECTROSCOPY AND MAPPING

39

proportional to the appropriate scattering cross-section. If t/λ exceeds unity, higher orders (n) of scattering are favored as a result of the nth power in Eq. 2.7. In the case of core loss scattering, t/λ is much less than unity because of the small ionization cross section (large λ) so the probability of a transmitted electron creating more than one inner-shell ionization can safely be ignored. But an electron that has caused inner-shell excitation may also generate n plasmons before leaving the specimen, with a probability Pn that is appreciable because of the much shorter mean free path for plasmon production, typically of the order of 100 nm. On the assumption that plasmon and core loss processes are independent, the scattering probabilities are multiplicative and the observed ionization-edge will be a simple convolution of its single-scattering proﬁle (seen in thin specimens) with the lowloss spectrum. The predicted effect at the carbon K-ionization edge is illustrated in Fig. 2.10. For t/λ > 1 (λ being the plasmon mean free path), plural scattering greatly modiﬁes the shape of the edge, delaying the maximum to typically 50–100 eV beyond the ionization threshold.

Fig. 2.10 Contributions of plasmon scattering (of order n) to the carbon K -edge, calculated as a convolution of a power-law edge proﬁle (BE −s where s = 5) with an idealized low-loss spectrum (a series of delta functions obeying Poisson statistics). On this simpliﬁed model, the contribution from each successive order of scattering is visible as a sharp step in the intensity; in reality, these steps are considerably rounded and may not be visible in some materials. Also shown are contributions to the total background (dashed curve) from N plasmons, calculated from the same model.

Plural scattering also contributes to the background underlying an ionization edge. In this case, the main contribution is expected to come from a single large energy

40

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

loss (such as the tail of a previous edge) and a small number of low-loss (plasmon) events [2.9]. Its effect can be modeled as a convolution of a single scattering powerlaw energy distribution (AE −r ) with the low-loss region, as illustrated in Fig. 2.10. In this case, however, simple convolution will be less accurate, as a result of the non-Lorentzian angular distribution of the background (see later, Fig. 2.12). Because the plural scattering intensity is ‘delayed’ relative to the ionization threshold, the core loss intensity (integrated over a limited energy window) increases with specimen thickness less rapidly than the pre-edge background, so the signal/background ratio becomes unacceptably small if the specimen is too thick; see Fig. 2.11. At the silicon K-edge, the decrease in signal/background is less dramatic (Fig. 2.11b) because of the lower slope (fractional decrease in intensity per eV) of the background at high energy loss and the lower plasmon energy. At high energy loss, there is also an appreciable background contribution from stray scattering in the spectrometer [2.9, 2.13], causing the signal/background ratio to be less than expected. Because this spectrometer background arises mainly from backscattering of the intense zero-loss beam, it is more noticeable in very thin specimens, for which the signal/background ratio may actually be lower than at moderate thickness; see Fig. 2.11b.

Fig. 2.11 Thickness dependence of the K -loss signal/background ratio (with 100 eV integration window) for (a) amorphous carbon and (b) crystalline silicon. Data points are experimental values, recorded with β = 10 mrad and E0 = 120 kV; the curves were calculated by convolving power-law edge and background proﬁles with a delta-function approximation of the low-loss spectrum.

EXPERIMENTAL CONDITIONS FOR SPECTROSCOPY AND MAPPING

2.7.2

41

Choice of Incident Energy

The previous discussion showed that plural scattering, involving one or more plasmon events in addition to a large energy loss, greatly affects the visibility and the shape of ionization edges. The extent of this scattering is dependent on the inelastic mean free path λ, whose value is determined mainly by plasmon-like processes visible in the low-loss region of the spectrum. An approximate expression for λ is [2.25] E0 F Em λ = (106 nm) (2.8) E0 ln 2β Em where Em = 7.6Z 0.36 is a material-dependent mean energy loss and F is a relativistic factor that is close to unity for E0 < 300 kV. As E0 is raised, λ increases and the scattering parameter t/λ, which determines the extent of plural scattering, decreases. Increased incident energy is therefore equivalent to a thinner specimen, and in most cases EELS analysis should be carried out with the TEM operated at its maximum accelerating voltage. If available, intermediate-or high-voltage microscopes will give better results from thicker specimens, provided the stray-scattering background is not excessive. 2.7.3

Choice of Collection Angle

We discussed earlier the effect of an angle-limiting aperture in terms of its control over lens and spectrometer aberrations. Besides inﬂuencing the spatial resolution of elemental analysis, this aperture determines the strength of the energy loss signal: the larger the aperture semiangle β, the larger the core loss intensity. Although maximum signal is desirable, it is actually the signal/noise ratio SNR (i.e., the fractional noise content) which determines the sensitivity and accuracy of core loss microanalysis [2.9]. Assuming a detector of high DQE, the noise component is predominantly shot noise, which increases only as the square root of the number of core loss electrons detected. On this basis, a large value of β might be expected to provide maximum SNR. However, we need to consider not only the core loss component but also its spectral background. Because this background arises from different physical processes, its angular distribution is different from that of the core losses. For energy losses just above the ionization threshold, inner-shell electrons are being excited with an energy transfer that is not much larger than their original binding energy. Under these dipole conditions, the intensity of core loss scattering per unit angle θ is given by dIc θ ∝ 2 2) dθ (θ + θE

(2.9)

42

EXPERIMENTAL TECHNIQUES AND INSTRUMENTATION

In Eq. 2.9, the parameter θE = E/mv 2 (where m is the relativistic mass and v the velocity of the incident electrons), called the characteristic scattering angle, is a measure of the width of the angular distribution; see Fig. 2.12. In contrast, the background to the edge arises from excitation of electrons of lower binding energy, where dipole conditions do not apply. In a simple case like the carbon K-edge, where there are no ionization edges at lower energy loss, the background arises entirely from excitation of valence electrons and its angular distribution is peaked at an angle of some tens of milliradians; see Fig. 2.12. Recorded electron intensities are proportional to the areas under the curves in Fig. 2.12, integrated up to an angle β. Clearly, a 5- or 10-mrad angle-limiting aperture will select a larger proportion of the core loss electrons, compared to those forming the preedge background. In general, the smaller the collection angle, the larger will be the edge/background ratio. Very small apertures can still be excluded on the basis of noise; typically the signal/noise ratio is optimum for β of the order of 10 mrad [2.9]. The above arguments ignore plural-scattering contributions to the spectral background. However, calculations of the angular distribution of plural inelastic scattering indicate that Poisson statistics still apply to the intensities recorded through an anglelimiting aperture [2.9], provided λ in Eq. 2.7 is replaced by an angle-dependent mean free path β2 ln 1 + 2 θE λ(β) = λ (2.10) θc2 ln 1 + 2 θE where θc ≈ (E/E0 )1/2 is an effective cutoff angle for the core loss angular distribution. As β is decreased, λ(β) increases and t/λ(β) decreases, providing a reduction in the plural-scattering component of the background. Because the angular width of plural scattering is larger than that of single scattering, the angle-limiting aperture preferentially rejects plural-scattering components, particularly for smaller values of β. 2.8 FUTURE INSTRUMENTATION The energy resolution in EELS (and energy-ﬁltered imaging) is limited by the energy spread of the electron source, which is typically within the range 1–2 eV for a tungsten or LaB6 ﬁlament, 0.5–0.7 eV for a Schottky source and 0.3–0.6 eV for an unheated ﬁeld-emission source. The energy width increases with emission current because of statistical Coulomb interactions between electrons (Boersch effect) that occur mainly when the electrons are close to each other: at a crossover in the optical system or at the source itself. Reduced energy spread can be achieved by placing a monochromator between the source and the specimen. The monochromator is an energy ﬁlter: some form of

FUTURE INSTRUMENTATION

43

Fig. 2.12 Angular dependence of the K -loss scattering (at 285 eV) and of the preceding background, for 80 kV incident electrons transmitted through a very thin carbon specimen.

spectrometer incorporating an energy-selecting slit. As the slit width w is reduced, the energy spread of the transmitted beam decreases, but only up to the point where w = s, where s is the beam diameter (for a single electron energy) at the slit plane. Smaller values of w would further decrease the electron intensity (or effective brightness of the electron source) without an appreciable reduction in energy width. The best energy resolution obtainable with the monochromator is therefore s/D where D is the energy dispersion at the slit plane. Wien ﬁlters are a popular choice of monochromator. In this device, perpendicular electric and magnetic ﬁelds provide opposing forces on an electron so that, for electrons of a particular energy, the net force is zero. Choosing this energy to be the average of the distribution ensures that the beam travels in a straight-line path through the spectrometer, a practical advantage for straight electron columns. Many years ago, Boersch, Geiger, and colleagues built a transmission spectrometer system that used Wien ﬁlters for the monochromator and analyzer, eventually achieving an energy resolution of 3 meV but only for 30 keV electron energy and with no attempt to achieve high spatial resolution [2.26, 2.27]. More recently, Terauchi et al. modiﬁed a TEM, incorporating Wien ﬁlters before and after the specimen, and obtained 80 kV energy loss spectra with 25 meV resolution from specimen areas 30–100 nm in diameter [2.28]. This system is being further developed (with JEOL) in the MIRAI 21 project. Recent designs place the monochromator within the electron gun, where high dispersion is possible because the electrons have undergone only a limited amount of acceleration (electron energy < 5 keV). The gun crossover is imaged onto an energyselecting slit so that energy selection is independent of the electron-emission angle [2.29]. The design of Mook and Kruit [2.30] uses a very short (4mm) Wien ﬁlter, thereby minimizing Boersch-effect broadening but providing only modest energy dispersion (4.2 µm/eV) so that a very narrow (150 nm) energy-selecting slit is re-

44

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quired. Such slits are readily made by silicon-fabrication techniques but will require very good local vacuum conditions to prevent beam-induced contamination. The monochromator has been tested on a 120 kV ﬁeld-emission STEM at IBM Watson Research Center [2.31]; the system has demonstrated 60 meV energy resolution and 0.14 nm spatial resolution, achieved through correction of spherical aberration. The Wien-ﬁlter monochromator design of Tiemeijer is more conventional, providing higher dispersion and avoiding the need for a submicron slit. Incorporated into an FEI 200 kV TEM, the energy resolution is ∼0.2 eV for beam currents < 30 nA; > 40 nA, the Boersch effect provides at least 0.1 eV of broadening [2.32]. The gun-monochromator design of Rose [2.33] is an electrostatic version of the omega ﬁlter. An energy-selecting slit is placed at the mid-plane (O2 in Fig. 2.4); the second half of the ﬁlter “compensates” for energy dispersion occurring within the slit (the overall dispersion is zero as a result of symmetry) and thereby optimizes the source brightness [2.29]. This design is being commercialized by CEOS GmbH and will be used together with a MANDOLINE analyzer [2.34] in a LEO microscope under the German SESAMe project [2.35]. Due to the loss of electrons at the energy-selecting slit, conventional monochromators reduce the source brightness and the illumination intensity, which is unfavorable for core loss spectroscopy. A preferable alternative would be a bright electron source with an inherently small energy width. Fransen [2.36] measured the properties of a multiwalled carbon nanotube used as a ﬁeld-emission tip, and obtained excellent current stability in a vacuum of 10−8 Pa. The energy width, full-width at half maximum was in the range 0.11–0.31 eV, dependent on emission current. The reduced brightness of the source was estimated to be 1.4 × 107 Am−2 sr−1 V−1 , somewhat lower than most measurements on tungsten ﬁeld emitters. Nevertheless, carbon nanotubes appear attractive as electron sources, if practical problems of characterization, reproducibility and accurate mounting can be overcome. However achieved, a small energy width is valuable for spectroscopy only if the electron spectrometer offers comparable resolution and stability. This has lead to a redesign of the traditional magnetic sector, with aberrations corrected up to 4th order, consideration of the spectrometer object size and improved stability of the spectrometer power supplies [2.37]. Energy loss systems which offer an energy resolution of ∼ 0.1 eV will be invaluable for low-loss spectroscopy, such as bandgap studies in semiconductors [2.38] and aloof measurements on nanotubes [2.39]. Because of the low energy loss, the spatial resolution is likely to be limited (by delocalization of the inelastic scattering) to ∼ 1 nm. Slightly better energy resolution would allow the detection of opticalphonon modes (e.g., in nanostructures), certainly of academic interest and possibly of practical beneﬁt. Acknowledgments This chapter is dedicated to the memory of Ludwig Reimer, a pioneer of electron-microscope techniques, who died on 29 April 2001.

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REFERENCES 2.1. P. J. Goodhew, “Thin foil preparation for electron microscopy” Practical Methods in Electron Microscopy, vol.11, ed. A. M. Glauert; Elsevier, Amsterdam, (1985). 2.2. Materials Research Society Symposium Proceedings, volumes 115 (MRS, Pittsburgh, 1988) and 199 (MRS, Pittsburgh, 1990). 2.3. Microscopy Research and Technique, 31, 265–310 (1995). 2.4. O. L. Krivanek, A. J. Gubbens, N. Dellby, and C. E. Meyer, “Design and ﬁrst applications of a post-column imaging ﬁlter,” Microsc. Microanal. Microstruct. 3, 187 (1992). 2.5. R. Castaing and L. Henry, “Filtrage magnetique des vitesses en microscopie electronique,” C.R. Acad. Sci. Paris B255, 76 (1962). 2.6. G. Zanchi, J.-P. Perez, and J. Sevely, “Adaptation of a magnetic ﬁltering device on a one megavolt electron microscope. Optik 43, 495 (1975). 2.7. S. Lanio, “High-resolution imaging magnetic energy ﬁlters free of second-order aberration,” Optik 73, 99 (1986). See also “Design and testing of Omega mode imaging energy ﬁlters at 200 kV,” J. Electron Micros. 46, 357 (1997). 2.8. A. V. Crewe, M. Isaacson, and D. Johnson, “A high resolution electron spectrometer for use in transmission scanning electron microscopy,” Rev. Sci. Instrum. 42, 411 (1971). 2.9. R. F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, second edition (Plenum Press, New York, 1996). 2.10. J. M. Titchmarsh and T. F. Malis, “On the effect of objective lens chromatic aberration on quantitative electron-energy loss spectroscopy (EELS),” Ultramicroscopy 28, 277 (1989). 2.11. Y. Y. Yang and R. F. Egerton, “The inﬂuence of lens chromatic aberration on electron energy loss quantitative measurements,” Micros. Res. Tech. 21, 361 (1992). 2.12. P. Crozier and R. F. Egerton, “Mass-thickness determination by Bethe-sumrule normalization of the electron energy loss spectrum,” Ultramicroscopy 27, 9 (1989). 2.13. A. J. Craven and T. W. Buggy, “Correcting electron energy loss spectra for artefacts introduced by a serial collection system,” J. Micros. 136, 227 (1984). 2.14. R. F. Egerton, Y.-Y. Yang, and S. C. Cheng, “Characterization and use of the Gatan 666 parallel-recording electron energy loss spectrometer”. Ultramicroscopy 48, 239 (1993).

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2.15. N. J. Zaluzec and M. G. Strauss, “Two-dimensional CCD arrays as parallel detectors in electron-energy loss and x-ray wavelength-dispersive spectroscopy,” Ultramicroscopy 28, 131 (1989). 2.16. J. N. Chapman, A. J. Craven, and C. P. Scott, “Electron detection in the analytical electron microscope,” Ultramicroscopy 28, 108 (1989). 2.17. O. L. Krivanek, C. C. Ahn, and R. B. Keeney, “Parallel-detection electron spectrometer using quadrupole lenses,” Ultramicroscopy 22, 103 (1987). 2.18. R. F. Egerton, “Parallel-recording systems for electron energy loss spectroscopy (EELS). “Journal of Electron Microscopy Technique 1, 37 (1984). 2.19. R. D. Leapman, “Scanning transmission electron microscope (STEM) elemental mapping by electron energy loss spectroscopy,” Ann. New York Acad. Sci. 483, 326 (1986). 2.20. C. Jeanguillaume and C. Colliex, “Spectrum-image: the next step in EELS digital acquisition,” Ultramicroscopy 28, 252 (1989). 2.21. J. A. Hunt and D. B. Williams, “Electron energy loss spectrum-imaging,” Ultramicroscopy 38, 47 (1991). 2.22. I. G. Salisbury, R.S. Timsit, S.D. Berger, and C.J. Humphreys, “Nanometer scale electron beam lithography in inorganic materials,” Appl. Phys. Lett. 45, 1289 (1984). 2.23. K. Varlot, J. M. Martin, C. Quet, and Y. Kihn, “Towards sub-nanometer scale EELS of polymers in the TEM,” Ultramicroscopy 68, 123 (1997). 2.24. S. Williams, X. Zhang, C. Jacobsen, J. Kirz, S. Lindaas, J. van’t Hof, and S. S. Lamm, “Measurements of wet metaphase chromosomes in the scanning transmission x-ray microscope,” J. of Microsc. 170, 155 (1993). 2.25. T. Malis, S. C. Cheng, and R. F. Egerton, “EELS log-ratio technique for specimen-thickness measurement in the TEM,” J. Electron Microsc. Technique 8, 193 (1988). 2.26. H. Boersch, J. Geiger and H. Hellwig, “Steigerung der Auﬂoesung bei der Elektronen-Energieanalyse,” Phys. Lett. 3, 64 (1962). 2.27. J. Geiger, “Inelastic electron scattering with energy losses in the meV-region,” 39th Ann. Proc. Electron Microsc. Soc. Am., ed. G.W. Bailey, Claitor’s Publishing, Baton Rouge, Louisiana, 182 (1981). 2.28. M. Terauchi, M. Tanaka, K. Tsuno, and M. Ishida, “Development of a high energy resolution electron energy loss spectroscopy microscope,” J. Microsc. 194, 203 (1999).

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2.29. F. Kahl and E. Voekl, “Present concepts and designs for gun monochromators,” Microsc. Microanal. 7 (Suppl. 2: Proceedings, Microsc. Soc. Am.), 922 (2001). 2.30. H.W. Mook and P. Kruit, “Construction and characterization of the fringe ﬁeld monochromator for a ﬁeld emission gun,” Ultramicroscopy 81, 129 (2000). 2.31. P. E. Batson, H. W. Mook, O. L. Krivanek and N. Delby, “Progress with the IBM very high resolution STEM,” Microsc. Microanal. 7 (Suppl. 2: Proceedings, Microsc. Soc. Am.), 234 (2001). 2.32. P. C. Tiemeijer, J. H. A. van Lin, and A. F. de Jong, “First results of a monochromatized 200 kV TEM,” Microsc. Microanal. 7 (Suppl. 2: Proceedings, Microsc. Soc. Am.), 1130 (2001). 2.33. H. Rose, “Electrostatic energy ﬁlter as monochromator of a highly coherent electron source,” Optik 85, 95 (1990). 2.34. S. Uhlemann and H. Rose, “The MANDOLINE ﬁlter — a new high-performance imaging ﬁlter for sub-eV EFTEM,” Optik 96, 163 (1994). 2.35. D. Krahl, S. Kujawa, A. Rilk, G. Benner, P. Hahn, G. Lang, E. Essers, and W. Probst, “Design and performance of the SESAME1”. Microsc. Microanal. 7 (Suppl. 2: Proceedings, Microsc. Soc. Am.), 924 (2001). 2.36. M. J. Fransen, Th. L. van Rooy, and P. Kruit, “Field emission energy distributions from individual multiwalled carbon nanotubes,” Appl. Surface Sci. 146, 312 (1999). 2.37. H. A. Brink, M. Barfels, B. Edwards, and P. Burgner, “A new high performance electron energy loss spectrometer for use with nonochromated microscopes”, Microsc. Microanal. 7 (Suppl. 2: Proceedings, Microsc. Soc. Am.), 908 (2001). 2.38. V. J. Keast, N. Sharma, M. Kappers, and C. J. Humphreys, “Electron energy loss spectroscopy (EELS) of GaN alloys and quantum wells,” Microsc. Microanal. 7 (Suppl. 2: Proceedings, Microsc. Soc. Am.), 1182 (2001). 2.39. B. W. Reed and M. Sarikaya, “Electronic properties of carbon nanotubes via aloof measurements using transmission-EELS,” Microsc. Microanal. 7 (Suppl. 2: Proceedings, Microsc. Soc. Am.), 192 (2001).

3

EELS QUANTITATIVE ANALYSIS Richard Leapman Division of Bioengineering and Physical Science, ORS National Institutes of Health, Bethesda, Maryland 20892

Abstract Quantitative elemental analysis by electron energy loss spectrometry (EELS) requires extraction of the characteristic inner-shell signals and determination of the appropriate ionization cross-sections. Accuracy of determining elemental concentrations depends on techniques for reliably separating the background from the signal, including the methods of linear-least-squares ﬁtting, digital ﬁltering and multipleleast-squares ﬁtting of reference spectra. In thick specimens the ideal spectral shape is distorted by plural inelastic scattering but such effects can be removed by deconvoluting the spectral intensity to yield single scattering distributions. Cross-sections for inner-shell excitations can be derived from quantum mechanical calculations or from measurements on standard specimens. Energy-differential cross-sections are determined for the range of scattering angles and for the incident beam energy used in the measurements; integrated partial cross-sections are then obtained for the appropriate energy windows above the core edge. Detection limits can be estimated from the signal/noise ratio for a given set of experimental conditions and for a particular element to be analyzed. The availability of techniques for acquiring EELS data point by point are now generating new approaches to providing quantitative elemental maps in both ﬁxed beam and scanned beam instruments. When coupled with highly optimized detectors, this spectrum-imaging technique provides greatly improved analytical sensitivity. It is now feasible to determine atomic concentrations < 100 parts per million (ppm) in a single pixel of an elemental map and to detect single atoms of certain elements.

49

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EELS QUANTITATIVE ANALYSIS

3.1 INTRODUCTION Electron energy loss spectrometry (EELS) has been developed over the past few years into a highly sensitive microanalytical tool, one that is capable, under suitable conditions, of detecting very small numbers of atoms [3.1–3.7]. The technique clearly offers an advantage over energy-dispersive X-ray spectroscopy (EDXS) for analyzing the light elements that have a low ﬂuorescence yield. Furthermore, EELS is also preferable to EDXS for analyzing certain heavier elements [3.8, 3.9]. In this chapter, we attempt to provide a basis not only for quantiﬁcation but also for deciding under what conditions EELS can be expected to yield favorable results. Two main steps are required to quantify the energy loss spectrum. First, it is necessary to separate the characteristic core edge intensities from the noncharacteristic background. Second, determination of the elemental composition requires information about the low-loss spectrum, knowledge of the acquisition parameters (e.g., collection geometry, beam energy, detector response) as well as the inner-shell ionization cross-sections. In addition, we must appreciate the importance of selecting suitably thin specimen regions for analysis and setting up the electron optics of the microscope in the appropriate way. We shall begin with an outline of the basic formulas that relate the measured spectrum to the specimen composition. Methods for determining the inner-shell inelastic scattering cross-sections as a function of energy loss and scattering angle are described. An understanding of these sections of the chapter provides the basis for quantitative microanalysis and for selecting the appropriate acquisition parameters. There is an account of the different approaches for extracting the core-edge signal from the spectral background and practical difﬁculties that can occur. This is followed by a discussion of limitations to accuracy and estimates of detection limits for typical experimental conditions and specimen compositions. Finally, there is a description of quantitative elemental mapping based on spectrum-imaging techniques in the energy-ﬁltering and scanning transmission electron microscopes. 3.2

BASIC FORMULAS FOR QUANTIFICATION

Consider an electron beam containing a total of J0 electrons incident on a thin specimen composed of nx atoms of element x per unit area. As shown in Fig. 3.1 the inelastically scattered electrons are collected inside semiangle β, which is usually greater than the probe convergence semiangle α. The characteristic inner-shell signal, Sx , used for microanalysis is then given by, Sx (∆, β) = J0 nx σx (∆, β)η

(3.1)

where σx (∆, β) is deﬁned as the cross-section per atom for ionizing an inner-shell electron x, and η is the detective quantum efﬁciency of the detector. If the probe current is j0 and the spectral acquisition time is τ then J0 = j0 τ /e, where e is the electronic charge. Because it is experimentally difﬁcult to integrate the core edge

BASIC FORMULAS FOR QUANTIFICATION

51

signal over all energy losses we choose a given energy range ∆ above the edge threshold inside which we integrate the signal. The partial cross-section σx (∆, β) is a measure of the ionization probability, whereby the fast electron loses energy from the threshold energy Ex to Ex + ∆ and with scattering angles less than β. The cross-section is given by an area per atom and, since nx is the number of atoms per unit area of the specimen, Eq. 3.1 is dimensionally correct. As we shall see later, for energy losses less than a few hundred electronvolt a large fraction of the inelastically scattered electrons lie inside angle β which is typically only ∼ 10−2 rad; this is not the case for the elastically scattered electrons that are deﬂected through larger angles.

Fig. 3.1 Scattering geometry in STEM mode and deﬁnition of specimen parameters used in Eqs. 3.1–3.4.

It is very important to estimate the noncharacteristic background intensity on which the core edges are superimposed. We shall examine this problem in more detail later but for the moment we observe that the background often follows an inverse power law as a function of energy loss E; this allows us to model the preedge background and extrapolate it into the edge region of the spectrum. The quantity J0 in Eq. 3.1 could be measured with a Faraday cage but this is not necessary if the entire energy loss spectrum is recorded because the low-loss spectrum (mainly unscattered and inelastically scattered electrons) provides a direct measure of the incident ﬂux. For a sample of ﬁnite thickness, we need to modify Eq. 3.1 slightly as prescribed by Egerton [3.10] for the following reason. Some of the scattered electrons that have excited inner-shells are also scattered elastically through angles greater than the spectrometer acceptance angle. Other electrons that have excited inner-shells lose energy again by excitation of valence electrons and are transferred outside the energy window ∆. However, a nearly equal number of electrons are also lost from the low loss spectrum by elastic and inelastic scattering. Therefore, provided the low loss spectrum is measured from the same specimen area and under the same experimental conditions we can write [3.10], Sx (∆, β) = J1 (∆, β)nx σx (∆, β)

(3.2)

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EELS QUANTITATIVE ANALYSIS

where the subscript l refers to the low-loss spectrum including the zero-loss peak. Note that the detective quantum efﬁciency factor η is now no longer required. Equation 3.2 thus enables us to determine the total number of atoms per unit area of the sample. If the analyzed area of the specimen is A, then the absolute number of atoms of element x contained in that region is given by, Nx =

Sx (∆, β)A J1 (∆, β)σx (∆, β)

(3.3)

The measured low-loss and core loss intensities (x and y) are indicated schematically in Fig. 3.2 for energy window ∆; integrated core loss signals are obtained by extrapolating the preedge background ﬁt over energy window Γ by means of an inverse power law.

Fig. 3.2 Deﬁnition of measured intensities in energy loss spectrum required for quantiﬁcation using Eqs. 3.1–3.4. Shaded areas correspond to numbers of counts in integration window ∆. Core edge signals are obtained by extrapolating background modeled using inverse power law over ﬁtting window Γ (see text).

An example of how the absolute number of atoms in a structure can be determined is shown in Fig. 3.3. The spectrum was recorded from a single molecule of ferritin using a ﬁeld-emission STEM with parallel-detection EELS. The beam energy was 100 keV, the probe current 0.8 nA, collection angle, β = 20 mrad, integration time = 1s for each of 10 read-outs. The area analyzed was ∼ 12 nm in diameter. The low-loss spectrum was acquired separately with attenuator switched on to avoid detector saturation (÷2000). The numbers of counts in the low-loss and Fe L23 -edge are indicated for an integration window ∆ = 50 eV (shaded areas) obtained using Γ = 50 eV. In most microanalytical problems we are more interested in relative numbers of atoms x and y rather than their absolute numbers. The atomic ratio is given simply by [3.5,3.10–3.14] Sx (∆, β) σy (∆, β) Nx = Ny Sy (∆, β) σx (∆, β)

(3.4)

Errors associated with atomic ratios are expected to be smaller than those associated with the absolute numbers; an accuracy of a few percent is sometimes achievable. The ratio determination is demonstrated by the analysis of a thin crystal of hexagonal

BASIC FORMULAS FOR QUANTIFICATION

53

Fig. 3.3 Example of determination of absolute number of atoms in a structure. Spectrum from single ferritin molecule (diameter ∼ 6 nm) showing Fe L23 -edge (708 eV). Calculated number of Fe atoms is 4100±500, which agrees with expected value of ∼5000 (error due to uncertainty in cross-section, not to counting statistics).

BN (Fig. 3.4). Measured spectral intensities in energy window are indicated for both the B and N K-edges. The estimated atomic fraction, B/N=1.03±0.05, is consistent with the known composition.

Fig. 3.4 Example of atomic ratio determination in a known material. Parallel-EELS from thin boron nitride sample recorded at 100-keV beam energy in 10 s with probe current ∼ 0.8 nA. Measured intensities in energy window ∆ = 50 eV, obtained with Γ = 50 eV, are indicated for both B and N edges. Estimated fraction of B/N=1.03±0.05 (error due to uncertainty in crosssection not to counting statistics) is consistent with the expected value, B/N=1.

In the two examples of Figs. 3.3 and 3.4, the accuracy of quantiﬁcation is limited by uncertainties in the ionization cross-sections rather than by counting statistics. Section 3.3, therefore deals with various methods that are available for estimating these quantities. Subsequently, we shall address in detail the problem of signal estimation and background subtraction, which becomes most important when edge overlap occurs, or when elemental concentrations are low.

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3.3 INNER-SHELL CROSS-SECTIONS 3.3.1

Generalized Oscillator Strengths and Angular Distributions

Quantum mechanical expressions for the inner-shell ionization cross-section can be derived either by using time-dependent perturbation theory for the transition rate between initial and ﬁnal states of the interacting electrons (i.e., Fermi’s Golden Rule), or by using a stationary description (scattering theory) where a continuous current of electrons incident on an atom gives rise to a continuous current of outgoing electrons [3.15,3.16]. It is assumed that the fast electrons are travelling with a kinetic energy that is high relative to the energies of the excited atomic states, and that the total wavefunction of the system can be written as the product of the atomic and plane wave states. The result obtained from both approaches is the same and is in the form of a cross-section σif , differential with respect to solid angle Ω, 2 kf ∗ 3 3 φf (r ) exp(i(ki − kf ) · r)VCoul (r, r )φi (r )d r d r ki (3.5) Here r are the coordinates of the fast electron that is considered to have an initial plane wave state exp(iki · r) and a ﬁnal state exp(ikf · r), where ki and kf are the initial and ﬁnal wavevectors, respectively. The fast electron interacts through the Coulomb potential VCoul with the atom that has coordinate system r and causes the initial state φi to go to ﬁnal state φf . The expression in the bracket is the matrix element for the transition. The quantity m is the rest mass of the electron and ¯h is Planck’s constant divided by 2π. We now set out a brief derivation of the core ionization cross-section differential with respect to energy loss E and scattering angle θ. This is given in Eqs. 3.6– 3.9. The scattering wavevector is given by q = ki − kf and the Coulomb potential by, VCoul = e2 /(4πε0 |r − r0 |). An analytical integration of Eq. 3.5 over the fast electron coordinates is then possible, dσif = dΩ

m 2π¯ h2

2

2 2 2 e m kf 1 dσif ∗ 3 =4 (r ) exp(iq · r )φ (r ) d r φ i f 2 4 dΩ ki q 4πε0 ¯ h

(3.6)

Since we are interested in transitions to continuum states rather than to discrete bound levels we normalize the ﬁnal states per unit energy loss E and label these states as φf (r ). We then obtain a result that is also differential with respect to energy, d2 σif = dEdΩ

4 a20

kf ki

1 q4

2 φ∗f (r ) exp(iq · r )φi (r ) d3 r

h2 /me2 is the Bohr radius. where a0 = 4πε0 ¯

(3.7)

INNER-SHELL CROSS-SECTIONS

55

The magnitude of q can be written in terms of the scattering angle θ by the relation q 2 = ki2 + kf2 − 2ki kf cos θ. By differentiating with respect to θ we can write the element of solid angle dΩ = 2π sin θdθ = 2πqdq/ki kf . We then obtain, h4 2π 1 dfx (q, E) ¯ d2 σx (q, E) = 2 2 dqdE m a0 q T E dE

(3.8)

where T = h ¯ 2 ki2 /2m, is the kinetic energy of the incident electrons and, 2 2mE dfx (q, E) ∗ 3 = φf (r ) exp(iq · r )φi (r ) d r 2 dE ¯q h is a property of the atom that is called the generalized oscillator strength (GOS) [3.15,3.17–3.19]. Notice that we have changed the subscript of the cross-section to x to be consistent with Eqs. 3.1–3.4. We can now obtain an expression for the cross-section differential with respect to 2 2 energy loss E and scattering angle θ by writing q 2 = ki2 θ2 + qmin = ki2 (θ2 + θE ), where θE = qmin /ki = E/2T , and qmin is the minimum momentum transfer in the scattering process, h4 1 ¯ θ dfx (q, E) d2 σx (θ, E) = 2π 2 2 (3.9) 2 2 dθdE m a0 T E θ + θE dE This equation can be shown to be valid at relativistic energies [3.5] provided corrected values are used for the fast electron kinetic energy and for the characteristic scattering rel angle θE , T rel = mv 2 /2 and θE = E/2γT , where v is the electron velocity and 2 2 −1/2 . The full relativistic formulation [3.5] including “retardation” γ = (1 − v /c ) is only important at beam energies above ∼ 300 keV and is not considered here. Figure 3.5 shows a plot of the GOS as a function of q for different energy losses above the carbon K-edge calculated using atomic Hartree–Slater wave functions for the initial and ﬁnal states [3.20,3.21]. At energies just above threshold (285 eV) the GOS falls off monotonically with q, but at large energy losses a maximum is observed 2 at q given by ¯ h2 q B /2m ∼ E corresponding to the ‘Bethe ridge’. The Bethe ridge can be interpreted classically as the momentum transferred to an unbound (free) electron in an inelastic collision with energy loss E. In the limit of small q the matrix element in the generalized oscillator strength tends toward the dipole limit (i.e., corresponds to the X-ray absorption spectrum). This is important in determining the selection rules for transitions to unoccupied bound states and is discussed further in Chapter 6. It also provides a basis for calculating electron-impact ionization cross-sections from X-ray absorption data [3.22]. The cross-section for inner-shell ionization by fast electrons can be estimated by assuming that the GOS is constant and equal to the optical oscillator strength, df (0, E)/dE, for scattering angles less than a cut-off angle deﬁned by the Bethe ridge, θB = qB /ki = (E/T )1/2 , and equal to zero for larger scattering angles. This approximation leads to the Bethe formula for the total ionization cross-section [3.5,3.15,3.22]:

56

EELS QUANTITATIVE ANALYSIS

σxTOTAL

¯2 h = 4π ma0

2

1 zx bx ln T Ex

cx T Ex

(3.10)

where the subscript x refers to the particular inner-shell level (K, L, M ....), Ex is the inner-shell binding energy, zx is the number of electrons in the core-shell, and bx and cx are constants. Typical values of these constants for K-edges are bx ∼ 1 and cx ∼ 4. The Bethe formula provides a useful parameterization of the total cross-sections that can be used for X-ray as well as EELS microanalysis.

Fig. 3.5 Differential generalized oscillator strength (GOS), dfx (q, E)/dE , as a function of momentum transfer q calculated from a Hartree–Slater program [3.20,3.21] for a carbon K -edge at different energies above threshold (285 eV). Dashed line is the approximate form of the GOS used to derive the Bethe formula (Eq. 3.10) with GOS having q = 0 ‘optical’ value for q less h. than a cut-off at qB = (2mT )1/2 /¯ At higher energies the Bethe ridge is evident near qB .

3.3.2

Partial Cross-Sections

In order to perform quantitative microanalysis using Eqs. 3.3 and 3.4 we must obtain values for the partial ionization cross-sections by integrating over suitable values of ∆ and β to give, β Ex +∆ σx (∆, β) = 0

Ex

d2 σx (θ, E) dθdE dθdE

(3.11)

where Ex is the inner-shell binding energy (i.e., edge-energy). A number of approaches have been used for estimating the partial ionization cross-sections. The matrix element in the GOS can be calculated from orthonormal

INNER-SHELL CROSS-SECTIONS

57

Hartree–Slater wave functions obtained by solving the Schr¨odinger equation for the bound and unoccupied continuum states in a self-consistent atomic potential [3.20,3.21,3.23]. The resulting GOS predicts quite well the different gross edge shapes, that is, the sharp hydrogen-like K-edge, and the delayed maxima for the L23 and M45 -edges (attributable to the centrifugal barrier in the radial wave equation for transitions to states with high angular momentum). Since the model is atomic it does not, of course, predict the near-edge ﬁne structure that arises from the distribution of bound states in the solid. Nevertheless the results provide the best computed estimates for partial and total ionization cross-sections. On the other hand the relative complexity of the calculations and the lack, so far, of readily available data bases for the GOS has led to the more widespread use of the much simpler hydrogenic model developed for EELS by Egerton [3.24,3.25]. The GOS for the hydrogen atom can be calculated analytically and the results scaled to give approximate values for atomic number Z. This is achieved by modifying the Coulomb potential to correspond to a charge of Ze, which is screened slightly by the remaining core electrons to give an effective charge Zs e. For the K-shell Zs ∼ Z − 0.3. The wave functions are thus concentrated closer to the nucleus but are are equivalent to hydrogen wave functions. Calculations of the hydrogenic GOS are easily performed on a small computer, and in addition to providing K-shell crosssections, the hydrogenic model has been adapted for the L-shell [3.26]; the programs for K- and L-shells are known as SIGMAK and SIGMAL, respectively. Resulting plots of dσ/dE for the K-edges (B to O) and the L23 -edges (Al to Cl, and Ti to Fe) are shown in Fig. 3.6(a–c) for E0 = 100 keV and β = 20 mrad. Sometimes it is useful to obtain a rough estimate of the partial ionization crosssection by assuming independent angle- and energy dependence [3.1,3.10,3.26,3.27]: σx (∆, β) = σxTOTAL F (β)G(∆)

(3.12)

σxTOTAL

where is the total ionization cross-section that can be obtained from the Bethe formula (Eq. 3.10), and F (∆) and G(β) are, respectively, the fractions of the cross-section lying within scattering semiangle β and within energy window of the edge. From Eqs. 3.9 and 3.11 we obtain, 1 + β2 ln θ2 E¯ for θ < θB and F (β) = ln 2 θE¯ (3.13) F (β) = 1 for θ > θB where θE¯ is the mean value of θE in the integration window. F (β) is plotted in Fig. 3.7a for E0 = 100 keV at different edge energies from 100 to 1600 eV. At an energy loss of 800 eV, approximately half the scattered electrons are collected inside a 20-mrad angle. If the core edge intensity falls off as ∼ E −r (the asymptotic form of the GOS at large E) we obtain from Eq. 3.11,

58

EELS QUANTITATIVE ANALYSIS

Fig. 3.6 SIGMAK [3.24] and SIGMAL [3.26] hydrogenic cross-sections for 100-keV beam energy and β = 20 mrad obtained using Gatan EL/P program: (a) K -shell excitation of B, C, N, and O; (b) L23 excitation of Al, Si, P, S, and Cl; (c) L23 excitation of Ti, V, Cr, Mn, and Fe.

59

INNER-SHELL CROSS-SECTIONS

G(∆) = 1 −

Ex Ex + ∆

r−1 (3.14)

G(∆) is plotted in Fig. 3.7b for E0 = 100 keV and for the same edge energy losses as in Fig. 3.7a; it is assumed that the exponent r = 4. For an edge energy of 400 eV approximately half the core edge intensity is contained within 100 eV of threshold. Eqs. 3.13 and 3.14 are useful for selecting appropriate spectrometer acceptance angles and energy integration windows for analyzing the spectra.

Fig. 3.7 (a) Fraction of energy loss electrons that are scattered inside semiangle β for different energy losses at E0 = 100 keV. β is deﬁned by an objective aperture in TEM mode and by a spectrometer acceptance aperture in STEM mode. (b) Fraction of core edge signal that lies inside integration window ∆ for different edge energies assuming edge intensity falls off as E −4 and E0 = 100 keV.

Another approach for deriving the partial cross-sections involves normalizing a measured edge proﬁle to the calculated cross-section or differential oscillator strength. Calculated cross-sections may be accurate to 5 or 10% for integration windows extending 100 or 200 eV above the edge, but may be very imprecise close to threshold because of solid-state or atomic effects. The method is illustrated in Fig. 3.8 for the iron L23 -edge recorded from a very thin layer of ferritin molecules. In this case the integrated intensity is matched to the hydrogenic SIGMAL cross-section over a range of 200 eV [3.26]. The large enhancement of the cross-section near the edge is due to the white-line resonance. Egerton has parameterized a wide range of inner-shell cross-sections by compiling dipole oscillator strengths for K, L, M, N, and O shells based on a combination of Hartree–Slater calculations, photoabsorption data and EELS measurements [3.29]. For certain inner-shell excitations (e.g., the M45 ), cross-section values may not be sufﬁciently reliable to perform accurate quantitative analysis. It can then be useful to measure cross-section ratios in a well-deﬁned scattering geometry using standard specimens [3.30,3.31]. This approach is illustrated by quantifying the spectrum from a single ∼ 100-nm particle of Y-Ba-Cu-oxide high Tc superconductor (Fig. 3.9). The elements detected are the O K-edge (532 eV), the Ba M45 -edge (780 eV), the Cu L23 -

60

EELS QUANTITATIVE ANALYSIS

Fig. 3.8 Cross-section for Fe L23 -edge for E0 = 100 kV and β = 20 mrad. Dashed line is hydrogenic calculation; solid line is edge shape measured from thin (∼ 10 nm) sample of ferritin and scaled to match hydrogenic cross-section integrated over 200 eV. This approach is useful for quantiﬁcation when only near-edge region can be used for analysis, for example, due to edge overlap or difﬁculty in background estimation.

Sample

SY /SO

Y2 O3 (standard) BaCO3 (standard) CuO (standard) Y-Ba-Cu-oxide

0.0403

SBa /SO

SCu /SO

nBa /nO

nCu /nO

0.667 1.08

0.0109

nY /nO

0.847

0.333 0.311 0.150

1.000 0.181 0.261 0.482 (±0.010) (±0.015) (±0.025)

Table 3.1 Ratios of signals and numbers of atoms in standards of yttrium, barium and copper oxides; values used to determine composition of Y–Ba–Cu– oxide particle. Beam energy = 100 keV, = 50 eV, β = 20 mrad.

edge (931 eV) and the Y L23 -edge (2041 eV). Integrated edge intensity ratios (∆ = 50 eV) are listed in Table 3.1 together with measurements from particles of Y2 O3 , BaCO3 and CuO which serve as standards. Assuming there are 6.5 oxygen atoms in the chemical formula, a composition of Y1.18±0.05 Ba1.70±0.10 Cu3.13±0.15 O6.5 is obtained, a result that is close to the nominal composition of YBa2 Cu3 O7−x . This specimen was in fact inhomogeneous and the particles were found to have a wide range of compositions. Hofer et al. employed systematically a similar experimental approach to a series of thin oxide ﬁlms and have thus obtained partial cross-sections for many of the K,

INNER-SHELL CROSS-SECTIONS

61

Fig. 3.9 Parallel-EELS from ∼ 100 nm diameter Y-Ba-Cu-oxide superconductor particle analyzed in the STEM mode with E0 = 100 keV and β = 20 mrad. Probe current was 1 nA and acquisition time 10 s. (a) O K -edge, Ba M45 -edge and Cu L23 -edge. (b) Y L23 -edge. Edge intensities shown for integration window ∆ = 50 eV, obtained with Γ = 50 eV (specimen courtesy of D. E. Newbury).

L, M, and N edges [3.32,3.33]. Since each of the metal ﬁlms contains oxygen in a known stoichiometric ratio, the inner-shell cross-sections of all the metals atoms can be determined from the oxygen K-shell cross-section, whose value is known reliably from the hydrogenic model.

62

3.4 3.4.1

EELS QUANTITATIVE ANALYSIS

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION Simple Least-Squares Fitting

The core edges that we wish to quantify are generally superimposed on a large background that can be produced by several processes as indicated in Fig. 3.10. There are contributions due to excitation of valence electrons to high energies in the continuum (the tail of the plasmon) and those due to lower energy core level excitations (tails of core-edges). In addition, there are plural scattering contributions to the background arising from multiple valence excitations and also a lower energy core excitation combined with one or more valence excitations. Finally there is the possibility of an instrumental background due to spurious scattering in the electron spectrometer or to detector noise. Despite these complications the background is often found to follow closely an inverse power law [3.34,3.35] of the form, Bj = AEj−r

(3.15)

where A and r are constants. Here j denotes the channel number with corresponding energy loss Ej . By taking the logarithm of Eq. 3.15 we can ﬁt the background with a straight line and extrapolate it into the region of the core edge [3.5,3.36]. We use simple linear regression and minimize the value of χ2 given by, χ2 =

ln Bj − ln A + r ln Ej 2 σj

(3.16)

where σj are the weights associated with channel j. If we write xj = ln Ej and yj = ln Bj and assume that the weights are constant in the ﬁtting region then the constants r and A are given by,

yj − m xj xj yj r = and 2

m x2j − xj

xj

yj +r ln A = m m where the sums are computed over all m channels in the ﬁtting region. The semi-empirical inverse power law has a basis in the theoretical form of the GOS [3.17,3.18]. From the hydrogenic calculation it can be shown that the high energy tail of the GOS at q = 0 has the form, dfx (0, E)/dE ∝ E −3.5 , so from Eq. 3.11 with β θE we can predict B ∝ E −6.5 . At intermediate values of β(θB β θE ) the spectrum is expected to have the form B ∝ E −4.5 , and for very large β and E Ex where most of the scattering is in the Bethe ridge we can predict B ∝ E −2 . Similarly it can be shown that the asymptotic form of the valence excitation spectrum in the Drude (quasi-free electron) model has the form B ∝ E −3 [3.5]. Even with plural inelastic scattering it is found that the background shape still often retains its inverse power law dependence. However, for thicker specimens the

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION

63

Fig. 3.10 Different contributions to the core edge spectrum: (1) background due to detector noise and spurious electron scattering in spectrometer, (2) singlescattering tails of valence or lower energy core excitations, (3) plural inelastic scattering involving high-energy tails of valence or lower energy core excitations combined with one or more ‘plasmon’ excitations, (4) single-scattering core edge intensity, (5) plural inelastic scattering involving core excitation combined with one or more ‘plasmon’ excitations.

parameter r can change its value signiﬁcantly as described by the detailed analysis of Su and Zeitler [3.37]. Any instrumental background such as detector dark current can affect the validity of the inverse power law signiﬁcantly and must subtracted before analysis. Before quantiﬁcation, it is also necessary to normalize the parallelrecorded energy loss spectrum by the detector response function, which is obtained by uniformly illuminating the detector array [3.38]. Accuracy in conventional EELS microanalysis is often limited by the systematic and statistical errors associated with the background ﬁtting procedure. The systematic variations are complex and difﬁcult to estimate [3.39,3.40]. They depend on: the validity of the inverse power law; the occurrence of plural scattering (discussed later); the magnitude of weak extended ﬁne structure oscillations; and on any ‘hidden’ minor edges that may be present in the spectrum. To minimize the statistical errors in the signal Sx obtained by subtracting the extrapolated background B, we should consider the choice of ﬁtting region Γ and the integration region ∆ carefully. The variance in Sx is given by the total number of counts in the edge region (Sx + B) plus the variance in the estimated value of B: var(Sx ) = Sx + B+Var(B). The errors produced by counting statistics in an ideal spectrum can be evaluated rigorously [3.41,3.42], but it also useful to consider a simpliﬁed expression for the signal to noise ratio, due to Egerton [3.43] Sx signal Sx = = noise [var(Sx )]1/2 (Sx + hB)1/2

(3.17)

where h = 1 + [var(B)]/B. The pre-edge region Γ should be as large as possible to minimize the variance in the estimated background. However there is a limit to the useful width of Γ because the inverse power law may break down resulting in undesirable systematic errors. For the same reason, the integration region ∆ should be smaller than Γ but should be optimized to include sufﬁcient signal. A complete error analysis [3.43] yields values for h that typically lie between 5 and 30, with h ∼ 15 when the pre-edge ﬁtting regions and the integration windows are equal (Γ = ∆). For K-edges in the range 300–1000 eV (carbon to sodium), optimal values are Γ ∼ 100 eV and ∆ ∼ 50 eV. In practice, the exact choice of Γ and ∆ is often not

64

EELS QUANTITATIVE ANALYSIS

critical for determining the statistical errors but may be more important for reducing systematic errors. 3.4.2

Plural Scattering Effects

Often the measured energy loss spectrum is complicated by plural inelastic scattering effects. For typical thicknesses of specimens in materials research, there is a high probability of the fast electron losing energy by excitation of valence electrons. Therefore when we measure the core- edge spectrum, a signiﬁcant fraction of the signal may consist of mixed inner-shell and valence excitations. Such processes change the shape of the spectrum transferring intensity away from the edge (see Fig. 3.10). There is also a transfer of pre-edge intensity towards higher energy losses and this increases the background intensity at the edge. In addition to reduction in signal/background ratio due to plural inelastic scattering, mixed elastic–inelastic scattering can also occur that scatters electrons out of the collection aperture; this will reduce both the signal and the background equally. Plural scattering can affect quantitative microanalysis in three ways. First, it may complicate the spectral shape so that edge misidentiﬁcation can occur. Second, it can invalidate the model used to ﬁt the background. Third, it can affect the accuracy of microanalysis based on measured or experimental values for the partial crosssections. Fortunately, it is very easy to assess the amount of plural scattering that is present by monitoring the low-loss spectrum. As the sample thickness increases, the fraction of electrons in the zero-loss peak falls off exponentially and this behavior is characterized by an inelastic mean free path λi . If Iz (E) is the zero-loss intensity, then, −t I(E)dE Iz (E)dE = exp λi where I(E)dE is the total spectral intensity. This gives a simple and useful expression for the relative sample thickness [3.44,3.45]: I(E)dE t (3.18) = ln λi Iz (E)dE It is the relative sample thickness t/λi that determines how the core edge spectrum is affected by plural scattering. Typical values for λi at 100-kV beam energy are ∼ 100 nm for carbon, 90 nm for aluminum and ∼ 70 nm for vanadium with a gradual decrease for higher atomic numbers [3.46]. The inelastic mean free path can be written in terms of the total inelastic cross-section σi for an element of atomic mass MA and density ρ as, λi = MA /N0 ρσi , where N0 is Avogadro’s number. Let us now estimate how plural scattering affects the visibility of core edges. The signal and background are given by,

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION

E x +∆

Sx (∆, β) =

Ix (E)dE Ex

65

E x +∆

and

B(∆, β) =

IB (E)dE Ex

Consider a small integration window (∼ 10 eV), a high-energy edge for which the background falls off relatively slowly with energy loss, and a noncrystalline specimen for which an elastic mean free path λe can be deﬁned. The signal is proportional to the thickness (i.e., number of atoms analyzed) and is attenuated by mixed elastic– inelastic as well as plural inelastic scattering. The background is also proportional to thickness but is attenuated only by elastic scattering; this is because the background extends to lower energy losses in front of the edge and is not attenuated by plural inelastic scattering. We can therefore write −t −t Sx (∆, β) ∝ t exp exp and λi λe −t B(∆, β) ∝ t exp (3.19) λe The signal/background ratio therefore falls off as exp(−t/λi ) and the maximum in the signal/noise = Sx (∆, β)/[B(∆, β)]1/2 occurs at an optimal sample thickness, topt = λe λi /(2λi + λe ). For an element such as silicon the inelastic and elastic mean free paths are approximately equal and the optimum sample thickness topt = λi /3 is only ∼ 30 nm at 100 keV beam energy. This explains why such thin samples must be used in EELS microanalysis. Theoretical curves for signal/background and signal/noise are plotted as a function of relative thickness in Fig. 3.11. The deterioration of signal/background is illustrated by the spectra in Fig. 3.12 from three thicknesses of boron nitride (t/λi = 0.21, 1.03, and 1.75). For the boron Kedge, where the background falls off sharply with energy loss, the signal/background actually decreases even faster than exp(−t/λi ).

Fig. 3.11 Signal/background (solid line) and signal/noise (dashed line) as function of relative sample thickness (t/λi ) in arbitrary units. Curves are estimated for sample such as silicon having approximately equal elastic and inelastic mean free paths, and for high energy core edge with slowly varying background (see text).

66

EELS QUANTITATIVE ANALYSIS

Fig. 3.12 Spectra recorded at E0 = 100 keV from BN sample for three different thicknesses: (a) t/λi = 0.21, (b) t/λi = 1.03, (c) t/λi = 1.75, where λi ∼ 70 nm (relative thickness determined from low-loss spectrum). At the boron K -edge the signal/background decreases by factor of 15 as t/λi increases to 1.75. This larger factor than predicted in Fig. 3.11 can be attributed to rapidly changing background ∼ 200 eV.

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION

67

Calculated Ratio B/N t/λi

∆ = 20 eV

∆ = 50 eV

∆ = 80 eV

0.21 1.03 1.49 1.75 2.55

0.98±0.05 0.96±0.05 0.88±0.05 0.79±0.05 0.58±0.05

1.04±0.05 1.06±0.05 0.90±0.05 0.82±0.05 0.57±0.05

1.05±0.05 1.07±0.05 0.95±0.05 0.79±0.05 0.59±0.05

Table 3.2 Estimated ratios of B/N in different thicknesses of BN for three values of ∆. Beam energy = 100 keV, Γ = 50 eV, β = 20 mrad.

The effect of sample thickness on the determination of elemental ratios has been described by Egerton [3.5], Zaluzec [3.47] and Su et al. [3.48]. It is found that there is a systematic change in the calculated atomic ratio with increasing thickness. This is demonstrated for the boron nitride sample by the data in Table 3.2. For t/λi < 1 the expected result is obtained but as t/λi approaches 1.5 the calculated ratio falls off sharply, dropping by ∼ 50% at t/λi = 2.5. This behavior is independent of integration window ∆ and appears to be due to effects of mixed elastic-inelastic scattering. If the thinnest regions of the specimen have a thickness greater than ∼ topt at a given beam energy (e.g., 100 kV) then the signal/background and signal/noise can be improved by increasing the microscope accelerating voltage since this increases the inelastic mean free path [3.49]. For a beam energy of 200 keV the total inelastic mean free path is approximately a factor of 1.6 greater than at 100 keV, and for 300keV electrons the factor is ∼ 2 greater than at 100 keV. The ‘intermediate-voltage’ analytical electron microscope therefore offers an important advantage for EELS. A more detailed description of how plural scattering affects the spectral shape can be obtained as follows. The measured intensity per unit energy loss I(E) is given by a Poisson distribution of multiple scattering events [3.50]: I(E)

=

exp

−t λi

δ(E) +

I0 (E) ∗

2 t t 1 Q(E) ∗ Q(E) + · · · (3.20) Q(E) + λi 2! λi

where, I0 (E) is the incident electron distribution broadened by the spectrometer resolution function and Q(E) is the single scattering distribution normalized to unity for the integral over all energy losses. Q(E) is the spectrum (less zero-loss peak) that would be measured in the limit of a very thin sample. Notice that the expression in the square-bracket sums to exp(t/λi ) when integrated over all E so that I(E)dE = I0 (E)dE = J0 , i.e., the total number of electrons in the spectrum

68

EELS QUANTITATIVE ANALYSIS

is equal to the total number of electrons incident on the sample. Equation 3.20 is exact if the spectrum is collected over all scattering angles. In practice we collect the spectrum inside angle β, so a proper description of the plural scattering would be much more complicated, involving the quantity Q(E, θ), and convolutions in angle and energy [3.50]. Nevertheless, Eq. 3.20 is found to hold approximately even for ﬁnite collection angle, which can be explained in part by the smaller size of θE for valence electron excitations relative to θE for core excitations. Equation 3.20 leads to a method for removing the effects of plural inelastic scattering through spectral deconvolution. By taking the Fourier transform (denoted by a line symbol), the convolutions become multiplications and we obtain [3.50]

I(E)

=

−t exp I 0 (E) × λi

2 t t 1 1+ Q(E) + Q(E) × Q(E) + · · · (3.21) λi 2! λi

By summing the series in the square bracket, we obtain an exponential, I(E) = exp

−t λi

I 0 (E) exp

t Q(E) = I 0 (E) exp{Q(E)} λi

Therefore the single scattering distribution is given by the inverse Fourier transform [3.50–3.52] I(E) −1 ln Q(E) = F T (3.22) I 0 (E) In practice, we must attenuate the high frequency Fourier coefﬁcients either by reconvolving by the normalized zero loss peak or by replacing the intensity in I0 (E) by a delta function [3.53,3.54]. Equation 3.22 is known as the “Fourier-logarithmic deconvolution” and it requires collection of the entire energy loss spectrum including the zero-loss peak. It is the method that must be used to unravel the core edge spectra at lower energy losses (less than ∼ 100 eV). This is the region where plural scattering tends to most complicate the background shape because of the proximity of the valence excitation peaks, e.g., background subtraction below the Li K-edge (∼ 55 eV). The logarithmic deconvolution can also be performed at higher energy losses where the characteristic scattering angle θE becomes large. Surprisingly the method is still found to work in practice in this regime even if a limited collection angle β is used, a fact that can be explained by the properties of the angular distribution [3.55,3.56]. For higher energy losses it is possible to simplify the single-scattering calculation provided we can ﬁrst subtract the background below a group of core edges that we wish to analyze. This measured intensity Ix (E) can be considered as the single-

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION

69

scattering core edge spectrum Qx (E) convolved with the measured low loss spectrum I(E). Therefore we can write [3.52,3.53,3.57]: I xE Qx (E) = F T −1 (3.23) IE which is known as the “Fourier ratio deconvolution”. Again it is necessary to attenuate high frequency Fourier coefﬁcients as mentioned for the Fourier ratio method. Spectral deconvolution is a very useful technique for interpreting spectra and is necessary as a preliminary step in the analysis of low-energy edges such as Li. However, it is not always required for quantiﬁcation of core-edges above ∼ 200 eV, even when the sample thickness exceeds λi . Most common core-edges can be analyzed satisfactorily with plural scattering present. It should also be emphasized that the standard deconvolution methods do not remove all artifacts, for example, systematic variations in estimated atomic ratios with increasing sample thickness, and they fail when the analyzed area has non-uniform thickness, producing additional artifacts. 3.4.3

Use of Filtering to Measure Low Concentrations

When measuring very low elemental concentrations, it is sometimes useful to apply digital ﬁltering techniques that enhance the visibility of weak core edges. For atomic concentrations < 10−3 , the signal to background ratio is typically < 0.01 so that small systematic errors created by ﬁtting the pre-edge background and extrapolating it into the post-edge spectrum may cause the standard ﬁtting and integration energy windows to become invalid even for quite small windows. Under these circumstances, it may be necessary to restrict the integration widow to a width comparable with the dominant edge feature such as a “white line” in the transition metals or rare earth elements. Two ﬁltering approaches have been used, one by adapting the acquisition procedure as ﬁrst described by Shuman [3.58,3.59], and the other by applying a digital ﬁlter to the spectrum after it has been acquired in the conventional way. The former approach has been very useful for measuring trace elemental concentrations using one-dimensional photodiode-array detectors whose pixels are typically 100 times longer than their width in the dispersion direction. This prevents exact correction of the channel gain function due to intensity variations that are perpendicular to the dispersion direction [3.60]. The ﬁrst-difference spectrum is obtained by acquiring two spectra shifted electrically, for example, by applying voltages −∆/2 and +∆/2 to the spectrometer drift tube. If the spectral intensity at channel n corresponding to energy En is I(En ) then the ﬁrst difference spectrum is given by, ∆ ∆ I (En ) = I En + − I En − (3.24) 2 2 This operation is similar to a ﬁrst derivative and it enhances small features situated on a slowly varying background. In order to optimize the signal/noise ratio in the

70

EELS QUANTITATIVE ANALYSIS

ﬁrst-difference spectrum, we should match ∆ to the energy width of the feature. In addition to enhancing edge visibility, the gain ﬂuctuations in I (En ) are greatly reduced because we are subtracting two nearly equal spectra each with the same channel-to-channel gain variations, which remain in a ﬁxed position on the array. Similarly we can obtain a second-difference spectrum [3.58,3.59] by collecting three spectra offset electrically by energies −∆, 0, and +∆, I(En + ∆) I(En − ∆) − (3.25) 2 2 This is similar to a negative second-derivative [3.61,3.62] and can be easier to interpret than the ﬁrst-derivative since a peak in the normal spectrum appears as a peak in the second difference spectrum. These difference spectra can be acquired easily and, although they may appear more complicated (e.g., contain negative intensities), quantiﬁcation can be accomplished by ﬁtting reference spectra (see below), provided the edge shape is similar in the unknown specimen and in the reference spectrum. With the availability of two-dimensional charge-couple device (CCD) arrays for electron energy loss spectroscopy, correction of the channel gain variations can be performed much more accurately and it is no longer necessary to use difference acquisition techniques to measure trace elements [3.63,3.64]. Nevertheless, it can still be useful to apply a ﬁlter such as the ﬁrst-difference or second-difference post facto in order to enhance the edge visibility and quantify the trace elemental concentrations. Use of the second difference method to detect low elemental concentrations is illustrated in Fig. 3.13, which shows second-difference spectra (∆ = 2 eV) from a sample of boron-implanted silicon recorded using a parallel-detection spectrometer and a ﬁeld-emission STEM. The analyzed areas were about 50 nm in diameter and the total acquisition time was 60 s with a 0.5-nA probe current. A sharp feature at ∼ 190 eV occurs only in the doped layer and can be attributed to the boron K-edge, which is present at ∼ 1 or 2 atom %. Quantiﬁcation was not possible in this sample however because no suitable reference spectrum was available (note, for example, that BN failed to give an appropriate reference spectrum because the near-edge structure was completely different). I (En ) = I(En ) −

3.4.4

Multiple Least-Squares Fitting

In order to quantify difference spectra or severely overlapping core edges conventional background ﬁtting techniques obviously cannot be applied. In these cases, an important alternative method is to ﬁt reference spectra recorded from standard specimens using a multiple least-squares (MLS) procedure [3.58,3.59,3.62]. First, we outline the principles of the MLS method and then discuss how it can be modiﬁed to take into account plural scattering and small changes in energy calibration. Let us consider a core edge spectrum I(En ) from an unknown specimen containing N channels with energies En . We assume the spectrum to be the sum of reference spectra, Xm (En ), where m = 1 to M with coefﬁcients am . Then the coefﬁcients can be determined by minimizing the value of χ2 ,

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION

71

Fig. 3.13 Second-difference spectra from ∼ 50-nm diameter regions of boron- implanted silicon specimen recorded at E0 = 100 kV, β = 20 mrad with ∆ = 2 eV. Probe current was 0.5 nA and total acquisition time 60 s. Small boron K -edge is observed only in boron-doped layer and not in silicon substrate; it is easily separable from the extended ﬁne structure oscillations. Boron was not detected in normal spectrum because signal is swamped by channel-to-channel gain variations in detector. The concentration of boron is approximately 1 atom% but could not be quantiﬁed because no suitable reference spectrum was available (specimen courtesy of R. F. Egerton and R. Kola).

χ2 =

N

2

M

I(En ) −

am Xm (En )

m=1

= minimum

2

[σ(En )]

n=1

(3.26)

where σ(En ) is the estimated standard error in the nth channel of the unknown spectrum. When the noise is limited by Poisson counting statistics, [σ(En )]2 is equal to I(En ). For a parallel detector it is very important that I(En ) be converted into numbers of incident electrons rather than detector counts [3.2], otherwise the estimated errors in the ﬁtting coefﬁcients will be incorrect. To solve Eq. 3.26 we write [3.65–3.67]:

αmj

=

N

Xj (En )Xm (En ) 2

n=1

βmj

=

[σ(En )]

N

I(En )Xm (En ) 2

n=1

[σ(En )]

The solution is then given by inverting the matrix [α],

and

(3.27)

72

EELS QUANTITATIVE ANALYSIS

M

aj =

[α]−1 jm βm

(3.28)

m=1

and the estimated variances in the coefﬁcients aj are given by the diagonal elements of the inverse matrix [α]−1 , σ 2 (aj ) = [α]−1 jj

(3.29)

The reference spectra Xm (En ) (ﬁrst-difference, second difference or normal) are generally measured from thin standard specimens where plural scattering is not signiﬁcant. Plural scattering can be taken into account by including P additional reference spectra consisting of the core-edge reference convolved once, twice, . . . , P times with the single-scattering low loss spectrum Q(E); these new reference spectra correspond to a core excitation + one plasmon, core excitation + two plasmons, and so on. Since the plural scattering effects are the same at each edge, we need only include the plural scattering terms for the major edges and those ﬁtted coefﬁcients will determine the ratios of plural scattering coefﬁcients for minor edges that are present [3.65]. The ﬁtted coefﬁcients are then given by, 2

χ =

N

I(En ) −

M P

2 amp Xmp (En )

p=1 m=1 2

[σ(En )]

n=1

= minimum

(3.30)

where, Xm0 (E) Xm1 (E)

= Xm (E)

Xm2 (E)

= Xm (E) ∗ Q(E) ∗ Q(E)

= Xm (E) ∗ Q(E)

... This method of spectral ﬁtting has the important advantage of not requiring uniform sample thickness within the analytical volume, that is, the coefﬁcients are not constrained to have the same ratios as in Eq. 3.20. Small drifts in energy calibration can also be taken into account by including an extra reference spectrum the ﬁrst derivative of the measured core edge intensity. The correction arises as the ﬁrst term in a Taylor expansion of the original reference spectrum. This technique was ﬁrst used to correct for small calibration variations in the analysis of X-ray spectra [3.68]. In the case of difference spectra acquired with parallel detection it can be helpful to add a ramp and an offset as two additional reference spectra to correct for small intensity shifts between successive read-outs. Quantitative results can be obtained easily from the MLS coefﬁcients if the reference spectra are normalized to the integrated counts for some energy window ∆. ref If Im (En , β) is the intensity in the spectrum used to generate the mth reference spectrum, the integrated signal in energy window ∆ for collection angle β is

73

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION

(Em +∆)/δE ref Sm (∆, β) =

ref Im (En , β)

Em /δE

where δE is the energy per channel. We set Xm (En ) =

ref Im (En , β) ref (∆, β) Sm

(3.31)

so that the ﬁtting coefﬁcients obtained from Eq. 3.28 are simply equal to the estimated unknown signal in the unknown spectrum: am = Sm (∆, β). This relation also holds for ref (∆, β) is obtained from ﬁrst or second difference spectra provided the quantity Sm the corresponding normal spectrum

1st−diﬀ (En ) = Xm

ref Im (En , β) ref (∆, β) Sm

and

2nd−diﬀ Xm (En ) =

ref Im (En , β) ref (∆, β) Sm

A demonstration of quantitative analysis using the MLS ﬁtting method is given by the spectra from a boron nitride sample in Fig. 3.14. Second-difference B and N K-edges with ∆ = 4 eV from a sample with t/λi ∼ 0.7 (solid lines) are ﬁt to reference spectra recorded from a thinner specimen with t/λi ∼ 0.2 (dashed lines). By normalizing the reference spectra to the corresponding integrated edge intensities (Eq. 3.31) an atomic fraction B/N= 1.05 ± 0.01 (statistical error) was obtained, in agreement with the known composition and the result obtained from the conventional method (Fig. 3.4). In this example the systematic errors in the cross-sections were greater than the statistical errors estimated from the MLS ﬁt. An example of quantiﬁcation of a low elemental concentration is given in Fig. 3.15a and b, which show spectra from a nickel oxide ﬁlm containing a trace concentration of manganese. Conventional analysis of the O K-edge and Ni L23 -edge (Fig. 3.15a) gives a Ni/O fraction of 1.07 ± 0.05 (error due to uncertainties in cross-sections) but the Mn L23 edge is barely visible in the normal spectrum. Nevertheless, the second difference spectrum (Fig. 3.15b) shows an easily-visible Mn L23 -edge. A reference spectrum was obtained from a potassium permanganate salt dried on a thin carbon ﬁlm (Fig. 3.15c and d). Although the L23 white line shape can vary signiﬁcantly depending on oxidation state [3.69] the second-difference KMnO4 reference spectrum ﬁts the Mn edge in the NiO ﬁlm satisfactorily (Fig. 3.15e). A value of Mn/Ni= 9.6 × 10−3 ±1.0×10−4 (estimated statistical standard deviation) is obtained demonstrating the feasibility of detecting atomic concentrations below 10−3 using parallel-EELS. In another application, it has been possible to measure concentrations of a few atomic parts per million of the rare earth elements in glasses by using the difference acquisition technique [3.60]; such a high analytical sensitivity is facilitated by the intense M4,5 white lines of the lanthanides. The main limitation of the MLS method for ﬁtting core edges is the requirement for suitable reference spectra. Signiﬁcant variations in near-edge ﬁne structure occur depending on the chemical environment of the atoms that are being probed. An edge is often very different in a metal and

74

EELS QUANTITATIVE ANALYSIS

Fig. 3.14 Demonstration of MLS ﬁtting procedure to quantify the B/N ratio in a BN sample using reference spectra also obtained from BN. (a) Second-difference spectrum (∆ = 4 eV) of B K -edge with MLS ﬁt (dashed line); (b) second-difference spectrum of N K -edge with MLS ﬁt (dashed line). B/N was 1.05 ± 0.01 (statistical error) which agrees well with the value obtained using the conventional method (Figure 3.4). The importance of the MLS approach is to detect very low concentrations and to separate overlapping edges for which conventional background extrapolation fails.

its oxide or carbide, for example, the Cu L23 -edge in copper metal and CuO. It is therefore necessary to have available several different reference spectra and to have some a priori knowledge about the specimen to decide which reference spectra is relevant. In some cases where this information does not exist, it is not possible to apply the MLS method. However for many applications it can give reliable and accurate results [3.58,3.59,3.65,3.70]. The MLS procedure not only provides us with an estimate of the errors in the coefﬁcients, but also gives us the quality of the ﬁt (χ2 ) so that a poor choice of reference spectra should become evident in the analysis.

BACKGROUND SUBTRACTION AND SIGNAL ESTIMATION

75

Fig. 3.15 Quantitative analysis of a specimen of NiO containing a small fraction of manganese; parallel-EELS recorded in STEM instrument at E0 = 100 keV with β = 20 mrad, probe current = 0.8 nA and acquisition time = 100 s. (a) O K -edge and Ni L23 -edge integrated signals for ∆ = 50 eV, obtained with Γ = 50 eV; conventional quantitative analysis gives Ni/O = 1.07; Mn L23 -edge is barely visible. (b) Seconddifference spectrum (∆ = 4 eV) now showing easily visible Mn L23 -edge superimposed on the O K -edge extended structure. (c) Normal reference spectrum for the Mn L23 edge obtained from a sample of KMnO4 . (d) Second-difference spectrum from KMnO4 . (e) MLS ﬁt to the spectrum in (b) using the reference spectrum in (d); a value of Mn/Ni = 9.6×10−3 ±1.0×10−4 (standard deviation) is obtained demonstrating the feasibility of detecting concentrations below 10−3 using parallel-EELS.

A variation of the multiple least-squares method developed by Steele et al. [3.71] involves ﬁtting the background and edge together using calculated core edge crosssection data instead of experimental reference spectra. This also allows difﬁcult analyses of overlapping edges to be attempted and has the further advantage of providing quantitative results directly.

76

EELS QUANTITATIVE ANALYSIS

3.5 FACTORS INFLUENCING ACCURACY OF QUANTIFICATION 3.5.1

Spatial Resolution (CTEM and STEM)

EELS microanalysis may be performed in the conventional transmission electron microscope (CTEM) as well as in the STEM. In the CTEM mode the image area of interest is moved to the center of the viewing screen and is selected by the spectrometer entrance aperture. The spectrometer is diffraction coupled, that is, has a diffraction pattern with short camera length as its object [3.5]. The objective aperture deﬁnes the scattering angles. An aperture situated after the imaging lenses of the microscope column selects the analyzed area, which means that aberrations can limit the spatial resolution. First, spherical aberration of the objective lens limits resolution to ∼ Cs β 3 , where Cs is the spherical aberration constant (typically ∼ 3 mm). For β = 20 mrad the resolution is degraded to ∼ 24 nm but if the objective aperture is removed to increase collection efﬁciency, the spatial resolution can be limited to several micrometers [3.5]. Second, chromatic aberration also produces a loss in spatial resolution in CTEM operation [3.5,3.72,3.73]. An image that is focused on the viewing screen will be degraded in resolution by Cc (E/E0 )β at energy loss E, where Cc is the chromatic aberration constant of the objective (also typically 3 mm). At 100-keV beam energy and β = 20 mrad, electrons that have lost an energy of 1000 eV will suffer a loss in resolution of ∼ 600 nm. In STEM (or convergent-beam diffraction) operation, the spectrometer is imagecoupled, that is, has the focused probe as its object. The spectrometer entrance aperture contains a diffraction pattern and therefore selects the scattering angle β according to the diffraction camera length. For many applications, probe currents of order 0.1–5 nA are required to obtain sufﬁcient signal and therefore the resolution in STEM mode is limited by the source brightness. The minimum useful probe diameter is typically ∼ 30 nm for conventional tungsten ﬁlament sources, and ∼ 10 nm for LaB6 sources. For the much narrower probes available with ﬁeld-emission sources, the specimen geometry may limit the useful spatial resolution. If an inﬁnitesimal probe is focused on the top surface of an amorphous specimen of thickness t, and electrons are collected through scattering angle β, the resolution will be degraded by an amount ∼ 2βt, corresponding to the divergence of the probe at the exit surface. For β = 20 mrad and t = 50 nm, the effective resolution is ∼ 2 nm. However, ﬁeldemission sources now provide probe diameters of 0.2 nm and recent developments in design of spherical aberration correctors for the STEM promise sub 0.1-nm diameter probes in the future. It is possible to exploit such high spatial resolution in the analysis of crystalline specimens, in well-deﬁned orientations, due to channeling of the electron probe down the atomic columns. Recent results indicate that electron energy loss spectra can be collected from single atomic columns that are imaged in high-angle dark ﬁeld.

FACTORS INFLUENCING ACCURACY OF QUANTIFICATION

3.5.2

77

Beam Convergence Correction

When EELS microanalysis is performed in the STEM, the probe convergence angle α is not necessarily small compared with the spectrometer collection angle β, particularly when a large objective aperture is used. Under these conditions the angular distribution of the core loss electrons is no longer given by Eq. 3.9, and if we use our previous estimates for the partial ionization cross-sections the quantitative results obtained from Eq. 3.3 will be in error (underestimated). A “beam-convergence” correction must therefore be applied to obtain a modiﬁed cross-section as described by Egerton [3.5]. The measured angular distribution of the core-edge can be obtained by 2 −1 a solid-angle convolution of the differential cross-section, dσx /dΩ ∝ (θ2 + θE ) with the incident probe angular distribution, dIprobe /dΩ = constant for θ < α, and dIprobe /dΩ = 0 for θ > α. This gives, dIx ∝ dΩ

α 2π θ0 =0 φ=0

dIprobe 2 2 −1 (θx + θE ) θ0 dθ0 dφ dΩ

(3.32)

where, θx2 = θ2 + θ02 − 2θθ0 cos φ. For angles α and β, Ix is then given by β Ix (α, β) = 0

dIx 2πθdθ dΩ

(3.33)

By integrating over an energy window ∆ we obtain E x +∆

Sx (α, β, ∆) =

Ix (α, β)dE Ex

If α < β, the corrected value for the number of atoms in the analytical volume is, Sx (0, β, ∆) (3.34) Nx = Nx Sx (α, β, ∆) If α > β, it is necessary to multiply Eq. 3.34 by an additional factor of β 2 /α2 to allow for a reduction of the low-loss spectral intensity. In practice the correction factor, Sx (α, β, ∆)/Sx (0, β, ∆) is typically ∼ 0.8 to 0.9 when α = β, and is usually very close to unity when α < β/2, in which case no correction is necessary [3.5]. 3.5.3

Crystalline Samples

In the CTEM a difﬁculty can occur in the quantitative EELS analysis of crystalline specimens if a strongly excited Bragg diffraction spot lies just outside or just inside the spectrometer collection angle β [3.5,3.74]. When the diffracted beam is just outside the aperture it contributes intensity to the core edge spectrum because of the relatively broad angular distribution (∼ θE ) of these electrons, but there is no

78

EELS QUANTITATIVE ANALYSIS

contribution to the zero-loss peak so the measured number of atoms obtained from Eq. 3.3 is overestimated. When the Bragg spot lies just inside the collection aperture the zero-loss peak will be higher relative to the core edge, so the number of atoms is underestimated. In CTEM operation one method of reducing these effects is to select a large collection angle (∼ 100 mrad) so that most of the diffracted beams are included [3.10]. Alternatively a correction factor can be applied [3.75]. When diffraction contrast is present in a CTEM bright-ﬁeld image there is another way that quantiﬁcation of core edges can be affected. The wider angular distribution of the electrons that have excited core levels is equivalent to an increased spread in the angles of the incident beam. Therefore the distribution of core edge intensity over the specimen area follows a smeared out diffraction contrast intensity. Since the lowloss spectrum is not smeared out the elemental composition obtained from Eq. 3.3 exhibits artifactual variations that can amount to ∼ 10–20%. In STEM operation the larger probe convergence normally prevents any signiﬁcant effect on quantitative microanalysis of crystalline specimens. Another factor that can cause errors in microanalysis of thin crystals is the dependence of inner-shell cross-section on sample orientation. In a crystalline specimen, the fast electrons can be considered as the sum of different Bloch waves each having the periodicity of the lattice; and the amplitudes of these Bloch wave coefﬁcients vary with sample orientation. Some Bloch waves have maxima that are localized on particular atoms and others have maxima localized between the atoms. If the spatial distribution of the scattering events that excite a particular core level is highly localized then the ionization cross-section may be strongly dependent on sample orientation. The effects will be strongest for more localized core levels having excitation energies above ∼ 1 keV [3.76,3.77]. For elements with edge energies of a few hundred eV, for example, the carbon K-shell, localization is not sufﬁcient to produce signiﬁcant effects. In the STEM mode, orientation dependence is greatly reduced because of the convergent probe (α >∼ 3 mrad), whereas in the CTEM the effects can be minimized by avoiding strong diffraction contrast conditions. 3.5.4

Errors in Cross-Section Estimation

There are two main sources of error in the estimated partial cross-sections needed for quantitative microanalysis. The ﬁrst involves the validity of theoretical calculations of the generalized oscillator strength described earlier. For K- and L-edges, and integration windows of ∼ 100 eV, an absolute accuracy of ∼ 5% is reasonable [3.23], although for M - and N -edges uncertainties are often considerably larger (> 20%). In general the smaller the integration window, the more the partial cross-sections are inﬂuenced by solid-state effects near the edge. The second source of errors in the estimated cross-sections involves the precision to which the collection angle β is known (see Fig. 3.7a). In the CTEM imaging mode β can be determined accurately from the objective aperture size and a calibrated diffraction pattern. In STEM β is determined by the spectrometer entrance aperture and the degree of angular compression produced by the post-specimen (projector) lenses. Even if there are no projector lenses, as is often the case in the dedicated

DETECTION LIMITS AND COMPARISON WITH EDXS

79

STEM, there is nevertheless signiﬁcant angular compression produced by the objective post-ﬁeld. For example, this is a factor of ∼ 4 for the STEM used to obtain the examples of EELS microanalysis described here; the exact compression factor depends on the objective lens excitation and may therefore vary with focus. The effective magnitude of β can be estimated from geometrical considerations, but a more precise value requires collection of a microdiffraction pattern under the same conditions used to record data from the unknown sample. In some TEM–STEM instruments, the projector lenses can be excited to give very large angular compressions so that scattering angles of ∼ 100 mrad can be collected [3.78]; a given uncertainty in β then gives a much smaller error in the partial cross-sections. 3.6 DETECTION LIMITS AND COMPARISON WITH EDXS Detection limits are generally governed by the counting statistics in the energy loss spectrum. The sensitivity can be described either in terms of the minimum detectable atomic fraction or the minimum detectable number of atoms of an element in a given matrix [3.4]. It is possible to estimate these quantities from an experimentally measured spectrum by using, for example, Eq. 3.29 to compute the standard deviation in the signal. It is also helpful to derive theoretical expressions for the detection limits that show their dependence on instrumental and specimen parameters [3.1,3.4,3.5,3.79]. The core edge signal Sx due to an atomic inner-shell excitation of element x in a sample of thickness t containing nx atoms/area is given by combining Eqs. 3.1 and 3.19, −t −t exp (3.35) Sx ≈ J0 ηnx σx exp λi λe We can write a similar expression for the background B under the edge, −t B ≈ J0 ηnσB exp λe

(3.36)

where J0 is the number of incident electrons, η is the detective quantum efﬁciency, n is the total number of matrix atoms per unit area of the sample contributing to the background intensity (n = N0 ρt/MA ), and σB is the mean background cross-section per atom. The energy window ∆ and the collection angle β have been dropped for convenience. By using Eq. 3.17 we obtain 1/2 J0 η −t Sx −t signal exp = ≈ n x σx exp noise hnσB λi 2λe [var(Sx )]1/2

(3.37)

where h is the parameter discussed earlier that depends on the widths of the ﬁtting and integration regions [3.37]. We consider a 99.7% conﬁdence for detection of

80

EELS QUANTITATIVE ANALYSIS

inner-shell excitation x corresponding to three standard deviations. Therefore the minimum detectable atomic fraction f is given by f=

n x

n

min

1/2 3 hσB −t −t ≈ exp exp σx nJ0 η λi 2λe

(3.38)

If the analysis is performed in a STEM at high magniﬁcation with a square raster of size d (or with a probe of diameter d) then the minimum detectable number of atoms is given by (Nx )min

1/2 −t 3d2 hσB −t exp ≈ exp σx J0 η λi 2λe

(3.39)

When analysis is performed on small specimen areas radiation-sensitivity may be an important limiting factor factor [3.70,3.80]. If the specimen can only tolerate a dose of Dmax electrons per unit area then Eq. 3.39 can be written as (Nx )min ≈

1/2 3d hσB n −t −t exp exp σx Dmax η λi 2λe

(3.40)

We illustrate the ultimate sensitivity that is achievable in EELS microanalysis by considering the detection of iron atoms in a thin 10 nm carbon matrix. We are interested in predicting the minimum number of atoms that are detectable in a probe of size d for different electron doses. We assume a beam voltage of 100 keV in a ﬁeldemission STEM with parallel-detection EELS (η = 1). The estimated cross-section for the Fe L23 edge (∆ = 6 eV, β = 20 mrad) is σFe = 6.2 × 10−26 m2 (Fig. 3.8) and the corresponding background cross-section (tail of C K-edge) is σB = 2.6 × 10−27 m2 . We further assume that the parameter h ∼ 8. The small value for ∆ ensures that only the Fe L3 white line is used to measure the signal and reduces the possibility of systematic errors being introduced in the background estimation. Using Eq. 3.40, we estimate that for a dose of 1010 e/nm2 into a 10-nm probe, ∼ 10 iron atoms are detectable at a 98% conﬁdence limit (corresponding to an atomic fraction of ∼ 10−4 ). Figure 3.16 shows the minimum number of iron atoms plotted against probe diameter on a log–log scale for electron doses ranging from 106 to 1012 e/nm2 (solid lines). Also indicated are the numbers of iron atoms versus probe diameter corresponding to different atomic fractions of iron from 10−1 to 10−5 (dashed lines). The prediction of near single-atom sensitivity for atoms in a thin carbon matrix has been conﬁrmed experimentally [3.70]. A more detailed method for estimating EELS detection limits has been developed by Natusch et al. [3.81]. In this approach, artiﬁcial spectra are computed for a particular specimen composition and electron dose by taking into account scattering probabilities for valence electron excitations, inner-shell excitations, plural inelastic scattering as well as elastic scattering. Random Poisson noise is then added to simulate a series of spectra, which can be analyzed to give signal/noise ratios that combine theory and experimental data.

DETECTION LIMITS AND COMPARISON WITH EDXS

81

Fig. 3.16 Microanalytical sensitivity illustrated by theoretical curves for iron detection (with 99.7% conﬁdence) in thin 10-nm carbon matrix assuming parallel-recording. Minimum detectable number of atoms is plotted on log–log scale as function of probe diameter, d, for different electron doses, D, (solid lines). Also indicated are corresponding atomic fractions, f , of iron in the analytical volume. Shaded regions of the graph are inaccessible, either because the probe diameter is too small or because the acquisition time is too long (e.g., a dose of 1012 e/nm2 into a region of 102 nm2 corresponds to an acquisition time of one hour with a 5-nA probe current)

Another important consideration regarding microanalytical sensitivity is the relative ﬁgure-of-merit between EELS and energy-dispersive X-ray spectroscopy (EDXS). This can be estimated by ﬁrst writing an expression for the ratio of EELS and EDXS signals, σx1 (∆, β) ηE −1 S EELS ≈ ω TOT EDXS S ηx x σx2

(3.41)

where σ TOT is the total cross-section for inner-shell excitation, ηE is the EELS detective quantum efﬁciency, ηx is the collection efﬁciency for X rays, and ωx is the X-ray ﬂuorescence yield. It should be noticed that the edge detected by EELS can originate from a different core level (x1) than the X rays detected by EDXS (x2). The relative sensitivity is given not by the ratio of signals but by the ratio of signal/noise. It is possible to estimate this quantity by making use of experimental values for the signal/background ratios for simultaneously acquired EELS and EDXS spectra. The relative sensitivity or ‘ﬁgure-of-merit’ is plotted in Fig. 3.17 as a function of atomic number, for trace concentrations of elements in a thin carbon matrix [3.66]. These

82

EELS QUANTITATIVE ANALYSIS

data correspond to analysis at 100 keV beam energy with parallel-detection EELS (β = 20 mrad) and an ultrathin window Si(Li) EDXS detector with a collection solid angle of 0.2 sterad.

Fig. 3.17 Calculated ratio of sensitivities for EELS and EDXS as a function of atomic number (Z < 30) for K and L23 edges using calculated values for ratios of EELS and EDXS signals and measured values for signal/background [3.66].

Figure 3.17 clearly indicates that EELS, using K-edge detection, gives higher sensitivity when elements in the ﬁrst row of the periodic table are embedded in a carbon matrix. For most elements with atomic numbers from 11 to 25, EELS retains an advantage over EDXS if the higher cross-section L23 -edge is used. The break in the curve at Z ∼ 20 corresponds to the occurrence of the ‘white line’ resonance at the beginning of the transition elements. The sensitivity of EELS relative to EDXS in general depends on the atomic number of the matrix. Similar plots could be established for detection of elements in, for example, silicon or iron matrices. Although EELS is indispensable for analysis of certain elements, some problems are more favorably solved by X-ray spectroscopy. Clearly, the two techniques must be regarded as complementary [3.82]. In certain applications even when EDXS has the advantage in terms of microanalytical sensitivity, EELS may still be the technique of choice because of the more detailed chemical information that is simultaneously available in the near-edge ﬁne structure. 3.7 QUANTITATIVE ELEMENTAL MAPPING 3.7.1

Spectrum Imaging

It is often much more useful to obtain two-dimensional spatial distributions of elements in a specimen rather than merely to perform a series of point analyses. Information obtained from such elemental maps can be treated in more sophisticated ways than is possible with point analyses. For example, we can plot all image pixels on a scatter diagram that indicates the relative concentrations of two or three ele-

QUANTITATIVE ELEMENTAL MAPPING

83

ments in a specimen. Clusters of pixels in the scatter diagram, reveal the presence of different phases that can be mapped back onto the specimen region [3.83–3.85]. The most straightforward approach for elemental mapping by EELS makes use of the energy ﬁltering transmission electron microscope (EFTEM) [3.86]. Typically, one or two pre-edge images and a post-edge image are acquired for each element, the so-called two-window or three-window methods, respectively [3.87–3.90]. In the two-window method, a single pre-edge image is scaled and then subtracted from the corresponding post-edge image for an element of interest until there is zero net intensity in areas where that element is known to be absent. Alternatively, a twowidow ratio map of the post-edge and pre-edge images can be calculated, which has the advantage of canceling out diffraction effects to ﬁrst order [3.91]. However, the two-window method does not consider changes in background shape from one region of the specimen to another. In the three-window method, two pre-edge images are recorded, which enables the background to be ﬁtted according to an inverse power law at each pixel in the image. The extrapolated background image is then subtracted from the post-edge image to give a more accurate elemental map, in which changes in background shape are taken into account. Compositional maps derived by the two- or three-window methods provide useful quantitative results for easily visible core-edges and for well-behaved spectral backgrounds. On the other hand, the techniques tend to be unsuitable for measuring low elemental concentrations (< 10−2 ) within a single pixel or if there are signiﬁcant edge overlaps. More accurate quantitative analysis can be achieved by acquiring multichannel spectral data pixel by pixel. This type of acquisition is known as a ‘spectrum-image’ [3.92–3.96] in the case of a two-dimensional elemental map or as a ‘spectrum-line’ in the case of a one-dimensional proﬁle across a boundary [3.97–3.99]. This technique enables the operator to take advantage of the analytical procedures that have been described in the previous sections of this chapter. Spectrum-imaging can be performed in the ﬁxed-beam EFTEM by collecting successively a series of parallel images at different energy losses, or in the scanned-beam STEM by collecting series of parallel-recorded spectra over an array of pixels. A schematic comparison of the two complementary methods for acquiring these threedimensional data sets (position coordinates x, y, and energy loss E) is shown in Fig. 3.18. Each approach has advantages and disadvantages, which make the optimal strategy for microanalysis dependent on the specimen composition and on the required analytical sensitivity. Some important factors include, (i) the size of specimen area to be mapped; (ii) the spatial resolution required; (iii) the elements to be analyzed; and (iv) the concentration range(s) of interest. Reliability of elemental maps is always a concern. In general, the more spectral channels that are contained in the spectrum-image the greater the conﬁdence in the result, since the detailed spectrum from a given region of the specimen can be analyzed post facto and a core edge signal can thus be subtracted from the underlying background. In this case, it is possible to compare the edge shapes with standard reference spectra, to quantify the core edge signal, and to extract weak signals by generating difference-spectra. An important advantage of the technique is that when the analytical sensitivity is insufﬁ-

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Fig. 3.18 Spectrum imaging is achieved by generating a three-dimensional data cube in position coordinates x, y , and energy loss E . In STEM-EELS, the data cube is generated column by column and the electron probe is digitally scanned in a rectangular pixel array over the specimen while the spectrum is collected in parallel. In EFTEM the ﬁlter’s energy selecting slit deﬁnes a narrow band of energy losses and a two-dimensional array detector collects the image in parallel. The energy loss is varied by successively decreasing the accelerating voltage of the microscope and a series of energy-selected images is acquired.

cient to quantify concentrations of speciﬁc elements within the volume contained by a single pixel, we can sum the spectrum-image over neighboring pixels to increase signal/noise ratio. It is thus possible to segment an elemental map into regions of arbitrary shape and to obtain accurate quantitative information from those regions. 3.7.2

Example of EFTEM Spectrum Imaging

The following example illustrates how spectrum imaging in a post-column energy ﬁlter can be used to map quantitatively the distribution of sulfur-containing organic compounds in a biological specimen [3.100,3.101]. Figure 3.19 shows the backgroundsubtracted nitrogen K and sulfur L2,3 images from a thin plastic-embedded section of a mouse pancreatic islet of Langerhans, which contains alpha and beta cells that secrete respectively the polypeptide hormones, glucagon and insulin [3.102]. These proteins are packaged into spherical granules that appear morphologically similar in the two cell types. However, insulin has six sulfur containing amino acids, which give the protein a high sulfur/nitrogen ratio of 0.097, whereas glucagon has a much lower sulfur/nitrogen ratio. The difference in sulfur content makes it possible not only to distinguish between the alpha and beta cells but also to estimate the insulin content of the beta granules. A series of 80 energy ﬁltered images was recorded with energy losses between 100 and 500 eV, and with a 5-eV slit width. The images in this data set only exhibit slight differences in contrast, as evident in Figs. 3.19a and b. But the complete series of EFTEM images carries detailed spectral information that can be extracted from regions of interest. For example, the 80-channel spectrum from a single beta granule is shown in Fig. 3.19c. Quantitative elemental maps are obtained by using the same methods as for standard spectral analysis, that is, ﬁtting the pre-edge background, subtracting the extrapolated spectrum above the edge, and summing the intensity over an appropriate integration window. Comparison of the derived sulfur and nitrogen maps (Figs. 3.19d and e, respectively) using an integration window

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85

Fig. 3.19 EFTEM spectrum-imaging of organic compounds in a thin plastic-embedded section of pancreatic islets of Langerhans. A series of 80 energy-ﬁltered images was recorded in the energy range 80 to 480 eV by using an energy slit width of 5 eV. Since the contrast is similar for most of the images in the series, only two are shown: (a) at 160 eV (below S L2,3 edge) and (b) at 190 eV (above the S L2,3 edge). (c) Integrated spectrum computed from the granule indicated by arrow and circle in Fig. 3.19a, where the inverse power-law background extrapolation is shown at the S L2,3 and N K edges, together with the subtracted core edge intensities. (d) Net nitrogen signal, and (e) net sulfur signal for integration window of 50 eV. It is evident that one cell contains granules of high sulfur content (beta) and another cell contains granules of low sulfur content (alpha). (f) Histogram of sulfur-to-nitrogen atomic ratio in the beta cell granules shows a peak at 0.093, indicating that the granule cores are composed entirely of insulin, which has a sulfur-to-nitrogen atomic ratio of 0.097. Bar = 1 µm in Fig. 3.19d. Spectrum images were acquired by means of a post-column imaging ﬁlter (Gatan GIF100) interfaced to a FEI/Philips CM120 transmission electron microscope operating at 120 kV beam voltage. The read-out of the 1024×1024 pixel cooled CCD array detector was binned to 256×256 pixels to optimize the signal to noise ratio. Images were acquired and processed by means of the Gatan Digital Micrograph software.

of 50 eV conﬁrms that one cell contains sulfur-rich beta granules and another cell contains alpha granules with low sulfur content. The atomic ratio of sulfur/nitrogen at each pixel in the image can be determined by multiplying the ratio of core edge intensities by the reciprocal of the ratio of partial cross-sections according to Eq.

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3.4. A complete analysis of an image area can then be made, as illustrated in Fig. 3.19f, which shows a histogram of the sulfur to nitrogen ratio in 25 beta granules. The distribution of sulfur to nitrogen ratios has a mean value of 0.093 and standard deviation of 0.015, which agrees well with the nominal value of 0.097 for insulin and indicates that the beta cell granules are composed of pure insulin. This demonstrates that quantitative microanalytical information can be obtained from EFTEM spectrum images even though the atomic concentration of sulfur in the plastic-embedded sulfurrich granules is only ∼ 1%. 3.7.3

Example of STEM Spectrum Imaging

The following example illustrates how STEM spectrum imaging can be used to map quantitatively the distribution of titanium and barium across grain boundaries in an ion-polished cross-section of barium strontium titanate ﬁlm (Ba1−x Srx TiO3 ) with 53.4% Ti, deposited on a Pt/SiO2 /Si substrate by metallorganic chemical-vapor deposition [3.103,3.104]. Spectra were acquired at each pixel in an energy-loss range from 400 to 800 eV, which includes the Ti L2,3 -, O K-, and Ba M4,5 -edges. The spectra were processed to remove the background, and the Ti L2,3 and Ba M4,5 core edge intensities were determined by integrating over 50-eV energy windows. Representative Ti L2,3 and Ba M4,5 core loss images are shown in Figs. 3.20a and b, respectively. The Ti/Ba ratio map was also derived as shown in Fig. 3.20c. Atomic ratios were calculated according to Eq. 3.4, using spectrum-images recorded from a pure barium titanate standard to determine experimentally the ratio of scattering cross-sections. The elemental intensity maps suggest that the grain boundaries are Ba deﬁcient, while only some of the boundaries have a higher Ti signal compared to that in the grain interiors. It is evident that the elemental signals are affected by both diffraction contrast and thickness variations across the analyzed area, whereas these artifacts are largely eliminated in the ratio image. The measurements demonstrate that the grain boundaries have a signiﬁcantly higher Ti/Ba ratio (1.57) than that in the grain interiors (1.35). This result suggests that excess Ti is accommodated through an increase in Ba vacancies near and at the grain boundaries, which is consistent with reported observations on bulk material, and with the formation of Ti-rich amorphouslike regions at some of the grain boundaries [3.104]. 3.7.4

Comparison of Quantitative Elemental Mapping in EFTEM and STEM

A comparison of sensitivities for spectrum imaging in the EFTEM and STEM for a pixel size d is obtained by expressing the relative signal/noise ratios in terms of the beam current I incident on the specimen, the total acquisition time τ , and the required image size D [3.105,3.106]. In the STEM a spectrum containing as many as 1024 channels can be obtained in parallel at each pixel, whereas in the EFTEM an image must be recorded sequentially at each of q energy channels. If F and S refer to the ﬁxed-beam and scanned-beam instruments, respectively, then,

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87

Fig. 3.20 STEM spectrum imaging of a barium strontium titanate ﬁlm. (a) Ti L2,3 map and (b) Ba M4,5 map obtained by subtracting the extrapolated pre-edge background and integrating within 50-eV energy windows above the respective core edges at each pixel. (c) Atomic ratio of Ti/Ba obtained by dividing image in Fig. 3.20a by image in Fig. 3.20b. Square box in Fig. 3.20b indicates region in center of a grain from which the spectrum (d) was recorded. Line across grain boundary is indicated in Fig. 3.20c. (e) Atomic ratio of Ti/Ba along line perpendicular to grain boundary. Bar = 20 nm. Data were recorded using a VG Microscopes HB501 STEM equipped with a cold ﬁeld-emission gun operating at a beam voltage of 100 kV and a Gatan parallel-detection EELS (DigiPEELS). The specimen was cooled to liquid-nitrogen temperature to prevent contamination and the probe diameter was set to 1.5 nm. Spectrum images containing 64×64 pixels and 1024 spectral channels per pixel were acquired with the Gatan Digital Micrograph program. This software automatically corrected for specimen drift by cross-correlating measured shifts in the annular dark-ﬁeld image recorded at the end of each scan line.

(S/N )F = (S/N )S

DS DF

dF dS

IF IS

τF τS

1 qF

(3.42)

where DF is the EFTEM spot size and DS is the required STEM image size. The much higher total beam current available in the EFTEM and the use of large cooled CCD detector arrays containing 1024 × 1024 pixels clearly give this instrument the advantage for imaging large areas of the specimen. For example, the beam current incident on the specimen in the EFTEM can be as high as 100 nA, compared with only ∼ 1 nA in the ﬁeld-emission STEM for a 1 nm diameter probe. When smaller

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areas of the specimen are to be mapped, STEM offers the best approach. Equation 3.42 indicates that for DS = DF , dS = dF , τS = τF , IF /IS = 100, and qF = 100, the two approaches have equivalent sensitivity. On the other hand, if an element can be mapped by using the three-window method (two energy windows below the edge and one above the edge), then the EFTEM is predicted to have the greater sensitivity by about a factor of 6. However, the choice between EFTEM and STEM may be complicated by other considerations. Specimen instabilities can be more easily compensated in the STEM than in the EFTEM, which allows longer total acquisition times, and hence provides even better relative sensitivity for STEM spectrum-imaging than predicted by Eq. 3.42. Moreover, for a given signal, the required total electron dose is always lower in STEM than in EFTEM because the spectrum is acquired in parallel, which gives STEM the advantage for mapping beam-sensitive specimens. 3.8

SUMMARY

In this chapter, we have described how electron energy loss spectroscopy provides quantitative high-resolution microanalytical information about thin specimens. The accuracy of EELS analysis is limited by uncertainties in ionization cross-sections and by errors in the background and signal estimation; accuracy can also depend on specimen thickness and diffraction effects. Single atom sensitivity is achievable for certain elements when EELS analysis is performed with a nanometer-sized focused probe. It is also feasible to quantify atomic concentrations as low as 10−4 within an analytical volume of the specimen if the elements of interest have favorable core edge shapes. The technique of spectrum imaging greatly extends the capabilities of EELS analysis and provides quantitative two-dimensional elemental distributions. Spectrum imaging in the STEM gives the highest spatial resolution and sensitivity but the image size is limited by the need to read out the spectrum at each pixel. The complementary technique of EFTEM provides larger numbers of image pixels but the spectral information must be acquired serially, which requires higher electron doses.

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Meeting of Electron Microsc. Soc. Am., ed. G.W. Bailey, Claitor’s Publishing, Louisiana, 114–115 (1980). 3.75. A. J. Bourdillon and W. M. Stobbs, “Elastic scattering in EELS—fundamental correction to quantiﬁcation.” Ultramicroscopy 17, 147–149 (1985). 3.76. J. C. H. Spence and J. Taftø, “ALCHEMI: a new technique for locating atoms in small crystals.” J. Microsc. 130, 147–154 (1983). 3.77. J. Taftø and O. L. Krivanek, “The combined effect of channelling and blocking in electron energy-loss spectroscopy.” In: 39th Proc. Ann. Meeting of EMSA, ed. G. W. Bailey, Claitor’s Publishing, Baton Rouge, Louisiana, 190–191 (1981). 3.78. R. F. Egerton, “Energy-loss spectrometry with a large collection angle.” Ultramicroscopy 7, 207–210 (1981). 3.79. P. Rez, “Detection limits and error analysis in energy-loss spectrometry”, In: Microbeam Analysis-1983, Ed, R.Gooley, San Francisco Press, San Francisco, 153–155 (1983). 3.80. P. A. Crozier, J. N. Chapman, A. J. Craven and J. M. Titchmarsh, “Some factors affecting the accuracy of EELS in determining elemental concentration in thin ﬁlms.” In: Analytical Electron Microscopy-1984, eds. D. B. Williams and D. C. Joy, San Francisco Press, San Francisco, 79–82 (1984). 3.81. M. K. H. Natusch, C. J. Humphreys, N. Menon, and O. L. Krivanek, “Experimental and theoretical study of the detection limits in electron energy-loss spectroscopy.” Micron 30, 173–183 (1999). 3.82. L. E. Thomas, “Light-element analysis with electrons and x-rays in a highresolution STEM.” Ultramicroscopy 18, 173–184 (1985). 3.83. D. S. Bright and D. E. Newbury, “Concentration histogram imaging–a scatter diagram technique for viewing two or three related images.” Anal. Chem. 63, 243–250 (1991). 3.84. W. Grogger, F. Hofer, and G. Kothleitner, “Quantitative chemical phase analysis of EF-TEM elemental maps using scatter diagrams.” Micron 29, 43–51 (1998). 3.85. P. J. Thomas, “Novel approaches in spectrum-image analysis.” Microsc. Microanal. 7 (suppl. 2), 1138–1139 (2001). 3.86. H. Rose and D. Krahl, “Electron optics of imaging energy ﬁlters.” In: EnergyFiltering Transmission Electron Microscopy, ed. L. Reimer, Springer Series in Optical Physics, Springer-Verlag, Berlin, Vol. 71, Chapter 2, 43–149 (1995). 3.87. F. P. Ottensmeyer, “Electron spectroscopic imaging: parallel energy ﬁltering and microanalysis in the ﬁxed–beam electron microscope.” J. Ultrastruct. Res. 88, 121–134 (1984).

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3.88. F. Hofer, W. Grogger, G. Kothleitner and P. Warbichler, “Quantitative analysis of EFTEM elemental distribution images.” Ultramicroscopy 67, 83–103 (1997). 3.89. F. Hofer and P. Warbichler, “Improved imaging of secondary phases in solids by energy-ﬁltering TEMs” Ultramicroscopy 63, 21–25 (1996). 3.90. F. Hofer, P. Warbichler, and W. Grogger, “Imaging of nanometer–sized precipitates in solids by electron spectroscopic imaging.” Ultramicroscopy 59, 15–31 (1995). 3.91. P. A. Crozier, “Quantitative elemental mapping of materials by energy-ﬁltering.” Ultramicroscopy 58, 157–174 (1995). 3.92. C. Jeanguillaume and C. Colliex, “Spectrum–imaging: the next step in EELS digital acquisition and processing.” Ultramicroscopy 28, 252–257 (1989). 3.93. J. A. Hunt and D. B. Williams, “Electron energy-loss spectrum-imaging.” Ultramicroscopy 38, 47–73 (1991). 3.94. R. D. Leapman and J. A. Hunt, “Compositional imaging with electron energy loss spectroscopy.” J. Microsc. Soc. Am. 1, 93–108 (1995). 3.95. M. Tenc´e, M. Quartuccio and C. Colliex, “PEELS compositional proﬁling and mapping at nanometer spatial resolution.” Ultramicroscopy 58, 42–54 (1995). 3.96. G. A. Botton and G. L’Esperance, “Development, quantitative performance and application of a parallel electron energy-loss spectrum imaging system.” J. Microsc. 173, 9–25 (1994). 3.97. J. M. Plitzko and J. Mayer, “Quantitative thin ﬁlm analysis by energy ﬁltering transmission electron microscopy.” Ultramicroscopy 78, 207–219 (1999). 3.98. J. Mayer, U. Eigenthaler, J. M. Plitzko, and F. Dettenwanger, “Quantitative analysis of electron spectroscopic imaging series” Micron 28, 361–370 (1997). 3.99. J. Bentley, J. E. Wittig, J. F. Al-Sharab, and N.D. Evans. “Elemental mapping of Co-based magnetic recording media: EFTEM and STEM spectrum imaging.” Microsc. Microanal. 7 (suppl 2), 1140–1141 (2001). 3.100. O. L. Krivanek, A. J. Gubbens, N. Dellby, and C.E. Meyer, “Design and ﬁrst applications of a postcolumn imaging ﬁlter.” Microsc. Microanal. Microstruct. 3, 187–199 (1992). 3.101. A. J. Gubbens, H. A. Brink, M. K. Kundmann, S. L. Friedman, and O. L. Krivanek, “Application of a post-column imaging ﬁlter in biology and materials science.” Ultramicroscopy 51, 146–159 (1993). 3.102. R. D. Leapman, C. M. Brooks, N. W. Rizzo, and T. L. Talbot. “Quantitative analysis of biological specimens by spectrum-imaging in the energy ﬁltering transmission electron microscope.” Microsc. Microanal. 6 (suppl 2), 160–161 (2000).

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3.103. I. Levin, R. D. Leapman, D. L. Kaiser, S. Bilodeau, R. Carl, and P. C. van Buskirk, “Accommodation of excess Ti in a (Ba,Sr)TiO3 thin ﬁlm with 53.4% Ti deposited on Pt/SiO2/Si by MOCVD.” Appl. Phys. Lett. 75, 1299–1301 (1999). 3.104. I. Levin, R.D. Leapman, and D.L. Kaiser, “Microstructure and chemistry of nonstoichiometric (Ba,Sr)TiO3 thin ﬁlms deposited by metalorganic chemical vapor deposition.” J. Mater. Res. 15, 1433–1436 (2000). 3.105. J. A. Hunt, G. Kothleitner and R. Harmon, “Comparison of STEM EELS spectrum imaging vs. EFTEM spectrum imaging.” Microsc. Microanal. 5 (suppl 2), 616–617 (1999). 3.106. R. D. Leapman and S. B. Andrews. “Strategies for optimizing detection limits in elemental maping of biological specimens by electron energy-loss spectrumimaging.” Proceedings 2nd Conference of the International Union of Microbeam Analysis Societies, Kailua-Kona, HI, (British) Inst. Phys. Conf. Ser. No. 165, 185– 186, (2000).

4

Energy Loss Fine Structure Peter Rez Department of Physics Arizona State University Tempe, Arizona 85287

Abstract One of the advantages of energy loss as an analytical technique is that the ﬁne structure can give information on bonding, local band structure and coordination, to supplement the elemental compositions from quantitaive analysis using inner shells. Although some progress can be made by resorting to ﬁngerprinting, an understanding of the theory enables ﬁne structure to be applied to a wider range of materials. The use of plasmons to estimate electron density, as well as the analysis of inner shell ﬁne structure is reviewed. In particular, the interpretation of near edge structure in terms of bonding according to band theory and the molecular orbital picture is discussed. To conclude there is a description of the use of white lines as a probe of charge state and d band occupancy. 4.1 INTRODUCTION In Chapter 1, the general form of the energy loss spectrum has been described, and in Chapter 3 the use of inner-shell edges for microanalysis has been outlined. For microanalysis, acceptable results can be obtained with relatively poor resolution, such as 5–10 eV. As the resolution of spectrometers improved to better than 1 eV, extra structure in addition to the broad low loss plasmons and the general atomic-like inner-shell edge shapes could be resolved. The promise and challenge of energy loss is whether this new information could be used to learn more about the properties of materials. It would be particularly useful if information on atomic charge, bonding or coordination could be extracted from a simple analysis of ﬁne structure. Energy loss would then provide unique capabilities in the study of materials on a nanometer scale. Many of the features that appear in energy loss are also present in X-ray spectra, and synchrotron radiation workers have made most of the advances in the theory of near edge structure. Early synchrotron studies concentrated on edges above ∼ 3 kV, so 97

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not all the methods that were developed are directly applicable for the analysis of energy loss edges between 50 eV and 2 kV. In this Chapter, we shall review the ﬁne structure that can be seen in energy loss spectra, and describe brieﬂy those theoretical ideas that can be used to understand the observed features. We shall show how the interpretations help in extracting new information, and where appropriate, we shall relate the energy loss analysis to X-ray synchrotron work. 4.2

THE LOW LOSS REGION

The low loss region is the intense part of the spectrum from 0 to ∼ 40 or 50 eV and is dominated by collective excitations known as plasmons that are discussed below. Optically it ranges from the visible to the ultraviolet (UV). The major contribution to the low loss spectrum comes from the excitation of electrons in the valence bands to empty states in the conduction bands. The probability of scattering with energy loss E can be derived from Fermi’s Golden Rule and is given by 2 8π d2 I ∗ 3 = 2 4 φc (r) exp(iq · r)φv (r)d r ρc (E) dEdΩ a0 q

(4.1)

where φ∗c (r) is the conduction band (ﬁnal state) wave function, φv (r) is the valence band (initial state) wave function, ρc (E) is the density of conduction band states and q is the momentum transfer. Unlike inner-shell spectroscopy the valence band covers a range of energies, and so the cross-section given above has to be convoluted by the density of initial valence band sates and the ﬁnal spectrum will be proportional to 2 |M (E)| ρc (E + E )ρv (E )dE (4.2) where ρv (E ) is the valence band density of states and M (E) the matrix element which is the term in straight brackets in Eq. 4.1. In the general case, the interpretation of this joint density of states multiplied by a square of the matrix element is not easy, especially as accurate wave functions for the conduction and valence bands are hard to calculate. There are some cases where peaks in the spectrum can be related to a peak in the joint density of states. A particularly simple analysis can be applied in semiconductors near the band gap, if one assumes that both the conduction and valence bands are parabolic [4.1]. The spectrum should then increase as E 1/2 where E is the energy above the band gap and there should be nothing between the zero loss and the band edge. In practice, it is not easy to observe this behavior as the interband scattering is swamped by the tail of the source function. Low loss spectra don’t usually have sharp peaks. An exception is alkali halides where many discrete excitation due to color centers can be observed. The most visible feature of the low loss spectrum are the peaks due to plasmons or collective excitations. The original paper [4.2] derived the results in terms of many body theory. A simpliﬁed explanation in terms of the oscillations of a free

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electron gas is given in standard solid state physics textbooks such as Kittel [4.3]. The equation of motion for a free electron is m¨ x = eE

(4.3)

and for a displacement x of N electrons/unit volume the ﬁeld by Gauss’s theorem is −N ex/ε0 . Equation 4.4 describes a simple harmonic motion of frequency ωp =

N e2 mε0

1/2 (4.4)

hωp the plasmon energy. Plasmons range where ωp is the plasmon frequency and ¯ in energy from a few eV to ∼ 30 or 40 eV. The width of the plasmon peak is a few eV and is a reﬂection of the plasmon lifetime. Another factor to consider is the variation of the plasmon energy as a function of momentum transfer called the plasmon dispersion. This manifests itself as an asymmetric shape with an extended tail on the high energy side, especially noticeable for large collection angles. Beyond a cut-off momentum transfer qc = mEp /¯ hqF , where qF is the Fermi wavevector, the plasmon cannot exist and decays into single electron excitations. A more sophisticated analysis is based on the theory of the dielectric response [4.4] ε(q, ω) of the solid, and gives 2 −1 dq d2 I = Im (4.5) 2 dEdq πa0 mv n ε(q, ω) q where v is the fast electron velocity and n is the number of atoms per unit volume. The plasmon energies are given by the poles (zeroes) of the complex frequency dependent (or energy dependent) dielectric function. Figure 4.1 shows the variation of 1/ε(ω, q) as a function of wave vector and energy loss for aluminium. The solid line shows the plasmon at 15 eV and its dispersion as a function of energy.

Fig. 4.1 Im[−1/ε(q, ω)] for aluminium showing the volume plasmon and its dispersion(change in energy as a function of momentum transfer q )(courtesy of P. E. Batson).

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An analysis of the electrodynamics of the response of the electron gas gives √ surfaces plasmons at energy ¯ hω/ 2 [4.5] and special modes at energy ¯hω(1 + ((L + 1)/L)1/2 for small spherical particles. Surface plasmons are very apparent in very thin specimens or in reﬂection energy loss spectroscopy. A comparison between energy loss spectra from MgO in transmission and reﬂection is given in Fig. 4.2 and clearly shows the enhancement of the 15-eV surface plasmon in the reﬂection spectrum.

Fig. 4.2 (a) Transmission EELS spectrum from MgO (EELS Atlas) showing volume plasmon at 20 eV. (b) Reﬂection EELS spectrum from MgO showing enhancement of surface plasmon at 15 eV (courtesy M. Gadarjiska and P. Crozier).

The main application of plasmons is determining the number of electrons/unit volume using Eq. 4.5. In early work, Williams [4.6] applied analysis of plasmons to aluminum alloys. More recent applications are separation of graphitic regions from a diamond matrix by Blake et al. [4.7]. Graphite has a plasmon at 11.0 eV due to the π electrons and 23 eV due to the joint (σ + π) electron band. Diamond, on the other hand, only has one plasmon peak at 24 eV. It is often useful to compare the energy loss results with optical and UV spectroscopy. As mentioned above the energy loss spectrum is proportional to Im[−1/ε(q, ω)] The real and imaginary parts can be extracted using the Kramers–Kronig relation as

∞ −1 2 E dE −1 Re = 1 − P Im ε(q, ω) π ε(q, ω) E 2 − E 2

(4.6)

0

where P denotes the principal part of the integral. In practice, it is easiest to use the Fourier transform method proposed by Johnson [4.8], which can be derived from the non-negative form of the dielectric function. If care has been taken to remove multiple scattering effects (see Chapter 3) the energy loss data can be more reliable

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than the optical data, as optical reﬂectance is easily affected by surface layers and contamination. In the analysis of the low loss region, it is sometimes convenient to use simple models for the dielectric constant. The Drude model is appropriate for metals and gives a lorentzian proﬁle of width 1/τ for the plasmon where τ is the plasmon lifetime. The dielectric function is ε(ω) = 1 −

ωp2 1 ω2 1 − 1 iωτ

(4.7)

where ωp is the plasmon frequency. Insulators are better described by a Lorentz model that considers a gas of bound electrons with different eigenfrequencies ωi . The dielectric function becomes ε(ω) = 1 +

ni e2 i

mε0

1

ωi2 − ω 2 +

iω τ

(4.8)

A particularly interesting effect in the low loss region is the changes in spectra as a small probe is moved across thin layers. The effects can be understood by a classical theory in which Maxwell’s equations are solved for the appropriate multilayer medium and the appropriate energy loss function derived [4.9]. This theory shows the surprising result that a beam running close to a specimen in the vacuum can lose energy and has a distinctive energy loss spectrum. 4.3 INNER-SHELLS As discussed in Chapter 1, the characteristic edges arise from the excitation of innershell electrons. The basic edge shapes are determined by atomic physics and so are independent of the environment or bonding of the atom. By far, the most important factor inﬂuencing edge shapes is the centrifugal barrier due to the angular momentum of the excited electron [4.10]. This potential barrier suppresses the ﬁnal state near threshold and results in delayed edge maxima. Higher angular momentum quantum numbers result in a greater separation of edge onset and maximum intensity. In the second row of the periodic table, elements Na to F, the maximum in the L23 -edge is 10–15 eV above threshhold. For the second-row transition elements, the M45 transitions to f states give rise to a maximum in the edge 60–80 eV above threshold. Superimposed on the basic atomic edge shape are oscillations often called ﬁne structure which are strongly dependent on either bonding, coordination or nearest neighbor distances. The ﬁne structure can be divided into strong oscillations up to 20–30 eV above threshold, usually called the near edge structure (NES) and weaker oscillations, beyond ∼ 30 eV, called the extended ﬁne structure, often abbreviated as EXAFS (Extended X-ray Absorption Fine Structure) or in energy loss EXELFS (Extended X-ray Edge Energy Loss Fine Structure). This division is shown schemat-

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ically in Fig. 4.3 for the oxygen K edge in NiO. At the threshold itself there are sometimes sharp peaks of high intensity frequently called white lines [4.11], as they originally appeared as lines on a photographic plate. White lines can be treated as an atomic effect since there is no requirement for a condensed phase for them to be detected. In the atomic picture they arise from transitions to unﬁlled bound states. Examples are the L23 transitions to empty 3d states in the transition metals or M45 transitions to empty 4f states in rare earth elements. A solid-state physicist would say that the Fermi level is in the 3d band for transition metals, or the f band for rare earths, and that these bands are narrow and maintain most of the features of the free atom wave function. The extended ﬁre structure, though harder to detect, is the easiest to understand and will be covered in the next section.

Fig. 4.3 Oxygen K -edge from NiO showing division between electron loss near-edge structure (ELNES) and extended ﬁne structure (EXELFS) (courtesy X. Weng).

4.4 EXTENDED FINE STRUCTURE Extended ﬁne structure analysis has proved to be very popular in X-ray absorption, partly because of the simplicity of the basic theory. The intensity in an edge is proportional to the differential scattering cross-section which is given by Fermi’s Golden Rule as [4.12] 2 4γ 2 d2 I = 2 4 f |exp(iq · r)| i ρ(E) (4.9) dΩdE a0 q where γ is the relativistic correction 1/ (1 − v 2 /c2 ), v is the velocity of the fast electron, a0 is the Bohr radius, q is the momentum transfer, ρ(E) the density of ﬁnal states, and |i and f | represent the initial and ﬁnal state wave functions respectively. In the atomic picture the initial state is an atomic wave function localized on a particular atom. The ﬁnal state is a solution of the Schr¨odinger equation at some energy ε in the continuum of states. Away from the atom it is like a spherical 1/2 outgoing wave, energy ε and wavevector k = (2mε/¯h) and can be mathematically

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represented as 1/|kr| exp(ik · r+δ) where δ is a phase shift. This outgoing spherical wave is reﬂected from neighboring atoms (type j), a process characterized by another phase shift δj , and the reﬂected wave interferes with the original outgoing wave. The strength of the reﬂection is given by an atomic scattering factor fj multiplied by a Debye–Waller term exp(−2σj2 k 2 ) to take account of atomic vibrations. The interference means that there is an oscillation of intensity above the edge given by [4.13,4.14]

χ(k) =

Nj fj (k) −2rj exp exp −2σj2 k 2 sin(2krj + δ0 + δj ) 2 rj k λ j

(4.10)

There will therefore be a different period of oscillation corresponding to different coordination shells at different distances. Contributions from shells a few atomic distances away are strongly attenuated in practice due to the exp(−2rj /λ) term and EXAFS is therefore mainly sensitive to 1st, 2nd and 3rd nearest neighbors. Although in principle, nondipole effects could distort energy loss extended structure in practice they have proven to be undetectable,and the analysis is identical to the X-ray case. More detailed explanations of the EXAFS procedures can be found in the books by Teo and Joy [4.15] and Bianconni [4.16]. The ﬁrst stage of the analysis is to strip the background from the edge. A low order polynomial or spline is then ﬁt to the edge in such a way that the extended ﬁne structure modulations oscillate around the ﬁtted line. The data is interpolated onto a scale of wave vector rather than energy. A region of the oscillations from about 20 eV above threshold to a few −1 −1 hundred eV (k = 2.5 ˚A to k = 12 ˚A ) is selected with a window that rounds off the sharp edges with a gaussian shape. The result is then Fourier transformed and the square of the modulus displayed (see Fig. 4.4). The rounding of the selection window is necessary to prevent spurious oscillations arising in the Fourier transformation process.

Fig. 4.4 Extended ﬁne structure from Si K -edge showing (a) background subtracted core loss with ﬁne structure, (b) χ(k) extracted after cubic spline ﬁt for core loss and (c) Fourier transform magnitude returned by transforming χ(k) × k 2 from 2.8 to 10.6 ˚A−1 .

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ENERGY LOSS FINE STRUCTURE

The ﬁnal result shows peaks corresponding to most probable interatomic separation and ﬁrst and second neighbor peaks can usually be distinguished. To translate the peak position into absolute distances a knowledge of the phase shifts is needed. If the phase shifts can be written as a power series as δ + δ k + δ k, the origin of the Fourier transformed plot is at δ0 + δj . Phase shift data can either be obtained by measurements on standards of similar materials [4.17,4.18] or from ﬁrst principles calculations [4.19]. Accuracies as high as 0.1 ˚A in position have been achieved. Debye–Waller factors and coordination data can also be derived. Coordination numbers from EXAFS should be reviewed with greater caution than interatomic separation results, as they depend strongly on accurate background subtraction and extraction of the ﬁne structure modulation. Another approach to EXAFS analysis is to use Eq. 4.10 directly, treating the interatomic distances and other unknown quantities as parameters to be ﬁttted from the experimental data [4.20]. The theory can also be extended by incorporating such effects as the curvature of the outgoing spherical wave and also some multiple scattering contributions [4.21,4.22]. It should then be classiﬁed with the other mufﬁn tin based theories for near-edge structure. EXAFS have proved to be most useful for investigation of the interatomic distances around a particular atom, especially when that atom is conﬁned to very speciﬁc sites in a crystal structure. In amorphous material they give, in effect, a local radial distribution function for a particular site. In transmission energy loss spectroscopy, the technique has not been as successful as in X-ray synchrotron studies. This is partly due to the difﬁculty in collecting data from what is a very weak effect. Although the basic counting rates are about the same, the main interest in electron microscopy is to gather data from small areas. Counting times are limited by drift and contamination in electron microscopy and runs of many hours are not feasible. A more serious difﬁculty is the restriction in the range of data that can be analyzed. Traditionally X-ray absorption studies have concentrated on edges with thresholds at 8 kV and above, as suitable monochromators for lower energies have not been available. There is usually ∼ 1 kV or more of extended ﬁne structure to analyze. In electron energy loss the edges are generally between 100 eV and 2 kV. In any real material this rarely leaves more than about 200–300 eV between edges and the range of data is restricted. Recent X-ray work in this region, mainly aimed at surface studies, also has this problem. It is not surprising that most of the reported EXELFS have been on Al K- and Si K-edges, both of which are above 1 kV. There have been some studies of early transition metal K-edges in aluminide compounds using a microscope operating at 400 kV [4.23]. A wider range of edges has been studied in reﬂection electron loss from bulk surfaces [4.24] but the interpretation of some of the lower energy edges is still the subject of controversy [4.25]. 4.5 NEAR-EDGE STRUCTURE In contrast to the interpretation of extended ﬁne structure, the theory for near-edge structure is not as well deﬁned and is still the subject of some controversy. Near-edge

NEAR-EDGE STRUCTURE

105

structure, however, is still of great interest in practical applications of energy loss spectroscopy in the electron microscope. This is partly due to the fact that it is relatively easy to observe, as the oscillations are large, and also because it promises to give information on ionic charge, bonding or coordination that cannot be obtained in other ways. In the absence of a complete theoretical framework, investigators have been forced to search for identical features (ﬁngerprinting) [4.26] that identify an element in a particular environment. This process feeds back into the theory and so helps understanding of near-edge structures. Fundamentally, all near-edge structure can be derived by considering Eq. (4.1). The initial state is the atomic inner-shell wave function and the ﬁnal state is the appropriate wave function for an empty conduction band state, as given by a band structure calculation. Over a short energy range the matrix elements do not change much and most of the structure comes from the changes in the density of states. For metals, with the exception of transition metals, the density of states is like that for free electrons and the near-edge structure is quite weak. Insulators and semiconductors show strong near-edge structure. It is also possible to probe states in the band gap, though these might be excitonic or many electron effects caused by the inner-shell excitation. The energy difference between an inner-shell and the conduction band onset changes when the element forms different compounds. The resulting change in edge position is called a chemical shift and usually is ∼ 2–3 eV. The largest value is 7 eV going from pure Si to SiO2 [4.27]. Caution should be exercised in the interpretation of chemical shifts as the inner-shell energy changes, as well as the conduction band position (see Fig. 4.5). This means that energy loss chemical shifts are not the same as those observed in XPS though, if many electron effects are excluded, it should be possible to put together results from different techniques to sort out core level shifts from conduction band shifts and valence band shifts.

Fig. 4.5 Schematic diagram showing contributions to the chemical shift.

As can been seen, the interpretation of near-edge structure is closely related to band theory. Band theory methods usually assume an inﬁnite perfect crystal lattice and are therefore unsuited to problems involving isolated atoms in different environments. Localized real space methods such as the XANES (X-ray Absorption Near Edge Structure) calculation method of Durham et al. [4.28] have proved popular, especially among X-ray absorption spectroscopists who are often interested in the near edge structure of the adsorbates on surfaces. These real space methods can be

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regarded as a generalization of EXAFS theory, but they can also be related to band theory methods. In our discussion so far, it has been implicitly assumed that the ejection of an electron from an inner-shell has no effect on the other electrons in the atom and that all electrons experience the same effective potential. This approximation is known as the one electron approximation; and if it were true, the near edge structure would directly probe the unoccupied states. At some stage the approximation will break down and the effects of the excitation process will have to be considered. One view is that the core hole creates a potential well in which there are bound states in the band gap called core excitons [4.29], analagous to the electron-hole excitons in semiconductors. Other theoretical physicists invoke more complicated many body effects [4.30,4.31]. Although the spectroscopist should be aware of core hole effects it is equally dangerous to assume that because they are signiﬁcant in one material, or class of materials, then all inner-shell spectra are dominated by them. If instrumental resolution were not a problem, the ﬁnest features would be given by the lifetime broadening. This can be estimated by calculating the transition rate due to deexcitation by Auger of X-ray emission and converting this to an energy broadening by the uncertainty principle. Estimates have been given by Bambynek et al. [4.32] and Fuggle [4.33] and range from ∼ 0.1 eV for edges at 100 eV, to 0.5 eV for edges at ∼ 10 kV. At one time there was much controversy on whether L23 -edges of free electron metals were rounded or cusp shaped due to many body effects. Later work showed that it would be impossible to detect this particular many electron effect due to the lifetime broadening [4.34]. 4.5.1 XANES The XANES theory mentioned above can be treated as a direct extension of the theory for EXAFS. Instead of being limited to a single reﬂection of the ejected electron by a neighboring atom, all possible multiple scattering paths are considered. The atoms are represented by spherical mufﬁn tins in a ﬂat potential well. Scattering from the mufﬁn tins is characterized by the scattering phase shifts which are determined by ﬁtting the spherical wave solutions to the calculated wave function at a given energy above the zero of the atom mufﬁn tins. This wave function is found by solving the Schr¨odinger equation for the atom, with appropriate modiﬁcations due to the nearest neighbors and the charge on the ion cores. The atomic potential is constructed from Hartree–Slater atomic wave functions [4.35]. Sometimes to take account of core hole effects a different potential is used for the excited atom. In many calculations it has been replaced with the next heavier element (Z +1 approximation) [4.36,4.37], the argument being that the outer electrons feel an equivalent nuclear charge increased by one electron. Other approximations include transition states, in which an extra electron is put in an unﬁlled orbital, or a combination of the Z + 1 approximation and the transition state description [4.38]. It is important to note that the material is represented as a superposition of atomic charges in nearly all XANES or similar cluster calculations, and no attempt is made to calculate the ground-state

NEAR-EDGE STRUCTURE

107

charge density self-consistently. Effects due to bonding and charge resistribution are therefore not described in the theory. In the Durham Pendry Hodges [4.28,4.39] method the scattering inside a coordination shell is calculated ﬁrst; and then the scattering between shells is considered. Scattering within a shell is called intrashell scattering, while scattering between shells is called intershell scattering. This separation can be very useful in practice, especially when dealing with different coordination numbers. A further development of the Durham Pendry Hodges codes includes a more sophisticated treatment of symmetry [4.40] that results in considerable savings in computer time. A multiple scattering method developed by Fujikawa [4.41] does not divide the neighboring atoms among the different shells, but is otherwise very similar. The f effective (FEFF) method developed by Rehr and colleagues [4.22,4.42] is a scheme for calculating extended ﬁne structure that selects scattering paths according to a maximum path length, the number of scattering events and an estimated amplitude. The XANES method has been applied extensively to deep inner-shell edges as observed in X-ray absorption spectroscopy. There has been less work on calculations for edges in the energy loss range. Lindner et al. [4.43] and Weng and Rez [4.44] have reported results on MgO and other oxides with the NaCl structure (see Fig 4.6). Whereas Lindner et al. used the Z + 1 approximation, Weng and Rez performed calculations with both neutral atoms and charged atoms. The neutral atom calculations agreed best with experiment.

Fig. 4.6 Experimetntal oxygen K -edges from MgO, CaO, and SrO (solid line) compared with XANES calculations (dotted line) (courtesy X. Weng and T. Manoubi).

XANES calculations show a good match to experimental measurements when the near edge structure is sensitive to coordination and interatomic dstances. In fact, the single scattering EXAFS theory can be used in an approximate form right up to the threshold, though not all the oscillations will be predicted. The work of Sette et al. [4.45], which shows that the position of the ﬁrst large peak (or shape resonance) can be related to interatomic bond length for diatomic molecules, is in agreement with this result. The effects of coordination can be considered simply in terms of the extra intrashell scattering. Taftø and Zhu [4.46] showed that Al, Mg, and Si K-edges appear different when coordination changes from octahedral to tetrahedral. Brydson and colleagues [4.47] also showed effects on the Al L-edge and Be K-edge in chrysoberyl when

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compared with Rhodosite (see Chapter 7). For higher coordination there is more possibility of intrashell scattering, and therefore extra peaks are more likely to be detected. This is in agreement with the above observations. The oxygen K-edge in oxides can also be understood in simple terms as the strong oxygen scattering dominates and the weak scattering from the metal cations can be neglected [4.48]. Two of the peaks (B and C in Fig. 4.6) are EXAFS single scattering features. The ﬁrst large peak (A in Fig. 4.6) comes from an intrashell sactttering in the ﬁrst oxygen coordination shell. It therefore seems likely that the relative height of the ﬁrst peak compared to the other features can be used as an approximate measure of coordination. XANES calculations have been less successful when applied to ceramics or semiconductors with open structures that have low coordination. Although in principle the calculation should converge the results have been disappointing in view of the calculation times needed. This is not surprising as covalent materials with directional bonds are poorly described by an array of spherical mufﬁn tin atoms. Vvedensky (unpublished), Kitamura [4.49] and Weng et al. [4.50] report results for diamond and graphite. The main features are reproduced in the correct positions but the relative heights are not in good agreement with experiment. Calculations for Si K have been more successful. A particularly interesting application is the decomposition of the near edge structure in terms of the order of scattering. This has then been applied to understanding the differences between the Si K-edge in crystalline and amorphous silicon [4.51]. It might be possible to use such an analysis to get information on distribution of nearest neighbor bond angles. It would seem that XANES methods are best suited to metals and oxides though without self-consistency it might be difﬁcult to interpret features at a scale of < 1–2-eV energy resolution. 4.5.2

LCAO or Molecular Orbitals

So far, it has been convenient to represent the excited electron state in terms of waves. In the near edge region a picture in terms of bonds between atoms is often more useful. A particularly simple method to derive electron bands is to take linear combinations of atomic orbitals. A symmetrized combination is appropriate for isolated molecules. In an inﬁnite solid, the wave function is best represented by a Bloch function

1 Ψjq (r) = √ exp(iq · r) Ψj (r − R) N R

(4.11)

where Ψj (r − R) are the atomic orbitals at site R. Alternatively, wave functions can be expressed as a sum of atomic wave functions for a cluster. This is then substituted into the Schr¨odringer equation, which is simpliﬁed by considering only nearest neighbor interactions. In matrix form, this can be represented as

Hij Cj = Sij λj Cj

(4.12)

where Hij = Ψ∗j HΨj d3 r is a matrix element for the Hamiltonian, Sij = Ψ∗j Ψj d3 r are overlap integrals, Cj are the eigen vectors and λj the energy eigenvalues. There

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are various semiempirical schemes for evaluating the matrix elements and the overlap integrals. It is instructive to consider a simple case such as the benzene ring. There are four outer electrons in the carbon atom, two 2s electrons and two 2p electrons. These four electrons rearrange as three sp2 orbitals at 120◦ to each other in a plane, leaving a loose p electron state perpendicular to the plane. The orbitals in the plane form the strong σ bonds and the corresponding σ ∗ antibonding states. The weaker overlap between the p electrons forms a low energy π state with corresponding π ∗ orbital. The π ∗ and σ ∗ states are empty and are available for electrons which have been excited from the K shell. Graphite is like an inﬁnite extension of benzene rings and so the spectrum of graphite can be analyzed in the same terms [4.52]. A similar analysis can be performed for other layer compounds such as hexagonal BN and π ∗ and σ ∗ features can be identiﬁed on both K-edges. A particularly interesting application has involved B in oxidized FeCr ﬁlms. Rowley et al. [4.53] were able to show that the bonding of B to oxygen was trigonal due to the presence of the π ∗ bond. As would be expected, the ratios of the π ∗ and σ ∗ features would vary depending on whether the scattering vector q lies along the plane or perpendicular to it. The orientation dependence was demonstrated in both BN and graphite by Leapman et al. [4.54,4.55], and was shown to extend into the EXELFS region by Disko et al. [4.56]. These experiments, which are the analog of polarization dependence in X-ray absorption, are not easy as the scattering wavevector has to be deﬁned precisely using collection apertures that are small compared with qE , which is 1.4 mrad for carbon K at 100 kV. In the case of diamond there are four symmetric sp3 orbitals in tetrahedral coordination, each containing one electron. There is only a σ ∗ peak, and this can be used to distinguish diamond from other forms of carbon [4.7,4.57]. In fact, the most studied specimen in electron microscopy, amorphous carbon, nicely illustrates the use of the π ∗ peak as an indication of the number of “loose" p electrons. The unattached p electron concentration as measured by the ratio of π ∗ to σ ∗ peak height is consistent with the electron spin resonance (ESR) measurements [4.58]. The low loss spectrum provides supporting evidence in the form of separate plasmon peaks for the π and σ + π electron oscillations. The plasmon energy, as shown above, can be used as a measure of the appropriate electron density. The molecular orbital approach has also been useful in understanding the ﬁne structure of the titanium oxides, rutile and anatase, and the related carbide and borides [4.38,4.59–4.62]. A molecular orbital energy level diagram can be constructed from the oxygen 2p levels and the metal 3d levels. The ﬁrst empty levels above the Fermi level are the three π bonded t2g levels. Slightly higher in energy are the two σ bonded eg levels where the orbitals are directed along the bonds. This also holds for other oxides with octahedral coordination, though the lower t2g level is ﬁlled for Fe2+ . The splitting between the levels is ∼ 1.6 eV for TiO2 and can clearly be seen in both the oxygen K-edge [4.62] and the individual L3 - and L2 -edges of titanium, as the orbitals have both oxygen p and metal d character. Spectra for both the oxygen K- and titanium L23 -edges are given in Fig. 4.7. In compounds with tetrahedral coordination the order of the t2g and eg states is reversed. It is also possible for the

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orbitals with different spin states to have different occupations. An example is Fe2 O3 and FeO as given by the calculations of Tossell [4.59]. The results have been used in the interpretation of the iron L23 - and oxygen K-edge ﬁne structure in various iron oxides. The same molecular orbital analysis has also been used by Krishan [4.63] to explain variation in L3 and L2 shapes for Fe2+ and Fe3+ in octahedral and tetrahedral sites in different spinels.

Fig. 4.7 (a) Titanium L23 and oxygen K -edge showing splitting between t2g and eg levels in rutile (courtesy J. Auerhammer). (b) High resolution spectrum of Ti L23 -edge in rutile (courtesy R. Brydson).

The main application of molecular orbital methods has been in understanding the spectra of gas molecules. The Xα method of Slater and Johnson [4.64], which was further reﬁned by Dill and Dehmer [4.65], is a powerful technique for calculating the energy levels of the molecules orbitals. In many ways it is similar to the XANES method. The atoms are again represented by spherical mufﬁn tins, and the scattering of spherical ingoing and outgoing partial waves is characterized by phase shifts. It is implicitly assumed that the atoms form a cluster and the computation assumes that the cluster behaves as a large spherical mufﬁn tin with its own average potential and an outgoing spherical wave. The Schr¨odinger equation is solved by matching the wave function and its derivative at the various boundaries. A key difference between this method and the XANES method is that an iteration is performed until self consistency is achieved, whereas XANES assumes that the ejected electron wave function is an independent solution of the Schr¨odinger equation with a speciﬁed potential. In both cases, an averaged exchange potential is used. (hence the name Xα for the Slater’s exchange formula). Tossell has applied the Xα method to both absorption and emission in various oxide systems [4.59,4.66,4.67]. Tanaka has used a self consistent molecular orbital method with the Xα exchange potential in calculations of near-edge structure in MgO and various silicon oxides [4.68]. The various molecular orbital or LCAO schemes have probably been least successful in describing the near edge structure in semiconductors and other silicon

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compounds. Skiff [4.69] has published calculations for Si3 N4 , Si, and SiC using an extended Huckel scheme [4.70] and Waki and Hirai [4.71] have interpreted the near edge structure on the Si L-edge in a range of Si compounds using a molecular orbital scheme. Although it seems the gross features can be explained, simple LCAO schemes do not reproduce the subtle edge shapes seen in these compounds. 4.5.3

Band Structure

The most detailed theoretical work on near edge structure has involved direct comparison of sophisticated band structure calculations with experimental results. Strictly XANES and molecular orbital or LCAO methods should be considered as band calculations; but as they are ammenable to simple physical interpretation, they have been covered separately. The XANES method is, after all, nothing more than a Korringa, Kohn, Rostoker (KKR) calculation. Band theory methods are all based on the separation of Eq. 4.7 into a matrix element term and a density of states term. 4γ 2 d2 I 2 = 2 4 |M (E)| ρ(E) dΩdE a0 q

(4.13)

To be strictly accurate, the dipole selection rules for small angle scattering or photon absorption should also be imposed. 2 2 4γ 2 d2 I = 2 4 ML+1 (E) ρL+1 (E) + ML−1 (E) ρL−1 (E) dΩdE a0 q

(4.14)

The band theory must therefore produce the density of states resolved into angular momentum components at a particular atomic site. As can be seen from Eq. 4.14 only the p density of states need be considered in K shell scattering. For L23 (2p) excitation, both the s and the d density of states have to be considered. The density of states is calculated directly from the energy bands as 1 dS (4.15) ρ(E) = |∇En (k)| 4π 3 Sn (E)

Portions where the band is ﬂat and the derivative is low near Brillouin zone boundaries or (000) will therefore make strong contributions to the density of states, and hence give rise to spectrum peaks. It is also assumed that in many cases the matrix elements vary slowly over the range of ∼ 10 eV, and are often neglected in comparing calculations with ﬁne structure. This can sometimes be dangerous and, where possible, the matrix element variation should be included in the calculation. The energy bands are calculated using a charge density that is varied in a selfconsistent manner to minimise the total energy. The contribution to the potential from exchange and correlation are given by functions of the local electron density (local density approximation) [4.72]. It can be shown that this procedure generates the exact ground-state charge density, though no theoretical claim can be made for

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the validity of the unoccupied states that are generated. Densities of states from the unoccupied bands do agree quite well with experiment and can provide insight into the origin of spectral features. The use of a self-consistent charge density is a signiﬁcant difference between band theory calculations and the simpliﬁed methods described above such as XANES. It would be expected that this would change the calculated near edge structure, especially close to the threshold. The KKR band structure method is the most similar to the XANES method described above. It is not as widely used as other methods as it can be computationally more demanding. The layered KKR method due to Maclaren [4.73,4.74] is especially well suited to problems in electron energy loss spectroscopy involving interfaces. It can be applied to the calculation of near edge structure in a wide range of materials. Figure 4.8 shows the results of a calculation for the oxygen K-edge in rutile. The t2g –eg molecular orbital splitting is correctly reproduced, as well as the weaker extened ﬁne structure. A XANES calculation would not show the molecular orbital splitting without the introduction of unphysical parameters, clearly demonstrating the importance of self-consistency in the region just above threshold.

Fig. 4.8 LKKR calculation of the oxygen K -edge in rutile TiO2 (solid line), compared with experiment (dashed line).

The Augmented Plane Wave (APW) method has also been applied in spectrum analysis. In an augmented plane wave calculation, the crystal is again divided into atomic mufﬁn tins and radially symmetric solutions are found inside the mufﬁn tins. These solutions are matched on to plane waves in the intersititial region between the mufﬁn tins. The solutions should be determined at each energy. A helpful simpliﬁcation for computational purposes is to make use of energy derivatives in a linearised APW (often called LAPW) to determine wave functions and densities of states over a range of energies. The Linearized Mufﬁn Tin Orbital (LMTO) or Linearized Spherical Wave (LSW) methods [4.75] are closely related to the linearized APW method. Spherical waves instead of plane wave are used in the region outside the mufﬁn tin spheres. Although in its original form the APW method assumed a constant interstitial potential, it is now customary to use a fourier series to represent spatially varying potentials arising from directional bonds between the mufﬁn tins. This enhancement, called the full potential method (FLAPW) [4.76], extends the

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range of applicability to semiconductors and other materials with low coordination. For spectroscopic calculations, an advantage of any method that uses mufﬁn tins is that it can directly give the appropriate angular momentum selected density of states at particular atomic sites. Early APW calculations of Painter and Ellis [4.77] were successfully used to explain Egerton’s [4.78] diamond spectra. Attempts by Fisher and Baun [4.79] to calculate near edge structure in TiO were less successful and they resorted to molecular orbital techniques [4.80]. However, APW calculations by Neckel et al. [4.81]show very good agreement with the metal L-edges in TiO, TiC, and VC. The corresponding p densities of states are also in good agreement with the anion ﬁne structure. Muller et al. [4.82] have also shown that the APW method reproduces the ﬁne structure on the L-edges of transition metals, and not just the oxides carbides and nitrides. The FLAPW method has recently been applied with great success to the calculation of the L-edges of transition metals in various inermetallic alloys [4.83]. However, these band theory calculations do not account for the anomalous L3 to L2 line height ratio which is mainly an atomic effect, and will be discussed at length in the following section. Pseudopotential methods have been very successful in calculating electronic properties of semiconductors. The results of total energy calculations are now accurate enough to permit theoretical determination of phase diagrams, interface structure and the atom positions in surface reconstructions. The pseudopotential methods have traditionally used a plane wave expansion, and have therefore not been well suited to determining the angular momentum resolved density of states at an atomic site, which is needed to understand inner-shell spectra. The relavant densities of states must therefore be projected from the calculated unocuppied states. A formulation of the pseudopotential method based on pseudoatomic orbitals has been developed [4.84]. which gives states that are resolved by angular momentum quantum number, and it has been applied with some success to the near-edge structure of semiconductors and ceramic materials by Weng and Rez [4.85]. They have reported results for the K- and L23 -edges of Si in elemental Si and SiC, the K-edge of carbon in diamond, beryllium carbide and silicon carbide, and the beryllium K-edge in beryllium carbide. The calculation for Si L23 in elemental silicon [4.86] (see Fig. 4.9) is of particular interest as it is in very good agreement with the high-resolution spectrum of Batson recorded with a resolution of ∼ 0.3 eV. The theory clearly shows that the ﬁrst peak is caused by p to s transitions and any core exciton contribution to the threshold must be of order 50 meV in energy or less. The silicon L23 -edge is a good example of the dangers of comparing with a total unoccupied density of states, which in this case is largely of p character, and therefore dipole forbidden. Another example is the calculation for the carbon K-edge in diamond given as Fig. 4.10. The technique has also been applied to hexagonal materials and results for the carbon K-edge in graphite [4.50] and the various edges in hexagonal boron nitride have been published. Comparison with the experiments of Batson for graphite shows an agreement with the measured separation of the π ∗ and σ ∗ peaks. The calculations can also help in simple interpretations of the experimental data. Peaks in the projected density of states arise from ﬂat energy bands, so any variations in band structure around defects

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should be detectable using small probe instruments with good energy resolution, such as a cold ﬁeld emission STEM equipped with a parallel recording spectrometer.

Fig. 4.9 Comparison between pseudo-atomic orbital calculation experiment for Si L23 -edge (theory due to X. Weng, experiment due to P. Batson).

Fig. 4.10 Pseudo-atomicorbital calculation for the carbon K -edge in diamond (dashed line), compared with experiment (solid line) (theory due to X. Weng, experiment due to P. Fallon).

The pseudo-atomic-orbital method does not provide for a complete description of the conduction wave functions and this becomes a serious problem > 10 eV above threshold. Recent results (Pickard, unpublished), based on a projection from a plane wave pseudo-potential wave function, show very good agreement with experiment for diamond and graphite up to 40 eV above threshold, and also give indications of the breakdown of the one electron theory. From these calculations it appears that the one electron picture can be used in the interpretation of near-edge structure, even for features as ﬁne as 0.5 eV. This should not be treated as a universal law; there are undoubtedly cases such as cations in ionic materials (e.g., LiF) [4.87], where core excitons are clearly present and the near edge region does not fully represent the unoccupied unexcited states. Many of these materials also damage easily in the electron microscope, so it is unlikely that the average electron microscopist will encounter spectra with these serious complications.

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4.5.4

115

White Lines

The white lines on the L23 absorption edges in the transition metals and the M45 edges of the rare earths have already been mentioned in the general introduction on inner-shell spectroscopy. They are the most prominent feature on the absorption edge and potentially can be used to give information on charge and spin state of the excited atom. The two white line components arise because of spin–orbit coupling. The spin (quantum number ±1/2) can couple in two ways to the angular momentum l giving a total quantum number j = l ± 1/2. In the case of the 2p excitation, l = 1 and j = 3/2 or j = 1/2. The subshell with quantum number j = l + 1/2 has a lower binding energy, and this peak marks the edge threshold. The peak from the j = l − 1/2 subshell appears at higher energy in the spectrum, and is sitting on the background due to transitions from the j = l + 1/2 subshell to continuum states. The separation between the peaks is the spin orbit splitting. The number of states (degeneracy) for each level is 2j + 1, so one would expect the lower binding energy j = 3/2 peak to be twice as intense as the higher energy j = 1/2 peak in the transition metals. Similarly for the rare earth M45 -edges one would expect an intensity ratio between the lower energy j = 5/2 peak and the higher energy j = 3/2 peak of 6:4. As ﬁrst shown by Leapman [4.11,4.88] for the L23 -edges in the ﬁrst row transition elements, the white line intensity ratio deviates dramatically from 2:1, being < 1:1 for elements near the beginning of the period, such as Sc and Ti, and reaching a maximum (reported to be about 5:1) for Fe then decreasing to 3:1 for Cu [4.89]. The result from the original paper is shown as Fig. 4.11. There was also a variation observed with charge state. The effects in the rare earths are even more pronounced, the M5 :M4 intensity ratio being ∼ 0.6 for Ce and Pr and increasing to 10:1 for Lu [4.90,4.91]. It is hard to see how such large changes could arise from solid-state effects and in fact the origin of the white line intensity ratio is mainly atomic in nature [4.92]. Another way of looking at this is to say that the d electrons in ﬁrst-row transition metals and the f electrons in rare earths are strongly correlated. This does not mean that solid-state effects, which generally assume a weak correlation between electrons according to the local density approximation, can be entirely neglected. To reproduce the white line intensity variations the electrostatic and the spin–orbit couplings between the hole and the possible distribution of electrons in d or f level in the ﬁnal state have to be considered. Matrix elements can then be taken between the allowed ground and excited states [4.93]. The result is a multiplicity of lines (multiplet) which will divide into two groups to form the two spin–orbit components [4.94,4.95]. In a practical calculation these individual lines can be broadened by a Lorentzian or Gaussian to represent appropriate lifetime and instrumental broadening. The ﬁrst results due to Sugar were obtained by treating the electrostatic and spin–orbit coupling as a perturbation and the method was applied to the La, Er, and Tm [4.93]. Composite lines from elements in the middle of the period can have many transitions (multiplets), and it is necessary to have a computer code to handle all the possibilities. The computer program can also ﬁnd the correct wave functions by diagonalizing the full Schr¨odinger equation with spin–orbit and electrostatic terms, rather than treating

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Fig. 4.11 Experimental variation of white line intensities for the L23 -edge in the ﬁrst row 3d transition elements (courtesy R. D. Leapman).

them as a perturbation [4.96]. Thole et al. [4.97] using such a program, published a complete survey of the M45 -edges in the rare earths that showed very good agreement with experiment. It is also possible to base the computer programs on the Dirac equation and Waddington et al. [4.98] published results for the ﬁrst row transition metal ions. Although there were disagreements with the exact white line ratio, they did show the general trend; and they showed that for Mn and Cr the white line ratio is sensitive to ionic charge [4.99]. The solid-state effects cannot be ignored in the 3d transition elements. Straightforward examples are the oxides of scandium and titanium where the L3 and L2 peaks are further split into subpeaks corresponding (approximately) to the t2g and eg molecular orbitals. The atomic multiplet structure is simple as the ﬁnal state has only 1 electron in the 3d level giving a total of 3 transition lines. Zaanen et al. [4.100] published an analysis of the ﬁne structure for Ca, Sc, and Ti oxide using a combination of the atomic multiplet theory and the results of band theory calculations. However, most of the solid-state efforts can be adequately taken into account by restricting the symmetry of the ﬁnal state according to the environment of the cation. Both Yamaguchi [4.101] and de Groot [4.102] have published results for the transition metal ions in cubic symmetry. A complete summary of all transition metal ions in

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both octahedral and tetrahedral symmetry has been published by Van der Laan and Kirkman [4.103]. These are in good agreement with high resolutions EELS [4.104] and X-ray absorption data (for examples and comparisons with experiment, see Fig. 4.12), and they show that with 0.5-eV energy resolution it is possible to distinguish low spin and high spin states and the different charge states of cobalt [4.102]. It is advantageous to work at an energy resolution of 0.5 eV or better when examining white lines as some of the multiplet structure then becomes distinguishable. Kaindl et al. [4.105,4.106] have reported that at this energy resolution it is possible to determine charge of rare earth ions such as Sm and Eu.

Fig. 4.12 Comparisons between theory (T) and experiment (E) for the multiplet structure in the L23 -edge in (a) MnF2 and (b) VF3 (courtesy F. M. de Groot).

The white line ratios for the second-row transition elements have approximately a 2:1 intensity ratio [4.107,4.108]. The wide variation of white line intensity ratios comes about when the exchange energy between the p and the partly occupied d shell is of the same order as the spin–orbit splitting. For the second-row transition elements, the spin–orbit splitting is very much greater than any exchange energies, and the two components can be treated separately. Other analyses of the transition element white line ratios have focused on a separation between the two spin states in the d band, and have tried to correlate the results with magnetic moment [4.109]. This approach ignores the important atomic effects that dominate the white line features. Another simple analysis takes the ratio of the intensity in the white lines to the continuum part of the edge and relates it to the total number of holes in the d band. A particularly striking example of this effect is the distinction between metallic and oxidized copper. The pure metal has a full d band and no white lines are seen, whereas the oxide has empty d states whose presence is indicated by a white line at thershold. There have been some attempts to analyse the Cu L23 spectrum to determine the valence of copper in the new oxide supeconductors [4.110]. However, the more signiﬁcant effect in these materials is the change in the oxygen near-edge structure that is strongly correlated with superconductivity [4.111,4.112]. Pearson et al. [4.113] have successfully used

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the white line to continuum ratio to investigate the variation of d band occupancy on alloying. In summary, white lines not only represent interesting physics, but also have immense practical application for the materials scientist. Even at relatively poor resolutions of 1–2 eV the total white line intensity can be used as a measure of d state occupancy for the transition metals, and the ratio of the L3 and L2 lines can give a measure of the charge state for Mn, Cr and Fe [4.99]. At resolutions of 0.5 eV or better, effects due to environment become more apparent, and by matching white line ﬁne structure with calculation it should be possible to differentiate technologically interesting atoms such as Co, Fe, and Sm [4.102,4.105] in different charge and spin states. 4.6 CONCLUSIONS There are not many simple prescriptions in ﬁne structure analysis, and the reconciliation of theory with experiment is still an active ﬁeld of research. Plasmons can give useful information electron density; and the low loss region can also be used to measure the dielectric constant from small regions of material. The inner-shell region can be analyzed in different ways to give information on bonding and local environment. The extraction of radial distribution functions from extended ﬁne structure is a wellestablished technique, though it might not be as useful in electron microscope energy loss experiments as it has been in X-ray synchrotron studies. Near-edge structure in the presence of strongly scattering anions gives information on coordination. An analysis in terms of empty molecular orbitals can often be used to identify peaks, or they can be related to features of a local projected band structure. It appears that the ﬁne structure truly represents a local probe and is not unduly affected by distant coordination shells. Finally, the white lines can be used to give data on the occupancy of partially ﬁlled bands and the charge of the excited atom, especially if spectra are recorded at energy resolutions of 0.5 eV or better. The white line structure can also be used to distinguish spin states in favorable cases.

REFERENCES 4.1. P. E. Batson, “Prospects for sub-bandgap electron energy-loss experiments with the scanning transmission electron microscope,” Ultramicroscopy, 18, 125 (1985). 4.2. D. Bohm and D. Pines, “A collective description of electron interactions II. Collective vs. individual particle aspects of the interactions,” Physical Review, 85, 338 (1951). 4.3. C. Kittel, Introduction to Solid State Physics (New York: J. Wiley, 1986).

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4.4. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (New York: Springer Verlag, 1980) 4.5. R. H. Ritchie, “Plasmon Losses by Fast Electrons in Thin Films,” Physical Review, 106, 874 (1957). 4.6. D. B. Williams and J. W. Edington, “High Resolution Microanalysis in Materials Science Using Electron Energy Loss Measurements,” Journal of Microscopy, 108, 113 (1976). 4.7. D. F. Blake et al., “The nature and origin of interstellar diamond,” Nature, 332, 611 (1988). 4.8. D. W. Johnson, “A Fourier series method for numerical Kramers-Kronig analysis,” Journal of Physics A, 8, 490 (1975). 4.9. A. Howie and R.H. Milne, “Excitations at Interfaces and Small Particles,” Ultramicroscopy, 18, 427 (1985). 4.10. R. D. Leapman et al., “K, L and M Shell Generalised Oscillator Strengths and Ionisation cross-sections,” Journal of Chemical Physics, 72, 1232 (1980). 4.11. R. D. Leapman and L.A. Grunes, “Anomalous L3 /L2 White-Line Ratios in the 3d Transition Metals,” Physical Review Letters, 45, 397 (1980). 4.12. M. Inokuti, “Inelastic Collisions of Fast Charged Particles with Atoms and Molecules-The Bethe Theory Revisited,” Journal of Modern Physics, 43, 297 (1971). 4.13. E. A. Stern, “Theory of the Extended X-Ray Absorption Fine Structure,” Physical Review B, 10, 3027 (1974). 4.14. P. A. Lee and J. B. Pendry, “Theory of the Extended X-Ray Absorption Fine Structure,” Physical Review B, 11, 2795 (1975). 4.15. B. K. Teo and D.C. Joy, EXAFS Spectroscopy, techniques and applications (New York: Plenum Press, 1981) 4.16. A. Bianconi et al.,ed., EXAFS and Near Edge Structure (Berlin: SpringerVerlag, 1983). 4.17. P. H. Citrin, et al., “Transferability of phase shifts in extended x-ray absorption ﬁne structure,” Physical Review Letters, 36, 1346 (1976). 4.18. E. A. Stern, et al., “Adsorbate and Substrate Characterization Using Extended X-Ray Absorption Fine Structure,” Physical Review Letters, 38, 767 (1977). 4.19. P. A. Lee and G. Beni, “New Method for the Calculation of Atomic Phase Shifts: Application to Extended X-Ray Absorption Fine Structure (EXAFS) in Molecules and Crystals,” Physical Review B, 15, 2862 (1977).

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4.49. M. Kitamura et al., “X-Ray-Absorption Near-Edge-Structure Study of Diamond: A Multiple-Scattering Approach,” Solid State Communications, 62, 663 (1987). 4.50. X. Weng et al., “Carbon K Shell Absorption Near Edge Structure: Multiple Scattering and Band Theory Calculations,” Physical Review B, 40, 4175 (1989). 4.51. A. Bianconi et al., “Multiple-Scattering Effects in the K-Edge X-Ray-Absorption Near-Edge Structure of Crystalline and Amorphous Silicon,” Physical Review B, 36, 6426–6433 (1987). 4.52. B. M. Kincaid et al., “Carbon K-edge in graphite measured using electron energy-loss spectroscopy,” Physical Review Letters, 40, 1296–1299 (1978). 4.53. P. N. Rowley et al., “Electron Energy Loss Studies of Fe-Cr-Mn Oxide Films,” Philosophical Magazine, 62 B, 229–238 (1990). 4.54. R. D. Leapman et al., “Orientation dependence of core edges from anisotropic materials determined by inelastic scattering of fast electrons,” Physical Review B, 28, 2361–2373 (1983). 4.55. R. D. Leapman and J. Silcox, “Orientation Dependence of Core Edges in Electron-Energy-Loss Spectra from Anisotropic Materials,” Physical Review Letters, 42, 1361–1364 (1979). 4.56. M. M. Disko et al., “Orientation-Dependent Extended Fine Structure in Electron Energy Loss Spectra,” Physical Review B, 25, 4252–4255 (1982). 4.57. S. D. Berger et al., “Applications of the Near-Edge and Low-Loss Fine Structure in the Analysis of Diamond,” Ultramicroscopy, 28, 43–46 (1989). 4.58. R. H. Jarman et al., “Determination of bonding in amorphous carbon ﬁlms: A quantitative comparison of core-electron energy-loss spectroscopy and 13C nuclear magnetic resonance spectroscopy,” Applied Physics Letters, 49, 1065– 1067 (1986). 4.59. J. A. Tossell et al., “The Electronic Structure of Rutile, Wustite, and Hematite from Molecular Orbital Calculations,” American Mineralogist, 59, 319–334 (1974). 4.60. D. W. Fischer, “Electronic Band Structure and the K and L X-Ray Spectra from TiO, TiN, TiC,” Journal of Applied Physics, 41, 3922–3926 (1970). 4.61. D. W. Fischer, “X-Ray Band Spectra and Molecular-Orbital Structure of Rutile TiO2 ,” Physical Review B, 5 4219–4226, (1972). 4.62. L. A. Grunes et al., “Oxygen K near-edge ﬁne structure: An electron energyloss investigation with comparisons to new theory for selected 3d transition-metal oxides,” Physical Review B, 25 7157–7173, (1982).

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4.79. D. W. Fischer and W. L. Baun, “Band Structure and the Titanium LII,III X-Ray Emission and Absorption Spectra from Pure Metal, Oxides, Nitride, Carbide, and Boride,” Journal of Applied Physics, 39, 4757–4776 (1968). 4.80. D. W. Fischer, “Molecular-Orbital Interpretation of the Soft X-Ray LII,III Emission and Absorption Spectra from Some Titanium and Vanadium Compounds,” Journal of Applied Physics, 41, 3561–3569 (1970). 4.81. A. Neckel et al., “Results of Self-Consistent Band-Structure Calculations for ScN, ScO, TiC, TiN, TiO, VC, VN and VO,” Journal of Physics C, 9, 579–592 (1976). 4.82. J. E. Muller et al., “X-Ray Absorption Spectra: K-Edges of 3d Transition Metals, L-Edges of 3d and 4d Metals, and M -Edges of Palladium,” Solid State Communications, 42, 365–368 (1982). 4.83. G. A. Botton et al., “Experimental and theoretical study of the electronic structure of Fe, Co and Ni aluminides with the B2 structure,” Physical Review B, 54, 1682–1691 (1996). 4.84. R. W. Jansen and O. F. Sankey, “Ab initio linear combination of pseudo-atomicorbital scheme for the electronic properties of semiconductors: Results for ten materials,” Physical Review, B 36, 6520–6531 (1987). 4.85. X. Weng et al., “Pseudo-atomic-orbital band theory applied to electron-energyloss near-edge structures,” Physical Review, B 40, 5694–5704 (1989). 4.86. X. Weng et al., “Single Electron Calculations for the Si L2,3 Near Edge Structure,” Solid State Communications, 74, 1013–1015 (1990). 4.87. S. T. Pantelides, “Electronic Excitation energies and the soft-x-ray absorption spectra of alkali halides.,” Physical Review B, 11, 2391–2402 (1975). 4.88. R. D. Leapman et al., “Study of the L23 edges in the 3d transition metals and their exides by electron-energy-loss spectroscopy with comparisons to theory,” Physical Review B, 26, 614–635 (1982). 4.89. J. Fink et al., “2p absorption spectra of the 3d elements,” Physical Review B, 32, 4899–4904 (1985). 4.90. C. Bonnelle et al., “Photoabsorption in the vicinity of 3d absorption edges of La, La2 O3 , Ce, and CeO2 ,” Physical Review A, 9, 1920–1923 (1974). 4.91. C. Bonnelle et al., “Photoabsorption in the vicinity of 3d Edges of Eu and Gd,” Journal of Physics B, 10, 795–801 (1977). 4.92. J. Garth et al., “Atomic nature of the LII,III white lines in Ca, Sc, and Ti metals as revealed by resonant photoemission,” Physical Review B, 28, 3608–3611 (1983).

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4.108. C. C. Ahn and O. L. Krivanek, EELS Atlas (Pleasanton, Ca: Gatan, Inc. and ASU, 1983) 4.109. T. I. Morrison, et al., “Iron d-Band Occupancy in Amorphous Fex Ge1−x ,” Physical Review B, 32, 3107–3111, (1985). 4.110. M. Grioni, et al., “Studies of copper valence states with Cu L3 x-ray-absorption spectroscopy,” Physical Review B, 39, 1541–1545 (1989). 4.111. P. E. Batson and M. F. Chisholm, “Anisotropy in the near-edge absorption ﬁne structure of YBa2 Cu3 O7−δ ,” Physical Review B, 37, 635–637 (1988). 4.112. N. Nucker et al., “Evidence for holes on oxygen sites in the high-Tc superconductors La2−x Srx CuO4 and YBa2 Cu3 O7−δ ,” Physical Review B, 37, 5158–5163 (1988). 4.113. D. H. Pearson et al., “Measurements of 3d state occupancy in transition metals using electron energy loss spectrometry,” Applied Physics Letters, 53, 1405 (1988).

5

Energy Filtered Diffraction Ludwig Reimer

5.1 INTRODUCTION Energy ﬁltering offers new possibilities for the application of electron diffraction since transmission electron microscopes with ﬁlter lenses or post-column imaging spectrometers are commercially available, and can be operated in the electron spectroscopic diffraction (ESD) mode. The ﬁltering of unscattered and elastically scattered electrons – zero-loss ﬁltering – will be the main application. It removes the strong background caused by inelastically scattered electrons and expands the useful thickness range. This allows us to obtain better quantitative information for all types of specimens, such as amorphous, poly- and single-crystalline foils, and for the different types of electron diffraction patterns (EDPs), such as small-angle electron diffraction, selected-area electron diffraction (SAED) with Bragg spots and Kikuchi lines and bands, and convergent-beam electron diffraction (CBED) containing the rocking curves of the dynamical theory of electron diffraction. EDPs of amorphous specimens contain diffuse diffraction rings. A Fourier transform allows us to get a radial density distribution or pair correlation of the nearestneighbor atoms. This method becomes more quantitative when the background from inelastically scattered electrons can be removed. EDPs of polycrystalline specimens consist of Debye–Scherrer rings. Increasing ﬁlm thickness results in an increasing background of inelastically scattered electrons. Zero-loss ﬁltering allows to get much better contrast and detectibility of high-indexed rings. EDPs of single-crystalline foils show better contrast after zero-loss ﬁltering. Weak reﬂections (e.g., from superlattices) can also be detected for thick foils. For example, the background between diffraction spots contains thermal-diffuse scattering caused by electron–phonon scattering, which results in thermal-diffuse streaks. These streaks are strongly masked by the inelastic contribution to the background and zero-loss ﬁltering offers the possibility of using this electron interaction for the characterization of materials at high spatial resolution. Whereas ﬁltering of inelastically scattered electrons gives no new information for amorphous and polycrystalline specimens with the exception of angular-resolved EELS patterns (see below), it will be very useful for single-crystalline foils. The main inelastic processes are plasmon losses, inter- and intraband transitions, innershell excitations, and Compton scattering. Continuous shifting of the energy loss 127

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window, not only gives information on how these processes contribute to the unﬁltered pattern, it also makes it possible to extract quantitative information from the change in intensity distributions. At large energy losses the diffraction pattern only consists of excess or defect Kikuchi bands. Convergent-beam electron diffraction and large-angle CBED is generated by a cone of primary electrons. The Bragg spots that are extended to circles, contain the pendell¨osung fringes described by dynamical theory, which give information about the foil thickness and the real and imaginary Fourier coefﬁcients, Vg and Vg , of the crystal lattice potential. A handicap in this analysis has been the need for background subtraction of inelastically scattered electrons. Therefore, zero-loss ﬁltering allows a more accurate ﬁt of the measured intensities, by varying the Fourier coefﬁcients in a calculation by the dynamical theory of electron diffraction. The difference between measured Vg and calculated values for single atoms results from different screening of the valence electrons therefore can be used for the construction of charge density distributions in chemical bonds, a method that formerly was the sole domain of X-ray crystallography. 5.2 INSTRUMENTATION The enhancement of contrast in EDPs by ﬁltering the elastically scattered electrons was proposed and performed long before the ﬁrst imaging ﬁlter was developed. Boersch [5.1] used a retarding grid at an intermediate image plane as a high-pass energy ﬁlter and showed a considerable decrease of the backgrond by inelastically scattered electrons in Debye–Scherrer patterns of an Al2 O3 ﬁlm. Later, elastically ﬁltered EDPs of amorphous and polycrystalline ﬁlms were investigated by scanning the EDP across an aperture in front of a retarding ﬁlter or an electron spectrometer [5.2–5.10] – a technique that can still be used with modern energy ﬁlter devices. The ﬁrst experiments with energy ﬁlter lenses of the Castaing type, once again allowed us to ﬁlter whole EDPs [5.11–5.14]. The development and performance of commercial TEMs with in-column ﬁlter lenses [5.15] and post-column imaging energy spectrometers [5.14,5.16] enables us to use electron spectrosopic imaging (ESI) and diffraction (ESD) as a routine method [5.15–5.18]. The principle of both types of energy ﬁltering can be described by the black-box diagram of Fig. 5.1 for the ESD mode. A selected-area or convergent-beam EDP is focused on the ﬁlter entrance plane by focusing the intermediate lens to the diffraction pattern in the focal plane of the objective lens. The focal plane of the projector lens in front of the ﬁlter unit then contains a demagniﬁed image of the selected area as a “point source”. The EDP is imaged behind the ﬁlter in a conjugate achromatic image plane. This means that electrons scattered with increasing energy losses to a point of an EDP pass this plane under increasing angles to the axis but still through the same point. An energy-selective plane containing the EELS is conjugate to the source plane. All electrons with the same energy loss from different points of an EDP now pass one point of the EELS. However, the intensity distribution in the

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energy-selective plane is convolved by the intensity distribution of the source plane. An energy-selecting slit in the energy-dispersive plane allows to select an energy loss window at an energy loss E with a width ∆ = 1–50 eV. The following projector lens can magnify either the energy-ﬁltered ECP in the achromatic image plane or the EELS in the energy-selecting plane onto the ﬁnal image plane, where they can be observed and recorded by a ﬂuorescent screen or a CCD camera coupled per lightpipe to a thin scintillator. [When the intermediate lens is switched to focus the ﬁrst image plane, the ﬁlter entrance plane contains a magniﬁed image of the specimen and the ﬁlter works in the electron-spectroscopic imaging (ESI) mode.]

Fig. 5.1 Principle of an energy ﬁltering lens or imaging spectrometer with different congugate planes.

Both types of ﬁlters show a second-order aberration which results in a parabolic shape of areas of constant energy loss in the ﬁltered image. This limits the diameter of the central area of nearly constant energy loss. Therefore, methods have been developed to correct this aberration by multipole elements [5.15]. When a slit of 1–5 µm is mounted in the ﬁlter entrance plane to select a line across the EDP, an angular-resolved EELS or energy-dispersed ESD, as we also can call it, can be recorded in the energy-dispersive plane where the dispersion is normal to the slit direction [5.22–5.25] (example in Fig. 5.3). This offers a direct overview of how the diffraction intensities are distributed in angle and energy loss. 5.3 SURVEY OF INELASTIC SCATTERING PROCESSES An electron diffraction pattern is the two-dimensional angular distribution of scattered electrons. When discussing energy ﬁltered EDPs at different energy losses E it is not only of interest to know the electron energy loss spectrum (EELS), which is normally measured with collection cones of 5–10 mrad, but also the distribution at larger angles. For details about inelastic scattering the reader is referred to [5.26,5.27].

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In Fig. 5.2 we only concentrate qualitatively on the most important contributions necessary to understand some of the phenomena that can be observed in inelastically ﬁltered EDPs.

Fig. 5.2 Schematic of EELS intensity distributions as a function of energy loss E and scattering angle θ with proﬁles E = const and θ = const showing a Compton maximum when crossing the Bethe ridge.

In the energy loss range E = 2–50 eV, plasmon, inter- and intraband transitions occur. The collective plasmon losses show a dispersion (parabolic increase of E with increasing scattering angle θ) and a cut-off angle θc . Beyond θc single-electron excitation dominates and this binary electron–electron collision results in a maximum at the Compton angle θC E [1 + (eU − E)/2m0 c2 ]−1 E/eU (5.1) eU (eU = primary electron energy), which is also called the Bethe ridge. At larger energy losses the ionization edges appear at E = EI (ionization energy) with a steep increase of the energy loss spectrum for K shells. This increase also extends to larger scattering angles. This schematic plot of scattered intensity over the E − θ plane can directly be observed by angular-resolved EELS (Fig. 5.3) with isodensities from a thin carbon ﬁlm, for example. From (a) to (c), the exposure time is increased stepwise by a factor 16. In (a), the plasmon loss and its dispersion can be recognized and in (c) the carbon K ionization edge at E = 279 eV and the Compton maximum (Bethe ridge) at large E and θ. A further advantage of angular-resolved EELS is to see also directly the inﬂuence of multiple scattering effects [5.25]. sin2 θC =

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131

Fig. 5.3 Isodensities of angle-resolved EELS of a 2.8 µg/cm2 carbon ﬁlm on a θ − E plane with exposures that increase from (a) to (c) by a factor 16 at each step.

5.4

5.4.1

DIFFRACTION OF AMORPHOUS AND POLYCRYSTALLINE SPECIMENS Amorphous Diffraction Patterns

Figure 5.4 shows the radial intensity distributions of a 15-nm amorphous germanium ﬁlm, unﬁltered, zero-loss ﬁltered (elastically scattered electrons only) and ﬁltered at an energy loss of E = 18 eV (plasmon loss). It demonstrates the increase of the relative intensities of the diffraction maxima and minima caused by zero-loss ﬁltering, whereas the oscillation between maxima and minima is much lower for inelastically scattered electrons with E = 18 eV. This shows that the inelastically scattered electrons are also diffracted. However, instead of the primary beam of a small illumination aperture associated with elastic scattering, the intensity distribution has to be convolved with the angular distribution of inelastically scattered 2 −1 electrons, proportional to (θ2 + θE ) . The characteristic angle θE /mv 2 increases with increasing energy loss E.

Fig. 5.4 Radial density distribution of the diffraction pattern of an evaporated amorphous 27-nm germanium ﬁlm: unﬁltered, zero-loss and plasmon-loss ﬁltered at E = 18 eV.

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The diffuse diffraction maxima of amorphous ﬁlms can be used to determine the radial density distribution ρ(r) of atoms where 4πr2 ρ(r)dr is the probability of ﬁnding neighboring atoms inside a shell between the radial distances r and r + dr. In a single-scattering approximation, the radial intensity distribution satisﬁes the equation ∞ dIin sin(2πqr) 1 dI 2 2 = N |f (q)| 1 + dr + N (5.2) 4πr ρ(r) I0 dΩ 2πqr dΩ 0 (I0 = incident electron current contributing to the diffraction pattern, dI = intensity recorded with a detector of solid angle dΩ, q = θ/λ = spatial frequency, θ = scattering angle); it oscillates around the curve N |f (q)|2 , which would be observed for independent elastic scattering at the N atoms contributing to the diffraction pattern with a scattering amplitude f (q) [5.28]. This distribution is superimposed on the angular distribution dIin /dΩ of inelastically scattered electrons. After forming the normalized function dIin dI 1 2 − N |f (q)| (5.3) − N i(q) = I0 N |f q|2 dΩ dΩ a reduced radial density distribution is obtained by an inverse Fourier transform [5.9,5.28]: ∞ 4πr[ρ(r) − ρ0 ] = G(r) = 8π i(q) sin(2πqr)q dq (5.4) 0

G(r) is also frequently referred to as the pair correlation function. It describes the deviation of the radial density ρ(r) from the average density ρ0 . Zero-loss ﬁltering of an amorphous diffraction pattern can remove the inelastic contribution, whereupon the maxima and minima of the diffracted intensities become more pronounced and it is easier to ﬁt the elastic contribution N |f (q)|2 , this will not, however, be exactly proportional to the differential elastic cross-section dσ/dΩ=|f (q)|2 because of multiple elastic scattering, which also occurs in thin ﬁlms. The search for a good average curve around which the maxima and minima oscillate is hence the largest problem for a quantitative analysis. For avoiding any inﬂuence of second-order aberrations on the recorded intensity distribution, the radial intensity distribution in the amorphous EDP should be recorded sequentially by using Grigson coils [5.5,5.6] to sweep the pattern across a small detector aperture with the recorded direction on-axis. 5.4.2

Debye–Scherrer Patterns of Polycrystalline Specimens

The Debye–Scherrer ring patterns from polycrystalline ﬁlms show an increasing background of inelastically scattered electrons with increasing ﬁlm thickness, though part of the background is also caused by thermal-diffuse scattering with negligible energy loss. Zero-loss ﬁltering therefore results in an increase of the ring intensities relative to the background, as ﬁrstly demonstrated by Boersch [5.1]. In the past,

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zero-loss ﬁltering was also used for quantitative work on the inﬂuence of temperature and dynamical theory on ring intensities [5.2–5.4] and Grigson coils were employed to record zero-loss ﬁltered amorphous and Debye–Scherrer patterns by scanning the pattern across a diaphragm of solid angle ∆Ω in front of an electron detector [5.7,5.8]. Figure 5.5a and c and the corresponding radial intensity records in Fig. 5.5b and d demonstrate the strong inﬂuence of zero-loss ﬁltering on the Debye–Scherrer diagram of a 230 nm evaporated aluminium ﬁlm.

Fig. 5.5 Diffraction patterns of a 62 µg/cm2 (230-nm evaporated aluminium ﬁlm (a) unﬁltered and (c) zero-loss ﬁltered with corresponding normalized radial intensity distributions (b) and (d) note the different scales of (1/I0 )dI/dΩ].

For a quantitative comparison of intensities in electron diffraction patterns from different ﬁlm thicknesses and elements, it is necessary to record a normalized intensity distribution (1/I0 ) ∆I/∆Ω. The current (intensity) ∆I passing through a solid angle element ∆Ω at a scattering angle θ is recorded in arbitrary units by a scintillator– photomultiplier combination. Though I0 will be the intensity of the primary beam spot in the absence of a specimen, direct measurement in the diffraction pattern overloads the photomultiplier. The transmissions

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I (αi ) 1 T (αi ) = = I0 I0

0

αi

∆I 2πθ dθ ∆Ω

(5.5)

(i=1,2) are therefore measured for two apertures α1 =9.2 and α2 =38.7 mrad (objective diaphragms of 50 and 200 µm). (The dash on I is introduced to distinguish intensities measured in the image mode from those in the diffraction pattern.) The difference T (α2 ) − T (α1 ) can be compared with the intensity ∆I between θ = α1 and α2 , which results in α2 ∆I 1 dI [T (α2 ) − T (α1 )] = 2πθ dθ. (5.6) I0 dΩ ∆Ω α1 Such normalized intensity records are shown in Fig. 5.5b and d demonstrating that the intensity of the zero-loss ﬁltered background is about 20 times weaker in Fig. 5.5d than the background of the unﬁltered pattern in Fig. 5.5b. As a function of mass thickness x = ρt, Fig. 5.6 shows (a) the background intensities below the 111-ring and (b) the peak-to-background ratios in unﬁltered and zero-loss ﬁltered EDPs. The beneﬁt of zero-loss ﬁltering can be quantiﬁed by calculating the ratio (gain) of peak-to-background intensities in Fig. 5.6b and d for zero-loss ﬁltered and unﬁltered diffraction patterns with increasing ﬁlm thickness (Fig. 5.7). The gain is largest for low atomic number and decreases as 1/Z owing to the decrease of the ratio ν (1.25) of inelastic-to-elastic total cross-sections, which can be described by ν 20/Z [5.30,5.31], Zero-loss ﬁltering is hence also of interest for diffraction experiments with carbon-rich organic sections, for example, apatite crystals in calciﬁed tissue sections where the resin or organic matrix causes a strong inelastic background [5.29].

Fig. 5.6 (a) Unﬁltered (open squares) and zero-loss ﬁltered (full squares) background intensities IB below the 111 rings of evaporated nickel ﬁlms as a function of mass thickness x = ρt and (b) the corresponding peak-to-background ratios I111 /IB .

The results in Figs. 5.6 and 5.7 are obtained with ﬁne-crystalline ﬁlms that satisfy the kinematical diffraction theory. When the ﬁlms contain coarse crystals, extending through the ﬁlm from top to bottom, the intensity of the Debye–Scherrer rings is

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135

Fig. 5.7 Gain of contrast G = (I111 /IB )f il /(I111 /IB )unf in Debye–Scherrer patterns of evaporated ﬁlms of Al, Ni, Pd, and Pt.

the result of averaging over dynamical reﬂection intensities, as demonstrated by the experiments of Horstmann and co-workers [5.2–5.4]. When recording the intensities in a Debye–Scherrer pattern with an energy window at higher energy losses E, the rings, including the primary beam, are blurred (convolved) by the angular distribution of multiple inelastic scattering, which increases with increasing E. Thus, unﬁltered diffraction patterns contain the elastic diffraction rings with a width equal to the illumination aperture superposed on a broader ring consisting of rings belonging to the angular distribution of energy losses. For narrow rings, the absence of zero-loss ﬁltering can make it difﬁcult to extrapolate the background below the rings. Zero-loss ﬁltering can also be of interest for diffraction from larger particles a few micrometres in size, to remove the inelastic background from thicker regions and from the carbon supporting ﬁlm. 5.4.3

Small-Angle Electron Diffraction

Large specimen periodicities Λ result in small scattering angles θ = λ/Λ. The largest periodicity detectable by electron diffraction depends on the illumination aperture αi , which can be reduced to ∼ 10−4 –10−5 rad. This results in a maximum periodicity Λmax = λ/2αi = 20–200 nm at 100 keV. Small-angle EDP can be recorded by strongly focusing the ﬁrst condenser lens, switching off the second one and using

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the largest possible camera length for diffraction [5.32–5.34]. The method can be employed to study periodicities in collagen and catalase [5.32,5.35], the mean distance of crystallites in evaporated ﬁlms with island structure [5.33], platinum aggregates in an annealed Pt/C ﬁlm [5.11,5.12], conglomerates of latex spheres and virus particles [5.36,5.37], or polymers [5.38], for example. Figure 5.8 shows the radial intensity distribution in a small-angle diffraction pattern of a thin Ag ﬁlm on a carbon substrate. Zero-loss ﬁltering (E = 0) shows the central primary beam and a better resolved halo than in an unﬁltered diffraction pattern. The most probable scattering angle of the halo appears at θ λ/d = 0.4 mrad, which is related to the mean distance d 10 nm of the crystallites. Filtering at E = 25 eV shows that the halo disappears. This absence of the halo can be explained in terms of the excitation volume of plasmons, which is of the order of 5 nm, and hence smaller than the mean distance between crystallites. Fig. 5.8 Radial densitometer record of a smallangle electron diffraction pattern of a Ag ﬁlm with island structure evaporated on a carbon ﬁlm, zeroloss and plasmon-loss ﬁltered with E = 25 eV.

When the illumination aperture can be decreased to the order of 10−6 rad, which should be possible with Schottky or ﬁeld-emission electron sources, the capabilities of small-angle X-ray diffraction, inclusively by analyzing the decrease of intensity very near θ = 0 can be transfered to electron diffraction. 5.5 DIFFRACTION AT SINGLE-CRYSTALLINE SPECIMENS 5.5.1

Energy Filtered Diffraction at Single Crystals

The contribution of inelastically scattered electrons increases with decreasing atomic number and increasing foil thickness in the diffraction patterns of single-crystalline

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137

specimens just as it does in those of polycrystalline ﬁlms. Zero-loss ﬁltering of elastically scattered electrons can be used to enhance the Bragg spot intensity, especially of weak reﬂections, and the quasi-elastic thermal-diffuse streaks caused by electron–phonon scattering; energy ﬁltering at high energy losses can separate the contributions from plasmon and inner-shell ionization losses to Kikuchi lines and bands [5.10,5.11,5.14,5.18,5.19,5.39,5.40]. Totally destructive interference of elastically scattered electrons between the Bragg diffraction spots, which are caused by constructive interference, can only be expected for ideal undisturbed lattices. Diffuse scattering caused by thermal vibrations can be treated as electron–phonon scattering using a Debye phonon model, for example [5.41,5.42]. The scattering is inversely proportional to the square of phonon frequency ν(q). The diffuse streaks are therefore mainly generated by transverse acoustic phonons of low frequency with wave vectors k perpendicular to one of the atomicchain directions and polarization vectors parallel to it [5.43–5.47]. Figure 5.9 shows an example of thermal-diffuse streaks in (b) a zero-loss ﬁltered diffraction pattern of a thin Si foil that cannot be seen in (a) an unﬁltered pattern, because of overlapping intensities of inelastically scattered electron, and also not in (c) a plasmon-loss ﬁltered diagramm because electron–phonon interactions only results in energy losses of the order of meV. In thin foils, therefore, the streaks only appear in the zero-loss pattern, whereas in thicker foils they can also be observed in plasmon-loss ﬁltered patterns due to elastic-inelastic scattering. The streaks then appear more diffuse as a result of the convolution with the angular distribution of the plasmon loss. This separation of the streaks by zero-loss ﬁltering shows that a quantitative investigation is almost impossible in a conventional diffraction pattern, which also contains the strong background of Kikuchi lines and bands formed by inelastically scattered electrons. The qualitative and quantitative use of thermal-diffuse streaks will therefore become of interest in the future.

Fig. 5.9 Thermal diffuse streaks in a single-crystal diffraction pattern of Si visible by (b) zero-loss ﬁltering but in the (a) unﬁltered and (c) zero-loss ﬁltered diffraction pattern (bar = 20 mrad).

An example of the enhancement of weak superlattice reﬂections from Al3 Li precipitates in an Al–7wt%Li alloy will be found in [5.48] together with electron spectroscopic images. Plasmon-loss scattering occurs at very small scattering angles

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and results in a broadening of the Bragg diffraction spots. This broadening can be particularly harmful if intensities and positions of the reﬂections are to be analysed quantitatively or if weak reﬂections surround very strong reﬂections, as is the case in large-unit-cell crystals or in crystals with a long-period super-cell. As an example, Fig. 5.10 shows the diffraction pattern of a decagonal quasicrystal along the 10-fold axis. In the unﬁltered pattern (recorded at 200 keV in a conventional TEM), the inelastic background due to plasmon losses broadens the reﬂections and in particular the primary beam. The details and weak reﬂections around the primary beam and the other strong reﬂections only become visible in the zero-loss ﬁltered pattern recorded at 120 keV in the Zeiss EM912 Omega [5.49].

Fig. 5.10 Diffraction patterns of a decagonal quasicrystal in CoNiAl obtained with (a) a conventional TEM at 200 keV and (b) an EFTEM at 120 keV. Note the reduction of the background and the much better visibility of the small reﬂections in the zero-loss ﬁltered pattern (by courtesy of J. Mayer).

The series of ﬁltered diffraction patterns in Fig. 5.11 of a 50-nm thick 111oriented silicon foil [5.40] shows the contributions of elastically and inelastically scattered electrons of increasing energy loss E to the conventional unﬁltered pattern (a), which is typical of all single-crystal diffraction patterns. The zero-loss ﬁltered pattern (b) of elastically scattered electrons shows sharp Bragg spots and Kikuchi lines as well as thermal diffuse streaks as discussed above. Near the plasmon loss (c), the Bragg spots Though Kikuchi lines are also generated by elastically scattered electrons, as shown in Fig. 5.11b, the main contribution is concentrated at low energy losses in the plasmon region. Because the excitation error sg of a Bragg spot can be read from the distance between the spot and the corresponding Kikuchi line (coincidence in case of exact Bragg condition), it can be an advantage to scan with an energyselecting window over the low-loss part of the EELS to obtain optimum contrast of Kikuchi lines. This is demonstrated in Fig. 5.11b, where the orientation of the Si foil was adjusted by the goniometer to exact 111 orientation, so that all the Kikuchi lines coincide with the six {220} Bragg reﬂections. and also the weak thermal diffuse

DIFFRACTION AT SINGLE-CRYSTALLINE SPECIMENS

139

streaks are already blurred by the angular distribution of the inelastically scattered electrons. This blurring increases and the intensity of the Bragg spots decreases with increasing E as shown for 100 eV in (d). Above a few hundred electronvolts [e.g., just below the Si K edge in (e)], the pattern consists only of excess Kikuchi bands. These high energy losses are concentrated near the nuclei and the intensity becomes proportional to the Bloch wave intensity at the nuclei. Beyond the Si K edge at E=1840 eV, as shown for E=2000 eV in (f), the intensity and the contrast of the excess Kikuchi bands increase and even the pattern of excess Kikuchi HOLZ lines from high-order Laue zones can be observed at the centre. These HOLZ lines are also found in conventional patterns only in the CBED mode (Fig. 5.14, for example).

Fig. 5.11 Series of electron spectroscopic diffraction (ESD) patterns of a 111-oriented Si foil (t 50 nm) with excess Kikuchi bands at higher energy losses: (a) unﬁltered, (b) E = 0 eV, (c) 16 eV, (d) 100 eV, (e) 1800 eV below and (f) 2000 eV beyond the Si K -edge.

When the foil thickness is increased, all diffraction patterns at different energy losses consist of defect Kikuchi bands as demonstrated in Fig. 5.12, which shows ESD patterns of a 800-nm Si foil. This can be explained by the broadening of the angular distribution of the scattered electrons due to multiple scattering at the entrance part of the foil [5.50]. A diffuse cone of scattered electrons then emerges from the foil. The numbers of electrons with k vectors on each of the Kossel cones are of equal magnitude and excess and defect Kikuchi lines are cancelled but the anisotropy of the anomalous absorption remains, resulting in defect Kikuchi bands. The same defect bands can be observed in Kossel patterns when the external aperture of the electron probe is increased, so that the CBED discs overlap and the intensity becomes proportional to Ig [5.51,5.52] or the incident electron beam becomes

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ENERGY FILTERED DIFFRACTION

diffuse by traversing a thick amorphous carbon foil in front of the single-crystal specimen [5.53].

Fig. 5.12 Series of ESD patterns of a thick 111-oriented Si foil t 800 nm with defect Kikuchi bands: (a) E = 100 eV, (b) 500 eV and (c) 1300 eV.

For semithin foils, diffraction patterns in front of the Si K-edge can also change to defect band patterns, whereas a pattern beyond the K-edge still can show the excess band character. This is explained schematically in Fig. 5.13. In the case (a) of thin foils, the electrons are directly scattered from the unscattered incident beam or the low-energy plasmon losses to high losses near the edge, which results in dominant excess bands as shown in Fig. 5.11e and f. In the case (b) of semithin foils, the electrons scattered below the K-edge contain a large fraction (D) of multiply and diffusely scattered electrons, resulting in defect bands, and only a small fraction (E) of electrons is directly scattered to high losses. This fraction (E) increases beyond the K-edge and the excess character again dominates. At medium energy losses, the excess and defect types of band contrast can result in a cancellation of band contrast. In case (c) of thick foils, all the electrons are multiply and diffusely scattered. The edge proﬁle is strongly blurred by multiple plasmon and higher energy losses and the pattern will only show defect bands, like those in Fig. 5.12. 5.6

CONVERGENT-BEAM DIFFRACTION METHODS

Convergent-beam electron diffraction patterns are obtained by producing an electron probe of 5–20 nm in diameter as a demagniﬁed image of the electron source. A convergent cone-shaped beam is thus formed. The convergence angle αc is deﬁned by the diameter of the condenser diaphragm but can be varied electron optically within certain limits. With this illumination geometry, the incident beam directions vary continuously within the cone. In the back focal plane of the objective lens, a pattern that consists of discs is formed (Fig. 5.14a) rather than the spot pattern obtained for parallel illumination. Each disc in the CBED pattern corresponds to one Bragg reﬂection g of the crystal. The continuous variation of incident beam directions is equivalent to a successive variation of the excitation error of the Bragg reﬂection and the observed intensity distribution corresponds to a two-dimensional

CONVERGENT-BEAM DIFFRACTION METHODS

141

Fig. 5.13 Schematical EELS for (a) thin, (b) semi-thin and (c) thick foils with contributions to defect (D) and (E) excess Kikuchi bands.

rocking curve (pendell¨osung) of the dynamical theory of electron diffraction. In the CBED discs (Figs. 5.15 and 16), the minima of the pendell¨osung can be seen as broad dark contours. In addition, sharp dark lines are visible in the central disc (Fig. 5.15). These HOLZ lines can be attributed to reﬂections at reciprocal lattice points in a high-order Laue zone, which satisfy the Bragg condition for the corresponding incident beam direction. In contrast to the low-index reﬂections, scattering into the HOLZ-reﬂections involves large scattering angles and is only possible for a very narrow angular range. The scattering into the HOLZ reﬂections is mostly investigated by examining the defect lines (the so-called HOLZ-lines) within the (000) disc. The arrangement of the HOLZ lines is very sensitive to small changes in accelerating voltage, local lattice parameter or composition. Energy ﬁltering allows thicker specimens to be investigated, thereby increasing the contrast and sharpness of the HOLZ lines. Energy ﬁltering also prevents the broadening and the shift of intensity minima of HOLZ lines by single or multiple plasmon losses. Furthermore, for an accurate quantiﬁcation of the HOLZ line positions, the dynamical shift of the HOLZ lines has to be taken into account [5.54–5.56].

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Fig. 5.14 Schematical generation of (a) CBED and (b) LACBED patterns.

The effect of zero-loss ﬁltering on CBED patterns is shown in Fig. 5.16b for the (000) and (220) discs of an excited systematic row of reﬂections in an Si foil. Both sets of discs contain parallel and complementary dark and bright pendell¨osung fringes and line-scans through the discs correspond to rocking curves. Without energy ﬁltering, the inelastic scattering background in Fig. 5.16a obscures most of the information. Figure 5.16c shows line-scans from the (220) discs in Fig. 5.16a and b. Energy ﬁltering is therefore indispensable for quantitative evaluation and comparison with theory. Further important applications of CBED are the study of crystal symmetries and the determination of point groups and space groups [5.57,5.58]. These applications are based on a qualitative interpretation of symmetries and dynamical extinctions in the CBED patterns and do not beneﬁt as much from energy ﬁltering as the applications above. In the standard CBED method (Fig. 5.14a), the maximum convergence angle αc of the incident beam corresponds to twice the Bragg angle θB of the reﬂection nearest to the (000) beam. Otherwise, the overlap of the discs obscures the information within them. This is a severe limitation in many applications, in particular for crystals with large unit cells, and hence closely spaced reciprocal lattice points. The ﬁrst technique that was proposed to overcome this problem was the beamrocking method [5.59,5.60]. In standard CBED, the whole pattern is obtained in parallel acquisition, whereas in the beam-rocking method it is acquired sequentially. The beam tilt coils are used to change the incident beam direction of a parallel beam in small steps across the angular range that is to be investigated. Below the specimen, a spot diffraction pattern is formed, which shifts as the incident beam direction changes. In order to select a particular beam in the diffraction pattern, the shift has to be compensated by means of a second set of deﬂection coils below the specimen.

CONVERGENT-BEAM DIFFRACTION METHODS

143

Fig. 5.15 Typical zero-loss ﬁltered CBED pattern of Si along the 111 zone axis (courtesy of J. Mayer).

The intensity in one beam can then be displayed on a monitor or recorded digitally as a function of the incident beam direction. In this method, energy ﬁltering and digital recording can be achieved very easily by passing the selected beam through a serial EELS spectrometer [5.61]. Ways of preventing overlap of the discs, while using large convergence angles in an incident cone of illumination, have been developed by Tanaka and co-workers [5.60,5.62] and are commonly referred to as “large angle CBED” (LACBED) or Tanaka patterns. The principle of this method is illustrated in Fig. 5.14b. An electron probe with a large convergence angle is focused onto the specimen and an image of this spot can be observed in the image mode. The specimen is now lifted by an amount

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Fig. 5.16 (a) Unﬁltered CBED pattern of Si in a two-beam orientation, (b) zero-loss ﬁltered, (c) line proﬁles across the (220) discs of (a) and (b) (courtesy of J. Mayer).

∆z, whereupon the single spot splits into a number of spots. The arrangement of these spots corresponds exactly to the reﬂections of a diffraction pattern, even though the spots are observed in the image mode! The magniﬁcation of this spot “diffraction” pattern increases as the specimen shift ∆z is increased. In other words, this pattern observed in the image plane in LACBED is formed by the tips of all the cones produced by Bragg diffraction, which is not the case in the standard CBED technique where the tips of all the cones coincide (Fig. 5.14a). In the LACBED technique, overlapping of the discs in the ﬁnal pattern is prevented by selecting one of the cones with an aperture, as illustrated in Fig. 5.14b. This can be done in any plane conjugate to the object plane at ∆z = 0. An obvious choice is the ﬁrst intermediate image plane, where the selected-area diaphragm is located. The centring of the diaphragm can be observed in the image mode. In a ﬁnal step, the microscope is switched to the diffraction mode, where the LACBED disc formed by the selected beam is observed. As an example, Fig. 5.17 shows an energy ﬁltered LACBED pattern of the (000) beam of a TiAl crystal in 110 zone axis orientation. A certain disadvantage of the LACBED technique is that by lifting the specimen within the cone of illumination, an increasingly large area of the specimen is illuminated (up to several hundred nm). One incident beam direction in Fig. 5.14b corresponds to one point in the specimen and produces a corresponding point in the LACBED disc. In the LACBED pattern, diffraction information is thus mixed with image information. This results in distortions and symmetry breaking in the pattern when the specimen is bent and/or inhomogeneous in thickness. On the other hand, this special feature of LACBED can be used to image lattice distortions, such as strain distributions, or to analyze dislocations, stacking faults and multilayers within the illuminated area [5.57].

CONVERGENT-BEAM DIFFRACTION METHODS

145

Fig. 5.17 A zero-loss ﬁltered LACBED pattern from TiAl along the 110 zone axis (courtesy of J. Mayer).

The required separation of the spots in the image plane depends on the diameter of the selected-area diaphragm that is used to select an individual spot. For smaller diaphragms the required shift ∆z is smaller and hence the illuminated areas are also smaller. This reduces the effect of thickness variations and specimen bending. Ideally a 5-µm diaphragm is used. If a very small selected-area diaphragm is used and the changes in specimen height are large, only a very narrow angular range of scattered electrons is allowed to pass through the aperture and to contribute to the LACBED pattern. Electrons inelastically scattered through angles larger than the selected aperture will also be intercepted by the diaphragm and can thus be removed from the pattern. The LACBED technique hence provides a kind of energy ﬁltering by the aperture being used [5.63]. Owing to the small changes of scattering vector during plasmon losses, large changes in specimen height or lens current would be necessary to provide a substantial amount of energy ﬁltering for plasmon loss electrons. The latter are therefore best removed by using an energy ﬁlter, even in LACBED. On the other hand, thermal diffuse

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scattering may lead to large scattering angles. It can be removed very efﬁciently in LACBED even for small changes in specimen height or lens current (while it cannot be removed by an imaging energy ﬁlter). Zero-loss ﬁltered LACBED thus provides information about the specimen free of inelastically scattered electrons and almost free of thermal-diffuse scattering. 5.6.1

Determination of Low-Order Structure Factors and the Charge Density Distribution

Most of the classical methods for structure-factor measurement by electron diffraction are based on the results of three-beam dynamical theory and in particular on the occurrence of degeneracies. Examples are the analysis of three-phase structure invariants [5.64], the critical voltage method [5.65] and the analysis of degeneracies in three-beam cases for both centrosymmetric crystals [5.66] and non-centrosymmetric crystals [5.67,5.68]. None of the techniques mentioned above requires the use of energy-ﬁltered diffraction patterns since they are all based on qualitative interpretation of the variations of the intensities. Novel techniques base on a quantitative analysis of the intensity distributions in CBED patterns by a comparison of experimental and simulated CBED patterns. The intensities in the diffraction patterns can be obtained by solving the eigenwert problem of the dynamical theory of electron diffraction and depend on specimen thickness, specimen tilt, accelerating voltage and the Fourier coefﬁcients Vg of the lattice potential, which are proportional to the structure factors The Fourier coefﬁcients of the lattice potential are composed of real parts Vg describing the elastic scattering and imaginary parts Vg describing the absorption by thermal-diffuse and inelastic scattering. For non-centrosymmetric crystals, both the elastic and absorptive part of the Fourier coefﬁcients are complex quantities:

|Vg | exp(iφg ) + i|Vg | exp(iφg ) · exp(−2π g · r) V (r) = (5.7) g

All these parameters can be determined from a single CBED pattern. However, there is no way of inverting the experimentally observed intensity distribution directly to obtain values of these parameters. Instead, simulations have to be made with a range of values of the parameters until a best ﬁt is obtained. The possibility to determine the Fourier coefﬁcient Vg and the phase φg with an accuracy of one degree allows to determine not only the content of an unit cell, like in conventional electron diffraction, but also the charge density distribution of the atomic electrons in chemical bonds. Previously this was only a domain for X-ray diffraction. The progress in simulation methods for CBED patterns now reached an accuracy that this technique is comparable with X-ray diffraction and shows the important advantage that it can be applid to very small crystal volumes. For measurement of the low-order structure factors, different methods have been proposed. They differ in the geometry of the CBED patterns, that is, whether the

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147

crystal is tilted to a zone axis [5.69], a systematic orientation [5.70], or a many-beam case with several strongly excited high-index reﬂections [5.71]. For the calculation of the intensities by the Bloch-wave formalism of dynamical theory, an accurate calculation of the intensities, a large number of reﬂections of the ZOLZ and HOLZ have to be included in the calculation. More than a hundred reﬂections can be at least weakly excited. About 30 reﬂections of the ZOLZ plus a few relatively strong reﬂections of the HOLZ are sufﬁcient to obtain accurate results. Strongly excited reﬂections are included in the matrix to be diagonalized; weakly excited reﬂections are treated by perturbation theory, which reduces the calculation time [5.70]. The numerical simulation has to be repeated for each new set of parameters and is hence very time-consuming. Even with very fast computers, the time required to simulate the whole pattern repeatedly for the whole parameter range would be much too high and calculations are restricted to one or two line proﬁles through the CBED discs. The reﬁnement of the calculated line proﬁles with respect to the experimental data is performed with the aid of the χ2 test; in statistical theory, χ2 is deﬁned as χ2 =

n

wi (cf theo − f exp )2 i

i=1

i

σi 2

(5.8)

The parameters fiexp and fitheo are the experimental and theoretically calculated intensity values of the point i within the linescan, c is a normalization constant. The sum runs over all points i within the line-scan. The difference between the experimental and calculated value can be weighted by a factor wi . The quantity σi is the standard deviation if each of the experimental values fiexp is obtained by averaging over several measurements. If only one measurement is made and Poisson statistics assumed for the experimental values, then σi2 is set equal to fiexp . For the minimization of the function χ2 , reﬁnement algorithms are used [5.70,5.72]. In the reﬁnement the whole set or a subset of the parameters described above is varied until a minimum of the function χ2 is found. The reﬁnement algorithms can be distinguished by whether they are able to escape from local minima and to ﬁnd the global minimum (global reﬁnement) or whether they are liable to get trapped in local minima (local reﬁnement). In general, a quadratic reﬁnement problem with n parameters may exhibit up to 2n local minima. In the structure-factor determination, the density of local minima increases with increasing specimen thickness since the number of oscillations in the intensity data increases. On the other hand, the accuracy in the structure-factor determination also increases with the number of intensity oscillations. For the given experimental ranges, several local minima frequently exist. Details of the technique are described in [5.56] and experimental results have been published on a number of low-index reﬂections in GaAs [5.73], MgO [5.70], BeO [5.74], NiAl and FeAl [5.75]. for example. Bonding charge densities maps of the latter constructed from the Fourier coefﬁcients 100, 110, 111, and 200 are shown in Fig. 5.18. The low-index reﬂections carry the information on the bonding within the crystal. This information can be extracted from the structure factors by calculating the charge

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Fig. 5.18 Bonding charge density of NiAl and FeAl constructed from the Fourier coefﬁcients 100, 110, 111, and 200. The diagrams show (110) sections through the unit cells. Contour line spacing is 0.025 electrons/˚A3 . The black solid lines indicate zero electron density, the white lines positive and the black lines negative electron density, compared to a reference crystal with neutral spherical atoms (courtesy of J. Mayer).

density distribution. The charge density distribution ρ is related to the electrostatic potential via Poisson’s equation: ∇2 V (r) = −

e (ρn − ρe ) ε0

(5.9)

The charge density can be separated into the contributions from the nucleus, ρn , and the contribution from the electrons, ρe . The former, ρn , can be expressed as a point charge of magnitude Zi at the lattice position ri of the atom i. The potential V (r) can be calculated by summation over all Fourier coefﬁcients Vg as in (5.7). Of the total electronic charge density, the bonding part is of interest. The contribution from the inner shells can be assumed to be spherical and does not directly inﬂuence any of the properties of the crystal. The charges forming, for example, the covalent bonds in a crystal can be composed of charges as small as 10−4 of the total electronic charge density. This contribution will thus not be visible in plots of the total charge density within a unit cell. The problem can be solved by plotting the difference charge density ∆ρ(r) = ρexp (r) − ρat (r)

(5.10)

where ρexp (r) is the charge density of the real crystal, which has been measured experimentally, and ρat (r) is the charge density of a hypothetical crystal with neutral spherical atoms at the lattice sites. The difference charge density map shows areas of positive and negative charge density, which indicate accumulation and depletion of electronic charges due to the formation of covalent or ionic bonds. Covalent bonds can be identiﬁed in this map by an accumulation of charge density between two atoms. The strength of the bond can be estimated by integrating over the three-dimensional area with a surplus of electronic charges. The presence of ionic bonds can be inferred from a positive or negative charge balance at the atom positions.

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149

For the calculation of the charge density ρat (r), the X-ray scattering amplitudes tabulated in the International Tables for X-ray Crystallography [5.76] are used. In the ﬁrst Born approximation the atomic scattering amplitude for electrons is related to the atomic scattering amplitude for X-rays via the Mott formula: Vg =

Zi − f x (s) e i exp(−Bi s2 ) exp(−2πig · r) 16π 3 ε0 Ω i s2

(5.11)

The ﬁrst term of the sum describes the Rutherford scattering of the electrons at the nucleus, which depends on the charge Zi of the nucleus i. This term does not occur in the atomic scattering amplitude for X-rays because the X-rays are scattered only by the electrons. Starting from the X-ray scattering amplitudes, the Vg are the potential coefﬁcients for an arrangement of neutral spherical atoms on the lattice sites. The X-ray structure amplitude Fgx is obtained by summation over all the scattering amplitudes f x of the atoms in the unit cell:

fix (s) exp(−Bi s2 ) exp(−2πig · r) (5.12) Fgx = i

with s = sinθB /λ = |g|/2. The inﬂuence of the thermal vibrations of the atoms is taken into account by including the Debye–Waller factor exp(−Bi s2 ), in which Bi is the temperature factor of atom i. The inﬂuence of the Debye–Waller factor increases quadratically with the magnitude of the scattering vector s. The Debye–Waller factors can be calculated from the tabulated values in the International Tables for X-ray Crystallography or, if available, taken from X-ray or neutron diffraction. As discussed above, it is also possible to measure the Debye–Waller factors by applying the CBED technique to high-index reﬂections. The number of reﬂections that has to be taken into account in the calculation of the charge density difference map like Fig. 5.18 depends strongly on the temperature. At higher temperatures, the decreasing Debye–Waller factor reduces the inﬂuence of reﬂections with large scattering vectors. Furthermore, the difference maps also depend strongly on the structure factors calculated for the neutral reference atoms. It has therefore been suggested [5.56] that all the research groups should use the same standards given in the International Tables. The structure factors, however, should not be calculated using the classical method of Doyle and Turner [5.77] but rather with the more accurate methods given in [5.78] or [5.79], especially for the high-index reﬂections and their absorption. The calculation of charge density maps requires that the crystallography of the phases being studied be known. Changes of the composition by doping, for example, are only possible if they do not lead to a change of the atom positions or to a distortion of the unit cell. If this is the case, high-index reﬂections have to be measured as well and in a ﬁrst step accurate atom positions have to be found. Since the determination of atom positions and of the charge density distribution are expected to become standard techniques in the near future, electron crystallography will be an important tool in many ﬁelds of materials science.

150

5.7

5.7.1

ENERGY FILTERED DIFFRACTION

IMAGING OF THE DISPERSION OF PLASMON AND COMPTON SCATTERING Compton Scattering

As shown in Sect. 3, scattering of electrons at quasi-free atomic electrons results in the Bethe ridge or Compton peak (Figs. 5.2 and 5.3) with a maximum at the Compton angle θC (5.1). The Bethe ridge can be recorded by different sections in the schematic diagram of Fig. 5.2. A quantitative record of the intensity across the ridge can be obtained either by a radial scan of the diffraction pattern at a ﬁxed energy loss E or by recording an EELS at a ﬁxed scattering angle of about 5◦ [5.19,5.25,5.80], which also contains the carbon K edge in Fig. 5.2. Another possibility of observation and quantitative analysis is the method of angle resolved EELS (Fig. 5.3) [5.25]. A vertical section in Fig. 5.2 at a constant energy loss E is an energy-dispersive diffraction (EDS) pattern and the Compton peak can be seen as a diffuse concentric ring with as diameter increasing with the square root of E (5.1) as shown in a series of EDS with increasing E for a graphite foil (Fig. 5.19).

Fig. 5.19 Series of electron spectroscopic diffraction patterns of a graphite foil with energy losses of (a) 0, (b) 200, (c) 400, and (d) 800 eV showing the diameters of the Compton maximum increasing as E 1/2 .

The analogous Compton scattering of X-rays [5.81,5.82] is in common use for testing calculations of atomic orbitals in solids and has also been applied to line-scans by electron diffraction [5.80]. The intensity proﬁle of the Bethe ridge is proportional to the projection of the momentum distribution of atomic electrons on the scattering direction (z): J(pz ) = n(px , py , pz )dpx dpy (5.13) where n(p) is the momentum probability density. The Fourier transform B(z) = J(pz ) exp(−ipz z/¯h)dpz

(5.14)

of a recorded intensity proﬁle is the autocorrelation function of the ground-state wave function. An advantage of electron Compton scattering is the shorter exposure time and the size of the selected area, a disadvantage is the need for thin specimens to

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151

avoid multiple scattering effects and the inﬂuence of Bragg reﬂections on the intensity proﬁle [5.83]. 5.7.2

Plasmon-Loss Scattering

Plasmon losses show a dispersion Epl (θ) = Epl (0) + 2eU αθ2

(5.15)

for angles smaller than the cutoff angle θc = λE/hvF

(5.16)

where the intensity of plasmon losses drops rapidly (vF = Fermi velocity). The agreement of the theoretical value of the constant 3 (5.17) EF /E 5 in (5.15) with experimental values is variable [5.84]. The dispersion and cutoff of plasmon losses can be imaged by angle resolved EELS (Fig. 5.3). Analog to Fig. 5.19, the dispersion can also be imaged in a series of ESD patterns with a narrow width ∆ 1 eV of the energy window, as demonstrated in Fig. 5.20 for a single-crystalline Sn ﬁlm with a plasmon loss at Epl (0) = 17 eV [5.48]. A disc of plasmon-loss scattered electrons can be observed around the primary beam and the Bragg reﬂections for losses near Epl (0) (Fig. 5.20a). The parabolic dispersion ∝ θ2 in (5.15) results in a diffuse ring increasing in diameter as the selected energy is raised by a few eV (Fig. 20b–d). On solving (5.15) for θ, the maximum of the ring is seen to be at a scattering angle α=

θ2 = [Epl (θ) − Epl (0)]/2αE0

(5.18)

Fig. 5.20 Series of electron spectroscopic diffraction patterns of a single-crystal evaporated Sn ﬁlm recorded with an energy window ∆ = 1 eV at (a) E = 17 eV, (b) 21, (c) 23, and (d) 28 eV showing a section through the dispersion surface of plasmon losses (scale bar = 10 mrad).

Beyond 28 eV the ring disappears due to cutoff. The measured value qc = θc /λ = 18 nm−1 agrees well with qc = 15 nm measured at 50 keV [5.85]. In diffraction

152

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patterns of a [111]-oriented Si foil, for example, the plasmon rings can also be observed around the {220}-type Bragg diffraction spots. This demonstrates that plasmon losses near the cutoff angle θc are also intraband-scattered, conserving the Bloch-wave ﬁeld. Plasmon losses and intraband transitions can show an anisotropy in anisotropic crystals when the dielectric tensor (ω, q) depends on the direction q of momentum transfer. When this tensor is transformed to its orthogonal main axes with diagonal components ii the intensity of the plasmon losses becomes proportional to Im{–1/ i ii qi2 }, where the qi are the components of q along the main axes [5.86]. The angle η between the momentum transfer hq and the primary electron direction is given by tanη = θ/θE

(5.19)

where θE = E/mv 2 . Thus, η approaches 90◦ for scattering angles θ θE . This anisotropy can be observed in ESD patterns as an azimuthal anisotropy of the angular distribution for scattering angles θ ≤ θc . Evidence for this anisotropy is shown in Fig. 5.21a and b by isodensities for the interband transition of π-electrons in graphite with the incident electron beam parallel to the c-axis [5.48]. The intensity contours of the plasmon scattering around the primary beam show hexagons with rounded corners directed towards the six {220} Bragg reﬂections at E = 7 eV (Fig. 5.21a) (ΓQ-direction of the Brillouin zone) but with the corners rotated azimuthally by 30◦ at E = 13 eV (Fig. 5.21b). In contrast to this, the intensity contours of the plasmon loss of graphite at E = 31 eV show an isotropic angular distribution (Fig. 5.21c). These results agree with direct measurement of the anisotropy of energy losses in graphite by sequential recording of EELS at different scattering angles and azimuths [5.87,5.88], whereas the contours in Fig. 5.20 form a parallel record of a two-dimensional map of anisotropy effects. Weaker anisotropies have also been observed with Si foils [5.48].

Fig. 5.21 Intensity contours (digital isodensities) in diffraction patterns of a graphite foil showing the azimuthal anisotropies for the interband transition at (a) E = 7 eV and (b) 13 eV and no anisotropy for (c) the volume plasmon loss at E = 31 eV (scale bar = 10 mrad)).

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5.43. G. Honjo, S. Kodera, and N. Kitamura: “Diffuse streak diffraction patterns from single crystals.” J. Phys. Soc. Jpn. 19, 351–367 (1964). 5.44. K. Komatsu and K. Teramoto: “Diffuse streak patterns from various crystals in x-ray and electron diffraction.” J. Phys. Soc. Jpn. 21, 1152–1159 (1966). 5.45. N. Kitamura: “Temperature dependence of diffuse streaks in single-crystal Si electron diffraction patterns.” J. Appl. Phys. 37, 2187–2188 (1966). 5.46. H. P. Herbst and G. Jeschke: “Diffuse streak-patterns from PbJ2 - and Bi-single crystals and their temperature dependence.” Electron Microscopy 1968, ed. by D. S. Bocciarelli (Tipograﬁa Poliglotta Vaticana, Rome 1968) Vol. 1, pp. 293–294. 5.47. E. M. H¨orl: “Thermisch-diffuse Elektronenstreuung in As-, Sb- und Bi-Kristallen.” Optik 27, 99–105 (1968). 5.48. I. Fromm, L. Reimer, and R. Rennekamp: “Investigation and use of plasmon losses in energy-ﬁltering TEM.” J. Micr. 166, 257–271 (1992). 5.49. J. Mayer and C. Deininger: “Omega energy ﬁltered convergent beam electron diffraction.” Electron Microscopy 1992, ed. by A. R´ios, J. M. Arias, L. Megias– Megias, A. L´opez-Gallindo (Secr. Publ. Universidad de Granada, Granada 1992) Vol. 1, pp. 181–182. 5.50. L. Reimer: “Electron diffraction methods in TEM, STEM and SEM.” Scanning 2, 3–19 (1979). 5.51. L. E. Thomas and C. J. Humphreys: “Kikuchi patterns in a high voltage electron microscope.” Phys. Status Solidi (a) 3, 599–615 (1970). 5.52. J. M. Cowley, D. J. Smith, and G. A. Sussex: “Application of a high voltage STEM.” Scanning Electron Microscopy 1970 (ITTRI, Chicago 1970) p. 11–16. 5.53. Y. Nakai: “Excess and defect Kikuchi bands in electron diffraction patterns.” Acta Crystallogr. A 26, 459–460 (1970). 5.54. J. Gjønnes and R. Høier: “Multiple beam-dynamic effects in Kikuchi patterns from natural spinels.” Acta Crystallogr. A 25, 595–602 (1969). 5.55. B. F. Buxton: “Bloch waves and higher order Laue zone effects in high energy electron diffraction.” Proc. R. Soc. (London) A 350, 335–361 (1976). 5.56. J. C. H. Spence and J. M. Zuo: Electron Microdiffraction (Plenum Press, New York 1992). 5.57. D. J. Eaglesham: “Applications of convergent beam electron diffraction in materials science.” J. Electr. Microsc. Techn. 13, 66–75 (1989). 5.58. J. W. Steeds and E. Carlino: “Electron crystallography.” In Electron Microscopy in Materials Science, ed. by. P. E. Merli and V. Antisari (World Scientiﬁc, Singapore 1992) pp. 279–313.

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5.74. J. M. Zuo, J. C. H. Spence, J. Downs, and J. Mayer: “Measurement of individual structure-factor phases with tenth-degree accuracy: The 00.2 reﬂection in BeO studied by electron and x-ray diffraction.” Acta Crystallogr. A 49, 422–429 (1993). 5.75. W. N¨uchter, A. L. Weickenmeier, and H. Mayer: “Determination of bonding charge density in NiAl by quantitative convergent beam electron diffraction.” Phys. Status Solidi (a) 166, 367 (1998). 5.76. C. H. Macgillavry and G. D. Riek (eds. ): International Tables for X-ray Crystallography (Kynoch, Birmingham 1968) Vol. III. 5.77. P. A. Doyle and P. S. Turner: “Relativistic Hartree–Fock x-ray and electron scattering factors.” Acta Crystallogr. A 24, 390–397 (1968). 5.78. A. Weickenmeier and H. Kohl: “Computation of absorptive form factors for high-energy electron diffraction.” Acta Crystallogr. A 47, 590–597 (1991). 5.79. D. M. Bird and Q. A. King: “Absorptive form factors for high-energy electron diffraction.” Acta Crystallogr. A 46, 202–208 (1990). 5.80. B. G. Williams, T. G. Sparrow, R. F. Egerton: “Electron Compton scattering from solids.” Proc. R. Soc. (London) A 393, 409–422 (1984). 5.81. B. G. Williams: Compton Scattering. The Investigation of Electron Momentum Distribution (McGraw-Hill, London 1977). 5.82. M. J. Cooper: “Compton scattering and electron momentum determination.” Rep. Prog. Phys. 48, 415–481 (1985). 5.83. P. Jonas, P. Schattschneider, and P. Pongratz: “Removal of Bragg–Compton channel coupling in electron Compton scattering.” Electron Microscopy 1990, ed. by L. D. Peachey, D. B. Williams (San Francisco Press, San Francisco 1990) Vol. 2, pp. 24–25. 5.84. H. Raether: Excitations of Plasmons and Interband Transitions by Electrons (Springer, Berlin, Heidelberg 1980). 5.85. P. Schm¨user: “Anregung von Volumen- und Oberﬂ¨achenplasmaschwingungen in Al und Mg durch mittelschnelle Elektronen.” Z. Phys. 180, 105–126 (1964). 5.86. J. Daniels, C. von Festenberg and H. Raether: “Optical constants of solids by electron spectroscopy.” In Springer Tracts in Modern Physics, ed. by G. H¨ohler (Springer, Berlin, Heidelberg 1970) Vol. 54, pp. 77–135. 5.87. K. Zeppenfeld: “Anisotropie der Plasmaschwingungen in Graphit.” Z. Phys. 211, 391–399 (1968). 5.88. K. Zeppenfeld: “Nichtsenkrechte Interband¨uberg¨ange in Graphit durch unelastische Elektronenstreuung.” Z. Phys. 243, 229–243 (1971).

6

Elemental Mapping Using Energy Filtered Imaging Ferdinand Hofer and Peter Warbichler Research Institute for Electron Microscopy Graz University of Technology, Steyrergasse 17, A-8010 Graz, Austria

Abstract Equipping a transmission electron microscope (TEM) with an energy ﬁlter offers extraordinary advantages for sample characterization for both materials science and biology. Besides improvements to TEM imaging and electron diffraction such as improved contrast and resolution, elemental mapping using inner-shell ionizations has become the main application of EFTEM. In this chapter, we discuss the principles of EFTEM elemental mapping, procedures for data acquisition, extraction of quantitative information, and some application examples mainly from materials science. 6.1 INTRODUCTION Energy ﬁltering transmission electron microscopy (EFTEM) also known as electron spectroscopic imaging (ESI) has emerged as a powerful tool for materials analysis. EFTEM explores the rich information provided by electron energy loss spectroscopy (EELS) in a spatially resolved manner [6.1]. This is best illustrated with the 3D information cube, where the ﬁrst two axes correspond to the x–y position on the specimen as with any image, and the third axis contains the energy-loss information (Fig. 6.1). There are two fundamentally different approaches for accessing the information cube: In the ﬁrst case a ﬁxed beam TEM is combined with an energy ﬁlter with which we can acquire entire x–y planes at once. This procedure can be repeated at a number of separate energy loss settings [6.2]. The alternative approach uses a focused probe in a scanning TEM (STEM) that has to be equipped with an EELS spectrometer to record sequentially EELS spectra for each point on the specimen; the entire dataset is known as a spectrum image [6.3–6.5].

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Fig. 6.1 The three dimensional (3D) data space that represents the intensity distribution at the bottom surface of the specimen; with schematic EELS spectrum [6.5].

EFTEM has become well established only during the last few years, due largely to the availability of high performance commercial energy ﬁlters. A TEM equipped with an energy ﬁlter offers extraordinary advantages for sample characterization for both materials science and biology [6.2]: The ability to select any spectral feature from an EELS spectrum opens up a variety of new imaging techniques: Zero loss imaging to improve contrast and resolution of TEM images and electron diffraction patterns by removing inelastic scattering; energy ﬁltered TEM imaging for “tuning” the contrast of heterogeneous specimens; relative thickness mapping to image a specimen’s thickness in units of mean free path; and chemical mapping to enable the imaging of the specimen’s local electronic structure. However, the most obvious advantage of EFTEM imaging is the measurement of compositional information through the use of element speciﬁc ionization edges in the form of elemental maps. In this Chapter, we will focus on ﬁxed beam EFTEM and its speciﬁc use in addressing questions regarding the micro- and nanochemistry of various materials. Typical examples taken both from our laboratory and other published work are used to illustrate the quality of data that are currently attainable. Additionally, we intend for this chapter to provide a literature survey including most materials science applications of EFTEM. 6.2

EFTEM INSTRUMENTATION

The ﬁrst experimental energy ﬁltering microscopes were built in the 1960s by Castaing and Henry [6.6,6.7] and were later modiﬁed by Ottensmeyer [6.8]. These

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microscopes incorporated a ﬁltering spectrometer consisting of two 90◦ magnetic sectors and an electrostatic mirror, but they had a maximum operating voltage of only 80 kV [6.9], which severely restricted their range of applications especially in materials science. However, with work of Senoussi [6.10] and Zanchi [6.11] this limitation was overcome by the development of in-column ﬁlters, with a spectrometer that consisted of four magnetic prisms arranged in the shape of a Greek Omega and located inside the projector lens system of the TEM [6.12,6.13] (Fig. 6.2, see also Fig 2.4). This type of ﬁlter can be used at accelerating voltages > 100 kV, and many of the principal optical aberrations can be eliminated resulting in excellent optical performance [6.14]. Several in-column microscopes are now commercially available; the LEO Ω-ﬁlter TEM can be operated with a maximum accelerating voltage of 120 kV [6.15] and more recently has been enabled to operate 200 kV [6.16]; and the JEOL Ω-ﬁlter TEM that can also operate at 200 kV [6.17]. In an alternative approach, post-column ﬁlters with a simpler, single prism geometry based on multipole lenses, bend the electron beam by 90◦ and are located after the main column of the TEM [6.18]. The only commercially available post-column instrument is the Gatan Imaging Filter (GIF), which can be attached and retroﬁtted to practically any conventional TEM and can also be corrected for optical aberrations [6.19–6.21] as shown in Fig. 6.2 and also Figs. 2.1 and 2.2.

Fig. 6.2 Different types of energy ﬁlters in TEM instruments: (a) magnetic sector attachment (Gatan Imaging Filter), (b) Omega-ﬁlter (LEO 912 W-Filter), (courtesy G. Kothleitner, Gatan).

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6.2.1

Operation of the Energy Filtering TEM

Energy ﬁltering TEMs ﬁrst form an unﬁltered image (or diffraction pattern), whereupon the magnetic prism transforms the image into an electron energy loss spectrum, within which a speciﬁc part of the spectrum is selected by an energy-selecting slit. Finally, the selected electrons are transformed back into an energy ﬁltered image (or diffraction pattern) by means of an electron optical transfer system and the image is recorded with a two-dimensional (2D) recording device. Both in-column and post-column ﬁlter types are operated in a similar manner: In order to form an energy ﬁltered image, the EELS spectrum is shifted relative to the energy-selecting slit that is positioned after the prism but before the ﬁnal lens system. The energy window ∆ can be varied from 0 to 60 eV by adjusting the width of the energy-selecting slit. The zero-loss peak of the EELS spectrum is adjusted on the optic axis and the energy shift is made by increasing the acceleration voltage of the microscope by +∆E in order to keep the energy-loss electrons of interest (∆E) on the optic axis. An unﬁltered image can be obtained by simply withdrawing the energy-selecting slit. Because the images are formed in parallel, elemental maps can be acquired in seconds to tens of seconds rather than tens of minutes to hours as is required by the pixel by pixel STEM approach. Alternatively, the strengths of the lenses may be adjusted to project the EELS spectrum onto the detector with variable dispersion, in a manner similar to conventional parallel EELS spectrometry. Images and spectra are digitally recorded by employing a TV camera or a CCD array. While the TV camera has to be adapted for quantitative measurements by introducing special ﬁlters [6.22,6.23], the CCD camera gives a practically linear response over a wide dynamic range and offers high sensitivity [6.24]. 6.2.2

Alignments and Corrections for Energy Filtering

Several speciﬁc microscope and energy ﬁlter alignments must be performed before EFTEM work. This will ensure that images are acquired with optimal resolution and may be used subsequently for quantitative analysis [6.25]. After acquisition, certain corrections must be applied to the images before they can be used in elemental analysis to ensure detector effects are removed effectively. In addition to the normal microscope alignment procedure for conventional TEM imaging, it is necessary to align the energy ﬁlter and to adjust the condenser and objective lens focus. At the start of each EFTEM session, some alignments on the ﬁlter should be performed to ensure optimum performance. The most important of these is the alignment of the ﬁrst- and second-order spectrum focus. The intensity distribution of an alignment ﬁgure, obtained by using a narrow energy-selecting slit, relates directly to aberrations to be corrected, allowing easy alignment. When correctly aligned, the entrance aperture used for imaging should be illuminated uniformly. Additionally, depending on the stability of the microscope it is necessary to reposition the zero-loss peak in the centre of the energy-selecting slit. An essential requirement for reliable quantitative elemental mapping is for the specimen illumination to remain constant as a function of energy loss. Since the

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energy loss is selected by changing the microscope acceleration voltage, this has the undesired effect of changing the illumination conditions with varying offset, resulting in a change of the beam convergence angle and consequently a change in specimen illumination area. This can be corrected by a condenser adjustment, wherein the strength of the condenser lens is adjusted as a function of energy loss, allowing the illuminated area and hence the current density to be kept constant. Although the energy-selecting slit always images electrons of approximately the same kinetic energy and focus, it may be necessary to readjust the objective lens strength at high energy losses to obtain the optimal focal setting. For example, if a different focus has been selected for the zero-loss image (e.g., Scherzer focus), then adjustment of the objective lens strength will be necessary for high energy loss imaging [6.26]. In practice, it is often sufﬁcient to adjust the focus directly in an energy-ﬁltered image, obtained at ∼ 100-eV energy loss, before the elemental maps are recorded. Before images read from the CCD detector can be used in compositional analysis, a number of corrections must ﬁrst be applied to remove detector artifacts and noise [6.24]. The dark current background must be removed by subtraction of a dark reference image and variations in detector response are eliminated by gain normalization which is achieved by dividing the ﬁnal image by a gain reference image. Sometimes, it is also necessary to correct the images for “blurring” within the CCD detector that occurs mainly as a result of multiple scattering of photons within the scintillator. This blurring leads to degradation of resolution which can be described by a point-spread function [6.27]. As a result of instabilities of the microscope or specimen, sample drift may occur during exposures. These artefacts can be corrected using cross-correlations, which occasionally, do not work well with contrast reversals in core-loss imaging. As an alternative, interactively minimizing the variance of ratio images works reliably and sensitively. Generally, specimen drift problems increase during longer exposure times. Additionally, the selected energy loss may change during long exposures due to drift of the high voltage or of the spectrometer prism current. Instabilities of these types can often be compensated by subdividing longer exposures into shorter ones, and correcting for the instability by post acquisition processing before summing the partial results [6.28]. Since the images in these sub-series are identical, alignment by cross-correlation is reliable. 6.2.3

Some Remarks about the Specimen

Large area mapping by EFTEM imposes stringent requirements on sample preparation. In order to extract reliable quantitative information, specimen thickness must be smaller than the mean free path of inelastic scattering (λ) which is ∼ 160 nm in biological sections and ∼ 90 nm in silicon foils for 200-keV electrons. Suitable techniques providing large thin areas are low-angle ion milling [6.29,6.30], focused ion beam micromachining [6.31,6.32] and ultramicrotomy [6.33]. Although electropolishing is often used for metals, it is less suited for EFTEM investigations if the metal contains other phases that are less easily thinned than the matrix. The thick-

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ness differences between matrix and precipitates can become too large for successful application of the EFTEM technique. In the case of insulating materials (many ceramics, catalysts, and catalyst supports) specimen charging presents many problems such as beam and image movement, which can be signiﬁcantly reduced by coating the specimen with a thin layer of carbon (e.g., 3–5 nm). Radiation damage is one of the most stringent limitations of energy ﬁltering TEM, because improving the noise in inelastic images requires large electron doses. During the sequential recording of these images the specimen can deteriorate between the ﬁrst and the last image of a series. Since inner-shell inelastic scattering is typically 102 – 106 times weaker than elastic scattering, the doses required for elemental mapping can be as high as 105 C/cm2 . Only a limited range of materials can withstand such a dose without undergoing changes. The inelastic scattering processes result in electron excitations that can destroy chemical bonds in inorganic and organic materials. As a consequence, atoms may be displaced from the irradiated area. This process is known as mass loss and it is of primary concern for elemental mapping, because some elements are displaced more rapidly than others, resulting in a change in chemical composition [6.34]. Radiation artifacts can be reduced by lowering the specimen temperature and ∼ 2–5 times higher doses can be applied at 77 K (liquid nitrogen cooling) [6.35,6.36]. However, the damage to the specimen is not changed when cooled, only the mobility of the fragments. Mass loss is thus reduced during the exposures but not eliminated. When the specimen returns to room temperature, gaseous atoms are released from the irradiated area [6.37]. Inorganic specimens like alloys or ceramics are normally more resistant than organic materials, which are irretrievably damaged by doses as low as 10−3 –10−1 C/cm2 . However, even ceramics can be damaged during EFTEM elemental mapping, because the doses required for elemental mapping with high energy-loss edges, for example, the Al or Si K-edges have to be very high. Under such conditions many materials tend to show blurring of their structural features due probably to knock-on displacements and subsequent local atomic mixing and sometimes even partial melting (e.g., the glass phase in a ceramic). Compared to STEM spectrum imaging, the total electron dose of the specimen is larger in EFTEM, but the current density of the probe is considerably smaller. Consequently, dose-rate dependent damage experienced in STEM such as movement of segregants and hole drilling can be reduced by using EFTEM [6.38]. 6.3

ENERGY FILTERED IMAGING

Since any feature of the EELS spectrum can be used for energy ﬁltering (see Fig. 6.1), a full understanding of the EELS spectrum is a prerequisite for the successful application of energy ﬁltering TEM. By allowing only zero-loss electrons through the energy-selecting slit (usually 5–10-eV wide) a zero-loss image or elastic image is formed that exhibits higher contrast and better resolution compared to an unﬁltered TEM image [6.8,6.39,6.40]. Zero-loss ﬁltering is important for quantitative high-resolution imaging, because it allows a better comparison with simulated images to be made [6.41,6.42].

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The low-loss region has particularly great potential for imaging of chemical phases or even visualization of variations in the local electronic structure. However, in metallurgical specimens identiﬁcation of particular phases is limited to systems that have reasonably sharp and well-separated plasmon-peaks or systems where a shift in plasmon energy occurs; for example, Be precipitates in Al [6.43], He bubbles in Al alloys [6.44], Ag precipitates in GaAs [6.1], AlLi3 precipitates in Al [6.45,6.46], and crystals of In and Sn on a carbon ﬁlm [6.47]. Plasmons can be useful for obtaining chemical contrast in very thick samples (t/λ > 1) even when the core edges are too weak for elemental mapping, because the plasmons are the lowest energy electronic excitations and the ﬁrst plasmon peak is the last feature to be distorted by multiple scattering. In addition, other features of the low-loss region such as surface and interface plasmons have been visualized, for example, surface plasmons on Al spheres [6.48,6.49] or MgO cubes [6.2]. Since various bonding states of organic or inorganic compounds are visible in the low-loss region, spectroscopic ﬁngerprint information is, in principle, also available; for example, imaging of sp2 -hybridized carbon with the π ∗ peak at 7 eV in polymer systems [6.50]. An advantage of low-loss images is that they can be recorded with short acquisition times and a high signal-to-noise ratio (SNR). However, a disadvantage is that diffraction or mass-thickness effects appear in a manner similar to that of zero-loss or unﬁltered TEM-images, due mainly to the preservation of elastic scattering in inelastic images [6.2]. From biological EFTEM investigations [6.51] it is already well known that contrast tuning can be used to enhance the contrast of thin, unstained sections. The reason is that EELS spectra from different parts of a specimen can intersect several times. This is due to differences in the decrease of the background intensity that accompanies increasing energy loss or overlapping ionization edges [6.40]. Consequently, local contrast changes and even contrast reversals are introduced by tuning the selected energy over a certain range of energy loss. Since this technique provides a rapid overview of the specimen, it can be very useful for imaging secondary phases in materials, for example, for visualization of different phases in a semiconductor device [6.52]; as shown in Fig. 6.3. Similarly to EELS, energy ﬁltering TEM can be used for measuring specimen thickness variations by dividing an unﬁltered TEM image by a zero-loss image and taking the logarithm of this image [6.53]. t/λ-values are obtained that can be converted to absolute thickness using the appropriate inelastic mean-free-path length λ [6.1]. In the case of heterogeneous specimens however, the t/λ-map cannot be converted simply to a thickness map, because the mean free paths λ vary between different phases. 6.4

ELEMENT DISTRIBUTION IMAGES

One of the most commonly used applications of EFTEM is to derive compositional information by recording energy ﬁltered images using element speciﬁc ionization edges. In general, EFTEM enables the mapping of the elements ranging from Li to

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Fig. 6.3 Contrast tuning for the visualization of phases in a semiconductor device; (a) unﬁltered TEM image; (b) energy ﬁltered image (50 eV).

U, because these elements give rise to inner-shell ionization edges in the energy loss region commonly accessible to energy ﬁltering TEMs (0–3000 eV) [6.54]. 6.4.1

Background Subtraction

Ionization edges are always superimposed on a background due to other energy losses, and extracting elemental information for elemental mapping ﬁrst necessitates their separation from the background. The large background signal present at core-loss energies, requires a careful choice of the background removal procedure in order to obtain reliable elemental maps that are free from artifacts and systematic errors. The commonly used approach to calculate the background is to estimate the background contribution under the edge from the preedge area. By making an assumption about the functional behavior of the background just below the ionization edge of interest, the background can ﬁrst be calculated and subsequently extrapolated to higher energies to subtract it from the core-loss region. In practice, the most usual way for background estimation in EFTEM elemental mapping is the three window technique [6.55], where the background is calculated from two pre-edge images in front of the ionization edge assuming a power law dependence, that is, a model of the form I = A · E −r ; where I is the intensity, E the energy loss, and A and r are two ﬁtting parameters (see Figs. 6.1 and 6.4) [6.1]. The beneﬁt of this technique is that it allows for variation in the shape of the background. Various other background models were proposed and checked, but in most cases the best ﬁtting results were obtained with the power law model [6.56]. Since the background may vary across the specimen due to changes in composition and thickness, the background must be subtracted for every single pixel of a core-loss image. Finally, it should be mentioned that the inverse power law only holds for relatively thin specimens, because plural

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scattering processes that are more probable in thicker specimens (t/λ > 1) lead to deviations of the background shape from the inverse power law [6.1].

Fig. 6.4 Background subtraction for EFTEM elemental mapping; example siliconnitride ceramics; EELS spectrum showing the N K -edge and pre- and post edge images; nitrogen K map calculated with A·E −r background function (three window method); nitrogen K jump-ratio image calculated by dividing post-edge and pre-edge 2 images (with histograms).

Several methods have been proposed to test the validity of elemental maps [6.57]. Not only should the presence of elemental quantities be demonstrated, but also the absence of detection in situations where no element is present. Bonnet et al. [6.56] showed the power of creating “ghost” images by applying a background correction and elemental detection on a data set known to contain no elemental contribution. Colliex et al. mention the possibility of evaluating a chi-square measure at each image point after the ﬁtting procedure to test the goodness-of-ﬁt of the power-law function to the data. In pixels, where no element is present, the grey level after background correction should be zero on average [6.58]. The three-window power law method leads to systematic errors for low-energy core-edges. Haking et al. described a generalized difference method, where the background under the core-edge can be described by one or two pre-edge windows

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as a third-order polynomial [6.59]. This function can be deduced from specimen areas that are known to not contain the element of interest, or from a second specimen used as a standard. The method was used for reliable background subtraction below the Fe M2,3 -edge in biological samples [6.59]. The disadvantage of the three window method is that the extrapolated background will generally be noisier than the measured images because the power-law ﬁt ampliﬁes statistical noise. Therefore, it has been proposed to take a map of the ﬁtted power-law exponent, r, then pass the map through a smoothing ﬁlter to eliminate much of the statistical noise and then substitute it back into the computation of the extrapolated background [6.60]. Another way of eliminating or suppressing the background and enhancing the signal over the noise has been proposed by simply dividing the signal by a background image, taken in front of the edge [6.61]. These so-called jump-ratio images offer the advantage of being less noisy and less susceptible to diffraction effects that occur in the elemental mapping of crystalline samples [6.61], see Fig. 6.4. Jump ratio images are, however, not quantitative, but instead only qualitatively indicative of elemental distributions. Furthermore, jump ratio images are particularly susceptible to artefacts as a result of their sensitivity to changes in the preceding background arising from thickness changes of the sample or from preceding ionization edges. Therefore, it is widely recommended to always acquire elemental maps by using the three window method and to calculate jump ratio images afterward. For elemental mapping with overlapping or closely spaced ionization edges, a four window method has been proposed and successfully used to map chromium with the L2,3 -edge in oxidized steel specimens [6.62]. Other methods for background subtraction have been proposed for special situations, for example, simple subtraction of pre- and post-edge images or the white line method [6.63,6.64], but these methods offer no advantage compared to the three window technique. More detailed image processing of elemental maps is often necessary, because, in addition to elemental contrast, contrast can also arise from mass thickness variations and diffraction effects (see also Section 6.4.6). Particularly, diffraction effects occurring in crystalline materials cause severe problems for reliable elemental mapping. However, by recording jump ratio images under rocking beam illumination, diffraction effects can be eliminated almost completely [6.65]. Because of the rocking of the electron beam the local diffraction conditions vary causing the bend contours to be locally displaced. Registration of the integrated images generated by the rocking process results in an image in which the strong diffraction contrast is smoothed out (Fig. 6.5). Another possibility to reduce diffraction effects is to record elemental maps under hollow cone illumination [6.66]. 6.4.2

Optimization of the SNR

In energy ﬁltered elemental mapping the achievable detection limit and spatial resolution are determined largely by the noise level in the recorded images. The small cross-sections for inner-shell excitations providing compositional information, along with the high background, result in a poor SNR, constituting a major practical prob-

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Fig. 6.5 Elimination of diffraction contrast in EFTEM elemental mapping; (a) TEM image of a steel specimen; (b) vanadium L2,3 elemental map; (c) schematic drawing of rocking beam geometry; (d) vanadium L2,3 elemental map recorded under rocking beam illumination.

lem for EFTEM. It is therefore important to use imaging parameters which yield an optimum SNR. The SNR is represented by SNR = √

Ik Ik + h · Ib

where Ik and Ib are the integral intensities (counts) of the core-loss signal and the background respectively, within an energy-window ∆ deﬁned by the energy-selecting slit. h is a measure of the quality of the background extrapolation procedure, and

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is < 30 for an acceptable ﬁt [6.1]. Berger and Kohl have derived an analytical approximation of h that is valid for the three-window method of elemental mapping [6.26]. Settings on a microscope that would inﬂuence the SNR are the high voltage of the electron source, the illumination cone angle, the objective aperture, the defocus, the width and the positions of the energy windows, and the pixel size of the detector in relation to the magniﬁcation [6.26,6.67]. According to the published recommendations [6.26], the SNR can be essentially improved by using greater accelerating voltages amd increasing the current density on the specimen; for example, by increasing the beam current and by using full cone illumination (instead of axial illumination) with the largest possible condenser aperture available on the microscope (e.g., 200 µm for 200-kV electrons). Berger and Kohl have shown that the four most important instrumental parameters for providing the best working conditions, that is, the optimum objective aperture, defocus and width of energy window, strongly depend on the ionization edge energy [6.26]. The objective aperture is chosen to accept a reasonably large fraction of the inelastically scattered electrons; it must be larger than the characteristic scattering angle θE for the relevant inner-shell ionization. For example, in case of 200-kV microscopes practical settings for lower energy losses are 20–40-µm objective diaphragms and for higher energy losses > 1000 eV, a 100-µm objective diaphragm. By adjusting the window width to the respective edge type and energy loss, high SNR with widths of 5–10 eV below 100 eV, 20–30 eV up to 1000 eV, and 50 eV for higher energy losses can be obtained. However, one has to take into account that broader energy windows decrease spatial resolution in energy ﬁltered images due to chromatic aberrations. Therefore the choice of the experimental parameters often involves a compromise between high resolution or low detection limits. The optimum width and position for the background ﬁtting region can be determined as the region where the parameter h is at a minimum. A few simple criteria may be followed to achieve this: The high energy end of the ﬁtting region should be placed as close to the edge threshold as possible whilst avoiding the edge onset, since this region has the greatest inﬂuence on the accuracy of the extrapolation procedure [6.26]. In addition the width of the ﬁtting region should be as large as possible so as to minimize the statistical error [6.68]. In the case of the three-window method the parameter h can be also minimized by increasing the interval between the two pre-edge windows, considering of course the limitations of the power-law model [6.26]. The optimum position and width for the post edge window can be determined as the region where the SNR is at a maximum: Recently, it was shown that the position of the post-edge window has to be adjusted to the overall edge shape, which in turn depends on the ionization process (Fig. 6.6). For saw-tooth type edges (K-edges) and edges with white lines (L2,3 -edges of K–Cu and M4,5 edges of Cs–Yb) the post-edge window should be positioned at the edge threshold, but for delayed edges (e.g., M4,5 edges of Rb–Xe) the optimum window position should be shifted > 50 eV toward higher energy losses [6.67]; see also Fig. 6.6.

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Fig. 6.6 The SNR optimum for different edge shapes as a function of post-edge window position and window width; calculated from experimental EELS-spectra. The bright areas correspond to the highest SNR (scaled to 100%) according to the optimal values for the post-edge window as displayed in the spectra below; (a) carbon K -edge; (b) molybdenum M4,5 -edge [6.67].

Another important factor is the point spread function of the CCD camera and the size of the feature of interest in the sample, which has to be properly adapted to the pixel size of the detector [6.69]. The SNR is independent of the pixel size if the lateral extension of the characteristic signal is smaller than the pixel size (Fig. 6.7). In order to approach the detection limit, it is therefore essential to choose the magniﬁcation of the microscope in such a way that the lateral extension of the signal on the pixel array does not exceed the pixel size. This is the optimum situation for detectability of, for example, an element in a grain boundary. Otherwise, in the situation for high spatial resolution, the signal is spread across several pixels and the SNR drops dramatically with decreasing pixel size. This is illustrated in Fig. 6.7 showing clearly that the conditions for optimum detectability are not identical with those for optimum spatial resolution (see also [6.69]). One important consequence is that it does not make sense to work at high magniﬁcations (which gives a small pixel size), if the resolution is not in accordance with the features in the elemental maps.

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Fig. 6.7 Optimum distribution of the current density on the pixel array of the image recording system (a) for detecting an element and (b) for resolving a given structure; d denotes the size of the pixels; (courtesy J. Mayer [6.69]).

Finally, it should be mentioned, that as the signal-to-background ratio decreases with increasing specimen thickness, the SNR decreases after passing a maximum as well [6.1]. Therefore thin specimen regions (t/λ < 1) should be chosen if quantitative imaging and low detection limits are desirable [6.67]. 6.4.3

Spatial Resolution

Spatial resolution in EFTEM is limited by a number of contributing factors similar to that of detection limits; both depend strongly on optimum imaging parameters, specimen thickness, energy-loss region and ionization edge shape. According to Krivanek [6.28], a number of factors limit the spatial resolution [6.28,6.69–6.71] in EFTEM including, instrumental broadening originating from the electron optics, delocalization effects arising from the impact parameter of inelastic scattering of electrons in the specimen, statistical noise resulting from the low ionization cross-sections, radiation damage and instrumental instabilities and detector effects. In the case of STEM, the main limitations to spatial resolution are probe size and delocalization effects. The maximum resolution loss d can be calculated by summing the individual contributions due to delocalization, diffraction limit and chromatic and spherical aberrations in quadrature [6.28,6.72], d2 = R 2 +

0.6λ β

2

2 ∆ + Cc β + (2Cs β 3 )2 E0

where R is the inelastic delocalization limit, λ is the wavelength, ∆ is the width of the energy interval deﬁned by the energy-selecting slit, β is the maximum scattering angle admitted by the objective aperture (collection angle), Cc and Cs are the coefﬁcients of the chromatic and spherical aberrations of the objective lens. In an alternative approach, Egerton and Crozier [6.73] describe the effect of lens aberrations on image contrast and resolution in terms of a point-spread function (PSF). The PSF representing blurring in an EFTEM image provides a more quantitative

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measure which, instead of describing the overall diameter of the aberration disk as in the Krivanek approach, describes the image-intensity distribution that would result from a point source scattering electrons within a speciﬁed angular and energy distribution. Egerton and Crozier obtained the total theoretical spatial resolution by calculating the individual factors such as delocalization, diffraction limit, chromatic and spherical aberrations and summing in quadrature. The theoretical resolution plots in Fig. 6.8. show the effects of varying each parameter within the Krivanek [6.28] and Egerton [6.73] approach, respectively. For a thin specimen (t/λ < 0.5), the point resolution predicted by Egerton is ∼ 0.3 nm for β in the range 7–15 mrad, being limited mainly by delocalization and chromatic aberration. This resolution is much lower than the value calculated by the Krivanek approach.

Fig. 6.8 Spatial resolution plots for EFTEM imaging plotted as a function of objective aperture semi-angle b, calculated following the approach of Krivanek et al. (a) and of Egerton & Crozier (b); for a 200 kV high performance TEM with CS = 0.47 mm, CC = 1 mm, ∆ = 20 eV, ∆E = 500 eV (courtesy R. F. Egerton [6.73]).

Delocalization of inelastic scattering arises because a fast electron can pass a distance away from an atom and still ionize it. For low energy losses, delocalization is the primary resolution limitation [6.1] and according to theoretical considerations the resolution scales as 1/∆E, suggesting a resolution limit on the order of 4–5 nm in the 5–50 eV range [6.26,6.74]. However, elemental maps have been recorded in the low-loss region displaying much better resolution of ∼ 1–2 nm [6.74–6.76]. Muller and Silcox showed that this effect is a consequence of the distribution of scattered intensity [6.77]: Resolution may be degraded in terms of full width at 10th maximum criterion d90 containing 90% of the inelastic intensity, but remains good for the diameter d50 containing 50% of the intensity [6.1]. While the diffraction limit of the objective aperture is only of concern for collection angles < 5 mrad, chromatic aberrations are one of the main limiting factors for EFTEM resolution. The chromatic aberrations of the objective lens can be kept low by using small objective apertures and small energy windows deﬁned by the energyselecting slit. However, this has to be weighed against the loss of useful signal and resolution due to larger delocalization (worse for smaller collection angles) as well

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as diffraction and noise limitations. In the Egerton approach, the chromatic limit to point resolution in an inelastic image is typically a factor of 3–8 times smaller than the diameter of the full aberration disk, as given by the above chromatic aberration formula [6.73]. Generally, the effect of spherical aberration is of less signiﬁcance, because the radius containing 50% of the total scattered intensity is typically between 2 and 10% of the total radius for spherical aberration given by the above formula [6.73]. In addition the effective pixel size of the CCD detector may limit the resolution in the ﬁnal image. The microscope magniﬁcation and/or the CCD binning should be selected such that the maximum resolution required corresponds to 3–4 pixels. The predictions of the models are in good agreement with experimental results, for example, the imaging of a 0.7-nm thin chromium-layer in a multilayer structure with chromium L2,3 loss electrons at E0 = 200 kV [6.28]. Recently, Freitag and Mader reported high-resolution imaging of Ba double layers with a distance of 0.38 nm in the high Tc superconductor NdBa2 Cu3 O7−δ by recording Ba N4,5 jump ratio images thus depicting a resolution considerably < 1 nm [6.78,6.79] (see also Fig. 6.18). 6.4.4

Detection Limits

When discussing analytical sensitivities, we should distinguish between two cases. For the case in which the element of interest occurs in low concentrations, the minimum detectable mass fraction (MMF) is of interest and therefore EELS spectroscopy (e.g., with STEM spectrum imaging) is the method of choice [6.80,6.81]. To detect an element at near-trace concentration, it is necessary to use special techniques such as difference spectrum imaging [6.82,6.83]. By recording a ﬁrst-difference spectrum image, detector artifacts can be eliminated and very low concentrations can be detected by summing the difference spectra from many pixels in arbitrarily shaped regions of the spectrum, for example, along grain boundaries. In cases where the atoms are located in small clusters, precipitates, grain boundaries or macromolecules, we should consider the minimum detectable mass (MDM) or the minimum detectable number of atoms. This second case is well suited for EFTEM elemental mapping, because we can expect a sufﬁciently high SNR. Based on the assumption that the edge signal should be three times the noise of the background, estimates of the MMF and MDM have often been discussed for EELS [6.81,6.83,6.84]. However, ﬁgures for EFTEM are generally worse, because only two or three images with relatively small energy windows are typically recorded. Analytical sensitivities of clusters of 150 phosphorus atoms have been observed in EFTEM investigations of biological samples [6.85]; Pd-particles with diameters of ∼ 2 nm can be detected thus representing ∼ 200 to 300 atoms [6.86], and with an EFTEM equipped with a FEG, gold particles as small as 1-nm diameter (30 atoms) were detectable with a SNR > 5 [6.87]. Mayer et al. could detect 0.7-nm thin grain boundary layers of silicon oxide in Si3 N4 ceramics [6.69].

ELEMENT DISTRIBUTION IMAGES

6.4.5

175

Correlation of Elemental Maps

If more than two or three elements have to be imaged, it is not enough to make a collage containing all elemental distribution images. However, it is even more important to show the spatial relationship between several elemental maps. The simplest way to correlate elemental maps of up to three elements is to form a color image (RGB image) by assigning each image a color (red, green, blue). These maps are easily calculated, but limited to three elements and often difﬁcult to interpret, because of the occurrence of mixed colors [6.88–6.90] (see also the enclosed CD-ROM). The more general approach relies on processing of the related elemental maps (= multivariate 2D data sets) by means of multivariate statistical methods [6.91,6.92]. These methods have been used in different microanalytical techniques (Auger electron spectroscopy, secondary ion mass spectroscopy, X-ray emission and X-ray ﬂuorescence spectroscopy) and examples of the processing of this type of data can be found in the literature [6.93–6.97]. Besides linear multivariate statistical tools [6.91], nonlinear methods such as scatter diagram analysis proves very useful [6.98–6.100]. Scatter diagrams are easily calculated and combined with other statistical techniques such as principal component analysis and phase classiﬁcation procedures [6.94]. A 2D scatter diagram usually consists of a square pixel array, where the abscissa corresponds to the gray levels of the ﬁrst image and the ordinate corresponds to the gray levels of the second image (see Fig. 6.9). Discrete clusters form at regions of well-deﬁned chemical ratios and a procedure known as intercorrelative partitioning (ICP) can be used to rebuild the corresponding chemical phase map. Each phase may be preferentially represented by a color, since it is easier to distinguish colors than gray levels. In principle, the number of elements used for the computation of scatter diagrams is not limited, although simple visualization is possible only in two or three dimensions. The extension to three or more dimensions requires sophisticated automated phase classiﬁcation procedures, due to the difﬁculties involved in interactively partitioning scatter diagrams with more than two dimensions. Scatter diagrams can be easily applied to quantitative concentration maps, for example, atomic ratio maps yielding quantitative chemical phase maps (see the example in Fig. 6.12) [6.88,6.89]. 6.4.6

Quantitative Analysis of Elemental Maps

Even if the background in EFTEM elemental maps is correctly subtracted, the intensities in these raw elemental maps depend, not only elemental concentrations, but also on mass thickness and diffraction effects. These contrast changes arise from variations in the amount of elastic scattering intercepted by the objective aperture. In the unﬁltered bright-ﬁeld image darker regions scatter the electrons elastically at scattering angles larger than the objective aperture and in a ﬁrst approximation this applies to the same degree for elemental maps on account of double elastic–inelastic scattering (Fig. 6.10). Consequently, quantitative elemental analysis requires special procedures to turn raw elemental images into true elemental maps.

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Fig. 6.9 Schematic for construction of a 2D scatter diagram from two elemental maps A and B. For every pixel the corresponding gray levels in both images (x, y) are used as the coordinates of a point in the scatter diagram P(x, y).

Fig. 6.10 Energy ﬁltered images of edge and bend contours in a Si foil imaged by (a) zero-loss ﬁltering, (b) energy ﬁltering at 300 eV, and (c) comparison of the intensity proﬁles of the line scans shown in the energy ﬁltered images.

6.4.6.1 Absolute Quantiﬁcation To a ﬁrst approximation, mass thickness effects can be removed, if the elemental map is divided by a low-loss image (including the zero-loss peak and low-loss electrons). Similarly to EELS spectrometry the number of atoms per unit area NA can be determined with the formula [6.1]: NA =

IA (β, ∆) Ilowloss (β, ∆) · σA (β, ∆)

IA (β, ∆) is the edge intensity collected inside a collection semiangle β and integrated within an energy range ∆. Ilowloss (β, ∆) is the signal from an energy range of equal width ∆ containing both the zero loss and plasmon loss. σA (β, ∆) is the partial ionization cross-section. Calculation of compositional images therefore, typically involves the division of the background subtracted elemental map by the low-loss image and the partial ionization cross-section for each pixel. Absolute quantiﬁcation has often been used for biological samples, because problematic mass thickness effects are corrected [6.18,6.22,6.23,6.101,6.102].

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Problems in performing an absolute quantiﬁcation arise from the high dynamic range of the EELS spectrum, the partial ionization cross-sections, specimen thickness and diffraction effects (in case of crystalline materials). The core-loss image and the low-loss image must be acquired under identical experimental conditions (window width and convergence angle). However, special care has to be taken to insure that the large intensity discrepancies between the elemental map and the bright ﬁeld image are properly considered [6.88]; the beam current, the acquisition times and the binning of the CCD detector should be adjusted [6.103] (see also Section 6.2.2.). The partial ionization cross-sections for EELS quantiﬁcation may be calculated with a reasonable degree of accuracy using the hydrogenic model (Li–Zn) [6.1,6.104–6.107] or the Hartree–Slater model (for all ionization edges and elements) [6.108,6.109]. However, several investigations have shown that, particularly for the heavier elements, the calculated cross-sections do not always agree with experimental values [6.105,6.110,6.111]. In order to improve EELS quantiﬁcation, cross-section ratios (k-factors) have been measured for ∆ = 50 and 100 eV using binary oxides; for example, K- and L2,3 -edges (Li–Zn) [6.112], M4,5 -edges (Sr–W) [6.113,6.114], N4,5 -edges (Ba–Tm) [6.115], M2,3 -edges (Ca–Cu) [6.116] and L2,3 -edges (Sr–Mo) [6.117]. In a conventional approach using one post-edge image at the ionization edge, ionization cross-sections cannot be used directly for elemental mapping, because energy windows often have to be smaller than 50 eV [6.88]. For such small energy windows the accuracy of the calculated cross-sections is limited due to near-edge ﬁne structure modulations. In order to get accurate k-factor values for optimum energy-windows, the k-factors have to be properly adapted [6.88]. An alternative approach has been described by Mayer et al. [6.5], where an energy ﬁltered image series was acquired over a broader energy range at the ionization edge (∆ > 50 eV) and extracted from the ionization edge signal from the whole image series. In most applications with this approach, the above mentioned ionization cross-section can then be used without further problems (see also Section 6.4.7.). Absolute quantiﬁcation based on the above approximation however, can give misleading results when the sample thickness exceeds the total inelastic mean free path λ. Thickness variations can be partially accounted for by normalization to specimen thickness (or with a t/λ map), which yields elemental concentrations in terms of atoms per unit volume [6.18,6.118]. However, this type of normalization only provides a ﬁrst-order correction for sample thickness variations and remaining diffraction contrast. It fails under strongly diffracting conditions, for example, across bend contours of crystalline specimens (e.g., as visible in Figs. 6.10 and 6.11) 6.8,6.88,6.119,6.120]. The problem is that diffraction contrast is always present to some degree in energy ﬁltered images [6.121]. The reasons are twofold: First, diffraction features like bend contours are blurred with increasing energy loss. This can be explained by the angular distribution of inelastic scattering that results in an incoherent superposition of images with a spectrum of excitation errors (Fig. 6.10) [6.122–6.124]. Second, it has been observed that thickness fringes are not only blurred with increasing energy losses, but that the fringes move with increasing

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Fig. 6.11 Quantitative elemental maps recorded from an ion-milled TiC-crystal: demonstration of the quantiﬁcation problems due to specimen thickness variations and diffraction effects; (a) TEM-bright ﬁeld image with line trace which has been used for the quantiﬁcations; (b) EELS spectrum; (c) t/λ plot, carbon map (C K -edge), titanium map (Ti L2,3 -edge), absolute quantiﬁcation of the titanium map in terms of atoms per unit area, absolute quantiﬁcation of the Ti map in terms of atoms per unit volume and atomic ratio map showing the Ti/C atomic ratio (from [6.125]).

energy loss as well [6.124]. Additionally, the necessary increase of beam current between low-loss and high-loss image may lead to a subtle bending of thin specimens, thus changing the diffraction features between the low-loss image and the elemental map. In order to demonstrate the relative merits of quantiﬁcation procedures, a TiC single-crystal wedge containing signiﬁcant diffraction contrast and thickness varia-

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tion has been studied [6.125]. Various line plots have been extracted from processed image data along the indicated path (Fig. 6.11a). Across bend contours, the sample appears to be thicker while intensities in the elemental maps are at the same time heavily attenuated (Fig. 6.11c). This consequently leads to incorrect ﬁgures for the quantitative titanium map. 6.4.6.2 Relative Quantiﬁcation It is common practice in microanalysis to cancel the above mentioned effects by calculating elemental ratios. For example, atomic ratios are typically measured in EELS spectra, rather than area elemental densities [6.126,6.127]. Similarly to EELS quantiﬁcation, atomic ratio maps can be calculated using the net elemental maps and the corresponding partial ionization cross-sections according to the formula IA (β, ∆) σB (β, ∆) NA · = NB IB (β, ∆) σA (β, ∆) This method was ﬁrst applied to the quantiﬁcation of elemental maps of biological specimens by Leapman [6.128] and to materials science specimens by Bonnet et al. [6.56]. In these initial investigations the elemental maps were acquired using a STEM equipped with a serial detection EELS spectrometer. The atomic ratio method is particularly useful for materials science specimens, because diffraction and thickness effects are signiﬁcantly reduced. As shown in Fig. 11c, diffraction contours vary less severely between energetically closely spaced images, for example, between the C K- and the Ti L2,3 -edge images than between the low-loss image and higher energy-loss images [6.88]. Therefore, by plotting the atomic ratio of titanium relative to carbon in Fig. 11c, the nominal value of 0.95 can be obtained with good accuracy and is largely consistent over the entire length of the plot despite thickness variations and diffraction effects [6.125]. A practical example for the relative quantiﬁcation method is shown in Fig. 6.12, which shows the interface region of TiC and Ti(C,N)-layers deposited onto a hard metal substrate with a specimen thickness varying from 0.3 < t/λ < 1.2. The interface is completely obscured in the TEM image. Although the image intensities in the raw elemental maps for Ti, C, and N are strongly inﬂuenced by thickness variations and diffraction effects (not shown here, for details see [6.88]), the chemical phases can be clearly visualized in the atomic ratio images which have been calculated for the C/Ti and N/C ratios (Fig. 6.12b and c). Between the TiC grains with a C/Ti ratio of ∼ 1.0, bright regions exhibiting very high C/Ti ratios can be detected that can be attributed to amorphous carbon enrichments as previously measured by EELS spectroscopy [6.129]. The Ti(C,N) layer exhibits a variation of the N/C ratios within the grains ranging from 0.94 to 3. Finally, it should be mentioned that in particular cases, the representation of an atomic ratio map of two elements (A/B) may be difﬁcult: If element B is missing in some specimen regions, the denominator goes to zero making the results sometimes difﬁcult to display, as the intensities will reach inﬁnity. Therefore, atomic ratios have to be chosen carefully in order to yield reliable quantiﬁcation results, especially if

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Fig. 6.12 Quantitative analysis of the interface region between a Ti(C,N) and a TiC layer on a hard metal; (a) TEM bright ﬁeld image; (b) C/Ti atomic ratio map calculated from C K and Ti L2,3 elemental maps with the histogram below; (c) N/C atomic ratio map calculated from N K and C K elemental maps with the histogram right; (d) scatter diagram constructed with the atomic ratio maps; (e) gray scale masks for masking chemical phases with certain composition ranges (f) quantitative chemical phase map revealing titanium carbide (bright), carbon at the grain boundaries (gray) and titanium carbonitride classiﬁed into regions of certain N/C atomic ratios, from [6.88].

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several elements are present. For such situations, it has been proposed to calculate the ratio A/(A+B) thus prohibiting the denominator from going to zero [6.56,6.130]. This approach also corrects for specimen thickness variations. 6.4.7

Image EELS

It is sometimes advantageous to not only record elemental maps of several elements, but to acquire a series of energy-ﬁltered images around one or two ionization edges (Fig. 6.13) and, if required for quantiﬁcation, also in the low-loss region. This technique, ﬁrst developed by Lavergne et al. for the analysis of biological specimens [6.131], is referred to as “image spectroscopy” or “EFTEM spectrum imaging” by analogy to “spectrum imaging” in the STEM. There are some advantages of image-spectroscopy in comparison to the simple three-window approach: From an extended EFTEM image series, it is possible to extract EELS spectra for any given image area, that is, each individual pixel or an array of pixels over which the signal is integrated [6.132–6.136]. Since the total number of signal in an extended EFTEM series is considerably greater than for the three window method, the background subtraction procedure is more reliable and the signal-to-statistical-noise content is considerably improved. Each image-spectrum contains energy loss information with the same spectral resolution as the acquisition slit-width, and may be acquired over multiple ionization edges, allowing analysis of a number of elemental species within a single acquisition. Further beneﬁts of this method are that the standard EELS procedures for single scattering deconvolution, least square ﬁtting (for overlapping edges) and quantiﬁcation already developed for EELS spectra can be used [6.5,6.101,6.137,6.138]. Recently, quantiﬁcation of a wedge shaped sample was improved substantially through Fourier deconvolution, effectively removing plural scattering over a wide range of sample thickness (0 < t/λ < 2) [6.25,6.138]. Disadvantages are that the acquisition times may be much longer than for the conventional elemental mapping approach. In order to achieve reasonable acquisition times, the image series has to be collected with energy windows of ∼ 5–10 eV which limits the energy resolution in the EELS spectra derived from the EFTEM spectrum image (Fig. 6.13b). 6.5

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EFTEM is a very powerful technique for the characterization of the micro- and nanochemistry of thin specimens. Elemental mapping is the preferred way to carry out microanalysis, since it tends to give the full picture rather than isolated spectra, often from operator biased selections of the region of interest. Consequently, previously overlooked features are easily and rapidly made visible by elemental mapping.

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Fig. 6.13 Image EELS: (a) A montage showing an EFTEM series taken from Nibase alloy with γ -precipitates (Ni3 (Al,Ti)) in a NiCr-alloy, illustrating the change in feature intensity with increasing energy loss; (b) with EELS spectra extracted from a precipitate and the matrix.

6.5.1

Secondary Phases in Steels and Alloys

One very common problem in materials science is the visualization of secondary phases embedded in a crystalline matrix, for example, precipitates and grain boundary phases in steels and alloys [6.139,6.140]. Since these phases play an important role in the mechanical behavior of metals, the characterization of their stereological parameters (diameter, distribution, volume fraction, etc.), chemistry and crystallography is a critical issue. Because of diffraction contrast such phases can be completely obscured in TEM bright ﬁeld images. In conventional TEM of crystalline specimens, precipitates in a matrix are most often visualized by dark ﬁeld imaging [6.141]. However, since the detectability of the particles strongly depends on their orientation, this method may not show all particles and gives no information about the chemical phase. Additionally, problems may occur in case of coherent precipitates [6.76]. EFTEM provides a good means to visualize all secondary phases as well as to obtain compositional information [6.65,6.142]. As an example Fig. 6.14a shows the TEM bright ﬁeld image of a 10% Cr steel, where the secondary phases are difﬁcult to see, due to unavoidable diffraction contours that mask the particles. By recording a jump ratio map of the matrix element iron with rocking beam illumination, all secondary phases with an iron concentration lower than the matrix are made visible as dark regions (Fig. 6.14c). Problematic diffraction effects are almost completely eliminated thus enabling unequivocal visualization of all secondary phases regardless of their crystallographic orientation [6.65].

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Fig. 6.14 Imaging of precipitates in a steel specimen; (a) TEM-bright ﬁeld image of a 10% Cr steel with Cr23 C6 , VN and Nb(C,N) precipitates; (b) EELS spectrum with the Fe M23 edge; (c) Fe M23 jump-ratio image recorded under rocking beam illumination; (d) Cr L23 jump ratio image; (e) V L23 jump ratio image; (f) Nb M45 jump ratio image.

In the analysis of steels, two different edges enable the detection of iron within the energy loss region accessible by EELS; the Fe M2,3 ionization edge occurring at 54 eV and the Fe L2,3 -edge at 708 eV. For very thin specimens the M2,3 -edge exhibits a better SNR and should be preferred for detecting small precipitates in steels, but since preservation of diffraction contrast can be a problem in the low-loss region, rocking beam illumination is always necessary for acquiring Fe M2,3 maps [6.65]. The Fe L2,3 edge should be used if the specimen is thicker (t/λ > 0.5) and if high resolution in the nanometer range is not essential. One advantage of the L2,3 -edge is that diffraction contrast is of less importance. If the jump ratio images of the matrix element are combined with EFTEM thickness maps (t/λ), the volume fraction of secondary phases can be measured with high accuracy [6.143,6.144]. Additionally, the chemical composition of the precipitates is revealed by acquiring elemental maps and/or jump-ratio images of the elements of interest, for example, the case of Fig. 6.14 for chromium, vanadium and niobium, revealing the occurrence of Cr23 C6 , VN, and Nb(C,N) precipitates [6.142,6.145]. Due to diffraction effects, these images were also recorded with rocking beam illumination. Detection of carbon or nitrogen is also possible [6.146] but limited to a thin specimen which does not contain elements such as molybdenum and niobium, because the M4,5 - and M2,3 -edges of these elements overlap with the carbon and nitrogen K-edges, respectively [6.147]. Edge overlap can only be properly managed

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by recording EELS and EDX spectra of the various precipitates [6.142] or by recording complete spectrum images or image EELS data of the precipitate and subsequent careful spectrum analysis using MLS ﬁtting techniques [6.148,6.149]. Energy ﬁltering TEM is extremely useful for the visualization of previously unseen and therefore unknown microstructural features in steels and alloys. For example, in an investigation of precipitates in a high-speed steel it could, for the ﬁrst time, be demonstrated that large primary M2 C-carbides are enveloped within a thin V(C,N) layer [6.150]. Recently, an EFTEM investigation of a tungsten modiﬁed 9% Cr steel showed the occurrence of a z-phase precipitate that was found for the ﬁrst time in this type of steel [6.151,6.152]. The mechanical properties of microalloyed steels and high-strength low-alloy (HSLA) steels are determined by dispersed nanosized precipitates of the type MX (M = Ti, Nb and V; X = C and N). These tiny precipitates are difﬁcult to detect in TEM bright- and dark-ﬁeld images, especially precipitates with diameters smaller than ∼ 10 nm. As shown in Fig. 6.15, EFTEM can visualize such precipitates [6.76]. While it is not possible to see any precipitate in the bright ﬁeld image (Fig. 6.15a), the Fe M2,3 jump-ratio image (Fig. 6.15b) clearly reveals regions of local iron deﬁciency. These dark particles correspond exactly to the bright regions in the jump ratio image that was recorded using the overlapping Nb N2,3 - and V M2,3 -edges. Obviously these nanometer sized precipitates are forming a “network”, that is, very small particles are presumably nucleated on dislocations which are also visible in the Fe M2,3 jump ratio image [6.76].

Fig. 6.15 Imaging of nanosized MX precipitates in a microalloyed steel; (a) TEM bright ﬁeld image of a 20-nm thin steel specimen; (b) Fe M23 jump ratio image; (c) jump ratio image recorded with the overlapping Nb N23 - and V M23 -edges [6.76].

The EFTEM investigation of a steel containing ∼ 1wt% copper and aged at 500◦ C for 24 h revealed in the Cu L2,3 jump ratio images, copper particles with diameters on the order of nanometers [6.59]. Rice et al. studied the affect of copper precipitation on the hardening of ferritic alloys with 0.9 wt% Cu and also detected copper precipitates by recording Cu L2,3 jump ratio images [6.153]. Mechanical properties of stainless steels are very sensitive to the structure and composition near the grain boundary. In particular, chromium depletion near the grain

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boundary is the direct cause of intergranular corrosion cracking. In a quantitative study, Bentley and co-workers found in a stainless steel that was sensitized by aging for 15 h at 600◦ C, grain boundary Cr depletion occurring between chromium-rich intergranular M23 C6 precipitates [6.154]. In this investigation Cr L2,3 elemental maps were normalized by the low-loss image and t/λ-image to yield chromium atoms per unit volume. Similar quantitative EFTEM studies were performed by Kimoto ﬁnding good agreement with the results obtained with X-ray microanalysis [6.155]. In one investigation of a sensitised stainless steel, the chromium depletion near the grain boundaries was combined with the growth of a 2-nm thin chromium carbide layer at the boundary [6.86,6.139]. Elemental mapping also proved useful for studying the interfacial composition in a complex Ni-base superalloy that contained coherent misﬁtting small and large γ Ni3(Al,Ti) particles in a γ Ni solid solution matrix [6.156]. The Ni intensity ratio for large γ relative to the γ matrix was determined from Ni L2,3 elemental maps which were quantiﬁed in terms of atoms per unit volume; it was found to be in good agreement with quantitative X-ray microanalysis. AlSn alloys deposited by sputtering are known to exhibit excellent tribological properties that might be associated with a peculiar lamellar structure of the alloy. EFTEM elemental maps revealed segregation of tin and oxygen to thin elongated inclusions at the grain boundaries with a width of ∼ 3 nm [6.157]. Csontos et al. could demonstrate that it is possible to obtain subnanometer compositional information at interfaces in an Al–Ag alloy during in situ heating studies [6.158]. Recently, partially crystallized amorphous Al–Ge alloys were investigated using the Al K- and Ge L2,3 edges; jump ratio images were compared with contrast in plasmon loss images arising from subtle differences in the plasmon peaks [6.159]. Oxide-dispersion-strengthened (ODS) alloys contain dispersed particles that inﬂuence considerably, the properties of several technologically important alloys. Due to the peculiar mechanism of dissolution and reprecipitation of very ﬁne-grained oxides through appropriate thermal treatment, understanding of the structure of these materials is essential and can be efﬁciently provided by EFTEM investigations. For example, nanometer sized titanium oxide particles were imaged in an ODS-niobium alloy designed for high load bearing medical implants [6.61]. In case of an ODS Ni/20%Cr alloy, several secondary phases could be detected: Y2 O3 and Y2 O3 –Al2 O3 precipitates, Cr at grain boundaries and TiN particles often intergrown with nanosized yttrium and aluminium oxides [6.90]. Recently, a Fe–Cr ferritic steel strengthened by disperse TiO2 particles was investigated by a combination of EFTEM and Auger spectroscopy in order to study the distribution of alloying elements in the vicinity of matrix-TiO2 interphase boundaries [6.160]. 6.5.2

Ceramics, Hard Metals and Composites

A primary aim in the development of structural ceramics and composites is to obtain control over the mechanical properties that are governed by the microstructure. It is thus desirable to be capable of inﬂuencing the microstructure, via the starting material and the processing conditions, and to know the result, via an efﬁcient

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characterization technique on a submicrometer scale that can be efﬁciently provided by EFTEM elemental mapping. In the last two decades, considerable research has been devoted to the evaluation of microstructures in Si3 N4 and SiC ceramics. For example, sintering of Si3 N4 ceramics is enabled by Al2 O3 and Y2 O3 as sintering additives and occurs via a liquid phase mechanism, where the intergranular ﬁlms play an important role. Consequently, one of the earliest applications of EFTEM in the ceramic ﬁeld was dedicated to the investigation of intergranular silicon oxide ﬁlms in hot-pressed Si3 N4 by recording an oxygen distribution map [6.161]. Homogeneous incorporation of SiC into a Si3 N4 matrix increases the hardness, creep and oxidation resistance [6.162]. For preparing such materials processing technologies based on the pyrolysis of polymer precursors have been successfully used. Although the microstructure of these SiC/Si3 N4 composites has been investigated by several methods, EFTEM elemental mapping proved to be a particularly powerful tool for rapidly imaging the distribution of SiC, Si3 N4 and oxygen-rich intergranular ﬁlms by recording the elemental maps of carbon, nitrogen and oxygen, respectively (Fig. 6.16) [6.163]. Recently, SiC-Si3 N4 nanocomposites have been developed that exhibit excellent mechanical properties (bending strength and fracture toughness) up to 1500◦ C [6.164]. These improvements are attributed to the existence of strength/toughening mechanisms operating on the nanometer level due probably to the presence of SiC nanoparticles included within the Si3 N4 micrograins. EFTEM investigations showed that not only are SiC particles included within the Si3 N4 grains, but that free carbon can sometimes be found, that may reinforce the effect of the SiC particles [6.165]. For Si3 N4 ceramics it has been predicted theoretically [6.166] and proven experimentally [6.167] that during sintering with oxide additives an amorphous grain boundary ﬁlm forms that possesses an equilibrium thickness. In a systematic study of EFTEM detection limits, Berger et al. investigated such materials with equilibrium ﬁlm thicknesses of 1.5, 1.0, and 0.7 nm [6.69]. Oxygen maps recorded under conditions optimized for low detection limits revealed the presence of grain boundary ﬁlms for all three materials (Fig. 6.17). In nanostructured, boron-doped Si3 N4 /SiC ceramics, interface layers consisting of B(N,C) could be detected [6.168]. Mayer et al. measured the Ca segregation in grain boundary ﬁlms and triple junctions of Ca-doped Si3 N4 ceramics [6.169]. A few investigations have been devoted to the EFTEM investigation of oxidic ceramic materials that are often used in electronic applications; for example, phase heterogenities in barium neodymium titanate ceramics [6.89,6.170] and Al2 O3 ceramics with nanometer sized Ti-oxide inclusions at the grain boundaries [6.86]. EFTEM has also been applied to the characterization of doped BaTiO3 ceramics exhibiting a core-shell structure (X7R dielectrics) [6.171]. The core-shell structure within single BaTiO3 grains was visualized by recording jump-ratio images at the Ti L2,3 -edge under rocking beam illumination. Although the concentrations of the doping elements zirconium, ytterbium and strontium were quite low (< 2 at%), their distribution could be imaged by recording jump-ratio maps with the Zr M4,5 - and Yb

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Fig. 6.16 Imaging of phases in a ceramics; (a) TEM-bright ﬁeld image of a Si3 N4 -SiC ceramics; (b) nitrogen K elemental map; (c) carbon K elemental map; (d) oxygen K elemental map, (specimen courtesy P. Sajgalik, Bratislava, Slovakia).

N4,5 -edges. In agreement with previous results [6.172,6.173], the doping elements could only be found in the shell region. Recently, Eibl investigated the intergrowth of (Bi,Pb)2 Sr2 CaCu2 O8+δ and (Bi,Pb)2 Sr2 Ca2 Cu3 O10+δ phases in high-Tc superconductors occurring at the nanometer scale [6.174]. In the high Tc superconductor NdBa2 Cu3 O7−δ Freitag and Mader have been able to detect barium layers with a spacing of 0.38 nm by recording Ba N4,5 jump ratio images [6.78,6.79] (Fig. 6.18). The properties of hard metals that are used in cutting tool applications are greatly inﬂuenced by a microstructure consisting of hard carbide or nitride grains embedded in a tough binder phase, and by the raw material powder mixture, as well as by the sintering conditions. For example, the EFTEM investigation of hard metals of the type WC–Co that had been doped with low concentrations of vanadium provided insight into the relationship between microstructure and improved wear-resistance [6.175]. By recording V L2,3 jump ratio images, nanosized VC-particles could be identiﬁed

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Fig. 6.17 Detection of silicon oxide layers in grain boundaries of a Si3 N4 ceramics; (a) oxygen distribution map of material containing grain boundaries with a thickness of 0.7 nm; (b) line scan of the image intensity across a grain boundary, marked in (a); (courtesy J. Mayer [6.5]).

that had been precipitated between the WC-crystals [6.176]. In a comparative study, cermets [Ti(C,N) based hard metals] have been investigated both by EFTEM, EELS, and atom probe ﬁeld ion microscopy (APFIM) ﬁnding good agreement between EELS and APFIM quantiﬁcations [6.177–6.179]. Furthermore, hard metal tools are often covered by wear resistant ceramic layers, for example, Al2 O3 , TiN, Ti(C,N), and TiC layers grown by chemical or physical vapor deposition (CVD and PVD). Cross-sectional TEM investigations of these materials provide information on the property–microstructure relationship [6.180]. For example, in the quantitative EFTEM study shown in Fig. 6.12 the microstructural features which lead to decreased wear resistance could be identiﬁed by calculating atomic ratio maps from elemental maps [6.88]: The TiC layers exhibited residues of free carbon between the TiC grains and in the Ti(C,N) layer the carbon nitrogen ratio varied even within single grains. Additionally, cobalt inclusions were found in the layers which had diffused from the hard metal matrix. Quantitative EFTEM was also useful to characterize complex hard layer systems of ∼ 10 layers consisting mainly of TiN, Al2 O3 , and Ti(C,N,O) of varying compositions [6.125]. Nanocrystalline hard coatings within the quasibinary systems TiN–TiB2 and TiC– TiB2 could be characterized with EFTEM that has been particularly useful for investigating the chemical composition of the interface region between the coatings and various substrate types [6.181–6.183]. During the last several years, hard metal tools have also been coated by thin diamond layers that are preferentially deposited by CVD processes. The layer adhesion may be strongly inﬂuenced by intermediate layers consisting of non-diamond carbon (e.g., amorphous carbon, diamond-like carbon). EFTEM is easily capable of distinguishing diamond and non-diamond carbon using the π ∗ -peak of the carbon K-edge (Chapter 7). Consequently, it is possible to map the distribution of non-

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Fig. 6.18 High-resolution EFTEM imaging of barium-layers in a NdBa2 Cu3 O7 -crystal in [100] orientation; (a) Ba N4,5 jump ratio image; (b) (pre-edge 1)/(pre-edge 2) ratio image; intensity proﬁles in (c) and (d) of the marked regions in (a) and (b), respectively; schematic of structure of NdBa2 Cu3 O7 in [100] is shown between the proﬁles; (courtesy W. Mader [6.79]).

diamond carbon on a nanometer scale by means of a 1–2-eV wide energy-selecting slit [6.75,6.184,6.185]. Interfacial phenomena are of concern in a wide variety of multiphase inorganic systems that include composites, coatings, electronic devices, and electrodes in electrochemical devices. In order to control the morphological evolution of interfacial layers between different materials at elevated temperatures, EFTEM elemental mapping provides important information about interfacial chemistry: For example, Brydson et al. investigated a ﬁber-reinforced metal matrix composite designed for use in high temperature aerospace applications [6.186,6.187]: A SiC ﬁber employs a developmental duplex coating based upon a compliant layer of gadolinium metal and a gadolinium boride reaction barrier both deposited by physical vapor deposition and embedded in a Ti-6Al-4V alloy. This material represents a complex system containing unexpected inhomogeneities with numerous reactions and diffusion processes

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occurring at the ﬁber-coating and coating-matrix interfaces. Important information on these features could be obtained by recording elemental maps and jump ratio maps using the Si L2,3 -, Gd N4,5 -, B K-, C K-, Ti L2,3 -, O K- and Gd M4,5 -edges (Fig. 6.19). Due to the complexity of the phases present, the conclusions drawn from the EFTEM investigation had to be conﬁrmed by additional EEL spectroscopic analyses on the regions of interest.

Fig. 6.19 Investigation of a ﬁber reinforced Ti-6Al-4V alloy designed for use in high temperature aerospace applications; (a) TEM image of the interface between the SiC-ﬁber (left) and a developmental duplex coating based upon a compliant layer of gadolinium metal and a gadolinium boride reaction layer (right); (b) Gd M4,5 elemental map; (c) O K jump ratio map; (d) Si L2,3 elemental map; (e) B K jump ratio map; (f) C K elemental map; (g) EELS spectra of the specimen regions shown in (c), the chemical phases determined by quantiﬁcation of the spectra are included; [6.187].

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EFTEM has been used to characterize previously unknown periodic layers in the interface between a Pt layer and NiO as part of a coulometric titration cell. In addition to elemental distribution maps using the Ni L2,3 -, O K- and Pt M4,5 -edges, EELS line scans were acquired in order to obtain quantitative composition proﬁles thus revealing the occurrence of thin Pt- and Ni-rich layers [6.188]. Surprisingly, EFTEM has not often been used in mineralogy. EFTEM elemental mapping has been applied to the study of ﬁne hornblende lamellae in cummingtonite [6.189] and Moore et al. characterized the local chemical distribution in exsolved pyroxenes [6.190]. In this study, the elements Ca, Fe and Mg were examined in two samples, one that contained intergrown augite and pigeonite and one that additionally contained orthopyroxene. The EFTEM results in combination with HREM imaging and EDX spectroscopy allowed the authors to conﬁrm that the transformation of augite to orthopyroxene occurs in two steps with pigeonite as an intermediate phase. 6.5.3

Semiconducting Materials and Devices

In the last 10 years there has been a large increase in the application of TEM to semiconductor device problems, because higher density chips and decreasing device dimensions have led to structural components that are frequently in the nanometer range. Much work has been devoted to morphological investigations and to the analysis of crystallography and crystal defects in devices, because defects such as dislocations are major yield detractors in many semiconductor technologies. However, energy ﬁltering TEMs are ideally suited for imaging semiconductor devices, making them invaluable for routine constructional analysis [6.52,6.191,6.192]. As shown in Fig. 6.20, EFTEM elemental maps provide an overview of a large specimen area that can be important for rapid defect identiﬁcation. In the TEM image (Fig. 6.20a), only particular structures can be visualized, but all structures and phases are visible in the elemental maps: The Si K jump ratio map reveals in the bright regions, the silicon wafer and a thin poly-Si layer; the gray regions show silicon dioxide, silicon oxynitride, and silicon nitride layers, respectively (Fig. 6.20b). It is even possible to detect a small gradient in Si concentration in the silicon dioxide layer, which was introduced by doping with boron and phosphorus. Additionally, a Si-rich inclusion in the conducting layer (Al-alloy) can be detected. The Ti L2,3 jump ratio map (Fig. 6.20c) clearly shows the bright Ti-nitride layer showing spikes in the conducting layer. The Al K jump ratio map in Fig. 6.20d shows the conducting layer; the silicon-rich defect and the titanium-rich spikes are visible as dark regions. Further application examples of EFTEM in semiconductor research are shown on the enclosed CD-ROM. A critical issue in device manufacture is the reliability of the Al-ﬁlled contact holes with respect to Al spiking in the silicon substrate. By using EFTEM elemental mapping Cerva et al. identiﬁed the Al-diffusion paths and the reaction mechanisms which may lead to contact failure [6.193]. They found that Al spiking takes place mainly via the Ti/TiN barrier sidewall. Also, as device dimensions decrease, the structure and composition of thin interfacial layers become increasingly important to device performance. EFTEM and

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Fig. 6.20 Investigation of chemical defects in a semiconductor device; (a) TEM bright ﬁeld image; (b) Si K jump ratio map revealing a silicon inclusion in the aluminium layer; (c) Ti L2,3 jump ratio map exhibiting Ti-spikes in the Al layer; (d) Al K jump ratio map; (e) N K jump ratio map; (f) O K jump ratio map; (specimen courtesy O. Leitner, Austria Mikrosysteme International AG).

EELS are proving to be uniquely suited to this task [6.52,6.194]. Recently, Grogger et al. investigated the chemistry of nanometer wide interfacial layers between an aluminium conductive layer, titanium nitride, and silicon oxide by quantitative EFTEM elemental mapping and EELS line scans perpendicular to the layers [6.52]. In this work, defective areas in and near the titanium nitride layer could be visualized and analysed quantitatively. Botton and Phaneuf studied the chemistry of a 14-nm wide oxynitride composite dielectric layers in a dynamic random access memory (DRAM) by recording elemental maps of oxygen and nitrogen [6.195]. Hetero-bipolar transistors consisting of SiGe layers with a graded proﬁle could be analysed by EFTEM and EELS spectrometry; a quantitative Si and Ge concentration proﬁle could be derived from Si K and Ge L2,3 elemental maps, respectively [6.196]. Si3 N4 /SiO2 /Si heterostructures have

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been studied by imaging the differences in the chemical shifts of the Si L2,3 -edge which occurs between Si3 N4 , SiO2 , and Si [6.197]. Depth proﬁles of SiO2 / Si3 N4 / SiOx Ny /Si layers have been measured at subnanometer spatial resolution by means of EFTEM-based spatially resolved EELS [6.198]. It was possible to show that the nitrogen within the SiOx Ny layer accumulated at the the Si interface, and it was found to be 1.6 nm in depth. The electronic properties of semiconductor multilayers such as quantum wells and superlattices are determined by composition, thickness, and interfaces of the individual layers. Hence improved techniques are required for a quantitative analysis of structure and composition on a nanometer scale, for which HREM and STEMspectrum imaging have already been used [6.199–6.201]. One of the ﬁrst applications of elemental mapping for characterizing semiconductor multilayers was carried out by J¨ager and Mayer [6.202] in which Si/Ge heterostructures comprising a range of alloy compositions and layer thicknesses were investigated. In this work, Si mapping of monolayers was performed with the Si L2,3 -edge using the three window technique, but attempts to image the Ge distribution were not reported. Recently, the microstructure and the chemistry of SiGe/Si(001) layers exhibiting quasi-periodic surface corrugations have been studied by EFTEM elemental mapping using the Ge L2,3 ionization edge [6.203] and Liu et al. investigated the concentration of As dopants in ultrathin As doped layers in InP with the As L2,3 -edge [6.204]. Group V transition metal nitride ﬁlms are important as diffusion barriers in microelectronic devices, for example, TiN in standard devices and TaN as the barrier material of choice in combination with copper metallization, recently introduced in subquarter micron technologies. Group V nitride layers on silicon that were synthesized by rapid thermal processing have been studied using EFTEM elemental mapping of cross-section samples [6.205,6.206]. 6.5.4

Multilayers and Thin Films

The physical properties of multilayer ﬁlms depend critically on both the microstructure and the compositional proﬁle across the layers, often at close to atomic scale due to the small thicknesses of the individual layers. For instance, magnetic devices employ a multilayer structure with thicknesses on the order of several nanometers. Since their magnetic properties are so sensitive to the thickness and ﬂatness of each layer, elemental mapping of cross-sectioned multilayer specimens is necessary in the analysis of processing controls that determine the properties of these materials. Kimoto et al. investigated multilayers consisting of Ni80 Fe20 and chromium layers with thicknesses ranging from 1 to 3 nm. The chromium layers could be clearly observed in Cr L2,3 elemental maps and the measured thicknesses were compared with results of X-ray reﬂectivity showing good agreement [6.207]. Many other element combinations have been realized in multilayer systems; a typical example exhibiting perpendicular magnetic anisotropy (3 nm Fe and 2 nm Tb grown on a Si wafer) is shown in Fig. 6.21 [6.208]. In this particular example, even the TEM image can yield information about the layer roughness. However, in many other multilayer systems where the materials are similar in atomic number, contrast differences can be

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poor, so that the interfaces are not simply recognizable. High resolution images may consequently, not yield information about the layer roughness. Therefore, Grogger et al. measured the roughness of thin magnetic layers quantitatively from EFTEM jump ratio images [6.209].

Fig. 6.21 (a) TEM-cross section of a FeTb multilayer system with magnetic anisotropy perpendicular to the layers; (b) 2-nm thin terbium layers are clearly resolved in the Tb N4,5 elemental map; (c) 3-nm thin iron layers in the Fe M2,3 elemental map [6.208].

Compositional variations in magnetic thin ﬁlms can occur not only perpendicular to the surface of the thin ﬁlms and multilayers, but also in the layer plane (parallel to the surface). These heterogenities are easily revealed by EFTEM elemental mapping. For example, Kimoto studied compositional variations in CoCrTa thin ﬁlms by recording elemental maps of cobalt and chromium [6.210]. Grogger et al. investigated the amount of Cr segregation in Co–Cr based thin ﬁlms by means of scatter diagram analysis of Co L2,3 and Cr L2,3 elemental maps [6.211]. The EFTEM results correlated well with both the magnetic properties and recording performance of the media, that is, a higher Cr content of the thin-ﬁlm alloy shows a higher amount of Cr segregation, most of it at the grain boundaries. Ristau et al. investigated magnetic CoPt ﬁlms that had been cosputtered with yttrium stabilized zirconia; these ﬁlms showed very high ferromagnetic coercivity [6.212]. Using the Zr N2,3 -, Co M2,3 and Pt O2,3 -edges, jump ratio images have been recorded under rocking beam illumination revealing a complex microstructure consisting of nanometer-sized zirconia particles embedded in the CoPt thin ﬁlm alloy (Fig. 6.22). Thin nanostructured Nd-Fe-B magnets show enhanced remnance caused mainly by effective exchange interactions between the hard magnetic Nd2 Fe14 B phase and soft magnetic α iron. Elemental mapping allows the visualization of the phase distribution with an average grain size of 20 nm [6.213]. Furthermore, EFTEM was shown necessary for the characterization of Fe-Nd-B alloys that had been tailored to approach theoretical coercivity limits [6.214]. Nd2 Fe14 B particles embedded in a nonmagnetic neodymium matrix could be imaged by recording either Nd M4,5 or Fe L2,3 jump-ratio images enabling the correlation of microstructure with observed magnetic behavior. In semiconducting thin ﬁlms, growth mechanisms have been extensively investigated and appropriate models have been established. In the case of complex ionic oxides, such as the superconducting cuprates, epitaxial growth is not com-

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Fig. 6.22 EFTEM investigation of a co-sputtered CoPt and zirconia thin ﬁlm annealed at 700◦ C for 10 min; (a) TEM bright ﬁeld image, (b) Zr N2,3 jump-ratio image and (c) Pt O2,3 jump-ratio image; (d) intensity proﬁle from Co and Pt images revealing the Zr-rich particles embedded in the CoPt thin ﬁlm [6.212].

pletely understood. In this respect, EFTEM has proven extremely useful in answering the fundamental questions of the growth mode of complex oxides and in complementing X-ray diffraction and resistivity measurements [6.215]. Epitaxial YBa2 Cu3 O7−δ /PrBa2 Cu3 O7 superlattice multilayer stacks grown on (100) SrTiO3 were characterized by recording Y and Pr jump ratio images revealing block-by-block growth mode, complementing results from X-ray diffraction. Multilayers may be also interesting for structural materials, because properties can be efﬁciently tailored by combining two or three components. Titanium and titanium alloys are attractive materials for aerospace applications as they offer good mechanical strength and stiffness which are retained at high temperature. These materials, however, exhibit an inherently low toughness especially at room temperature. In the case of the Ti-Al system, lamination offers a route by which toughness can be

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enhanced [6.216]. Ti-Al nano-laminates with repeat periods of nanometers and an overall deposition thickness of several mm have been produced. EFTEM allowed a rapid assessment of layer integrity in cross-sectional TEM specimens at high spatial resolution by recording Al L2,3 and Ti L2,3 jump ratio maps [6.217,6.218]. 6.5.5

Polymers

Since variations in polymer microstructures are often a manifestation of spatial variations in composition and bonding, the ability to distinguish microstructural features based on spatial differences in composition and chemical nature plays a central role in polymer characterization. Many block copolymers and blends contain phases that have similar densities and similar scattering cross-sections and thus have little contrast difference in TEM images. Staining with strong oxidizers such as RuO4 or OsO4 is often used to distinguish polymer phases. However, staining has several disadvantages in that changes in morphology occur due to swelling of one of the polymeric components and the change of chemical composition from staining may prohibit quantitative elemental analysis. One main problem arises due to the limited stability of polymers under electron irradiation. In order to reduce mass loss, analytical investigations of polymers have to be performed at low temperatures (−180◦ C) [6.34,6.35]. Because the rate at which the chemical structure changes during analysis increases with the current density for constant total doses [6.219], investigations should be performed at the lowest possible dose. Therefore, several low dose exposures that are summarized afterwards are preferable. In situations where statistically signiﬁcant core-loss maps cannot be obtained, a plasmon ﬁngerprint based on valence electron excitations is useful provided characteristic features are present in the spectrum (see Fig. 6.23). Phase separation and morphology of polymer blends and block copolymers have an immense inﬂuence on material properties and are therefore of great theoretical, experimental and industrial interest. With regard to the length scales of phase separation, TEM is ideal for such characterization. Often though, similarity of component composition renders contrast insufﬁcient for imaging. Structural information can then only be supplied by energy ﬁltering TEM. Energy ﬁltered imaging experiments using oxygen distribution maps have been successful for identifying phases in a polymer blend [6.220] and for characterizing the morphology of polyamide/polycarbonate blends [6.221]. A typical example of this kind of investigation is shown with the characterization of an unstained polyamide/polyphenylene-ether-blend (Fig. 6.23). The distribution of the polyamide phase could be visualized by elemental mapping with the N K-edge and that of the polyphenylene–ether, by using the characteristic 7-eV π-π ∗ valence transition peak found in phenyl rings but which is absent in polyamide [6.222]. Du Chesne has investigated a poly(vinyl acetate) latex stabilized with sodium dodecyl sulfate and copolymerized sodium ethenesulfonate [6.223]. The emulsiﬁer could be located at the particle boundaries by recording S L2,3 elemental maps. The chemical and mechanical properties of natural rubber or rubber-like polymers can be enhanced by adding various ﬁllers and additives. EFTEM can distinguish

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Fig. 6.23 Energy ﬁltered imaging of polyamide/poly(phenylene ether) blend, (a) TEMbright ﬁeld image; (b) EELS spectra of PA and PPE; (b) π ∗ map in the low loss region revealing the PPE distribution; (c) carbon K elemental map; (d) nitrogen K elemental map revealing the polyamide; from [6.222].

between the different ﬁllers, for example, soot, TiO2 , Fe2 O3 , ZnO, SiO2 , and so on, simply by recording elemental maps or jump ratio images [6.224,6.225]. Additionally, energy-ﬁltering was successfully applied to the investigation of rubber-brass adhesion, which plays an important role in tire production. The distribution of nanocrystalline Cu and Zn sulﬁdes in the bonding layer was visualized by elemental mapping using the L2,3 -edges of copper, zinc, and sulfur [6.226]. A topic of growing importance is structuring of inorganic materials by their interaction with polymers. Here EFTEM has the potential to play a central role in the characterization of these materials. A typical system was investigated by Du Chesne [6.223]: Unstained ultrathin sections of a diblock copolymer of polyimideblock-poly(ethylene oxide) treated with an inorganic precursor-material were studied with EFTEM. Si and Al L2,3 elemental maps were used to visualize the alumino-

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silicate phase with incorporated poly(ethylene oxide) chains covalently connected to the polyimide matrix. However, it is important to note that the energy-ﬁlter captures the sample in different states of radiation damage, spectrum imaging in a STEM records a complete spectrum at one time at a single radiation dose. Therefore, previous studies of polymer blends have often been performed with a STEM probe and EELS spectroscopy, for example, investigation of a rubber/Nylon blend [6.227], the characterization of a polyethylene/polystyrene blend [6.228] and a nylon 6/polyethylene blend [6.229]. 6.5.6

Heterogeneous Catalysts, Colloids and Clusters

Accurate knowledge of catalyst composition and microstructure at the nanometer scale is critical to a fundamental understanding of the chemistry occurring during heterogeneous catalysis. In many cases, there is a direct relationship between microstructure and catalytic properties that may depend on the presence and distribution of promoter elements. Additionally, it is often essential for metal particles to be highly dispersed over a supporting substrate. Consequently, direct visualization of heterogeneous catalyst morphology on the nanometer scale can help in identifying critical factors in catalyst synthesis and design. Characterizing modern industrial catalysts can be difﬁcult though, due to the complex nature of their morphologies. There are often many different morphologies and varying particle sizes that make up the catalyst and it is important to characterize relatively large areas and relatively large numbers of nanometer sized particles to obtain information that is statistically signiﬁcant. Moreover, it is not uncommon for modern industrial catalyst to have 6–10 elements in widely differing concentrations. Crozier demonstrated that EFTEM is well suited for the characterization of industrial heterogeneous catalysts [6.221,6.222]. In this work an unused NiCuCr amination catalyst consisting of NiCuCr particles dispersed over a γ-alumina substrate was investigated (Fig. 6.24). In order to obtain uniformly ﬂat specimens, a prerequisite for using EFTEM, the samples were prepared for TEM by ultramicrotomy of resin embedded pellets that were cut to a nominal thickness of 50 nm. By recording elemental maps of the elements Cu, Ni and Cr and calculating atomic ratio maps, Crozier could show that 90% of the Cu was found in particles with a Cu/Ni ratio of ∼ 0.16 and 10% of the Cu was found in particles with a measured Cr/Ni concentration ratio close to unity. The Cr was found to exist in the form of small oxide particles 10–20 nm in size decorating the surface of Ni particles (see Fig. 6.24). Nobbenhuis et al. characterized titania supported vanadia catalysts (V2 O5 /TiO2 ) which are used in various reactions [6.223]. Vanadium maps revealed that the titania support particles are covered by nanosized vanadium oxides, but a thin amorphous vanadia layer, a so-called monolayer as suggested by other authors could not be detected. A more favorable situation for EFTEM occurs when nanosized particles, for example, transition metal or oxides, are distributed on thin supporting ﬁlms (e.g., 2nm carbon); they can then be easily detected by using the N2,3 - and the M2,3 -edges, respectively [6.86]. Recently, Pt/Sn catalysts were studied by employing EFTEM

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Fig. 6.24 CuNiCr catalyst on γ -Al2 O3 support; (a) TEM bright ﬁeld image of an ultramicrotome section; (b) nickel distribution; (c) copper distribution; (d) chromium distribution; (e) overlay of nickel (gray) and chromium (bright) distributions showing that the chromium nanoparticles are located on the surface of the nickel grains; (courtesy P. Crozier [6.230]).

[6.233]. It was possible to show by recording Pt O2,3 elemental maps that Pt and Sn are unevenly distributed, but it was not possible to map tin unequivocally due to the overlap of the Sn N4,5 -edge with the plasmon peak. By employing cryo-EFTEM, Bovin et al. could image the aggregation behavior of palladium nanoparticles while still in solution [6.234]. This technique allowed the study of the arrangement of colloidal palladium particles in solution (as a function of pH and ionic strength) by preparing the specimen by the plunge freezing technique. Although EFTEM has been very useful for the characterization of catalysts, one disadvantage of EFTEM has to be taken into account: In many catalysts, heavy

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elements can be present that are not always easily accessible by EELS or EFTEM elemental mapping. For example, the metals hafnium to bismuth can be analyzed by using either the O2,3 -edges ranging from 30 to 92 eV or the M4,5 -edges (from 1661 to 2580 eV). In the ﬁrst case, the SNR is low due to the vicinity of the plasmon-peak and in the second case, the intensities are too low thus requiring very high electron doses which very often lead to melting of the reactive nanoparticles. For these elements, EDX-spectrometry in a STEM seems to be more appropriate for elemental mapping. For example, Walther et al. investigated gold nanoparticles on titanium oxide by recording elemental maps using the Au O2,3 -, N4,5 -, and M4,5 -edges [6.87]. It was found that the M4,5 -edge could be used to map the gold distribution, using a series of cross-correlated images acquired over several minutes. Small gold particles down to 1-nm diameter (30 atoms) were detectable. 6.5.7

Powder Materials

The above mentioned problems in catalyst characterization may also play a signiﬁcant role in the investigation of powder specimens with particle sizes ranging from a few nm to some 100 nm. Since particles with diameters of > 100 nm lead to multiple inelastic scattering that prohibits elemental mapping, specimen preparation is again a very important step. In order to get uniformly ﬂat specimens resin embedding of the powders and ultramicrotomy or ion milling have proven very useful in this respect. Kurata et al. investigated PbCrO4 particles that were covered with amorphous SiO2 ﬁlms [6.235]. The ﬁne particles could be discriminated from the SiO2 ﬁlm in the oxygen map by means of ﬁne structure features of the O K-edge. Binder et al. characterized titanium-doped electrolytic MnO2 , which is used in primary Zn-MnO2 batteries [6.236]. Ti L2,3 elemental maps revealed that the majority of titanium was uniformly distributed in MnO2 , but segregated areas with signiﬁcantly higher titanium concentrations were detected as well [6.237]. EFTEM elemental mapping is a valuable and complementary technique on the nanometer scale to X-ray diffraction and microprobe analysis for evaluation of phase purity and homogeneity of all solid-state materials (see Fig. 6.25). It is particularly valuable for studying products of sol–gel or other alternative synthetic methods, especially where elemental homogeneity is a major driving force for the synthesis choice. Recently, EFTEM has been used to characterize the purity and phase composition of perovskite- and spinel-type oxides [6.238] (see also Fig. 6.25) and of Sn/SnSb composite electrodes for lithium ion batteries [6.239]. Silver halide microcrystals are the basic material in photographic science. Their design has become more and more complex and recently it was found that the introduction of iodide increases sensitivity. Lavergne et al. could show that these crystals exhibit a core-shell structure, that is, a center part of pure AgBr and an external shell of AgBrI by recording element maps with the M4,5 edges of silver and iodine [6.240]. EFTEM can also be of use for the investigation of environmental particulate samples like for the characterization of acquatic iron oxyhydroxide particles [6.241]. Furthermore, the layer structure of vanadium oxide nanotubes was studied by using

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Fig. 6.25 Checking the homogeneity of the complex spinel CdCr1.5 In0.5 O4 prepared from nitrate precursors; (a) TEM image and (b) In M4,5 jump ratio image revealing the occurrence of two phases with different indium concentration.

V L2,3 elemental maps thus revealing that the bent vanadium oxide layers inside the tube walls are “scrolls” rather than concentric cylinders [6.242]. Inorganic particles in biological samples like lung tissue are of special interest both for medical diagnostic purposes and for exploring the environmental situation of recent human beings and ancient mummies. EFTEM is playing a pivotal role in the characterization of the ﬁne deposits in human lung tissue, because elemental mapping enables the rapid analysis of extended specimen areas [6.243,6.244]. Once a particle is recognized, it can be easily analyzed using standard electron microscopic techniques such as EDXS, electron diffraction and EELS. In the case of the 5300 year old glacier mummy found in the Tyrolean Alps, many different deposits could be found in the lung tissue that would have been overlooked with conventional EM-techniques, for example, soot, muscovite, plagioclase, silica, illite, iron oxides, apatite, iron phosphates and even corn threshing residues [6.245]. Various examples are shown on the enclosed CD-ROM. 6.6 CONCLUSIONS AND OUTLOOK We have seen that EFTEM elemental mapping is a powerful tool for the advanced characterization of all solid-state materials. However, we have to consider the limitations as well. Although simple elemental maps are very useful and can even be quantiﬁed to give concentration maps, they often need to be supplemented by recording either complete image EELS series or EELS spectra from distinct specimen regions. What can we expect in the future? We will probably see rapid progress in the development of subnanometer analysis by means of EFTEM, which will be mainly

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catalysed by signiﬁcant improvements in instrumentation, that is, microscopes with correction of spherical aberrations [6.246], improved energy ﬁlters with correction of higher order aberrations and hence higher transmissivity [6.247] and the introduction of monochromators aimed at EELS with high energy resolution [6.248,6.249]. Additionally, we already have improvements in EFTEM data processing [6.250–6.252] and interpretation of EFTEM images at high resolution, that is, new theoretical approaches for computing the contrast in lattice images formed by inelastically scattered electrons [6.253]. Consequently, element speciﬁc imaging and high-resolution electron microscopy will grow together thus presenting exciting new insight into speciﬁc problems of materials science. For example, high resolution EFTEM will open up a whole new ﬁeld of experimental interface and grain boundary chemistry at the atomic level. Acknowledgments The work presented in this paper is the result of a fruitful collaboration with our colleagues around the FELMI–EFTEM: Werner Grogger, Gerald Kothleitner, Ilse Papst, Martina Albler, Albert Brunegger, Irmgard Rom, and Wolfgang Geymayer. Thanks are to our visitors for helpful discussions and supplying interesting specimens: Joachim Mayer, Rik Brydson, Louise Coast-Smith, Mark Rainforth, Helmut Kohl, Kathy Barmak, Josef Zweck, Hugo Ortner, Peter Schattschneider, Jenni Zackrisson, P Sajgalik, Horst Cerjak, Christian Mitterer, Maria Anna Pabst, Othmar Leitner, Harald P. Fritzer, Leo Binder, Bernd Kolbesen, Alois Popitsch, and Otto Fruhwirth. We are grateful for ﬁnancial support from the Forschungsf¨orderungsfonds der Gewerblichen Wirtschaft, Vienna and the Forschungsf¨orderungsfonds der Wissenschaftlichen Forschung, Vienna (SFB-project “Elektroaktive Stoffe”).

REFERENCES 6.1. R. F. Egerton, “Electron Energy-Loss Spectroscopy in the Electron Microscope” (Plenum, New York, 1996). 6.2. L. Reimer (ed.) “Energy-Filtering Transmission Electron Microscopy” (Springer Verlag, Berlin, 1995). 6.3. C. Jeanguillaume and C. Colliex, “Spectrum image: the next step in EELS digital acquisition and processing,” Ultramicroscopy 28, 252 (1989). 6.4. J. A. Hunt and D.B. Williams, “Electron energy-loss spectrum-imaging,” Ultramicroscopy 38, 47 (1991). 6.5. J. Mayer, U. Eigenthaler, J. M. Plitzko, and F. Dettenwanger, “Quantitative analysis of electron spectroscopic imaging series,” Micron 28, 361–370 (1997).

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6.249. S. Uhlemann and M. Haider, “Experimental set up of a fully electrostatic Monochromator for a 200 kV TEM,” ICEM-15 Durban 2002, Vol. 2, 327. 6.250. G. Kothleitner, F. Hofer, and C. Trevor, “Difference spectrum images: numerical ﬁlters applied to EELS 3D data sets“, Microsc. Microanal. 7 (Suppl. 2: Proceedings) Microscopy Society of America (2001) pp. 1160–1161. 6.251. P. J. Thomas and P. A. Midgley, “Image-spectroscopy I: the advantages of increased spectral information for compositional EFTEM analysis,” Ultramicoscopy 88, (2001) 179–186. 6.252. P. J. Thomas and P. A. Midgley, “Image-spectroscopy II: the removal of plural scattering from extended energy-ﬁltered series by Fourier deconvolution,” Ultramicroscopy 88, (2001) 187–194. 6.253. T. Navidi-Kasmai and H. Kohl, “Computation of contrasts in atomic resolution electron spectroscopic images of planar defects in crystalline specimens,” Ultramicroscopy 81, (2000) 223–233.

7

Probing Materials Chemistry Using ELNES Richard Morgan Drummond-Brydson, Hermann Sauer∗ and Wilfried Engel∗ Department of Materials School of Process, Environmental and Materials Engineering University of Leeds Leeds LS2 9JT, U.K. ∗

Fritz-Haber-Institut der Max-Planck-Gesellschaft Faradayweg 4-6 14195 Berlin Germany

Abstract Qualitative and semiquantitative analysis of the electron energy loss near-edge ﬁne structure (ELNES) associated with a particular core loss edge can, in favorable cases, provide information about the structural and chemical properties of the atom undergoing excitation. It is demonstrated that, in many cases, the observed ELNES exhibits a shape that represents a ﬁngerprint of the nearest-neighbor coordination and, where two differing coordinations coexist in a material, it is often possible to quantify the relative site occupancies. Furthermore, it is shown that often the ELNES is also sensitive to the valence state of the excited atom; this may take the form of a chemical shift of the edge onset arising due to changes in the relative energetic positions of the initial and ﬁnal states of the transition or, alternatively, a redistribution of the ELNES intensity in situations where the resultant near-edge structure is sensitive to electron–electron and core hole–electron interactions in the ﬁnal state. Coordination and valency are simple chemical concepts which allow a clear description of the local bonding of a particular atom and, investigating the orientation dependence of such effects, permits the anisotropy of such bonding in crystalline systems to be elucidated. However, the methodology behind such techiques needs to be supported by both systematic measurements on suitable reference materials such as minerals, which possess a wide range of structural and electronic properties, as well as relatively simple electronic structure calculations. ELNES measurements may then be applied to determine the local bonding in unknown phases existing within bulk materials as 223

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well as that associated with spatially localized features such as interfaces and defects which may be ultimately property-determining on the macroscopic scale. 7.1 INTRODUCTION In the study of solid-state samples, electron energy loss spectroscopy (EELS) in the transmission electron microscope (TEM) provides a valuable method of quantitative nanochemical analysis that may be combined with the results of energy dispersive X-ray (EDX) analysis to give a comprehensive elemental determination across the whole periodic table [7.1–7.3]. However, additional insight into the chemical and structural properties associated with the atom undergoing excitation may be gained via both qualitative and semiquantitative analysis of the electron energy loss nearedge structure (ELNES) associated with each core loss ionization edge [7.4,7.5]. In favorable cases, analysis of the ELNES allows the determination of two features of signiﬁcant interest: ﬁrst, nearest-neighbor coordinations in complex structures and secondly, the valence state of the atom undergoing excitation. This approach facilitates the rapid identiﬁcation of unknown phases in complex microstructures, the semiquantitative determination of atom site occupancies, as well as the determination of the local bonding at interfaces and defects using spatially resolved ELNES measurements [7.6]. In order to develop a sound methodology for the use of such techniques it is necessary to perform measurements on reference compounds with known structural and ‘chemical’ properties. Minerals constitute an extremely good class of diverse materials: they have often been well characterized by a range of complementary techniques and, in many cases, are sufﬁciently resistant to electron beam induced damage as to allow accurate measurements to be made. Garvie et al.,[7.7] provides a good review of this subject area. This methodology for the extraction of bonding is in essence, semiempirical, but can be supported by electronic structure calculations ranging from the complex to the relatively simple. 7.2 EXPERIMENTAL CONSIDERATIONS In order to investigate and characterize the modiﬁcations in the ELNES associated with changes in coordination and valence state of a particular atom, the importance of recording spectra with good, accurate statistics and good energy resolution can not be stressed too highly. Parallel detection systems, while providing good detective quantum efﬁciency, inﬂuence the data by adding additional noise and signal to the raw data as well as having a non-uniform response across the length of the detector as discussed in the chapter by Egerton. A number of routine corrections are needed prior to further signal processing. The ﬁnite detector dark count, which is the detector read-out signal in the absence of illumination, is best measured immediately prior to, or after, data acquisition using the same acquisition parameters. This avoids the problem of “ghost peaks" caused by overexposure of the detector system. The non-uniform

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channel-to-channel gain appear as both high and low frequency variations. The high frequency components appear as noise on the as-recorded spectrum and correcting these variations is crucial for optimizing detection sensitivity. Uncorrected, the lower frequency components may be misidentiﬁed as ELNES, but may be readily identiﬁed by displacing the spectrum across the detector array. Two methods of gain correction are frequently employed, one involving uniform illumination of the detector to produce a gain normalization function for subsequent correction of spectra, and the second involving shifting of the spectrum across the array by applying small voltage adjustments to the spectrometer drift tube [7.8]—separate spectra are then realigned and summed. Energy resolution is determined by a number of factors but depends largely on the energy spread of the electron source. For cold ﬁeld emission sources a resolution of 0.3 eV is attainable, thermally assisted ﬁeld emission sources routinely provide 0.8 eV, while LaB6 sources result in resolutions of between 1 and 1.5 eV. Wien ﬁltering and monochromation is also possible but not yet commercially available. It is important to realize that for a given spectrometer system focused at the zeroloss peak, the energy resolution degrades with increasing energy loss. One option for the recording of high energy loss edges is to focus the spectrometer using a sharp feature such as a white line, although care should be exercised so as not to introduce apparent features in the measured ELNES. For a given energy resolution, for ELNES measurements it is generally best to use the highest possible dispersion during measurement, typically 0.1–0.3 eV/channel, provided this does not unduly degrade statistics. Further corrections to the data which are necessary in high resolution systems are due to the apparent degradation of the resolution due to the spreading electron beam and light in the scintillator. The system point spread function, measured by reducing the dispersion such that all the elastically scattered electrons in the zeroloss peak are focused to a spot smaller than the physical dimension of a single diode, reﬂects the spreading of the beam by the scintillator and may be used to routinely deconvolve all spectra using Fourier techniques. Additionally ELNES data may be sharpened by deconvolving the measured zero-loss peak, but this may be limited by noise considerations. An accurate means of absolute energy calibration is also desirable if the small energy shifts arising due to changes in the charge distribution are to be measured and compared. One method is to monitor the position of an edge relative to a second edge measured in the same spectrum; advantitious carbon contamination can be useful in this respect. Another way is to apply a known voltage to the electron drift tube of the spectrometer (or, in some machines, change the HT of the microscope) and calibrate by reference to a known edge energy (e.g., the Ni L3 -edge in NiO) [7.9]. The voltage required to displace the ELNES structure of interest to the zero loss peak can then be used to determine edge positions to an accuracy of around ± 0.3 eV. Future methods may rely on a self-compensating feedback system between the HT supply and the spectrometer drift tube [7.10]. The majority of the spectra shown here were measured using a dedicated scanning transmission electron microscope (STEM) ﬁtted with a cold ﬁeld emission gun, a sector ﬁeld spectrometer and parallel recording system based on either a silicon

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intensiﬁed target detector [7.11], a CCD system or a diode array. At best the resultant energy resolution can be as low as ∼ 0.3 eV and absolute energy values may be determined to within ± 0.2 eV by direct measurement of the high voltage using a precision high voltage divider. In practice, once such differences in near-edge structure have been identiﬁed and catalogued, it may be possible to employ a lower energy resolution and statistical accuracy in routine use. Inelastic scattering results in a Lorentzian angular distribution with a characteristic angle given by ΘE = ∆E/2E0 . To optimize sensitivity for a given edge an aperture size (either objective aperture in CTEM image mode, spectrometer aperture and camera length in CTEM diffraction mode or collector aperture in STEM) should be chosen that accepts the majority of inelastically scattered electrons. However, increasing the collection angle, β, increases both the background beneath the edge and the fraction of electrons that have undergone non-dipolar transitions. It is important to collect electrons that have transferred small amounts of momenta. This ensures that the resultant ELNES are then dominated by dipole allowed transitions. ELNES spectra may then be directly compared with those obtained using X-ray absorption spectroscopy (XAS and XANES) and more easily modeled using theoretical electronic structure calculations since the ﬁnal states of the excitation process are then given, in the atomic picture, by ∆l = ±1, where ∆l is the change in angular momentum quantum number between the core level and the unoccupied ﬁnal states. Overall a good compromise is to employ a collection angle of ∼ 2–3 times ΘE . Processing of ELNES data should be performed on data that has been background subtracted. This can be difﬁcult, especially for low energy edges and various strategies for background subtraction are discussed in the chapter by Egerton. An optimum sample thickness, t, for detection sensitivity and signal/noise ratio, while avoiding the effects of excessive multiple inelastic scattering, is ∼ 0.5 t/λ. However, even if multiple inelastic scattering effects are small, wherever possible spectra should always be deconvoluted using the standard techniques [7.1,7.12] if accurate comparisons and modeling of near-edge features is to be achieved. The latter is particularly true if quantitative measurements of site occupancies are to be obtained. Finally, if possible, ELNES measurements should be performed on polycrystalline sample areas so as to avoid orientation-dependant effects—unless, as is discussed later, such information is required for the assignment of near-edge features and the extraction of directional bonding information. There are various techniques for extraction of orientation dependent ELNES. The simplest in the CTEM is to use parallel illumination and suitably orient the sample with respect to the electron beam, use of a small collection angle then isolates the parallel component of the scattering vector [7.13]. Other methods involve measurements at two different collection angles followed by subsequent retrieval of the two separate components [7.14] and ﬁnally, orientation of sample with, say, the c-axis at 45◦ to the electron beam and measurements of the two separate components by displacing the collector aperture in the STEM [7.15,7.16]. The spatial resolution of ELNES measurements depends critically on the probe size, probe current, and the radiation sensitivity of the material under investigation. Certain studies in optimized experimental conﬁgurations have shown the feasibility

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of sampling single atomic columns [7.17,7.18], however, the general applicability of such an approach will undoubtedly be limited by the sample itself, both in terms of its sensitivity to electron-beam induced damage and drift during measurement. An alternative approach is to employ the spatial difference technique that is essentially a difference spectrum formed by monitoring changes in ELNES intensity with respect to changes in (a larger) beam position or spatial coordinate. It is generated by numerically determining the difference between two (or more) spectra recorded at different locations in the sample and can reveal changes in composition and bonding [7.19]. A schematic diagram showing the procedure for extracting the ELNES signal at a metal-ceramic interface is shown in Fig. 7.1; the techique is discussed in more detail in Section 6. Such an approach has revealed low levels of segregant at grain boundaries and defects [7.20,7.21] and been extended to monitor changes in bonding across a boundary at nanometre resolution with better than monolayer sensitivity [7.22,7.23]. Because the ‘difference’ approach highlights small changes in ELNES, care has to be taken to exclude possible artifacts such as damage and instabilities in the energy axis and beam position. Such changes appear as derivative components. Most artifacts can be identiﬁed by repeating measurements in a different sequence, as well as acquiring spectra as a series of time, to ensure sytematic instabilities are not a cause for concern. Spectrum imaging, either as one-dimensional line proﬁles or two-dimensional maps across sample regions, can also provide information on the spatial dependence of ELNES features although datasets tend to be relatively large. The application of statistical tools such as multivariate statistical analysis (MSA) and neural networks are ideally suited to a series of linearly superimposed datasets representing the different bonding environments dataset of n spectra are represented as a matrix that can be transformed into a set of n eigenvectors, each with an eigenvalue. These are then separated into those m components that are statistically signiﬁcant and those nm that reﬂect noise in the data. Each spectrum in the original dataset can then be decomposed into the m principal components, each of which simply represent variations in ELNES relative to a mean spectrum. These principal components can then be transformed into a physically interpretable dataset. The spatial difference technique outlined above can essentially be regarded as reduced dataset suitable for MSA analysis, consisting of two or possibly three spectra. Finally, both 2D spectrum imaging and electron spectroscopic imaging, see Chapter 6, can be used to map out the variation in intensity of a particular ELNES feature within a given energy window, as a function of position within the microstructure so as to extract chemical bond maps. An example is given in [7.26]. 7.3

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The theoretical background behind ELNES is throughly discussed in Chapter 4 of this book. Essentially ELNES probes the local symmetry-projected unoccupied density of states (DOS) at a particular atomic site within the solid (which may be modiﬁed

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Fig. 7.1 A schematic diagram showing the procedure for extracting the ELNES signal at a metal–ceramic interface using the spatial diffence procedure.

to a greater or lesser extent by many-body effects often associated with the creation of the core hole in the inner-shell). In many cases, it is found that the observed ELNES exhibit a structure speciﬁc to the arrangement and type of atoms in the ﬁrst coordination shell—a so-called coordination ﬁngerprint [7.27]. This principally arises when the local DOS available to the excited electron is dominated by the interaction of the central atom with the nearest neighbors. In terms of a (elastic) scattering picture of the excited electron, an ELNES ﬁngerprint may be thought of as a characteristic diffraction pattern (with varying energy, and hence wavelength rather than angle) created by the local atomic environment. The reasons for the existence of such coordination ﬁngerprints may

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be understood by consideration of the following points. In solids, molecules or ions can interact with each other to form bands of orbitals that extend throughout the structure. The degree of intermolecular or inter-ion interaction directly governs the energy widths of these bands and therefore it is often found that the band structure of most molecular and ionic solids can be directly related to the molecular orbital (MO) structure of the individual molecules or ions [7.28]. Stohr et al. [7.29] have shown how the XANES (the X-ray equivalent of ELNES) of macromolecules and polymers may be understood by use of a simple building block picture. The spectra of complex molecules may be regarded as a superposition of the spectra of diatomic and pseudodiatomic building blocks. Similarly, once the ﬁngerprints of functional groups have been identiﬁed they may be used to interpret the spectra of macromolecules, and hence molecular solids. More explicitly, in complex structures with large, complicated unit cells, there exists very little order outside the ﬁrst coordination shell. This lack of long and medium range order means that the resultant ELNES is dominated by scattering from nearest neighbor atoms. Furthermore, certain atoms/ions are relatively strong backscatterers, a notable example being electronegative species such as the O2− ion [7.30], and where such species occur in the ﬁrst coordination sphere then this will give rise to a potential cage or barrier for electron scattering [7.31]. Useful counter examples of coordination ﬁngerprints are provided by the O KELNES in many oxides [7.32] and the N K-ELNES in metal nitrides [7.33]. Here the central atom (e.g., oxygen or nitrogen) is surrounded by weakly scattering cation species which means that the excited electron probes a much greater range resulting in information from next-nearest neighbors and beyond. Scattering from a large number of atoms at much larger distances has to be considered in order to adequately model spectra. Another consideration important for the understanding of coordination ﬁngerprints involves the effect of the unscreened core hole potential produced during the excitation process. In certain cases, especially in insulators where the screening is considerably less than in metals, the effect of the core hole potential will be to relax the energy levels of the available ﬁnal states to lower energies particularly near the edge-onset. This in turn causes the resultant ELNES in this region to be dominated by transitions to ‘bound states’ formed from the interaction of the central atom with the nearest neighbors [7.34]. For ﬁnal states with a large value of the angular momentum quantum number, l, the potential is dominated by the centrifugal term and this results in a large transfer of oscillator strength to higher energies [7.1]. Furthermore, Colliex [7.2] has discussed the plasmon-like proﬁles of edges in the 20–50-eV domain that show little sensitivity to the environment of the atom undergoing excitation. Consequently, spectra originating from deeper lying core levels (e.g., K- or L-edges) generally exhibit more characteristic differences in near-edge structures between different coordinations than those from higher lying initial states (e.g., M - or N -edges), which have a larger overlap with continuum wavefunctions. ELNES coordination ﬁngerprints may be determined by the comparison of EELS spectra obtained from a range of reference materials possessing the same coordi-

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nation of a particular atom. As already mentioned, minerals provide a wealth of well-documented examples in this respect. Alternatively, ELNES measurements, recorded under the correct experimental conditions, may be directly compared with XAS measurements, especially those made on gaseous or solution phase species where the gross contribution from the nearest neighbor shell is clearly apparent. Measurements on amorphous materials also provide an indication of the contribution from the nearest-neighbor shell since the lack of long-range order will minimize the contribution from second and higher neighbor shells. An advantage of XAS measurements arises due to the relative ease with which polarized measurements on single crystals may be made. Similar experiments may be performed in EELS using a number of techniques outlined in Section 2 such as suitable choice of the scattering angle and orientation of a single-crystal specimen [7.15]. The orientational dependence of a particular near-edge feature allows the determination of the symmetry of the resultant ﬁnal state and this greatly aids subsequent interpretation; this aspect is discussed in Sections 2 and 5.3. Once the coordination ﬁngerprint of a particular atom has been identiﬁed, interpretation follows via comparison with the results of theoretical electronic structure calculations [7.35]. The large range of available techniques are throughly reviewed in Chapter 4, however, the important thing to note here is that one is looking for common patterns in the electronic structure which reﬂect the local atomic environment. By far the simplest and often most illuminating approach is the assignment of the various near-edge features to transitions to speciﬁc molecular orbitals formed by the interaction of atomic orbitals on the central atom in the polyatomic cluster with those from the nearest-neighbor atoms. MO calculations may be performed by a number of methods at different levels of approximation. The traditional approach is the Self-Consistent Field Linear Combination of Atomic Orbitals (SCF–LCAO) method. Such calculations have been successfully used to interpret the X-ray emission (XRE) spectra of a variety of solids [7.36,7.37]. However, for the calculation of unoccupied orbitals the SCF-Xα multiple scattered wave method is inherently superior [7.38]. This approach allows the calculation of transitions to both bound and continuum states and has been shown to give good agreement with XAS data from solids as well as molecules [7.39,7.40]. A very similar approach is the discrete variational Xα which has been adapted to ELNES modeling [7.41]. The nature of the ﬁnal state MOs is essentially governed by the symmetry of the polyatomic cluster and the number of valence electrons [7.42]. It therefore follows that a group of isoelectronic species which possess the same symmetry will exhibit very similar ELNES (e.g., the central atom L2,3 -edges of PO43− , SO42− and ClO4− [7.5]). An alternative approach is to employ the results of multiple scattering (MS) calculations performed using the non self-consistent ICXANES computer code of Vvedensky et al. [7.43] or the self-consistent FEFF8 code of Rehr et al. [7.44]. This method allows the real space cluster size (number of shells) to be progressively increased, and consequently it becomes possible to identify the importance of speciﬁc scatterers and scattering paths in determining the resultant near-edge features. While formally equivalent, in the limit, to a Korringa–Kohn–Rostocker (KKR) band structure calculation, this approach is considerably more ﬂexible allowing the cluster

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geometry to be changed and effect of the core hole potential to be included by use of various approximations for the central atom [7.45,7.46]. The scattering properties of the atoms are described by phase shifts that can now be calculated self-consistently by imposing a mufﬁn tin potential on the structure in question, although problems may arise in covalent systems that possess directional bonding. It is also possible to employ the symmetry projected DOS derived from band structure calculations to describe the origin of the features present in a coordination ﬁngerprint [7.47]. However, such an approach requires considerable computational effort in order to determine common electronic structure patterns for related systems. Many previously published band structure calculations do not include the higher unoccupied levels as well as the site and symmetry projections necessary for comparison with experiment. However, recent advances in pseudopotential [7.48] and full linear augmented plane wave (FLAPW) codes [7.49] and, most importantly, in accessible, high-perfomance computing power have made these techniques more attractive including the ability to handle large supercells containing excited atoms so as to account for the effect of the core hole. Finally, for certain edges, such as the “white line” edges of transition metals and rare earths and their compounds, due to the strong interaction between the core hole and the highly localized ﬁnal state the most appropriate description is in terms of short range interactions described by atomic multiplet theory [7.50]. 7.3.1

Review of ELNES Fingerprints

A number of speciﬁc examples of coordination ﬁngerprints accessible to EELS are now presented. These have been determined by both EELS and XAS measurements on a variety of materials that include both naturally occurring and synthetic compounds. While this subjective review covers many of the commonly found coordinations of light elements, it is not intended to be exhaustive. 7.3.1.1 Cation Fingerprints Extensive XAS [7.51,7.52] and EELS [7.53–7.56] measurements on a range of boron compounds have revealed the existence of a boron coordination ﬁngerprint which provides a good illustration of the technique. Figures 7.2a and 7.3a show the boron K-edges measured from the borate minerals vonsenite and rhodizite, respectively. In vonsenite, Fe3 BO5 , boron is in trigonal planar coordination to oxygen while in rhodizite the coordination is tetrahedral. The marked differences arise due to the way the MOs containing boron 2p character are constructed in the two differing symmetries [7.37]. In tetrahedral symmetry they remain energetically degenerate and the main peak present in Fig. 7.3a corresponds to transitions to σ ∗ (σ-antibonding) MOs. However, in planar trigonal symmetry they separate into a π ∗ (π-antibonding) and two σ ∗ MOs. The transition to the π ∗ MO is evident as a sharp peak at 194 eV in Fig. 7.2a, while the transitions to σ ∗ MOs occur some 10 eV higher in energy. The postulate that these spectra are representative of BO3 and BO4 units is conﬁrmed by the results of multiple scattering calculations for BO3 and BO4 clusters shown in Figs. 7.2b and 7.3b, respectively.

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Fig. 7.2 (a) Boron K -ELNES of the mineral vonsenite after background subtraction. (b) The results of ICXANES calculations for a single shell of oxygen atoms trigonally coordinated to a central boron atom, core hole effects have been included.

Fig. 7.3 (a) Boron K -ELNES of the mineral rhodizite after background subtraction. (b) The results of ICXANES calculations for a single shell of oxygen atoms tetrahedrally coordinated to a central boron atom, core hole effects have been included.

Furthermore, for each coordination, the general shape of the B K-edge spectrum is observed in other compounds where the coordination is known to be similar. Garvie et al. [7.56] have studied the B K-ELNES from a large variety of boron minerals, containing both BO45− and BO33− ions, and have conﬁrmed these original ﬁndings presented here. The B K-XANES of NaBF4 and KBF4 [7.52], both of which possess boron in tetrahedral coordination, exhibit the same structure as that in Fig. 7.3a, while the B K-XANES of gaseous BF3 , which is isoelectronic with BO33− , also has the same form as Fig. 7.2a [7.57]. The B K-ELNES of hexagonal and cubic BN also show similar forms to those in Figs. 7.2a and 7.3a reﬂecting the respective trigonal and tetrahedral bonding arrangements [7.56]. As a general rule, changing the ligand type affects the energy positions of the various features in the coordination ﬁngerprint, that is in hexagonal BN the π ∗ feature is at ∼ 192 eV, which is 2 eV lower than in B2 O3 , reﬂecting the lower electronegativity of the ligand species. The π ∗ feature present in Fig. 7.2a, which is indicative of trigonally coordinated boron, actually corresponds to a bound state lying below the original position of the conduction band edge at 195.7 ± 0.5 eV (derived from XPS measurements together with the band gap obtained from low loss measurements) [7.58]. Consequently,

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this feature is also seen in boron Kα XRE (X-ray emission) spectra of trigonally coordinated boron minerals [7.59] and also provides a means of determining the nearest-neighbor coordination. The B K-ELNES of many metal borides show a similar π ∗ -σ ∗ structure owing to the fact that the boron coordination is often trigonal prismatic [7.13,7.56,7.60], however, in this case the structure is considerably broadened and shifted to lower energy which allows the two environments to be readily distinguished. There exists a large number of minerals containing magnesium, silicon and/or aluminium in either octahedral or tetrahedral coordination to oxygen. Tossell [7.61, 7.62] has calculated the resultant MO structure of these elements in the two coordinations. Experimentally, Taftø and Zhu [7.63] have studied the K-ELNES of compounds containing these elements in an attempt to identify a coordination ﬁngerprint. They observed that the metal K-ELNES exhibited a more prominent peak at the edge in the case of octahedral as opposed to tetrahedral coordination. This has been found to be a general effect found in many metal K-ELNES spectra (e.g., Li, Mg, Al, Si,...) for the two coordinations and appears to arise due to the increased number of nearest-neighbor scatterers in octahedral coordination giving rise to a much more well deﬁned intrashell multiple scattering peak at the edge onset. Detailed ELNES shapes are found to depend on not only distortions from perfect symmetry (see Section 5.2), but also on the arrangement of next nearest neighbors and beyond, however, the gross underlying structure can invariably be classiﬁed according the the nearest neighbor arrangement. Owing to its high energy, the majority of high resolution studies of the Si K-edge have been performed by XAS [7.64–7.66] and these have conﬁrmed this general ﬁngerprint for octahedral and tetrahedral coordination. Using EELS, Hofer [7.67] performed extensive measurements on the Si L2,3 -ELNES of a range of silicate minerals and identiﬁed a ﬁngerprint for the SiO4 tetrahedral unit that is clearly different from the Si L2,3 -XANES of stishovite [7.68] where silicon is octahedrally coordinated to oxygen. This was later conﬁrmed by the higher resolution measurements of Batson [7.69], Garvie [7.7] and McComb et al. [7.70]; the latter studied a range of nesosilicates and provided a MO interpretation for the Si L2,3 -ELNES ﬁngerprint. Modeling of the Si L2,3 -ELNES by Hansen [7.71] identiﬁed the concept of the dipole-allowed “p → p -like transition” in noncentrosymmetric tetrahedral coordination implying both the hybridization of p and d atomic orbitals in this symmetry as well as the formation of sp3 hybrid orbitals. In certain cases, notably zircon [7.72], this ELNES ﬁngerprint can be somewhat modiﬁed due to the inﬂuence of next nearest neighbor cations [7.73]. XAS studies of Mg K-ELNES in minerals have been performed by Ildefonse et al. [7.74], Mottana et al. [7.75] and Cabaret [7.76]. Generally, the ﬁngerprint for octahedral coordination of Mg at the Mg K-, Mg L1 - and Mg L2,3 -edges can be derived by the seminal work on MgO by Lindner et al. [7.30]. Waychunas et al. [7.77] investigated the aluminium K-XANES of a variety of minerals and their results are shown in Fig. 7.4. They concluded that tetrahedrally coordinated aluminium exhibits a peak at 1566 eV corresponding to transitions to a t2 MO, while octahedral aluminium displays prominent peaks at 1568 and 1572 eV

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corresponding to transitions to t1u MOs. Further work on a large range of Al KXANES has been performed by McKeown [7.78], Cabaret [7.76,7.79] and Ildefonse et al. [7.80]. Spatially resolved Al K-ELNES measurements have been used to identify the variation in Al environment within the microstructure of hydrated portland cement pastes and the results combined with bulk MASNMR data [7.81,7.82].

Fig. 7.4 Aluminium K -XANES of various aluminium containing minerals measured by Waychunas et al. [7.77]. Jadeite, corundum, kyanite, and topaz all contain Al in octahedral coordination, while albite and the three glasses all contain tetrahedral Al. Silliminite contains equal amounts of both octahedral and tetrahedral Al.

Using the lower energy Al L2,3 -edge, the octahedral coordination of aluminium in the mineral rhodizite has been conﬁrmed via comparison of the observed Al L2,3 - and K-ELNES with those measured from the minerals chrysoberyl and corundum (where the aluminium coordination is known to be octahedral) together with the results of multiple scattering calculations for AlO6 and AlO4 clusters [7.4,7.27]. The results for the Al L2,3 -ELNES are presented in Fig. 7.5 and clearly show the existence of an aluminium coordination ﬁngerprint. The results of the calculations indicate that

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in the octahedral case, transitions to s-like ﬁnal states are dominant in the region near the edge-onset and these give rise to the characteristic, strong initial peak in Fig. 7.5. Conversely, for the tetrahedral case, the Al L2,3 -edge onset is shifted to lower energies, due to the decreased positive effective charge on Al in tetrahedral as opposed to octahedral coordination, furthermore transitions to s-like ﬁnal states now appear as a low energy shoulder. Further work by Hansen et al. [7.83] investigated 4-fold, 5-fold, and 6-fold coordinated Al K- and L2,3 -ELNES in minerals and made attempts to extract mixed site occupancies. The Al L2,3 -edge ﬁngerprint has been used to study zeolites [7.84] and to characterize oxide ﬁlms formed on aluminium–chromium ferritic steels in simulated coal gasiﬁcation atmospheres [7.85]. The mineral rhodizite also contains beryllium tetrahedrally coordinated to oxygen. Figure 7.6 shows the Be K-ELNES of rhodizite compared with that of chrysoberyl which also contains the BeO46− unit together with results of multiple scattering calculations for a BeO4 tetrahedral unit [7.27]. It is apparent that the twin-peaked structure, which is also observed in BeO and Be2 SiO4 [7.67], is characteristic of beryllium tetrahedrally coordinated to oxygen. In the case of the 3d transition metals, the accessible EELS edges of interest are the L2,3 -edges that originate from the 2p core level, although much work has been done on the high energy K-edges using XAS that also exhibit coordination ﬁngerprints [7.86-7.88]. As discussed in Chapter 4, there are a variety of interactions which give rise to a very rich multiplet structure that may be most easily used to determine the valence of the transition metal ion. However, simplistically one may regard the L2,3 -ELNES from an atomic viewpoint as reﬂecting the ligand ﬁeld splitting of the metal 3d band [7.89] (this is also reﬂected to some extent in the structure observed at the ligand edge due to hybridization [7.90]). The ligand ﬁeld splitting is considerably different in octahedral as opposed to tetrahedral or eightfold coordination [7.91] and should provide a means of distinguishing between the metal environments. Figure 7.7 shows the titanium L2,3 -ELNES measured from Ba2 TiO4 (tetrahedral Ti) and BaTiO3 (octahedral Ti). The distinct differences observed offer a means of coordination ﬁngerprinting [7.67,7.92] and in principle, this technique should be readily extendable to other 3d transition metal L2,3 -edges. Brydson et al. [7.93] have presented data for isoelectronic series of d0 XO4n− anions (X = Ti, V, Cr, and Mn) in tetrahedral coordination that revealed a common pattern due to the crystal ﬁeld splitting which is very different from the octahedral data. Kurata et al. [7.94] have determined the coordination of chromium in a CrO3 graphite intercalation compound (GIC) via comparison of the observed Cr L2,3 - and O K-ELNES with those measured from K2 CrO4 and Cr2 O3 . The spectrum derived from the GIC more closely resembled that from K2 CrO4 , where chromium is tetrahedrally coordinated to oxygen, than that from Cr2 O3 where the coordination is octahedral. The results agree well with the 9− predictions of MO theory for CrO2− 4 and CrO6 clusters [7.40, 7.95]. Similarly, Krishnan [7.96] has explained the differences observed between the Fe L2,3 -ELNES of Fe3+ in octahedral and tetrahedral coordination in terms of the differing ligand ﬁeld splittings and available ﬁnal states predicted from a simple ligand ﬁeld approach. Garvie [7.97] has presented Ca L2,3 -edge data from 6-fold, 8-

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Fig. 7.5 Aluminium L2,3 -ELNES from (A) chrysoberyl and (B) rhodizite after background subtraction. (C) The results of ICXANES calculations for a single shell of oxygen atoms octahedrally coordinated to a central aluminium atom.

Fig. 7.6 Beryllium K -ELNES from (A) chrysoberyl and (B) rhodizite after background subtraction. (C) The results of ICXANES calculations for beryllium tetrahedrally coordinated to oxygen, core hole effects have been included.

fold and 12-fold-coordinated Ca, which reveals a gradual reduction in the observable crystal-ﬁeld splitting. Van der Laan and Kirkman [7.98] have performed a large range of atomic multiplet calculations of 3d transition metal L2,3 –spectra comparing the results in both octahedral and tetrahedral coordination for a range of crystal-ﬁeld strengths; these theoretical spectra permit a detailed comparison with experiment. 7.3.1.2 Anion Fingerprints Carbon exhibits a range of coordination ﬁngerprints. It has long been noted that the two allotropes of carbon: graphite and diamond may be distinguished from their C K-ELNES. The presence of sp2 bonding in graphite results in a π ∗ peak at 285 eV at the C K-edge, whereas this is absent in the case of sp3 -bonded carbon atoms [7.99]. Between these two extremes there are a whole range of carbonaceous materials with intermediate bond types including evaporated amorphous C and glassy C. In graphitic materials, the degree of graphene ordering

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Fig. 7.7 Titanium L2,3 -ELNES from (a) BaTiO3 and (b) Ba2 TiO4 after background subtraction.

is reﬂected in the sharpness of the σ ∗ peak at 291 eV [7.100]. Fullerenes have similar C K-ELNES to graphite but with differences due to the molecular nature of the C60 and C70 units [7.101]. In recent years carbonaceous alloys have become important materials, carbon alloyed with one of more of the following elements: boron, nitrogen and silicon, have been produced either as particulates or nanotubes by a variety of routes. Studies of the C K- and the B K-, N K- and Si L2,3 -ELNES [7.102–7.105] have revealed the extent of solubility or otherwise of these elements within the graphite network. Different carbides also show pronounced differences [7.46]. Those containing the acetylide ion, C22− show a π ∗ -σ ∗ structure shown in Fig. 7.8 while in those possessing isolated C atoms the π ∗ peak is absent [7.106]. Craven has presented a detailed comparative study of metal carbides with the rocksalt structure [7.107] and these have been extensively modeled by Scott et al. [7.108,7.109] who revealed the sensitivity of ELNES features to lattice parameter and stoichiometry. The C22− ion is isoelectronic with the molecule N2 , and the N K-ELNES of gaseous N2 has a similar form to that in Fig. 7.8 [7.110]. Berger et al. [7.111] have employed this N K-ELNES ﬁngerprint in their analysis of voidites in diamond. The measured N K-ELNES more closely resembles that from N2 rather than NH3 in both the shape and the energy position that leads them to the conclusion that the voidites consist of solid N2 inclusions. The planar trigonal CO32− anion is isoelectronic with the BO33− anion and the C K-ELNES of carbonates all show a similar π ∗ -σ ∗ structure to that in Figure 7.2a [7.5, 7.7]. This can not be confused with the ﬁngerprint for the C22− anion since the latter occurs some 11 eV lower in energy owing to the less positive effective charge on the carbon atom [7.46]. Smith et al. [7.112] have employed the C K-edge ﬁngerprint for carbonates to determine the presence of BaCO3 at grain boundaries in the Yttrium barium copper oxide superconductor system.

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Fig. 7.8 Carbon K -ELNES from Li2 C2 after background subtraction. The poor statistics are due to the radiation sensitivity of the material.

There is considerable interest in the detection of C-H bonding, Stohr [7.113] has postulated that the presence of a feature at 288 eV is indicative of C-H σ bonds. Another isoelectronic polyanion with the same D3h symmetry is the nitrate, NO3− , anion and Hofer [7.67] has shown that the N K-ELNES coordination ﬁngerprint of this species has the same π ∗ -σ ∗ form. Hofer has also identiﬁed two other N K-ELNES ﬁngerprints [7.67]. The linear OCN− ion exhibits a familiar π ∗ -σ ∗ structure, whereas the azide ion N3− , which is isoelectronic with OCN− , possesses two chemically inequivalent sites owing to the different effective charges on the central and outer nitrogen atoms. The N K-ELNES of N3− therefore consists of two such π ∗ -σ ∗ structures separated by ∼ 5 eV. Metal nitrides have been studied by Craven et al. [7.114] who concentrated on transition metal nitrides with the rocksalt structure and revealed many common patterns with the associated carbides, meanwhile Serin et al. [7.33] who extensively modeled the N K-ELNES of AlN. The characteristic triple-peaked structure observed at the N K-edge of both AlN and GaN appears to be a ﬁngerprint for the wurtzite structure. The isoelectronic series of polyanions PO43− /SO42− /ClO4− all exhibit tetrahedral symmetry. The L2,3 -XANES of phosphorus/sulfur/chlorine displayed in Fig. 7.9 all show a characteristic twin peaked structure separated by ∼ 10 eV followed at higher energy loss by a much broader peak [7.5,7.115]. This may be easily distinguished from the corresponding ELNES of the element or the phosphide/sulﬁde/chloride, which displays a much broader, featureless near-edge structure [7.5]. Additional studies have concentrated on the S K- and S L2,3 -ELNES of various metal sulﬁdes [7.116,7.117]. XAS measurements have been performed on the isoelectronic series SF6 , PF− 6, all of which possess octahedral symmetry [7.118]. The L -XANES and SiF2− 2,3 6 of sulfur/phosphorus/silicon all show a common gross structure that is considerably

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Fig. 7.9 L2,3 -XANES of Na3 PO4 , Na2 SO4 and NaClO4 measured by Sekiyama et al. [7.118].

detailed but can be assigned to transitions to the various MOs of the appropriate cluster. Measurements on the O K-edges from a range of titanium (IV) oxides have been performed [7.119]. In all cases oxygen is coordinated to titanium and the ﬁrst 5 eV of the O K-ELNES is dominated by transitions to O 2p - Ti 3d hybridized orbitals [7.90]. This leads to a twin peaked structure which reﬂects transitions to pdπ and pdσ antibonding MOs formed from the interaction of oxygen with the titanium nearest neighbors. The relative intensities of these two peaks vary with changes in the coordination symmetry, and in the case of cubic perovskites there appears to be a common edge structure due to the linear, twofold coordination of titanium to oxygen. To illustrate this, Fig. 7.10 shows the O K-ELNES of two cubic perovskites, BaTiO3 and SrTiO3 together with the results of multiple scattering calculations. In order to reproduce all the observed structure at least seven shells are required. However, the ﬁrst 5 eV of the observed ELNES may be adequately modeled using only the nearestneighbor shell, indicating the existence of a coordination ﬁngerprint in this region. By aligning the various edges in SrTiO3 , Guerlin et al. [7.120] have extended the analysis of the unoccupied electronic states in this material. Different oxide polymorphs have also been studied and, while the O K-edge spectra do not unambigiously deﬁne the local coordination they canprovide a means of distinguishing between the diffrent polymorphs; such systems include TiO2 [7.121], ZrO2 [7.122], Al2 O3 [7.123] and SiO2 [7.124,7.125] as well as Mg-Al spinels [7.126].

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Fig. 7.10 Oxygen K -ELNES of SrTiO3 and BaTiO3 after background subtraction together with the results of ICXANES calculations with the inclusion of seven shells.

7.4

DETERMINATION OF VALENCIES

Experimentally observed ELNES are sensitive to the valence of the atom undergoing excitation in two distinct, but inherently related ways. First, it is well known from X-ray photoelectron spectroscopy (XPS) that a change in the effective charge on a particular atom leads to shifts in the various core level binding energies [7.127] and therefore in the onset of core to conduction band transitions at a particular core loss edge. This description considers only the effect of charge on the initial core level state, however associated shifts would also be expected in the energy position of the ﬁnal state. The combination of these two effects are collectively known as a chemical shift of the edge. A simple example would be the change in going from a metal to an insulator where the introduction of a band gap leads to a shift of the edge on-set to higher energies (e.g. the Al L2,3 -edge onset energies of aluminium and alumina are ∼ 73 and 78 eV, respectively). However, this shift does not necessarily correspond to the magnitude of the band gap. Second, the valence of an excited atom can affect the intensity distribution in the edge structure. This effect would be expected to occur in systems where the ELNES is sensitive to electron–electron interactions in the ﬁnal state as well as interactions between the core hole and the excited electron. This effect is extremely important in determining the L3 /L2 white line intensity ratios of the L2,3 -edges of the 3d transition metals and their compounds that depend speciﬁcally on the number of d electrons and therefore the valence of the particular transition metal ion. Chemical shifts measured using EELS are complicated phenomena involving two qualitatively different orbitals, the core level and the lowest energy unoccupied level. This is a very different situation from the processes occuring in XPS, where the core

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photoelectron is excited to states far above the Fermi level and has a large kinetic energy causing the XPS ﬁnal state to be essentially independent of the sample band structure. An XPS chemical shift thus provides a measure of the difference in energy between the respective core levels in two compounds. The energy shift of the core level, which will be a component of the EELS chemical shift, is inﬂuenced by both the charge on the particular atom in question as well as the charge present on neighboring ions. However, in a compound the deep lying core orbitals of the atom undergoing excitation should be less inﬂuenced by external factors in comparison with the outer orbitals that have signiﬁcant overlap with the wave functions of the ligand atoms and participate directly in the formation of chemical bonds. Although the nature of the bonding will have an effect on the energy of the initial state, one would expect that the predominant contribution to any observed energy shift of the edge onset will be due to a modiﬁcation of the ﬁnal state distribution. Furthermore, from the viewpoint of simple MO theory, changing the valence of the central atom in a polyatomic cluster will alter the valence electron conﬁguration and hence this might change the lowest unoccupied MO. The energy shift of the ﬁnal state, which is exclusive to EELS, will be inﬂuenced by many direct bonding effects and will therefore be dependent on the nature of the ﬁnal state, which may be an excitonic or a continuum level and in some cases may involve atomic multiplet or many electron effects. Attempts have been made in XAS to extract the position of the main edge due to transitions to the continuum and so circumvent the problem of the energy shift due to the ﬁnal state. This is extremely difﬁcult to perform accurately. However, for the few cases where it is possible the results show good agreement with XPS data [7.128]. The preceding description has ignored the effect of core hole production on the respective energies of the initial and ﬁnal states. When a core hole is created, the remaining one-electron orbitals relax adiabatically and cause a reduction in the XPS core level binding energy equal to the relaxation energy [7.127]. The remaining electrons do not necessarily experience the full potential of the core hole since some of them (particularly the valence electrons) may be able to screen this potential; this being especially true if the sample is a conductor with relatively free valence electrons centred at the particular atomic site in question. A similar effect will be operative in EELS, however it will affect both the initial and ﬁnal states. In XPS, the excited electron is ejected to energies above the vacuum level and is not available for screening, however, since the excited electron is promoted to a localized state in EELS, it can also contribute to the screening of the core hole and the perturbation on the remaining electrons can therefore be reduced. Relaxation effects would therefore be expected to be somewhat greater in XPS than in EELS. Despite the differences between the two processes it is interesting to try and correlate XPS chemical shifts with those measured by EELS. Garvie et al. [7.73] have performed an interesting comparison of Si L2,3 -edge onsets with XPS and NMR data for a range of silicates. Experimentally, valence dependent chemical shifts and edge intensity redistributions may be studied by the accurate measurement of suitable reference materials; these may be naturally occuring minerals, inorganic or organic compounds. In practice, this is a similar procedure to that employed in the routine characterization of

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solids by XPS for the detrmination of atomic charge states. Many minerals possess atoms in different chemical environments and hence these atoms exhibit differing effective charges. Such samples provide a means of investigating the magnitude of chemical shift effects, while keeping other effects, such as the magnitude of the band gap, constant. As already mentioned, since EELS chemical shifts are of the order of a few eV the spectra must be measured accurately on an absolute energy scale. Measurements on the L2,3 -edges of the 3d transition metal compounds and M4,5 -edges of the rare earths also require good resolution if the subtle changes in intensity are to be observed. Processing of spectra is also an important aspect. If intensity ratios are to be measured and compared to literature values, the use of standardized data processing techniques is essential. Ultimately, chemical shift data cannot be interpreted empirically and some means of correlating the observed shifts with the effective charge on a particular ion is required. The effective charge takes into account a number of parameters such as valency, coordination number, ionicity and the nature of the chemical bond in the compound under study. Charges may be determined or calculated by a variety of experimental and theoretical methods [7.129]. These range in complexity, however, one of the simplest is the use of electronegativities and a simple charge potential (CP) model that may be applied to virtually all systems [7.4]. One major problem is that the charges derived by different experimental and theoretical methods cannot always be directly compared because of the different physical sense inherent in them. This is particularly true in solids, where the chemical concept of “charge transfer” between atomic species is extremely difﬁcult to deﬁne as it critically depends on how the volume within the unit cell is divided up and apportioned between the various atoms. At best, for a range of similar compounds, the division of space (e.g., into mufﬁn tin spheres) should be performed in a consistent manner and the resulting charge density distributions for the variuos solids compared. Scott et al. [7.109] have recently compared the “charge transfer” in a range of transition metal carbides and nitrides using the results of FLAPW calculations; the magnitude of “charge transfer" being determined by the difference in the charge density distributions between that derived from a simple superposition of neutral atom potentials, and that found from the ﬁnal (iterated) self-consistent potential. Theoretical prediction of edge onset energies from the difference between initial and ﬁnal state energies dermined from either band structure or MS-type calculations is indeed possible [7.109], however the important and differing effects of the core hole on the two different symmetry states need to be taken into account perhaps using a Slater transition state picture (which employs half a hole in the inner-shell and the other half in the ﬁnal state) to account for the excitation process. 7.4.1

Review of Chemical Shift Data

It has been well established in XAS that the position of the edge is inﬂuenced by the physico-chemical environment of the absorbing atom. A good overview of XAS chemical shift data is provided by Mande and Sapre [7.130]. Much of this work has been performed in energy ranges inaccessible to EELS, and it is not the

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intention to provide a review of such data. However, it is instructive to consider one example in order to bring out the salient points. Wong et al. [7.87] have investigated the V K-edges of selected vanadium compounds using XAS. They found a linear correlation between the energy positions of various near-edge features (including the edge-onset) and the formal valence of the absorbing vanadium atom in a series of vanadium oxides. This relationship is known as Kunzl’s law [7.131], and has been shown to be useful for systems having the same ligand type. Final states which were more tightly bound were found to be less sensitive to changes in valence than the less tightly bound states. More generally, chemical shifts are due to a combination of the effects of valence, ligand electronegativity, coordination number and other structural features. These inﬂuences may be accounted for in the concept of coordination charge as deﬁned by Batsonov [7.132]. Wong et al. [7.87] observed a good correlation of the energies of various features with the coordination charge for a wide range of vanadium compounds. Very little systematic work on chemical shifts EELS has been performed. However, the creation of a band gap on going from a metal to an insulator provides a simple example of the qualitative use of chemical shifts. The Al L2,3 -edge shifts from 73 eV in the metal to 77 eV in alumina [7.2], while the C K-edge of graphite at 284 eV shifts to 289 eV in diamond due to the presence of a 5-eV band gap [7.99]. The Si L2,3 -edge exhibits a progressive increase in the edge onset energy in going from the semiconductor Si (99.5 eV) through SiO (103 eV) to the insulator SiO2 (106 eV) [7.2]. In principle, the study of a set of isostructural materials should provide a good demonstration of the phenomenon of EELS chemical shifts. As all materials possess the same crystallographic arrangement (e.g. transition metal carbides or nitrides of Group IVA and VA [7.107,7.114]), the form of the electron energy band structure should be identical (a so-called “rigid-band” model), thus changes in the edge onset energy should be a function of the shift in the Fermi level, which will be governed by the total number of electrons in the system. Leapman et al. [7.133] studied the L2,3 -edges of the 3d transition metals and their oxides and attempted to relate the EELS chemical shifts between the oxide and the metal with the corresponding XPS chemical shifts. However, using a simple one electron scheme they had little success and explained the discrepancies as due to differences in the many-body relaxation effects between the two techniques. The ALCHEMI (Atom Location by CHanneling Enhanced MIcroanalysis) technique is extremely useful for the determination of site occupancies [7.134]. It involves maximizing the standing-wave intensity pattern at certain atomic sites by suitable orientation of the crystal and is ideally suited to structures with large unit cells such as minerals. Taftø and Krivanek employed this technique to study the distribution of Fe ions in a chromite spinel [7.135]. They concluded that Fe3+ ions occupied octahedral and Fe2+ ions tetrahedral sites within the unit cell. The shift between Fe2+ and Fe3+ was measured to be 2 eV to higher energy loss, 1.3 eV of which they attributed to the change in valence while the remainder was assigned to the change in site. The Fe L2,3 -edge of the mineral vonsenite, which contains both Fe2+ and Fe3+ ions in octahedral sites coordinated to oxygen, exhibits a splitting of the main

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L3 peak of 1.3 eV in agreement with their ﬁndings [7.58]. Otten et al. [7.136] studied the chemical shifts of the L2,3 -edges of Ti, Mn, and Fe as a function of oxidation state in various minerals and observed appreciable effects even at relatively poor resolution. In the case of manganese, this work was subsequently repeated by Rask et al. [7.137] and Patersen et al. [7.138] with improved resolution, while Kurata et al. [7.139] have recently measured the Fe L2,3 -ELNES of compounds containing Fe octahedrally coordinated to various ligands. Garvie et al. have presented similar data for Fe, Cr and Mn [7.7,7.97], while Hoche et al. [7.140] have employed the technique to study the valency of Ti in a fresnoite glass. Such results suggest the possibility of transition element oxidation state determination in samples. As is subsequently discussed, a more reliable procedure might be the high resolution measurement of the ﬁne structure and L3 /L2 ratio at the transition metal L2,3 -edge. Auchterlonie et al. [7.141] have measured the Si L2,3 -ELNES of different amorphous silicon alloys and demonstrated that the shape and energy positions of the features depend on the ligand type (e.g., B, C, N, O and P). They demonstrated a linear correlation between the energy of the ﬁrst peak in the ELNES and the Pauling electronegativity of the ligand that gives a measure of the charge transfer from the silicon to the ligand atom. Their results are displayed in Fig. 7.11.

Fig. 7.11 Graph of the energy shift (relative to amorphous Si) of the ﬁrst peak in the Si L2,3 -ELNES of amorphous silicon alloys as a function of the Pauling electronegativity (relative to Si) of the ligand atom (from [7.141]).

Isaacson [7.142] studied the carbon K-edges of the nucleic acid bases adenine, thymine and uracil and related the positions of the observed peaks to the charges on the various carbon atoms calculated by semiempirical methods. The good agreement achieved is shown in Fig. 7.12 and this supports the use of a simple CP model for the determination of EELS chemical shifts. Such a model has been employed to describe the features observed at the oxygen K-edge of the mineral rhodizite [7.4] which contains oxygen in three distinct chemical environments, O(1), O(2), and O(3) with relative site occupancies 1:3:3 respectively [7.143]. The observed O K-edge (Fig. 7.13a) exhibits a very broad peak at the edge onset and taking the ﬁrst derivative of the spectrum it is clear that this is comprised of at least three peaks. Using a Gaussian ﬁtting procedure that involved ﬁtting the derivative of the spectrum with derivatives of Gaussians (in order to isolate the contribution of transitions to the continuum), it was found that three peaks of relative

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Fig. 7.12 C K -ELNES of (a) adenine, (b) uracil and (c) thymine. The right-hand side shows the relationship between the ELNES peak positions and the calculated charge on each carbon atom (from [7.142]).

intensities 3:3:1 were required for a satisfactory ﬁt in the near-edge region (Fig. 7.13b). The energy positions and relative intensities of these peaks represent the chemical shifts and number of oxygen atoms in the different chemical environments. By using Pauling electronegativities, it is possible to calculate the ionicity of each oxygen– metal bond and so determine the net charge transferred to each oxygen site. Assuming atoms may be approximated as hollow nonoverlapping electrostatic spheres, the potential felt at each site will be a superposition of two potentials. One arising from the charge on the particular atom, the second due to the charge distribution on neighboring atoms. The potential energy Ei , experienced by a core electron on atom i may then be written as Ei = ke(qi /ri + Σqj /rij ) + V where qi is the charge transferred to atom i, ri is the radius of the valence shell of atom i, qj is the charge transferred to atom j and rij is the distance of atom j from atom i. The sum is over all nearest neighbors, e is the electron charge and k and V are constants. Assuming that a similar relationship applies to the energy position of the ﬁnal state, the factor k allows for the differing effects of charge transfer on the initial and ﬁnal states. V represents a reference binding energy that will depend on the band gap and the relaxation due to core hole production. Plotting the peak positions derived from the Gaussian ﬁt versus the calculated effective core potential, the linear relationship shown in Fig. 7.13c was obtained.

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Fig. 7.13 (a) The oxygen K -edge in rhodizite after background subtraction. (b) The derivative of the spectrum shown in (a) together with the ﬁtted curve consisting of three Gaussians. (c) The energies of the three ﬁtted peaks at the O K -edge plotted against the effective core potential arising from the charge transferred between each oxygen atom and its nearest neighbors.

The ﬁtted line gives values of k = 0.49 ± 0.01 and V = 531.6 ± 0.3 eV. The value of V corresponds to the binding energy of an oxygen 1s level in rhodizite for an oxygen atom with zero charge and this is consistent with the magnitude of the band gap in rhodizite and the relaxation energy of the oxygen K-edge [7.4]. It should be noted that this simple model correctly placed the O(1) site with a relative occupancy of 1, for which the core potential is most positive, at highest energy loss in agreement with previous studies [7.143]. Further support for this explanation was provided by the results of XPS and Auger measurements on rhodizite [7.4].

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This analysis has also been applied to other minerals that also contain oxygen atoms in distinct chemical environments. The mineral wollastonite essentially contains three oxygen sites which differ in the types and number of nearest-neighbor atoms. In this case, the EELS O K-edge actually exhibited three peaks in the near-edge region. Using a Gaussian derivative ﬁtting procedure it was possible to ﬁt the experimental spectrum and correlate the positions and relative intensities of the ﬁtted peaks with the predictions of a charge potential model. Plotting the ﬁtted peak position versus the effective core potential again gives a slope, k, of approximately 0.5 and a value of V of 529.5 ± 1.5 eV which was consistent with the magnitude of the band gap and the O K-edge relaxation energy [7.58]. Furthermore, the relatively complex shape of the O K-edge of the mineral titanite has been qualitatively explained in terms of inequivalent oxygen sites and the effective core potential experienced at each of these sites [7.119]. The oxygen K-edge of the zeolite mordenite displays a 2-eV shift to lower energy loss upon dealumination with mineral acid [7.144]. There are a number of proposed mechanisms for the dealumination process all of which essentially predict a relative change in the oxygen site occupancies. However, analysis of the oxygen K-edge shift in terms of the CP model [7.41] combined with the results of low loss analysis [7.84,7.144] appears to suggest one particular reaction scheme. Employing this mechanism, the magnitude and sign of the theoretically predicted chemical shift (after taking into account the change in band gap) is in good agreement with experiment. Furthermore, XPS results indicate that the binding energy of the O 1s level actually increases with decreasing aluminium content for a wide range of zeolites [7.145]. If the EELS chemical shift was due solely to a change in binding energy of the initial state, one would then expect the O K-edge onset to shift to higher energies upon dealumination. However, since the observed shift is to lower energy loss, this conﬁrms that the effect of charge transfer on the ﬁnal state is the dominant term in determining the resultant EELS chemical shift. In effect, the simple CP model described above is attempting to simplify the band structure of these complex materials with atoms in differing sites and possessing different effective charges into the idealised contributions from the separate sites. Such an approach will obviously have its limitations, however it does provide an insight into the origin of ELNES that may allow common patterns to be drawn amongst different materials. 7.4.2

Review of Valence-Dependent Intensity Redistributions

As already alluded to, another route to the determination of valencies in the case of the 3d transition metals and their compounds is the measurement of the whole L2,3 ELNES rather than simply the position of the edge onset. In the case of copper, the presence of a white line at the Cu L2,3 -edge onset reﬂects the presence of an unﬁlled 3d band and hence Cu2+ (e.g., CuO) as opposed to elemental copper, which possesses a full d band and exhibits no such feature [7.133]. This is also accompanied by a shift of the Cu L3 -edge to lower energy loss, which results from a number of competing factors: the lowering of the Fermi level into the 3d band, the creation of a band gap

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in the oxide and the shift to higher energy loss with increasing metal valency. Cu+ (Cu2 O) meanwhile exhibits a small white line at an energy higher (933 eV) than that of Cu2+ (∼ 931 eV). This valence ﬁngerprint is relatively well deﬁned and has been used to investigate oxidation states in various minerals [7.146], however, for other elements in the transition metal series, the valence dependent changes in the spectra are much less obvious. Leapman and Grunes [7.147] ﬁrst observed departures from the statistical L3 /L2 white line intensity ratios of 2:1 in their studies of the 3d transition metals, they suggested that many body effects were important. This work was later extended to 3d transition metal oxides [7.133] and similar effects were observed. Here they argued that the deviations arose due to a breakdown of j-j coupling. Sparrow et al. [7.148] demonstrated the variation of the L3 /L2 ratio with d electron occupancy and observed that both the L3 and L2 white line intensities decrease linearly with the number of d electrons [7.149]. Waddington et al. [7.150] performed multiconﬁgurational Dirac– Fock calculations for the L2,3 -edges of the 3d transition metal ions. While ignoring solid state effects, they stressed the importance of atomic many electron effects and achieved good agreement for the variation of the L3 /L2 ratio with cation charge state. They also provided a convenient tabulation of experimental results. A full treatment using an atomic multiplet description with the inclusion of the crystal ﬁeld due to the ligands was eventually provided by Thole and van der Laan [7.151]. As well as explaining the observed trends in the L3 /L2 ratios they were able to model the detailed shapes present in high resolution XAS [7.89] and EELS [7.121,7.152] measurements on the L2,3 -edges of 3d transition metal compounds. By using the same approach, de Groot et al. [7.50] published extensive calculations of the L2,3 edges for the commonly observed valencies of 3d transition metal ions in cubic crystal ﬁelds of varying strengths. Consequently, it should be possible to use these theoretical spectra as a ﬁngerprint of the valency and the magnitude of the cubic crystal ﬁeld for a particular ion by comparison with experimental EELS L2,3 spectra. In principle, the L3 /L2 white line intensity ratio can provide a means of determining the 3d state occupancy. This approach has been employed in the study of magnetic alloys by Morrison et al. [7.153] as well as in the determination of transition metal ion valencies in Fe–Cr–Mn oxide ﬁlms [7.53]. However, the relationship between the L3 /L2 ratio and d-band occupancy does not exhibit a simple behavior. Consequently, Pearson et al. [7.154] have developed an alternative procedure that involves examination of the sum of the L3 and L2 white line intensities normalized to the continuum contribution. The correlation of this normalized sum with 3d-state occupancy was shown to be linear and their results are shown in Fig. 7.14. They have used this relationship to study charge transfer during the formation and ordering of Cu alloys [7.154]. The L2,3 -edges of the 4d transition metals also exhibit L3 /L2 ratios that deviate from the statistical value. Pearson [7.155] has studied the behavior of the normalized sum of the L3 and L2 white line intensities with 4d occupancy. The behavior is considerably more complex than for the 3d case, however, it does appear possible to determine the degree of charge transfer in alloys for the second half of the 4d series where the relationship is linear.

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Fig. 7.14 Graph of the normalized sum of the white line intensities versus nominal d band occupancy for the L2,3 -edge of the 3d transition metals after [7.154].

The M4,5 -edges in rare earth compounds have been studied by both EELS [7.156,7.157] and XAS [7.158]. These features originate from 3d → 4f transitions and the M5 /M4 white line ratios also show deviations from the statistical value of 3:2. The general trend is a monotonic increase along the 4f series and the strong dependence of this ratio on the f count should provide a means of identifying heterogeneous mixed valent materials. Although understanding of the detailed white line shapes of the 3d and 4d transition metal L2,3 -edges and the M4,5 -edges of the rare earths has steadily improved over recent years, standardization of data processing techniques is also an important aspect. Manoubi et al. [7.159] have developed a method for quantitative extraction of the important parameters associated with white line edges. These include background subtraction, accurate separation of the continuum contribution and ﬁtting of the various resonant transitions, all performed under the constraint of a least squares ﬁt. Clearly, once such ﬁtting procedures have been universally agreed upon, then this should greatly aid the use of these techniques in the determination of electron orbital occupancies together with related spin and chemical bonding effects. A practical example of the use of such techniques has been provided by Levefre in the study of metal-support interactions in Ni/CeO2 catalysts [7.160]. 7.5 7.5.1

FURTHER APPLICATIONS Quantitative Determination of Coordination- and Valence-State Site Occupancies

When the ELNES ﬁngerprints of two different coordinations of the same element have been determined, it is often possible to apply this to compounds where both coordinations coexist and so determine the relative site occupancies via an algorithm or ﬁtting procedure. This arises due to the localized nature of the core level excitation process which results in the ELNES from two different sites within a structure combining in a simple linear superposition, greatly aiding subsequent interpretation.

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Berger et al. [7.161] have employed this technique in the determination of the proportion of sp2 -bonded carbon atoms in amorphous diamond ﬁlms. Since sp2 -bonded carbon gives rise to a π ∗ peak in the C K-ELNES, measurement of the relative area of this peak (normalized to the intensity in a given energy window) gives a measure of the contribution of this species to the total ELNES. This was later extended by Bruley et al. [7.162]. Similar algorithms have been applied to B K-edge spectra in order to determine the proportion of BO3 and BO4 groups in various boron-oxygen compounds [7.54,7.56]; the validity of the method was conﬁrmed by performing the analysis on various boron minerals where the structure, and hence the relative site occupancy, was known . These techniques have been used to study decomposition of BO4 sites in minerals during electron beam irradiation [7.54] and to study the structure of boron-doped oxide ﬁlms formed on an Fe–Cr alloy in superheated steam [7.60]. An alternative approach is to use ELNES reference spectra, representing the differing coordinations, and to perform a “best-ﬁt linear combination” to the ELNES spectrum of the unknown. Hansen et al [7.83] have investigated the possibilities for using Al K- and Al L2,3 -ELNES ﬁngerprints for the extraction site occupancies in mixed coordination materials. Bruley et al. [7.163] has contributed an elegant study of the relative occupancy of Al in tetrahedral sites in spinels and hence degree of spinel inversion. In a similar fashion, the relative proportions of differing valence states in a material can be determined by ﬁtting suitable reference spectra to experimental spectra (usually white line spectra). This has been demonstrated by Cressey et al. [7.164] using XANES from a number of Fe-containing minerals having mixed valency and coordination, as well as Garvie et al. [7.165] and van Aken et al. [7.166] using ELNES data. 7.5.2

Determination of Bond Lengths and Bond Angles

Small variations in coordination ﬁngerprints can often be related to differing degrees of distortion from perfect coordination (i.e., variations in bond lengths and bond angles) [7.167]. Comparison of experimentally observed spectra with those derived from standards, together with the results of various trial calculations, can often yield a measure of the amount of distortion. Measurements of the Ti L2,3 -ELNES from a series of titanium (IV) oxides, chosen so that the distortion in the Ti coordination octahedron gradually increased, indicated that the broadening and splitting of the near-edge features, especially those corresponding to transitions to orbitals directed at the oxygen ligands, increased as the Ti site symmetry was lowered [7.121,7.168]. The results are presented in Fig. 7.15. Patersen and Krivanek [7.138] have observed differences in the Fe L2,3 -ELNES between the α and γ forms of Fe2 O3 that arise due to the slight differences in the octahedral environments of the Fe3+ ions in the two systems. These differences are also reﬂected at the corresponding O K-edges. The conventional way to determine bond lengths in EELS is to analyze the weak EXELFS oscillations beginning some

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251

Fig. 7.15 Titanium L2,3 -ELNES of various Ti (IV) oxygen compounds. The distortion of the Ti coordination octahedron increases from SrTiO3 through titanite.

40–50 eV above the edge onset [7.169]. However, in the high energy ELNES region (some 20–40 eV above the edge onset) it is possible to observe the presence of broad peaks well above the ionization threshold, known as scattering shape resonances. These shape resonances may be approximated as arising from single scattering from

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nearest and, in some cases, next nearest neighbor coordination shells. Bianconi et al. [7.170] have postulated a general rule, which states that the energy of a scattering shape resonance (a transition to a σ ∗ MO) above the edge onset is proportional to 1/R2 , where R is the bond length. Identiﬁcation of such shape resonances permits a semiquantitative determination of the corresponding bond lengths and Kurata et al. [7.32] have demonstrated this effect in the O K-edges of transition metal oxides, while Craven and co-workers [7.107,7.114] has also conﬁrmed their presence in the C K-edges of transition metal carbides. 7.5.3

Anisotropy in Bonding Via Orientation Dependent Measurements

Orientation dependency in EELS arises from any anisotropy in the density of unoccupied electronic states and hence transitions to these empty ﬁnal states. In the case of isotropic materials, there is no directional dependence, however, for the case of anisotropic materials there is an inﬂuence on the direction of the scattering vector, q, relative to the beam direction. These effects may be used to study the directionality of bonding character within a given structure. Orientation dependent energy loss measurements have been recorded for anisotropic structures such as graphite [7.14], boron nitride [7.15], high Tc superconductors [7.170] and TiB2 [7.13]. TiB2 is a hexagonal material with trigonal sheets of boron and titanium arranged in layers perpendicular to the c-axis. Measurements of the B K-ELNES with q parallel and perpendicular to the c axis are shown in Fig. 7.16 and reveal clear differences related to the metallic bonding within the boron sheets and the covalent B–Ti bonding between sheets. These effects have been modeled with both band structure and multiple-scattering calculations and revealed the differing effects of the core hole in the two directions which is related to the degree of metallic character, and hence screening of the core hole as well as the different nature of the ﬁnal states predominant in each of the two directions [7.13]. 7.6 APPLICATION TO INTERFACES AND DEFECTS A rapidly expanding area in EELS that utilizes the high spatial resolution attainable in TEM and STEM is the investigation of bonding at nonperiodic features such as interfaces and defects [7.6]. The importance of these studies in modernday materials analysis cannot be over-emphasised as such features are often propertydetermining on the macroscopic scale, whether it be in terms of overall mechanical, thermal or electronic properties. Clearly the elucidation of the bonding and localized electronic structure at such features is of great interest and spatially resolved ELNES measurements can complement and often inform high resolution structural imaging either by phase contrast in HRTEM or high resolution annular dark ﬁeld images in the STEM for the determination of the atomistic and bonding arrangements at such non-periodic features. Spatially resolved ELNES measurements are commonly obtained in dedicated STEMs, making full beneﬁt of the small probes of high current density obtainable with

APPLICATION TO INTERFACES AND DEFECTS

253

Fig. 7.16 Boron K -ELNES from TiB2 measured with a small collection angle (to isolate the parallel component of scattering vector) with the scattering vector, q , oriented along three different directions: a general direction q [2423], q⊥ to c and ﬁnally q to c-axis.

a cold ﬁeld emission source. However, the coming years should see these techniques being more routinely applied in ﬁeld emission conventional TEM/STEMS [7.171]. As discussed in Section 2, measurements fall into two categories: those making use of focused probes of atomic dimension and those employing spatial difference methods where measurements are made over a larger irradiated area positioned both on and off the feature of interest. The spatial difference procedure was ﬁrst applied by Berger and Pennycook [7.172] in their detection of nitrogen segregation at platelet defects in diamond. This work was later extended by Bruley [7.173] and Fallon [7.21] who showed that the detection sensitivity using this method was ∼ 0.1 monolayers of segregated nitrogen. N KELNES measurements showed that the ﬁne structure was largely similar to that of diamond with the additional presence of a pre-peak at the edge onset. Extensive MS modeling of the N K-ELNES using various clusters representing the nitrogen environment at the platelet [7.174] appear, when combined with HRTEM and electron spin resonance data, to suggest that nitrogen platelets consist of N2 dimers or ‘Acenters’ at antiphase boundaries in the diamond lattice. The spatial difference method has been extensively applied to homophase and heterophase interfaces and a number of references are listed below. An excellent example, which highlights the power of the technique, is provided by studies of model

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Cu/α-alumina interfaces grown by MBE [7.76]. HRTEM imaging had determined the interfacial orientation, however image simulations could not unambiguously distinguish the chemistry of the interface, essentially whether there was Cu–O bonding or Cu–Al bonding between the basal plane of alumina and the (111) plane of Cu [7.176]. EELS spatial difference measurements were made using a reduced area scan, typically 3 × 4 nm2 , positioned both on the interface and in the neighboring matrix (typically 10 nm away in a region of comparable thickness). These spectra are shown in Fig. 7.17a–c for the Al L2,3 , Cu L2,3 and O K-edges. Because of the ﬁnite analysis volume, the interface spectrum actually consists of independent localized signals from both interfacial atomic species and those in the bulk. To isolate the contribution from the interface, the matrix spectrum is normalized and subtracted from the interface spectrum to give the spatial difference spectrum; procedures for determining normalization factors are given in reference [7.19]. For each of the various edges shown in Fig. 7.17a–c, these spectra represent the contribution from interfacial atomic environments which are substantially different from those in the neighboring bulk material, that is, differences in local coordination or valency of atoms at or near to the interface. These reveal a zero Al L2,3 spatial difference spectrum, implying that the local coordination of Al at or near the interface remains predominantly octahedral as in bulk alumina — that is, the octahedral Al L2,3 coordination ﬁngerprint remains predominantly unchanged. The Cu L2,3 -ELNES reveal a L3 white line feature at an energy corresponding to that observed in Cu2 O — implying the existence of Cu+ , that is, oxidized copper, at the interface. These results both imply the existence of Cu–O bonding at this interface, indicating that the alumina basal plane is terminated by a layer of oxygen anions. This chemical information was then fed into HRTEM image simulations that then could adequately reﬁne an atomistic model for the interfacial atomic arrangement [7.176]. Using this model, the interfacial environments of oxygen atoms (15 in total!) were calculated and used to simulate the O K-ELNES spatial difference spectrum using MS calculations. As previously mentioned, the overall shape of the O K-ELNES in oxides is generally much more sensitive to the longer range environment than say, for instance, the Al L2,3 -ELNES, hence up to eight shells extending to ∼ 0.5 nm from the central oxygen atom were used for modeling purposes. Excellent agreement between the simulations and the measured O K-ELNES spatial difference spectrum conﬁrmed the validity of the proposed atomistic model over other possibilities [7.175]. This work demonstrates the synergy between HRTEM and spatially resolved ELNES measurements for the determination of the local chemistry and structure associated with interfaces and defects. Similar spatial difference studies have been performed on grain boundaries in polycrystalline alumina [7.177], segregated species at alumina grain boundaries [7.178], SrTiO3 [7.179] and BaTiO3 [7.180] grain boundaries, intergranular glassy ﬁlms of thickness 1–2 nm at silicon nitride grain [7.181] and alumina boundaries [7.182]. The latter study demonstrated a direct link between the intergranular ﬁlm chemistry/ composition and the macroscopic erosive wear rate [7.182]. Other heterophase interfaces studied include: Nb–alumina interfaces grown by both MBE and diffusion bonding which resulted in distinctly different interfacial

APPLICATION TO INTERFACES AND DEFECTS

255

Fig. 7.17 EELS spatial difference spectra from an MBE-grown Cu/alumina interface: (a) region containing the Al L2,3 - and Cu M2,3 -edges, (b) region containing the Cu L2,3 -edge and (c) region containing the O K -edge. The nonzero signal for the Cu L2,3 - and O K -edges imply that the local electronic structure at the interface is modiﬁed for these elements.

chemistries [7.23] evidenced by EELS measurements, diffusion bonded Cu–alumina interfaces [7.171], metal–alumina interfaces in composites materials [7.183,7.184] as well as interfaces in ceramic compostites [7.163,7.185,7.186]. In metallic systems, ELNES spatial difference spectroscopy has revealed the modiﬁcation in electronic properties caused by segregation of impurity species to homophase grain boundaries, these include Bi in Cu [7.7, 7.187], S in Ni [7.188], P in Fe [7.189] and B in Ni3 Al [7.190] — in this latter work Muller elegantly demonstrated a means of relating results to a measure of grain boundary cohesion relative to the bulk [7.191]. Difference techniques have also been used to study the electronic structure and bonding at defects, as well as the work on N in platelets [7.174], interest has focused on oxygen at inversion domain boundaries in AlN [7.192] and defects in gallium nitride [7.193]. Recent specialized developments in instrumentation, particularly the ability to form subnanometer probes, have opened the possibility of direct measurement of ELNES from single atomic columns as evidenced by work on cobalt silicide–silica interfaces [7.17], SrTiO3 grain boundaries [7.194] and dislocations in silicon [7.195].

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7.7 CONCLUSIONS This chapter has attempted to highlight the possibilities of using EELS as a probe of the local electronic structure around a particular atomic species in a sample. In principle, EELS can yield information about a very wide range of properties of both physical and chemical interest. Although quantitative microanalysis has now become a routine technique, the extraction of information from the ELNES still remains a somewhat empirical process. Systematic studies of the effects of changes in the local coordination and charge transfer on the resultant near-edge structure should remedy this to some extent. A deeper theoretical understanding of the relevant excitation processes is also required for the technique to become predictive. Once these goals have been achieved near-edge structure analysis using EELS should become a very powerful tool in the armoury of material scientists. Acknowledgments This chapter is an extensive modiﬁcation (by RB) of an original text written with H. Sauer and W. Engel of the Fritz-Haber Institute in Berlin, where much of the original research was performed. In this respect, the authors are deeply indebted to the help and guidance of Professor E. Zeitler. Many others have indirectly contributed to this work, notably, Prof. Sir J. M. Thomas, Prof. L. M. Brown, Prof. M. Ruhle, Prof. C. Colliex, Prof. D. Vvedensky, Dr B.G. Williams, Prof. P. Rez, Dr. J. Bruley, Dr. H. Mullejans, Dr. C. Scheu, Prof. A. Craven, Dr. L. Garvie, Dr. D. McComb, Dr. A. Scott, Dr. K. Lie, Dr. V. Serin, and Prof. F. Hofer.

REFERENCES 7.1. R. F. Egerton, Electron Energy Loss Spectroscopy in the Electron Microscope , 2nd edition (Plenum Press: New York 1996). 7.2. C. Colliex, “Electron energy loss spectroscopy in the electron microscope,” in Advances in Optical and Electron Microscopy, vol. 9, eds. R. Barer and V. E. Cosslett (Academic Press: London 1984). 7.3. B. G. Williams, “Electron energy loss spectroscopy,” Prog. in Solid State Chem. 17 , 87 (1987). 7.4. R. Brydson, B. G. Williams, H. Sauer, W. Engel, R. Schlogl, M. Muller, E. Zeitler, and J. M. Thomas, “EELS and the crystal chemistry of Rhodizite. Part 2 - Near-edge structure,” J. Chem. Soc. Faraday Trans. 1 , 84, 631 (1988). 7.5. F. Hofer and P. Golob, “New examples for near-edge ﬁne structures in EELS,” Ultramicroscopy 21 , 379 (1987).

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7.6. R. Brydson, “Probing the local structure and bonding at interfaces and defects using EELS in the TEM,” J. Microsc. 180, 238 (1995). 7.7. L. A. J. Garvie, A. J. Craven, and R. Brydson, “Use of electron energy loss nearedge ﬁne structure in the study of minerals,” Am. Mineralog. 79, 411 (1994). 7.8. C. B. Boothroyd, K. Sato, and K. Yamada, “The detection of 0.5 atom% boron in Ni3 Al usimg PEELS ” Proc. 12th ICEM, vol. 2, p. 80 (San Francisco Press: San Francisco 1990). 7.9. P. E. Batson, S. J. Pennycook and L. G. P. Jones, “A new technique for the scanning and absolute calibration of EEL spectra,” Ultramicroscopy 6, 287 (1981). 7.10. A. Bleloch, L. M. Brown, M.J. Marsh, D. McMullen, J. J. Rickard, and V. Stolojan, “Cancelling energy drift in a PEELS spectrometer”, Proc. Conf. EMAG ’99, Inst. Phys. Conf. Ser. 161, 195–198 (IOP Publishing: Bristol 1999). 7.11. W. Engel, H. Sauer, E. Zeitler, R. Brydson, B. G. Williams and J. M. Thomas, “EELS and the crystal chemistry of Rhodizite. Part 1 - Instrumentation and chemical analysis,” J. Chem. Soc. Faraday Trans. 1, 84, 617 (1988). 7.12. R. F. Egerton, B. G. Williams and T. G. Sparrow, “Fourier deconvolution of EEL spectra,” Proc. R. Soc. London A398, 395 (1985). 7.13. K. Lie, R. Brydson and H. Davock, “Band structure of TiB2 : Orientation dependent ELNES and the effect of the core hole at the B K-edge,” Phys. Rev. B 59, 5361 (1999). 7.14. N. D. Browning, J. Yuan and L. M. Brown, “Real space determination of anisotropic electronic structure by EELS,” Ultramicroscopy 38, 291 (1991). 7.15. R. D. Leapman and J. Silcox, “Orientation dependence of core edges in electron energy loss spectra from anisotropic materials,” Phys. Rev. Lett. 42, 1361 (1979). 7.16. R. D. Leapman, P.L. Fejes and J. Silcox, “Orientation dependence of core edges from anisotropic materials by inelastic scattering of fast electrons,” Phys. Rev. B 28, 2361 (1983). 7.17. N. D. Browning, M.F. Chisholm and S.J. Pennycook, “Atomic resolution chemical analysis using a scanning transmission electron microscope,” Nature 366, 143 (1993). 7.18. P. E. Batson, “Simultaneous STEM imaging and electron energy loss spectroscopy with atom column sensitivity,” Nature 366, 727 (1993). 7.19. H. Mullejans and J. Bruley, “ELNES of internal interfaces by spatial difference spectroscopy,” J. Microsc. 180, 12 (1995). 7.20. J. Bruley, T. Hoche, H.J. Kleebe, and M. Ruhle, “Recent attempts to detect magnesium in a heavily doped sapphire bicrystal by spatially resolved EELS”, J. Am. Cer. Soc. 77, 2273 (1994).

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7.21. P. J. Fallon, L. M. Brown, J. C. Barry, and J. Bruley, “Nitrogen determination and characterization in natural diamond platelets,” Philos. Mag. A 72, 21 (1995). 7.22. J. Bruley, “Spatially resolved ELNES analysis of a near Σ11=tilt boundary in sapphire”, Microsc. Microanal. Microstruct. 4, 23 (1993). 7.23. J. Bruley, R. Brydson, H. Mullejans, J. Mayer, G. Gutekunst, W. Mader, D. Knauss, and M. Ruhle, “Investigations of the chemistry and bonding at niobium sapphire interfaces”, J. Mat. Res. 9, 2574 (1994). 7.24. C. Gatts, G. Duscher, H. Mullejans, and M. Ruhle, “Analysing linescan EELS data with neural pattern recognition”, Ultramicroscopy 59, 229 (1995). 7.25. J. M. Titchmarsh, “Detection of electron energy-loss near edge shifts and ﬁne structure variations at grain boundaries and interfaces”, Ultramicroscopy 78, 241 (1999). 7.26. L. Ponsonnet, B. Vacher, and J. M. Martin, “ELNES mapping of carbon and oxygen by Image EELS method”, Electron Microscopy 1998, 3, 663 (IOP Publishing: Bristol 1998). 7.27. R. Brydson, H. Sauer, W. Engel, J. M. Thomas, and E. Zeitler, “Coordination ﬁnger-prints in electron loss near-edge structures: Determination of the local site symmetry of aluminum and beryllium in ultraﬁne minerals,” J. Chem. Soc. Chem. Comm. 15, 1010 (1989). 7.28. J. A. Tossell, “A comparison of silicon-oxygen bonding in quartz and magnesium Olivine from X-ray spectra and MO calculations,” Am. Mineralog. 62, 136 (1977). 7.29. J. Stohr and D. A. Outka, “Near-edge X-ray absorption ﬁne structure studies of molecules and molecular chains bonded to surfaces,” J. Vac. Sci. Tech. A5, 919 (1987). 7.30. Th. Lindner, H. Sauer, W. Engel, and K. Kambe, “Near-edge structure in electron energy loss spectra of MgO,” Phys. Rev. B 33, 22 (1986). 7.31. A. Bianconi, M. Dell’Ariccia, P. J. Durham, and J. B. Pendry, “Multiple scattering resonances and structural effects in the X-ray absorption near-edge spectra of Fe II and Fe III hexacyanide complexes,” Phys. Rev. B 26, 6502 (1982). 7.32. H. Kurata, E. Lefevre, C. Colliex, and R. Brydson, “ELNES structure in the oxygen K-edge spectra of transition metal oxides,” Phys. Rev. B 47, 13673 (1993). 7.33. V. Serin, C. Colliex, R. Brydson, S. Matar, and F. Boucher, “EELS investigation of the electron conduction band states in wurtzite AlN and oxygen doped AlN(O),” Phys. Rev. B 58, 5106 (1998). 7.34. A. Bianconi, “One electron multiple scattering resonances and splitting of ﬁnal state conﬁgurations at threshold,” in EXAFS and Near-Edge Structures III, eds.

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7.172. S. D. Berger and S. J. Pennycook, “Detection of nitrogen at 100 platelets in diamond”, Nature 298, 635 (1982). 7.173. J. Bruley and L. M. Brown, “Quantitative EELS microanalysis of platelet and voidite defects in natural diamond”, Philos. Mag A 59, 247 (1989). 7.174. R. Brydson, L. M. Brown, and J. Bruley, “Characterizing the local nitrogen environment at platelets in type IaA/B diamond”, J. Microsc. 189, 137 (1998). 7.175. C. Scheu, G. Dehm, M. Ruhle, and R. Brydson, “EELS studies on Cu/αalumina interfaces grown by MBE”, Philos. Mag A 78, 439 (1998). 7.176. G. Dehm, C. Scheu, G. Mobus, R. Brydson, and M. Ruhle, “Synthesis of analytical and HRTEM to dermine interface structure of Cu/α-alumina” Ultramicroscopy 67, 207 (1997). 7.177. J. Bruley, “Spatially resolved ELNES analysis of a near Σ11=tilt boundary in sapphire”, Microsc. Microanal. Microstruct. 4, 23 (1993). 7.178. K. Kaneko, T. Gemming, I. Tanaka, and H. Mullejans, “Analytical investigation of random grain boundaries of Zr-doped alumina by TEM and STEM”, Philos. Mag. A 77, 1255 (1998). 7.179. I. Tanaka, T. Nakajima, J. Kawai, H. Adachi, H. Gu, and M. Ruhle, “Dopant modiﬁed local chemical bonding at a grain boundary in SrTiO3 ”, Philos. Mag. Lett. 75, 21 (1997). 7.180. A. Recnik, J. Bruley, W. Mader, D. Kolar, and M. Ruhle, “Structural and spectroscopic investigation of (111) twins in barium titanate”, Philos. Mag B 70, 1021 (1994). 7.181. J. Bruley, I. Tanaka, H.J. Kleebe, and M. Ruhle, “Chemistry of grain boundaries in calcia doped silicon nitride studied by spatially resolved EELS”, Anal. Chim. Acta 297, 97 (1994). 7.182. R. Brydson, S. Chen, X. Pan, F. L. Riley, S. J. Milne, and M. Ruhle, “Microstructure and Chemistry of intergranular glassy ﬁlms in liquid phase sintered alumina”, J. Am. Cer. Soc. 81, 369 (1998). 7.183. R. Brydson, H. Mullejans, J. Bruley, P. A. Trusty, X. Sun, J. A. Yeomans, and M. Ruhle, “Spatially resolved EELS studies of metal-ceramic interfaces in transition metal/alumina cermets”, J. Microsc. 177, 369 (1995). 7.184. C. Scheu, G. Dehm, H. Mullejans, R. Brydson, and M. Ruhle, “ELNES of metal-alumina interfaces”, Microsc. Microanal. Microstruct. 6, 19 (1995). 7.185. K. Suenaga, D. Bouchet, C. Colliex, A. Thorel, and D. G. Brandon, “Investigations of Alumina/spinel and alumina/zirconia interfaces by sptially resolved electron spectroscopy”, J. Eur. Cer. Soc. 5, 2194 (1998).

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7.186. X. Jiang, R. Brydson, S. P. Appleyard and B. Rand “Characterisation of the ﬁbre-matrix interfacial structure in carbon-ﬁbre-reinforced PCS-based SiC matrix composites using STEM/EELS”. J Microsc. 196, 203-212 (1999). 7.187. V. J. Keast, J. Bruley, P. Rez, J. M. Maclaren and D. B. Williams “Chemistry and Bonding Changes Associated with the Segregation of Bi to Bonding Changes Associated with the Segregation of Bi to Grain Boundaries in Cu”, Acta Mat. 46, 481 (1998). 7.188. V. J. Keast and D. B. Williams, “Investigation of Grain Boundary Embrittlement in the STEM” Inst. Phys. Conf. Ser. 153, 299 (1997). 7.189. D. Ozkaya, J. Yuan, L. M. Brown and P. E. J. Flewitt, “Segregation induced hole drilling at grain boundaries”, J. Microsc. 180, 300 (1995). 7.190. D. A. Muller, D. J. Singh, P. E. Batson, S. L. Sass and J. Silcox, Phys. Rev. Lett. 75, 4744 (1995). 7.191. D. A. Muller, D. J. Singh and J. Silcox, “Connections between EELS spectra, the local electronic structure, and the physical properties of a material: a study of nickel aluminium alloys”, Phys Rev B 57, 8181-8202 (1998). 7.192. J. Bruley, A. D. Westwood, R. A. Youngman, J. C. Zhao and M. R. Notis, in “Structure and Properties of Interfaces in Ceramics”, MRS Symposia Proceedings 357, 265 (MRS: Pittsburgh 1995). 7.193. M. Natusch, G. Botton and C. J. Humphreys, “Evidence for charged defects in wurtzite GaN from spatially resolved EELS”, Electron Microscopy 1998, Vol III, 391 (1998). 7.194. N. D. Browning, M. M. McGibbon, A. J. McGibbon, M. F. Chisholm, S. J. Pennycook, V. Ravikumar and V. P. Dravid, “Atomic resolution characterization of interface structure and interface chemistry in the STEM”, Electron Microscopy 1994, Vol. 1, p 735 (Les editions de physique: Paris 1994). 7.195. P. Batson, “Advanced spatially resolved electronic structure analysis”, Inst. Phys. Conf. Ser. 161, 15 (1999).

8

Application of EELS to Ceramics, Catalysts, and Transition Metal Oxides J. Bentley1 and J. Graetz2 1

Metals and Ceramics Division Oak Ridge National Laboratory P.O. Box 2008 Oak Ridge, TN 37831-6376

2

Division of Engineering and Applied Sciences California Institute of Technology Pasadena, CA 91125

Abstract The application of electron energy loss spectrometry (EELS) in a transmission electron microscope to ceramics, catalysts, and transition metal oxides is reviewed with illustrative examples. Special experimental concerns such as insulating specimens, contamination, radiation damage, and specimen preparation artifacts, are discussed. Composition determination of supported catalysts and multiphase and solid solution ceramics is described. Examples include B4 C in SiC and liquid-phase sintered TiB2 and Si3 N4 . Limitations such as background ﬁtting, aberration effects, and detection limits are described. The characterization of multiple valencies, colloid formation, hybridized bonding, oxidation states, coke formation, dealumination, and structural characterization in glass, illustrate the use of plasmon and energy loss near-edge structure (ELNES) analyses. The electronic structure of transition metal (TM) oxides and Li intercalated TM oxides is characterized by quantitative analysis of the O K- and TM L2,3 -near-edge structure. In applications of extended energy loss ﬁne structure (EXELFS), the methodology for accurate, reproducible results is highlighted with examples from MgO, SiC, and Al2 O3 . Different forms of amorphous alumina are shown to reﬂect the bond structure of the respective crystalline forms into which they transform. Energy ﬁltered imaging is applied to mixed phase TM oxides for compositional and valence state mapping. Time-resolved studies and analyses in the reﬂection geometry, are also brieﬂy discussed. 271

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8.1 INTRODUCTION It is not intended for this chapter to provide a full literature review of applications of electron energy loss spectrometry (EELS) to ceramics, catalysts, and transition metal (TM) oxides but to provide a summary of the kind of work that has been done with illustrative examples from Oak Ridge National Laboratory (ORNL), Caltech, and other published work. Some examples are treated more as case studies to bring out important practical aspects that are typical of those still likely to be encountered in such work. Perhaps the most obvious use of EELS in studies of oxides, ceramics, and catalysts is (elemental) composition determination. Elements with low atomic number (Z) are often major constituents of such materials but are difﬁcult to detect by conventional energy-dispersive X-ray spectrometry (EDS) even when equipped with ultra-thin-windowed EDS detectors. Since quantitative analysis of low-Z elements by EELS is readily achieved, it has become the technique of choice for such microanalyses in analytical electron microscopes. But beyond information on elemental composition, EELS can provide information on the chemistry or atomic bonding from analysis of the ﬁne structure present in high-resolution spectra. The low-loss and near-edge regions provide information on valence electrons, charge states, and bond character. Analysis of the extended ﬁne structure can provide information on near-neighbor bond lengths for speciﬁc atomic species. However, ﬁrst it will be useful to consider some of the special limitations faced in experimental work. 8.2

GENERAL EXPERIMENTAL POINTS

Many ceramics and catalysts or catalyst supports are electrical insulators; a few, such as SiC, are semiconducting; even fewer are good conductors (e.g., TiB2 ). Insulating specimens are prone to suffer from charging in the electron beam. Instabilities caused by charging and discharging result in beam and image movement that preclude imaging and microanalysis at even moderate spatial resolution. With the insertion of an objective aperture, the average charge on the specimen often changes, because of the effects of backscattered and secondary electrons from the aperture mechanism, leading to further difﬁculties in imaging, in using the objective aperture to deﬁne the collection angle for EELS, and in correctly aligning the beams into the spectrometer. The usual “cure” to prevent or signiﬁcantly reduce the problems of specimen charging is to coat the transmission electron microscope (TEM) specimen with a thin layer of carbon. Whereas this has only minor consequences for most imaging studies, for EELS it means that the carbon edge is often the strongest in the spectrum and the low loss region is strongly distorted. In some cases the presence of a carbon ﬁlm may preclude quantitative or even qualitative analysis. Imaging and analysis of noncoated insulating specimens at temperatures of several hundred degrees in a heating holder has been suggested as a way to avoid charging effects, but this is not always possible without changing the microstructure of the sample. Another source of carbon on specimen surfaces is hydrocarbon contamination. Gross effects are instantly recognizable from the dark spot or ring produced with

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focused probes. However, more subtle manifestations can be equally as disastrous. EELS analysis is typically performed in the thinnest regions of the specimen where even thin layers of hydrocarbon contamination that are not apparent in TEM images can result in huge carbon edges and major distortions of the low loss region. In modern instruments the main source of contamination is from mobile adsorbed molecules present on the specimen as it is inserted into the analytical electron microscope (AEM). There are four commonly used cures. The ﬁrst is to “ﬂood” the specimen at low magniﬁcation with a large electron ﬂux to “ﬁx” or polymerize the hydrocarbons in the form of a thin uniform layer, thus depleting (for a limited time) the source for diffusion to the analyzed region. In ultra-high vacuum (UHV) instruments, simply leaving the specimen in the microscope overnight can be effective (presumably the responsible molecules desorb). This approach can also be helped by a mild bake. Perhaps the most common cure is to cool the specimen. This reduces the mobility of the offending species. Molecules that would be mobile at the lower temperatures are not present because they would have desorbed at room temperature. An important advantage of cooling is that it works in AEMs with non-UHV quality vacuums. However, one common problem when cooling the specimen to near liquid nitrogen temperatures is the formation of ice on the specimen. Intermediate temperatures of from −130 to −50◦ C prevent contamination but usually avoid problems of ice formation. Another practical limitation posed by cooling holders is that specimen drift is often degraded. Unlike the majority of metallic specimens, which do not suffer radiation damage effects at accelerating voltages of 100–300 kV, most ceramics and catalysts are strongly affected by the incident beam. Although displacement damage can occur, ionization damage and sputtering are of most concern. Major changes in electronic states or valence, crystalline structure, and composition can occur rapidly. Such changes may completely invalidate otherwise high quality measurements. For example, crystalline phases can be amorphized [8.1], amorphous phases can crystallize [8.2], material can be oxidized or reduced [8.3,8.4], and beam damage (such as preferential sputtering observed in Cr3 C2 , TiC, Cr2 N, Fe2 O3 , NiO, and TiN) can change the local composition [8.5,8.6]. The problems are most severe in ﬁeld emission gun (FEG) instruments where focused probes with extremely high current densities can be produced, leading in some cases to “hole drilling” (described in more detail below). Another area of concern in performing EELS experiments involves specimen preparation artifacts. Undesirable surface ﬁlms again ﬁgure prominently. Ion milling is perhaps still the most widely employed method for thinning ceramic specimens to electron transparency. Thin surface layers “damaged” by the ions often have a different composition (because of differential sputtering) or have a different structure (e.g. they may be amorphized) from the “bulk” material. Composition differences can also occur by the redistribution of material sputtered from other parts of the specimen. The effects of surface layers are illustrated in Fig. 8.1, where parallel-detection EELS (PEELS) results for increasing specimen thickness from a large intergranular particle of B4 C in sintered a-SiC are shown. For the thinnest regions, the boron edge is undetectable and the carbon edge is characteristic of amorphous carbon. In spectra

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from regions of increasing thickness the carbon edge structure changes and the boron-to-carbon edge intensity ratio increases, eventually approaching that of B4 C.

Fig. 8.1 A series of PEELS results from B4 C for increasing specimen thickness showing the effect of an amorphous carbon surface ﬁlm.

Material sputtered from the ion miller specimen holder can also be deposited on the specimen. Gross contamination is easily recognized but a specimen quality that appears to be satisfactory for TEM (no apparent additional structure) can present major problems for microanalysis, as illustrated in Fig. 8.2 [8.7]. The choice of support materials (e.g., tantalum instead of stainless steel) may help, but possible interferences still need to be avoided. For backthinned specimens (ion milled from only one side), the prior deposition on the unmilled surface of a thin layer of salt (that can be easily dissolved in warm water after thinning) has proven to be an effective way of controlling sputter deposited contamination [8.7]. Another important example of a specimen preparation artifact is that of grain boundary grooving. The analysis of the structure and local composition of grain boundaries receives much attention because of the importance in controlling materials properties. Amorphous grain boundary phases are of particular concern in structural and electrical ceramics. The work of Simpson et al. [8.8,8.9] showed that the accumulation of amorphous sputtered material in the grooves formed at the intersection of grain boundaries with the specimen surface can be highly misleading for both imaging and microanalysis. The presence of an intergranular amorphous phase or grain boundary segregation can be incorrectly indicated. Careful microscopy is needed to arrive at the correct conclusion. Imaging or analysis of inclined as well as edge-on boundaries is useful in revealing the true situation [8.8,8.9]. A ﬁnal point to be emphasized in this section is that EELS is usually performed on the thinnest areas of the specimen where many of the undesirable affects described above (hydrocarbon contamination, ice formation, disturbed surface layers,

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Fig. 8.2 Contamination on Al2 O3 from sputtering during ion milling. (a) Image and (b) EDS results from a grossly contaminated specimen. (c) Image and (d) corresponding EDS results showing Mo, Fe, and Ar on an apparently clean specimen [8.7].

sputter deposited impurities) are most signiﬁcant since they are a large fraction of the specimen thickness. Higher voltage microscopes have an advantage because of their increased penetration. In EDS, an effective ﬁrst-order correction to remove the contribution of surface ﬁlms on wedge specimens is to subtract from the spectrum of interest a spectrum obtained under identical conditions from a thinner region of the specimen [8.10,8.11]. It is possible that this procedure could work also for EELS, but perhaps only after recovering single scattering data. 8.3

COMPOSITION DETERMINATION

The basis of methods for quantiﬁcation of elemental concentrations are well established [8.12]. Partial ionization cross-sections calculated with the SIGMAK and SIGMAL programs are widely used [8.12]. Although the absolute number of atoms of a particular element in the excited volume can be determined, a ratio method (similar to that commonly used in quantitative EDS analyses) is more frequently employed: σy (β, ∆)Ix (β, ∆) Nx (8.1) ≈ Ny σx (β, ∆)Iy (β, ∆)

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where N is the number of atoms, σ is the partial ionization cross-section, I is the core-loss intensity in an energy window ∆ for a collection angle β, and the subscripts refer to elements x and y [8.12]. In this section, several representative examples, some that expose further experimental complications, are described. The examples are classiﬁed into the analysis of intergranular ceramic phases, intragranular ceramic phases, dilute solid solutions, and catalysts. The emphasis is on low-Z constituents and the complementary nature of the technique to EDS and other AEM methods. 8.3.1

Intergranular Ceramic Phases

One of the earliest materials science applications of high spatial resolution coreloss microanalysis by EELS was to the analysis and identiﬁcation of intergranular phases that form as a consequence of the addition of sintering aids (particularly magnesia) to Si3 N4 [8.13,8.14]. The intergranular phase was known to control the high temperature strength, fracture mechanisms, compressional creep, and oxidation resistance, but its composition was not known in detail. Both amorphous (SiO2 ) and crystalline (MgSiO3 and Si2 N2 O) regions were identiﬁed largely on the basis of the quantitative analysis of K-edges. Energy dispersive X-ray detectors capable of detecting light elements were not available at the time. The results were important for a better understanding of the composition of the intergranular material and the possibilities for modiﬁed processing to mitigate its deleterious effects. The high hardness and melting point of TiB2 make it attractive for cutting tools, valve components for particle-laden slurries, and similar specialized applications. The fabrication of dense TiB2 ceramics with high fracture strength and toughness requires that grain growth and secondary phases be controlled. Liquid-phase sintering with nickel allows densiﬁcation at temperatures 600◦ C lower than for pure TiB2 , thus inhibiting TiB2 grain growth and consequent microcracking. However, reactions with the nickel result in complex multiphase microstructures. In material containing 10– 20 wt% Ni hot-pressed at 1720 K and 12 MPa, an Ni3 B intergranular phase was identiﬁed by the combined AEM techniques of EDS, CBED, and EELS (see Fig. 8.3) [8.15,8.16]. In contrast, materials pressureless sintered with 5–20 wt% Ni contained an intergranular Ni20.3 Ti2.7 B6 τ phase [8.17]. Hot-pressed TiB2 –Ni3 Al contained an intergranular Ni–Al–B τ phase [8.17,8.18]. An understanding of why different intergranular phases formed in the TiB2 –Ni system was reached only after analysis of the exuded material where the important role played by oxides was revealed [8.19]. Although other AEM techniques were also used, EELS played an important role in the above work. First it allowed an unambiguous identiﬁcation of the phases as borides. Second, it helped clarify the sometimes misleading results from EDS. The titanium content of the intergranular phases was vital for understanding the reaction between Ni and TiB2 (e.g., in the formation of Ni3 B, where did the extra Ti go?). Secondary excitation of remote regions by hard X rays or backscattered electrons produced by the interaction of the primary beam with the specimen, contributes a component (10% is not unusual) to the EDS signal that is characteristic of the average composition of the specimen [8.20, 8.21]. Thus, the Ti content of the intergranular

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Fig. 8.3 EEL spectra from (a) TiB2 grain and (b) Ni3 B intergranular phase in TiB2 liquid phase sintered with Ni.

phases was always overemphasized by EDS [8.21]. No such problem occurs for EELS and more reliable compositions were obtained. However, not everything about the EELS analyses was straightforward. The usual inverse power law ﬁt to the background failed completely for the boron edge in spectra from the intergranular phases because of the presence of the large Ni M -edges. With a polynomial instead of the usual linear ﬁt to a log–log plot of the data, the spectra could be processed satisfactorily [8.22,8.23]. Accurate quantitative analysis of Ni3 B and Ni20.3 Ti2.7 B6 single-crystal standards validated the procedure. The large Ni M -edges not only caused problems in background ﬁtting, but also resulted in very low peak-to-background (P/B) ratios for boron in Ni3 B (Fig. 8.3b) and τ phases. The low P/B was important in exposing the inadequacy of inverse power law background ﬁtting; with more pronounced boron edges the problem would have been much less apparent. With the traditional spectral analysis procedures employed, the detectability limit of boron in nickel was estimated at 5–10 at.% as a result of the low P/B ratios. The use of second difference methods now available would be expected to overcome the background ﬁtting problems and increase the sensitivity for boron detection. 8.3.2

Intragranular Ceramic Phases

Although the addition of sintering aids is a common practice for structural ceramics, not all additions result in the formation of new intergranular phases. In sintered α-SiC containing additions of B4 C, no boron has been detected at grain boundaries with the use of surface analysis techniques on fractured specimens [8.24, 8.25]. However, large particles of B4 C several mm in diameter were detected and “mapped” by SIMS. They were also observed in AEM examinations; quantitative EELS analyses (see Fig. 8.1), along with CBED, conﬁrmed their identiﬁcation [8.25]. The AEM studies also revealed a distribution of ﬁne (< 10 nm dia.) intragranular precipitates in both annealed and crept specimens [8.24,8.25]. Attempts were made to identify these precipitates by CBED and EELS. Two serious experimental difﬁculties were encountered. The ﬁrst problem to be overcome, which is generic to this type of analysis, was matrix overlap. The small particles did not usually extend all the way through the foil. In CBED, this situation

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can result in double diffraction effects that produce extra reﬂections equivalent to artiﬁcial d-spacings that can lead to difﬁculties in identiﬁcation. In EELS, at best only qualitative analysis is possible. Besides edge overlap from elements common to both phases, some minor constituents of the precipitate may not be detected because of their reduced effective concentration in the analyzed volume. The second problem, which is increasingly being recognized as more-or-less universal in nonmetallic specimens, was beam damage with the high current density focused probes achieved in FEG instruments. For the SiC–B4 C specimen [8.26] such beam damage was severe after even only a few seconds (Fig. 8.4). To avoid the problems of a focused probe, (serial) EELS was performed at high magniﬁcation in the image mode, with the EELS entrance aperture used for area selection. The measurements were made at 300 kV in a LaB6 -equipped instrument to avoid the problems of high frequency ﬂicker in the probe current for the FEG [8.25, 8.26]. A spectrum from a precipitate overlapped by the matrix, a condition detected by the presence of moir´e fringes, is shown in Fig. 8.5a. Note the large Si L2,3 -edge. A spectrum from a precipitate that completely penetrated (the thinnest region of) the specimen, as judged by the absence of moir´e fringes, is shown in Fig. 8.5b. The Si L2,3 -edge is much reduced but is still present. Assuming part of the carbon signal (in proportion to the Si L signal) was from SiC, the particle was shown to be a boron-rich carbide, but the composition obtained was not exactly the expected B4 C stoichiometry. A surface contamination/disturbed layer was present (mostly from ion milling) and this could have been more carbon rich than SiC. However, another possibility is that part of the Si L and C K signals were due to chromatic aberration effects [8.27]. The intensities in the different edges would then originate from different regions of the specimen, the largest lateral displacement (into the matrix) being for the carbon edge. Thus the EELS results can be rationalized. Together, the semi-quantitative EELS results and limited CBED results were enough to identify the small intragranular precipitates as B4 C.

Fig. 8.4 Intragranular B4 C precipitate in a-SiC. (a) Before and (b) after focusing the probe for a few seconds in a 100-kV FEG TEM.

Chromatic aberration effects should be minimized by operating with the AEM in the diffraction mode (so-called image-coupled mode) and with area selection by the

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extent of the electron probe rather than with the use of a selected area aperture. The selected area aperture method has also been shown to be susceptible to chromatic aberration effects [8.28]. When radiation damage precludes the use of a focused probe, the effects of chromatic aberration will have to be recognized and tolerated where high spatial resolution is required. The proposal that EELS on a high resolution TEM can provide high spatial resolution microanalysis without the need to use ﬁne probes or EDS detectors that might be incompatible with objective lenses optimized for imaging performance, is thus seen to be impracticable. An imaging ﬁlter, such as a corrected omega ﬁlter, may be the best way to add high spatial resolution EELS capability to a “dedicated” high-resolution electron microscope (HREM), and has the additional advantage of being able to remove the valence loss component from HREM images.

Fig. 8.5 EEL spectra for B4 C in SiC. (a) Particle overlapped by matrix. (b) Particle completely penetrating the foil.

8.3.3

Dilute Solid Solutions

The use of EELS for the measurement of the composition of solid solutions has been less frequent than for analysis of second phases as described in the previous sections. One illustrative example on a technologically important material is the work of the Toulouse group on carbon ﬁbers. Nitrogen can be introduced from the precursor materials and can affect the mechanical properties of the ﬁbers when used in composites. Serin et al. [8.29] were able to measure the nitrogen proﬁle through the ﬁber by serial EELS measurements on ultramicrotomed sections. In an outer 2-mmthick shell, nitrogen levels of 6–7 at% were measured, whereas in the core region a uniform 2–3 at% N was present. The data were useful in understanding the behavior during processing to produce high tensile strength ﬁbers. The authors estimated a detectability limit of ∼1% nitrogen. A useful comparison to this carbon ﬁber work is the detection and characterization of nitrogen at 100 platelets and voidites in diamond [8.30,8.31].

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8.3.4

Catalysts

There have been surprisingly few published studies of composition determination by EELS applied to catalysts. Wang et al. have reported results from a comparison of intermetallic LaCo5 in two conditions with Co/La2 O3 and Co/CeO2 supported catalysts, each examined after use for the hydrogenation of carbon monoxide into methane and other hydrocarbons [8.32]. Besides EELS, EDS and HREM were used in the electron microscopic characterization. Features in the low-loss region and particularly higher energy core losses were used to distinguish components such as La2 O3 , cobalt metal, and carbon. Quantitative EELS was used to determine the cobalt/oxygen ratios for (partially) oxidized cobalt particles. In both the LaCo5 -based and Co/La2 O3 materials the cobalt was not as strongly oxidized as in the Co/CeO2 catalyst and carbon separated the cobalt from the lanthanum oxide. The results were used to rationalize the much lower activity of the Co/CeO2 catalyst. Further work on the Co/CeO2 system was subsequently reported [8.33], where again the importance of electron-beam-induced changes was discussed. 8.4

LOW-LOSS/ELNES STUDIES

The ability to obtain information on atomic bonding and chemistry, rather than just the elemental composition, is one of the features of EELS that distinguishes it from EDS and one that is ﬁnding increased use in the characterization of materials (see the chapters by Brydson and Batson). Of course, good energy resolution is required for such studies, but with modern instrumentation this is not a particular problem. However, some applications require extreme resolution, dictating speciallyconstructed spectrometers and the use of ﬁeld emission guns. In many cases, both near-edge ﬁne structure and low-loss spectra are used together to good advantage. 8.4.1

Transition Metal Oxides

The transition metal oxides exhibit a variety of crystalline and electronic properties, many of which are reﬂected in the near-edge structure of the energy loss spectrum [8.34–8.36]. Although a more thorough summary of ELNES is given in the chapter by Brydson, this section is intended to illustrate some spectroscopic features and trends in the near-edge structure of the TM oxides, which may offer insights into more complex systems. To a ﬁrst approximation, the O K-edge represents the p-projected density of unoccupied states for the O atom. Figure 8.6 shows the experimental O K-edge from several TM oxides [8.37]. In these materials the O K-edge can be partitioned into two main regions. The ﬁrst, 5–10 eV above the edge onset (peak a), is dominated by transitions into unoccupied states of O 2p character. The strong hybridization of the O 2p and the highly localized TM 3d states creates holes in the O 2p band [8.38]. The division of peak a in MnO2 and TiO2 is attributed to a ligand ﬁeld splitting (the energy gap between 2t2g and 3eg states) and exchange splitting (the spin-up and

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spin-down energy difference). Intensity contributions to the region 10–30 eV above the threshold (peaks b–d) is often attributed to transitions into O 2p states hybridized with the TM 4s and 4p bands [8.37–8.39]. However, the O 2p density of states (DOS) is minimal 20 eV above the threshold and the TM 4sp states are free-electron-like and do not form localized bands like the TM 3d states. Therefore, the O 2p - TM 4sp band will contribute little to the structure at energy losses greater than around 15 eV above the edge onset. The peak structure is best explained by electronic scattering off of various O shells. Speciﬁcally, peak b can be attributed to intrashell multiple scattering within the ﬁrst O shell [8.37,8.40,8.41], peak c is due to intershell multiple scattering from outer lying O shells, and peak d is the result of single scattering from the ﬁrst O coordination [8.37].

Fig. 8.6 Experimental O K -edges from several TM oxides after deconvolution and a background subtraction.

The sensitivity of the O K-edge to the ligand ﬁeld and exchange splitting is shown in the near-edge structure of the manganese oxides (Fig. 8.7a)[8.39]. The O K-near-edge structure manifests a single prepeak. The 1.4 eV separation between the unoccupied 2t2g and 3eg states is sufﬁciently large tospli this peak however, transitions into the 2t2g band have a much higher probability owing to the abundance of 2p holes when compared with the eg band. In contrast, exchange and ligand-ﬁeld interactions in Mn2 O3 divide the near-edge structure into three peaks. Peak a1 is the result of transitions into 3eg (spin-up) states, which are separated by 1 eV from peak a2 , which is due to transitions into 2t2g (spin-down) states. 3.4 eV separates peak a2

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from a3 , which is attributed to transitions into 3eg (spin-down) states. The near-edge structure of MnO2 is similar to that of Mn2 O3 , however, the effect of the ligand-ﬁeld and exchange interactions combine such that the 2t2g (spin-down) and 3eg (spin-up) states overlap in energy to form peak a1 while peak a2 is the result of transitions into higher energy 3eg (spin-down) states [8.39]. The Mn L2,3 -edges from various Mn oxides are shown in Fig. 8.7b [8.39]. The obvious trend in these data is the direct dependence of the energy onset with the formal oxidation state of the Mn ion. The ﬁne structure of the Mn L2,3 white lines is due to the atomic multiplet structure that result from 3d–3d and 2p–3d interactions as well as a cubic-crystal-ﬁeld coupling [8.42].

Fig. 8.7 O K -edge and Mn L2,3 -edge from a series of manganese oxides after deconvolution and background subtraction.

In addition to the Mn oxides, CoO and Co3 O4 are another set of compounds that are chemically similar and yet exhibit very different ionization edges. A plot of the O K-edges from CoO and Co3 O4 is shown in Fig. 8.8a, from our recent work, which replicates previous results [8.43]. In CoO the Co ions have a 2+ valence and are octahedrally coordinated by O atoms. Co3 O4 has a spinel structure and consists of mixed valent Co with one third of the ions occupying tetrahedral sites (Co2+ ) and the remaining two thirds in octahedral sites (Co3+ ). A large peak is observed in the nearedge structure (530 eV) in Co3 O4 . This structure is absent in materials containing ionically bonded O such as in CoO. In ionic TM oxides the bound transition 1s → 2p is prohibited due to the six electrons localized in the 2p band. The availability of this transition requires a strong O 2p - TM 3d hybridization. The intensity of this peak

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is a measure of the covalency of the TM - O bond [8.38,8.44,8.45] and can therefore be used to characterize bonding in these materials on the nanometer scale.

Fig. 8.8 (a)O K -edge and (b) Co L white lines for CoO and Co3 O4 .

The Co L2,3 white lines are similar for both phases with the differentiating factor being a slightly higher onset energy of the L3 line in Co3 O4 (Fig 8.8b). This energy shift conforms to the trend of the L3 onset energy scaling directly with the TM valence mentioned earlier [8.39]. Using a 50 eV normalization window 50 eV above the L3 edge onset [8.46] the integrated white line intensity was measured to be 0.53 ± 0.02 for both Co oxides suggesting equivalent 3d occupancies [8.43]. 8.4.2

Lithium Transition Metal Oxides

Lithium transition metal oxides (LiTMO2 ) are currently a group of technologically interesting materials due to their widespread use as cathodes in rechargeable Li batteries. Recent studies of the electronic structure of LiTMO2 at different states of lithiation have produced some surprising results. First-principles calculations have shown that in LiCoO2 the total charge density about the Co ion is virtually unaffected by Li concentration [8.47–8.50]. Despite the nearly complete charge transfer from the Li to the TMO2 host, these calculations suggest the Co valence remains in the trivalent state during Li intercalation. Therefore, the incoming charge from the Li 2s electron is almost entirely accommodated by the O ion. This behavior is noteworthy since conventional chemistry states that the O valence is generally 2 minus and invariant. An experimental investigation into the valence changes occurring during lithiation of a TMO2 host, began with a quantitative analysis of the core edges in several delithiated (charged) LiTMO2 s. The O K-edges for ﬁve samples of Li1−x Ni0.8 Co0.2 O2 are shown in Fig. 8.9 [8.51]. The edge onset is characterized by a large prepeak at 530 eV corresponding to a transition into the eg band (σ ∗ ). As mentioned earlier, the prepeak is the consequence of transitions into holes that emerge from the O 2p - TM 3d hybridization. The ﬁne structure of the broad peak ∼15 eV above the edge onset is attributed to intrashell multiple scattering. In this study, the spectral intensities

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Fig. 8.9 EELS spectra of the O K -edge from Li1−x Ni0.8 Co0.2 O2 labeled by value of x.

were normalized to a 10 eV window 40 eV above threshold and the deintercalation value, x, was measured by comparing the intensity of the Li K-edge with known standards. The sum of the p densities of states at oxygen atoms was calculated from the eigenvalues and the projection operators given by the ab initio total energy and molecular dynamics program VASP (Vienna ab initio simulation program). To compare these calculations to EELS spectra, the atomic cross-sections were calculated to account for the differences in probabilities between the O 1s → 2p transition and the O 1s → continuum transition. The cross-sections were calculated with hydrogenic [8.12] and Hartree–Slater [8.52,8.53] wave funcions, which gave similar results. A plot of simulated O partial density of states (PDOS) for Li1−x CoO2 is shown in Fig. 8.10a. The O PDOS were normalized to the 2p band ∼15 eV above the Fermi energy (EF ) and the PDOS was plotted relative to EF (E − EF ). Figure 8.10b shows a plot of the O PDOS calculated with VASP and the energy loss spectrum for a sample of Li1.0 CoO2 . In this plot the EELS spectrum was corrected for the inelastic form factor by dividing out the computed cross-sectional intensity. The core binding energy (E1s − E2p ≈ 529 eV) has been subtracted in Fig. 8.10b. The peak located just above EF at ∼ 1.5 eV is the lowest unoccupied state, and corresponds to a σ ∗ transition in the O 2p–Co 3d band. Similar to the experimental spectra, the calculations of the O PDOS show that with Li extraction, this peak just above the Fermi energy grows in intensity. The O K-edges for the delithiated series of Fig. 8.10 show an increase in the integrated intensity of the prepeak at 530 eV. This suggests that delithiation changes the electron density about the anion. This dependence is quantiﬁed in Fig. 8.11,

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Fig. 8.10 (a) Calculated O PDOS for Li1−x CoO2 for four compositions and (b) Simulated and experimental O PDOS of Li1−x CoO2 . The cross-sectional intensity has been divided out of the EELS spectrum and a core energy of 529 eV was removed.

Li1-xCoO2 (VASP) Li1-xCoO2 (EELS) Li1-xNi0.8Co0.2O2 (EELS)

Pre-peak Intensity

0.50

Fig. 8.11 Normalized intensity of the antibonding peak in calculated O PDOS (VASP) for Li1−x CoO2 and prepeak of the O K -edge (EELS) for Li1−x CoO2 and Li1−x Ni0.8 Co0.2 O2 . The ionization cross sections were divided out of the experimental data.

0.45 0.40 0.35 0.30 0.0

0.2

0.4

0.6

Li Concentration (x)

which compares the integrated intensity of the near-edge structure in LiCoO2 [8.54] and LiNi0.8 Co0.2 O2 [8.51], normalized to the continuum, with Li concentration. A linear ﬁt to the Li1−x Ni0.8 Co0.2 O2 data gives a 60% increase in the intensity of the O K near-edge structure. The experimental energy loss data used in this plot were corrected for their inelastic form factor, which were calculated using Hartree–Slater wavefunctions. The simulated data was quantiﬁed by normalizing the integrated intensity of the lowest unoccupied peak to the p DOS 15 eV above EF . The plot shows that the intensity of the unoccupied antibonding state is correlated to the Li concentration through a linear relationship. The slope for the Co and (Ni,Co) systems are ∼ 0.26 and 0.28, respectively. The difference in the initial intensity is likely due to the different number of O 2p holes present in the stoichiometric material of the two systems. The Li(Ni,Co)O2 material has fewer O 2p holes than does LiCoO2 .

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Changes in charge compensation by the transition metal ion were also measured using ionization edges. There are a number of means by which to extract valence information from the TM L2,3 -edge. These include measurements of the the L3 /L2 white line ratio [8.57–8.59] or L3 /(L2 + L3) branching ratio, and the L3 onset energy [8.39,8.55,8.56], as mentioned earlier for the Mn oxides. Another, possibly more direct course, is to intergrate the total white line intensity (normalized to the continuum) to obtain a 3d occupancy [8.46,8.60]. Although each method has advantages, it should be noted that small energy ﬂuctuations in the incident beam can make it difﬁcult to determine the L3 onset energy accurately. One way to overcome the resulting edge drift is to calibrate each set with another edge in the spectrum such as the C K-edge from the TEM grid. Using core edges from other elements within the sample such as O or another TM should be done with some reservation since the charge density about each ion will likely change with composition. The edge onset of even the most stable elements, such as O, can shift with small variations in composition. In this study the total white line intensity was used to determine the TM 3d occupancy. The Co and Ni L2,3 -edges for the delithiated series are shown in Fig. 8.12. The general shape and line intensities appear to be invariant with changes in Li concentration. To maximize the energy resolution, the spectra were acquired in two intervals and matched subsequent to the acquisition. These spectra were aligned along the Co L2,3 white and therefore the edge onset energies are not reliable. The Ni and Co white line intensities and their respective L3 /L2 ratios, shown in Fig. 13 a–c, conﬁrm the nearly constant white line intensity over a wide range of Li concentration. In measuring the integrated white lines, the free-electron-like contributions to the core edges were approximated as step functions, and subtracted from the total intensity (Fig. 8.13d window 1). The white lines were then normalized to a 50 eV window just above the L2 tail (Fig. 8.13d window 2). In Li1−x CO2 and LiNi0.8 Co0.2 O2 the linear ﬁts to the Co L2,3 intensities are invariant over the delithiated series. The essentially constant Co white line intensity indicates a constant 3d occupancy during delithiation. In contrast, the Ni L2,3 intensity from LiNi0.8 Co0.2 O2 increases by ∼ 6% with the removal of two-thirds of the Li ions. This implies a depletion of the 3d band, which suggests that the Ni ions may compensate for some of the Li charge in this material. Observations of decreasing white line intensity with increasing atomic number (increasing d occupancy) are well documented for 3d transition metals [8.46,8.60]. The calibration of the relationship of white line intensity and 3d occupancy in transition metal oxides was originally performed by Stolojan et al. [8.61]. The Ni L2,3 intensities of the delithiated samples were compared with our own analysis of the transition metal oxide standards based on the original study. The 6% increase in the Ni white line intensity corresponds to a change in the Ni valence of < 0.24 electrons over the delithiated series. An extrapolation of the linear ﬁt from 0 ≤ x ≤ 1 provides a total Ni charge compensation of < 0.33 electrons. The relative invariance of the Co white lines suggests a constant net valence in the Co ions during delithiation. It should be noted however, that the total intensity difference over the delithiated series (6% in Ni) is near the detectable limit, given the ﬂuctuations between points of similar Li concentration. Therefore,

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Fig. 8.12 (a) Co L2,3 -edge from Li1−x CoO2 and (b) Co L2,3 -edge and Ni L2,3 -edge from Li1−x Ni0.8 Co0.2 O2 , labeled by x.

Fig. 8.13 (a) Co L2,3 white line intensity and L3 /L2 ratio from Li1−x CoO2 . (b) Ni and (c) Co L2,3 white line intensities and the respective L3 /L2 ratios from Li1−x Ni0.8 Co0.2 O2 . (d) Co L2,3 -edge showing the step function subtracted from the white line intensity (window 1) and the 50 eV continuum used for normalization (window 2).

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the total charge compensation measured is not absolute but should be regarded as an upper bound. 8.4.3

Valence Determination and Atom Location in Catalysts, Ceramics, and Other Metal Oxides

The near-edge ﬁne structure has recently been used to extract valence information in rare earth materials [8.62]. The Ce M4,5 white lines were characterized using a series of Ce oxide standards containing Ce3+ and Ce4+ ions. The appearance of high-energy satellite peaks just above the L2 and L3 white lines in the M4,5 -edge is a feature of Ce4+ that is absent in the Ce3+ ion, and is therefore an excellent signature of the Ce valence state. The amount of trivalent and tetravalent Ce can be quantiﬁed with the appropriate standards. In this study the near-edge ﬁne structure was used as a ﬁngerprint to identify the valence state in two compounds of cerium stabilized zirconia (Fig. 8.14). The cubic phase Zr0.31 Ce0.69 O2 was found to have predominately Ce4+ ions while the Ce in the monoclinic phase appeared to be primarily trivalent. The tetravalent Ce ions at high dopant concentrations were found to contribute to the bonding through a hybridization of the Ce 4f and O 2p states. As a result of this overlap, the changes in the Ce valence states were also observed in the O K-near-edge structure. The contribution of tetravalent Ce to the bonding energy was found to be an important part of the stabilization of zirconia.

Fig. 8.14 Ce M4,5 -edge from cubic and monoclinic cerium zirconia. The peak 6 eV above the L2 line in c-Zr0.31 Ce0.69 O2 is a property of the tetravalent Ce and is absent in the compound containing trivalent Ce.

The recent discovery of superconductivity in MgB2 has aroused much interest in the electronic structure of this material [8.63, 8.64, 8.65]. The arrangement of the B atoms on hexagonal planes is believed to be a play a crucial role in the superconductivity of MgB2 (similar to the role of CuO2 planes in other high temperature superconductors). Zhu et al. have used orientation-dependent EELS to characterize the DOS just above the Fermi energy in the px py and pz bands [8.65]. They exploited the fact that when the momentum transfer (q) is normal to the c-axis in a uniaxial system (e.g., TiB2 [8.66] or MgB2 [8.65]), the electrons will probe predominately px py states. Similarly, when q is parallel to the c-axis, the scattering is dominated by pz states. Figure 8.15 reveals an enhancement of the near-edge structure in the spectrum acquired perpendicular to the c-axis. These results conﬁrm early predictions [8.64]

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that the boron σ(spx py ) band dominates the DOS just above the Fermi energy. The σ holes are believed to be the progenitors of superconductivity in this material. The intensity of lowest lying peak in the B K-edge is a function of the hole concentration in the σ band and may therefore be a means to probe the superconducting properties of MgB2 on the nanometer scale.

Fig. 8.15 B K -edge acquired with momentum transfer q (a) perpendicular to the c-axis and (b) parallel to the c-axis.

Atom location by channeling enhanced microanalysis (ALCHEMI) EDS technique [8.67] has been widely applied to determine sublattice occupancies in ceramics and minerals (as well as metals and semiconductors), the equivalent EELS technique of ELCE (energy losses by channeled electrons) [8.68], which was developed at the same time as ALCHEMI, has been used hardly at all, despite its great potential for providing unique information. The early demonstration of the method is shown in Fig. 8.16. In the MgAl2 O4 spinel used, the oxygen and aluminum occupy octahedral sites and line up in one set of alternate (800), whereas magnesium occupies tetrahedral positions and lines up in the other (800). For s400 < 0 (Fig. 8.16a), where sg is the deviation parameter for reﬂection g, the signal from the more densely packed

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octahedral sites is emphasized and for s400 > 0 (Fig. 8.16b) the tetrahedral signal is emphasized. In Fig. 8.16b, note the signiﬁcantly higher Mg K-edge and lower O and Al K-edges than in Fig. 8.16a. To ensure sufﬁcient localization of the inelastic scattering, it was necessary to displace the collection aperture, but parallel to the Kikuchi lines so as not to change the diffracting conditions relative to the (400).

Fig. 8.16 ELCE of MgAl2 O4 spinel [8.68]. (a) s400 < 0, octahedral sites emphasized. (b) s400 > 0, tetrahedral sites emphasized. The insets indicate the position of the incident beam (solid) and collection aperture (open) with respect to the Kikuchi lines.

The ELCE technique was extended to demonstrate site-speciﬁc valence determination by EELS [8.69]. The technique was demonstrated by locating Fe2+ and Fe3+ ions in a naturally occurring Cr1.1 Fe0.7 Al0.7 Mg0.5 O4 chromite spinel. In Fig. 8.17a, the octahedral signal is emphasized for s < 0 and the tetrahedral for s > 0; thus the oxygen and chromium signals are stronger for s < 0. In Fig. 8.17b, the corresponding Fe L2,3 -edges are shown in greater detail, where it is clear that the Fe L3 and L2 white lines are split into two components separated by ∼ 2 eV. By examining other materials, 1.3 eV of the shift to higher binding energies was attributed to increased valence and 0.7 eV to location on octahedral over tetrahedral sites. The spectra thus demonstrate that Fe3+ occupies octahedral sites whereas Fe2+ occupies tetrahedral sites. The Fe L2,3 -ELNES has been used in an attempt to characterize the state of iron implanted into silicon carbide [8.70] as part of a large effort on ion-implanted ceramics [8.71]. Clustering or precipitation of the iron in the amorphized implanted layer was

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Fig. 8.17 Site-speciﬁc valence determination in chromite spinel [8.69]. (a) Enhanced oxygen and chromium signals for s400 < 0 (octahedral sites emphasized). (b) Corresponding Fe L2,3 -edges in greater detail. Splitting indicates Fe3+ occupies octahedral sites and Fe2+ occupies tetrahedral sites.

suspected, but not detected by TEM imaging. By comparison with metallically bonded iron, the white lines in spectra from the implanted iron were broader and less intense relative to the level of excitation to continuum states (the region above ∼ 735 eV). While not deﬁnitive, and with a clear need for comparisons with Fe–Si and Fe–C compounds, the preliminary results demonstrated that the implanted iron was not primarily in small metallic clusters and supported complementary M¨ossbauer spectroscopy results that indicated a possible covalent character. A number of studies have recently been performed on interfaces in ceramics. One example from Muller et al. illustrates the formation of metal-induced gap states (MIGS) at a 222 MgO–Cu interface [8.72]. The local electronic structure at an interface between two dissimilar materials such as a metal and a ceramic is difﬁcult to predict. Using a focused probe, the composition and electronic structure were measured across the interface on the atomic scale. Figure 8.18 shows the O K-edge from bulk MgO and from various distances from the interface. The formation of a prepeak ∼ 530 eV suggests the formation of metal-induced gap states at the interface. The prepeak decays rapidly with the distance from the interface and is completely absent ∼ 4 ˚A from the interface where the bulk MgO O K-edge is observed. These MIGS have recently been shown to be a general feature of the metal–ceramic interface [8.73]. 8.4.4

Colloids, Hole drilling, and Hybridized Bonding

In a detailed study of MgAl2 O4 spinel implanted at 650◦ C with 2 MeV Al+ ions, primarily to study radiation damage effects in support of applications for fusion reactors, small features in the implanted region near end-of-range (Fig. 8.19a and b) were thought to be metallic aluminum colloids, but could not be conclusively identiﬁed by imaging and diffraction. Parallel-collection plasmon spectrometry performed at 100

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Fig. 8.18 O K -edges taken at various distances from the 222 MgO/Cu interface. The crosshatch indicates the unoccupied region of the band gap.

and 300 kV, having a better sensitivity than the Al L-ELNES, was used to conﬁrm and quantify the presence of metallic aluminum [8.74]. Fourier-log deconvolved spectra from the implanted region were further deconvolved into the components due to metallic aluminum (with a bulk plasmon at 15 eV) and spinel (with a valence loss maximum at ∼24 eV) by linear least squares multiple regression analysis with carefully quantiﬁed reference spectra (see Fig. 8.19c). The measured metallic aluminum depth proﬁle with a maximum of ∼4 vol % was in general agreement with calculated proﬁles. A high spatial resolution PEELS system with a 100-kV FEGAEM producing probes with ∼ 2 nm dia. and 0.8 nA produced spectra with large metallic aluminum components from individual clusters, but beam damage (hole drilling) was observed. Beam damage is also known to proceed by reduction to the metallic components in other oxide ceramics (see below). Spectra were therefore obtained at lower current densities in the TEM image mode at high magniﬁcation with area selection by a 2-mm spectrometer entrance aperture. Chromatic aberration effects did not limit the spatial resolution for this low-loss work. Two spectra from the regions indicated in Fig. 8.19b, but obtained under weaker diffracting conditions, are shown in Fig. 8.19d, and clearly conﬁrm the metallic aluminum character of the clusters. The ability to distinguish between metallic aluminum in clusters and Al3+ ions in the spinel at a spatial resolution of < 10 nm is illustrative of the remarkable power of modern analytical electron microscopy. The above example was based to some extent on the pioneering EELS studies by Berger et al. [8.3] of “hole drilling” in aluminas, a process that has potential for nano-lithography. During drilling of amorphous alumina (a-Al2 O3 ), a sharp peak at 9 eV and a broad asymmetric feature at ∼ 22 eV (Fig. 8.20b) replaced the bulk plasmon at ∼ 24 eV that was originally present (Fig. 8.20a). The peak at 9 eV disappeared as the hole was pierced (Fig. 8.20c). During drilling of Na β-Al2 O3 , a sharp peak appeared at 15 eV with a shoulder at 9 eV superimposed on a broad feature peaked at ∼ 22 eV (Fig. 8.20e). After perforation the spectrum (Fig. 8.20f) was similar to that for a-Al2 O3 . The oxygen K-edge for a-Al2 O3 changed dramatically during drilling and exhibited a sharp peak at 532 eV (Fig. 8.21a and b). In Na β-Al2 O3 , the oxygen

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Fig. 8.19 MgAl2 O4 implanted with aluminum. (a) Cross-section bright-ﬁeld image of implant layer. (b) {222} spinel dark-ﬁeld image near end of range. (c) Fourier-log deconvoluted low-loss spectra from (1) Al metal, (2) undamaged spinel, (3) material at ∼1.6 mm implant depth, (4) best ﬁt from regression analysis indicating 3.7 ± 0.6 vol.% metallic Al. (d) High spatial resolution spectra from regions 5 and 6 in (b).

K-edge slowly disappeared during drilling but maintained its original shape (similar to Fig. 8.21a). The aluminum L-edge, however, showed a dramatic shape change (Fig. 8.21c and d). In a-Al2 O3 , the aluminum L-edge was similar to Fig. 8.21c, but disappeared during drilling. Energy loss images were also obtained and helped the interpretation, which is as follows. For a-Al2 O3 , drilling proceeds by removal of aluminum. The peaks at 9 and 532 eV are consistent with molecular oxygen, the results suggesting that a high pressure bubble(s) forms and eventually bursts, forming a hole. For Na β-Al2 O3 , material is removed atom plane by atom plane from both surfaces with some of the aluminum forming small metal particles that give rise to the bulk (15 eV) and surface (9 eV) plasmons and an aluminum L-edge characteristic of aluminum metal. Furthermore, the authors explain both of these apparently different schemes on the basis of the model of Knotek and Feibleman [8.75] that involves desorption or displacement due to an interatomic Auger process following ionization. Berger et al. have also reported similar studies on TiO2 , Ti2 O3 , and TiO, all of which appeared to damage by reduction with a mechanism involving ionization [8.76]. Combined plasmon and near-edge spectrometry have also been used to good advantage by Berger et al. in EELS analyses of vacuum arc-deposited diamond-

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Fig. 8.20 Low-loss EEL spectra (a,d) before, (b,e) during, and (c,f) after hole drilling of (a–c) amorphous Al2 O3 and (d–f) Na β -Al2 O3 [8.3]. See text for details.

Fig. 8.21 ELNES changes during hole drilling [8.3]. (a,b) oxygen K -edge in amorphous Al2 O3 and (c,d) aluminum L-edge in Na β -Al2 O3 (a,c) before and (b,d) during hole drilling.

like ﬁlms [8.77]. Spectra from this amorphous diamond-like carbon (a-D) were compared with ones from diamond, graphitized carbon, and amorphous carbon (a-C) (Fig. 8.22). A small amount of sp2 -bonded material was detected in the a-D, but the high plasmon frequency (energy) and the shape of the K-edge showed that the material was essentially an amorphous form of diamond. The fraction of sp2 -bonded carbon in a-D was quantiﬁed to be ∼15% from the size of the normalized 1s to π ∗

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peak at ∼286 eV (corrected for multiple scattering) compared to that for graphitized carbon.

Fig. 8.22 Low-loss EEL spectra from (a) diamond, (b) amorphous diamond-like carbon, (c) graphitized carbon and (d) amorphous carbon. Core-loss EEL spectra from (e) graphitized carbon (dotted line) and amorphous carbon (solid line) and (f) diamond (dotted line) and amorphous diamond-like carbon (solid line) indicating ∼15% sp2 bonding [8.77].

8.4.5

Catalysts

Turning now to catalysts, it is again surprising that comparatively little EELS ﬁne structure work has been published. The work described above on Co/La2 O3 and Co/CeO2 also made use of plasmon and near-edge ﬁne structure, particularly at the oxygen K-edge, in the characterization of the catalysts [8.32, 8.33]. An earlier study by Lyman et al. [8.4] on a copper–zinc oxide methanol synthesis catalyst attempted to determine, from differences in the copper L-near-edge structure, whether the active species of copper in this system was dissolved Cu+ or Cu0 (metal). Unfortunately, the results (see Fig. 8.23) were ambiguous. Evidence for small copper particles (Fig. 8.23b) and copper oxide (Fig. 8.23c, where the L3 white line is present) was obtained. Regions of ZnO free of copper (Fig. 8.23a) suggested that any dissolved Cu+ was not homogeneously distributed. The work did show, however, that the problems of secondary excitation in EDS are not present for EELS, and that consequently composition determination is more reliable. One of the problems encountered in the work, which was performed in a FEG dedicated STEM, was beam damage that

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caused the CuO to be reduced to copper metal. This problem was studied further by Long et al. [8.78] who reported early time-resolved spectrometry results showing both the rapid reduction of the L3 , L2 white line intensity and an overall mass loss as the oxide was reduced to metal under the action of the electron beam. More recently, the safe limits of electron beam dose rate and total electron dose to which CuO can be exposed before electron-beam-induced reduction occurs has been examined in more depth [8.79].

Fig. 8.23 EEL spectra from a Cu/ZnO catalyst [8.4]. (a) Cu-free ZnO area. (b) Small Cu particle. (c) Copper oxide particle.

The utility of EELS to locate and characterize carbon deposits on catalysts has been demonstrated by Gallezot et al. [8.80]. External coke deposits produced by n-heptane cracking on an ultra-stable zeolite (USHY), proton-exchanged offretite (H-OFF), and H-ZSM-5 zeolite were studied with a FEG dedicated STEM and serial EELS. The local structure of the coke was determined from carbon K-ELNES by comparisons with reference materials such as amorphous carbon, graphite, coronene, and pentacene (Fig. 8.24). In H-ZSM-5 and H-OFF, the coke forms an external envelope around the zeolite crystal and its structure is similar to that of coronene (polyaromatic-pregraphitic). In USHY, part of the coke is in the form of 1-nm carbon ﬁlaments protruding from the zeolite mesopores and micropores and its structure is more like that of pentacene (linear polyaromatic). The same group have extended their work to alumina-supported platinum catalysts coked with cyclopentane where they have concluded that the coke covering the support surrounding each metal particle probably consists of a disordered arrangement of polyaromatic molecules [8.81].

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Fig. 8.24 Carbon K -ELNES from coke deposits on (a) USHY, (b) HOFF, and (c) H-ZSM-5 zeolites compared with (d) graphite, (e) amorphous carbon, (f) coronene, and (g) pentacene reference materials [8.80].

In slightly different work on zeolites, McComb and Howie applied combined valence loss spectrometry and oxygen K-ELNES to characterize the changes that occur in H-mordenite catalysts on dealumination [8.82]. Small energy shifts (−2 eV for the oxygen K-edge and a 1 eV decrease of the band gap) were observed, with control experiments showing that the possible presence of water within the structure did not appear to affect the analyses. A detailed analysis of valence spectra by dielectric excitation theory, in which four Lorentz oscillators were used to ﬁt the experimental data, allowed detailed insight into changes in the electronic structure due to different amounts of dealumination. To better overcome electron beam damage problems, experiments with a parallel-acquisition system were begun. 8.4.6

Glass

The understanding of short-range order in glassy materials can be a complex and difﬁcult task, but a structural characterization is essential to our understanding of these materials. Real-space information such as atom coordinates or a radial distribution function is necessary for the computation of macroscopic properties. Unlike crystals, where the detailed position of every atom can easily be determined by diffraction,

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atomic arrangements in glasses are disordered and therefore require an investigation into the short and medium-range order.

Fig. 8.25 (a) Energy loss spectra from CaO-Al2 O3 -2SiO2 and (b) LDOS calculations of inequivalent Al atoms in the crystalline material [8.83].

Spatially resolved ELNES has recently been used to investigate the long-range structural ﬂuctuations in silicate glasses [8.83]. Figure 8.25a shows the Al and Si Ledges from four different regions in CaO-Al2 O3 -2SiO2 . The general features labeled A1-A4 and S1-S4 are similar in all four regions indicating the existence of short-range order. Local density of states (LDOS) calculations were performed for a number of inequivalent Al atoms within the crystalline material (Fig. 8.25b). Although the general near-edge features in the glass and the LDOS are similar, speciﬁc features in the various regions of the glass were found to correspond with a unique Al site [e.g., the ﬁrst spectrum in Fig. 8.25a and Al (1) in Fig. 8.25b]. Instead of these features being broadened, due to contributions from each of the unique Al sites, the features remain sharp but vary from region to region within the glass. These results suggest the presence of short and medium-range order in CaO–Al2 O3 –2SiO2 glass and demonstrate the existence of long-range structural ﬂuctuations.

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8.5 EXELFS ANALYSIS 8.5.1

Procedures

Analysis of extended electron energy loss ﬁne structure (EXELFS) can provide, for speciﬁc atomic species, information on near-neighbor bond lengths and coordination number as represented by the radial distribution function (RDF). Brieﬂy, the extended ﬁne structure arises from an interference effect due to the ejected core electron wave backscattering from neighboring atoms. The wavelength (λ) of the ejected electron and the path length determine the constructive/destructive character of the interference: thus with increasing energy loss (E) the interference is alternately constructive and destructive, leading to maxima and minima in the scattering cross section, the oscillatory part being χ(E) =

J 1 (E) − A(E) A(E)

(8.2)

where J 1 (E) is the “observed” single scattering intensity distribution and A is the intensity distribution from isolated atoms (no backscattering). The theoretical treatment and data analysis procedures (e.g., [8.12]) follow those of the more widely used equivalent technique of extended X-ray absorption ﬁne structure (EXAFS). Thus,

χ(k) =

Nj fj (k) −2rj exp( ) exp(−2σj2 k 2 ) sin[2krj + φ(k)] 2 r k λ i j j

(8.3)

where the terms on the right hand side represent the RDF, backscattering, “absorption” due to inelastic scattering, radial broadening due to thermal (and static) disorder, and the interference condition with components due to both path length and also the phase change of the electron wave in the ﬁelds of the emitting and backscattering atoms, with k the electron wavevector, Nj the number of atoms in the jth shell at radius rj , fj the backscattering amplitude or form factor, λi the inelastic mean free path (∼1 nm at 100 eV), and σj2 the mean square displacement. The data analysis procedure is illustrated in Fig. 8.26 for the magnesium K-edge from MgO [8.84] and involves the following steps: 1. Obtain J 1 (E) by deconvolution, usually by the Fourier ratio method following background subtraction (Fig. 8.26a). 2. Obtain τ (E) by curve-ﬁtting a polynomial (e.g., fourth order) or cubic spline (Fig. 8.26b and c). 3. Convert χ(E) to χ(k) with k=

2π = [2m0 (E − Ek )]1/2 λ

(8.4)

where m0 is the electron rest mass and Ek is the ionization edge threshold energy taken as the energy midpoint of the edge rise or as a variable parameter such that

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peaks in the imaginary part and modulus of the Fourier transform of τ (k) occur at the same radius. 4. Interpolate χ(k) to give equally spaced data points. 5. Obtain k n χ(k) to correct for the k dependence of backscattering (Fig. 8.26d); usually, the exponent 1 < n < 3. 6. Truncate χ(k) at appropriate zero crossings or with a smooth-edged “window” function [8.12, 8.85]: a low k cut-off (20 − 40 nm−1 ) to avoid near-edge effects and a noise-limited high k cut-off (60 − 120 nm−1 ) are usual. 7. Perform a fast Fourier transform to yield (Fig. 8.26e) |χ(r)| =

φ 1 Nj δ(r − rj − ) 2π j rj2 2

(8.5)

8. Determine |χ(r)| from standards or calculation [8.86] to “correct” interatomic distances.

Fig. 8.26 EXELFS spectral processing steps illustrated for the magnesium K -edge from MgO [8.14]. (a) Deconvolution, (b) fourth order polynomial ﬁt, (c) oscillation function, χ(E), (d) conversion to k -space and weighting, kχ(k), (e) modulus of Fourier transform (uncorrected RDF.)

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301

Since the procedures are complex, the ﬁnal results can be inﬂuenced not only by the collection methods used, but also by the choice of parameters used during the data analysis [8.12]. Therefore, in the two examples of ORNL work that follow, the data were collected and analyzed following a set of standardized procedures which had previously been shown to produce the most consistent and reliable results. Once an optimum procedure was determined for a given edge, it was used for analysis of all spectra for that particular edge. 8.5.2

Silicon Carbide

Ion implantation of ceramics can produce amorphous implanted layers (e.g., [8.71]). The amorphous state is difﬁcult to characterize except by the RDF. Angelini et al. performed EXELFS analyses on SiC amorphized by chromium ion implantation at room temperature and on SiC amorphized by high-ﬂux 300 kV in situ electron irradiation at < 100 K [8.1,8.87,8.88]. The data were collected with a serial EELS at 300 kV and the carbon and silicon K-edges were analyzed. Figure 8.27 shows the pronounced EXELFS oscillations on the background-subtracted silicon K-edge from crystalline SiC and the much weaker oscillations on the edge from amorphous material. Following careful processing of the data according to the procedure outlined above, the RDFs were obtained and showed possible slight elongations of the Si–C and Si–Si bonds and a clear decrease in the second nearest neighbor (Si–Si) coordination for the amorphous material [8.89]. The phase shifts of Teo and Lee [8.86] produced excellent agreement with the known bond lengths in the crystalline SiC. The RDF results were consistent with the density decrease accompanying amorphization that was measured by proﬁlometry [8.71]. Recently, a similar EXELFS analysis comparing amorphous and crystalline SiC has been made by Martin and Mansot [8.90] who used a friction-induced amorphous SiC.

Fig. 8.27 (a) EXELFS oscillations on Si K -edge from crystalline and amorphous SiC recorded at 300 keV[8.88]

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8.5.3

Alumina

A detailed EXELFS study of ion implanted Al2 O3 produced some fascinating results [8.2,8.91,8.92]. Two amorphous materials produced by ion implantation at <100 K were compared; one amorphized by implantation with iron (a-Al2 O3 –Fe) and the other by implantation with aluminum and oxygen in such a way as to preserve the Al2 O3 stoichiometry (a-Al2 O3 –Al,O). The a-Al2 O3 –Al,O material crystallized to mostly γ-Al2 O3 (a cubic transitional form) when annealed for 1 h at 960◦ C, and was electron-beam-sensitive, crystallizing to γ-Al2 O3 under a focused probe. The a-Al2 O3 –Fe was stable under the electron beam and formed epitactic α-Al2 O3 after an equivalent anneal. These observations suggested structural differences might exist in the two amorphous aluminas and these were sought by EXELFS analyses of aluminum and oxygen K-edges, the data again collected serially at 300 kV. Great care was exercised in the data collection and analysis procedures to ensure identical treatment for the different materials and crystalline standards. Only because of this were reliable comparisons considered possible. The ﬁrst peak in the RDFs derived from the aluminum K-edge data (representing the Al-O nearest neighbor bonds) are shown in Fig 8.28. The peak positions and peak shapes for the a-Al2 O3 –Fe and crystalline α-Al2 O3 are very similar and clearly different from those for a-Al2 O3 – Al,O and crystalline γ-Al2 O3 . Thus, the EXELFS analyses conﬁrmed the existence of underlying structural differences in the amorphous aluminas. Further, and quite remarkably, the Al–O bonds in the two amorphous aluminas clearly reﬂect the bond structure of the two respective crystalline forms into which they ﬁrst crystallize. The implications for understanding amorphous structures and competing crystalline forms are far-reaching.

Fig. 8.28 The ﬁrst peak in the RDFs derived from 300 kV EXELFS analyses of aluminum K -edge for α-Al2 O3 , γ -Al2 O3 , and aluminas amorphized by implantation with iron or a stoichiometry-preserving ratio of aluminum and oxygen [8.92].

OTHER TOPICS

303

There have been other EXELFS studies of alumina (e.g., [8.85,8.93,8.94]). Comparison with the results of Sklad et al. [8.92] is complicated by the role of hydrogen and hydroxyl groups in “aluminas” fabricated from various precursors and even by anodic oxidation. In fact, the attempts at correlating Al–O bond length with coordination number appear to be compromised by differences in material stoichiometry and impurity content. 8.6 8.6.1

OTHER TOPICS Time-Resolved EELS

With the advent of parallel-detection EELS, time-resolved studies have become a much more practicable proposition than with serial collection spectrometers, although some attempts were made [8.78]. Analysis of beam-sensitive materials and indeed studies of the beam damage process itself are important and obvious applications. The radiation damage characteristics of silicon oxynitride (Si2 N2 O), a common intergranular phase in Si3 N4 ceramics that forms as a result of sintering aid additions or from oxygen on the original powder particles, have been described by Das Chowdhury et al. [8.95]. Time-resolved PEELS was used to follow changes in elemental concentration with irradiation time in a 100 kV FEG TEM. Interestingly, whereas the silicon and nitrogen signal decreased monotonically, the oxygen signal exhibited pronounced maxima. The same group adapted the instrumentation to perform line scans across interfaces in SiC ceramics [8.96]. Oxygen segregation at grain boundaries was detected and the use of an astigmatic probe aligned parallel to the interface was suggested as a way of overcoming beam damage problems. 8.6.2

Reﬂection EELS

Reﬂection electron energy loss spectrometry (REELS) combines the techniques of grazing-incidence reﬂection electron microscopy (REM) and EELS to provide a surface-sensitive analysis technique for bulk crystals in a TEM or STEM [8.97]. The technique has been applied to metals, semiconductors, and particularly ceramics [8.98] and has recently been reviewed [8.99]. A central concept is that, under the surface resonance conditions employed, the electrons travel a certain distance along the surface before being reﬂected. In the low-loss region, surface plasmon excitation is usually dominant, with plasmon-to-zero loss intensity ratios considerably larger than in conventional thin-ﬁlm transmission EELS. Core-loss P/B ratios and overall intensities are much lower than in equivalent transmission EELS experiments, so that parallel detection is almost a necessity for work involving core losses above ∼1 keV, but acceptable spectra can be obtained (see Fig. 8.29). The optimum experimental conditions for quantitative surface microanalysis by REELS have been examined with reference to MgO [8.100]. Since strong dynamical scattering effects cannot be avoided in REELS, the expected deviations from a

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Fig. 8.29 Parallel-detection REELS data for (a) oxygen and (b) magnesium K edges from a MgO (100) surface acquired at 300 kV with the (600) specular reﬂection and a beam azimuth near [001].

Lorentzian distribution of inelastic scattering and atomic site discrimination effects have to be accounted for in attempts at quantiﬁcation. One way of handling such effects is to include them in an effective ionization cross section. Ratios of effective ionization cross sections can be measured from a thin specimen standard by duplicating the diffracting conditions used for REELS [8.100-8.102]. Errors of a factor of two can easily arise if the familiar transmission EELS cross sections are used (see Fig. 8.30).

Fig. 8.30 Comparison of REELS background-subtracted (a) oxygen and (b) magnesium K -edges from a MgO (100) surface acquired at 300 kV with beam azimuths near [100] and [110]. Note the strong effect of channeling discrimination for the [110] azimuth where the magnesium and oxygen atoms are aligned in separate rows along [110].)

The technique has been effective in measuring near-surface compositions for MgO and Al2 O3 including adsorbed surface layers, and for understanding beam damage of Al2 O3 surfaces [8.98,8.99], a process that appears to be related to that in “holedrilling” described earlier.

OTHER TOPICS

8.6.3

305

Energy Filtering

An alternative TEM method to EXELFS for measuring RDFs of amorphous thin ﬁlms involves Fourier transforming the diffraction pattern intensity proﬁle [8.103]. The intensity data are collected by scanning an electron diffraction pattern across the entrance aperture of an energy loss spectrometer operating as an energy ﬁlter. The technique has been applied to several semiconductor and ceramic thin ﬁlms in amorphous or polycrystalline forms (see [8.104] for further references). Serial collection was originally used [8.103] but the technique has recently been adapted to use a parallel-detection EELS [8.104]. Unlike EXELFS, the technique is not speciesspeciﬁc, but also is not limited to elements producing strong K-edges, and bond lengths are given directly without the need of a phase-shift “correction.” Although application to structural materials not in thin-ﬁlm form has been attempted, with some success in bond-length measurements, plural scattering is a severe limitation to the accuracy of the derived coordination numbers [8.105]. The same technique of measuring energy ﬁltered electron diffraction intensities has been used to measure accurate structure factors from convergent beam electron diffraction patterns [8.106,8.107]. Such measurements in turn can be used to derive charge density difference maps, which indicate the covalency or ionicity of the bonding. Applications have been to semiconductors and ceramics such as MgO. A more optimum energy ﬁltering arrangement for the measurement of such diffraction intensities involves an imaging ﬁlter, such as an omega ﬁlter, and a two-dimensional detector, such as a slow-scan CCD camera. In addition to simple energy ﬁltering, other energy loss spectrometry is possible with such instrumentation [8.108]. In particular, electron spectroscopic imaging to produce elemental maps has been demonstrated in a variety of systems such as structural ceramics [8.109] and metal/metal interfaces [8.110]. In addition to chemical mapping, valence state mapping has recently been performed on various transition metal oxides [8.111, 8.112]. A group of energy ﬁltered images from a specimen containing a mixture of Mn oxides is shown in Fig. 8.31. The sample was prepared by reducing MnO2 to create a mixed oxide with Mn valences of 2+, 3+, and 4+. The conventional bright-ﬁeld image is shown in Fig. 8.31a. Figure 8.31b and c exhibit energy ﬁltered images using the Mn L2 and L3 white lines, respectively. Figure 8.31d shows the image of the background subtracted white line ratio L3 /L2 and directly below is a line scan across this image. The distribution of valences is clearly visible from this line scan which illustrates the reduction of the Mn near the surface (Mn2+ ) with the bulk remaining predominately Mn4+ . Figure 8.31e shows the normalized image of the L2 + L3 white lines and displays little contrast variation across the specimen. Although the spatial resolution of an energy ﬁltered image is contingent upon both the instrument and the specimen quality, an impressive example of high spatial resolution was recently achieved in the Co oxide system [8.111]. A partially oxidized specimen containing grains of CoO and Co3 O4 was used to generate the energy ﬁltered images in Fig. 8.32. Figure 8.32a shows the valence distribution map while 8.32b illustrates the chemical (O/Co) map. Figure 8.32b clearly demonstrates the

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Fig. 8.31 Images from a mixed phase Mn oxide (a) bright-ﬁeld TEM image (b) energy ﬁltered image of Mn L2 and (c) L3 white lines. (d) Image of Mn L3 /L2 ratio with an attached line scan across this image where the presence of Mn2+ , Mn3+ , and Mn4+ are identiﬁed. (e) Combined L2,3 intensity normalized to the continuum.

presence of two different stoichiometries of Co oxide. A line scan across the grain boundary in the valence distribution map reveals a spatial resolution of 2 nm.

Fig. 8.32 Energy ﬁltered images from mixed phase Co oxide (a) Co L3 /L2 ratio showing the contrast from the different Co valences. (b) O/Co distribution from O K -edge/Co L-edge.

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307

8.7 CONCLUSIONS The special experimental problems of working with ceramics, catalysts, and metal oxides notwithstanding, EELS has already been proven to be an immensely valuable tool in materials science investigations of such materials, with state-of-the-art techniques providing unique and sometimes unexpected results to help understand materials properties. A most important recent development is the availability of commercial PEELS systems. Not only is operator convenience improved, thus helping to overcome the reluctance of some investigators, but short acquisition times and efﬁcient collection of information mean that the ubiquitous problem of beam damage is ameliorated. In the near future, more applications to catalysts are expected, as well as a continued high level of work on ceramics and metal oxides. Acknowledgements J. Bentley acknowledges the support of the Division of Materials Sciences, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. The author gratefully acknowledges Dr. Mark Disko and the author’s colleagues at ORNL, particularly Drs. Neal Evans, Ed Kenik, Phil Sklad, and Z. L. Wang, for helpful discussions; Drs. Steve Berger, Ondrej Krivanek, Charles Lyman, and P. Gallezot for permission to use ﬁgures from their work; and Drs. Mark Disko and Channing Ahn for their exceptional patience. J. Graetz acknowledges the support of the Department of Energy through Basic Energy Sciences Grant No. DE-FG03-00ER15035.

REFERENCES 8.1. P. Angelini, J. C. Sevely, K. Hssein, and G. Zanchi, “EXELFS of Amorphous and Crystalline SiC,” Analytical Electron Microscopy-1987, p. 267, ed. D. C. Joy (San Francisco Press, San Francisco, 1987). 8.2. P. S. Sklad, P. Angelini, C. J. McHargue, and C. W. White, “An Analytical Electron Microscopy Investigation of Amorphous Structures in Ion-Implanted Al2 O3 ,” Analytical Electron Microscopy-1987, p. 45, ed. D. C. Joy (San Francisco Press, San Francisco, 1987). 8.3. S. D. Berger, I. G. Salisbury, R. H. Milne, D. Imeson, and C. J. Humphreys, “Electron Energy-Loss Spectroscopy Studies of Nanometre-Scale Structures in Alumina Produced By Intense Electron-Beam Irradiation,” Philos. Mag. B 55, 341 (1987). 8.4. C. E. Lyman, A. Ferretti, and N. J. Long, “Analysis of a Cu/ZnO Catalyst by Electron Energy Loss Spectroscopy,” Analytical Electron Microscopy-1984, p. 209, eds., D. B. Williams and D. C. Joy (San Francisco Press, San Francisco, 1984).

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8.5. L. E. Thomas, “Light-Element Analysis with Electrons and X-rays in a High Resolution STEM,” Ultramicroscopy 18, 173 (1985). 8.6. A. J. Craven and M. M. Cluckie, “Mass Loss from Titanium Nitride Observed by EELS,” Electron Microscopy and Analysis 1991, p. 135, ed. F.J. Humphreys, Inst. Phys. Conf. Ser. No. 119, (Institute of Physics, Bristol, UK, 1991). 8.7. A. T. Fisher and P. Angelini, “Preparation of Backthinned Ceramic Specimens,” Proc. 43rd Ann. Meet. Electron Microscopy Soc. Am., p. 182, ed. G. W. Bailey (San Francisco Press, San Francisco, 1985). 8.8. Y. Kouh Simpson, C. B. Carter, K. J. Morrissey, P. Angelini, and J. Bentley, “The Identiﬁcation of Thin Amorphous Films at Grain-Boundaries in Al2 O3 ,” J. Mat. Sci. 21, 2689 (1986). 8.9. Y. Kouh Simpson, C. B. Carter, P. S. Sklad, and J. Bentley, “Grain Boundary Glassy-Phase Identiﬁcation and Possible Artifacts,” Materials Problem Solving with the Transmission Electron Microscope, p. 427, eds. L. W. Hobbs, K. H. Westmacott, and D. B. Williams, Mater. Res. Soc. Symp. Proc. 62 (Mater. Res. Soc., Pittsburgh, 1986). 8.10. E. A. Kenik, M. J. Godbole, D. H. Lowndes, A. J. Pedraza, and D. F. Pedraza, “Structure of Rapidly Solidiﬁed Ni-Ti,” Analytical Electron Microscopy-1987, p. 76, ed. D. C. Joy (San Francisco Press, San Francisco, 1987). 8.11. J. Bentley and P. Angelini, “X-ray Microanalysis of Ion-Implantation Proﬁles in Backthinned Specimens,” Proc. 42nd Ann. Meet. Electron Microscopy Soc. Am., p. 578, ed. G.W. Bailey (San Francisco Press, San Francisco, 1984). 8.12. R. F. Egerton, “Electron Energy-Loss Spectroscopy in the Electron Microscope,” (Plenum Press, New York, 1986). 8.13. D. R. Clark, N. J. Zaluzec, and R. W. Carpenter, “The Intergranular Phase in Hot Pressed Silicon Nitride: I. Elemental Compositions,” J. Am. Cer. Soc. 64, 601 (1981). 8.14. D. R. Clark, N. J. Zaluzec, and R. W. Carpenter, “The Intergranular Phase in Hot Pressed Silicon Nitride: II. Evidence for Phase Separation and Crystallization,” J. Am. Ceram. Soc. 64, 608 (1981). 8.15. P. S. Sklad, J. Bentley, A. T. Fisher, and G. L. Lehman, “Phase Identiﬁcation in TiB2 -Ni Composites by Analytical Electron Microscopy,” Proc. 39th Ann. Meet. Electron Microscopy Soc. Am., p. 138, ed. G. W. Bailey, (Claitor’s Publishing Div., Baton Rouge, 1981). 8.16. P. S. Sklad and J. Bentley, “Analytical Electron Microscopy of TiB2 -Ni Ceramics,” Scanning Electron Microscopy/1981/I, p. 176, (SEM Inc., AMF O’Hare (Chicago), 1981).

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8.17. P. Angelini, P. F. Becher, J. Bentley, C. B. Finch, and P. S. Sklad, “Processing and Microstructural Characterization of TiB2 Liquid Phase Sintered With Ni and Ni3 Al,” Defect Properties and Processing of High-Technology Nonmetallic Materials, p. 299, eds. J. H. Crawford, Jr., Y. Chen, and W. A. Sibley, Mat. Res. Soc. Symp. Proc., 24 (Elsevier, New York, 1984). 8.18. P. Angelini, J. Bentley, C. B. Finch, and P. S. Sklad, “Microstructure of TiB2 Liquid Phase Sintered with Ni3Al,” Proc. 41st Ann. Meet. Electron Microscopy Soc. Am., p. 62, ed. G. W. Bailey, (San Francisco Press, San Francisco, 1983). 8.19. P. Angelini, P. F. Becher, J. Bentley, J. Brynestad, M. K. Ferber, C. B. Finch, and P. S. Sklad, “Processing and Microstructural Development in Liquid Phase Sintered and Pure TiB2 Ceramics,” Proc. 2nd Int. Conf. Science Hard Mater., Inst. Phys. Conf. Ser. No. 75, p. 1019, (Institute of Physics, Bristol, UK, 1986). 8.20. P. Angelini and J. Bentley, “Secondary Fluorescence Effects on X-ray Microanalysis,” Proc. 42nd Ann. Meet. Electron Microscopy Soc. Am., p. 582, ed. G. W. Bailey (San Francisco Press, San Francisco, 1984). 8.21. J. Bentley, P. Angelini, and P. S. Sklad, “Secondary Fluorescence Effects on X-ray Microanalysis,” Analytical Electron Microscopy-1984, p. 315, eds. D. B. Williams and D. C. Joy (San Francisco Press, San Francisco, 1984). 8.22. J. Bentley, G. L. Lehman, and P. S. Sklad, “Background Fitting for Electron Energy Loss Spectra,” Analytical Electron Microscopy-1981, p. 161, ed. R.H. Geiss (San Francisco Press, San Francisco, 1981). 8.23. J. Bentley, G. L. Lehman, and P. S. Sklad, “Alternative Background Fitting for Electron Energy Loss Spectra,” Proc. 40th Ann. Meet. Electron Microscopy Soc. Am., p. 496, ed. G. W. Bailey (Claitor’s Publishing Div., Baton Rouge, 1982). 8.24. R. F. Davis, J. E. Lane, C. H. Carter, Jr., J. Bentley, W. H. Wadlin, D. P. Grifﬁs, R. W. Linton, and K. L. More, “Microanalytical and Microstructural Analyses of Boron and Aluminum Regions in Sintered Alpha Silicon Carbide,” Scanning Electron Microscopy/1984/III, p. 1161, (SEM Inc., AMF O’Hare, Chicago, 1984). 8.25. K. L. More, C. H. Carter, Jr., J. Bentley, W. H. Wadlin, L. LaVanier, and R. F. Davis, “Occurrence and Distribution of Boron-Containing Phases in Sintered a-Silicon Carbide,” J. Am. Ceram. Soc. 69, 695 (1986). 8.26. K. L. More, J. Bentley, C. H. Carter, Jr., and R. F. Davis, “Analytical Electron Microscopy Characterization of Boron in Sintered a-SiC,” Analytical Electron Microscopy-1987, p. 299, ed., D. C. Joy (San Francisco Press, San Francisco, 1987). 8.27. J. M. Titchmarsh and T. F. Malis, “On the Effect of Objective Lens Chromatic Aberration on Quantitative Electron Energy Loss Spectroscopy (EELS),” Ultramicroscopy 28, 277 (1989).

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8.28. Y-Y. Yang and R. F. Egerton, “On Mode Selection for Quantitative Electron Energy Loss Spectroscopy,” Proc. 12th Int. Cong. for Electron Microscopy, p. 48, eds. L. D. Peachey and D. B. Williams (San Francisco Press, San Francisco, 1990) Vol. 2: Analytical Sciences. 8.29. V. Serin, M. Guigon, R. Fourmeaux, Y. Kihm, and J. Sevely, “Nitrogen Concentration Proﬁle by EELS in Carbon Fibers,” Proc. Inst. Phys. Electron Microscopy and Analysis Group and Royal Microscopical Society Conf. (EMAG-MICRO 89), p. 79, eds., P. J. Goodhew and H. Y. Elder, Inst. Phys. Conf. Ser. No. 98, (Institute of Physics, Bristol, UK, 1989) Vol. 1. 8.30. S. D. Berger and S. J. Pennycook, “Detection of Nitrogen at 100 Platelets in Diamond,” Nature 298, 635 (1982). 8.31. S. D. Berger, J. Bruley, L. M. Brown, and D. R. McKenzie, “Applications of the Near-Edge and Low-Loss Fine Structure in the Analysis of Diamond,” Ultramicroscopy 28, 43 (1989). 8.32. Z. L. Wang, C. Colliex, V. Paul-Boncour, A. Percheron-Guegan, J. C. Achard, and J. Barrault, “Electron Microscopy Characterization of Lanthanum-Cobalt Intermetallic Catalysts,” J. Catal. 105, 120 (1987). 8.33. C. Colliex, D. Ugarte, Z. L. Wang, M. Gasgnier, and V. Paul-Boncour, “High Spatial Resolution Analytical Electron Microscopy Studies on the Co/CeO2 System,” Surf. Inter. Anal., 12, 3 (1988). 8.34. M. T. Otten, B. Miner, J. H. Rask, and P. R. Buseck, “The Determination of Ti, Mn and Fe Oxidation States in Minerals by Electron Energy Loss Spectroscopy,” Ultramicroscopy 18, 285 (1985). 8.35. O. L. Krivanek and J. H. Paterson, “ELNES of 3d Transition-Metal Oxides I. Variations Across the Periodic Table,” Ultramicroscopy 32, 313 (1990). 8.36. J. H. Paterson and O. L. Krivanek, “ELNES of 3d Transition-Metal Oxides II. Variations with Oxidation State and Crystal Structure,” Ultramicroscopy 32, 319 (1990). 8.37. H. Kurata, E. Lefevre, C. Collies, and R. Brydson, “Electron-Energy-Loss NearEdge Structures in the Oxygen K-Edge Spectra of Transition-Metal Oxides,” Phys. Rev. B 47, 13763 (1993). 8.38. F. M. F. de Groot, M. Grioni, J. C. Fuggle, J. Ghijsen, G.A. Sawatzky, and H. Petersen, “Oxygen 1s X-Ray-Absorption Edges of Transition-Metal Oxides,” Phys. Rev. B 40, 5715 (1989). 8.39. H. Kurata and C. Colliex, “Electron-Energy-Loss Core-Edge Structures in Manganese Oxides,” Phys. Rev. B 48, 2102 (1993).

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8.80. P. Gallezot, C. Leclercq, M. Guisnet, and P. Magnoux, “Coking, Aging, and Regeneration of Zeolites. VII. Electron Microscopy and EELS Studies of External Coke Deposits on USHY, H-OFF, and H-ZSM-5 Zeolites,” J. Catal. 114, 100 (1988). 8.81. P. Gallezot, C. Leclercq, J. Barbier, and P. Marecot, “Location and Structure of Coke Deposits on Alumina-Supported Platinum Catalysts by EELS Associated with Electron Microscopy,” J. Catal. 116, 164 (1989). 8.82. D. W. McComb and A. Howie, “Characterisation of Zeolite Catalysts Using Electron Energy Loss Spectroscopy,” Ultramicroscopy 34, 84 (1990). 8.83. N. Jiang, J. Qiu, and J. C. H. Spence, “Long-Range Structural Fluctuations in a CaO-Al2 O3 -2SiO2 Glass Observed by Spatially Resolved Near-Edge Spectroscopy,” Phys. Rev. B 66, 054203 (2002). 8.84. P. Angelini and P. S. Sklad, unpublished research (1988). 8.85. D. E. Johnson, S. Csillag, and E. A. Stern, “Analytical Electron Microscopy using Extended Energy Loss Fine Structure (EXELFS),” Scanning Electron Microscopy/1981/I, p. 105, (SEM Inc., AMF O’Hare (Chicago), 1981). 8.86. B. K. Teo and P. A. Lee, “Ab Initio Calculations of Amplitude and Phase Functions for Extended X-ray Absorption Fine Structure (EXAFS) Spectroscopy,” J. Am. Chem. Soc. 101, 2815 (1979). 8.87. J. C. Sevely, P. Angelini, K. Hssein, and G. Zanchi, “EXELFS Analyses at 300 kV and 1 MV,” Analytical Electron Microscopy-1987, p. 106, ed. D.C. Joy (San Francisco Press, San Francisco, 1987). 8.88. P. Angelini, P. S. Sklad, J. C. Sevely, and D. K. Hssein, “Evaluation of Amorphous and Crystalline SiC by Extended Energy Loss Fine Structure (EXELFS) Analysis,” Proc. 46th Ann. Meet. Electron Microscopy Soc. Am., p. 466, ed. G.W. Bailey (San Francisco Press, San Francisco, 1988). 8.89. P. Angelini, J. C. Sevely, K. Hssein, and G. Zanchi, “EXELFS of Amorphous and Crystalline SiC,” Oak Ridge National Laboratory Report, ORNL/TM-10601, p. 21, (1988). 8.90. J. M. Martin and J. L. Mansot, “EXELFS Analysis of Amorphous and Crystalline Silicon Carbide,” J. Microsc. 162, 171 (1991). 8.91. P. S. Sklad, P. Angelini, and J. C. Sevely, “The Characterization of Amorphous Al2 O3 by Extended Energy Loss Fine Structure (EXELFS) Analysis,” Proc. 46th Ann. Meet. Electron Microscopy Soc. Am., p. 468, ed. G.W. Bailey (San Francisco Press, San Francisco, 1988). 8.92. P. S. Sklad, P. Angelini, and J. Sevely, “Extended Electron Energy Loss Fine Structure (EXELFS) Analysis of Al2 O3 ,” Philos. Mag. A 65, 1445 (1992).

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9

EELS Analysis of the Electronic Structure and Microstructure of Metals J. K. Okamoto, D. H. Pearson, A. Hightower, C. C. Ahn, and B. Fultz Division of Engineering and Applied Science California Institute of Technology Pasadena, California 91125

Abstract We review applications of bulk plasmons, chemical shifts in threshold energy, and near- and extended-edge structure in the study of metals and alloys. Because of the free electron nature of metals, plasmons are useful in determining electron densities in sp metals with only moderate elaboration of simple models required in order to interpret changes that result from hydriding or alloying. Our systematic study of near edge structure in 3d and 4d transition series reveal that white line intensities reﬂect the occupancies of outer d states. This observation, which makes possible the determination of d electron transfers upon alloying, especially when one of the constituents is a nearly full d-band metal. Extended ﬁne structure can be interpreted from K, L, and M edges. It is applied to the study of temperature-dependent differences in vibrational behavior, seen as changes of intensity of nearest-neighbor Fourier transform magnitude peak heights. 9.1 INTRODUCTION Applications of EELS in metals research range from studies of the electronic structure of metals and alloys to quantitative analyses of the light element chemistries of multiphase alloys in engineering service. Tractable models of varying sophistication are available for every feature of the energy loss spectrum, and we will discuss the applications of each feature individually. Arranged in terms of increasing energy loss these features are (1) plasmon excitations, used to determine free electron densities; (2) chemical shifts in threshold energy in metallic alloys; (3) electron near-edge 317

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structure (ELNES), useful for studying the chemistry of the metallic bond; and (4) extended energy loss ﬁne structure (EXELFS) above the core edges, considered a technique primarily for measuring local atomic structure. The discussion of plasmon experiments is a general review, with some examples taken from our own work, but the discussions of chemical shifts, ELNES and EXELFS are strongly oriented toward our studies of energy storage materials and transition metal alloys. The EXELFS discussion emphasizes our work with Fe3 Al, which undergoes an order–disorder transformation that may be accompanied by a change in local lattice dynamics. Chemical shifts are normally related to differences that arise in screening of outer electrons that alter the binding energy of core-loss transitions. In a metal like Li that has only 3 electrons, the binding energy of 1s electrons should be particularly sensitive to the state of bonding, extent of electron charge transfer and near neighbor distance. Li also represents one of the important metals from the standpoint of energy storage, as compounds based on Li have the highest electrochemical energy densities. The near-edge structure of metals and alloys is directly related to the nature of their chemical bonding. In simple metals the energy bands responsible for cohesion are quite broad and are adequately described by a nearly free electron model for the density of states. As a result, there is little near-edge structure. Bonding in transition and rare earth metals, on the other hand, is better described in terms of a tight binding model in which the energy states of the outer electrons form a narrow band. This tight binding model works best for the d and f electrons of the transition and rare earth metals. In these metals, there may be a high density of unoccupied states above the Fermi level yielding intense peaks known as “white lines” near EELS absorption edges. With adequate energy resolution it is possible to study the widths and shifts of the white lines to learn about the nature of the unoccupied band states. However, such measurements are difﬁcult and their interpretation is not direct. More typically, one measures the intensities or ratios of the white lines in order to measure charge transfers in alloys and the strengths of magnetic exchange interactions, respectively. The two other categories of EELS experiments of metals that we will discuss are not particularly dependent on the nature of the metallic bond. The ﬁne structure above an absorption edge extends to energies well beyond those of the bonding electrons in the metal. Consequently, interpretations of these EXELFS experiments is the same for different metals, and even the same for semiconductors and insulators. Elemental chemical analysis is another application where it is possible to consider the core electrons to be atomic-like. The elemental abundance in a material can be related to the strength of the absorption edges. This approach is straightforward, but for quantiﬁcation it may be risky to neglect “solid state” effects such as white lines. 9.2

PLASMONS

Next to the zero-loss peak of the energy loss spectrum the dominant inelastic feature is the bulk plasmon. The foundation for our discussion of plasmons in materials was presented in Chapter 5, and there are several comprehensive references on this topic [9.1–9.3]. Understanding of plasmon behavior developed from work in the 1950s on

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319

free or nearly free electron metals. Equation 4.4 deﬁnes the plasma oscillation and energy as a function of electron density for free-electron metals, otherwise known as the “jellium” model. This equation works well for the sp band metals like the alkalis and other free electron metals like Al, as shown below in Fig. 9.1. Plasmon peaks in free electron metals have Poisson distributions from which the plasmon energy can be easily determined.

Fig. 9.1 Low loss regions of Li, Al, and a LiAl alloy showing the sharp bulk plasmons, which are typical of free-electron metals. The effect of multiple scattering in this data is signiﬁcant.

Unfortunately, 3d and 4d metals and alloys pose more of a problem in interpretation because the binding energies for these electrons are close to those of bulk plasmon energies. In the 3d series, there are M4,5 transitions of 3d electrons on the order of ∼ 2–6 eV, which complicate the shape of the bulk plasmon on the low energy loss end of the plasmon distribution. The bulk plasmons are themselves broad in energy. On the tail or high energy side of the bulk plasmon peak are M2,3 -edges (3p transitions) with energies from 35 to 70 eV. Some of the ﬁrst experimental work on the 3d transition metals was done by Robins and Swan [9.4]. While the data were obtained in reﬂection geometry, the dominant effects seen in the data are due to bulk plasmons, with energies from 22 to 36 eV. Weaker surface plasmon peaks √ that generally appear at 1/ 2 of the bulk plasmon energy are also present. Using a M¨ollenstedt analyzer and recording the spectra on photographic plates, Misell and Atkins [9.5] systematically examined the transition metals by transmission EELS. Their results, obtained with an AEI EM6 TEM operated at 60 keV and with an EELS spectrometer having an energy resolution of 1 eV, are shown below in Fig. 9.2. For the 3d series, the maxima of the broad plasmon peaks, attributable to collective oscillations of 3d and 4s electrons, occurs between 18 and 22 eV. These energies are lower than that predicted by theory. Losses from 9 to 12 eV seen in these data result from losses of bonding electrons from 4s wave function overlap. In the Ni and Cu data, losses at energy higher than the main plasmon peak are related to the ﬁlling of the 3d shell. There are fewer systematic studies on plasmon behavior of the 4d metals [9.6,9.7] but such studies have demonstrated core polarization near the end of the series. Core

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Fig. 9.2 Bulk plasmons from 3d transition series metals from the work of Misell and Atkins[9.5].

polarization yields a decreasing shift in ωp , and results from the movement and narrowing of the 4d band relative to the free electron conduction band. As with the 3d metals where the binding energy of 3d electrons is on the order of free electron plasmon energies, 4d-5sp interband transitions will generally complicate the low loss structure of the 4d series. For any material in which free electron behavior is not expected, interpretations of plasmon spectra are best made by the use of the dielectric response to a longitudinal wavevector and frequency dependent dielectric constant (Eq. 4.5). For example, the inversion of the dielectric function has been used within the framework of density functional theory to produce a good theoretical match to the plasmon of Pd [9.8]. Zn is a metal that poses special difﬁculties in the interpretation of low loss behavior due to the overlap of the lower part of the conduction band with 3d bands. Decoupling of the features of this region was done via momentum transfer dependent measurements which showed that the core excitations could be suppressed at high q, revealing the Zn plasmon to exist at 14.2 eV [9.9]. Part of the interest in plasmon features arises from their replication on core losses resulting from multiple inelastic scattering as can be seen in Fig. 9.1, especially for Li and Al. The ﬁrst treatment of the multiple scattering problem was by Landau [9.10] although an expansion of this original theory using an analytical differential cross-section generates better ﬁts to empirical data [9.11]. While the prospects are poor for seeing direct evidence of ionizations associated with hydrogen (13.6 eV) in the plasmon region, there are indirect manifestations of

CHEMICAL SHIFTS IN THRESHOLD ENERGY OF METALLIC ALLOYS

321

hydride formation in the form of shifts in bulk plasmon energies, ﬁrst observed by Colliex [9.12] and Zaluzec [9.13], and discussed previously by Zaluzec [9.14]. These studies show plasmon shifts in Mg, Ti, and Zr hydrides that result from changes to the free-electron density, and no changes between V and FeTi hydrides, due presumably to differences in effective mass. Zr hydrides were also studied by Woo and Carpenter [9.15] who noted that differences in plasmon energies were effective in quickly identyfying differences between γ and δ phases of this hydride. Brown and co-workers [9.16,9.17] studied hydrides of several transition series metals. They observed in the case of the partially full d-band metals, V and Nb, hydrogenation induces an additional small energy loss peak at very low energies, well below that due to the plasmon loss of the pure metals. They interpret this peak as due to a band of states induced by hydrogen and located several eV below the conduction bands of the metal. In the case Pd with a full d-band, the change in plasmon behavior resulting from hydrogenation is to shift the spectral feature at 7.4 eV to 5.4 eV, which they concluded to be a small change to the electronic structure of Pd. It is not clear that the feature at 7.4 eV is in fact a plasmon as it does not appear to exhibit dispersion [9.18], a fundamental attribute of bulk plasmons. Equation 4.4 has been used to obtain the difference in plasmon energies between alkali hydrides and metals and between alkaline earth hydrides and alkaline earth metals. In these studies, the metallic forms were obtained by decomposition of the hydrides under the electron beam of the microscope [9.17,9.18]. In an application of plasmon analysis that takes advantage of spatially resolved EELS in a STEM, Hunt and Williams [9.20] probed shifts in bulk plasmon position in an Al-10.5 at% Li alloy in order to obtain a quantitative elemental map of lithium rich δ precipitates. Differences in plasmon peak position in this alloy arise from differences in free electron density. Al has three electrons available for conduction while Li has one, and the bulk plasmon energy in the alloy changes nearly linearly with Li concentration (at least to a 25 at% Li concentration). Acquiring a spectrum from each point of an 80×80 pixel ﬁeld, the plasmon centroid of each spectrum was determined to an accuracy of better than 5 meV using a nonlinear least-squares ﬁt. The concentration of Li was then determined using the equality: [Li] = (15.18−E)/4 for each plasmon peak energy, E. In spite of the broad features generally associated with plasmons, we see that it is possible to determine differences in free or nearly free electron behavior in more complex alloy systems than simple sp metals with a modest amount of effort. Quantiﬁcation to a high degree of accuracy is obtainable by such analysis. 9.3

CHEMICAL SHIFTS IN THRESHOLD ENERGY OF METALLIC ALLOYS

Spectral features > 50 eV are associated with ionization of atoms by excitation of their core electrons into higher unoccupied energy states. The threshold energy of these spectral features is sensitive to the changes in the outer-shell and core electron wave functions. There has been much work quantifying the chemical shifts in disordered

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alloys and modeling these results by “potential models” and density-functional-theory local-density-approximation [9.21–9.25]. The spectra of lithium alloys display large chemical shifts relative to other metallic alloys. Core-level binding energies have traditionally been measured by X-ray photoelectron spectroscopy (XPS), formally called electron spectroscopy for chemical analysis (ESCA). This technique measures the energy of photoelectrons that have been excited from a sample surface (nanometer depths) by a monochromatic X-ray beam (1253.6 eV Mg Kα (or 1486.6 eV Al Kα)). The core binding energy, BE, can be determined from the measured electron energy, KE, the incident photon energy, hν, and the spectrometer work function θs . KE = hν − BE − θs

(9.1)

A chemical shift refers to any increase in the core-level binding energy relative to the value in the pure element. In the case of metal oxides, oxidation reduces the valence electron charge density about the metal atom. This results in reduced screening of its nuclear ﬁelds and a positive shift in the edge onset. Chemical shifts in XPS measurements have been found to correlate to charge transfer [9.24], cohesive energy [9.26], heat of mixing [9.21,9.27] and segregation energy [9.28]. In contrast to XPS, the edge energy onset measured by EELS, or x-ray absorption spectroscopy (XAS) corresponds to the energy difference between the initial corelevel state and the lowest unoccupied ﬁnal state. The ﬁnal state energies of the excited electron also shift with the effective charge of the central atom. XAS studies have shown this effect is most prominent in metal oxides and halides (up to 20 eV) [9.29,9.30], though metal alloys can also display chemical shifts (typically <1 eV). The chemical shifts of lithium alloys are especially prominent due to lithium’s large electronegativity and limited nuclear screening. The many-bodied relaxation effect also plays an important role in the chemical shifts of lithium alloys. When a core hole is created by an inner-shell excitation, the surrounding electron orbitals are pulled inward. The screening by the surrounding orbitals reduces the magnitude of the measured binding energy by an amount equal to the “relaxation energy”. Leapman [9.31] found that a difference in relaxation energy between a metal and its compounds has a signiﬁcant effect on the measured chemical shift. In XPS, excited electrons leave the solid resulting in relaxation energies of tens of eV. In EELS and XAS, the core electron is excited to the lowest unoccupied state. In contrast, the relaxation energy in XAS and EELS is reduced by the screening of the excited core electron which is in a state just above the threshold energy and in close proximity to the core-hole [9.32]. Lithium, with its small atomic number (Z = 3), is particularly susceptible to relaxation effects. Figure 9.3 displays the Li K-edge obtained from intercalated LiC6 , LiF, and metallic Li. We identify the onset of the Li K edge in the metallic sample at 55 eV, consistent with results of [9.33]. The broad proﬁle of the metallic Li K-edge is consistent with the promotion of core electrons into a continuum of free electron states. The Li K-edge onset of LiC6 is shifted by only 0.2 eV from the K-edge of metallic Li and has a similarly broad proﬁle. Our observed Li K-edge for LiC6

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lacks the peaks at 59 and 63 eV previously observed by Grunes et al. [9.34]. These peaks evidently originate from oxidized Li as they are found in samples exposed to atmosphere, or after long times in the microscope. The Li K-edge of LiF, similar to that reported by Chen et al. [9.35], shows a strong chemical shift and sharp features of well-deﬁned unoccupied states. Such large differences are expected with the large electron transfer from Li to F. The EELS spectra of the Li K-edge indicate that Li in LiC6 has a local electronic structure more similar to Li metal than Li+ , contrary to previous XPS studies. Extensive XPS results conﬁrm a Li K-edge chemical shift of about 3 eV in LiC6 with respect to metallic Li [9.36,9.37] easily resolved from the 1.3 eV chemical shift of Li2 O [9.38]. The mean free path of photoelectrons measured by XPS is on the order of 1 nm. Thus, assuming that the surface Li remains unreacted, we expect XPS to be more sensitive to Li surface states.

Fig. 9.3 Lithium K-edges of metallic Li, LiF, and LiC6 .

9.4 NEAR-EDGE STRUCTURE IN THE STUDY OF BAND STRUCTURE The relationship between the band structure of a solid and its corresponding nearedge structure was described in Chapter 4. Speciﬁcally, Eq. 4.1 and the discussion of Section 4.5, show that the near-edge structure is governed by the product of the density of ﬁnal states and the matrix element that couples the initial and ﬁnal state of the excited electron. In transition metals there may be a large density of unoccupied d states at the Fermi level, yielding prominent “white lines” in the near edge structure. In simple metals, on the other hand, the density of states is free electron-like, and the resulting near-edge structure is largely devoid of any prominent features. We will therefore concentrate on the transition metals, and discuss how the white lines may be used to obtain information about the 3d and 4d states of transition metal atoms. In the energy loss spectra of transition metals, white lines are found at the onsets of the L2 and L3 absorption edges and result from excitations of 2p1/2 and 2p3/2 electrons to unoccupied outer d states. Their relative intensities are therefore proportional

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to the number of outer d electron holes. L23 -edge spectra for the 3d transition metals are shown in Fig. 9.4 and have been scaled such that their background intensities are nearly equal. The relative intensities of the white lines decrease with increasing atomic number, reﬂecting the systematic electron ﬁlling of the 3d band. Although not shown here, a similar trend is observed for the 4d transition metals. White lines also occur at the M45 -edges of rare earth metals, and arise from 3d to 4f excitations.

Fig. 9.4 Background-subtracted L23 -edges for some of the 3d transition metals. The spectra were collected in diffraction mode with a collection semi-angle of about 20 mrad and have been deconvoluted to remove multiple scattering. Relative positioning of edge onsets is arbitrary.

One type of information that may be obtained from white lines is the electron transfer that occurs between outer d states of transition metal atoms upon alloying or during solid state phase transformations. The general principle is quite straightforward: changes in the white line intensity (relative to the trailing background) correspond directly to changes in the number of d electron holes. In order to make this statement quantitative one must consider a number of factors including the relative strengths of the transitions from the 2p1/2 and 2p3/2 to the nd3/2 and nd5/2 states (n = 3, 4, 5), the number of d electron holes in the elemental metal, the method by which the white lines are isolated from the continuum excitations in the edge region, and the method by which the spectra are normalized. These details have been addressed in both x-ray absorption studies of white lines of late 4d and 5d transition metals [9.39–9.41] and in EELS studies of the white lines of Fe–Ge alloys [9.42]. In most of these studies, the general approach has been to establish the proportionality between white line area and d electron holes for the pure transition metal of interest by ﬁrst empirically isolating the white lines from the continuum excitations and by then associating their areas with the number of d electron holes obtained from a band structure calculation. Changes in the white line intensity may then be associated with changes in the number of d electron holes through this proportionality. This method may suffer from signiﬁcant systematic errors for the late transition metals, however,

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due to the uncertainties in isolating the relatively small white line areas and from inaccuracies of band structure calculations. Pearson et al. [9.43] have approached the problem by studying the systematic decrease in white line intensity across the 3d transition series, thereby obtaining a proportionality between white line area and the number of d electron holes. A more detailed study of the white line systematics across the 3d and 4d series of use in electron-transfer measurements has recently been performed by Pearson [9.44], and a brief outline of that analysis follows. To establish the numerical proportionality between white line intensity and d holes, the white lines must be isolated from the background intensity and normalized. Figure 9.5 illustrates a method for isolating and normalizing the white line intensities for the 4d transition metals that is somewhat different from methods used in the X-ray absorption studies mentioned above, but that has proven successful. First, the white line intensity is isolated from the continuum intensity. Unfortunately, the edge shape due to continuum excitations in the threshold regions cannot be measured and is difﬁcult to calculate accurately. Consequently, an empirical approach must be used to isolate the white lines. Modeling the continuum by a step function is a reasonable choice because the edge shapes for silver, which has no white lines, resemble step functions. A straight line is ﬁt to the background intensity immediately following each white line over a range of ∼ 50 eV. This line is then extrapolated into the threshold region and set to zero at energies below that of the white line maximum. The L2 white line is further isolated by smoothly extrapolating the L3 background intensity under the L2 white line. The area in each white line above these step functions is then divided by the area in a normalization window 50 eV in width beginning 50 eV past the L3 white line onset. The sum of these normalized areas deﬁnes the total normalized white line intensity.

Fig. 9.5 L23 -edge for niobium showing the method for the isolating and normalizing the white line intensities for the 4d transition metals.

A plot of this normalized white line intensity versus 4d occupancy is shown in Fig. 9.6. The occupancy of the 4d states is determined by assuming a valence electron conﬁguration of 4dN−1 s1 where N is the total number of valence electrons, as this is approximately the conﬁguration in the solid [9.45,9.46]. A resulting linear correlation is obtained, and a least-squares ﬁt to the data yields

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Fig. 9.6 Plot of normalized white line intensity versus 4d occupancy. The straight line is the least-squares ﬁt to the data.

I = 0.94(1 − 0.092n)

(9.2)

where I is the normalized white line intensity and n is the 4d occupancy. That this correlation reﬂects the occupancy of the 4d band can be shown by considering the equation given by the linear ﬁt. For this correlation to accurately reﬂect the occupancy of the 4d band, it should have a maximum value at n = 0 and should go to zero at n = 10. That is, the correlation should have the form I = K(1 − 0.1n)

(9.3)

where K is a constant. The plot in Fig. 9.6 is useful because it enables one to determine changes in 4d occupancy upon alloying or during solid-state phase transformations if corresponding changes in the normalized white line intensity are observed. That is, assuming that the radial matrix element between the 2p and 4d states stays constant during alloying, a change in the normalized white line intensity directly corresponds to a change in the 4d occupancy through the slope of the line in Fig. 9.6. Furthermore, since the 2p states are atomic and the 4d states are tightly bound about a given atom in the solid, the information obtained is largely local to given atom species. An advantage of using this approach for measuring electron transfers is that it suffers neither from uncertainties in isolating small white line intensities for the late transition metals nor requires accurate band calculations for a speciﬁc metal. The proportionality between white line area and 4d occupancy is not solely dependent upon one metal but is determined by analyzing the systematics across the entire series. A similar approach may be used in the analysis of the 3d metals, although the situation is somewhat more complicated due to the overlapping white lines in the spectra of the early 3d metals. The approach is illustrated in Fig. 9.7. In this case, the white lines are isolated by modeling the continuum intensity in the threshold region using a double step function. Again, a straight line is ﬁt to the background immediately following the L2 white line over a region of ∼ 50 eV and then extrapolated into the

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threshold region. This line is then modiﬁed into a double step of the same slope with onsets occurring at the white line maxima. The ratio of the step heights is chosen as 2:1 in accordance with the multiplicity of the initial states (four 2p3/2 electrons and two 2p1/2 electrons). The white line area above this step function is then divided by the area in the normalization window again deﬁned to be 50 eV in width beginning 50 eV past the onset of the L3 white line.

Intensity

Fig. 9.7 L23 -edge for vanadium showing the method for isolating and normalizing the white line intensities for the 3d transition metals.

500

520

540

560

580

600

620

640

Energy Loss (eV)

A plot of the normalized white line intensity versus 3d occupancy assuming a 3dN−1 s1 valence electron conﬁguration is shown in Fig. 9.8.

Fig. 9.8 Plot of normalized white line intensity versus 3d occupancy. The straight line is the least-squares ﬁt to the data.

A linear correlation is again obtained with a least-squares ﬁt yielding I = 1.06(1 − 0.094n)

(9.4)

Using Eqs. 9.4 and 9.6 for measuring changes in d occupancy ignores the contribution of the matrix elements to the normalized white line intensity. Speciﬁcally, using Eqs. 9.4 and 9.6 as they stand assumes that the ratio of the white line matrix element to the continuum (window) matrix elements is constant across each transition series. Atomic calculations by Pearson [9.44] suggest that this ratio is not constant, and that

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as a result, changes in d occupancy measured by this method may be overestimated. Corrected measurements for the late 3d transition metals appear to be about 40% smaller than those given by Eq. 9.6. The magnitude of the correction is less serious for the early 3d elements. To date, however, no detailed solid state calculations that explicitly address this effect are as of yet available. As a speciﬁc application of these correlations, we consider the Cu L23 -edges shown in Fig. 9.9 for pure Cu metal and amorphous CuZr. The enhancement of the white lines in the alloy indicates that electrons have left the Cu 3d band upon alloying. By scaling the pure Cu metal spectrum to that of the alloy in the edge region, we may isolate the white line intensity in the spectrum of the alloy. After normalizing this white line intensity and applying the correlation in Fig. 9.8, we ﬁnd that approximately 0.3 electrons/atom have left the Cu 3d band upon alloying with Zr. In comparison, a charge transfer of 0.31 electrons/atom are predicted by the band structure calculations of Manh et al. [9.47].

Fig. 9.9 Cu L23 -edge in pure Cu and amorphous CuZr. The enhancement of the white lines in the alloy indicates that some electrons have left the 3d band of Cu upon alloying.

This result suggests that white line measurements might be useful in studying the electron transfer effects that accompany alloying between transition metals. Intuitively, one expects the electron transfer during alloying to scale with the heat of formation of an alloy. This result was found experimentally for a comparison of three alloys [9.43]. A simple band model of alloying in transition metals due to Pettifor [9.48], however, suggests that the details of the alloying process may be more complicated. Pettifor’s model predicts expressions for electron transfers and heats of formation that are not simply proportional to one another. They are instead sensitive to the energy band parameters of the constituent metals. We expect that further measurements of electron transfers using white lines will furnish data for comparison to theories of alloying in transition metals leading to a better understanding of the alloying process. Empirical data from the CuPd system show that effects of short range order that are otherwise unrelated to alloying can be discerned as an enhancement to white line intensity of the Cu L23 -edge [9.49]. The recent availability of beam lines with energy

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ranges capable of measureing data in the soft X-ray region has made X-ray absorption near edge structure experiments comparable to EELS measurements possible. In another application of white lines, the ratios of their intensities are used to determine the relative numbers of j = 3/2 d holes and j = 5/2 d holes in the valence d band. As discussed by Dooley in Chapter 11, this type of information has been used in studies of the magnetic moment in FeGe alloys. The reader is referred to that section for further information. In addition to providing information about changes in the numbers of 3d and 4d electrons, the white lines can potentially provide information about the width and shape of the unoccupied part of the bands, though the interpretation is by no means straightforward. A dramatic example is the distortion of the Ni L3 white line that occurs in NiAl. Figure 9.11 shows our previously unpublished spectra for the Ni L23 -edge in both pure Ni and NiAl which are in agreement with the X-ray absorption spectra of Pease and Azaroff [9.50] and the energy loss data of Zaluzec [9.51]. Pease and Azaroff have interpreted the near-edge structure in the alloy as being due to the ﬁlling of the 3d band in conjunction with the creation of a double-peaked structure in the s-like density of states above the Fermi level. They noted that this interpretation is consistent with the density of states calculations of Connoly and Johnson [9.52], and of Colinet et al. [9.53]. The increased use of high-resolution EELS will aid in understanding these types of effects as well as uncovering others.

Fig. 9.10 Ni L23 -edge in pure Ni and NiAl. The data have been background-subtracted but not deconvoluted to remove multiple scattering.

Any analysis involving white lines requires some care in data processing as well as in interpretation. Speciﬁcally, since multiple scattering can modify the intensity following an absorption edge, it must be removed from the spectrum using deconvolution techniques such as those described in Chapter 3. This is particularly important in the analysis involving electron transfers because the white line intensity is normalized to the trailing background intensity. In addition, the analysis itself should be applied judiciously as it may be inappropriate to interpret the white lines solely in terms of d electrons in cases where s–p electrons are known to contribute signiﬁcantly to the bonding, as in the case of NiAl. In spite of such difﬁculties in interpretation,

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however, white lines can provide information of fundamental importance in the study of transition metals and their alloys. 9.5 EXTENDED ENERGY LOSS FINE STRUCTURE We will now turn our attention to the extended ﬁne structure, which begins beyond ∼30 eV past the edge onset energy and is superimposed on the smooth atomic edge shape of core losses. While the features associated with the near edge come predominantly from the electronic structure of a material, the contribution to extended energy-loss ﬁne structure oscillations comes from the crystallographic or structural short range order about the ionized element. Both the near and extended ﬁne structure oscillations are clearly visible on the K-edge of Al shown below in Fig. 9.11.

Fig. 9.11 Al K-edge after background subtraction.

The origin of extended ﬁne structure is well understood [9.54–9.60] and is illustrated schematically in Fig. 9.12. A core electron is excited from the so-called central atom and becomes an outgoing spherical wave (solid rings). The energy of the outgoing wave corresponds to the energy loss in excess of the ionization energy. Some of the outgoing wave is elastically scattered (dashed rings) from neighboring atoms. One can visualize the interference between outgoing and scattered waves as varying periodically with the wavelength of the excited electron (i.e., with the distance between concentric rings in Fig. 9.13). From Fermi’s Golden Rule, Eq. 4.1, we know that it is only the interference in the region of the initial state (i.e., at the central atom core) which changes the excitation probability, and hence modiﬁes the edge shape. If constructive interference occurs at the central atom, as in Fig. 9.12a, then the excitation probability increases, creating positive extended ﬁne structure. For destructive interference, shown in Fig. 9.12b, the extended ﬁne structure is negative. Thus extended ﬁne structure is a quantum interference phenomenon which depends upon the atoms surrounding the ionized atom. Both X-rays and high energy electrons can ionize core electrons. If X-rays are used, then the extended ﬁne structure on the core edge is called EXAFS (extended

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Fig. 9.12 Schematic illustration of (a) constructive and (b) destructive interference at the central atom. Rings represent electron-wave crests.

X-ray absorption ﬁne structure). If high energy electrons are used, then it is called EXELFS (extended energy-loss ﬁne structure). EXAFS and EXELFS are similar in physical origin but there are many signiﬁcant differences between the experimental techniques which exploit them [9.61–9.64]. EXAFS is a photoabsorption process in which a complete transfer of X-ray photon energy to the excitation of a photoelectron occurs, while EXELFS involves the analysis of small momentum transfers of the incident electron beam and is thus more analogous to a Raman scattering experiment. Important advantages of EXELFS are its increased spatial resolution and the ability to measure extended ﬁne structure in elements with very low atomic number. Disadvantages of EXELFS include a limited energy range, a greater likelihood of overlapping edges, and the need for very thin samples in order to minimize problems with multiple scattering. In contrast to the vast amount of EXAFS research that has been done [9.65–9.71], relatively few EXELFS experiments have been performed to date. The number of EXELFS experiments is likely to increase in the future, however, as parallel electron energy-loss detectors become more widely used. 9.5.1

EXELFS Data Processing

Much of the challenge in obtaining useful EXELFS data is caused by the vagaries of data processing. Consequently, in this section we will discuss some of the procedures that one should follow, as well as the pitfalls that one might avoid, in order to obtain quantitative radial distribution functions (RDF). Note that diffraction is far superior for determining average interatomic distances in crystalline solids, which have longrange order. Extended ﬁne structure is useful, however, in the determination of short-range order. Even though EELS core loss data collection might now be considered to be relatively efﬁcient when compared to data collection from emission processes in

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the TEM, the EXELFS oscillations on these core losses are comparatively weak. Consequently, data of high statistical quality are required. Serial detection EXELFS studies generally suffered from inadequate signal-to-noise ratios in energy loss data, resulting in very limited data ranges in k-space [9.72]. The advent of parallel detectors has greatly improved the statistical quality of EELS data [9.73]. As discussed in Chapter 3. Linear photo-diode arrays for which corrections needed to be made for diode nonuniformities [9.74] have been replaced by two-dimensional (2D) arrays. These 2D arrays average out nonlinearities in the nondispersive direction and obviate the need for channel-to-channel gain variation correction. Normalization to an instrument response is still necessary though, but correction can typically be applied through the acquisition software. The EXELFS signal, χ, is the oscillatory part of the edge intensity, ∆J(E), normalized to the nonoscillatory part, J0 (E) and is expressed as χ(E) =

J(E) − J0 (E) ∆J(E) = J0 (E) J0 (E)

(9.5)

where J(E) is the experimental edge intensity, and J0 (E) is the smooth edge intensity which would be observed in the absence of backscattering. J(E) and J0 (E) are both single-scattering intensities [9.32]. Unless the sample is very thin, plural scattering from the low-loss region will alter the shape of a core edge. Plural scattering effects are removed by deconvolution of the EELS data (see Chapter 3 and Ref 9.54). Because modulations in the near-edge region are altered drastically by plural scattering, deconvolution is almost always necessary in quantitative studies of near-edge structure. In contrast, the oscillating intensity in the extended region, ∆J(E), is not very sensitive to plural scattering. Deconvolution prior to EXELFS analysis is still necessary, however, if the correct energy dependence of the normalizing intensity, J0 (E) is to be determined. Unfortunately, as pointed out by Spence [9.75], deconvolution also results in noise ampliﬁcation. Figure 9.13 shows a pre-edge background subtracted edge without and with deconvolution.

Fig. 9.13 Background-stripped Al K -edge in a moderately thick sample of Fe3 Al without deconvolution (upper trace) and with deconvolution (lower trace).

In principle, pre-edge background subtraction removes the counts that are not due to the particular atomic edge of interest. Usually, an energy range preceding the

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edge is ﬁt to a power law energy dependence, AE −B , where A and B are the ﬁtting parameters [9.32]. The power law is then extrapolated through the edge, as shown in Fig. 9.14 for the Al K-edge.

Fig. 9.14 Power law extrapolation (broken line) of pre-edge background for EELS at K -edge of Al.

In practice, however, the extrapolation into the extended region is not always satisfactory, especially when the ratio of the edge jump to the background is very small. Since pre-edge background subtraction serves primarily to deﬁne the normalizing intensity, J0 (E), other methods besides power law extrapolation can be used if J0 (E) is determined by other means. A common practice in EXAFS is to use the pre-edge subtraction to deﬁne the height of the edge jump, while assuming a functional form for the energy dependence of J0 (E) [9.76]. The most popular method used to isolate the ∆J(E) superimposed upon the core edge is with a polynomial spline ﬁt. A polynomial spline function is composed of a series of consecutive intervals, each containing a polynomial of some order. The intervals are “tied together” by making the function and its ﬁrst derivative continuous across the boundaries or “knots” [9.76]. If the spline intervals and the orders of their polynomials are well chosen, a spline ﬁt can remove the low frequency components due to the smooth atomic edge shape without affecting the higher frequency EXELFS signal. Too many intervals, or polynomials of too high an order will result in the removal of part of the EXELFS oscillations. Too few intervals, or polynomials of too low an order will result in a large peak in the low-r region (r below ∼ 1.0 ˚A) in the Fourier transform [9.77]. A reasonable ﬁrst attempt may be to span the EXELFS region with knots spaced about 2.5 ˚A−1 apart in k-space. Figure 9.15 shows a good spline ﬁt to the Al K-edge. Transformation from energy loss, E, to the wavevector of the outgoing electron, k, is accomplished by the equation E − E0 =

¯ k2 h 2 = (3.81˚A eV)k2 2me

(9.6)

where E0 is the edge onset energy. Finally, to compensate for its attenuation at high-k values, χ(k) is usually multiplied by k n , where n = 1, 2, or 3 [9.78]. This prevents the low-k data from dominating the high-k data in the determination of interatomic

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Fig. 9.15 Spline ﬁt (broken line) to K -edge of Al.

distances that depend only on the frequency, and not the amplitude, of the oscillations [9.77]. In general, when the neighboring atoms are light elements, n = 3 should work well [9.78]. Heavier neighbors require smaller n values. 9.5.2

EXELFS of Al K-, Fe L23 - and Pd M45 -Edges

EXELFS occurs above all the ionization edges in a condensed matter sample, but the analysis of EXELFS is usually performed only for K-edge data. K-edge data are simple to analyze because they correspond to transitions from 1s core states to only those unbound ﬁnal states with p symmetry (assuming the dipole selection rule holds, as is typical in these cases). The L- and M - edge data, on the other hand, are complicated by the variety of possible initial and ﬁnal angular momentum states. The L-edges, for instance, have both 2s and 2p initial states, and transitions from the 2p initial states can result in ﬁnal states with either s or d symmetry. In this section, we will show that in spite of the variety of possible transitions, useful EXELFS information can be extracted from L- [9.79] and M -edges. In particular, we will show that information about static and thermal disorder in the ﬁrst nearest neighbor (1 nn) shell can be obtained from Fe L23 and Pd M45 EXELFS data. In addition, we show that reasonable 1 nn distances can be derived from Fe L23 -edge data. EXELFS data from the K-edge of Al, the L23 -edge of Fe, and the M45 -edge of Pd are shown in Fig. 9.16. Fourier transformation of the EXELFS data is the most common method of extracting structural information. The peaks in the magnitude of the transform correspond to shells of nearest-neighbor atoms, but their positions are shifted slightly from the actual radial distances because of phase shifts in the backscattered wave which are element and k-dependent. Figure 9.17 shows Fourier transform magnitudes of the Al K, Fe L23 , and Pd M45 EXELFS at different temperatures [9.80]. Nondipole transitions are not strictly forbidden in EELS because of the ﬁnite momentum transfer in electron scattering. Nevertheless, theoretical calculations [9.61–9.64] show that in the region of the experimental EXELFS the dipole allowed 1s to p transition dominates in the Al K-edge by a factor of over 100. It is therefore

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Fig. 9.16 kχ(k) from Al K -, Fe L23 -, and Pd M45 -edges from samples at 97 K. Spectra are gain averaged over 30, 60, and 400 eV, respectively.

Fig. 9.17 Fourier transforms of k3 χ(k) for Al K (2.5
a very good approximation to assume that the ﬁnal state of the ionized electron is an outgoing wave of p symmetry. We now turn our attention to the analysis of the EXELFS of the Fe L23 -edge. While more complicated than K edge analysis, transformation of L-edge data is possible for all of the 3d series if done over regions sufﬁciently far from the ionization threshold. L23 ionizations in the energy loss regime have spin–orbit splitting from several to several tens of eV, as well as closely lying L1 -edge overlaps. It is possible to remove the L2 contribution by deconvolution and to minimize the L1 -edge step [9.61] if transformation of regions close to threshold are required. However, if a sufﬁciently large energy range of data beyond the L1 -edge is available for analysis, then these steps are not necessary. For Fe, the spin-orbit splitting between the L3 - and L2 -edges of ∼ 13 eV can be neglected because it is small compared with the spacing between the EXELFS maxima above the L1 ionization threshold energy (6.5
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Fig. 9.18 Magnitude of FT of χ(k) for theoretical 1nn shell L23 (solid line) and L1 (dashed line) EXELFS signals.

Moreover, transforming the data from energy loss space to the k-space corresponding to the L23 -edge effectively raises the frequencies of the L1 EXELFS oscillations, and furthermore makes them incoherent. Figure 9.19 shows the Fourier transform (FT) of the theoretical L1 EXELFS signal from the 1 nn shell in Fe metal after being mapped to the k-space of the L23 -edge, and compares it with the FT of the theoretical 1 nn shell Fe L23 EXELFS [9.80]. The 1 nn shell L1 EXELFS was weighted by a factor of 1/4 to take into account the smaller differential cross-section of the L1 edge. The L1 EXELFS should not greatly affect the 1 nn shell L23 EXELFS signal, although it may signiﬁcantly distort the more distant neighboring shells.

Fig. 9.19 Magnitude of FT of χ(k) for theoretical 1nn shell L23 (solid line) and L1 (dashed line) EXELFS signals.

Having dealt with the complications arising from the presence of different initial states in the Fe L-edge, we now consider the variety of ﬁnal states possible in transitions from the 2p core state. Figure 9.20 contains calculated partial differential cross sections for the excitation of 2p electrons in Fe into ﬁnal states of s, p, d, or f character [9.80]. In the EXELFS region, the 2p to d transition dominates over the sum of all others by a factor of ∼25. This domination of the 2p to d transition makes possible the

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Fig. 9.20 Partial differential cross sections for the excitation of 2p electrons in Fe. Numbers are orbital angular momentum quantum numbers of ﬁnal states.

interpretation of the 1 nn shell Fe L23 EXELFS using a generalization of the standard planewave EXAFS equation for polycrystalline samples: N f (k) χ(k) = exp kR2

−2R λ

exp(−2σ 2 k 2 ) sin(2kR + δ2 (k) + η(k))

(9.7)

Here N is the number of atoms in the 1 nn shell situated a distance R from the central atom. The sine term reﬂects the periodic interference between outgoing and backscattered waves. The function f (k) is the backscattering amplitude per 1 nn atom, η(k) is the backscattering phase shift, and δ2 (k) is the central atom phase shift for an outgoing d wave. The factor exp(−2R/λ) is a phenomenological term to account for the ﬁnite lifetime of the excited state, where λ is the mean free path of the outgoing electron. Finally the term exp(−2σ 2 k 2 ) is a Debye–Waller type factor due to vibrations between atoms, where σ 2 is the mean-square relative displacement (MSRD) between the central atom and an atom in the 1 nn shell. From Eq. 9.7 we see that the theoretical shift of the 1 nn peak in the Fourier transform of the Fe L23 EXELFS can be determined simply from the k dependence of the phase shifts due to the interaction between the outgoing d-like electron wave with the potentials of the central and neighboring atoms. The EXELFS analysis of the Pd M45 -edge follows closely that of the Fe L23 -edge. The spin–orbit splitting between the M5 - and M4 -edges of ∼ 5 eV is ignored because it is much smaller than the spacing between the EXELFS maxima, especially for energies far above the edge onset energy. The M23 - and M1 -edge jumps are removed from the M45 EXELFS signal by transforming only data sufﬁciently past the M1 edge jump. Figure 9.21 shows the differential cross sections of the Pd M45 -, M23 -, and M1 -edges [9.80]. In the experimental EXELFS region, the differential cross section of the Pd M45 -edge is about twice as large as that of the Pd M23 -edge and about 8 times as large as that of the Pd M1 -edge.

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Fig. 9.21 Differential crosssections for M45 -, M23 -, and M1 -edges.

Figure 9.22 presents the Fourier transforms of the theoretical M23 and M1 EXELFS signal from the 1 nn shell in Pd metal after being transformed to the k-space of the M45 -edge, and compares them to the Fourier transform of the experimental Pd M45 EXELFS [9.80]. To make the comparison reasonable, the theoretical 1 nn shell EXELFS from the Pd M23 -edge was weighted such that its magnitude was roughly 1/2 that of the experimental Pd M45 EXELFS from the 1 nn shell. The theoretical 1 nn EXELFS from the M1 -edge was weighted by a factor of 1/4 with respect to that of the M23 -edge. We see that the presence of the M1 EXELFS primarily affects the data beyond the 1 nn shell. The 1 nn shell M23 EXELFS, on the other hand, overlaps in frequencies with the 1 nn shell M45 EXELFS. This probably precludes the reliable determination of 1 nn shell distances from Pd M45 EXELFS. It is still reasonable, however, to determine changes in the static or thermal disorder from the damping of the 1 nn Pd M45 EXELFS signal.

Fig. 9.22 Magnitude of the Fourier transform of χ(k) for theoretical 1 nn shell M23 (dashed line) and M1 (dotted line) EXELFS. Fourier transform of experimental M45 χ(k) (solid line) is shown for comparison.

Although the interference from the M23 EXELFS probably precludes the quantitative determination of 1 nn distances from Pd M45 EXELFS data, we show in Fig. 9.23, the calculated partial differential cross-sections for the excitation of 3d

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339

electrons in Pd into ﬁnal states of various angular momenta [9.80]. The 3d to f transition is seen to dominate over the sum of all others by a factor of ∼20.

Fig. 9.23 Partial differential cross-sections for the excitation of 3d electrons in Pd. Num-bers are orbital angular momen-tum quantum numbers of ﬁnal states.

By comparing the experimental Fourier transforms in Fig. 9.17 with Fourier transforms of ab initio calculations [9.78] shown in Fig. 9.24, ﬁrst nearest-neighbor (1 nn) distances are found in Al and Fe that agree with X-ray diffraction results [9.85] to within ±0.05 ˚A. Distances to the 2 nn shell and beyond may not be reliable, even from the Al K data, because similar bond lengths exist in the surface oxide. The uncertainty in the proper choice of edge onset energy, E0 , contributes to the error in the absolute determination of distances from our EXELFS data. Lee and Beni [9.86] suggest choosing E0 by requiring that the imaginary part and the absolute value of the Fourier transform should peak at the same distance.

Fig. 9.24 Fourier transforms of 1 nn shell ab initio calculations of k 3 χ(k) for Al K (2.5
The cross-section for high energy electron scattering limits EELS experiments to core edges at energy losses below ∼ 5 keV [9.87]. Only elements lighter than

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titanium (Z = 22) have K-edges below 5 keV. This should not, however, limit EXELFS experiments to only those elements of low atomic number. As we have shown, useful EXELFS information can be extracted from L23 - and M45 -edges as well. The use of L23 - and M45 -edges opens up most of the periodic table to EXELFS experiments. 9.5.3

Applications to Metals

The extended ﬁne structure lets each atom act as a “localized probe” of its own immediate environment. As a localized probe technique, EXELFS is useful for determining the identity and positions of nearest-neighbor atoms. An important feature of the technique is its ability to probe independently the environments of different atomic species. This feature makes EXELFS appropriate for studies of the atomic structure of alloys, especially alloys with high concentrations of the probe species. To our knowledge, no EXELFS studies of dilute alloys have been made; such experiments would require extremely good signal-to-noise ratios, which are more easily achieved with EXAFS using a synchrotron source rather than with EXELFS. Research has begun, however, on using EXELFS to determine the chemical shortrange order (CSRO) of nondilute alloys [9.88, 9.89]. Such studies seek to measure the average number of solute species in the lattice sites of the ﬁrst and perhaps second nearest-neighbor shells around the probe atoms. Extended ﬁne structure measurements are sensitive to the static displacements of atoms from perfect lattice positions. Such displacements are common in alloys, especially alloys having atomic species with large differences in their atomic sizes. Such alloys usually prefer to form intermetallic compounds in their state of equilibrium, which have only a few atomic environments for the probe atoms. In recent years, however, a wide variety of experimental techniques have been used to synthesize alloys far-removed from thermodynamic equilibrium. Nonequilibrium alloys such as metallic glasses, nanocrystalline materials, and chemically disordered alloys all possess a high degree of disorder over atomic-scale distances, and this is the regime where EXELFS structural studies are most powerful. Some features and problems of using EXAFS for studies of atomic-scale disorder have been discussed previously [9.68,9.90,9.91]. In addition to static information about atomic environments, extended ﬁne structure can determine features of the local lattice dynamics. The vibrations of the probe atom with respect to its nearest-neighbor atoms are known as thermal mean-square relative displacements (MSRD). From the temperature dependence of the MSRD, local Debye–Waller factors can be obtained. As discussed below, these are similar, but not identical to the Debye–Waller factors measured in X-ray diffraction. EXELFS has been applied to the investigation of thermal MSRDs in metals and alloys [9.88,9.89,9.92].

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9.5.4

341

Chemical Short-Range Order

Local atomic arrangements in the nearest-neighbor shells of the core-loss element can be determined by extended ﬁne structure because the backscattering of the outgoing electron depends on the chemical species of the neighboring atoms. Table 9.1 lists the number of 1 nn Fe atoms surrounding both Al and Fe atoms in chemically disordered and D03 ordered Fe3 Al. In Fig. 9.25 are Fourier transforms of ab initio calculations [9.78] from the disordered and ordered alloys. Comparing Table 9.1 with Fig. 9.25, we see that the magnitude of the 1 nn peaks depend on the number of Fe atoms in the 1 nn shell. In this way, states with different chemical short-range order can be distinguished by EXELFS.

disordered Fe3 Al ordered Fe3 Al

Number of 1 nn Fe atoms Al central atom Fe central atom 6 6 8 5.333

Table 9.1 Number of 1 nn Fe atoms surrounding Al and Fe atoms in fully disordered and fully ordered Fe3 Al.

Fig. 9.25 Fourier transforms of 1 nn shell ab initio calculations of kχ(k) for Al K in Fe3 Al (4
Fourier transforms of experimental data from chemically disordered (piston and anvil quenched) and ordered (annealed) Fe3 Al are shown in Fig. 9.26. Piston-anvil quenching cools liquid metal at a rate of ∼ 106 K/s. The annealing was performed for over a day at 300◦ C, which is below the critical temperature for D03 ordering. Comparison with Fig. 9.25 indicates that our piston–anvil quenched sample is signiﬁcantly disordered. This is in qualitative agreement with M¨ossbauer spectrometry studies

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[9.93]. Quantitative measurements of short-range order parameters by EXELFS, however, have yet to be made.

Fig. 9.26 Fourier transforms of experimental kχ(k) for Al K in Fe3 Al (4
9.5.5

Thermal Mean-Square Relative Displacements

The Debye–Waller factors in extended ﬁne structure depend on mean-square relative displacements between the central atom and its neighbors. There is a different MSRD for each nearest-neighbor shell surrounding the central atom. Changes in the MSRD (denoted ∆MSRD) are measured by varying the temperature of the sample. Einstein temperatures for the different atoms can be readily determined, and Debye temperatures can be extracted from the ∆MSRD data by using the correlated Debye model [9.94]. It should be noted that the Debye temperatures derived from ∆MSRD data are expected to be slightly different from the Debye temperatures obtained by X-ray diffractometry or calorimetry. The Bragg peaks in X-ray diffraction are characteristic of long-range periodicities in the material, and the vibrational correlations between atoms are lost over these long distances. The Debye–Waller factors in X-ray diffraction are therefore usually understood in terms of the mean-square displacements (MSD) of atoms from their average positions (although the thermal diffuse scattering must be considered in terms of MSRDs). Using a force constant model for aluminum [9.95] deduced from inelastic neutron scattering data [9.96], we have calculated the density of vibrational modes and the projected density of modes [9.91, 9.97] for the 1 nn shell. These are shown in Fig. 9.27. The ∆MSRD derived from the projected density of modes is compared with experimental data in Fig. 9.28. Finally, Table 9.2 compares the Al force constant model predictions of Debye temperatures with Debye temperatures extracted from experimental measurements of heat capacity, X-ray diffractometry (∆MSD), and EXELFS (∆MSRD).

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Fig. 9.27 Normalized density of vibrational modes in Al (solid line) and normalized, projected density of modes for 1 nn shell in Al (broken line).

Fig. 9.28 1 nn shell ∆MSRD in Al from projected density of modes (line) and from temperature-dependent EXELFS data [9.88] with arbitrary constant offset (dots).

Debye temperatures for Al Quantity Measured

Force Constant model predictions

heat capacity ∆MSD ∆MSRD (1 nn shell)

393 K 394 K 419 K

Experimental Measurements 394 K [9.85] 395 K [9.98] 430 K [9.88], 415 K [9.92]

Table 9.2 Comparison of force constant model predictions of Debye temperatures for Al with Debye temperatures extracted from experimental measurements.

9.5.6

Vibrational Entropy

Traditional theoretical treatments of phase transformations in alloys [9.99,9.100] typically approximate conﬁgurational entropy and neglect vibrational entropy. Some

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recent ﬁrst-principles calculations of phase diagrams have included vibrational entropy through a Debye–Gr¨uneisen model. Experimentally measuring the importance of vibrational entropy to phase transformations in alloys is an interesting application for temperature-dependent EXELFS. ∆MSRD data can be easily ﬁt to Einstein models. Assuming isotropic vibrations, each atomic species in the alloy can be assigned the Einstein temperature, θE , of its atomic edge. For T θE , the vibrational entropy per atom in the Einstein model [9.101], neglecting zero-point vibrations, is approximately T (9.8) Sv = 3kB ln θE Experimental 1 nn ∆MSRD data for Al and Fe atoms in piston–anvil quenched and D03 ordered Fe3 Al are presented in Fig. 9.29. Also included are 1 nn ∆MSRD data from pure Al metal and pure Fe metal. If the crystal is body-centered cubic, then the 2 nn peak overlaps with the 1 nn peak. Therefore, for bcc crystals, we make the assumption that the 2 nn ∆MSRD is the same as the 1 nn ∆MSRD. Because changes in sample thickness affect the normalization of the EXELFS oscillations, oscillations in the ∆MSRD data are attributed to the drifting of the electropolished samples in the microscope. The experimental data are ﬁt to Einstein models with Einstein temperatures given in Fig. 9.29. Notice that as the number of Fe–Al bonds increases, such as in going from disordered to D03 ordered Fe3 Al, the ∆MSRD becomes smaller. This indicates that, as expected, the bonds become “stiffer” as the alloy orders. Unfortunately, the uncertainty in our present Fe3 Al ∆MSRD data is still rather large. The accuracy of our future ∆MSRD data will be improved by minimizing the effect of thickness variations and by gain averaging over larger energy ranges.

Fig. 9.29 1 nn shell ∆MSRD data for Al and Fe atoms in chemically disordered and ordered Fe3 Al, as well as in pure Al and pure Fe metals [9.57]. Values are relative to data at 97 K. Einstein temperatures are 315 K for pure Al, 350 K for Al in disordered Fe3 Al, 460 K for Al in ordered Fe3 Al, 300 K for pure Fe, 400 K for Fe in disordered Fe3 Al, and 440 K for Fe in ordered Fe3 Al.

CONCLUSIONS

345

With our measured difference between the Einstein temperatures of disordered and ordered Fe3 Al, the difference in vibrational entropy is 0.4 kB per atom at the critical temperature. In comparison, the difference in conﬁgurational entropy calculated in the mean-ﬁeld approximation is 0.56 kB per atom. The present temperaturedependent EXELFS measurements suggest that vibrational entropy is important in the thermodynamics of the order–disorder transformation in Fe3 Al. Recent advances in parallel detection systems will promote further EXELFS investiga-tions. The increased spatial resolution of EXELFS is one of its advantages over EXAFS. EXELFS experiments are not limited to elements of low atomic number, because useful information can be extracted not only from K-edges, but from L23 -, and M45 -edges as well. In many materials, however, edge overlap problems can still restrict the applicability of EXELFS. 9.6 CONCLUSIONS We have surveyed a number of important applications of transmission EELS to the study of metals and alloys. Almost all features of the energy loss spectrum provide important information about electronic structure and microstructure. For instance, in spite of the general complexity of near-edge structures that originates from details of the local chemical bonding, we have seen that the integrated intensities of white lines can be used to measure charge transfers in alloys. Even more subtle are the EXELFS oscillations found above core edges. Interpretations of these oscillations are reasonably straightforward for K-edges, and are useful for obtaining RDFs and local lattice dynamical behavior. The EXELFS above L- and M -edges can be detectable for many elements, and these data are useful for temperature dependence studies, but their use in deriving RDFs awaits further work. Electron energy loss spectrometry has been, and will continue to be, an important tool for physical metallurgy, as new alloys are synthesized, characterized, and applied. While accurate quantitative measurements will probably rely upon empirical data, the use of calculated data will have to await advances in theory that can adequately predict the intensities of near-edge structure. Acknowledgments J. K. Okamoto, D. H. Pearson, and B. Fultz acknowledge the support of the U.S. Department of Energy grant #DEFG0386ER45270. C. C. Ahn and B. Fultz acknowledge the support of the National Science Foundation DMR 8811795.

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9.74. H. Shuman and P. Kruit, “Quantitative data processing of parallel recorded energy-loss spectra with low signal to background,” Rev. Sci. Instrum. 56, 231 (1985). 9.75. J. C. H. Spence, “Uniqueness and the inversion problem of incoherent multiple scattering,” Ultramicroscopy, 4, 9 (1979). 9.76. D. E. Sayers and B. A. Bunker, “Data analysis,” in X-ray Absorption, ed. by D. C. Koningsberger and R. Prins, Vol. 92 of Chem. Anal. (Wiley, New York, 1988). 9.77. B. K. Teo, EXAFS: Basic Principles and Data Analysis (Springer, Berlin, 1986). 9.78. B. K. Teo and P.A. Lee, “Ab initio calculations of amplitude and phase functions for extended x-ray absorption ﬁne structure spectroscopy,” J. Am. Chem. Soc. 101, 2815 (1979). (Software implementation by R.A. Scott et al.) 9.79. R. D. Leapman, L. A. Grunes and P. L Fejes, “Study of the L23 edges in the 3d transition metals and their oxides by electron-energy-loss spectroscopy with comparisons to theory,” Phys. Rev. B 26, 614 (1982). 9.80. J. K. Okamoto, C. C. Ahn and B. Fultz, “EXELFS analysis of Al K, Fe L23 , and Pd M45 edges,” Microbeam Analysis–1991, ed. D. G. Howitt, p. 273. 9.81. These calculations of electron scattering cross-sections are done according to Bethe [9.82]. Atomic Hartree-Slater wave functions from the algorithm of Herman and Skillman [9.83] continuum states with a free electron density of states [9.84]. The collection angles chosen correspond to the conditions under which the experimental data were acquired. The calculations ignore spin-orbit splitting. 9.82. H. A. Bethe, “Zur theorie des d¨urchgangs schneller korpuskularstrahlen durch materie,”Ann. Phys. 5, 325 (1930). 9.83. F. Herman and S. Skillman, Atomic Structure Calculations (Prentice Hall, New Jersey, 1963). 9.84. R. D. Leapman, P. Rez, and D. F. Mayers, “K, L, and M shell generalized oscillator strengths and ionization cross sections for fast electron collisions,” J. Chem. Phys. 72, 1232, 1980. 9.85. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976). 9.86. P. A. Lee and G. Beni, “New method for the calculation of atomic phase shifts: Application to extended x-ray absorption ﬁne structure (EXAFS) in molecules and crystals,” Phys. Rev. B 15, 2862 (1977). 9.87. C. C. Ahn and O. L. Krivanek, EELS Atlas, 1983. Center for Solid State Science, Arizona State University, Tempe, Arizona 85287.

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9.88. J. Okamoto, C. C. Ahn and B. Fultz, “Temperature-dependent EXELFS of chemically ordered and disordered Fe3 Al,” Proceedings of the XIIth International Congress for Electron Microscopy (San Francisco Press, 1990), p. 50. 9.89. J. K. Okamoto, C. C. Ahn and B. Fultz, “Temperature-dependent EXELFS of Al K and Fe L23 edges in chemically disordered and DO3 ordered Fe3 Al,” Microbeam Analysis–1990 , J. R. Michael and P. Ingram, eds. (San Francisco Press, 1990), p. 56. 9.90. P. Einsenberger and G. S. Brown, “The study of disordered systems by EXAFS: limitations,” Solid State Commun. 29, 481 (1979). 9.91. E. D. Crozier, J. J. Rehr, and R. Ingalls, “Amorphous and liquid systems,” in [9.71], Ch. 9. 9.92. M. M. Disko, G. Meitzner, C. C. Ahn and O. L. Krivanek, “Temperaturedependent transmission extended electron energy-loss ﬁne-structure of aluminum,” J. Appl. Phys. 65, 3295 (1989). 9.93. B. Fultz, Z. Q. Gao and H. H. Hamdeh, “Short-range ordering in undercooled Fe3 Al,” Hyperﬁne interactions 54, 521 (1990). 9.94. G. Beni and P.M. Platzman, “Temperature and polarization dependence of extended x-ray absorption ﬁne-structure spectra,” Phys. Rev. B 14, 1514 (1976). 9.95. E. R. Cowley, “A Born-von Karman force constant model for aluminum,” Can. J. Phys. 52, 1714 (1974). 9.96. R. Stedman, L. Almqvist and G. Nillson, “Phonon-frequency distributions and heat capacities of aluminum and lead,” Phys. Rev. 162, 549 (1967). 9.97. E. Sevillano, H. Meuth and J. J. Rehr, “Extended x-ray absorption ﬁne structure Debye-Waller factors. I. Monatomic crystals,” Phys. Rev. B 20, 4908 (1979). 9.98. E. A. Owen and R. W. Williams, “The effect of temperature on the intensity of x-ray reﬂexion,” Proc. R. Soc. London Ser. A 188, 509 (1947). 9.99. F. Seitz, Modern Theory of Solids (McGraw, New York, 1940), Section 124. 9.100. J. W. Christian, The Theory of Transformations in Metals and Alloys (Pergamon, New York, 1975), p. 172. 9.101. D. L. Goodstein, States of Matter (Dover, New York, 1985), p. 147.

10

Electron Energy Loss Studies in Semiconductors Philip E. Batson IBM Thomas J. Watson Research Center Yorktown Heights, New York 10598

Abstract Atomic bonding and local electronic structure are obtainable from subnanometer regions of semiconductors, using electron energy loss spectroscopy (EELS) in the scanning transmission electron microscope (STEM). This chapter reviews work on direct interband scattering and soft x-ray core loss ﬁne structure. Using CCD detection, we can obtain spectra having 0.1% statistical quality and 0.2-eV energy resolution in a few minutes. Low loss scattering identiﬁes the direct, or indirect, nature of the semiconductor or insulator gap. The atomic bonding environment can then be identiﬁed from the near-edge ﬁne structure of the core-loss spectra. Conduction band electronic structure can also be obtained provided core excitonic effects are considered. On the scale of 0.1 eV, EELS spectra are demonstrably different from soft X-ray absorption spectra, leading to some difﬁculties in interpretation. However, in the GeSi alloy series, very detailed information about band offsets and strain splitting is clearly evident and of great use. Spatially resolved EELS of Si at the 0.2 nm resolution has revealed conduction band electronic structure in separate regions of a single dislocation structure. New instrumental developments promise to extend this technique to the 35–50meV and sub-0.1 nm regime, where EELS will be possible as a function of impact parameter relative to an individual atomic column. 10.1 INTRODUCTION Semiconductor devices are dependent on electrical properties that are often dominated by the behavior of atomic level structures. For instance, insulating-gate oxide layers in modern ﬁeld-effect transistors are now as thin as 2.5 nm. When these oxides reach 1–1.5 nm thickness, physical limits imposed by overlap of conduction electron wavefunctions are expected to signal the need for alternative technologies 353

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for further device improvements [10.1]. On the other hand, nanometer scale design of semiconductors can be expected to produce new, desirable behavior. The srained Gex Si1−x alloy series, for instance, has now been exploited to produce transistor circuits operating above 100GHz, beating even GaAs for very high speed devices. Finally, individual devices of atomic dimensions can be expected to exhibit new behavior that can be used for computation. A good example of this might be a single electron transistor that operates at room temperature [10.2]. In principle, electron energy loss spectroscopy (EELS) is sensitive to atomic bonding and electronic structure because the probe electron strongly interacts with the charge density within the material, causing charge density ﬂuctuations that are characteristic of possible electronic motion within the solid. In so doing, the probe electron loses energy in response to these ﬂuctuations, giving us a detailed picture of the electronic behavior. A common experimental setup is to scatter electrons having well deﬁned energy and momentum in such a way that their energy loss and angles through which they scatter convey information about the energy levels and possible conduction channels of electrons within the semiconductor. If we combine EELS with a scanning transmission electron microscope (STEM), we speak of spatially resolved EELS (SREELS) with two ideas in mind. First, in the simplest case, we can reproduce the traditional angle resolved experiment in small areas. In regions that are several nanometers across, this is a straightforward extension of conventional EELS ideas. Well-deﬁned energy losses and scattering angles, or momentum transfer can be obtained. The utility in using a small probe is to obtain this information in a limited area of a heterogeneous sample. Second, in subnanometer sized regions, the physics of both the electronic structure and of the EELS experiment can be modiﬁed due to the small size of the probed region. For instance, when the size of the probed volume is smaller than the characteristic wavelength of a particular electronic excitation, then the excitation will be strongly damped or can become quantized within the small volume. Also, when a probe electron passes nearby an electrically polarizable object, energy loss will be zero, but will depend on the object properties in a very complex manner. This type of measurement requires extensive thought and effort to begin relating the observed results to quantities of interest to the materials designer. For the most part, current results are merely extensions of the bulk experiments to areas that are small, but not so small as to require extensive rethinking of the scattering theory. Exceptions to this occur in small Si spheres, where strong shifts of conduction band levels can be observed. Also, treatment of core excitonic distortion in the presence of the incident swift electron requires a spatially extended, time-dependent view of the scattering. Finally, as probe sizes ultimately become comparable to the interatomic distances within a material, the precise positioning of the probe relative to speciﬁc atoms will strongly affect the observed scattering. There are several types of measurement that can be attempted for semiconductors. These can be classiﬁed according to the amount of energy lost in the scattering process. In the very low energy loss regime (0.1–3 eV), direct, single-electron, interband excitations will occur. These carry information about both valence and conduction electron levels within a few eV of the Fermi energy. In an intermediate

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energy range (5–30 eV) we observe collective excitations, surface and bulk plasmons. These carry information about average bulk electronic properties but may be made to yield information about local properties under certain circumstances. In the soft X-ray energy loss regime (50–1000 eV), we observe core excitations, that is, direct, single-electron excitations from the discrete core levels to excited-state conduction bands of the material. The core excitations are useful for determining the amount and type of elements that are present, as is described in other chapters. In typical semiconductor devices, this use has been limited because the composition is usually easily inferred from the observed density and morphology, due to the limited number of elements that are known to be present. TEM analysis is then adequate to identify regions of varying composition. EELS is beginning to make a contribution in identifying local compositions when the atomic density is similar. A good example of this is in mixed Si3 N4 /SiO2 /Si2 N2 O materials used for electrical isolation in semiconductors. At a more detailed level, EELS can detect local changes in band structure that defects and interfaces. These changes may then be related to electrical properties that are relevant to device operation. The following discussion, therefore, is largely limited to determining composition and electronic structure from the shape of EELS features when the atomic structure is deliberately varied to obtain enhanced electrical properties. Each of the three energy regimes has unique experimental requirements. We ﬁrst, discuss general considerations when using a probe-forming instrument together with EELS. Next, we cover interband and core losses with emphasis on the requirements for typical (usually silicon based) semiconductors. A few examples include: an analysis of composition and electrical properties in a trench capacitor structure, EELS conduction band results for small Si particles, and some results from the Gex Si1−x alloy system. We will not discuss collective excitations or core excitations for elemental analysis because they are covered in other chapters. In Section 10.9, we comment on the difﬁculties of putting together an EELS experimental apparatus that can address these requirements. 10.2 GENERAL CONSIDERATIONS Electron energy loss scattering is a function of both position on the sample and scattering angle. In a large-probe instrument, for a uniform material, the probe position is unimportant and the measurement scheme is optimized to obtain good angular resolution. In a small-probe instrument, the probe position on a heterogeneous sample is very important. Since the electron optical methods for producing a stable, controllable probe necessarily restrict the total beam current available, it becomes impractical to design for a small angular acceptance. In addition, for a diffractionlimited probe size, the angular spread of the scattering is dominated by the angular content of the probe — the scattering distribution is limited by the uncertainty principle applied to the small probe. Therefore, SREELS is restricted by both signal levels and the basic scattering physics to be integrated over angle. In the results

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described here, the the probe size is 2–4× greater than the uncertainty principle limit. Therefore, an adequate description may be given within the conventional scattering theory. Figure 10.1 summarizes the scattering kinematics for inelastic scattering in the STEM [10.3,10.4]. Prior to scattering, the incident beam consists of a coherent bundle of waves having a range of wavevectors, Ki , deﬁned by an aperture subtending an input angle, α. After scattering, the waves are characterized by wavevector, Kf , with f − K i = q. The detector sums over a range of ﬁnal wavevectors, Kf , deﬁned K by an axial aperture subtending an angle β. In the TEM the electron detector can lie in an image plane, conjugate to the probe, so that the collection summation may be coherent. This is true for energy ﬁltered imaging using in-column energy ﬁlters and for the imaging EELS spectrometers sited underneath the TEM. In the STEM, the detector lies in a Fraunhofer diffraction plane, so that electrons scattered in different directions are separated, and the summation is incoherent. Typical detectors will lie somewhere between the two extremes, but tend toward incoherent summation.

Fig. 10.1 Scattering diagram for energy loss in the STEM. The incoming beam is made up of a coherent sum over input angles. The outgoing wave is usually summed incoherently by the spectrometer collection optics. Momentum transfer is largely within a plane perpendicular to the incoming beam direction.

The momentum given up to the sample, q, is approximately related to the scattering angle by, q ≈ Ki θ and is mostly pointed perpendicular to the incident beam direction. It also has a parallel part, qE ≈ Ki E/2E0 , which is controlled by the energy loss, E and the incident beam energy, E0 . Parenthetically, for energy losses below 1000 eV, it is feasible to use a collection angle, β, which is small enough (perhaps < 50 mR at 100 keV) to avoid conﬂict with a high angle dark ﬁeld detector (HAD). This range of elastic scattering is the basis for the current imaging technique of choice in the STEM, providing a high resolution, directly interpretable image[10.5,10.6]. Recent advances in instrumentation is now also facilitating sub-angstrom spatial resolution with this technique for use with EELS [10.7].

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Thus, EELS results in the small-probe geometry are usually integrated over scattering angle, or equivalently, momentum transfer. Usually, this integration is incoherent, but may be coherent under some circumstances. This can be turned to advantage when we consider that the electronic structure of a small object — a defect, an impurity atom, or an interface — also encompasses a large momentum uncertainty by the uncertainty principle. Thus, an angle-integrated experiment provides the most efﬁcient use of the scattering that results from excitation of localized states. This becomes of the highest concern for heterogeneous semiconductor systems, whose macroscopic electrical properties can be dominated by atomic level structure. Semiconductor devices commonly depend for operation on sub-electronvolts energy scales. Typical band gaps range from 0.1–5.0 eV; spin–orbit splitting of core excitations in the soft X-ray region are ∼ 0.3–2.0 eV; core hole lifetime energy widths are 0.1–0.5 eV; and shifts of band edges in the presence of defects or impurities are on the order of 0.0–0.5 eV. Thus, practical applications of EELS to these materials requires both high energy resolution system and high stability. The system at IBM [10.8,10.9] currently delivers ∼ 70-meV resolution at 8 mR semiangle collection. To allow practical use of this resolution, the energy scale and stability of the instrument need to be several times better than the resolution. The present system delivers an energy scale accuracy and stability of ±20 meV over the range 0–1000 eV. This is a bare minimum for studies involving variation of band-structure with position. For instance, the addition of 5% Ge to Si in the alloy shifts the Si L2,3 edge by ∼ 10 meV [10.10]. Somewhat less accuracy is required for bonding changes that typically shift features by 1–2 eV. 10.3 INTERBAND EXCITATIONS Direct excitation of electrons from the valence to conduction bands is observable in EELS. First, we discuss the scattering in terms that are common for angle resolved spectroscopy. Results for the STEM within this framework will be predicted by incoherent summations over probe and collection aperture angles. At spatial resolutions close to the diffraction limit of the probe-forming optics, this approach will not be valid. However, for many situations it provides a useful way of extending the accepted analysis that is valid for the small probe work. In this approach, the wavevector-dependent scattering cross-section is given by Fermi’s Golden Rule [10.11,10.12], ∂2I ∂Ω∂E

= =

4γ 2 | f | exp(iq · r)|i|2 ρf,i (E) a20 q 4 γ2 −1 Im 2π 2 e2 a20 q 2 ε(q, E)

(10.1)

with symbols deﬁned as in Chapter 4, using the macroscopic dielectric constant, ε, and the joint density of initial and ﬁnal states, ρf,i (E). In the large-angle collection

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geometry, we integrate over the various apertures to obtain a result that does not depend on scattering angle [10.13], γ2 ∂I qM −1 = 2 2 2 Im ln (10.2) ∂E π e a0 ε(0, E) qE with the maximum wavevector accepted by the collection aperture being qM . This result assumes that ε changes slowly with q so that the integration over scattering angle is dominated by the behavior of the 1/q 2 term in Eq. 10.1. It also assumes that single electron transitions are allowed for small q. If Re(ε) is much greater than Im(ε), then ∂I/∂E will be proportional to Im(ε) and the EELS results should be very similar to optical absorption. Also, because qE ≈ 0.01 ˚A−1 is relatively small for the low energy region, most of the scattering deﬁned by Eq. 10.2 may be classiﬁed as zero-angle scattering. Thus we can compare optical and EELS results by evaluating ε(E) for each case. This provides a convenient way to predict expected scattering cross-sections for particular materials. As an example, direct interband scattering at 1.5 eV in GaAs occurs with a scattering cross section, ∂σ/∂E = 0.84 × 10−22 cm2 eV−1 [10.14]. This is a small quantity, roughly comparable to the scattering for a soft X-ray core excitation at a few hundred eV. This is the case because the joint density of states(JDOS), ρf,i (E), above, is very small for transitions between the band edges. Most of the electron density of states is concentrated 1–2 eV away from the band edges. This is true in silicon as well, where there are only 0.05 states atom−1 eV−1 at the conduction band edge [10.15]. Since both the initial and ﬁnal states are close to the Fermi level, neither are simply described. However, a couple common situations do occur that can be evaluated. Figure 10.2a shows the situation for two parabolic bands located at the Brillouin zone center. Different effective masses characterize the valence and conduction bands. Dipole vertical transitions are allowed because these bands are typically formed by symmetric (bonding, valence band) and antisymmetric (antibonding, conduction band) combinations of atomic orbitals. If we sum vertical transitions suggested by the ﬁgure, we ﬁnd that the JDOS is given by, ρf,i (E) ≈

mv mc mv + mc

3/2 (E − Eg )1/2

(10.3)

where mv,c are the effective masses for the valence and conduction bands, and Eg is the gap energy [10.16]. This expression will not be precisely correct for EELS, because some momentum transfer is allowed, indicated by the spread of vertical vectors indicated in Fig. 10.2. If however, the momentum transfer is small with respect to the extent of the Brillouin zone, then Eq. 10.3 should describe the EELS case as well. If the conduction band minimum and the valence band maximum are not spherical and located at the Brillouin zone center, the scattering integration becomes more complicated. One situation in which two spherical bands are centered at different places, but not too far apart, in the Brillouin zone, is summarized in Fig. 10.2b. For scattering intensity near the gap energy, scattering occurs near some qave , deﬁned by

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Fig. 10.2 Model band structure for the near-threshold interband absorption. (a) Situation for vertical transitions between two parabolic bands at the Brillouin zone center. (b) When one of the bands is not at the zone center, direct optical transitions are forbidden, but energy loss scattering can still occur by transfer of some large average momentum.

the displacement of the band edges. This situation has been studied by Bruley and Brown [10.17], who deduced a square-root energy dependence as in Eq. 10.3, but with a mass m = mv + mc rather than the reduced mass. The treatment has been extended to include well separated bands by Rafferty and Brown [10.18]. They evaluated the angular convolution for scattering from states near a valence band maximum to states near the conduction band minimum, ﬁnding two types of scattering: Idirect ∝ (E − Eg )1/2 , Iindirect ∝ (E − Eg )3/2

(10.4)

The new understanding here is that they found, in agreement with privous treatments, that the JDOS continues to show an E 1/2 dependence, but that the matrix element for indirect transitions also shows a linear dependence on E, so that the scattering intensity takes on an E 3/2 dependence. As a practical result, Rafferty and Brown further found that they could use this understanding to deduce the indirect or direct character of interband losses by appropriate ﬁtting of theoretical shapes to data for diamond, BN, GaAs and ZnO. Interpretation of optical interband absorption encounters severe difﬁculties due to the attractive interaction of the electron and hole in the excited state. In typical semiconductors, a loosely bound Wannier exciton occurs. This distorts the spectrum and adds sharp, pre-onset peaks corresponding to bound electron–hole pairs. Elliott considered this problem in a parabolic-band, effective mass approximation [10.16]. He constructed a localized excited state using a linear combination of crystalline Bloch states associated with the conduction band. The spatial limits of this localized state are described by an envelope function which is the solution to an effective-mass, Schr¨odinger equation for an electron bound to a screened hole. The scattering is modiﬁed by insertion of the square of this envelope function, |ψ(0)|2 , evaluated at the origin, into Eq. 10.1:

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4γ 2 ∂2I = 2 4 |ψc (0)|2 |Mf,i |2 ρf,i (E) + |ψn (0)|2 ∂Ω∂E a0 q n

(10.5)

where we replaced the matrix element with Mf,i . The envelope function occurs in two forms resulting from the two types of ﬁnal states that are possible. There is a distortion of the scattering by nonbound continuum states, ψc , and an added summation over discrete hydrogenic bound states, ψn . In the absence of strong carrier screening, experience with optical data suggest that the exciton makes the measurement of ρf,i (E) impossible, because |ψc (0)|2 can have a strong energy dependence, and because it is difﬁcult in practice to separate the distortion caused by |ψc (0)|2 from the added intensity due to the bound-state summation, |ψn (0)|2 . However, there are some recent experimental results which suggest that this may not be as difﬁcult a problem for EELS. Figure 10.3, shows results for the bulk absorption in Si, GaAs and Ga0.85 In0.15 As. The data are compared with the prediction of Eqs. 10.2 and 10.3 convoluted with an instrumental resolution of 0.35 eV. Also included are the optical data [10.19] and prediction of Eq. 10.5 for GaAs. We can see that over the indicated energy range with the broad energy resolution, the excitonic analysis does not grossly distort the energy dependence. EELS measurements interband scattering in diamond [10.17], BN, ZnO, [10.18] and GaN [10.20] appear to be consistent with this. On the other hand, the Si results [10.21] in Fig. 10.3 appear consistent with an E 1/2 behavior, contradicting Eq. 10.4. This disagreement will no doubt be resolved as more of these types of experiments become possible.

Fig. 10.3 Direct interband absorption for Si, GaAs (E) and GaIn0.15 As0.85 compared to the GaAs optical results (O). The solid lines show the predicted behavior for the JDOS. The dashed line shows the predicted behavior including an excitonic interaction for GaAs. At 0.35 eV resolution, the simple JDOS interpretation appears adequate.

In the STEM, we can follow the variation of the gap as a function of position. This has allowed the identiﬁcation of interband scattering associated with a single

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misﬁt dislocation at the GaAs/Ga0.85 In0.15 As interface [10.14]. Consideration of that result also has led to a couple of interesting ideas. The ﬁrst idea is related to the ﬁnding above, that the cross-section for bulk scattering near the band edge is necessarily small because most of the bonding electronic states lie well away from the band edges. A direct result is that one electronic defect state can dominate the scattering in or near the band gap. For the GaAs example, the defect scattering was found to be almost 100 times stronger on a per atom basis than the bulk scattering [10.14]. Thus, defect signals can be strong in a scattering volume that comprises a limited number of atoms. The second idea is that the scattering selection rules can be grossly modiﬁed. If the defect state is highly localized, it will be spread out into a fairly ﬂat band throughout the Brillouin zone. Thus, transitions to ﬁnal states near the zone boundaries are possible. As a result, the ﬁnal state for defect scattering can be quite different from the ﬁnal state for normal bulk scattering or for the optical data. Each experiment must therefore be examined in detail to determine the extent that simple ideas can describe the results. These types of experiments have been very difﬁcult to perform in the past. Figure 10.4 shows a summary of the measured intensity in the no-loss region for GaAs. Several results are plotted. First, a ﬁeld emission gun energy distribution having a full-width at half-maximum of 0.35 eV is plotted (solid line is the theoretical proﬁle and open dots are the measured proﬁle, ×1). Data, in transmission through GaAs, is overlaid on this (black dots, ×1). Above 1.3 eV, these traces are multiplied by 100 to show the GaAs scattering (black dots), and they are compared with the measured (open dots) and calculated ﬁeld emission distribution (dashed line). Thus we see that the measured ﬁeld emission distribution is well described by the calculation. The troubling problem is that the ﬁeld emission distribution can be fully 100 times bigger at 1.5 eV than the GaAs interband scattering [10.14]. Thus, the experiment is very difﬁcult with the unmodiﬁed electron source. As equipment for monochromatization of the incident beam energy has recently become available, it will be interesting to see the advances on this analysis that soon will become possible. 10.4 SOFT X-RAY CORE EXCITATIONS: FINGERPRINTS In earlier chapters, core excitations have been treated thoroughly from the point of view of atomic density and ﬁne structure for homogeneous materials. In particular, it is well known that the shape of the near-edge region depends strongly on the local environment of the excited atom. A generalized spectral shape, or “family resemblance” can even be attributed to a particular local environment, irrespective of the precise identity of the central atom. It follows that variations in the shape of an edge can signify variations in the local environment. Figure 10.5 shows a set of Si L2,3 spectra for Si, SiO, and crystalline and amorphous SiO2 . These spectra were obtained from 1 nm sized areas with an energy resolution of ∼ 0.4 eV [10.22]. Oxidation of the Si results in large energy shifts and changes in the near-edge shape. In the oxides, the near-edge shape is attributable to a bound core exciton (105.7 eV) and “inner-well resonances” (108 and 115 eV)

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Fig. 10.4 The interband absorption raw data for GaAs using a room temperature ﬁeld emission source. A severe experimental limitation is imposed by the large tunneling tail of the ﬁeld emission process even at 1.5-eV energy loss. The background is 100 times bigger than the intensity at the absorption edge.

Fig. 10.5 Comparison of the Si L2,3 -edge for various oxides of Si with thin Si. Oxidation is easy to detect. The resemblance of the mono- and dioxide results suggest that the shape is sensitive to local environment in this case.

[10.23,10.24]. The latter lie above the conduction band edge, but are still localized within the cage of oxygen atoms surrounding the silicon. None of these peaks bear much relationship to the ground-state band structure in the absence of the excited-state core hole. The presence of the highly localized excitonic character of this structure, however, ensures a very local probe of the atomic environment. This is the reason that the amorphous and crystalline SiO2 spectra are so similar. The spectral shape signiﬁes a full fourfold coordination of silicon by oxygen. In the SiO spectrum, it

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thus appears that most of the Si atoms are fully coordinated by oxygen. In X-ray photoemission, it is well known that the Si core level shifts by varying amounts for environments consisting of different numbers of Si and O [10.25]. This has been observed also in soft X-ray partial photo-yield experiments [10.26]. This information leads us to assign the small pre-edge ramp in the SiO spectrum to Si atoms that have less than four oxygen neighbors. Because the shape of this ramp is different from the bulk Si-edge, we can distinguish between a uniform substoichiometric oxide and a mixture of SiO2 and Si having the same average composition. 10.4.1

Application to Substoichiometric Silicon Oxide

Figure 10.6 shows an example of a practical application of this ﬁnding [10.22]. Figure 10.6a shows a bright ﬁeld image of an Al/SiOx /Si structure. The probe size for this image and spectra was about 1 nm. The SiOx layer was about 25 nm thick. The intention was to construct an oxide layer consisting of three distinct regions: (1) a 10-nm thick layer next to the Al having a local structure of Si islands in an SiO2 matrix and a stoichiometry, Si:O of 1:1, (2) a 5-nm layer of SiO2 at the center, and (3) a second 10-nm layer next to the Si, of Si islands within an SiO2 matrix having an average stoichiometry of 1:1. Quantitative measurements of the Si L2,3 - and O K-edges compared to well characterized materials established that the desired atomic ratios were obtained. Analysis of the shapes of the L2,3 spectra, however, show that regions (1) and (3) did not have the same local structure. While region (3) displayed a shape that is readily separable into SiO2 and a Si spectra, the results from region (1) were characteristic of a homogeneous substoichiometric mixture. After an annealing procedure, the spectra in region (1) became identical to region (3), indicating that the suboxide phase separated to form the desired Si islands in SiO2 . In region (2), where the 1:2 stoichiometry is desired, this is obtained, but some Si scattering was still evident. This may be due to some scattering from the adjacent regions (1,3) due to the roughness of the boundaries. It may also be due to undesired inhomogeneities in layer (2). This type of measurement has been taken to the atomic level by Muller [10.1], who showed that the one or two layers of substoichiometric oxide, on either side of an SiO2 gate layer, does not have a fully developed SiO2 bandgap. This places a fundamental lower limit on practical gate oxide thickness’s of ∼ 1 nm. 10.5 CORE EXCITATIONS: JDOS INTERPRETATION From the discussion above, we can see that it is possible to make fairly detailed judgments about the local structure in very thin layers in semiconductor systems using a “ﬁngerprint” analysis. In the end, we need to use ﬁrst principles evaluations of the scattering to understand the origin of various near-edge ﬁne structure, and to use them to derive a measured symmetry projected density of states appropriate to the semiconductor ground state.

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Fig. 10.6 Use of the Si L-edge for a local bonding determination. A choice between a uniform suboxide and a mixture of Si and the di-oxide can be made by examination of the shape of the pre-edge structure.

Figure 10.7 shows a high resolution Si L2,3 -edge for bulk silicon compared with an s, d-projected single electron ﬁnal density of states by Weng [10.27]. The ﬁgure includes the background subtracted raw data, a sharpened spectrum obtained by unfolding the ﬁeld emission distribution [10.28], and an edge with the L2 scattering stripped away. The spin–orbit splitting was removed by deconvolution in Fourier space of two d-functions having a splitting of 0.608 eV and a relative weighting of 2:1. The deviation of the calculated DOS from the measured spectrum is small, suggesting that EELS core excitonic distortion is small for this material. Thus, the single electron calculations are quite successful at predicting, on an ab initio basis, the near-edge ﬁne structure of Si. Weng’s work was the ﬁrst to show a positive correspondence of measured ﬁne structure with calculated ﬁne structure for the Si 2p absorption. Since then, there has been some discussion about whether the ﬁne structure observed at the edge of Si is really a good indication of ground-state DOS, or whether distortion of the absorption edge by core excitonic effects is expected to obscure the DOS ﬁne structure [10.29]. In Fig. 10.7, we show a more recent LDA calculation for a P donor atom within a 64 atom Si matrix [10.30]. This result supports Weng’s calculation, reproducing the positions of edge features within 2–3eV of onset, but improving the agreement with the relative magnitudes of those features. For instance, the hole interaction enhances

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Fig. 10.7 A comparison of the Si L3 edge at 0.25 eV resolution with the calculated s, dprojected density of states in the conduction bands for Si and for a P donor atom within a Si matrix. A reasonably good matching is possible, giving reason to hope that ab initio calculations may be useful in the future for interpretation of the near edge structure.

the onset intensity, bringing it into better agreement with the measured result, but not totally obscuring the conduction band structure. As will be discussed later, and was noticed above in the discussion about interband scattering, spatially resolved EELS results appear to be less inﬂuenced by excited state electron–hole interactions than are X-ray photon absorption experiments, due to the presence in the scattering problem of the extra charge of the swift electron [10.31,10.32]. The SiO2 and Si examples illustrate two commonly encountered extremes. In SiO2 , we believe the near edge ﬁne structure bears no simply recognizable relationship to the single electron density of states. But for Si the projected DOS calculation describes the results very nicely. It is crucial to understand the details of this behavior if we expect to have any success in using EELS for determination of conduction bandstructure. The two examples can be distinguished in a general way by considering their dielectric constant, ε0 . In the simple Wannier exciton picture, a core or valence exciton consists of a weakly bound electron–hole pair. The attractive potential between these carriers is simply the Coulomb potential, screened by the dielectric constant of the material, e2 /ε0 r. Thus, a hydrogenic system is created, superposed onto the periodic background of the crystal. If ε0 is large, the Coulomb interaction is weak, and very little excitonic distortion occurs. If ε0 is small, the core or valence hole is poorly screened, the excited electron–hole pair can be tightly bound, and the excitonic effects are strong. Thus we expect SiO2 with ε0 = 2 to show strong excitonic effects. Silicon, with ε0 = 11, should exhibit only a weak excitonic distortion. Metals are expected to show no effects at all. This view appears to hold fairly well with the following caveat: The appropriate dielectric constant can be

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Fig. 10.8 The diamond K edge provides a vivid demonstration of the differences which exist between the EELS and partial photo-yield data. Beyond about 4 eV from the onset, the spectra are identical. Near the edge, in the region inﬂuenced by the core exciton, the two results are different. The calculated p-projected DOS is from [10.33].

spatially dependent on a very local scale. Thus, in SiO2 , the Si L2,3 -edge shows a strong excitonic distortion, but the oxygen K-edge in the same material does not appear to be distorted. Apparently, valence band charge density, the major screening agent, is physically depleted around the Si atoms in the material, and concentrated near the oxygen. Thus, holes on the the Si atoms are largely unscreened, while holes on the oxygen atoms are well screened. In the intermediate range of dielectric constants, ε0 ≈ 4–7, spectra are distorted by varying amounts. Figure 10.8 shows spectra for diamond (ε0 = 5.5) obtained by EELS and by soft X-ray partial photo-yield (XAS)[10.32]. These are compared with the calculated p-projected DOS [10.33]. Several interesting things are evident. First, the XAS and the EELS spectra are nearly identical beyond about 4 eV from the edge, but are not the same in the near edge region. Second, the p-projected DOS matches the measurement away from the edge on a scale of order 1 eV. But close to the edge, the p-DOS underestimates the intensity and does not predict the small peak at the absorption edge. It turns out that the small peak is a core exciton, and the edge distortion can be understood using the Elliott exciton formalism alluded to above. According to conventional notions about EELS within the dipole limit and optical absorption, the two experimental results in Fig. 10.8 should be the same. It appears, however, that the excitonic part of the edge intensity is different for XAS and EELS. Recently, we developed some ideas as to why this might be so [10.32]. In the EELS case, the swift electron is present near the core hole for a signiﬁcant part of the time that it takes the core electron to make the transition from the core level to the exciton orbital. Thus, the central hole charge is reduced by some fraction related to a time average of the swift electron charge. This reduces the distortion caused by the core exciton. Figure 10.9 shows the near edge region for XAS and EELS diamond spectra. Using Elliott’s formalism, modiﬁed to include the swift electron and it’s associated charge density correlation wake, theoretical edge shapes can be calculated

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as shown, and a p-DOS can be derived which is consistent with both the EELS and photo-yield results. The JDOS is remarkably similar to the ρf,i (E) = (E − Ec )1/2 predicted in Eq. 10.3.

Fig. 10.9 A detailed comparison between the EELS and photo-yield results compared to the results of theory. It appears that the EELS results are different because the scattering electron inﬂuences the exciton distortion. In effect, it provides more complete screening of the core hole, partially suppressing the exciton formation.

Clearly, the excitonic distortion cannot be ignored. It appears to be less of a problem for EELS, however, than it is for photon absorption spectroscopies. In the semiconductors, where the excitonic binding energies are typically small (0.005– 0.5 eV), ab initio ground-state JDOS calculations can give results that are very close to the observed spectral shapes. This is likely to be the reason that the results for Si in Fig. 10.7 follow the calculated JDOS so closely. 10.6 CORE EDGE INTRA-GAP SCATTERING Given a good understanding of the excitonic problem, we may readily inspect part of the region below a core edge for scattering involving empty defect states within the gap of a semiconductor. In order to do this well, we must work with as thin a specimen as possible to minimize scattering due to unmodiﬁed bulk material. When this is attempted, scattering due to surface electronic states are visible within the gap region. In Fig. 10.10, the in-gap region for diamond is shown at about 0.4-eV resolution for a single dislocation in a thin region, and for the bulk, beside the dislocation [10.34]. In passing, it can be pointed out that the core exciton is totally obscured at this energy resolution. A high energy resolution is absolutely necessary for serious investigation of the near-edge shape of these edges. Without it, evaluation of the magnitude of excitonic distortion is impossible. Both spectra show signiﬁcant scattering into ﬁnal states that reside within the band gap. In addition, as veriﬁed by calculations [10.34], the in-gap intensity exists down to the surface related valence band maximum. Thus in the thin sample, signiﬁcant surface related scattering is present. In other studies, it

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was veriﬁed that this scattering was independent of thickness. When the STEM beam intersected a single dislocation, extra intensity appeared in the upper-half of the gap. After subtracting the bulk and surface related signals, scattering due only to the single dislocation can be extracted. The result shows relatively uniform scattering in the upper half of the gap. A Fermi level position, based on calculation, is shown. Clearly, we obtain electronic structure information which is unique to the dislocation. But the statistical quality of the obtained signal is still poor. These results were obtained with a diode array detector of fairly crude design [10.8,10.9], and with a 0.5-nm probe size. With a sub-angstrom probe this kind of data will be striking and unavoidable.

Fig. 10.10 Application of the carbon K -edge interpretation to allow probing of single dislocations in diamond. In regions thin enough to allow detection of ingap signal from single dislocations, surface related scattering must be considered. It can be subtracted from the total scattering to reveal only the scattering due to the embedded dislocation.

Thus, spatially resolved EELS has the sensitivity to obtain in-gap defect scattering, provided care is taken to thoroughly understand the scattering from both the bulk and nearby surfaces. An excitonic analysis may not be necessary, but should at least be considered so as to gauge the extent of excitonic distortion. 10.7 SI L-EDGE EXCITONIC ANALYSIS Given the large distortion of the diamond results from the free electron expectation, we have to wonder whether the comparison with the DOS in Fig. 10.7 for Si is a valid exercise, in spite of the positive comparison with the P donor calculation. If the exciton screening ideas described here are valid, then this comparison should be good because the exciton distortion should be suppressed given the relative weakness of the effect in a well-screened system. In Fig. 10.11 the EELS and photo-yield results were plotted for Si. Then we extracted a DOS that is consistent with the data using the Wannier exciton analysis described above. This is accomplished by estimating a DOS, and then using it in expression 10.5 above to calculate an expected absorption shape. Figure 10.11 shows how the various pieces of the calculation affect the ﬁnal

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result. There are bound states that add a small peak to the very low edge of the data. There is a continuum enhancement, that multiplies the DOS to produce the continuum part of the edge. The result of this analysis compares well with two sets of measured data [10.15,10.35] This treatment is also successful using an identical DOS for comparison with the EELS results.

Fig. 10.11 Application of the exciton theory to the Si L-edge to allow an extraction of the s, d-projected DOS, which is consistent with both the photo-yield absorption data. The results support the ﬁnding that the measured Si data closely follow the DOS calculation. This analysis shows that the excitonic distortion is almost completely suppressed in the Si case.

This supports the suggestion above, that materials having high dielectric constants will give EELS core loss shapes that are close to the true DOS. It should be emphasized strongly that this complex analysis is necessary if electronic structure information is desired. As we have seen, structure on the scale of 0.2–0.5 eV can be signiﬁcantly distorted by excitonic effects. It is crucial, therefore, to estimate the expected amount of the distortion to allow interpretation. The exciton analysis gives a further prediction that has a direct impact on spectral interpretation for conﬁned volumes. Figure 10.12 shows an intensity plot of the exciton well potential for both the X-ray absorption and the EELS cases in Si, taken from [10.32]. The full-width at half-maximum of the envelope function for the exciton in each case is also shown. This has been obtained by an analytical solution of Elliott’s effective mass Schr¨odinger equation using an exciton well potential as given in the ﬁgure. The photon absorption case recovers Elliott’s analysis. The exciton in the EELS case is distorted by the cylindrically symmetric elongated well potential. In the photon absorption case, the orbit is spherical with about a 2 nm radius. In the EELS case, the orbit is grossly elongated in the direction antiparallel to the swift electron trajectory. In the limit of excitation to the conduction band continuum, this orbit becomes very long and thin. Standard scattering theory assumes that the excited continuum wave function is spherical and conﬁned close to the site of the excitation – an example is the multiple scattering EXAFS treatment. If the excited electron wave function envelope is elongated, we expect that it should interact with the top and bottom surface of the sample, producing damping or

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Fig. 10.12 Distortion of the excitonic orbital in the presence of the scattering electron for Si. Notice that it is elongated along the path of the electron. Thus the volume sampled is delocalized anisotropically.

smearing of the ﬁne structure. Figure 10.13 shows several Si spectra obtained as a function of thickness in a wedge shaped sample [10.36]. Clearly, the ﬁne structure is suppressed when the specimen becomes thin. This is of very practical signiﬁcance because the length scale over which this occurs is very long, of order 20 nm for the Si example. We therefore expect that core edge ﬁne structure obtained in very conﬁned volumes will be suppressed irrespective of the intrinsic band structure of the material. For example, we do not expect to observe bandstructure quantum conﬁnement to occur until the volume size becomes comparable to a unit cell of the material. But the core-loss ﬁne structure will show ﬁnite volume effects on a scale of several tens of unit cells. It is easy with the STEM to probe volumes that are much smaller than this. In the SiOx example above, the identiﬁcation of subnanometer crystalline Si islands relied on a spectral comparison not with thick Si but with spectra from Si islands embedded in SiO2 . These were large enough to show diffraction contrast, but small enough to display broadening due to their ﬁnite size. Thus, the observation of indistinct ﬁne structure is not, in itself, sufﬁcient to infer a noncrystalline behavior. However, knowing this, we can do a more detailed analysis, and we ﬁnd that observation of the shape at the level of 0.2 eV is sufﬁcient to tell between amorphous Si and nanometer-sized crystalline Si. We should next be concerned as to whether the ﬁne structure will be sensitive to surface in the lateral direction. Considering Fig. 10.12 again, we would guess not, because the excited electron wave function is narrow in the lateral direction.

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Fig. 10.13 Thickness dependence of the Si L-edge. The ﬁne structure is sensitive to the thickness in a manner consistent with the analysis based on the excitonic theory.

Experiments near cleaved edges of Si conﬁrm that this is the case. No smearing of the absorption structure is observed until the probe actually overlaps the physical edge of the sample. Thus, we are ﬁnally led to the picture that the information in the absorption edge is averaged over the specimen thickness within a region sharply deﬁned by the probe intensity. 10.7.1

An Example in Si Nano-Particles

A very practical example of this behavior can be seen in small Si particles [10.37], summarized in Figs. 10.14 and 10.15. The particles were formed by chemical decomposition of Si bearing gas. They were single crystal, and often clustered in chains, as shown in Fig. 10.14. Importantly, their surfaces were terminated with hydrogen to prevent the formation of a surface oxide. Their sizes ranged from about 20 ˚A up to of order 500 ˚A. Figure 10.15 summarizes the spectral results. At 88 ˚A, the absorption edge clearly retains bulk-like behavior. At 55 ˚A, this behavior is almost completely suppressed by the scattering physics, but no change in the onset energy of the edge is observed. Below 50 ˚A, the edge loses all resemblance to the bulk shape and begins shifting to higher energies, a signature of quantization of conduction band states within the particle. 10.8 BAND STRUCTURE INTERPRETATION OF SI ELNES It remains to establish that the Si 2p3/2 absorption is an accurate probe of the local s-,d-symmetry projected Density of States (LDOS). This was suggested by investigations of Gex Si1−x alloys, [10.38,10.39] which provide a continuous range of band

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Fig. 10.14 Annular dark ﬁeld image of a chain of elongated Si spheres. Fig. 10.15 EELS spectra from Si spheres of varying diameter. The spectra ﬁrst lose sharp features, and then shift due to quantization of CB states.

structure intermediate between Si and Ge, following very closely predictions using the “virtual crystal” approximation [10.40]. It became apparent that inﬂection points in the Si absorption edge shifted continuously as the alloy composition changed, providing evidence for a close relationship to the band structure. It is also interesting to compare the measured shape with the known Si band structure. In Fig. 10.16, the Si 2p3/2 absorption edge is plotted over the Si band structure [10.41]. It is obvious from the comparison that changes in absorption intensity align well with band edges, provided one assumes that the ﬁrst absorption edge coincides with the band edge at ∆1 . In the past, theoretical results did not help to relate the experimental structure with the bandstructure, because the calculations identiﬁed LDOS having particular angular momentum (s-,d-symmetry), but did not distinguish by location within the Brillouin zone (BZ). In order to go forward, it became necessary to model the LDOS so that the data could be parameterized. This method picks out 3–4 important conduction bands having known symmetry and non-zero overlap with s-,d-atomic orbitals and then estimates their DOS contribution using simple shapes. A threedimensional parabolic band will therefore give a DOS having a shape proportional to k = (2me ∗ /¯h2 )1/2 (E − ECB )1/2 , where k is the electron momentum and me is the electron effective mass [10.42]. A decomposition of the absorption edge using this idea is shown in Fig. 10.17. There are contributions from ∆1 , L1 , L3 , Γ1,5 , and Γ2 . Motivation for the choice of which bands to include is provided by optical dipole selection rules, limiting contributions to s-,d-like ﬁnal states for transitions from the 2p core level. The symmetry of a wave function identiﬁed with a particular position in the BZ is well known from group theory [10.43]. Whether or not a BZ critical point includes s-,d-like symmetries has been cataloged [10.44,10.45]. From this

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Fig. 10.16 Si 2p3/2 →CB experimental intensity compared with the accepted Si band structure. Inﬂection points in the data line up with dipole allowed regions having large effective mass.

analysis, we ﬁnd that the CB zone center for Si is mostly p-like and so does not contribute to the spectral results, leaving three prominent shapes (∆1 , L1 , and L3 ) which comprise most of the model LDOS in Fig. 10.17. Next, it must be also realized that the bands are conﬁned below an upper energy. If the band remains dipole allowed throughout its range, then the upper edge also has an E 1/2 dependence. Otherwise it goes to zero with a ﬁnite slope that can be small. Thus, the ∆1 contribution is s-like and has E 1/2 dependence at the CB edge and at some maximum energy. L3 is interesting, because it is host to a nearly ﬂat saddle-point, and so has a shape that varies like E −1/2 . These shapes are all straightforward extractions from simple BZ constructions [10.46]. Finally, the modeled LDOS can be used in the core excitonic scattering theory above to include modest edge distortion, core hole lifetime damping, and an instrumental resolution. The solid line ﬁt to the measured data in Fig. 10.17 is the result [10.32]. This approach can be validated by a theoretical calculation that partitions the local DOS by symmetry (s, d, p, d,T etc.), and by k-point, or, position within the BZ. This has been accomplished and is summarized in Fig. 10.18 [10.30]. This particular result does not include consideration of the core hole interaction, so the precise shapes of the contributions are certainly not ﬁnal. However, ∆1 and L clearly make separable contributions to the ﬁnal spectrum. Also, the shapes of the calculated contributions are not grossly different from the model shapes. As shown in Fig. 10.7, a better correspondence to the details of the onset can be made using a P donor to approximate a core-hole interaction.

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Fig. 10.17 Si 2p3/2 intensity from three regions, compared with the sum of model band shapes. Contributions from dipole allowed (∆1 , L1 , L3 , Γ1,5 ) and dipole forbidden (Γ2 ) contributions are present.

10.8.1

Fig. 10.18 Symmetry- and k-point partitioned DOS for Si. It can be seen that ∆1 and L produce distinct features that correspond to features in Fig. 10.17. Compare with Fig. 10.7.

Gex Si1−x Alloy System

These ideas can be illustrated in practice by using the Gex Si1−x alloys grown in thin buffer layers for strain relief under GeSi multilayer structures [10.38]. The samples used for these measurements were grown by stoichiometric co-evaporation of Si and Ge onto Si(100) substrates in a molecular beam epitaxy (MBE) chamber. Each sample consisted of many distinct alloy layers with speciﬁed Ge content. Twenty one different alloy compositions were available, that is, 0–100% in 5% increments. The thickness of the different alloy layers (5–200 nm) and their composition were deliberately chosen to produce epitaxial alloys. Cross-sections of the samples were prepared by mechanical thinning followed by low temperature ion milling. STEM images showed well-deﬁned, smooth interfaces between the layers. Figure 10.19 shows background stripped Si 2p3/2 → CB loss spectra for alloys containing 0–95% Ge. The inﬂection point at the absorption edge absorption onset was measured to be 99.86±0.02 eV for Si, rising smoothly to 100.16 eV for the 95% Ge alloy. The dots are 2p3/2 edges obtained in the various alloys, and modeled as described above. The solid line ﬁts through the data were obtained from an EELS exciton analysis using the JDOS shown below each data set. These ﬁts show several clear trends. First, the peak L3 and the onset ∆1 largely remain separated by a constant amount. Thus, they shift together under the inﬂuence of changes in lattice potential caused by the presence of Ge. This behavior is predicted by band structure calculations, and is probably a result of a common symmetry element between ∆1 and L3 . For instance, the wave functions associated

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Fig. 10.19 Set of measured 2p3/2 core absorption data (dots) compared with excitonic ﬁts (lines through data) based on the shown JDOS (lines below each data set). Movement of conduction band critical points can be inferred from the ﬁts. This JDOS is comprised of simple analytical expressions for the expected JDOS associated with three dipole allowed ﬁnal states: the largely s-like ∆1 and L1 band minima, and the largely d-like L3 saddle point. The exciton analysis was applied separately to the ∆1 , L1 , and L3 contributions. As the Ge content was smoothly varied, the positions of the critical points (noted in the ﬁgure) were adjusted to ﬁt the measured data.

with these points are symmetric with respect to reﬂection through the x–z plane. The position of L1 , however, rapidly shifts downward relative to ∆1 as the Ge content is increased. Around 85%, it is coincident with ∆1 . Above 85%, L1 has moved below ∆1 and we can see that the absorption edge onset begins to move downward. In addition to the movement of L1 , we can infer a movement of Γ2 , because the shape of the absorption in the 0–2-eV range cannot be reproduced otherwise. Finally, we must also include a peak associated with the Ge-like saddle point in the Λ1 symmetry

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line. This saddle point is either slightly above, or coincident with, Γ1,5 in Si. As the Ge content is increased, Γ2 shifts rapidly downwards and pulls the Λ1 saddle point with it. Details of the absorption shape between 0.5 and 2.5 eV are thus sensitive to the precise positions of Γ2 , Λ1 , and L1 . 10.8.2

Application to Misﬁt Dislocation Structure

One can legitimately ask whether these types of measurements are possible using atomic resolution. The main difﬁculty is experimental: The statistical accuracy needed to do this analysis is very high. Each of the spectra in Fig. 10.19 were a sum of about ten 30 s long exposures. Maintaining the probe on one atom location during this time is very difﬁcult. Second, the 2 ˚A probe size used in these studies is a little big for true atomic resolution. The best that can be claimed is that it is possible that data can be obtained from an area as small as, say, 4 ˚A in diameter. This introduces some uncertainty when a truely atomic level structure is addressed. Figure 10.20 shows an ADF image of a 60◦ misﬁt dislocation at the substrate interface of a Ge30 Si70 /Si/Ge30 Si70 quantum well structure. This structure was deliberately grown to induce the injection of misﬁt dislocations to measure their effect on electron mobility in the quantum well and has been described in detail elsewhere [10.47,10.48]. The dislocation is dissociated into three parts: (1) a 30◦ partial dislocation at the interface, (2) a 10 unit long intrinsic stacking fault (ISF), and (3) a 90◦ partial dislocation embedded in the GeSi alloy. On either side of the ISF, there are regions of strain – tension on the upper left and compression on the lower right. Figure 10.21 summarizes results for Si 2p3/2 absorption for six different regions: (a) bulk relaxed, (b) in tension, (c) in compression, (d) at the mid-point of the ISF, (e) at the core of the 30◦ partial, and (f) at the core of the 90◦ partial. In the regions of tension and compression, the most notable change is to shift the spectra lower or higher in energy. A secondary difference is an apparent broadening of the L1 contribution in the compressed region. In order to understand these results, we need to consider two kinds of effects. First, volume strain shifts both the core level and the CB levels, but the sign of the shift depends on the symmetry of the electron states. For instance, bonding orbitals shift downwards in energy for negative strain: They become more energetically favorable at smaller interatomic distances. The ∆1 CB of Si and the SiGe alloys also has bonding character and therefore also shifts to lower energies when the lattice constant is made smaller [10.39]. The L1 CB, on the other hand, shifts upward with smaller lattice constant, more in keeping with the behavior of antibonding symmetry. The second effect of strain is to change the shape of each band contribution for nonisotropic strains. An example of this is the biaxial strain splitting of the ∆1 CB within the strained Si well where a twofold piece of ∆1 shifts downward, forming the high mobility conduction channel, while the lower mobility fourfold piece shifts upward. The second obvious type of change is the apparent splitting of the L1 peak at the ISF and at the 30◦ partial dislocation. This one is very interesting. It is well known that the ISF consists of a rotation through 180◦ about the a [111] type direction. In doing this, some Si third neighbors are rotated into second neighbor positions relative

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Fig. 10.20 ADF image of a dissociated 60◦ dislocation at the substrate interface of a 15 nm thick Si quantum well. EELS spectra were acquired at the positions (a–f). The 30◦ and 90◦ partial dislocations are clearly located, separated by a 10-unit intrinsic stacking fault.

Fig. 10.21 Si 2p3/2 spectra obtained from the various positions in Fig. 10.20. Many mechanisms are operating here: strain on either side of the ISF push the edge up in compression (c) or down in tension (b). At the ISF the L1 contribution is split into two pieces (d). At the partial dislocations, intensity extends below the edge into the gap (e,f). Finally, the L1 splitting is present at the 30◦ partial dislocation but not at the 90◦ partial.

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to an atom at the ISF. This places orbitals having L1 symmetry within interaction range. The result is a bonding/antibonding splitting, apparent in the spectra [10.48]. A close inspection of the most likely model for the 30◦ partial dislocation reveals that a similar interaction occurs there. The spectra, therefore, provide evidence for likely structural models in the absence of direct imaging data. The absence of this splitting at the 90◦ partial dislocation indicates that the third neighbor behavior there is similar to the bulk material. This rules out the standard 5–7 atom ring structure for the core of the dislocation. A more recent model, that includes many kinks in the core, appears to provide third neighbor interactions more in line with those inferred from this data. Finally, the clear evidence for in-gap intensity at both dislocation cores argues for small distortions in the bonds that pull states out of the CB. Full one-electron states in the gap are not likely, since reconstruction of the core is expected to clear this type of state out of the gap. What remains are small shifts caused by bond rotation as the frustrated lattice near the defect relaxes into a low-energy equilibrium structure. 10.9 SPECIALIZED EQUIPMENT We require EELS information, which relates to electrical properties of electronic devices, we must deal with the relevant spatial and energy scales. As was apparent above with the misﬁt dislocation, the interesting regions may exist only within 0– 2 ˚A of an interface, a surface or a defect. The spatial resolution requirement can only be met in a ﬁeld emission equipped machine. The work described here uses a VG Microscopes HB501 STEM that routinely produced a probe size of 0.5 nm as delivered in 1983. In the 1990s new pole-piece designs brought the probe size to 2 ˚A at 120 kV. Recently, the IBM instrument has been ﬁtted with spherical aberration correction optics to bring the system probe size to below 1 ˚A[10.7]. Figure 10.22 shows an example of the present system imaging capability.

Fig. 10.22 High-resolution ADF image of a thick GeSi alloy using spherical aberration in the IBM STEM at 120kV. The “dumbbell” structures are clearly imaged.

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Energy loss equipment that has the relevant resolution and stability has started to become available commercially, but the work described here was performed with a high-resolution spectrometer constructed at IBM. As shown in Fig. 10.23, it consists of a Wien ﬁlter spectrometer mounted within a high-voltage electrode situated at the top of the HB501 column. Electrons exiting the specimen are transferred to the spectrometer environment by a doublet quadrupole lens, they are decelerated, on entering the high voltage electrode, to about 70 eV for energy analysis. Thus, the quality of the spectrometer is relatively modest — 0.07/70 = 10−4 . After analysis, the electrons are reaccelerated to 120 keV and strike a thin YAG screen that is mounted on a ﬁber optic bundle. The bundle, in turn, carries the spectral information out to a CCD array. Because the deceleration voltage is derived from the microscope, instabilities in the gun voltage are canceled within the spectrometer electrode. An absolute calibration of the energy axis is obtained by applying a known voltage between the gun and spectrometer electrode. This calibration has recently been improved to about 5 meV. The CCD array is utilized as a multiple slit detector. When the energy spectrum is swept across the array, we generate many similar spectra, shifted by the physical separation of each CCD pixel. These spectra are then shifted to a common energy axis and averaged for the ﬁnal result. This is an important operation, because the differences that are relevant to the electronic structure are small. For instance, it has been possible with this system to detect the signal from 0.5 wt% carbon in YBa2 Cu3 O7−δ . The signal amounted to less than 1% above the background at 300 eV [10.49]. The array averaging process allows us to eliminate channel to channel gain and background variations. In addition, the scanning process allows us to calibrate the energy axis, converting energy axis distortions, at the array, into a slightly degraded energy resolution. A major weakness of this operation is that many read operations are required, leading to signiﬁcant noise problems. It was apparent above that, even with a 0.2 eV resolution, the current instrument cannot do routine work on direct interband transitions. This major weakness means that the important valence band information is lacking. The IBM instrument, therefore, is also being improved to include an electron monochromator in the gun [10.50]. The major difﬁculty here is that the monochromator must not degrade the brightness of the electron source any more than that implied by the energy resolution change. We believe that this problem is solved with the present design, proven by a direct measure of the monochromated brightness. Figure 10.24 shows the improvement in energy resolution that the monochromator gives. The instrumental limit will be improved from the present 0.2 eV, with deconvolution, to about 80 meV without deconvolution calculations. 10.10 CONCLUSIONS This chapter has covered details that are needed to begin the task of extracting electronic structure and bonding information from Si related semiconductor systems. Electronic structure information due to strain, bonding defects, and composition

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Fig. 10.23 Line diagram of the decelerating Wien ﬁlter. This ﬁlter is located within a high-voltage electrode that is held about 70 V from the gun potential. Electrons are thus decelerated to 70 eV for energy analysis.

variation, is sometimes subtle, requiring very high performance. However, the required performance is clearly within reasonable expectations. Bonding information is readily available with today’s equipment. For this use, most of this chapter provides conﬁdence that the basis for interpretation — which must include ideas such as core excitons, scattering physics, and electronic structure distortion — is adequately developed for Si-based systems. For the future, higher resolution systems, both in spatial and in energy resolution, are certain to bring out new detail. It will be interesting to see how well current understanding can handle the new information.

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Fig. 10.24 The performance of an electron monochromator for the IBM STEM compared with that obtained without the monochromator. A resolution of 80 meV has been obtained.

REFERENCES 10.1. D. Muller, T. Sorsch, S. Moccio, F. Baumann, K. Evans-Lutterodt, and G. Timp, The electronic structure at the atomic scale of ultrathin gate oxides, Nature 399, 758 (1999). 10.2. E. Solodov, V. Kanin, A. Trifanov, D. Presnov, S. Yakovenko, G. Khomutov, C. Gubin, and V. Kolesov, Single electron transistor based on a single cluster molecule at room temperature, JETP Lett. (USA) 64, 556 (1996). 10.3. A. Crewe, M. Isaacson, and D. Johnson, A high resolution electron spectrometer for use in transmission scanning microscopy, Rev. Sci. Inst. 42, 411 (1971). 10.4. P. Batson, STEM, In K. Tu and R. Rosenberg, editors, Treatise on materials science and technology volume 27 pp. 337–384. Academic Press (1988). 10.5. S. Pennycook and D. Jesson, High resolution incoherent imaging of crystals, Phys. Rev. Lett. 64, 938 (1990). 10.6. D. Shin, E. Kirkland, and J. Silcox, Annular dark ﬁeld electron microscope images with better than 2˚A resolution at 100 keV, Appl. Phys. Lett. 55, 2456 (1989). 10.7. N. Dellby, O. Krivanek, P. Nellist, P. Batson, and A. Lupini, Progress in aberration-corrected scanning transmission electron microscopy, J. Electron Microscopy 50, 177 (2001). 10.8. P. Batson, High resolution electron energy loss spectrometer for the scanning transmission electron microscope, Rev. Sci. Inst. 57, 43 (1986).

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10.9. P. Batson, Parallel detection for high resolution electron energy loss studies in the scanning transmission electron microscope, Rev. Sci. Inst. 59, 1132 (1988). 10.10. J. Morar and P. Batson, Measurements of critical point energies in the conduction bandstructure of Gex Si1−x , J. Vac. Sci. Tech. 10, 2022 (1992). 10.11. M. Inokuti, Inelastic collisions of fast charged particles with atoms and molecules - the bethe theory revisited, Rev. Mod. Phys. 43, 297 (1971). 10.12. D. Pines and P. Nozieres, The theory of quantum liquids, W.A. Benjamin New York (1966). 10.13. R. Egerton, Electron energy loss spectroscopy in the electron microscope, Plenum Press New York (1986). 10.14. P. Batson, K. Kavanagh, J. Woodall, and J. Mayer, Observation of defect electronic states associated with misﬁt dislocations at the gaas/gainas interface, Phys. Rev. Lett. 57, 2729 (1986). 10.15. F. Brown, R. Bachrach, and M. Skibowski, L2,3 threshold spectra of doped silicon and silicon compounds, Phys. Rev. B 15, 4781 (1977). 10.16. R. Elliott, Intensity of optical absorption by excitons, Phys. Rev. 108, 1384 (1957). 10.17. J. Bruley, L. Brown, and S. Berger, A study of the electronic structure of diamond by stem, Inst. Phys. Conf. Ser. 78, 561 (1985). 10.18. B. Rafferty and L. Brown, Direct and indirect transitions in the region of the band gap using electron energy loss spectroscopy, Phys. Rev. B 58, 10326 (1998). 10.19. M. Sturge, Optical absorption of gallium arsenide between 0.6 and 2.75 eV, Phys. Rev. 127, 768 (1962). 10.20. U. Bangert, A. Harvey, J. Davidson, R. Keyse, and C. Dieker, Correlation between microstructure and localized band gap of GaN grown on SiC, J. Appl. Phys. 83, 7726 (1998). 10.21. P. Batson, Spatially resolved interband spectroscopy, In Johari, editor, Proc. 5th Pfefferkorn Conference, pp. 189–195 Chicago (1987). Scanning Microscopy International. 10.22. L. Dori, J. Bruley, D. DiMaria, P. Batson, J. Tornello, and M. Arienzo, Thinoxide dual-injector annealing studies using conductivity and electron energy-loss spectroscopy, J. Appl. Phys. 69, 2317 (1991). 10.23. A. Bianconi, Core excitons and inner well resonances in surface soft x-ray absorption (SSXA) spectra, Surface Science 89, 41 (1979).

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10.24. J. Neaton, D. Muller, and N. Ashcroft, Electronic properties of the si/sio2 interface from ﬁrst principles, Phys. Rev. Lett. 85, 1298 (2000). 10.25. G. Hollinger and F. Himpsel, Probing the transition layer at the SiO2 -Si interface using core level photoemission, Apply. Phys. Lett. 44, 93 (1984). 10.26. G. Harp, Z. Han, and B. Tonner, Spatially resolved x-ray absorption near edge spectroscopy of silicon in thin silicon dioxide ﬁlms, Phys. Scr. T 31, 23 (1989). 10.27. X. Weng, P. Rez, and P. Batson, Single electron calculations for the Si L2,3 near edge structure, Solid State Commun. 74, 1013 (1990). 10.28. P. Batson, D. Johnson, and J. Spence, Resolution enhancement by deconvolution using a ﬁeld emission source in electron energy loss spectroscopy, Ultramicroscopy 41, 137 (1992). 10.29. R. Buczko, G. Duscher, S. J. Pennycook, S. T. Pantelides Excitonic effects in core-excitation spectra of semiconductors, Phys. Rev. Lett. 85 2168 (2000). 10.30. P. Batson and J. Neaton, Symmetry and k-position partitioned si dos, unpublished (2002). 10.31. P. Batson and J. Bruley, Dynamic screening of the core exciton by the swift electron in electron energy loss scattering, Phys. Rev. Lett. 67, 350 (1991). 10.32. P. Batson, Distortion of the core exciton by the swift electron in electron energy loss scattering, Phys. Rev. B 47, 6898 (1993). 10.33. X. Weng, P. Rez, and H. Ma, Carbon k-shell near edge structure: Multiple scattering and band theory calculations, Phys. Rev. B 40, 4175 (1989). 10.34. J. Bruley and P. Batson, Electron energy loss studies of dislocations in diamond, Phys. Rev. B 40, 9888 (1989). 10.35. W. Eberhardt, G. Kalkoffen, C. Kunz, D. Aspnes, and M. Cardona, Photoemission studies of 2p core levels of pure and heavily doped silicon, Phys. Stat. Sol. (b) 88, 135 (1978). 10.36. P. Batson, Silicon L2,3 near edge ﬁne structure in conﬁned volumes, Ultramicroscopy 50, 1 (1993). 10.37. P. Batson and J. Heath, EELS of single silicon nanocrystals: The conduction band, Phys. Rev. Lett. 71, 911 (1993). 10.38. P. Batson and J. Morar, Conduction bandstructure of Gex Si1−x using spatially resolved electron energy loss scattering, Appl. Phys. Lett. 59, 3285 (1991). 10.39. J. Morar, P. Batson, and J. Tersoff, Heterojunction band lineups in Si-Ge alloys using spatially resolved electron energy loss spectroscopy, Phys. Rev. B 47, 4107 (1993).

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10.40. C. Van de Walle, Band line ups and deformation potentials in the model-solid theory, Phys. Rev. B 39, 1871 (1989). 10.41. J. Chelikowsky and M. Cohen, Nonlocal pseudopotential calculations for the electronic structure of eleven diamond and zinc-blend semiconductors, Phys. Rev. B 14, 556 (1976). 10.42. P. Batson, Atomic resolution electronic structure in silicon-germanium alloys, J. Electron Microsc. 45, 51 (1996). 10.43. V. Heine, Group Theory, McGraw Hill New York (1960). 10.44. W. Eberhardt and F. Himpsel, Dipole selection rules for optical transitions in fcc and bcc lattices, Phys. Rev. B 21, 5572 (1980). 10.45. W. Eberhardt and F. Himpsel, Erratum: Dipole selection rules for optical transitions in fcc and bcc lattices, Phys. Rev. B 23, 5650 (1980). 10.46. N. Ashcroft and N. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York (1976). 10.47. P. Batson, Atomic and electronic structure of a dissociated 60◦ misﬁt dislocation, Phys. Rev. Lett. 83, 4409 (1999). 10.48. P. Batson, Structural and electronic characterization of a dissociated 60◦ dislocation in gesi, Phys. Rev. B 61, 16633 (2000). 10.49. T. Shaw, D. Dimos, P. Batson, A. Schrott, D. Clarke, and P. Duncombe, Carbon retention in YBa2 Cu3 O7−δ and its effect on the superconductivity transition, J. of Mat. Res. 5, 1176 (1990). 10.50. H. Mook, P. Batson, and P. Kruit, Operation of the fringe ﬁeld monochromator in ﬁeld emission STEM, In: Electron Microscopy and Analysis 1999 volume 161, pp. 223–226, C. Kiely, editor, Institute of Physics, Bristol (1999).

11

Electron Energy Loss Spectroscopy of Magnetic Materials Jennifer A. Dooley Jet Propulsion Laboratory Pasadena, California 91109

Abstract The application of EELS to studies of magnetic materials will be discussed in this chapter, with special attention given to techniques involving imaging EELS. In many cases, the applications will be described in the context of case studies that make use of these techniques in conjunction with complimentary analysis methods. Examples include the study of nanostructured materials prepared by carbon arc, and conﬁrmation of density of states (DOS) calculations for amorphous and crystalline transition metal-boron (TM–B) alloys. In addition to the discussion of analytical techniques for the determination of chemical information as applied to magnetic materials, new techniques under development that directly measure the magnetic characteristics on a ﬁne scale will be addressed. One of the initial applications to magnetic materials was the use of L-edge electron energy loss spectroscopy as a direct probe of the d-band occupancy of transition metal alloys and examples will be given here. The feasibility of performing magnetic linear dichroism (MLD) and magnetic circular dichroism (MCD) experiments by means of EELS will be discussed. Finally, a method for energy ﬁltered Lorentz microscopy at high spatial resolutions as applied to rare earth–transition metal (RE–TM) intermetallics will be discussed. 11.1 INTRODUCTION Virtually all engineering parameters of magnetic materials (coercivity, magnetostrictive coefﬁcients, . . . ) can be strongly affected by chemical and microstructural features. Transmission electron microscopy (TEM) has been a standard tool for the determination of microstructure and phase information for some time. Used in con385

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junction with energy dispersive X-ray spectroscopy (EDXS), which can record the integrated intensities of the X-ray emission from the specimen, it has been possible to study chemical composition at high resolution. By making use of standard Lorentz microscopy techniques, this information may be correlated with the magnetic domain structure of a material [11.1]. As the length scale of modern electronic and magnetic devices continues to decrease, it must be accompanied by improvements in the observation techniques used to study these micro- and nanostructures. The advent of EELS detectors made it possible to study light elements (Z < 11) in the analytic electron microscope which at the time were unaccessible by EDXS [11.2]. Since then EDXS detectors have reduced the lower limit of detectable X-ray energies to allow the detection of C, N, and O [11.3]. EELS retains advantages, however, such as the information contained in the ﬁne structure of the edges, which is related to bonding and electronic structure and its greater energy resolution. An additional advantage of EELS is extremely large collection efﬁciencies since the inelastically scattered electrons tend to have small scattering angles [11.1]. The introduction of the Imaging EELS detectors has made it possible to produce energy ﬁltered images with high resolution. The application of EELS and Imaging EELS to studies on magnetic materials has proven extremely useful both to obtain chemical and microstructural information as well as to study directly the magnetic properties of the material. 11.2 CHEMICAL AND MICROSTRUCTURAL INFORMATION 11.2.1

Nanoparticles

EELS is very useful as a tool for identifying phases and local chemical concentrations and one application has been to the study of nanoparticles, that is particles on the order of 2–1000 nm in size. Nanoparticle ferromagnets have become of great practical interest due to the dependence of the magnetic properties on the size of the particles. For instance, the coercivity of recrystallized amorphous magnetic materials has been shown to vary greatly with the grain size distribution [11.4]. As the grain size decreases, the coercivity increases until a critical size is reached, and then the coercivity decreases rapidly. Ferromagnetic materials exhibit short range exchange forces that act to align neighboring spins parallel to one another. Magnetostatics dominate at larger distances so that domains form spontaneously in the bulk to minimize the total external ﬁeld. Below the monodomain threshold it is energetically favorable for the particle to contain a single domain and such particles are said to be superparamagnetic [11.5]. The alignment of the particle moment with an external magnetic ﬁeld occurs by rotation of the collection of moments as opposed to domain wall motion. Consequently, there has been much interest in producing very hard and very soft magnetic materials by preparing nanoscale particles of, for example SmCo5 and FeCo, respectively, for use in composite materials or as precursors to bulk sintered material [11.6].

CHEMICAL AND MICROSTRUCTURAL INFORMATION

387

Kirkpatrick et al. have used a modiﬁed Huffman–Kratchmer carbon arc process to produce carbon coated TM and RE–TM nanoparticles with a variety of compositions [11.7]. The phases produced depend on the preparation conditions and may be metastable. Included in their studies are Sm-Co-C, Fe-Co-C, and Nd-Fe-B-C, all of which are coated with graphitic shells that form by virtue of the phase segregation that occurs during cooling [11.8]. Although these carbon shells can protect the nanoparticles from environmental degradation, oxidation of the alloys remains problematic. Imaging EELS which combines high-resolution TEM and electron energy loss spectroscopy, also referred to as electron spectroscopic imaging (ESI), discussed in detail in Chapter 6, has been employed in these studies to determine the nanoparticle composition and homogeneity, as well as to characterize the oxygen content. Brieﬂy, high-resolution elemental maps of the sample are generated by combining energy-selected images above and below the core-loss ionization edge for the element. Energy-selected images are acquired at two pre-edge energies and are ﬁt to a power law model that describes the background intensity near the edge region. The model is then used to calculate the background image intensity at an energy just above the ionization energy. This background is then subtracted from an energy selected post-edge image to produce an elemental map. To a good approximation the intensity in this map then results only from the core-loss scattering events of the selected element. This method of acquiring elemental maps is often referred to as the three window method. Performed for example on a JOEL 4000EX with a Gatan Imaging Filter attachment it allows one to obtain high resolution elemental maps. Chemical mapping has been used in conjunction with X-ray diffraction (XRD) to determine the phases present and to look at the chemical homogeneity of the nanoparticles. Nanoparticles of Fex Co1−x alloys were prepared for various compositions (x = 0.0, 0.2, 0.4, 0.5, 0.6, and 0.8) [11.9]. FeCo alloys have high permeability, large saturation induction, and high Curie temperature and have been used in industrial soft magnet applications for some time. The need for soft magnets suitable for use at elevated temperatures has led to the investigation of nanoparticulate FeCo. While amorphous materials have no magnetocrystalline anisotropy, they are unstable and may easily recrystallize. In contrast FeCo nanoparticles are crystalline with low magnetocrystalline anisotropy. Exchange forces decrease the coercivity in nanocrystalline soft magnets because the exchange length is much greater than the grain size and the magnetic properties are averaged over many grains. FeCo magnetic nanoparticles prepared by their carbon arc method were all shown to be disordered bcc phase by XRD, except for the x = 0.2 alloy for which some fcc Co was seen, with a typical particulate size of 200 nm. ESI indicates that the distributions of Fe and Co within the nanoparticles are homogeneous as shown in Fig. 11.1. The ﬁgure contains four images, including conventional bright ﬁeld TEM, an elemental Co, C, and O maps. The elemental Fe map was not included as it was very similar to the Co map, which indicated that the particles were homogeneous. There are some mapping artifacts in these images which are seen most clearly for the elemental carbon map. In particular, some of the edge intensity in the maps probably arises from drift that occurred between acquisition of the energy-selected images. It was not possible to correct this drift completely since the particles actually moved relative to each other during the

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experiment as opposed to a uniform translation vector, due possibly to charging of the particles. The magnetization of the separated nanoparticles was measured with a SQUID magnetometer for various compositions, and show large magnetizations with the predicted dependence on composition.

Fig. 11.1 GIF images of Fe–Co nanoparticles indicating a homogeneous distribution of Fe and Co. Left frame shows a bright ﬁeld image. Middle and right are Fe and Co maps, respectively.

In a related study SmCo particles were prepared by the same carbon arc method in a helium atmosphere. The resulting soot was scraped from the walls of the reaction vessel, sonicated in methanol, and drop mounted on a carbon coated TEM grid. In the initial studies, it was possible to determine by XRD that several phases were present including SmCo5 , Sm5 Co2 , fcc Co, and graphite [11.7]. Phases with greater Sm content were likely missing due to the much higher partial pressure of the Sm. XRD was used to identify phases and EDS was used to verify that the ratio of Sm to Co varied from region to region, but because the electron beam was large enough to excite atoms in multiple nanocrystals simultaneously, it could not be used to verify individual phases found by XRD. With imaging EELS however, phase separation on the nanometer scale may be detected. Shown in Fig. 11.2 are an elastic-only bright ﬁeld TEM image and a composite elemental map of a Sm-Co-C samples produced by carbon-arc evaporation. The maps were made using the three-window method. The contrast and color saturation in the map have been stretched to improve visibility, but this has made the image nonquantitative. The red channel is the histogram equalized elemental Sm map, the blue channel is the histogram equalized elemental Co map, and the green channel is the histogram equalized elemental C map. Kauffeldt et al., have used EELS to determine the chemical nanostructure of Fe particles ∼ 100 nm in size that were produced using a condensation technique [11.10]. The FeCO5 is heated to between 200 and 600 ◦ C with pure nitrogen ﬂowing across the surface of the liquid. Fe vapor is formed when the compound decomposes and condenses into Fe particles. The size distribution is determined by the partial pressure of FeCO5 , and when very low, minimizes agglomeration of the particles. The size distribution in this case is centered around 20 nm, small enough so that each particle has a single domain. A STEM with 0.5-nm probe diameter and equipped with EELS and nanodiffraction capabilities was used to study phase and composition

CHEMICAL AND MICROSTRUCTURAL INFORMATION

389

Fig. 11.2 GIF images of SmCoC.

of individual particles, which tend to form into chains. The diffraction patterns obtained were consistent with single crystals of bcc type, as were diffraction ring patterns taken with a broader electron probe of an agglomeration of particles. For chemical analysis EELS spectra were obtained from the core and surface regions of individual particles. The core regions exhibited strong Fe peaks with very weak oxygen peak, excluding the possibility of compositions such as Fe2 O3 . In addition the white lines in the ﬁne structure of the Fe peak were seen to be very weak, which is consistent with having minimal oxygen content. The white lines are much stronger for Fe oxides than for pure Fe. At the surface of the particles the oxygen content is seen to be much higher, and correspondingly the white lines appear stronger. EELS has been used to study the elemental composition and the volume fraction of superparamagnetic γ-Fe2 O3 nanoparticles dispersed in a silica gel matrix [11.11]. If a magnetic ﬁeld is applied during solidiﬁcation, it acts to produce a magnetic texture. This study sought to determine the bulk magnetic properties of such composite materials and compare one solidiﬁed with an in-plane ﬁeld and one with no ﬁeld applied during solidiﬁcation. Conventional TEM was used to determine the particle size and shape distribution and also to verify the separation. In order to determine particle elemental composition accurately, EELS and EDXS were performed simultaneously using a 10-nm diameter electron probe. Over larger areas EELS and EDXS were used to determine the volume fraction of particulate. EDXS was calibrated using the OK , SiK , and FK characteristic signals from standard samples of SiO2 , Fe2 O3 , and Fe2 SiO4 . EELS was calibrated using the same standards by considering the Si L-,O K-, and Fe L-edges. The thickness of the specimen as estimated by EELS and the experimental value of the mean free path were used in the photon absorption correction made for the EDXS. This study illustrates the close agreement of the two methods, as shown in Table 11.1. The EELS ﬁne structure was characteristic of SiO2 and γ-Fe2 O3 , which provided additional evidence. The particles were seen to be dispersed dilutely enough to avoid the agglomeration of particles or the formation of chains and to warrant their treatment as magnetically isolated. The TEM analysis allows the determination of microstructural and morphological information that is essential for interpretation of

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Element

EELS

EDXS

Si O Fe

21.5 ± 1.0 62.0 ± 3.0 16.5 ± 1.2

21.9 ± 0.6 62.3 ± 5.0 15.8 ± 0.5

Table 11.1 Comparison of the elemental compositions in the composite material in at% measured by EELS and by EDXS [11.11] .

magnetic properties of the composite. This information includes particle size, shape and distribution, prevalence of agglomerates, as well as the volume fraction. 11.2.2

Thin Films

Thin-ﬁlm magnets lend themselves to study by EELS as well. For example, AgCo thin ﬁlms thought to consist of nanoparticles of Co in a metallic Ag matrix and which exhibit giant magnetoresistance were studied by Scott and Majetich [11.12]. Magnetoresistance is a dependence of the electrical resistivity on an externally applied magnetic ﬁeld. ESI revealed that larger particles, on the order of 50 nm, were Ag rich. But as the spatial resolution of the ESI was limited to 1–2 nm, it was necessary to couple ESI with high resolution TEM to determine whether ﬁne Co nanocrystals were present within the matrix. Instead of well deﬁned Co nanocrystals within a matrix, regions of crystallinity with continuously varying d-spacings were observed. This suggests that the if Co clusters exist they are not sharply segregated and interdiffusion remains. Another study of thin-ﬁlm materials exhibiting giant magnetoresistance, Cr/Fe multilayers, combined EELS, EDXS and conventional TEM. The magnetoresistive behavior of the multilayers depends heavily on the layer thickness and the character of the boundary between layers. The multilayers studied by Rennekamp et al., were very thin with typical layer thicknesses between 5 and 10 nm in stacks of either 20 or 40 double layers [11.13]. Cross-sectional specimens were prepared with thicknesses less than 50 nm, to minimize multiple scattering. They were able to observe both the L- and M -edges of Cr and Fe. One hundred twenty eight EELS spectra were taken along a line parallel to the multilayer ﬁlm normal, with a probe diameter of 0.7 nm and a step size of 0.32 nm. Since the Fe and the Cr L-edges are at 708 and 574 eV energy loss, respectively, it is necessary to remove the background from the Cr-edge from the Fe spectra. It is possible to see a difference of only about 1 nm in the thickness of the Cr as opposed to the Fe layers. The lack of scattering contrast due to the similar Z numbers of the elements was problematic, however, the defects were imaged with strong contrast in chemical maps made by selecting the M -edge of either element that retain the Bragg contrast in the plasmon losses.

DETERMINATION OF DOS

11.3

391

DETERMINATION OF DOS

Another common application of EELS, used in conjunction with other spectroscopic methods such as ultraviolet photoelectron spectroscopy (UPS), X-ray photoelectron spectroscopy (XPS), and Auger electron spectroscopy (AES), has been conﬁrmation of calculations of the electronic structure and the associated magnetic properties. These techniques are extremely sensitive to both the chemical and electronic structure in a thin layer at the surface of the material. In XPS and UPS, relatively low energy (< 2 keV) X-ray and ultraviolet radiation, respectively, are used to excite electron emission from the specimen [11.2]. The measured electron energy is then the difference between the binding energy and the energy of the incident photon. Both spectroscopies are a direct measure of the occupied density of states below the Fermi level [11.14], in contrast to EELS, which maps out the unoccupied density of states above the Fermi level. In AES the surface is bombarded with electrons and since emitted electrons result from multiple transitions making it somewhat more difﬁcult to interpret the density of states information from the secondary electron energies [11.3]. One example of this approach is the study of the electronic and magnetic structures of SrRuO3 , which is ferromagnetic, and the closely related CaRuO3 , which is nonmagnetic [11.15]. These materials exhibit complicated magnetic and electronic behaviors that have attracted considerable experimental attention, in part because of the potential suitability of SrRuO3 both for ferroelectric applications and, due to small lattice mismatch with yttrium–barium–copper–oxide, superconducting multilayers. Santi and Jarlborg [11.15], describe the details and results of the calculations along with comparisons to experimental EELS, UPS, and XPS observations performed by Cox et al. [11.16]. Local spin density functional approximation (LSDA) calculations predict that both are ferromagnetic with moments of about 2 µB for the orthorhombic structure while only SrRuO3 is ferromagnetic in the cubic structure. The general features of the DOS calculations are conﬁrmed by experimental methods, moreover the sharply peaked features of the DOS near the Fermi level suggest the variability in the properties of these similar compounds. An additional point of interest is the method presented in their paper for the calculation of UPS peak widths from EELS data. This same combination of techniques has been used by Fujisawa et al., to study the electronic structure of CuFeS2 and CuAl0.9 Fe0.1 S2 , which have interesting optical, electrical and magnetic properties [11.17]. The magnetic behavior of Fe–B alloys has been of considerable fundamental as well as engineering interest. This compound is one of a family of alloys of the type TMx A1−x , where TM is a transition metal (Fe, Ni, Co) and A is a metalloid (B, Si, Ge, Sn), which can be solidiﬁed in an amorphous state by rapid quenching from the melt. These metallic glasses have been studied extensively by means of UPS, XPS, AES, M¨ossbauer spectroscopy, and resistivity, however, controversy over the complex magnetic behavior remains [11.2]. For the Fe-rich amorphous Fe–B glasses some experimental evidence shows a maximum in magnetic moment near a composition of x = 0.8 as opposed to the pure iron [11.18– 11.20]. This may be explained partly by difﬁculties in sample preparation that led to the presence of additional phases and impurities along with variations in the surface

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chemistry. Since the moment loss occurs in the amorphous phase, this must be due to short-range order effects, and has been attributed both to charge transfer affecting the 3d band of the transition metal and orbital hybridization leading to a broadening of the d band [11.21]. More recently considerable theoretical work has been performed; in one such study the tight-binding linear mufﬁn-tin-orbital method has been used to calculate the electronic structure of the crystalline and amorphous Fen B (n = 1, 2, 3) alloys [11.20,11.22] for comparison with EELS, UPS, and AES experimental data collected by Paul and Neddermeyer [11.18]. This study suggests the magnetic moment of the transition metal atom decreases monotonically to zero from a maximum for pure Fe with increasing B content for both crystalline and amorphous Fe–B alloys. The calculated EELS spectra for all crystalline iron compounds studied were almost indistinguishable for the paramagnetic and ferromagnetic phases, in close agreement with experimental evidence that EELS is not sensitive to the state of magnetic ordering. The experimental studies focused on Fe3 B that crystallizes from an amorphous state to a single phase. Although signiﬁcant differences were found between the Fe3 B and the Fe spectrum, EELS spectra for crystalline and amorphous Fe3 B were seen to be very similar by Paul and Neddermeyer, in contrast to earlier studies which had shown marked differences. Differences seen in other concentrations on crystallization were attributed to precipitation of α-Fe into the matrix. 11.4

WHITE LINE PROBE OF D-BAND OCCUPANCY

The excitation of electrons from the 2p1/2 and the 2p3/2 to the outer unﬁlled d states for the 3d and 4d transition metals give rise to prominent features termed “white lines” for historical reasons at the onset of the L2 and L3 absorption edges [11.2]. As mentioned elsewhere in the text, it has been shown that the relative intensities of the white lines in the L2 and L3 near edge ﬁne structure decrease with increasing occupation of the outer d shells. This has proven a valuable tool for the investigation of electron transfer occurring between outer d states of the transition metal atoms upon alloying. Methods for making quantitative determination are discussed in Chapter 9. A study by Morrison et al., of the decreasing transition metal moment in another TMx A1−x binary alloy focused on the Fe-Ge system. The ratio of the white lines was measured as a function of x, with the sum of the amplitudes of the L2 and L3 edges directly proportional to the number of holes in the band [11.21,11.23]. Further, the ratio of the integrated amplitudes of the L2 - and L3 -edges can be directly related to the ratio of the number of d5/2 to d3/2 holes as hd5/2 1 5IL3 EL3 = −1 hd3/2 6 2IL2 EL2

(11.1)

hωLn ] is the energy of the transition. where ILn is the Ln -edge amplitude and ELn = [¯ Figure 11.3 shows a sequence of Fe L shell spectra for the Fex Ge1−x system with increasing Fe content. The data is processed ﬁrst to remove multiple scattering effects and subtract the background [11.24] and then by deconvolving the peaks arising from

WHITE LINE PROBE OF D-BAND OCCUPANCY

393

Fig. 11.3 Experimental L2 - and L3 -edges for Fex Ge1−x normalized to the total number of Fe atoms, that is the L shell cross-section, after [11.21]. Data is plotted with open squares while the line gives the best ﬁt to data.

transitions to the bound states from those arising from transitions to continuum states by one of the several possible methods [11.25, 11.21]. The L2 - and L3 -edges are then ﬁt to Lorentzians and once normalized to the total number of Fe atoms, the amplitude of the edges arising from bound state transitions is obtained. From these amplitudes, the total number of holes and the ratio of the holes with 5/2 symmetry to those with 3/2 is determined as given in Table 11.2.

Table 11.2

x

htotal / htotal (F e)

hd5/2 / hd3/2

1.0 0.6 0.5 0.4 0.3 0.2

1.0 1.3 1.1 1.1 1.3 1.0

0.68 0.69 0.72 0.4 0.29 0.45

The variation of the d-band occupancy with composition in Fex Ge1−x [11.2].

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From inspection one can see that the total number of holes remains the largely unchanged, which suggests charge transfer between the Ge and Fe is not responsible for the reduced magnetic moment from 2 to nearly zero Bohr magnetons across the composition range, however, the hole ratio varies dramatically. Since the Fe d-band is more than half-full, the redistribution of holes has been attributed to an increase in spin-pairing due to hybridization which would explain the dramatic change in magnetic behavior. A similar study by Pease et al., concerned the increase in the magnetic moment of Cr from 0.6 to nearly 4 Bohr magnetons upon alloying with Au [11.26]. The ratio of the L3 to L2 peak areas was seen to change from 1.57 in elemental Cr to 2.55 in Cr20 Au80 . This study provided further evidence that there is a correlation between the L3 - to L2 -edge ratio and the band occupancy, and therefore the spin–pairing of the 3d transition metals [11.23]. Further efforts focused on the development of a linear relationship between the peak ratio and the local magnetic moment have shown promise for a number of cases [11.27]. The application of white line ratio to study of electronic structure includes studies of Mn in yttria-stabilized zirconia [11.28], Ni-based intermetallics [11.29] and a number of transition metals (Cu, Ni, Fe, Cr, Ti) and their oxides [11.30]. Due to the high spatial resolution achievable with EELS, it is possible to study the behavior of transition metals at grain boundaries and interfaces. Efforts to integrate magnetic data storage technology with semiconductor integrated circuit technology have focused attention on Fe thin ﬁlms grown epitaxially on GaAs (100) wafers. One complicating factor is that thin ﬁlm properties often vary substantially from those of the bulk showing a strong dependence on ﬁlm thickness. Yuan et al. investigated the variation of the magnetic moment per atom from the bulk value as the thickness is reduced from 150 ˚A for Fe thin ﬁlms on GaAs [11.31]. As shown in Fig. 11.4, the Fe L-edge shifts to higher energy losses as the thickness of the thin ﬁlms is decreased. The ratio of the L3 - to L2 - edge amplitudes, which increase from 2.9 in the bulk to 5 in the 70-˚A ﬁlm, suggest a substantial increase in the moment per Fe atom in spite of the decrease in average magnetization. It is possible to obtain additional information from the width of the white lines that is determined by the lifetime of the core holes, which in turn is determined by the density of the conduction electrons. It can be seen as the ﬁlms become thinner the width of the white lines decreases, indicating that the density of the conduction band is decreasing. Both of these observations could be explained by the diffusion of As into the Fe layer, which would tend to form the antiferromagnetic species Fe2 As. A quantitative valence state mapping technique has been demonstrated by Wang et al., for a specimen composed of a mixture of MnO2 and Mn3 O4 , and another composed of CoO and Co3 O4 [11.32.11.33]. Previous experiments have shown that the ratio of the L3 to the L2 peak intensities vary strongly with the valence state of the cation, but are relatively insensitive to the thickness of the specimen. A ﬁve window method analogous to the spatial elemental mapping technique is employed and by referencing standards of known composition the valence state of the cation may be deduced. The results were conﬁrmed by comparison with measurements of the chemical composition. This technique is applicable for study of many magnetic

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Fig. 11.4 Fe 2p spectra from thin ﬁlms of varying thickness. The reference spectrum taken from a 2000-˚A thick particle.

transition metal and rare earth oxides of engineering interest at high spatial resolutions and ∼2 nm has been demonstrated. 11.5

MAGNETIC DICHROISM EFFECT IN EELS

X-ray absorption spectroscopy (XAS), like EELS, provides information about the unoccupied density of states, in this case by measuring the probability of photon absorption at a particular photon energy as electrons in the material are promoted into states above the Fermi level [11.2]. As discussed elsewhere in this book Xray absorption near edge ﬁne structure (XANES) and extended X-ray absorption ﬁne structure (EXAFS) both have analogous techniques in EELS, energy loss near edge ﬁne structure (ELNES) and extended energy loss ﬁne structure (EXELFS), respectively. Consequently, there has been interest in developing the EELS analogues to X-ray magnetic circular dichroism (XMCD), termed dichroic electron energy loss spectroscopy (DEELS), and X-ray magnetic linear dichroism (XMLD) [11.34]. Magnetic circular dichroism (MCD) is deﬁned as the difference in absorption crosssection for incident electric ﬁelds with left and right polarization, while magnetic linear dichroism (MLD) is the difference for incident electric ﬁelds polarized parallel and perpendicular to the magnetization in the material. The dichrosim effect in solids arises from a splitting of the J levels that may be caused by the interaction with the local magnetic ﬁeld B or from electrostatic interactions that result in a symmetrybreaking anisotropic crystalline ﬁeld. The different dipole selection rules for different polarizations give different probabilities of transitions to the ﬁnal state. 11.5.1

Magnetic Circular Dichroism in EELS

The demonstration conducted by Harp et al., was a reﬂection electron energy loss spectroscopy (REELS) experiment and although we are primarily concerned with transmission EELS, it is included here for completeness [11.35]. MCD is based

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on the dependence of absorption on the relative orientation between photon spin and sample magnetization for which the difference is proportional to the average value of the magnetization. Their simple model in which electrons undergoing a 90◦ scattering event are considered, suggests that MCD will result in similar signals from DEELS as from XAS. The transfer of energy from the incident electron to the core electron is considered as the emission and absorption of a virtual photon. Using this framework, Harp et al., have selected the incident electron energies such that the energy loss sufﬁcient to excite a 2p electron will fall at the maximum of the difference in right and left circularly polarized virtual photons produced in this process. Since this maximum is about 60%, one might expect to see differences in cross-section of this order. Calculations by van der Laan and Thole predict strong circular and linear dichroism for the 3d transition metals [11.35]. Polycrystalline transition metal (Fe, Co, and Ni) thin-ﬁlm specimens magnetized in the plane of the specimen were chosen to provide a favorable test case for demonstration of the effect. The scattering geometry of the MCD reﬂection EELS experiment must be chosen such that the dipole selection rules are largely valid. The specimen is positioned and a spectrum collected for the net circular polarization parallel to the magnetization and then antiparallel. Difference spectra for both Fe and Co show a very low signalto-noise ratio. Harp et al., concluded that the circular DEELS effect was less than 10% of that seen for XMCD, and that the EELS technique was unlikely to provide a workable alternative method. This reduced sensitivity has been attributed primarily to multiple scattering of the electron before and after the energy loss event. 11.5.2

Magnetic Linear Dichroism in EELS

Magnetic linear dichroism (MLD) is the difference in absorption cross section for photons linearly polarized parallel and perpendicular to the magnetization in the material. Unlike in the case of MCD in EELS that relies on large angle inelastic scattering processes, MLD takes advantage of the more common small angle scattering available in the transmission microscopy setup. Yuan and Menon have observed MLD in EELS, which depends on M 2 (as opposed to M for MCD) of the ions in the solid, and developed a framework for simulating the contrast [11.36–11.38]. They chose the compound hematite α-Fe2 O3 whose magnetic and chemical structure has been well characterized to illustrate the method. Hematite has a trigonal crystal structure with 6 formula units per cell. Above the Morin temperature Tm = −10 ◦ C the moments are aligned perpendicular to the c-axis, with spins on any one basal plane oriented parallel to each other, while those on neighboring basal planes are oriented in opposite directions. Below Tm the spins lie parallel to the c-axis. Since the character of the magnetic structure is such that M = 0, while M 2 is nonzero, this presents a convenient test case. The experiment was performed on a VG-HB501 STEM operated in the normal imaging mode with a 1–2-nm probe size. The condensed electron beam used to achieve high spatial resolution contains electrons with a range of incident wave vectors spanning several mrad convergence semiangle. The detector collects a range of ﬁnal wave vectors with a collection semiangle of several mrad as well. The

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momentum-transfer q is the difference between the initial and ﬁnal wavevectors with the perturbing waves longitudinally polarization parallel to the direction of momentum transfer. The EELS edge then contains a range of momentum transfer vectors, and can be decomposed into those with q parallel and those with q perpendicular to the initial wavevector. The dichroic information can then be obtained by collecting two different spectra with varying weighting of the components. The weightings of the wavevectors in the probe can be varied by changing the convergence angle, which is accomplished practically by selecting objective apertures with different diameters.

Fig. 11.5 (a) The normalized EELS spectra of Fe 2p absorption in Hematite obtained using 25 µm (I1 - dotted line) and 50 µm (I2 - solid line) objective apertures. Also plotted is the difference I2 − I1 that is proportional to the magnetic dichroism effect. (b) Theoretical L-edges for Fe in Hematite with local symmetry corresponding to spin arrangement perpendicular to trigonal axis, incorporating weighting factors for the 25 µm (dotted line) and 50 µm (solid line) objective apertures. Also shown is the difference spectrum. After [11.37].

The L2 and L3 absorption of the Fe3+ ion is considered because it was expected to show a signiﬁcant dichroic effect. The orientation of the crystallites was adjusted in order to align the c-axis parallel to the incident beam. Two spectra were obtained for 25 and 50 µm, corresponding to 3.7 and 7.4 mrad, respectively, with collection times of 20 s. The spectra along with the difference spectrum, produced by aligning and then subtracting the two spectra, is shown in Fig. 11.5a. It can be shown that the difference spectrum is proportional to the MLD effect by writing the normalized spectral intensities as Ii = αi Iparallel + βi Iperpendicular , i = 1, 2

(11.2)

where Iparallel and Iperpendicular are the components of intensity for ﬁeld polarization parallel and perpendicular to the incident wavevector [11.36]. With the normalization

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condition that αi + αj = βi + βj , Idiﬀ = I2 − I1 = (α2 − α1 )(Iparallel − Iperpendicular )

(11.3)

This detected intensity difference is then proportional to the difference in cross section for ﬁeld polarizations parallel and perpendicular to the spin of the ions in the solid. The difference spectrum has a peak intensity that is 5% that of the acquired spectra as shown in Fig. 11.5a. Simulations based on Hartree–Fock calculations incorporating crystalline ﬁeld effects and the magnetic superexchange as laid out by van der Laan and Thole [11.35] and plotted in Fig. 11.5b agree well with the observed spectra in amplitude and sign. The sign of the difference spectra shown is consistent with the spins perpendicular to the c-axis, giving additional conﬁrmation that the observed effect is MLD as opposed to a misalignment error. Details of the calculation of the dichroic difference spectrum are given in [11.38]. This EELS technique shows great promise for the study of chemically speciﬁc magnetic information on a very ﬁne scale, for example, the study of magnetic domain boundaries in antiferromagnetic materials. 11.6 11.6.1

ENERGY FILTERED LORENTZ MICROSCOPY Introduction

For several decades electron optical tools have been used to probe the micromagnetic structure of various magnetic materials. The resolution of these techniques has steadily increased from the early bitter decoration observations [11.39] to the latest coherent Foucault [11.40] and differential phase contrast (DPC) methods [11.41]. Many engineering applications of magnetic materials require information about the micro-magnetic conﬁgurations at the highest available spatial resolution. For instance, the desired increase of the information storage density of magnetic recording media to 1.55 Gbit/cm2 by the end of this century will require a detailed understanding of the relation between the magnetic domain structure and the underlying microstructure. Similarly, increased use of single-crystal magnetic actuators necessitates a clear insight in the interactions between magnetic domains and crystal growth defects. Conventional Lorentz microscopy suffers from a reduced ﬁnal image magniﬁcation as compared to conventional transmission electron microscopy, caused by the need for a low-ﬁeld sample environment and the consequent increase in the focal length of the objective lens. This is a serious limitation for the quantitative study of nanoscale magnetic structures. Furthermore, inelastic scattering in the sample causes an overall blurring of the images, which further limits the attainable resolution of the standard Lorentz modes. In this section a new combination of existing tools, which converts a 400-kV high resolution TEM into an energy-ﬁltered Lorentz microscope (EFLM), is presented. The functionality of a JEOL 4000EX HRTEM (top-entry, Cs =1 mm) has been extended by the addition of a post-column Gatan Imaging Filter

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(GIF) with a 1024×1024 14-bit CCD camera. It has been possible to perform the standard Lorentz modes as well as extend the Foucault method to produce induction maps. All examples focus on Terfenol-D, a magnetostrictive alloy of composition Tb0.73 Dy0.27 Fe1.95 . This work was motivated by an interest in characterization of the interfaces at twin boundaries between different lamellar domain sequences as described by the James and Kinderlehrer model [11.42]. Speciﬁcally, looking for a difference, if one exists, between the exactly compatible interfaces and those which are only compatible in an average sense. Lorentz microscopy was employed in a quantitative way allowing determination of the speciﬁc variant present. The ultimate goal of micromagnetic observations is to fully document the magnetic induction as a function of location in the sample. This includes a study of domain conﬁgurations and the character of the domain walls. There are very few techniques capable of producing direct maps of the (in-plane) magnetic induction in a thin foil. Differential Phase Contrast (DPC) microscopy in a STEM [11.41] is perhaps the most successful method but it requires instrument modiﬁcations (installation of descanning coils and a four-quadrant detector) and is not generally accessible in an average research laboratory. Daykin and Petford-Long [11.43] have recently proposed an equivalent technique for the TEM (equivalent to DPC by the reciprocity theorem). They have demonstrated that for thin foils a TEM equipped with a low- ﬁeld Lorentz pole piece can be used to obtain maps of the components of magnetic induction at right angles to the trajectories of the incident electrons (which, for brevity, we will refer to as the in-plane components). They also point out that inelastic scattering limits the resolution of DPC images both STEM and TEM so that any form of inelastic scattering ﬁlter is desirable. 11.6.2

Classical Lorentz Microscopy

Classical Lorentz microscopy relies on the fact that when electrons traverse a sample with an in-plane magnetic induction component they experience a Lorentz deﬂection. Electrons passing through magnetic material with regions of different in-plane magnetization direction are deﬂected by an angle ε in the direction of the Lorentz force causing splitting of the central and diffracted beams. The transmitted electrons form, in the back focal plane of the objective lens several distinct spots or overlapping disks depending on the imaging conditions. There will be one spot for each set of domains with a single magnetization direction. Two ways of taking advantage of this interaction are the Foucault and Fresnel modes. In the Fresnel mode, the image is out-of-focus and contrast arises only where the magnetization direction is changing within the ﬁlm, so that mainly domain walls are observed. In magnetic ﬁlms with domains having in-plane magnetization and 180◦ walls, deﬂected electrons form convergent or divergent domain wall images, which are seen as regions of increased or decreased intensity. The Foucault mode is an in-focus technique in which domains are imaged by selecting one portion of the split central beam (and suppressing the others) in the back focal plane. The Foucault technique causes domains with different magnetic induction directions to show different contrast in the electron image.

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In-plane directions of the magnetic induction of the material can be obtained in a well characterized system. Most applications of Lorentz microscopy require that the sample be mounted in a low-ﬁeld or, preferably, ﬁeld-free region in the microscope column. The sample can be located above the objective lens pole piece, although the increased focal length associated with such a conﬁguration results in a reduced ﬁnal magniﬁcation as compared to conventional microscopy. In order to achieve the high magniﬁcations required for many applications, the standard objective lens pole piece is typically replaced by a dedicated Lorentz pole piece. There is clearly a need for a setup which, while still providing sufﬁcient magniﬁcation in the Lorentz mode, allows for direct conversion between Lorentz and conventional high resolution imaging modes without column modiﬁcation. Additionally, digital image acquisition is superior to the more standard photographical methods for extraction of quantitative information from Lorentz images. 11.6.3

Experimental Conﬁguration

A novel Lorentz microscopy setup combines a high-resolution top-entry JEOL 4000EX TEM with a post-column Gatan imaging ﬁlter (GIF) [11.44]. When the main objective lens in this microscope is turned off, the sample can be mounted in an essentially ﬁeld-free region. The objective mini-lens, which sits well below the plane of the sample, then provides sufﬁcient ﬁeld strength to focus the electrons into a diffraction pattern at the selected area (SA) aperture plane. The selected area aperture can then serve as the contrast forming (CF) aperture in place of the objective aperture. The contrast forming aperture, 30 µm in diameter and located 20 cm below the sample, provides a reciprocal diameter of 0.4 nm−1 . Fresnel and Foucault modes are available by changing the minilens current and moving the SA aperture, respectively. This conﬁguration provides a maximum nominal magniﬁcation of 3000× at the plane of the negative, insufﬁcient for the study of domain structure in Terfenol. A GIF (Gatan Imaging Filter) positioned below the microscope column improves the situation. The GIF is used for two purposes : (1) it provides an additional magniﬁcation of approximately 20× ; and (2) it allows for the acquisition of zero-loss images, increasing the signal-to-noise ratio by at least an order of magnitude, using a slit width of around 10 eV [11.44]. The internal magniﬁcation of the GIF is ∼ 20× (19.88× measured value), resulting in a total magniﬁcation at the 1.0-in.2 surface of the CCD camera of 60,000× (57,522×). The internal GIF magniﬁcation is thus of crucial importance for high-resolution Lorentz observations at 400 keV. An additional beneﬁt of the use of the GIF is that most of the inelastically scattered electrons can be removed from the beam. For standard Lorentz work, imaging conditions are usually similar to those used for standard two-beam observations, although it is preferable to minimize regular diffraction contrast by using a large excitation error. The illumination system of the TEM is thus used mostly in a converged mode where the electron beam ﬁlls the entire viewing screen. The CF aperture is used to select the transmitted beam for bright ﬁeld observations. For Fresnel (out-of-focus) observations the objective minilens current

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can be varied, while the Foucault mode is accessible by shifting the CF aperture (or, in practice, tilting the beam) to exclude portions of the split transmitted beam. An additional beneﬁt of the GIF is that, by inserting a 7–20 eV slit width to remove most of the inelastically scattered electrons, energy-ﬁltered Lorentz images can be formed. These images exhibit a better signal-to-noise ratio than unﬁltered images, allowing study of thicker regions, in both Fresnel and Foucault modes. An undercondensed beam with a beam divergence angle, θc , of less than 10−7 rad can be easily produced in order to obtain coherent images. The condensor lens assembly of the JEOL 4000EX high resolution TEM has been designed to provide a reasonably parallel incident beam. Much longer exposure times are required in this case and specimen drift becomes a problem, so that a compromise must be struck between the beam coherence and the total acquisition time. The GIF-4000EX combination is well suited to coherent Lorentz microscopy, and examples of coherent Fresnel and Foucault contrast will be shown in the next section. All experimental images in this section were obtained from a thin foil of TerfenolD, or Tb0.73 Dy0.27 Fe1.95 . This material has a MgCu2 cubic Laves phase structure, with a lattice parameter a = 0.732 nm [11.45]. Terfenol-D has a room temperature magnetostriction coefﬁcient λ111 ≈ 2, 200 ppm, which makes it a material well suited for low-frequency, high power sonar, micro-positioning and anti-vibration. For a detailed description of the microstructure of this material the reader is referred to [11.42,11.44,11.46]. For the purposes of this discussion, it will be sufﬁcient to state that Terfenol-D, at room temperature, has the 111 directions as soft magnetic directions, resulting in the presence of 8 different magnetic induction directions in an unpoled sample. At room temperature the cubic unit cell is rhombohedrally distorted along the magnetically soft 111 directions. Since the rhombohedral distortion is extremely small, we will continue to refer to the cubic reference frame. During singlecrystal growth, 180◦ rotational growth twins are formed on the (111) plane, resulting in an additional 6 induction directions. Domain walls are either of the zigzag 180◦ type, or of the 71◦ (109◦ )type, which is the angle between cubic 111 directions. James and Kinderlehrer [11.42] have shown that the micromagnetic structure consists of lamellar sequences of alternating domains, intersected by 180◦ domain walls, and have predicted the existence of two types of growth twin laminates : 1. Compatible laminates, in which the thickness of the transition layer at the twin boundary vanishes in the limit of an inﬁnitely ﬁne laminate (i.e., γ → 0) 2. Exactly compatible laminates, in which there is no transition layer at the twin boundary, even for a ﬁnite laminate size. Figure 11.6 shows a Foucault image of one such characteristic lamellar domain arrangement, with the domain wall character indicated on the ﬁgure. 11.6.4

Lorentz Observation Modes

As mentioned in the discussion of the experimental conﬁguration, the Fresnel mode is readily accessible through the objective minilens controls. Images of a particular

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Fig. 11.6 Foucault image of lamellar domain with wall character as indicated.

region in a Terfenol-D crystal oriented close to the [0¯11] zone axis are used to illustrate the various Lorentz modes. All images were acquired by binning the 1024×1024 CCD camera into 512×512 pixels, and applying a standard gain normalization. The area of interest is shown in the unﬁltered image of Fig. 11.7b. The coffee-bean shaped contrast corresponds to Widmanst¨atten precipitates of the RFe3 phase, with R = Tb, Dy. They form platelets and are responsible for the rather high coercivity in these materials [11.46]. The image in Fig. 11.7a is a zero-loss image of the same region, taken with a slitwidth of ∆E = 7 eV. The signal-to-noise ratio of the ﬁltered image is substantially better than that of the unﬁltered images, as is apparent from the improved contrast. The images shown in Fig. 11.7c and d are zero-loss overfocus and underfocus, respectively, Fresnel images of the same region (at slightly lower magniﬁcation). The convergent and divergent wall images are clearly delineated and one set of walls parallels the (100) planes. Note how these walls also parallel the strings of second phase particles; this is most likely caused by a pinning action of the Widmanst¨atten platelets. The long wall running from the top to the bottom of the image is a 180◦ domain wall, the others are 71◦ or 109◦ walls. Coherent zero-loss Fresnel images are obtained simply by underfocusing the second condensor lens; the image shown in Fig. 11.8 was obtained with the same objective minilens overfocus setting as the one in Fig. 11.7c by spreading the beam, with a resulting acquisition time of 17 s and a beam divergence of approximately θc = 10−7 rad. Clear fringes are present in the 180◦ domain wall, and faint fringes can also be seen in the rightmost convergent wall. The visibility of such fringes depends sensitively on the thickness of the sample, on the amount of defocus necessary to produce them (usually in the range of 0.1–3 mm), and on the aperture radius. The spacing between the fringes provides information on the product of the in-plane magnetic induction component and the crystal thickness. For the short segment in the middle of Fig. 11.8 the fringe spacing is 20.8 ± 0.5 nm, which translates into a

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Fig. 11.7 (a) Zero-loss ﬁltered and (b) unﬁltered in-focus images of a region containing multiple domain walls; (c) and (d) are overfocus and underfocus zero-loss Fresnel images of the same area.

Lorentz deﬂection angle of β = 39.5 µrad, and Bt = 99 T nm. Since the saturation induction of Terfenol-D is ∼ 1.2 Tesla, the sample thickness is ∼ 82 nm.

Fig. 11.8 Coherent zero-loss (slit width 7 eV) Fresnel image of the central region of Fig. 11.7.

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Figure 11.9 shows four zero-loss images for approximately equal CF aperture shifts in four directions as indicated. All images were taken at zero-focus, with a slightly converged electron beam; the in-focus zero-loss bright ﬁeld image of the same region is shown in the center of the ﬁgure. From these images, it can be shown that there are four different domain variants in this region of the sample; a detailed analysis of the contrast patterns and the corresponding aperture shifts shows that the magnetization directions are most likely the ones shown in the lower part of the ﬁgure.

Fig. 11.9 Zero-loss Foucault images for aperture shifts in four orthogonal directions; the central image in an in-focus bright ﬁeld image of the same region.

The beam splitting associated with the Lorentz deﬂection in neighboring domains can be observed in diffraction mode. In general, for a domain arrangement that varies on a ﬁne scale, there will be several overlapping diffraction disks. The GIF interface gives the operator independent access to a number of the post-specimen lens currents, a conﬁguration similar to a free lens control mode. It has been possible to set the

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lenses to form a diffraction pattern with sufﬁcient angular range at the plane of the negative. It is also possible to demagnify the diffraction pattern so that it ﬁts on the CCD camera, although there are substantial electron optical distortions present which to date have not been possible to correct. Access to additional lens currents in the GIF itself may improve matters substantially. An example of a digitally acquired zero-loss diffraction pattern is shown in Fig. 11.10. The crystal zone axis is close to [0¯11], which is also the foil normal. The image distortion is clearly visible in the distances between the origin and the 111 and ¯ 1¯ 1¯ 1 reﬂections. When the average distance for those reﬂections is used to calibrate the angular range, the spot splitting due to the Lorentz deﬂection from a 180◦ domain wall is measured to be 2β ≈ 200 µrad, corresponding to Bt ≈ 500 T nm. This in turn results in a crystal thickness of ∼ 416 nm for B = 1.2 T.

Fig. 11.10 Zero-loss demagniﬁed [0¯ 11] diffraction pattern captured on the GIF CCD camera; note the presence of substantial image distortions caused by aberrations in the projector lens system.

The improvement in signal-to-noise ratio obtained by removal of inelastically scattered electrons is illustrated in Figs. 11.11 and 11.12. Figure 11.11a shows an unﬁltered, incoherent (focused beam) Foucault image corresponding to the −x image of Fig. 11.8. When the incident beam is defocused to a beam divergence of ∼ 10−7 rad, the image contrast is markedly improved, and weak fringes appear at the edges of the bright domains. Figure 11.11b shows an unﬁltered coherent Foucault image, while Fig. 11.11c shows the corresponding zero-loss ﬁltered image. Note how the intensity in the dark domain is rather noisy in the unﬁltered image, but almost perfectly zero (black) in the ﬁltered image. Figure 11.12a shows a 900 nm line scan along the line AB in Fig. 11.11c, for both unﬁltered and ﬁltered images. The intensity scale refers to counts on the CCD camera after gain normalization. The ﬁltered Foucault image shows a greatly improved signal-to-noise ratio, and most of the inelastically scattered electrons contributing to the dark region have been removed. The effect of zero-loss ﬁltering becomes even more pronounced for the line scan CD; Fig. 11.12b shows the averaged intensity for 90 consecutive line segments parallel to CD. The unﬁltered and ﬁltered averaged line scans exhibit fringes parallel to the domain wall and up to three fringes can be identiﬁed in the ﬁltered proﬁle. The peak-to-valley intensity ratio increases from 4.3 for the unﬁltered image to 47 for the ﬁltered one,

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that is, zero-loss ﬁltering can improve the signal-to-noise ratio in coherent Foucault images by a factor of 10.

Fig. 11.11 (a) Unﬁltered incoherent, (b) unﬁltered coherent, and (c) zero-loss coherent Foucault images; line segments AB and CD indicate line scans shown in the next ﬁgure.

Fig. 11.12 (a) Line scan corresponding to the segment AB in the previous ﬁgure for both unﬁltered and ﬁltered coherent Foucault images. (b) line scan CD integrated over 90 pixels parallel to the boundary shows a marked improvement in peak-to-valley intensity ratio upon zero-loss ﬁltering.

11.6.5

Magnetic Induction Mapping in the TEM

As laid out by Chapman and Morrison, quantitative magnetic induction maps can be produced in a STEM equipped with post specimen lenses and a quadrant detector by the method of differential phase contrast microscopy (DPC) [11.47]. In the absence of an in-plane component of magnetic induction, the descan coils maintain the position of the electron bean at the center of the detector, regardless of the position of the probe on the specimen. For a specimen with in-plane magnetic induction components, the additional Lorentz deﬂection is not compensated for by the descan coils, and hence the beam will move off of the center of the detector. The deﬂection can be measured

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by taking the difference of the signals from opposite quadrants. These difference signals are proportional to the deﬂection of the electrons due to the magnetic ﬁeld in the sample at two orthogonal directions (±U and ±V ). The intensity (+U ) − (−U ) is proportional to a component of magnetic induction in a direction 90◦ clockwise from the +U direction referred to as the BU direction, and similarly for (+V ) − (−V ). In this way the in-plane components of magnetic induction can be measured at a single point and by scanning the probe over the sample, a two-dimensional map can be produced. This technique allows the direct determination of domain wall proﬁles, limited in resolution primarily by the probe diameter. A technique equivalent to DPC has recently been developed by Daykin and Petford-Long for the TEM and implemented on a JEOL 4000EX ﬁtted with a Lorentz lens. The method can be implemented on a conventional TEM, provided that the objective lens has been replaced by a low-ﬁeld Lorentz pole piece [11.43], and for details the reader is referred to their paper. In the standard Foucault method, a low ﬁeld Lorentz lens focuses a parallel incident electron beam into sharp spots in the back focal plane and an aperture is then inserted to select a deﬂected beam and obtain a “dark ﬁeld” image. Comparison of the beam deﬂection direction with the bright domains in the Foucault image allows one to determine relative differences in the local magnetic induction direction and magnitude. While this is a suitable method for studying simple domain structures, more complex or ﬁner magnetic structures can not readily be analyzed from Foucault images alone. Furthermore, the Foucault image contrast is very sensitive to the position of the aperture due to the complex nonlinear relationship between the object function and the observed intensity in the image [11.43]. Magnetic induction mapping in the TEM is accomplished by extending the Foucault mode. It is necessary to obtain four series of Foucault images taken with a regular beam tilt step in the positive and negative of each of two orthogonal directions. Two orthogonal components of the magnetic induction, BU and BV , are determined from four series of images recorded at regular intervals of beam tilt in the ±U and ±V directions. The sum of the images in a single series gives an intensity related to the deﬂection of the electron beam, approximating an integration over tilt angle. For each point in the image the intensity in this Foucault sum is comparable to the signal in one quadrant of the detector in the STEM setup; the Foucault sum can be thought of as the current obtained from a line detector, the main difference being that the current is obtained for each point in the image simultaneously. A difference image is then obtained by subtracting the −U (V ) summed image from the +U (V ) summed image. The intensity in the ∆U (∆V ) difference map is proportional to the in-plane component of magnetic induction, BU (BV ). Daykin and Petford-Long have constructed induction maps from Foucault images in this way. The two difference images can then be combined to produce a vector map of the local magnetic structure. The equivalence of the two methods is a consequence of the reciprocity theorem, which states that if the source and observation point are exchanged that the intensity recorded must be, for purely elastic scattering, the same for any system of lenses. The information required for unique determination of the variant type present near a twin boundary in Terfenol-D can be provided by the

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induction mapping technique. This technique has been implemented on the JEOL 4000EX-GIF Lorentz setup. The adaptation of the induction mapping technique to the EFLM is a straightforward extension of the Foucault technique and requires no additional modiﬁcation of the column. One important difference between the DPC technique and the TEM technique is that for regions of rapidly varying magnetic induction, for instance, domain walls, the limits on the resolution take a different form. As mentioned previously, the main limitations on the resolution of DPC induction maps is the probe diameter, that is, the condensor aperture size. For the TEM technique, the images collected for a series are randomly phase shifted with respect to each other, degrading the resolution of the ﬁnal map. In addition the resolution is strongly linked to both the aperture size and the orientation of the tilt directions to the domain wall normal. If the higher frequency components lie outside the contrast forming aperture, they will not contribute to the image. The reciprocal space information related to the domain wall proﬁle is located along the line perpendicular to the domain wall. If the beam tilt direction coincides with the wall normal, the spatial frequencies over a large tilt range may contribute to the reconstruction. On the other hand, if the tilt direction is far from the wall normal, only the lower spatial frequencies will contribute to the reconstructed wall proﬁle and the domain wall width will be overestimated. These two factors depend sensitively on each other; the smaller the aperture is the more critical it is for the tilt direction to coincide with the domain wall normal. Because of these complicated relationships, the contrast in the induction maps near areas of rapidly varying magnetization cannot be straightforwardly interpreted. In an effort to extract meaningful information, the image contrast has been simulated for comparison with the experimentally acquired micrographs and will be discussed in the next section. The Daykin–Petford-Long technique has been adapted to the EFLM setup described previously by extending the standard Foucault technique in a similar way. For each map it is necessary to acquire four series of Foucault images, taken for tilts in the ± U and ± V directions, and to determine the U and V tilt directions relative to the crystallographic directions in the sample. It is possible to use the bright ﬁeld (i.e., prespecimen) tilts, however, there will be a contribution from diffraction contrast that cannot easily be accounted for. The JEOL 4000EX has two sets of image shift coils that can be used in conjunction to provide a post-specimen beam tilt. This is critical when the specimens are not amorphous thin ﬁlms, but strained, relatively thick sections of bulk material, such as Terfenol. The ﬁrst image shift coils that sit just above the objective minilens coil provides the tilt. The accompanying shift of the image can be compensated for in large part by the second image shift coils, which sit just below the objective minlens and therefore produce a much smaller tilt relative to a shift of the image. This ensures that the change in contrast throughout the series arises from the magnetic splitting of the central beam, as opposed to a change in the exit wavefunction. Image shift one sits above the objective mini-lens, while image shift two sits below, so that when the beam is tilted in this way to obtain succesive Foucault images, the beam travels through a slightly different part of the lens. It is therefore possible to have an additional phase shift from image to image that cannot easily be accounted for.

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The determination of the direction of the tilt relative to the crystallographic reference frame of the sample is a simple matter and can be accomplished by taking a long exposure of a diffraction pattern while simulataneously tilting ﬁrst in the +U direction and then the +V direction. The only other information required to determine the beam tilt directions is then the angle of rotation between the diffraction pattern and the image at a given magniﬁcation. As most of Terfenol’s more interesting domain features occur near twin boundaries, it has been convenient to take patterns showing this feature, which can then be overlaid directly onto the image. By removing the sample from the path of the electron beam, the aperture can be centered in a relatively ﬁeld free region where the beam is taken to be undeﬂected. Magnetization vectors are determined relative to this position of the incident beam. Inelastic scattering remains a serious problem, even though the aperture diameter is 0.4 nm−1 , very small with respect to the diffraction pattern. The improvement in signal-to-noise ratio of energy-ﬁltered images becomes more important when several images are algebraically combined. Figure 11.13a and b show the sums of seven images acquired with a regular +U tilt. The images for Fig. 11.13a were taken without a slit inserted and those for Fig. 11.13b were taken with a slit inserted to remove the inelastically scattered electrons. The images were acquired for the same number of total counts, with the ﬁltered image requiring 3.2 s integration times as opposed to 1.6 s for the unﬁltered images. In Fig. 11.13c, a line trace is shown that corresponds to an average of 48 pixels taken parallel to the line AB shown in Fig. 11.13a. The sharp peak present in the ﬁltered sum is completely absent in the unﬁltered sum. It is possible that detailed simulation coupled with more exact knowledge of the imaging conditions may lead to the meaningful interpretation of the contrast present at the wall in summed Foucault series and induction difference images. It is clear that ﬁne features in the contrast are overwhelmed by the inelastically scattered electrons and that zero-loss images are required to achieve the highest resolution.

Fig. 11.13 (a) Unﬁltered and (b) ﬁltered summed Foucault series of Terfenol-D. (c) Plot of the line trace for both (a) and (b) averaged over 48 pixels taken parallel to line AB. Note that the bright fringe at the domain wall in the zero loss sum is completely absent from the unﬁltered image.

In order to investigate further the effect of energy ﬁltering, both zero-loss and unﬁltered mapping experiments were conducted for identical illumination conditions

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on the region shown in Fig. 11.6. A long exposure diffraction pattern is shown in Fig. 11.14; the transmitted beam was originally centered on the negative. Since a tilt corresponds to a shift of the pattern in that direction, a spot that was deﬂected in the opposite direction would be allowed by the aperture. The tilt directions are shown in Fig. 11.14, and by rotating 90◦ clockwise the corresponding magnetization directions can be determined.

Fig. 11.14 Long exposure diffraction pattern (reverse contrast) of a twinned region of Terfenol. A converged beam was used to illuminate the sample and the L shape is marked out by tilting ﬁrst in the +U direction and then in the +V direction.

Figure 11.15 is a Foucault image of a twin boundary nearby the region shown in Fig. 11.6 showing the induction directions. Four series of six Foucault zero-loss images were obtained and are shown in Fig. 11.16. The fringes perpendicular to the 109◦ result from thickness variations. Unﬁltered images were obtained under identical conditions except for the removal of the energy slit in order to investigate the effect of energy ﬁltering. The Foucault images in each of the series collected for this induction map were aligned using another interactive program based on Powell minimization of the differences between successive images; this procedure is thought to give alignment to within a fraction of a pixel. The sums for the ﬁltered series are shown in Fig. 11.17, along with the difference images for both the ﬁltered and unﬁltered series. The BU and BV directions are shown on the ﬁgure. The difference images can then be rebinned and displayed as a vector map, shown in Fig. 11.18 for both the ﬁltered and the unﬁltered case. Note that only minor differences, primarily occuring near the domain walls, are apparent between the two vectormaps, indicating that the contribution of inelastically scattered electrons to the images has only a small effect on the reconstruction in this case. Although the advantages of ﬁltering are not striking in the case of Terfenol, it will undoubtedly be required for induction mapping studies of materials with smaller

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Fig. 11.15 Foucault image of the region corresponding to the diffraction pattern in Fig. 11.14. The beam tilt directions are indicated with respect to the twin plane.

Fig. 11.16 Foucault series taken with beam tilts of ±U and ±V . A slit width of 15 eV was used for each.

saturation magnetization. The large magnetization of Terfenol-D leads to relatively large splitting of the central beam, and therefore strong Foucault contrast. Zero-loss ﬁltering improves the signal-to-noise ratio in the image; when the signal is already very strong, this improvement will be less critical than when it is weak. In addition it is difﬁcult to produce Terfenol samples which have large regions of a relatively constant thickness, and this leads to contrast changes between inelastic and elastic images. Since the Lorentz deﬂection is proportional to the sample thickness, it is necessary to use a sample with a nearly constant thickness across the region of interest

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Fig. 11.17 Left two rows show summed images for the four tilt series shown in Fig. 11.16. The right two rows show the difference images for both tilt direction, for the ﬁltered and unﬁltered tilt series. Linear features near the edges of the images are artifacts of the image alignment procedure.

Fig. 11.18 Magnetic induction vector maps for the in-plane components are shown for the central 256×256 region of the difference images of Fig. 11.17, for both ﬁltered and unﬁltered mapping experiments.

to perform accurate magnetic induction mapping experiments. For a wedge shaped crystal, electrons which travel trough the thicker part of the wedge will have a higher probability of undergoing inelastic scattering events, and will also experience a larger Lorentz deﬂection. The induction mapping technique produces information which is proportional to the product of the in-plane components of magnetic induction and the sample thickness. Consequently, irregularities in the sample thickness may produce artifacts in the induction maps. Care must be taken to orient the sample such that there is a minimum of diffraction contrast. This is particularly important if quantitative

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comparisons with simulations (which are based on a pure strong phase object) are needed. 11.6.6

Compatibility of Domain Conﬁgurations

After studying many domain wall conﬁgurations near growth twins it became apparent that there are indeed two different types of conﬁgurations, as shown in Fig. 11.19 (note the difference in scale between a and b). In Fig. 11.19a the zero-loss Fresnel and Foucault images show a set of tapered domains in the laminate on the left. The wide dark band corresponds to a pair of growth twin planes with the twin variant in a strongly diffracting orientation (hence the absence of domain contrast between the twin planes). Figure 11.19c is an edge-enhanced version of the central region of the Foucault image, clearly outlining the domain walls (the edge enhancement procedure results in dark features wherever the contrast in the original image changes, i.e., at boundaries). In Fig. 11.19b three growth twins are present, two of which are at a spacing of only 5 nm. In both the zero-loss Fresnel and Foucault images one can clearly observe domain contrast inside the region between the two twin planes, thereby establishing that the magnetic resolution of our system is better than 5 nm. Furthermore, it is obvious from the images that none of the domains are tapered; all domain walls continue straight up to the twin planes and there is a 1:1 correspondence between domains on either side of each twin plane. The domain wall locations are again clearly delineated in the edge-enhanced image of Fig. 11.19d. 11.6.7

Discussion

Energy ﬁltered Lorentz microscopy (EFLM) allows one to use a standard highresolution TEM with an attached GIF in Lorentz imaging modes, at a total ﬁnal magniﬁcation of around 60,000 X at the CCD camera. Operation of the energy ﬁlter in zero-loss mode, with a slit width of around 10 eV, provides substantial contrast improvements, in particular for the coherent Foucault mode, since inelastic contributions of “dark” domains are completely eliminated. Magnetic induction mapping becomes a viable technique when the microscope is operated with the energy ﬁlter control program in free-lens control mode. Magnetic domain contrast in both Fresnel and Foucault mode has revealed image detail at better than 5 nm resolution, for a slit width of 7 eV. Induction map resolution has been set in the lower limit at 60 nm. By assuming a magnetic structure, computing the Aharanov– Bohm phase shift [11.48], taking into account the microscope-transfer function, the defocus (Fresnel mode) and/or the aperature positions (Foucault), it is possible to produce a matrix of simulated results which compare well with experimental images. The methods are described in detail in [11.44]. Simulated images for a particular laminate have been produced by implementing the Mansuripur algorithm [11.49] in combination with FFT techniques for both Fresnel and Foucault contrast. The resulting images compare well with Lorentz micrographs obtained for comparable domain structures. Detailed simulation of the induction mapping technique may lead

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Fig. 11.19 (a) Zero-loss Fresnel and Foucault images of tapered boundaries near two growth twins. (b) zero-loss Fresnel and Foucault images of perfectly straight domain walls near three growth twins. (c) and (d) are edge-enhanced enlargements of the central sections of the Foucault images in (a) and (b) respectively, illustrating the difference between the compatible and exactly compatible domain conﬁgurations. Note the scale difference between (a) and (b).

to techniques for the interpretation of magnetic domain structures which vary on a ﬁne scale in analogy to the methods used in the interpretation of high-resolution TEM image contrast. For example, comparison of wall proﬁles derived from computed induction maps with those from the experimental maps will give a ﬁrst order estimate of the experimental wall width. 11.7 CONCLUSION The introduction of the imaging energy ﬁlters has made it possible to obtain elemental and induction maps for correlation with microstructural features and bulk properties as can be seen from examples mentioned here. There are many additional studies on

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magnetic materials in which EELS plays a role, and those discussed here are by no means a comprehensive list. MLD shows considerable promise for probing magnetic properties that require chemical speciﬁcity. An energy ﬁltered Lorentz microscopy (EFLM) setup allows one to switch easily from standard high-resolution and elemental mapping studies to Lorentz imaging modes, at a total ﬁnal magniﬁcation of around 60,000× at the CCD camera. Acknowledgments The author gratefully acknowledges the many contributions made by M. De Graef in numerous discussions. The author wishes to thank N. T. Nuhfer who ﬁrst suggested the combined use of the JOEL 4000EX and the post column imaging ﬁlter for Lorentz microscopy and N. Biery who the image alignment software used to compensate for image shifts during beam tilting. The author would also like to thank J. H. Scott for providing descriptions of the imaging conditions for his elemental maps. The author is indebted to B. Fultz and C. C. Ahn for their technical guidance over the years.

REFERENCES 11.1. L. Reimer, Transmission Electron Microscopy, Springer, Berlin (1993). 11.2. N. J. Zaluzec, “Electropn Energy Loss Spectroscopy in Advanced Materials,” in Transmission Electron Energy Loss Spectroscopy in Materials Science, ed. by M. M. Disko, C. C. Ahn, and B. Fultz (Minerals, Metals and Materials Society, Warrendale, PA 1992). 11.3. M. H. Loretto, Electron Beam Analysis of Materials, Chapman & Hall, New York (1994). 11.4. G. Herzer J. Magn. Magn. Mat. 112, 258 (1992). 11.5. B. D. Cullity, Introduction to Magnetic Materials, Addison-Wesley, Menlo Park, California (1972). 11.6. S. A. Majetich, J. H. Scott, E. M. Kirkpatrick, K. Chowdary, K. Gallagher, and M. E. McHenry, “Magnetic Nanoparticles and Magnetocrystalline Anisotropy,” Nanostruct. Mat. 9, 291 (1997). 11.7. S. Kirkpatrick, M. E. McHenry, M. DeGraef, P. A. Smith, Y. Nakamura, and D. E. Laughlin, “Magnetic Properties of Carbob-Coated Sm-Co-C and Mn-Al-C Alloy Nanoparticles,” Scr. Metallurg. Mat., 33(10/11), 1703 (1995). 11.8. E. M. Brunsman, J. H. Scott and S. A. Majetich, “Magnetic properties of monodomain Nd-Fe-B-C nanoparticles,” J. Appl. Phys. 79(8), 5293 (1996).

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12

Electron Energy Loss Spectroscopy of Polymers Matthew R. Libera* and Mark M. Disko** *Dept. of Chemical Engineering Stevens Institute of Technology Hoboken, New Jersey 07030 **ExxonMobil Research and Engineering Company 1545 Route 22 East Annandale, New Jersey 08801-3059

12.1 INTRODUCTION The need to better understand the morphology of polymeric materials is a compelling one. Polymers are ubiquitous, and their use throughout society continues to grow at an explosive rate. They have, for example, long been used in a great variety of structural applications, and there is a vast literature on homopolymer blends and composites for such uses. The materials-substitution paradigms practiced for decades in the transportation industry have now moved into major infrastructure applications, and today, for example, we see polymer–matrix composites in bridge building and in chemical processing where their corrosion resistance and formability make them increasingly attractive relative to steels. Likewise, because of their controllable barrier properties, polymers have been extensively used in the food packaging industry, often in applications simply taken for granted. The 12-oz carbonated beverage bottle, for example, represents a multibillion dollar market currently dominated by aluminum which continues to be approached by novel polymers with highly engineered barrier properties. In electronic applications, polymeric LEDs have been developed and are extremely attractive because of their ease of manufacture and cost effectiveness in many applications that do not need, or cannot use, silicon technology. There have also been impressive advances in the development and use of polymeric materials in biomedical applications, particularly in tissue scaffolding and drug delivery, where morphology must be controlled at the length scales of proteins and cells. This list of applications, and the concomitant need to study polymer morphology, will no doubt continue to grow as novel paradigms emerge that link macromolecular synthesis to nano- and mesoscale morphology. 419

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Electron-optical methods are needed to study polymer morphology for the same reason they are needed to study the microstructures of metals, ceramics, and semiconductors. Combinations of imaging, diffraction, and spectroscopy provide a level of spatial resolution not available using other types of radiation such as light, X-rays, or neutrons. However, because most polymers are composed principally of carbon and have similar densities, contrast based on elastic electron scattering is typically poor. In the most traditional method of imaging polymer morphology in the TEM, many polymers are thus exposed to differential heavy-element stains to induce contrast based on differential elastic scattering. Inelastic scattering and the measurement of inelastically scattered electrons by energy loss spectroscopy open an alternate means of contrast based on the intrinsic interactions between the incident electrons and the specimen without the need for stains. Signiﬁcantly, polymeric materials are well suited for study by energy loss techniques for two important reasons. First, polymers are largely composed of light elements such as C, N, and O whose K-edge energy losses lie in the sub-1000 eV range that is especially well suited for study using the spectrometers commercially available for modern TEMs and STEMs. Second, largely because of the variety of both sigma and pi bonding states characteristic of carbon, there is a rich valence structure that can be exploited to differentiate otherwise similar polymers. This chapter describes the application of electron energy loss spectroscopy [EELS] to the study of polymers and, particularly, the application of spatially resolved EELS measurements to the determination of polymer morphology. This chapter begins with a brief overview of typical morphological questions presented by homopolymer blends and self-assembling block copolymers and then goes on to explain how these questions are addressed using conventional imaging methods in the transmission electron microscope (TEM). Conventional TEM imaging of polymers is largely based on principles of elastic electron scattering coupled with heavy element staining to induce differential elastic scattering between different morphological features. The discussion next turns to opportunities in characterizing polymers and their morphology based on inelastic electron scattering using energy loss spectroscopy. Typical energy loss features characteristic of both the low-loss and core-loss regions are described as well as how these features have been exploited in spectrum imaging, spectrum proﬁling, and ﬁltered imaging. Imaging based on inelastic scattering does not require heavy element stains to induce contrast and, thus, this alternate imaging method brings a number of important advantages. Finally, the chapter addresses the radiation sensitivity of most polymers and how this sensitivity deﬁnes limits on incident electron dose and, hence, on achievable spatial resolution. A very important take-home message is that much can be learned about polymer morphology when working carefully within the bounds of dose-limited constraints. This chapter is designed to address two somewhat different audiences. On the one hand, there is a community of highly skilled microscopists who concentrate primarily on problems associated with inorganic materials. This chapter introduces that community to the relevant length scales and characteristics associated with polymer morphology and how some of the spectroscopic features associated with polymers can be exploited to study morphology. Polymer EELS experiments have

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much in common with similar experiments done on inorganic materials, the greatest distinction being the radiation sensitivity of most polymers and the limit this places on spatial resolution. On the other hand, there is a community of excellent polymer scientists and engineers who increasingly need the spatial resolution associated with electron-optical methods to understand polymer morphology in the broader context of synthesis, processing, and properties. For this community the chapter outlines some of the important basic aspects of both elastic and inelastic electron scattering in the context of polymer microscopy emphasizing, in particular, the methods and opportunities associated with spatially resolved chemical imaging. 12.2 12.2.1

ELEMENTS OF POLYMER MORPHOLOGY Macromolecular Architecture

The features of morphological interest in a polymeric material most relevant to electron-optical studies range in their characteristic length scale from ∼ 1 nm to 1 µm. To a very substantial degree, these features are deﬁned by the architecture of individual polymer molecules. One should note that polymer chemists often refer to a molecular architecture as the polymer microstructure in contrast to how the word microstructure is traditionally used by the inorganic materials community where it denotes mesoscale morphological structure. There are two general categories of polymer molecular architecture that broadly deﬁne the length scales associated with morphological features in polymers: homopolymers and block copolymers. Both are based upon covalently connected monomeric units. A brief listing of monomeric repeat units for some of the polymers mentioned in this chapter is given in Fig. 12.1. Comprehensive tabulations can be found in Mark [12.1] and Brandrup [12.2]. Homopolymers are macromolecules that consist of n repeats of a single monomeric unit. Figure 12.2 illustrates the nature of homopolymers A and B with schematics of possible molecular conformations for each. These schematics assume that each polymer molecule has a high degree of ﬂexibility such as that characteristic of poly(ethylene). Macromolecular ﬂexibility and, hence, conformation become constrained due to: (i) steric hinderance imposed by bulky side groups such as the phenyl ring in poly(styrene); (ii) branching; (iii) cross-linking; and (iv) reduced rotational freedom due bonds along the main chain with nonsigma character such as that in poly(dienes) [–C=C–] and in rigid-rod polymers such as poly(HBA–HNA) liquid crystal polymers (LCPs), which contain main-chain aromatic rings. Block copolymers consist of homopolymer sections linked together by a strong covalent bond along the main chain (Fig. 12.2). Diblock copolymers, usually denoted AB, consist of one [A]n section with a degree of polymerization n linked to one [B]m section with a degree of polymerization m. Higher order block copolymers such as triblocks (e.g., ABA) and multiblocks (ABABAB...) are also possible. Random copolymers consist of a random sequence of monomeric units all covalently connected (e.g., ABAABABBBAAB). Block copolymers can also be synthesized using three or more different monomer units (e.g., ABC triblocks) as well as in

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Fig. 12.1 Molecular structures of some common monomeric units.

nonlinear architectures such as stars where different blocks emanate from a common point. Proteins are natural polymers whose monomeric repeat units are derived from twenty different amino acids [12.3,12.4]. Most proteins can be categorized as random copolymers that involve a subset of the 20 possible monomeric structures. Synthetic poly(amino acid) homopolymers have been synthesized, and many are commercially available.

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Fig. 12.2 A homopolymer consists of n repeats of a single monomeric unit (e.g., A). Interactions between different homopolymer macromolecules are largely deﬁned by secondary forces. A block copolymer consists of homopolymeric sections([A]n – [B]m ) connected together by primary bonds along the polymer main chain.

12.2.2

Macro- and Microphase Separation

A polymer blend, occasionally referred to as a polymer alloy, involves the mixing of two or more dissimilar homopolymer constituents [12.5,12.6]. The enthalpic component of the interaction between different molecules of the different species, typically quantiﬁed by the so-called chi (χ) parameter in the Flory–Huggins formulation [12.7,12.8], is usually, though not always, a repulsive one. A repulsive interaction gives rise to macrophase separation, where a phase rich in one polymer species forms a dispersion in a continuous matrix phase rich in the other polymer species. Under certain conditions, the two phases can be cocontinuous [12.9]. The length scale associated with macrophase separation is typically of order 1 µm or more. This length scale is largely deﬁned by macroscopic coarsening phenomena which act to minimize the total surface energy associated with the dispersed phase and is limited principally by the high viscosity and low mobility characteristic of polymeric materials. The questions one typically asks concerning the morphology of a homopolymer blend center on the size distribution of the dispersed phase, the shape of the dispersed phase, and the spatial distribution of the dispersed phase as a function of various polymer-processing variables. As one example, Fig. 12.3 shows a bright-ﬁeld TEM image of a melt-mixed poly(styrene)–poly(ethylene) [PS–PE] homopolymer blend where the dispersed PS phase has been preferentially stained by RuO4 . In contrast to blends, phase separation in block-copolymer systems is subject to the additional constraint that the different sections (blocks) of homopolymer within a given molecule are covalently bonded together. As in blends, a repulsive χ parameter favors phase separation, but the covalent bond linking the A and B blocks in any given

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Fig. 12.3 Bright-ﬁeld TEM image of a melt-mixed poly(styrene)– poly(ethylene) blend stained by exposure to RuO4 vapor that preferentially stains the aromatic rings in PS.

copolymer molecule prevents the chemically dissimilar species from moving far away from each other. The length scale characteristic of phase separation in a block copolymer system is consequently deﬁned by the characteristic size of the individual blocks, typically quantiﬁed by their radius of gyration, Rg , and is of order 10–100 times ﬁner than the length scale characteristic of macrophase-separated homopolymer blends. Block-copolymer systems give rise to a range of beautiful morphologies depending on composition and molecular weight. Block-copolymer morphology has been particularly well studied in the model system of poly(styrene)–poly(isoprene) (PS–PI) diblock copolymers, and Fig. 12.4 shows a schematic of possible microphaseseparated morphologies in this system as a function of composition. Research on this class of polymer has been extensively reviewed (e.g., [12.10–12.12]), and it continues very actively worldwide. 12.2.3

Polymer Interfaces

Just as in metals, ceramics, and semiconductors, interfaces in multiphase polymers play a signiﬁcant role determining properties and performance. Consequently, a tremendous amount of science and engineering effort has been invested in studying polymer-polymer interfaces. Driven largely by entropic forces, two immiscible homopolymers entangle at an interface (Fig. 12.5). Many important bulk properties — strength, fracture toughness, . . . — are in great measure determined by the nature and extent of the entangling at interfaces. The spatial variation in composition across an interface can be modeled in a number of ways [12.13–12.15]. Figure 12.5 illustrates the Helfand–Wasserman hyperbolic tangent model originally developed for block copolymer interfaces [12.16], which gives the density of component A, ΦA (x), as a function of spatial coordinate, x, as: 2x 1 ΦA (x) = (12.1) 1 − tanh 2 aI where the interfacial width, aI , is related to the statistical√segment length of polymer A, b, and the Flory interaction parameter, χ, by aI = 2b/ 6χ. Interfacial widths can vary from ∼1 nm to tens of nanometers depending on χ and the molecular weight.

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Fig. 12.4 Schematic description of characteristic morphologies in poly(diene)poly(styrene) diblock copolymers as a function of composition (courtesy of E. L. Thomas).

Notably, even polymer interfaces that are considered to be narrow are larger than interfaces in most inorganic systems where interfaces can be atomically sharp. Practically speaking, interfacial control is often accomplished using so-called compatibilizers, often a diblock copolymer, with thermodynamic afﬁnity for both of the polymers on either side of the interface. A good analogy can be drawn to the constitution of salad dressing. Salad dressing is typically based on a physical mixture of two immiscible liquids – water and oil. Homopolymer blends are thermodynamically similar to water and oil. They don’t want to mix, and the minor phase forms a dispersion of droplets in a continuous matrix of the major phase. The important distinctions between polymers and salad dressing are largely kinetic ones. The manner in which mixing occurs is thus different — polymer processing employs powerful machinery. And, the time scale over which mixing and demixing occurs – minutes/hours rather than seconds – is also different. A dispersion of oil droplets in a water matrix can be kinetically stabilized by the introduction of an emulsiﬁcation agent which preferentially localizes itself to the oil–water interfaces and lowers the

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Fig. 12.5 A hyperbolic-tangent compositional proﬁle across an interface between entangled A– B homopolymers.

surface energy there. This is the same motivation driving the use of compatibilizers in polymeric systems. The structural nature of polymer–polymer interfaces, as well as interfaces between polymers and inorganic adherends, has largely been studied using model one-dimensional (1D) geometries by X-ray and neutron scattering techniques. Comparatively little work on such interfaces has been done using the electron microscope. The complex geometries typical of many engineering applications, however, are well suited to the high spatial resolution techniques afforded by electron optics, and there is continued room for TEM-based research in this area. 12.3 ELASTIC SCATTERING AND STAINING METHODS 12.3.1

Elastic Scattering

While there is certainly a great number of both natural and synthetic polymers with sufﬁcient crystallinity to enable electron-diffraction studies, bright-ﬁeld imaging based on Bragg contrast, and high-resolution imaging (e.g., [12.17,12.18]), many polymers cannot be studied by these techniques because they are amorphous or semicrystalline. These polymers thus lack strong Bragg peaks required for good diffraction contrast. Furthermore, most polymers have similar compositions and densities. They thus exhibit similar elastic electron scattering behavior, so using an objective aperture to remove electrons scattered to high angles typically leads to poor image contrast between different polymer phases. These properties have substantially determined how polymer microscopy has been executed in the past, namely, that heavy element stains are employed to induce strong differential contrast by elastic scattering.

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Models describing the elastic electron-scattering characteristics of various atomic species have been developed which closely resemble the model established by Rutherford [12.19] and Bohr [12.20] for the scattering of alpha particles by gold. The partial differential cross-section for elastic scattering based on a Rutherford model modiﬁed to account for screening of the nuclear potential, dσe /dΩ, is given as [12.21,12.22]: 4γ 2 Z 2 dσe 1 (12.2) = 2 2 dΩ a0 k0 θ2 + θ02 where Z is atomic number, γ is the relativistic correction factor, a0 is the Bohr radius, θ is the scattering angle, and θ0 is the so-called characteristic scattering angle given as θ0 = 1/(k0 r0 ), where k0 is the incident wave vector and r0 = a0 /Z 1/3 is the atomic radius. One can use this expression, for example, to show that the increase in elastic scattering expected by exchanging a nitrogen atom for the number 2 aromatic carbon in poly(styrene) [PS] to create poly(2-vinyl pyridine) [PVP, see Fig. 12.1] increases the elastic scattering strength of PVP by less than ∼ 5% relative to PS. While some morphological information can be gleaned at this level of contrast, image quality tends to be relatively poor, and ﬁne details are usually unresolvable. 12.3.2

Heavy Element Staining

To enhance the elastic scattering strength of one polymer phase relative to another, heavy element stains are frequently used to label some morphological feature. For example, OsO4 is routinely used to preferentially label unsaturated carbon double bonds such as those found in poly(butadiene) (PB) and poly(isoprene) (PI) [12.23]. Ruthenium tetroxide, RuO4 , is used to preferentially label aromatic rings [12.24]. Energy loss spectroscopy has recently provided better insights into both the OsO4 and RuO4 staining mechanisms [12.25,12.26]. Iodine is used to label PVP [12.27,12.28]. Uranyl acetate labels amine groups such as those in poly(amidoamine) (PAMAM) dendrimers [12.29]. Negative staining methods, commonly practiced in the biological community, can also be applied to polymeric systems [12.30]. Other protocols are reviewed by Sawyer and Grubb [12.31]. The signiﬁcance of heavy element staining is illustrated by Fig. 12.6, which plots the differential cross-section for elastic scattering described by Eq. 12.2 as a function of scattering angle for C, Ru, and Os. Os elastically scatters electrons approximately sixteen times more strongly than carbon, and the introduction of even a small amount of Os preferentially into one phase of a multiphase polymer morphology can lead to signiﬁcant image contrast. Figure 12.7 shows that the preferential labeling of unsaturation by OsO4 in the poly(butadiene) phase of a microphase-separated block PS-PB-PS copolymer [12.32] gives rise to substantial amplitude contrast. Heavy element staining is a highly successful cornerstone for much polymer microscopy. Staining methods coupled with bright-ﬁeld TEM imaging have been practiced since the earliest days of electron microscopy and will no doubt continue to play a very signiﬁcant role in the research and development of polymeric materials. These methods, however, are not without their shortcomings, and there are

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Fig. 12.6 Partial differential cross-sections for elastic electron scattering from osmium, ruthenium, and carbon.

several reasons why one would like to use imaging methods that do not rely on stains. First, despite the fact that a great variety of candidate stains has been identiﬁed and used, there are many situations where a good stain, which preferentially labels a speciﬁc morphological feature in a polymeric material, cannot be identiﬁed. This arises when different phases or features have similar chemical functionality and,

Fig. 12.7 OsO4 preferentially stains poly(butadiene) in a PS–PB–PS triblock copolymer. Os-stained PB elastically scatters electrons to higher angles than unstained PS. In the objective lens back focal plane, electrons scattered to high angles are blocked by the objective aperture. The continuous PB phase thus has dark image contrast (lower left). At focus, an unstained specimen (lower right) gives no contrast between PS and PB domains.

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hence, have comparable afﬁnities for the staining agent. For example, poly(lactic acid) [PLA] and poly(glycolic acid) [PGA] are used extensively and increasingly in biomedical applications (e.g., resorbable suture materials). They are thermodynamically incompatable and tend to phase separate when blended or microphase separated when used as copolymers. They differ only in the nature of a single side group, however. PLA replaces a hydrogen atom with a methyl group (CH3 ). While this change affects many properties, it does not open dramatic chemical differences that lend themselves well to staining, and the morphology of many PLA–PGA based materials thus remains incompletely understood. A second limitation of stains is that there are situations where more than one morphological feature reacts with a given stain, which can cause confusion when trying to interpret image contrast. An example is given by Fig. 12.8, which shows: (a) a homopolymer blend of 30 wt% poly(ethylene glycol) [PEG] dispersed in a poly(pseudo amino acid) known as poly(DTE carbonate); and (b) a random blocky copolymer of poly(DTE carbonate) – co –40 wt% PEG 1000. Poly(DTE carbonate) is an aromatic polyester, whereas PEG is purely aliphatic. These specimens were exposed to RuO4 to preferentially label the aromatic phase. The homopolymer blend specimen of Fig. 12.8a shows contrast that is easy to interpret. The poly(DTE carbonate) phase reacts with the RuO4 and it elastically scatters electrons to higher angles, leading to dark contrast in the bright-ﬁeld image of Fig. 12.8a. The PEG phase thus has the light contrast and makes up the minor phase in this morphology. The copolymer specimen is made from the same chemical species as the blend, and one might anticipate similar staining behavior when exposed to RuO4 . However, Fig. 12.8b shows three levels of contrast and not two. One can offer several speculations concerning the meaning of this contrast, but understanding the origin of such ambiguous contrast can become a research activity in itself and such research is often difﬁcult to pursue in the broader context of materials-development activities.

Fig. 12.8 RuO4 staining of: (A) a poly(DTE carbonate) – PEG homopolymer blend; and (B) a poly(DTE carbonate)–co– PEG1000 random blocky copolymer. The two levels of contrast in image (A) can be easily interpreted whereas origin of the three levels of contrast in (B) is less obvious.

A third shortcoming of staining techniques is that staining can lead to artifactual structure that complicates image interpretation, particularly at high resolution (e.g., 1–5 nm). Staining by exposure to RuO4 vapor, for example, can create nanosized RuO2 particles on the specimen surface [12.25]. This effect is true of a number of other stains as well. Particularly in applications involving inorganic nanoparticles

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integrated into polymeric matrices [12.33,12.34], such nanoscale staining artifacts must be avoided. Finally and importantly, the majority of staining agents used in electron microscopy are toxic. OsO4 , for example, can cross-link unsaturated hydrocarbons in physiological structures such as lipid membranes just as easily as it can cross-link unsaturated hydrocarbons in synthetic polymers. It is known for its reactivity with the human cornea and can lead to blindness if improperly handled. In general, the stains used in electron microscopy are usually both highly reactive and made of high atomic number elements. While handling protocols have been developed to deal with the safety issues associated with stains, one would prefer to avoid them altogether, if possible. 12.4 INELASTIC SCATTERING AND SPATIALLY RESOLVED SPECTROSCOPY Inelastic electron scattering provides an alternate signal with which to determine polymer morphology without the need for stains. The opportunity is also there to make such measurements fully quantitative both in terms of chemistry and composition. The partial cross-section for inelastic scattering differential with respect to scattering angle, including valence and core excitations, can be modeled as [12.21,12.22]: 1 4γ 2 Z (12.3) σi = 2 4 1 − a0 q {1 + (qr0 )2 }2 2

where q 2 = k02 (θ2 + θE ) and θE = E/(γm0 v 2 ) is the characteristic scattering angle for mean energy loss E. The ratio of the inelastic scattering cross-section, σi , to the elastic scattering cross-section scales as 1/Z [12.35] and exceeds unity for elements with Z less than ∼20. As illustrated by Fig. 12.9, the inelastic scattering for 200-keV electrons by carbon is more signiﬁcant than the elastic scattering up to scattering angles of order 10 mrad. This trend continues for lower energy (e.g., 100 keV) incident electrons as well. The fact that inelastic scattering is so signiﬁcant in carbon suggests that energy loss methods should be well suited to studies of polymers that predominantly consist of carbon and other low-Z elements such as nitrogen, oxygen, and sulfur, among others. Energy loss spectroscopy has long been used with low spatial resolution to study the electronic structure of materials in condensed solids [12.36], gases [12.37], and at surfaces [12.38]. Applications to the measurement of materials morphology have necessarily linked energy loss spectroscopy to the high spatial resolution afforded by electron-optical systems. The basic principles of electron energy loss spectroscopy in the transmission (TEM) and scanning-transmission (STEM) electron microscopes are now very well established [12.21,12.39,12.40]. Signiﬁcant conceptual advances were made with the advent of spectrum imaging and spectrum proﬁling [12.41,12.42] and energy ﬁltered imaging [12.43]. The effectiveness of these methods continues to grow with advances in instrumentation such as: (i) more efﬁcient microscope-spectrometer

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Fig. 12.9 The partial differential cross-section for elastic and inelastic scattering of 200-keV electrons from carbon. For a large range of angles, inelastic scattering from carbon is substantially more probable than elastic scattering from carbon.

coupling to digital control, data acquisition, and data processing; (ii) higher quantum efﬁciency electron detectors; and (iii) more effective spectrometer/ﬁlter designs. Enhanced user interfaces have also rendered electron spectroscopic methods more accessible to applications-oriented researchers. The essential qualitative concept associated with spatially resolved energy loss spectroscopy is illustrated by a three-dimensional (3D) data cube (Fig. 12.10). Two dimensions correspond to lateral position coordinates x and y. The third dimension corresponds to electron energy loss. All or part of the 3D (x, y, ∆E) data cube can be generated either by the STEM-based technique called spectrum imaging or by the TEM-based technique called energy ﬁltered imaging (often using the acronym EFTEM). Figure 12.10 illustrates schematically how a 3D data cube is generated by spectrum imaging in a STEM. A focused electron probe, with a Gaussian intensity distribution characterized by a full width at half maximum (FWHM) typically of order 0.2–5 nm, is digitally rastered across a thin (∼10–100-nm thick) specimen. The dwell time at each pixel and the inter-pixel spacing are controllable parameters. At each speciﬁc pixel, for example, (x1 , y1 ), in the raster an entire energy loss spectrum I(∆E) is collected. The result is a 3D cube of spatially resolved spectroscopic data. If the data are collected along only one spatial dimension, for example, in a line across an interface, one speaks of a spectrum proﬁle. If the data are collected along two spatial dimensions, one speaks of a spectrum image. Substantial post-acquisition processing of the data is typically required to extract speciﬁc chemical information characterizing each pixel position. Spectrum imaging/proﬁling lends itself relatively well to such processing, since both pre-edge and post-edge data are collected and provide for optimal background modeling. While each spectrum can be collected relatively quickly by a parallel spectrometer (PEELS), generation of the 2D and 3D datasets is a serial process and can require relatively long acquisition times. Modern acquisition systems are now fully automated, however, and they can correct for the effects of specimen drift over these extended acquisition times.

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Fig. 12.10 The 3D spectrum data cube is generated by digitally rastering a focused electron probe across a thin specimen such that an entire energy loss spectrum is collected at each pixel.

The alternate methods of energy ﬁltering based on the Castaing–Henry, Omega, and most recent alpha in-column ﬁlters as well as on the Gatan post-column imaging ﬁlter (GIF) collect data in parallel at a speciﬁc energy loss. In Fig. 12.10, this approach corresponds to collecting a horizontal slice from the data cube at an energy loss ∆E2 for all x and y. A larger subset of the 3D data cube can be generated by collected ﬁltered images at a series of different energy loss values. The spectrum-imaging and energy ﬁltered imaging approaches each have their merits [12.22,12.40]. In choosing one over the other, one must consider tradeoffs among data acquisition time, completeness of spectral acquisition, precision of background modeling and subtraction, and energy resolution, among other variables. One must be aware when reading the literature that spectrum imaging and energy ﬁltered imaging refer to two related but different methods for collecting spatially resolved energy loss data. 12.5

CORE-LOSS SPECTRA, PROFILES, AND IMAGES

Core-loss data can be used to assess both compositional and chemical information characteristic of polymeric materials. Compositional information is given by the

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Element Edge Edge Energy (eV) C K 284 N K 401 O K 532 F K 684 99 Si L2,3 165 S L2,3 Cl L2,3 200 Table 12.1 Core-loss edges for various elements common in polymeric materials (after [12.44])

position and net integrated intensity of an edge, and chemical information is given by the ﬁne structure within an edge (see, e.g., Chapter 3). Most polymers are composed of low atomic number elements that lend themselves well to core-loss spectroscopy. Some relevant core-loss edges are summarized in Table 12.1. In general, the K edges tend to be more distinct above the background and easier to work with than the L edges. Early core-loss studies from polymeric materials using commercial instruments were limited, because the ﬁrst commercially viable spectrometer from Gatan (model 607) collected data in a serial fashion. Egerton [12.45, 12.46] was among the ﬁrst to pursue careful energy loss studies of polymers and dose-dependent chemical changes response to the ionizing radiation of the TEM. Briber and Koury [12.47, 12.48] used a serial spectrometer to study nylon 6,6 and poly(chlorotriﬂouroethylene) [PCTFE]. They were able to resolve core-loss edges associated with C, O, and N in the nylon and C, F, and Cl in the PCTFE. They were also able to extract atomic concentrations of these various elements in reasonable agreement with anticipated stoichiometric values. One of the main limitations of a serial spectrometer is that it makes relatively inefﬁcient use of the electron dose to which the specimen is exposed. This exacerbates the problems associated with the susceptibility of polymeric materials to radiationinduced chemical change. Serial spectrometers typically scan an energy dispersed spectrum across a detector and record only a fraction of the spectrum at any given time. Consequently, many of the incident electrons that carry valuable spectral information are not collected by the spectrometer. In part because of the nature of the commercially available instrumentation, relatively little work was thus done during the 1970s and 1980s applying EELS to polymers in the TEM, particularly on problems requiring spatial resolution. The development of energy loss spectrometers with parallel recording (PEELS), reviewed by Egerton [12.21], substantially reduced the total dose needed to collect a useful spectrum. A parallel spectrometer uses a photodiode array or a 2D CCD chip to record all channels of a spectrum, typically of order 1024, simultaneously. Together with enhancements in computer technology as well as in more approachable

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user interfaces, the 1990s saw increasing application of energy loss spectroscopy to polymeric problems in applications both with and without the need for spatial resolution, where the constraints on electron dose are greater than those associated with a defocused incident beam. The fact that chemical information can be resolved within the details of the carbon K edge using a relatively common microscope-spectrometer conﬁguration has been nicely demonstrated by Varlot et al. [12.49]. One example is illustrated by Fig. 12.11. This shows background-subtracted carbon core-loss spectra for a series of different polymers. The ﬁne structure within these peaks can be attributed to different chemical functionalities. Among these are [12.49,12.50]: C=C π ∗ (284.8 eV); C=O π ∗ (288.0 eV); C–O σ ∗ (290.4 eV), and C–C σ ∗ (291.1 eV). In principle, these differences can be exploited to distinguish spatial variations in local compositions involving mixtures of two or more of these polymers.

Fig. 12.11 Background-subtracted carbon core-loss spectra from a series of different polymers showing variations in near-edge ﬁne structure associated with different types of carbon bonding in each polymer. PMMA = poly(methyl methacrylate); PET = poly(ethylene terepthalate); PS = poly(styrene); PP = poly(propylene); PE = poly(ethylene); bd = low density PE; hd = high density PE; PB = poly(butadiene). (after [12.49]).

Figure 12.12 illustrates an example from Varlot et al. [12.49–12.51] showing differences in near-edge ﬁne structure in both the oxygen and carbon K-edges characteristic of PET and PMMA. The background-subtracted integrated intensities under these edges can be used [12.49] to show that the atomic oxygen/carbon ratio in each polymer is 0.40 with a precision of ∼ 2.5% imposed by the dose constraints of the PMMA. This experimentally derived ratio is in good agreement with the stoichiometric O/C ratio of 4/10 in PET and 2/5 in PMMA. A second example, discussed again later in this section, involving quantitative compositional analysis is given in Fig. 12.13. This presents background-subtracted carbon and nitrogen K-edges characteristic of poly(2-vinyl pyridine) where the nominal C/N stoichiometric ratio is 7:1 [12.52]. When the integrated counts from each signal is combined with scattering cross-sections calculated based on a hydrogenic model [12.21] for the appropriate collection angle (β = 10 mrad) and energy window

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Fig. 12.12 Core-loss spectra showing carbon and oxygen edges in poly(ethylene terepthalate) [PET] and poly(methyl methacrylate) [PMMA] (after [12.49]).

(∆E = 30 eV), these data predict a C/N ratio of 7.05 ± 0.2. Similar analysis of nylon 6,6 by Ban et al. [12.53] found a nitrogen concentration of 12 ± 2 at% in comparison to the stoichiometric value of 14.3 at%. The difference was attributed to contributions to the collected signal from nonnylon phases. Spatially resolved core-loss data are increasingly being used to map distributions of one or more chemically distinct polymer phases in another. These studies are being done most frequently using TEM-based methods with energy ﬁltering using both in-column (e.g., Omega and related ﬁlters) and post-column (e.g., GIF) spectrometer systems. As illustrated schematically by Fig. 12.10, ﬁltered imaging essentially maps in two spatial dimensions the intensity of the core loss signal integrated over an energy window typically on the order of 10–30 eV. The background can be subtracted by jump-ratio and three-window schemes, both of which model the background using pre-edge spectral data. An example from a polymer blend is shown in Fig. 12.13. These particular data were collected using a Zeiss 912 TEM with a Castaing–Henry ﬁlter and recorded on image plates. The specimen is a homopolymer blend of poly(carbonate) [PC] dispersed in poly(styrene)-poly(acrylonitrile) [SAN] copolymer. The spectra show that these two polymers can be readily distinguished on the basis of their oxygen and nitrogen contents (Fig. 12.14), and the ﬁltered images clearly show that the oxygen-containing PC phase is dispersed in the nitrogen-containing SAN matrix. A number of comparable studies have recently appeared in the literature addressing both overall morphology [12.33,12.54–12.60] as well as polymer interfaces [12.61,12.62]. Studies of polymer morphology by core-loss spectrum imaging in STEM mode have thus far been less commonly practiced than core-loss energy ﬁltering in TEM mode. This is in part due to the fact that spectrum imaging requires longer dataacquisition times to collect spectra that contain adequate counting statistics in coreloss edges. Long acquisition times present a problem partly from the point of view of user convenience but, more signiﬁcantly, because of specimen drift during the acquisition. Uncorrected drift can lead to substantial distortion of the spatial coordinates in a 3D spectrum datacube. The implementation of various schemes for

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Fig. 12.13 Background-subtracted core-loss spectra of the carbon K (a) and nitrogen K (b) edges from poly(2 vinyl pyridine) [PVP]. The lower ﬁgure is a HAADF STEM image of a PVP nanosphere on a holey carbon support grid. The inset shows N and C core loss images of the nanosphere (after [12.52]).

online drift correction has been attempted on and off with varying degrees of success over the past 20 years. Substantial progress in this and other online processing methods was made during the late 1990s with the integration of fast, inexpensive, and user-friendly computer systems into microscopes. One can consequently anticipate more applications of energy ﬁltering based on spectrum imaging. Figure 12.13 presents an example of polymer spectrum imaging from the Stevens group [12.63] using a Philips CM20 FEG TEM/STEM interfaced to an EmiSpec Vision data acquisition/control system and a Gatan 666 PEELS spectrometer. The specimen is a PVP sphere on a holey-carbon TEM grid. The specimen was cooled

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Fig. 12.14 Energy ﬁltered nitrogen and oxygen maps using a Zeiss 912 with an Omega ﬁlter of an SAN-PC blend [poly(styrene-co-acrylonitrile) - poly(carbonate)] showing part of a PC particle in the SAN matrix (courtesy of S. Horiuchi).

to approximately −125◦ C in a single-tilt cryo stage. Cooling to a temperature in this vicinity is an effective way to reduce contamination because it reduces the diffusivity of hydrocarbons on a specimen surface. In addition to collecting spectral data from a 70 × 70 pixel area centered on the polymer nanospheres, drift correction was implemented by collecting HAADF STEM images from an adjacent subimage area at 70-spectra intervals. Based on an autocorrelation with the initial subimage region, electronic shifts were imposed to correct for drift after each interval. The drift correction was thus imposed 70 times during the experiment. A 3D spectrum dataset was generated from the 70×70 pixel subregion around the PVP particle itself at an interpixel spacing of 5 nm and a pixel dwell time of 1.5 s. Figure 12.13 shows the resulting maps of nitrogen and carbon derived from spatially resolved background-subtracted core-loss data. Despite the fact that the specimen drifted at a rate of almost 6 nm/min for ∼ 120 min, integrity of undistorted spatial coordinates is largely preserved by the drift correction. One should recognize that such long-time data collection is entirely automated, and post-acquisition processing routines are increasingly accessible that extract compositional maps over time scales of minutes once the spectrum datasets have been acquired. In addition to the spectral data which can be collected using electron spectroscopy, valuable information for interpreting the ﬁne structure of EELS core-loss edges can

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be gotten from X-ray absorption experiments. The energy resolution afforded by absorption measurements using synchrotron sources is on the order of 0.1 eV. This is substantially better than that of present-day commercially-available TEM-based electron energy loss spectroscopy systems, where the energy resolution is limited by the energy spread of the electron source [12.64–12.66]. Cold ﬁeld-emission electron sources limit energy resolution to 0.4–0.5 eV. Schottky sources have a resolution of order 0.8–1.0 eV. Thermionic sources have a resolution of ∼ 1.4 eV or more. Consequently, X-ray absorption experiments typically resolve greater detail in core-loss ﬁne structure than do electron energy loss studies, at least as presently practiced. Figure 12.15 [12.67] illustrates the differences by comparing core-loss spectra collected from poly(ethylene terepthalate) [PET] using both thermionic and cold ﬁeld-emission electron sources as well as two different synchrotron sources.

Fig. 12.15 Comparison of carbon core-loss spectra from PET collected by electron energy loss spectroscopy (top two curves) and X-ray absorption spectroscopy (lower two curves). (Courtesy of H. Ade).

While the electron energy loss spectra show many of the basic features of the nearedge structure, the X-ray data provide more detail with clearly identiﬁable peaks and subpeaks. This level of detail enables accurate identiﬁcation of speciﬁc spectral features with particular chemical functionalities. Much of the methodology behind such approaches is reviewed by Storr [12.68]. The scanning transmission X-ray microscope (STXM) couples the chemical analysis capabilities of X-ray absorption spectroscopy with spatial resolution to provide a powerful tool for studying spatially resolved structure with extremely high chemical speciﬁcity. The method is well suited to the study of polymers and biological materials [12.67,12.69,12.70]. The spatial resolution is currently limited to ∼ 40 nm by the fabrication of appropriate diffractive optics [12.70,12.71].

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Electron monochromators can substantially improve the energy resolution associated with EELS measurements, and prespecimen energy ﬁltering schemes to monochromate the incident electron energy have been used successfully in electron energy loss experiments since the 1960s [12.72,12.73]. In studies of both low-loss and core-loss structure in polymers [12.74,12.75], prespecimen ﬁltering has provided energy resolution of order 0.1–0.2 eV where substantially greater spectral detail can be seen and related to speciﬁc electronic structure. Figure 12.16 presents an example from Fink et al. [12.74] that shows how the ﬁne structure of both the carbon and nitrogen K edges vary in poly(pyrrole)-based polymers as various sidegroup and main-chain moieties are varied. These spectra were collected with 0.180-eV spectral resolution.

Fig. 12.16 The near-edge ﬁne structure in a series of pyrolebased polymers can be effectively resolved at 0.180 eV spectral resolution using prespecimen energy ﬁlters to monochromate the incident electron beam. Left: carbon K -edge; Right: nitrogen K -edge. (after [12.74]).

Prespecimen electron monochromators are now beginning to emerge on commercial instruments using technologies similar to those developed for post-specimen ﬁltering. Robust drift-compensation schemes will grow ever more signiﬁcant in monochromated electron systems, since prespecimen ﬁltering will reduce the incident beam current and necessitate even longer acquisition times. 12.6

LOW-LOSS SPECTRA, PROFILES, AND IMAGES

Low-loss data, in an energy loss regime of less than ∼ 50 eV, principally provide information concerning valence electron states. Because of the variety of different bonding conﬁgurations associated with carbon, the low-loss regime characteristic of polymers can be rich with spectral features and ﬁngerprints. Furthermore, because

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the cross-section for inelastic scattering decreases with increasing energy loss, one can anticipate an inelastic signal per unit incident electron dose that is approximately two orders of magnitude higher at low losses than at high losses. The low-loss regime thus makes more efﬁcient use of the incident electron dose than the core-loss regime. Empirically one also ﬁnds that the effects of multiple inelastic scattering are less complicating at low losses than at high losses. Such multiple scattering typically depletes intensity in the low-loss regime rather than convolute spectral features as occurs in core-loss data. Consequently, when working in the low-loss regime, one can often use specimens thicker than needed for good core-loss spectroscopy. Among the earliest low-loss studies of polymers are measurements by Swanson and Powell [12.76] to establish the presence of a spectral feature at ∼ 7-eV energy loss in poly(styrene). Such a spectral feature was at that time anticipated based on observations of the absorption of ultraviolet (UV) light at wavelengths near 180 nm. Low-loss data subsequently collected by Ritsko and Bigelow [12.75] characterizing PS and PVP are shown in Fig. 12.17. These spectra were collected using a spectrometer with pre-specimen ﬁlter to achieve 0.100-eV energy resolution. Substantial valence-electron ﬁne structure is shown. The peak at ∼ 21 eV is attributed to a bulk plasmon. It is also sometimes identiﬁed as a σ − σ ∗ transition. The peak at approximately 7 eV is attributed to a π − π ∗ transition characteristic of the aromatic ring of polystyrene. The appearance of an additional smaller peak at slightly lower energy (4.75 eV) in PVP is attributed to the fact that the nitrogen on the aromatic ring in PVP produces an asymmetry in the delocalized wavefunction relative to that in polystyrene [12.75].

Fig. 12.17 Low-loss spectra of PVP and PS (after [12.75])

Valence-electron structure has more recently been observed by a number of researchers using commercially available instrumentation. The energy resolution is typically of order 0.6–1.5 eV, depending on the speciﬁc type of electron source. This is easily sufﬁcient to distinguish aromatic structures from aliphatic ones on the basis of π − π ∗ transitions near 7-eV loss. Poly(styrene) has been studied by several researchers [12.77–12.79]. Du Chesne [12.77], for example, shows detailed results on the nature of the recorded π − π ∗ peak as a function of specimen thickness and

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scattering angles. Similar features are observed in other aromatic polymers such as poly(ethylene terepthalate) [PET], poly(ethyl ether ketone) [PEEK], and poly(ether sulphone) [PES], among others [12.50,12.80,12.81]. Siangchaew and Libera [12.82] have shown that unsaturated C=C bonds characteristic of dienes can be distinguished from aromatics. The fact that an aliphatic polymer such as poly(ethylene) can develop a π − π ∗ type spectral feature in response to electron irradiation is also well established [12.78,12.83,12.84]. This is attributed to the loss of hydrogen with subsequent reaction between adjacent main-chain carbon atoms. Spectrum imaging and energy ﬁltered imaging using low-loss signals have both been applied to the stain-free imaging of two-phase polymers. Thus far, this type of application has relied principally on the presence or absence of the ∼ 7 eV π − π ∗ peak to differentiate aromatic structures from aliphatic ones. The 7 eV peak is often a very useful spectral ﬁngerprint in low-loss studies. Figure 12.18 presents a result from one of the earliest studies of polymers by spectrum imaging involving a homopolymer blend of polystyrene dispersed in a continuous matrix of low-density poly(ethylene) [12.78]. These image data were extracted from a 120 × 120 pixel ×1024 channel spectrum data set with an interpixel spacing of 75 nm and a pixel dwell time of 0.05 s. The dataset was collected over a period of 17.8 min. The π − π ∗ intensity in each spectrum of the dataset was quantiﬁed by multiple leastsquares ﬁtting [12.78] to reference spectra characteristic of pure poly(ethylene) and pure poly(styrene). Alternate methods of signal extraction involving background modeling and subtraction are possible [12.79]. The data represented by Fig. 12.18 are fully quantitative in the sense that the numeric value associated with each pixel represents a quantitative measure of the poly(styrene) or poly(ethylene) concentration at that particular position. Similar methods have recently been used to map the spatial distribution of an aromatic epoxy resin and an aliphatic curing agent in an epoxymatrix composite material using STEM spectrum proﬁling with ∼ 5 nm spatial resolution [12.85,12.86].

Fig. 12.18 Quantitative map of PS distribution in a PS–PE blend derived from a 3D spectrum dataset. Pure white in the image corresponds to the maximum and pure black corresponds to the minimum through-thickness PS concentrations (after [12.78]).

Energy ﬁltering has been applied to imaging polymers using low-loss spectral signals. Unlike core-loss signals, where the decaying pre-edge background can be

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estimated and removed as part of either a two-window or three-window algorithm, the rapidly changing background between the zero loss and plasmon peaks is less easily modeled based only on a small number of measurements. Nevertheless, in a number of cases the ∼ 7 eV peak is sufﬁciently strong to provide good contrast between aromatic and aliphatic moieties with no background subtraction at all. Figure 12.19 shows one such example collected using a LEO 912 with an Omega ﬁlter. The specimen was a blend of an aromatic poly(phenylene ether) dispersed in an aliphatic poly(amide) [12.57]. The aromatic-containing dispersed phase appears with bright contrast when an image is formed using a 2-eV window centered on a loss of 6.4 eV. Using a similar instrument with a 4-eV window centered on a 7-eV energy loss, York [12.87] was able to clearly resolve aromatic copolymer microdomains of order 30–50 nm in size which were dispersed in an aliphatic matrix. Based on the π − π ∗ signal, Varlot et al. [12.88] have used energy ﬁltering in a Gatan GIF system to image PS localized at interfaces in a three-component polymer blend. Most recently, Horiuchi et al. have imaged aliphatic domains in an aromatic matrix (Fig. 12.20) using an Omega ﬁlter in a LEO 200 keV TEM [12.33].

Fig. 12.19 Energy-ﬁltered image (Zeiss 912 with Omega ﬁlter) mapping the distribution of π − π ∗ intensity in a blend of aromatic poly(phenylene ether) dispersed in an aliphatic polyamide matrix. The images was collected using a ﬁltering window of 2 eV centered on 7 eV with no background subtraction. (courtesy of W. Heckmann, BASF).

12.7

CONSTRAINTS IMPOSED BY RADIATION DAMAGE

The resolution with which one can study the structure of a material that changes in response to ionizing radiation is fundamentally limited. There is a critical radiative dose to which the material can be exposed before it is changed signiﬁcantly by radiation chemistry. This critical dose sets an upper bound to the amount of radiative exposure which can be used in any given scattering experiment. At the other extreme one must also bear in mind that there is a lower bound to the exposure determined by a need to achieve statistically signiﬁcant scattered signal above the noise. These two bounds deﬁne a window, often very broad, of electron exposure within which meaningful data can be collected.

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Fig. 12.20 Energy ﬁltered π − π ∗ image of an asymmetric diblock copolymer of poly(styrene) and poly(hydroxylated isoprene) where the spherical PUIOH domains have dark contrast in a continuous matrix of PS. The inset low-loss spectrum indicates the 4 eV-wide window around the π − π ∗ peak used for imaging (after [12.33]).

The lower bound of exposure can be described in terms of an expression known as the Rose criterion [12.89] which dates back to the late 1940s in the context of television technology. The Rose criterion relates the minimum resolvable feature size to incident electron dose. It has been revisited in the context of electron scattering from polymers the minimum dose of radiation so that the signal in a pixel with a characteristic dimension of d can be distinguished from the statistical noise. A derivation is outlined below. Assume that the average number of electrons per pixel is N where an array of pixels is mapped over a specimen with a characteristic pixel dimension of d. In a detection system limited by shot noise, there will be an average ﬂuctuation in the dose per pixel that scales as N −1/2 . In order to distinguish one pixel from another, there must be a difference in the number of electrons recorded in each pixel which can be represented as ∆N . The contrast, C, between a particular pixel and the average signal is C = ∆N/N . To be detectable, the contrast must exceed the signal to noise ratio (SNR) by a factor k that Rose argues should be ∼ 5. One can make statistical arguments that the factor k should be at least 3. The signal in a particular channel is N , and, for a system dominated by shot noise, the noise scales as N 1/2 . To distinguish a signal from the noise one considers the case where ∆N = N 1/2 , so C = kN 1/2 /N = k/N 1/2 . The number of counts is given by the product of the incident electron dose D, in units of electrons/unit area, and the irradiated area associated with an individual pixel d2 . The contrast C can thus be described by the relation: k C= f Dd2

(12.4)

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where f is a factor representing the efﬁciency with which incident electrons are scattered and collected. The scattering efﬁciency can be quantiﬁed in terms of cross-sections for various inelastic processes and is discussed by Egerton [12.21] and Weiss and Carpenter [12.93]. The collection efﬁciency is quantiﬁed in terms of the Detective Quantum Efﬁciency (DQE) and typically ranges from approximately 0.01 to close to one depending upon the particular spectrometer being used. Figure 12.21 shows the relation predicted by the Rose criterion (Eq. 12.4) between incident electron dose and achievable resolution. This particular calculation assumes that k = 3, C = 0.1, and f = 0.5. The speciﬁc relation between D and d is sensitive to these parameters, but the trend that resolution decreases as dose decreases remains perfectly valid.

Fig. 12.21 Constraints imposed by the Rose criterion and the critical dose deﬁne an exposure window within which adequate signal above noise can be collected without introducing catastrophic damage to the specimen.

Spatial resolution in beam-sensitive materials such as polymers is rarely limited by the ultimate resolution of the microscope. Instead, resolution is limited by the critical dose, Dcrit , to which a specimen can be subjected before it suffers some threshold level of radiation-induced change. The critical dose corresponds to the upper bound of exposure permissible with a radiation-sensitive material. It is usually deﬁned experimentally by determining the dose at which some signal of interest falls to 1/e of its initial value [12.94]. The critical dose depends on the type of polymer system being studied as well as the speciﬁc signal being used to create contrast. One can, for example, distinguish three broad categories of specimen damage: loss of crystallinity (manifested by the fading of Bragg peaks in an electron diffraction pattern); loss of mass (manifested by a reduction in the total number of counts associated with a core-loss edge in an EELS spectrum); and change in chemical structure (manifested by degradation in ﬁne structure associated with core-loss edges or with valence peaks/edges in low-loss spectra). The critical doses characteristic of these three categories of structure are not necessarily the same. A crystalline polymer, for example, can amorphize at doses far less than those required to induce signiﬁcant mass loss or change in chemical structure (bonding). Careful spatially resolved energy loss spectroscopy must include some consideration of critical dose and how this limits resolution.

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445

Fig. 12.22 Intensity of the background-subtracted π –π ∗ intensity characteristic of an aromatic epoxy decays exponentially with increasing incident electron dose (after [12.95]).

Figure 12.22 presents one example illustrating the effects of radiation damage within the context of a spectroscopic ﬁngerprint measurable by energy loss spectroscopy. It shows the dose-dependent decay in background-subtracted π − π ∗ intensity characteristic of an aromatic epoxy [12.85,12.86,12.95]. The specimen, ∼ 50-nm thick, was prepared by room-temperature microtomy. The energy loss data were collected while the specimen was being irradiated by an incident beam approximately 50 nm in diameter. On a linear scale (Fig. 22a) the curve decays roughly as an exponential function of incident dose. On a logarithmic scale (Fig. 22b), the curve is essentially constant over many orders of dose until it drops abruptly. Deﬁning the critical dose as the dose at which these curves fall to 1/e = 0.37 of the maximum ﬁnds a critical dose Dcrit = 2 C/cm2 . This value then gives a guide to the upper limit of dose to which this particular material can be exposed and still collect spectroscopic information concerning the π − π ∗ intensity that is representative of the virgin material. Introducing the value of 2 C/cm2 onto Fig. 12.21 illustrates the window of exposure (shaded) within which enough counts can be collected to distinguish real signal from noise while simultaneously not exposing the material to so much radiation that it is irrevocably changed. The shaded dose-window described by Fig. 12.21, suggests that one should be able to achieve at least 1-nm spatial resolution, if not better, when studying this particular polymer system (DGEBA epoxy) by following this particular spectroscopic signal (π − π ∗ ). A wealth of information exists on the effects of ionizing radiation on polymer structure. Throughout the broader community studying radiation effects in materials, so-called G values are used to quantify the extent of chemical degradation as a function of energy deposited in a specimen [12.96]. This formalism dates back to the 1950’s [12.97]. Much of the data have been determined in the context of MeV irradiation of bulk specimens. G values have been discussed in the narrower context of polymer microscopy by Sawyer and Grubb [12.31]. They give useful guidelines to the relative sensitivities of different chemical moieties. One can, for example, expect

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that hydrogen evolution would be slightly higher from poly(ethylene) relative to poly(propylene) where the relevant G values are 3.7 and 2.8, respectively, in units of events per 100 eV absorbed by the specimen. G values also give some insight into the consequences of chemical change. One can thus get some sense of whether a polymer would be more likely to undergo chain scission or crosslinking as a consequence of high-energy electron irradiation. Despite their utility, however, G values are rarely referred to in the polymer microscopy community. This is due in part due to the fact that electron microscopy typically involves kV radiation of thin specimens where dose is most commonly quantiﬁed in terms of the number of electrons incident per unit area (e.g., C/cm2 ) rather than in terms of the energy deposited per unit volume. Much of the literature by the microscopy community on damage effects in polymeric specimens is experimental and follows how one or more features such as crystallinity, mass, or chemical structure change in response to dose, dose rate, and specimen temperature, among other variables. Reimer [12.22] provides a good review of much of this literature. Relatively little is known about the speciﬁc microscopic mechanisms of interaction between energetic electrons and the various possible chemical moieties within a polymer which lead to the macroscopic manifestations of damage. There continues to be research on the relative roles of valence and core excitations in generating specimen damage [12.79,12.82,12.98,12.99]. There is also ongoing debate whether specimen heating during irradiation is sufﬁcient to cause primary damage [12.50,12.79,12.100]. Understanding these mechanisms goes well beyond the interests of studies of polymers in the electron microscope. 12.8

CONCLUSION AND OUTLOOK

For approximately four decades polymer morphology has been studied in the transmission electron microscope using the conventional method of differential heavy element staining to induce contrast between morphological features based on elastic scattering. This approach has had substantial success. TEM imaging based on differential staining methods has contributed signiﬁcantly to many of the tremendous advances made in polymeric materials over the past few decades, and there is no doubt that this approach will continue to play a major in polymer-morphology studies in the years to come. Nevertheless, morphological problems exist where phase-speciﬁc stains are not known or are inadequate for particular problems. Moreover, there is a rapidly increasing number of polymer-morphology problems involving, for example, interfaces or dendrimer-like mesoscale structures that require nanometer levels of resolution. Stains can lead to artifactual structure at these ﬁne length scales and should be avoided if possible. Inelastic electron scattering and its measurement using electron energy loss spectroscopy in the TEM or STEM are providing an alternate approach to studying polymer morphology without the need for stains. This chapter has reviewed some of the opportunities for EELS-based work in both the core-loss and low-loss regions. Because they consist principally of carbon, nitrogen, and oxygen, most polymers are well suited to studies in the core loss. Because they have rich valence structure due

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to the variety of bond conﬁgurations afforded by organic materials, many polymers are also well suited to studies in the low loss. While energy loss studies of polymeric materials date back to the early 1960s, the past decade has seen an accelerating increase in the number of polymer-based groups practicing energy loss spectroscopy. This trend is almost sure to continue. There is not yet any clear consensus concerning which of the two approaches the parallel data acquisition methods, such as the EFTEM methods practiced with the Omega ﬁlter and the Gatan imaging ﬁlter (GIF), or the serial acquisition methods such as spectrum imaging (SI) - will emerge as preferred for studies of polymer morphology. There is certainly room for both. Spectrum imaging uses less total electron dose than EFTEM, and it also produces substantially more information, since an entire energy loss spectrum is collected at each pixel. Spectrum imaging, however, requires far greater time for data acquisition as well as more involved postacquisition analysis. One might anticipate, then, that the parallel-acquisition EFTEM methods may be used in polymer studies where precise quantitation can be traded for speed in solving a particular problem. Perhaps the biggest single challenge facing continued development of this ﬁeld is the need to recognize, measure, and understand the constraint imposed on data acquisition when studying a radiation-sensitive material such as a polymer. While the following may seem a trivial comment to many readers, others need to be reminded that beam sensitive polymers can be very effectively studied using quantitative electron-scattering methods. The fact that these materials are radiation sensitive does not impose an impenetrable barrier to their study. It instead imposes a constraint within which a microscopist must work. Such constraints are well recognized within the biological community practicing quantitative electron crystallography as well as in the catalyst community studying beam-sensitive ceramics such as zeolites. Depending on the polymer and the feature being studied, many polymers have critical doses ranging from ∼ 10−3 –100 C/cm2 . Even when relaxing the assumptions made for k, f , and C, relations such as that illustrated by Fig. 12.21 suggest that nearnanometer spatial resolution can be achieved under such doses. One should further recognize that most bulk polymer-morphology problems do not require spatial resolution greater than approximately one nanometer, and many polymer-morphology questions can be adequately answered with substantially poorer resolution. For example, the sharpest polymer–polymer interfaces have widths of order ∼ 1 nm. Most polymer interfaces are much wider than that. Microphase separation of block copolymers typically occurs at length scales of order 10–50 nm. Phase separation in homopolymer blends typically occurs at even larger length scales. Problems such as these are well suited for study using electron-optical techniques and will no doubt continue to be the focus of techniques based on energy loss spectroscopy and spectral imaging. Compelling challenges will remain in trying to image single molecules within a critical-dose limit.

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Acknowledgments The authors would like to thank H. Ade, W. Heckmann, S. Horiuchi, D. J. Lohse, D. N. Schulz, E. Thomas, S. Urquhart, and K. Varlot, among others, for generously providing ﬁgures and information to this chapter. M. Libera would like to thank the many students and post-docs who have pursued polymer-morphology and energy loss research at Stevens. Libera’s research in this ﬁeld has been supported by a series of grants, ably monitored by the now-retired Dr. Robert Reeber, from the Army Research Ofﬁce (the most recent being DAAD19-00-1-0481) and the NIH- National Institute for Biomedical Imaging and Bioengineering (grant No. P41 EB 000922-01).

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Index

α (convergence angle), 30 (convergence angle), 50, 77–78, 140, 142 β, 14 (collection angle), 14, 17, 26–27, 29–30, 41–42, 51, 56–57, 59, 62, 68, 72, 76–79, 172–173, 226, 356, 434 Aberration, 28 chromatic, 28–29, 31, 76, 170, 172–173, 278–279, 292 spherical, 29, 44, 76, 172, 174, 201, 378 lens, 28, 172 Accuracy, 11, 52 background, 170 CBED, 146–147 edge energy, 225 energy scale, 357 microanalysis, 41, 63–64 quantiﬁcation, 53, 78, 177, 321 ALCHEMI, 243, 289 Amorphous material, 76, 104, 108–109, 127, 131–133, 140, 179, 186, 188, 198, 200, 230, 236, 244, 250, 271, 273–274, 276, 292–293, 296, 301–302, 305, 361–362, 385–387, 392, 426 Angular distribution, 40, 42, 68, 78, 129, 131, 135, 137, 139, 152, 177, 226 background, 41–42 core loss, 42, 77 Angular momentum, 11, 101, 113, 334, 372 quantum number, 11, 101, 111, 113, 115, 226, 229 Auger, 106, 175, 185, 246, 293, 391 Augmented Plane Wave (APW), 112–113 full potential, 112–113, 231, 242 linearised, 112 Background, 3, 42, 50 detector, 34–35, 163 ELNES, 226 imaging, 83–84, 127–128, 132, 166–168, 170, 387, 441 MLS, 70 preedge, 37, 42, 51–52, 69, 103–104, 332

spectral, 32, 39–42, 51, 62, 64, 68, 83, 115, 165 spectrometer, 32, 34, 40, 62–63 stray, 40–41 Backgroundspectral, 50 Bethe formula, 55–57 Bethe ridge, 20, 55, 62, 130, 150 Bloch function, 108 Bloch wave, 78, 139, 147, 152, 359 Boersch effect, 42, 44 Bohr radius, 16, 54, 102, 427 Brillouin zone, 111, 152, 358, 361, 372 Castaing–Henry ﬁlter, 25, 128, 160, 431, 435 Channel-to-channel gain variation, 34–35, 70–71, 332 Characteristic scattering angle, 42, 55, 68, 170, 427, 430 Charge transfer, 242, 244–245, 247–248, 283, 318, 322, 328, 345, 392, 394 Charge-coupled device (CCD), 23, 25, 33–34, 70, 85, 87, 129, 162–163, 170, 174 Chemical shift, 105, 193, 240–245, 247, 317–318, 321–323 Collection aperture, 17, 64, 78, 109, 290, 357 Compton angle, 130, 150 Compton maximum, 130 Compton peak, 150 Compton scattering, 128, 150 Conduction band, 98, 105, 232, 240, 320–321, 354–355, 357–359, 362, 365, 369, 371–372, 375, 394 Contamination, 36, 44, 87, 101, 104, 225, 272–274, 278, 436 Coordination number, 104, 107, 242–243, 299, 303, 305 Core exciton, 106, 113–114, 354, 362, 364, 366–367, 373 Core hole, 17, 106, 223, 228–229, 231, 240–242, 245, 252, 322, 357, 362, 366–367, 373, 394 Core level, 38, 62, 78, 81, 226, 229, 240–241, 249, 355, 363, 366 Crystalline specimen, 30, 76–78, 127, 137, 177, 182 Debye temperatures, 342

455

456

INDEX

Debye–Waller factor, 103–104, 149, 337, 340, 342 Deconvolution, 34, 68–69, 181, 299, 329, 332 Delocalization, 44, 172–173 Density of states (DOS), 227, 231, 284, 288–289, 298, 358, 364–366, 368–369, 372–373, 385, 391 Detection limit, 79–80, 168, 171–172, 186 Detective quantum efﬁciency (DQE), 36, 50, 52, 79, 81, 444 Dielectric constant, 7, 101, 320, 357, 365–366, 369 Dielectric function, 99–101, 320 Differential cross-section, 18–19, 49, 77, 335–337, 427 Diffraction coupling, 14 Dipole transitions, 19, 42, 55, 59, 111, 113, 226, 233, 334, 358, 372–373, 395–396 angle resolved, 129 energy, 12, 14, 33, 43 plasmon, 99, 130, 151, 321 Drude model, 62, 101 Dynamic range, 35–36, 162, 177 Elastic scattering, 20, 131–132, 146, 164, 356, 420, 427, 430 Energy dispersive X-ray (EDX), 1, 50, 81–82, 184, 200 Energy ﬁlter, 2, 23, 25, 37, 43, 83–84, 127–128, 141, 159–160, 162, 164, 305, 356, 409 Energy losses by channeled electrons (ELCE), 289–290 Energy resolution, 23, 26–28, 32, 36, 42–44, 108, 117, 225–226, 357, 361, 437–438 Fermi level, 102, 109, 241, 243, 247, 318, 358, 391, 395 Fermi’s Golden Rule, 54, 98, 102, 330, 357 Fourier transform, 68, 100, 127, 132, 150, 333–334, 339, 341 Free electron, 4 Free electron density, 5, 321 Free electron metals, 106, 319 Generalized oscillator strength (GOS), 19, 55–57, 62 Hartree–Fock, 398 Hartree–Slater calculation, 59, 177 Hartree–Slater wave functions, 55, 57, 106, 284–285 High-angle annular dark ﬁeld (HAADF), 76, 252, 356, 437 Hydrogenic model, 57, 61, 177, 434 Hydrogenic wave functions, 11, 284 Image coupling, 14 Incident electrons, 42, 55, 71, 79, 399, 420, 433, 443 Insulating speciments, 6, 164, 271–272, 353 Interatomic distance, 104, 300, 331, 334, 354, 376 Interband scattering, 98, 353, 358, 360–361, 365 Intershell scattering, 107

Intrashell scattering, 107, 233, 281, 283 Ion milling, 163, 273, 278, 374 Ionization cross-section, 8, 39, 50, 53–57, 77–78, 176–177, 179, 275–276, 304 total, 16, 55, 57, 81, 134 Ionization damage, 273 Ionization edge, 28, 41–42, 130, 165–166, 168, 170, 172, 177, 181, 183, 224, 286, 299, 334, 387 Korringa-Kohn-Rostoker (KKR) calculation, 111–112, 230 Kunzl’s law, 243 Lifetime broadening, 6, 106 Lorentz microscopy, 386, 398–400 Lorentz model, 101 Low loss spectrum, 51, 69, 72, 98, 109 Magnetic prism, 22–25, 161–162 elemental, 2, 37–38, 83, 160, 162–163, 165, 168, 177 induction, 407, 412–413 valence, 394 elastic, 65 inelastic, 38, 41–42, 64–65, 67, 160, 163, 165, 177, 299 photoelectron, 323 plasmon, 39 Mean-square relative displacement (MSRD), 337, 340, 342 Minimum detectable mass fraction, 79–80, 174 calculations, 230 structure, 229, 233 theory, 235, 241 Molecular orbitals, 230–231, 239 Mufﬁn tin, 104, 106, 108, 110, 112, 231 Multiple least-squares (MLS), 70, 75, 441 Multiple scattering, 106–107, 230 calculations, 231, 234–235, 239 intrashell, 104 plural, 67, 100, 130, 139, 151, 165, 295, 329, 390 Nondipole effects, 103 Nondipole transitions, 334 Objective aperture, 14, 27, 76–78, 170, 172–173, 226, 272, 400, 426 Omega ﬁlter, 25, 44, 279, 305, 442, 447 approximation, 106, 114, 243 atom, 9 equation, 10 Optical oscillator strength, 55 Order–disorder transformation, 318, 345 Orientation dependence, 78, 109, 223, 226, 252 Parallel detection, 72, 224, 303, 345 Partial cross-section, 49, 51, 59–60, 64, 78, 85, 430 Pauling electronegativity, 244–245 Peak-to-background ratio (P/B), 134, 277, 303 Photomultiplier tube (PMT), 31–32, 133

INDEX

Plasmon, 2–6, 8, 39–40, 62, 72, 97–101, 130, 136–138, 145, 151, 165, 176, 185, 196, 291, 293, 303, 317–318, 355, 441 Poisson statistics, 6, 38, 42, 67, 71, 80, 147, 319 Quadrupole lens, 23, 33, 378 Radial distribution function (RDF), 299, 301–302, 305, 331 Radiation damage, 32, 36–37, 164, 172, 198, 273, 279, 291, 303, 444 Reference spectrum, 70, 72–73 Reﬂection electron loss, 104, 303, 395 Resolution limit, 29, 173 Rutherford scattering, 149 Sample thickness, 2, 6, 64–65, 67, 69, 72, 177, 181, 226, 344, 411–412 Scanning-transmission electron microscope (STEM), 23, 26, 37, 44, 76, 78, 80, 86, 164, 172, 193, 198, 200, 226, 303, 321, 354, 356, 378, 388, 399, 406 Scattering wavevector, 54, 109 Schottky source, 42, 136, 438 Second difference method, 70, 72–73, 277 Selection rules, 55, 111, 334, 361, 372, 395–396 Serial recording, 13–14, 31–32, 34, 36–37, 143, 179, 278, 296, 301, 303, 305, 332, 433 Signal-to-background ratio, 40, 64–65, 67, 81, 171 Signal-to-noise ratio (SNR), 41, 165, 168, 170–171, 181, 332, 340, 400–402, 405–406, 409 Single scattering distribution, 40, 67–68, 181 Single scattering resonances, 252 Site occupancies, 223, 226, 235, 244, 249–250 Solid-state effects, 12, 59, 78, 115–116, 326 Spatial resolution, 2, 31, 41, 43–44, 76, 170–172, 252, 292, 356, 441, 444–445

колхоз 6:45 pm, 11/1/05

457

Specimen charging, 164, 272, 388 Spectrometer resolution function, 67 Spectrum image, 85–87, 159, 174, 181, 184, 431 Spin-orbit splitting, 335, 351 Standards, 104, 250, 284, 288, 300, 302, 394 Systematic errors, 63, 69, 73, 80, 166–167, 324 charge density, 149 diffraction, 133 EXELFS, 334 magnetostriction, 401 sample, 87, 164, 196, 272–273 Thermal diffuse scattering, 138, 145, 342 Thickness variation, 86, 145, 165, 168, 177–179, 181, 344 Time-resolved studies, 296, 303 Uncertainty principle, 6, 106, 355, 357 Valence determination, 290 Valence electrons, 128, 230, 241, 272, 325 excitation, 42, 51, 62, 64 Wannier exciton, 359, 365, 368 White line intensity, 118, 283, 286, 296, 324–325 White line intensity ratio, 115, 117, 240, 248 White lines, 3, 12, 73, 97, 102, 115, 117, 282, 318, 323–324, 329, 389 Wien ﬁlter, 43–44, 225, 378 X-ray Absorption Near Edge Structure (XANES), 105–108, 229 X-ray absorption spectroscopy(XAS), 226, 230–231, 233, 238, 242, 249, 322, 366 X-ray photoelectron spectroscopy (XPS), 105, 232, 240–241, 243, 246–247, 322–323, 391 deconvolution, 68 imaging, 127–128, 132, 134–135, 137–138, 142, 160, 163, 165, 402, 404–405, 409 peak, 2, 52, 64, 68, 78, 98