978-1-4939-6676-9.pdf

Joseph I. Goldstein Dale E. Newbury Joseph R. Michael Nicholas W.M. Ritchie John Henry J. Scott David C. Joy Scanning Electron Microscopy and X-Ray Microanalysis Fourth Edition Scanning Electron Microscopy and X-Ray Microanalysis Jose...

Joseph I. Goldstein Dale E. Newbury Joseph R. Michael Nicholas W.M. Ritchie John Henry J. Scott David C. Joy Scanning Electron Microscopy and X-Ray Microanalysis Fourth Edition Scanning Electron Microscopy and X-Ray Microanalysis Joseph I. Goldstein Dale E. Newbury Joseph R. Michael Nicholas W.M. Ritchie John Henry J. Scott David C. Joy Scanning Electron Microscopy and X-Ray Microanalysis Fourth Edition Joseph I. Goldstein Dale E. Newbury University of Massachusetts National Institute of Standards and Technology Amherst, MA, USA Gaithersburg, MD, USA Joseph R. Michael Nicholas W.M. Ritchie Sandia National Laboratories National Institute of Standards and Technology Albuquerque, NM, USA Gaithersburg, MD, USA John Henry J. Scott David C. Joy National Institute of Standards and Technology University of Tennessee Gaithersburg, MD, USA Knoxville, TN, USA ISBN 978-1-4939-6674-5    ISBN 978-1-4939-6676-9 (eBook) https://doi.org/10.1007/978-1-4939-6676-9 Library of Congress Control Number: 2017943045 © Springer Science+Business Media LLC 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is con- cerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, com- puter software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media LLC The registered company address is: 233 Spring Street, New York, NY 10013, U.S.A.  V Preface This is not your father’s, your mother’s, or your is that a reader seeking specific information can grandparent’s Scanning Electron Microscopy and select a topic from the list and obtain a good X-Ray Microanalysis (SEMXM). But that is not to understanding of the topic from that module say that there is no continuity or to deny a family alone. While each topic is supported by informa- resemblance. SEMXM4 is the fourth in the series tion in other modules, we acknowledge the like- of textbooks with this title, and continues a tradi- lihood that not all users of SEMXM4 will “read tion that extends back to the “zero-th edition” in it all.” This approach inevitably leads to a degree 1975 published under the title, “Practical Scanning of overlap and repetition since similar informa- Electron Microscopy” (Plenum Press, New York). tion may appear in two or more places, and this is However, the latest edition differs sharply from entirely intentional. its predecessors, which attempted an encyclope- dic approach to the subject by providing extensive In recognition of these fundamental changes, the details on how the SEM and its associated devices authors have chosen to modify SEMXM4 exten- actually work, for example, electron sources, lenses, sively to provide a guide on the actual use of the electron detectors, X-ray spectrometers, and so on. instrument without overwhelming the reader with the burden of details on the physics of the opera- In constructing this new edition, the authors have tion of the instrument and its various attachments. chosen a different approach. Modern SEMs and the Our guiding principle is that the microscopist- associated X-ray spectrometry and crystallography microanalyst must understand which parameters measurement functions operate under such exten- can be adjusted and what is an effective strategy to sive computer control and automation that it is select those parameters to solve a particular prob- actually difficult for the microscopist-microanalyst lem. The modern SEM is an extraordinarily flex- to interact with the instrument except within care- ible tool, capable of operating over a wide range fully prescribed boundaries. Much of the flexibility of electron optical parameters and producing of parameter selection that early instruments pro- images from electron detectors with different sig- vided has now been lost, as instrumental operation nal characteristics. Those users who restrict them- functions have been folded into software control. selves to a single set of operating parameters may Thus, electron sources are merely turned “on,” with be able to solve certain problems, but they may the computer control optimizing the operation, or never know what they are missing by not explor- for the thermally assisted field emission gun, the ing the range of parameter space available to them. electron source may be permanently “on.” The user SEMXM4 seeks to provide sufficient understand- can certainly adjust the lenses to focus the image, ing of the technique for a user to become a com- but this focusing action often involves complex petent and efficient problem solver. That is not to interactions of two or more lenses, which formerly say that there are only a few things to learn. To would have required individual adjustment. More- help the reader to approach the considerable body over, the nature of the SEM field has fundamentally of knowledge needed to operate at a high degree changed. What was once a very specialized instru- of competency, a new feature of SEMXM-4 is the ment system that required a high level of training summary checklist provided for each of the major and knowledge on the part of the user has become areas of operation: SEM imaging, elemental X-ray much more of a routine tool. The SEM is now sim- microanalysis, and backscatter-diffraction crystal- ply one of a considerable suite of instruments that lography. can be employed to solve problems in the physical and biological sciences, in engineering, in technol- Readers familiar with earlier editions of SEMXM ogy, in manufacturing and quality control, in fail- will notice the absence of the extensive material ure analysis, in forensic science, and other fields. previously provided on specimen preparation. Proper specimen preparation is a critical step in The authors also recognize the profound changes solving most problems, but with the vast range of that have occurred in the manner in which peo- applications to materials of diverse character, the ple obtain information. The units of SEMXM4, topic of specimen preparation itself has become whether referred to as chapters or modules, are the subject of entire books, often devoted to just meant to be relatively self-contained. Our hope one specialized area. VI Preface Throughout their history, the authors of the Finally, the author team sadly notes the passing in SEMXM textbooks have been closely associated 2015 of Professor Joseph I. Goldstein (University as lecturers with the Lehigh University Summer of Massachusetts, Amherst) who was the “found- Microscopy School. The opportunity to teach and ing father” of the Lehigh University Summer interact with each year’s students has provided a Microscopy School in 1970, and who organized very useful experience in understanding the com- and contributed so extensively to the microscopy munity of users of the technique and its evolution courses and to the SEMXM textbooks throughout over time. We hope that these interactions have the ensuing 45 years. Joe provided the stimulus to improved our written presentation of the subject the production of SEMXM4 with his indefatigable as a benefit to newcomers as well as established spirit, and his technical contributions are embed- practitioners. ded in the X-ray microanalysis sections. Dale E. Newbury Nicholas W.M. Ritchie John Henry J. Scott Gaithersburg, MD, USA Joseph R. Michael Albuquerque, NM, USA David C. Joy Knoxville, TN, USA The original version of this book was revised. Index has been updated.  VII Scanning Electron Microscopy and Associated Techniques: Overview Imaging Microscopic Features location on the sample is digitized and recorded into computer memory, and is subsequently used The scanning electron microscope (SEM) is an to determine the gray level at the corresponding instrument that creates magnified images which X-Y location of a computer display screen, form- reveal microscopic-scale information on the size, ing a single picture element (or pixel). In a con- shape, composition, crystallography, and other ventional-vacuum SEM, the electron-optical physical and chemical properties of a specimen. column and the specimen chamber must operate The principle of the SEM was originally demon- under high vacuum conditions (1 million total counts but that intensity is also modified by all other ele- from threshold to E0) now possible with the sili- ments present in the interaction volume through con drift detector EDS (SDD-EDS), enables this their influence on the electron scattering and level of accuracy to be achieved for major and retardation (“atomic number” matrix effect, Z), minor constituents even when severe peak inter- X-ray absorption within the specimen (“absorp- ference occurs and there is also a large concen- tion” matrix effect, A), and X-ray generation tration ratio, for example, a major constituent induced by absorption of X-rays (“secondary flu- interfering with a minor constituent. Trace con- orescence” matrix effects, F, induced by charac- stituents that do not suffer severe interference teristic X-rays and c, induced by continuum can be measured to limits of detection as low as X-rays). The complex physics of these “ZAFc” C = 0.0002 (200 parts per million) with spectra matrix corrections has been rendered into algo- containing >10 million counts. For interference rithms by a combined theoretical and empirical situations, much higher count spectra (>100 approach. The basis of quantitative electron- million counts) are required. excited X-ray microanalysis is the “k-ratio proto- An alternative “standardless analysis” protocol col”: measurement under identical conditions uses libraries of standard spectra (“remote stan- (beam energy, known electron dose, and spec- dards”) measured on a different SEM platform trometer performance) of the characteristic with a similar EDS spectrometer, ideally over a intensities for all elements recognized in the wide range of beam energy and detector parame- unknown spectrum against a suite of standards ters (resolution). These library spectra are then containing those same elements, producing a set adjusted to the local measurement conditions of k-ratios, where through comparison of one or more key spectra (e.g., locally measured spectra of particular ele- k = I Unknown / I Standard  (1) ments such as Si and Ni). Interpolation/extrapola- tion is used to supply estimated spectral intensities for each element in the unknown. Standards are for elements not present in or at a beam energy not materials of known composition that are tested to represented in the library elemental suite. Testing be homogeneous at the microscopic scale, and of the standardless analysis method has shown preferably homogeneous at the nanoscale. that an RDEV range of ±25 % relative is needed to Standards can be as simple as pure elements—e.g., capture 95 % of all analyses. C, Al, Si, Ti, Cr, Fe, Ni, Cu, Ag, Au, etc.—but for High throughput (>100 kHz) EDS enables col- those elements that are not stable in a vacuum lection of X-ray intensity maps with gray scale rep- (e.g., gaseous elements such as O) or which resentation of different concentration levels (e.g., degrade during electron bombardment (e.g., S, P,. Fig. 7a). Compositional mapping by spectrum and Ga), stable stoichiometric compounds can be imaging (SI) collects a full EDS spectrum at each used instead, e.g., MgO for O; FeS2 for S; and GaP pixel of an x-y array, and after applying the quanti- for Ga and P. The most accurate analysis is per- tative analysis procedure at each pixel, images are formed with standards measured on the same created for each element where the gray (or color) instrument as the unknown(s), ideally in the same level is assigned based on the measured concentra- measurement campaign, although archived tion (e.g.,. Fig. 7b). XII Scanning Electron Microscopy and Associated Techniques: Overview Al Fe 20µm Ni Al Fe Ni a BSE Al 20 mm BSE MAG: 1000 x HV: 20.0 kV WD: 11.0 mm Ni Fe 20µm 0.001 0.01 0.1 1.0 b 0.1 1.0 10 100 wt%..      Fig. 7 a EDS X-ray intensity maps for Al, Fe, and Ni and color overlay; Raney nickel alloy; E0 = 20 keV. b SEM/BSE (sum) image and compositional maps corresponding to a  XIII Scanning Electron Microscopy and Associated Techniques: Overview Measuring the Crystal Structure  ual-Beam Platforms: Combined D Electron and Ion Beams An electron beam incident on a crystal can undergo electron channeling in a shallow near-surface layer A “dual-beam” instrument combines a fully func- which increases the initial beam penetration for tional SEM with a focused ion beam (FIB), typi- certain orientations of the beam relative to the cally gallium or argon. This combination provides crystal planes. The additional penetration results in a flexible platform for in situ specimen modifica- a slight reduction in the electron backscattering tion through precision ion beam milling and/or coefficient, which creates weak crystallographic ion beam mediated material deposition with contrast (a few percent) in SEM images by which sequential or simultaneous electron beam tech- differences in local crystallographic orientation nique characterization of the newly revealed can be directly observed: grain boundaries, defor- specimen surfaces. Precision material removal mations bands, and so on (e.g.,. Fig. 8). enables detailed study of the third dimension of a The backscattered electrons exiting the speci- specimen with nanoscale resolution along the men are subject to crystallographic diffraction depth axis. An example of ion beam milling of a effects, producing small modulations in the intensi- directionally solidified Al-Cu is shown in. Fig. 9, ties scattered to different angles that are superim- as imaged with the SEM column on the dual- posed on the overall angular distribution that an beam instrument. Additionally, ion-beam amorphous target would produce. The resulting induced secondary electron emission provides “electron backscatter diffraction (EBSD)” pattern scanning ion microscopy (SIM) imaging to com- provides extensive information on the local orienta- plement SEM imaging. For imaging certain speci- tion, as shown in. Fig. 8b for a crystal of hematite. men properties, such as crystallographic EBSD pattern angular separations provide mea- structure, SIM produces stronger contrast than surements of the crystal plane spacing, while the SEM. There is also an important class of stand- overall EBSD pattern reveals symmetry elements. alone SIM instruments, such as the helium ion This crystallographic information combined with microscope (HIM), that are optimized for high elemental analysis information obtained simultane- resolution/high depth-of-field imaging perfor- ously from the same specimen region can be used to mance (e.g., the same area as viewed by HIM is identify the crystal structure of an unknown. also shown in. Fig. 9). a b 40 mm BSE MAG: 400 x HV: 20.0 kV WD: 11.0 mm..      Fig. 8 a Electron channeling contrast revealing grain boundaries in Ti-alloy (nominal composition: Ti-15Mo-3Nb-3Al- 0.2Si); E0 = 20 keV. b Electron backscatter diffraction (EBSD) pattern from hematite at E0 = 40 keV XIV Scanning Electron Microscopy and Associated Techniques: Overview Field of view Dwell Time Mag (4x5 Polaroid) 50.00 um 5.00 um 50.0 us 2,540.00 X mag HV WD HFW curr 20 µm Working Dist Image Size Blankar Current Detector 5 000 x 15.00 kV 3.9 mm 51.2 µm 86 pA 12.1 mm 1024x1024 0.7 9A PrimaryETDetector..      Fig. 9 Directionally-solidified Al-Cu eutectic alloy after ion beam milling in a dual-beam instrument, as imaged by the SEM column (left image); same region imaged in the HIM (right image)  odeling Electron and Ion M References Interactions Knoll M (1935) Static potential and secondary emission of bodies under electron radiation. Z Tech Physik 16:467 An important component of modern Scanning Knoll M, Theile R (1939) Scanning electron microscope for Electron Microscopy and X-ray Microanalysis determining the topography of surfaces and thin is modeling the interaction of beam electrons layers. Z Physik 113:260 and ions with the atoms of the specimen and its Oatley C (1972) The scanning electron microscope: part 1, environment. Such modeling supports image the instrument. Cambridge University Press, Cambridge von Ardenne M (1938) The scanning electron microscope. interpretation, X-ray microanalysis of challeng- Theoretical fundamentals. Z Physik 109:553 ing specimens, electron crystallography methods, and many other issues. Software tools for this pur- pose, including Monte Carlo electron trajectory simulation, are discussed within the text. These tools are complemented by the extensive database of Electron-Solid Interactions (e.g., electron scat- tering and ionization cross sections, secondary electron and backscattered electron coefficients, etc.), developed by Prof. David Joy, can be found in chapter 3 on SpringerLink: http://link.springer. com/chapter/10.1007/978-1-4939-6676-9_3.  XV Contents 1 Electron Beam—Specimen Interactions: Interaction Volume 1 1.1 What Happens When the Beam Electrons Encounter Specimen Atoms? 2 1.2 Inelastic Scattering (Energy Loss) Limits Beam Electron Travel in the Specimen 2 1.3 Elastic Scattering: Beam Electrons Change Direction of Flight 4 1.3.1 How Frequently Does Elastic Scattering Occur? 4 1.4 Simulating the Effects of Elastic Scattering: Monte Carlo Calculations 5 1.4.1 What Do Individual Monte Carlo Trajectories Look Like? 6 1.4.2 Monte Carlo Simulation To Visualize the Electron Interaction Volume 6 1.4.3 Using the Monte Carlo Electron Trajectory Simulation to Study the Interaction Volume 8 1.5 A Range Equation To Estimate the Size of the Interaction Volume 12 References 14 2 Backscattered Electrons 15 2.1 Origin 16 2.1.1 The Numerical Measure of Backscattered Electrons 16 2.2 Critical Properties of Backscattered Electrons 16 2.2.1 BSE Response to Specimen Composition (η vs. Atomic Number, Z) 16 2.2.2 BSE Response to Specimen Inclination (η vs. Surface Tilt, θ) 20 2.2.3 Angular Distribution of Backscattering 22 2.2.4 Spatial Distribution of Backscattering 23 2.2.5 Energy Distribution of Backscattered Electrons 27 2.3 Summary 27 References 28 3 Secondary Electrons 29 3.1 Origin 30 3.2 Energy Distribution 30 3.3 Escape Depth of Secondary Electrons 30 3.4 Secondary Electron Yield Versus Atomic Number 30 3.5 Secondary Electron Yield Versus Specimen Tilt 34 3.6 Angular Distribution of Secondary Electrons 34 3.7 Secondary Electron Yield Versus Beam Energy 35 3.8 Spatial Characteristics of Secondary Electrons 35 References 37 4 X-Rays 39 4.1 Overview 40 4.2 Characteristic X-Rays 40 4.2.1 Origin 40 4.2.2 Fluorescence Yield 41 4.2.3 X-Ray Families 42 4.2.4 X-Ray Nomenclature 43 4.2.5 X-Ray Weights of Lines 44 4.2.6 Characteristic X-Ray Intensity 44 4.3 X-Ray Continuum (bremsstrahlung) 47 4.3.1 X-Ray Continuum Intensity 49 4.3.2 The Electron-Excited X-Ray Spectrum, As-Generated 49 4.3.3 Range of X-ray Production 50 4.3.4 Monte Carlo Simulation of X-Ray Generation 51 4.3.5 X-ray Depth Distribution Function, ϕ(ρz) 53 XVI Contents 4.4 X-Ray Absorption 54 4.5 X-Ray Fluorescence 59 References 63 5 Scanning Electron Microscope (SEM) Instrumentation 65 5.1 Electron Beam Parameters 66 5.2 Electron Optical Parameters 66 5.2.1 Beam Energy 66 5.2.2 Beam Diameter 67 5.2.3 Beam Current 67 5.2.4 Beam Current Density 68 5.2.5 Beam Convergence Angle, α 68 5.2.6 Beam Solid Angle 69 5.2.7 Electron Optical Brightness, β 70 5.2.8 Focus 71 5.3 SEM Imaging Modes 75 5.3.1 High Depth-of-Field Mode 75 5.3.2 High-Current Mode 78 5.3.3 Resolution Mode 80 5.3.4 Low-Voltage Mode 81 5.4 Electron Detectors 83 5.4.1 Important Properties of BSE and SE for Detector Design and Operation 83 5.4.2 Detector Characteristics 83 5.4.3 Common Types of Electron Detectors 85 5.4.4 Secondary Electron Detectors 86 5.4.5 Specimen Current: The Specimen as Its Own Detector 88 5.4.6 A Useful, Practical Measure of a Detector: Detective Quantum Efficiency 89 References 91 6 Image Formation 93 6.1 Image Construction by Scanning Action 94 6.2 Magnification 95 6.2.1 Magnification, Image Dimensions, and Scale Bars 95 6.3 Making Dimensional Measurements With the SEM: How Big Is That Feature? 95 6.3.1 Calibrating the Image 95 6.4 Image Defects 98 6.4.1 Projection Distortion (Foreshortening) 98 6.4.2 Image Defocusing (Blurring) 100 6.5 Making Measurements on Surfaces With Arbitrary Topography: Stereomicroscopy 102 6.5.1 Qualitative Stereomicroscopy 103 6.5.2 Quantitative Stereomicroscopy 107 References 110 7 SEM Image Interpretation 111 7.1 Information in SEM Images 112 7.2 Interpretation of SEM Images of Compositional Microstructure 112 7.2.1 Atomic Number Contrast With Backscattered Electrons 112 7.2.2 Calculating Atomic Number Contrast 113 7.2.3 BSE Atomic Number Contrast With the Everhart–Thornley Detector 113 7.3 Interpretation of SEM Images of Specimen Topography 114 7.3.1 Imaging Specimen Topography With the Everhart–Thornley Detector 115 7.3.2 The Light-Optical Analogy to the SEM/E–T (Positive Bias) Image 116 7.3.3 Imaging Specimen Topography With a Semiconductor BSE Detector 119 References 121  XVII Contents 8 The Visibility of Features in SEM Images 123 8.1 Signal Quality: Threshold Contrast and Threshold Current 124 References 131 9 Image Defects 133 9.1 Charging 134 9.1.1 What Is Specimen Charging? 134 9.1.2 Recognizing Charging Phenomena in SEM Images 135 9.1.3 Techniques to Control Charging Artifacts (High Vacuum Instruments) 139 9.2 Radiation Damage 142 9.3 Contamination 143 9.4 Moiré Effects: Imaging What Isn’t Actually There 144 References 146 10 High Resolution Imaging 147 10.1 What Is “High Resolution SEM Imaging”? 148 10.2 Instrumentation Considerations 148 10.3 Pixel Size, Beam Footprint, and Delocalized Signals 148 10.4 Secondary Electron Contrast at High Spatial Resolution 150 10.4.1 SE range Effects Produce Bright Edges (Isolated Edges) 151 10.4.2 Even More Localized Signal: Edges Which Are Thin Relative to the Beam Range 152 10.4.3 Too Much of a Good Thing: The Bright Edge Effect Can Hinder Distinguishing Shape 153 10.4.4 Too Much of a Good Thing: The Bright Edge Effect Hinders Locating the True Position of an Edge for Critical Dimension Metrology 154 10.5 Achieving High Resolution with Secondary Electrons 156 10.5.1 Beam Energy Strategies 156 10.5.2 Improving the SE1 Signal 158 10.5.3 Eliminate the Use of SEs Altogether: “Low Loss BSEs“ 161 10.6 Factors That Hinder Achieving High Resolution 163 10.6.1 Achieving Visibility: The Threshold Contrast 163 10.6.2 Pathological Specimen Behavior 163 10.6.3 Pathological Specimen and Instrumentation Behavior 164 References 164 11 Low Beam Energy SEM 165 11.1 What Constitutes “Low” Beam Energy SEM Imaging? 166 11.2 Secondary Electron and Backscattered Electron Signal Characteristics in the Low Beam Energy Range 166 11.3 Selecting the Beam Energy to Control the Spatial Sampling of Imaging Signals 169 11.3.1 Low Beam Energy for High Lateral Resolution SEM 169 11.3.2 Low Beam Energy for High Depth Resolution SEM 169 11.3.3 Extremely Low Beam Energy Imaging 171 References 172 12 Variable Pressure Scanning Electron Microscopy (VPSEM) 173 12.1 Review: The Conventional SEM High Vacuum Environment 174 12.1.1 Stable Electron Source Operation 174 12.1.2 Maintaining Beam Integrity 174 12.1.3 Stable Operation of the Everhart–Thornley Secondary Electron Detector 174 12.1.4 Minimizing Contamination 174 12.2 How Does VPSEM Differ From the Conventional SEM Vacuum Environment? 174 XVIII Contents 12.3 Benefits of Scanning Electron Microscopy at Elevated Pressures 175 12.3.1 Control of Specimen Charging 175 12.3.2 Controlling the Water Environment of a Specimen 176 12.4 Gas Scattering Modification of the Focused Electron Beam 177 12.5 VPSEM Image Resolution 181 12.6 Detectors for Elevated Pressure Microscopy 182 12.6.1 Backscattered Electrons—Passive Scintillator Detector 182 12.6.2 Secondary Electrons–Gas Amplification Detector 182 12.7 Contrast in VPSEM 184 References 185 13 ImageJ and Fiji 187 13.1 The ImageJ Universe 188 13.2 Fiji 188 13.3 Plugins 190 13.4 Where to Learn More 191 References 193 14 SEM Imaging Checklist 195 14.1 Specimen Considerations (High Vacuum SEM; Specimen Chamber Pressure < 10−3 Pa) 197 14.1.1 Conducting or Semiconducting Specimens 197 14.1.2 Insulating Specimens 197 14.2 Electron Signals Available 197 14.2.1 Beam Electron Range 197 14.2.2 Backscattered Electrons  197 14.2.3 Secondary Electrons  197 14.3 Selecting the Electron Detector 198 14.3.1 Everhart–Thornley Detector (“Secondary Electron” Detector) 198 14.3.2 Backscattered Electron Detectors 198 14.3.3 “Through-the-Lens” Detectors 198 14.4 Selecting the Beam Energy for SEM Imaging 198 14.4.1 Compositional Contrast With Backscattered Electrons 198 14.4.2 Topographic Contrast With Backscattered Electrons 198 14.4.3 Topographic Contrast With Secondary Electrons 198 14.4.4 High Resolution SEM Imaging 198 14.5 Selecting the Beam Current 199 14.5.1 High Resolution Imaging 199 14.5.2 Low Contrast Features Require High Beam Current and/or Long Frame Time to Establish Visibility 199 14.6 Image Presentation 199 14.6.1 “Live” Display Adjustments 199 14.6.2 Post-Collection Processing 199 14.7 Image Interpretation 199 14.7.1 Observer’s Point of View 199 14.7.2 Direction of Illumination 199 14.7.3 Contrast Encoding 200 14.7.4 Imaging Topography With the Everhart–Thornley Detector 200 14.7.5 Annular BSE Detector (Semiconductor Sum Mode A + B and Passive Scintillator) 200 14.7.6 Semiconductor BSE Detector Difference Mode, A−B 200 14.7.7 Everhart–Thornley Detector, Negatively Biased to Reject SE 200 14.8 Variable Pressure Scanning Electron Microscopy (VPSEM) 200 14.8.1 VPSEM Advantages 200 14.8.2 VPSEM Disadvantages 200 15 SEM Case Studies 201 15.1 Case Study: How High Is That Feature Relative to Another? 202 15.2 Revealing Shallow Surface Relief 204 15.3 Case Study: Detecting Ink-Jet Printer Deposits 206  XIX Contents 16 Energy Dispersive X-ray Spectrometry: Physical Principles and User-Selected Parameters 209 16.1 The Energy Dispersive Spectrometry (EDS) Process 210 16.1.1 The Principal EDS Artifact: Peak Broadening (EDS Resolution Function) 210 16.1.2 Minor Artifacts: The Si-Escape Peak 213 16.1.3 Minor Artifacts: Coincidence Peaks 213 16.1.4 Minor Artifacts: Si Absorption Edge and Si Internal Fluorescence Peak 215 16.2 “Best Practices” for Electron-Excited EDS Operation 216 16.2.1 Operation of the EDS System 216 16.3 Practical Aspects of Ensuring EDS Performance for a Quality Measurement Environment 219 16.3.1 Detector Geometry 219 16.3.2 Process Time 222 16.3.3 Optimal Working Distance 222 16.3.4 Detector Orientation 223 16.3.5 Count Rate Linearity 225 16.3.6 Energy Calibration Linearity 226 16.3.7 Other Items 227 16.3.8 Setting Up a Quality Control Program 228 16.3.9 Purchasing an SDD 230 References 234 17 DTSA-II EDS Software 235 17.1 Getting Started With NIST DTSA-II 236 17.1.1 Motivation 236 17.1.2 Platform 236 17.1.3 Overview 236 17.1.4 Design 237 17.1.5 The Three -Leg Stool: Simulation, Quantification and Experiment Design 237 17.1.6 Introduction to Fundamental Concepts 238 17.2 Simulation in DTSA-II 245 17.2.1 Introduction 245 17.2.2 Monte Carlo Simulation 245 17.2.3 Using the GUI To Perform a Simulation 247 17.2.4 Optional Tables 262 References 264 18 Qualitative Elemental Analysis by Energy Dispersive X-Ray Spectrometry 265 18.1 Quality Assurance Issues for Qualitative Analysis: EDS Calibration 266 18.2 Principles of Qualitative EDS Analysis 266 18.2.1 Critical Concepts From the Physics of Characteristic X-ray Generation and Propagation 266 18.2.2 X-Ray Energy Database: Families of X-Rays 269 18.2.3 Artifacts of the EDS Detection Process 269 18.3 Performing Manual Qualitative Analysis 275 18.3.1 Why are Skills in Manual Qualitative Analysis Important? 275 18.3.2 Performing Manual Qualitative Analysis: Choosing the Instrument Operating Conditions 277 18.4 Identifying the Peaks 278 18.4.1 Employ the Available Software Tools 278 18.4.2 Identifying the Peaks: Major Constituents 280 18.4.3 Lower Photon Energy Region 281 18.4.4 Identifying the Peaks: Minor and Trace Constituents 281 18.4.5 Checking Your Work 281 18.5 A Worked Example of Manual Peak Identification 281 References 287 XX Contents 19 Quantitative Analysis: From k-ratio to Composition 289 19.1 What Is a k-ratio? 290 19.2 Uncertainties in k-ratios 291 19.3 Sets of k-ratios 291 19.4 Converting Sets of k-ratios Into Composition 292 19.5 The Analytical Total 292 19.6 Normalization 292 19.7 Other Ways to Estimate CZ 293 19.7.1 Oxygen by Assumed Stoichiometry 293 19.7.2 Waters of Crystallization 293 19.7.3 Element by Difference 293 19.8 Ways of Reporting Composition 294 19.8.1 Mass Fraction 294 19.8.2 Atomic Fraction 294 19.8.3 Stoichiometry 294 19.8.4 Oxide Fractions 294 19.9 The Accuracy of Quantitative Electron-Excited X-ray Microanalysis 295 19.9.1 Standards-Based k-ratio Protocol 295 19.9.2 “Standardless Analysis” 296 19.10 Appendix 298 19.10.1 The Need for Matrix Corrections To Achieve Quantitative Analysis 298 19.10.2 The Physical Origin of Matrix Effects 299 19.10.3 ZAF Factors in Microanalysis 299 References 307 20 Quantitative Analysis: The SEM/EDS Elemental Microanalysis k-ratio Procedure for Bulk Specimens, Step-by-Step 309 20.1 Requirements Imposed on the Specimen and Standards 311 20.2 Instrumentation Requirements 311 20.2.1 Choosing the EDS Parameters 311 20.2.2 Choosing the Beam Energy, E0 313 20.2.3 Measuring the Beam Current 313 20.2.4 Choosing the Beam Current 314 20.3 Examples of the k-ratio/Matrix Correction Protocol with DTSA II 316 20.3.1 Analysis of Major Constituents (C > 0.1 Mass Fraction) with Well-Resolved Peaks 316 20.3.2 Analysis of Major Constituents (C > 0.1 Mass Fraction) with Severely Overlapping Peaks 318 20.3.3 Analysis of a Minor Constituent with Peak Overlap From a Major Constituent 319 20.3.4 Ba-Ti Interference in BaTiSi3O9 319 20.3.5 Ba-Ti Interference: Major/Minor Constituent Interference in K2496 Microanalysis Glass 319 20.4 The Need for an Iterative Qualitative and Quantitative Analysis Strategy 319 20.4.1 Analysis of a Complex Metal Alloy, IN100 320 20.4.2 Analysis of a Stainless Steel 323 20.4.3 Progressive Discovery: Repeated Qualitative–Quantitative Analysis Sequences 324 20.5 Is the Specimen Homogeneous? 326 20.6 Beam-Sensitive Specimens 331 20.6.1 Alkali Element Migration 331 20.6.2 Materials Subject to Mass Loss During Electron Bombardment— the Marshall-Hall Method 334 References 339  XXI Contents 21 Trace Analysis by SEM/EDS 341 21.1 Limits of Detection for SEM/EDS Microanalysis 342 21.2 Estimating the Concentration Limit of Detection, CDL 343 21.2.1 Estimating CDL from a Trace or Minor Constituent from Measuring a Known Standard 343 21.2.2 Estimating CDL After Determination of a Minor or Trace Constituent with Severe Peak Interference from a Major Constituent 343 21.2.3 Estimating CDL When a Reference Value for Trace or Minor Element Is Not Available 343 21.3 Measurements of Trace Constituents by Electron-Excited Energy Dispersive X-ray Spectrometry 345 21.3.1 Is a Given Trace Level Measurement Actually Valid? 345 21.4 Pathological Electron Scattering Can Produce “Trace” Contributions to EDS Spectra 350 21.4.1 Instrumental Sources of Trace Analysis Artifacts 350 21.4.2 Assessing Remote Excitation Sources in an SEM-EDS System 353 21.5 Summary 357 References 357 22 Low Beam Energy X-Ray Microanalysis 359 22.1 What Constitutes “Low” Beam Energy X-Ray Microanalysis? 360 22.1.1 Characteristic X-ray Peak Selection Strategy for Analysis 364 22.1.2 Low Beam Energy Analysis Range 364 22.2 Advantage of Low Beam Energy X-Ray Microanalysis 365 22.2.1 Improved Spatial Resolution 365 22.2.2 Reduced Matrix Absorption Correction 366 22.2.3 Accurate Analysis of Low Atomic Number Elements at Low Beam Energy 366 22.3 Challenges and Limitations of Low Beam Energy X-Ray Microanalysis 369 22.3.1 Reduced Access to Elements 369 22.3.2 Relative Depth of X-Ray Generation: Susceptibility to Vertical Heterogeneity 372 22.3.3 At Low Beam Energy, Almost Everything Is Found To Be Layered 373 References 380 23 Analysis of Specimens with Special Geometry: Irregular Bulk Objects and Particles 381 23.1 The Origins of “Geometric Effects”: Bulk Specimens 382 23.2 What Degree of Surface Finish Is Required for Electron-Excited X-ray Microanalysis To Minimize Geometric Effects? 384 23.2.1 No Chemical Etching 384 23.3 Consequences of Attempting Analysis of Bulk Materials With Rough Surfaces 385 23.4 Useful Indicators of Geometric Factors Impact on Analysis 386 23.4.1 The Raw Analytical Total 386 23.4.2 The Shape of the EDS Spectrum 389 23.5 Best Practices for Analysis of Rough Bulk Samples 391 23.6 Particle Analysis 394 23.6.1 How Do X-ray Measurements of Particles Differ From Bulk Measurements? 394 23.6.2 Collecting Optimum Spectra From Particles 395 23.6.3 X-ray Spectrum Imaging: Understanding Heterogeneous Materials 400 23.6.4 Particle Geometry Factors Influencing Quantitative Analysis of Particles 403 23.6.5 Uncertainty in Quantitative Analysis of Particles 405 23.6.6 Peak-to-Background (P/B) Method 408 23.7 Summary 410 References 411 XXII Contents 24 Compositional Mapping 413 24.1 Total Intensity Region-of-Interest Mapping 414 24.1.1 Limitations of Total Intensity Mapping 415 24.2 X-Ray Spectrum Imaging 417 24.2.1 Utilizing XSI Datacubes 419 24.2.2 Derived Spectra 419 24.3 Quantitative Compositional Mapping 424 24.4 Strategy for XSI Elemental Mapping Data Collection 430 24.4.1 Choosing the EDS Dead-Time 430 24.4.2 Choosing the Pixel Density 432 24.4.3 Choosing the Pixel Dwell Time 434 References 439 25 Attempting Electron-Excited X-Ray Microanalysis in the Variable Pressure Scanning Electron Microscope (VPSEM) 441 25.1 Gas Scattering Effects in the VPSEM 442 25.1.1 Why Doesn’t the EDS Collimator Exclude the Remote Skirt X-Rays? 446 25.1.2 Other Artifacts Observed in VPSEM X-Ray Spectrometry 448 25.2 What Can Be Done To Minimize gas Scattering in VPSEM? 450 25.2.1 Workarounds To Solve Practical Problems 451 25.2.2 Favorable Sample Characteristics 451 25.2.3 Unfavorable Sample Characteristics 456 References 459 26 Energy Dispersive X-Ray Microanalysis Checklist 461 26.1 Instrumentation 462 26.1.1 SEM 462 26.1.2 EDS Detector 462 26.1.3 Probe Current Measurement Device 462 26.1.4 Conductive Coating 463 26.2 Sample Preparation 463 26.2.1 Standard Materials 464 26.2.2 Peak Reference Materials 464 26.3 Initial Set-Up 464 26.3.1 Calibrating the EDS Detector 464 26.4 Collecting Data 466 26.4.1 Exploratory Spectrum 466 26.4.2 Experiment Optimization 467 26.4.3 Selecting Standards 467 26.4.4 Reference Spectra 467 26.4.5 Collecting Standards 467 26.4.6 Collecting Peak-Fitting References 467 26.4.7 Collecting Spectra From the Unknown 467 26.5 Data Analysis 468 26.5.1 Organizing the Data 468 26.5.2 Quantification 468 26.6 Quality Check 468 26.6.1 Check the Residual Spectrum After Peak Fitting 468 26.6.2 Check the Analytic Total 469 26.6.3 Intercompare the Measurements 469 Reference 470 27 X-Ray Microanalysis Case Studies 471 27.1 Case Study: Characterization of a Hard-Facing Alloy Bearing Surface 472 27.2 Case Study: Aluminum Wire Failures in Residential Wiring 474 27.3 Case Study: Characterizing the Microstructure of a Manganese Nodule 476 References 479  XXIII Contents 28 Cathodoluminescence 481 28.1 Origin 482 28.2 Measuring Cathodoluminescence 483 28.2.1 Collection of CL 483 28.2.2 Detection of CL 483 28.3 Applications of CL 485 28.3.1 Geology 485 28.3.2 Materials Science 485 28.3.3 Organic Compounds 489 References 489 29 Characterizing Crystalline Materials in the SEM 491 29.1 Imaging Crystalline Materials with Electron Channeling Contrast 492 29.1.1 Single Crystals 492 29.1.2 Polycrystalline Materials 494 29.1.3 Conditions for Detecting Electron Channeling Contrast 496 29.2 Electron Backscatter Diffraction in the Scanning Electron Microscope 496 29.2.1 Origin of EBSD Patterns 498 29.2.2 Cameras for EBSD Pattern Detection 499 29.2.3 EBSD Spatial Resolution 499 29.2.4 How Does a Modern EBSD System Index Patterns 501 29.2.5 Steps in Typical EBSD Measurements 502 29.2.6 Display of the Acquired Data 505 29.2.7 Other Map Components 508 29.2.8 Dangers and Practice of “Cleaning” EBSD Data 508 29.2.9 Transmission Kikuchi Diffraction in the SEM 509 29.2.10 Application Example 510 29.2.11 Summary 513 29.2.12 Electron Backscatter Diffraction Checklist 513 References 514 30 Focused Ion Beam Applications in the SEM Laboratory 517 30.1 Introduction 518 30.2 Ion–Solid Interactions 518 30.3 Focused Ion Beam Systems 519 30.4 Imaging with Ions 520 30.5 Preparation of Samples for SEM 521 30.5.1 Cross-Section Preparation 522 30.5.2 FIB Sample Preparation for 3D Techniques and Imaging 524 30.6 Summary 526 References 528 31 Ion Beam Microscopy 529 31.1 What Is So Useful About Ions? 530 31.2 Generating Ion Beams 533 31.3 Signal Generation in the HIM 534 31.4 Current Generation and Data Collection in the HIM 536 31.5 Patterning with Ion Beams 537 31.6 Operating the HIM 538 31.7 Chemical Microanalysis with Ion Beams 538 References 539 Supplementary Information 541 Appendix 542 Index 547  1 1 Electron Beam—Specimen Interactions: Interaction Volume 1.1 What Happens When the Beam Electrons Encounter Specimen Atoms? – 2 1.2 Inelastic Scattering (Energy Loss) Limits Beam Electron Travel in the Specimen – 2 1.3 Elastic Scattering: Beam Electrons Change Direction of Flight – 4 1.3.1 How Frequently Does Elastic Scattering Occur? – 4 1.4 Simulating the Effects of Elastic Scattering: Monte Carlo Calculations – 5 1.4.1 What Do Individual Monte Carlo Trajectories Look Like? – 6 1.4.2 Monte Carlo Simulation To Visualize the Electron Interaction Volume – 6 1.4.3 Using the Monte Carlo Electron Trajectory Simulation to Study the Interaction Volume – 8 1.5 A Range Equation To Estimate the Size of the Interaction Volume – 12 References – 14 © Springer Science+Business Media LLC 2018 J. Goldstein et al., Scanning Electron Microscopy and X-Ray Microanalysis, https://doi.org/10.1007/978-1-4939-6676-9_1  2 Chapter 1 · Electron Beam—Specimen Interactions: Interaction Volume 1.1  hat Happens When the Beam W atomic electrons (binding energy of a few eV) to form sec- 1 Electrons Encounter Specimen Atoms? ondary electrons; ejection of tightly bound inner shell atomic electrons (binding energy of hundreds to thousands of eV) By selecting the operating parameters of the SEM electron which subsequently results in emission of characteristic gun, lenses, and apertures, the microscopist controls the X-rays; deceleration of the beam electron in the electrical characteristics of the focused beam that reaches the speci- field of the atoms producing an X-ray continuum over all men surface: energy (typically selected in the range 0.1– energies from a few eV up to the beam’s landing energy (E0) 30 keV), diameter (0.5 nm to 1 μm or larger), beam current (bremsstrahlung or “braking radiation”); generation of waves (1 pA to 1 μA), and convergence angle (semi-cone angle in the free electron gas that permeates conducting metallic 0.001–0.05 rad). In a conventional high vacuum SEM (typi- solids (plasmons); and heating of the specimen (phonon pro- cally with the column and specimen chamber pressures duction). While energy is lost in these inelastic scattering reduced below 10−3 Pa), the residual atom density is so low events, the beam electrons only deviate slightly from their that the beam electrons are statistically unlikely to encounter current path. The energy loss due to inelastic scattering sets any atoms of the residual gas along the flight path from the an eventual limit on how far the beam electron can travel in electron source to the specimen, a distance of approximately the specimen before it loses all of its energy and is absorbed 25 cm. by the specimen. To understand the specific limitations on the distance kThe initial dimensional scale traveled in the specimen imposed by inelastic scattering, a With a cold or thermal field emission gun on a high- mathematical description is needed of the rate of energy loss performance SEM, the incident beam can be focused to 1 nm (incremental dE, measured in eV) with distance (incremen- in diameter, which means that for a target such as gold (atom tal ds, measured in nm) traveled in the specimen. Although diameter ~ 288 pm), there are approximately 12 gold atoms in the various inelastic scattering energy loss processes are the first atomic layer of the solid within the area of the beam discrete and independent, Bethe (1930) was able to sum- footprint at the surface. marize their collective effects into a “continuous energy loss At the specimen surface the atom density changes approximation”: abruptly to the very high density of the solid. The beam elec- trons interact with the specimen atoms through a variety of dE / ds ( eV / nm ) = − 7.85 ( Z ρ / AE ) ln (1.166 E / J ) (1.1a) physical processes collectively referred to as “scattering events.” The overall effects of these scattering events are to where E is the beam energy (keV), Z is the atomic number, ρ transfer energy to the specimen atoms from the beam elec- is the density (g/cm3), A is the atomic weight (g/mol), and J is trons, thus setting a limit on their travel within the solid, and the “mean ionization potential” (keV) given by to alter the direction of travel of the beam electrons away from the well-defined incident beam trajectory. These beam ( ) J ( keV ) = 9.76 Z + 58.5 Z −0.19 x 10−3 (1.1b) electron–specimen interactions produce the backscattered electrons (BSE), secondary electrons (SE), and X-rays that The Bethe expression is plotted for several elements (C, Al, convey information about the specimen, such as coarse- and Cu, Ag, Au) over the range of “conventional” SEM operat- fine-scale topographic features, composition, crystal struc- ing energies, 5–30 keV in. Fig. 1.1. This figure reveals that ture, and local electrical and magnetic fields. At the level the rate of energy loss dE/ds increases as the electron needed to interpret SEM images and to perform electron- energy decreases and increases with the atomic number of excited X-ray microanalysis, the complex variety of scatter- the target. An electron with a beam energy of 20 keV loses ing processes will be broadly classified into “inelastic” and energy at approximately 10 eV/nm in Au, so that if this rate “elastic” scattering. was constant, the total path traveled in the specimen would be approximately 20,000 eV/(10 eV/nm) = 2000 nm = 2 μm. A better estimate of this electron “Bethe range” can be made by explicitly considering the energy dependence of 1.2 I nelastic Scattering (Energy Loss) dE/ds through integration of the Bethe expression, Eq. 1.1a, Limits Beam Electron Travel from the incident energy down to a lower cut-off energy in the Specimen (typically ~ 2 keV due to limitations on the range of appli- cability of the Bethe expression; see further discussion “Inelastic” scattering refers to a variety of physical processes below). Based on this calculation, the Bethe range for the that act to progressively reduce the energy of the beam elec- selection of elements is shown in. Fig. 1.2. At a particular tron by transferring that energy to the specimen atoms incident beam energy, the Bethe range decreases as the through interactions with tightly bound inner-shell atomic atomic number of the target increases, while for a particu- electrons and loosely bound valence electrons. These energy lar target, the Bethe range increases as the incident beam loss processes include ejection of weakly bound outer-shell energy increases. 1.2 · Inelastic Scattering (Energy Loss) Limits Beam Electron Travel in the Specimen 3 1..      Fig. 1.1 Bethe continuous energy loss model calculations for Bethe continuous energy loss model dE/ds in C, Al, Cu, Ag, and Au as a 25 function of electron energy Au 20 Energy loss rate (eV/nm) Ag 15 Cu 10 Al 5 C 0 5 10 15 20 25 30 Incident beam energy (keV)..      Fig. 1.2 Bethe range calcula- tion from the continuous energy Bethe range loss model by integrating over the 14000 range of energy from E0 down to a cut-off energy of 2 keV 12000 10000 C Bethe range (nm) 8000 Al 6000 4000 Cu, Ag 2000 Au 0 5 10 15 20 25 30 Incident beam energy (keV) kNote the change of scale cross section and the Bethe range as its altitude, the volume The Bethe range for Au with an incident beam energy of of a cylinder 1 nm in diameter and 1200 nm deep would be 20 keV is approximately 1200 nm, a linear change in scale of approximately 940 nm3, and the number of gold atoms it a factor of 1200 over an incident beam diameter of 1 nm. If contained would be approximately 7.5 × 104, which can be the beam–specimen interactions were restricted to a cylin- compared to the incident beam footprint surface atom count drical column with the circular beam entrance footprint as its of approximately 12.  4 Chapter 1 · Electron Beam—Specimen Interactions: Interaction Volume 1.3  lastic Scattering: Beam Electrons E a 1 Change Direction of Flight Simultaneously with inelastic scattering, “elastic scattering” events occur when the beam electron is deflected by the elec- P1 trical field of an atom (the positive nuclear charge as partially shielded by the negative charge of the atom’s orbital elec- trons), causing the beam electron to deviate from its previous path onto a new trajectory, as illustrated schematically in. Fig. 1.3a. The probability of elastic scattering depends strongly on the nuclear charge (atomic number Z) and the energy of the electron, E (keV) and is expressed mathemati- b cally as a cross section, Q: ( ) Qelastic ( >φ 0 ) =1.62 ×10−20 Z 2 / E 2 cot 2 (φ0 / 2 ) P1  ( [events > φ0 / electron atom / cm 2   ) (1.2) where ϕ0 is a threshold elastic scattering angle, for example, 2°. Despite the angular deviation, the beam electron energy is effectively unchanged in energy. While the average elastic scattering event causes an angular change of only a few degrees, deviations up to 180o are possible in a single elastic scattering event. Elastic scattering causes beam electrons to deviate out of the narrow angular range of incident trajecto- c ries defined by the convergence of the incident beam as con- trolled by the electron optics. P1 P2 1.3.1  ow Frequently Does Elastic H Scattering Occur? P3 The elastic scattering cross section, Eq. 1.2, can be used to estimate how far the beam electron must travel on average to experience an elastic scattering event, a distance called the “mean free path,” λ:..      Fig. 1.3 a Schematic illustration of elastic scattering. An energetic electron is deflected by the electrical field of an atom at location P1 λelastic ( cm ) = A /  N 0 ρ Qelastic ( >φ 0 )  (1.3a) through an angle ϕelastic. b Schematic illustration of the elastic scatter- ing cone. The energetic electron scatters elastically at point P1 and can land at any location on the circumference of the base of the cone with λelastic ( nm ) =107 A /  N 0 ρ Qelastic ( >φ 0 )  (1.3b) equal probability. c Schematic illustration of a second scattering step, carrying the energetic electron from point P2 to point P3 where A is the atomic weight (g/mol), N0 is Avogadro’s num- ber (atoms/mol), and ρ is the density (g/cm3).. Figure 1.4 is of the order of nm. Elastic scattering is thus likely to occur shows a plot of λelastic for various elements as a function of hundreds to thousands of times along a Bethe range of sev- electron energy, where it can be seen that the mean free path eral hundred to several thousand nanometers. 1.4 · Simulating the Effects of Elastic Scattering: Monte Carlo Calculations 5 1..      Fig. 1.4 Elastic mean free path as a function of electron kinetic Elastic scattering mean free path (f0 = 2°) energy for various elements C 10 Al Elastic mean free path (nm) 1 Cu Ag Au 0.1 0.01 5 10 15 20 25 30 Beam energy (keV) 1.4  imulating the Effects of Elastic S shown in. Fig. 1.3c, creating an increasingly complex path. Scattering: Monte Carlo Calculations Because of the random component of scattering at each of many steps, this complex behavior cannot be adequately Inelastic scattering sets a limit on the total distance traveled described by an algebraic expression like the Bethe continu- by the beam electron. The Bethe range is an estimate of this ous energy loss equation. Instead, a stepwise simulation of distance and can be found by integrating the Bethe continu- the electron's behavior must be constructed that incorpo- ous energy loss expression from the incident beam energy E0 rates inelastic and elastic scattering. Several simplifications down to a low energy limit, for example, 2 keV. Estimating are introduced to create a practical “Monte Carlo electron the effects of elastic scattering on the beam electrons is much trajectory simulation”: more complicated. Any individual elastic scattering event 1. All of the angular deviation of the beam electron is ascribed can result in a scattering angle within a broad range from a to elastic scattering. A mathematical model for elastic threshold of a fraction of a degree up to 180°, with small scat- scattering is applied that utilizes a random number (hence tering angles much more likely than very large values and an the name “Monte Carlo” from the supposed randomness of average value typically in the range 5–10°. Moreover, the gambling) to select a properly weighted value of the elastic electron scattered by the atom through an angle ϕ in scattering angle out of the possible range (from a threshold. Fig. 1.3a at point P1 can actually follow any path along the value of approximately 1° to a maximum of 180°). A second surface of the three-dimensional scattering cone shown in random number is used to select the azimuthal angle in the. Fig. 1.3b and can land anywhere in the circumference of base of the scattering cone in. Fig. 1.1c. the base of the scattering cone (i.e., the azimuthal angle in 2. The distance between elastic scattering events, s, which the base of the cone ranges from 0 to 360° with equal proba- lies on the surface of the scattering cone in. Fig. 1.3b, is bility), resulting in a three-dimensional path. The length of calculated from the elastic mean free path, Eq. 1.3b. the trajectory along the surface of the scattering cone 3. Inelastic scattering is calculated with the Bethe depends on the frequency of elastic events with distance continuous energy loss expression, Eq. 1.1b. The specific traveled and can be estimated from Eq. 1.3a for the elastic energy loss, ΔE, along the path, s, in the surface of the scattering mean free path, λelastic. The next elastic scattering scattering cone,. Fig. 1.3b, is calculated with the Bethe event P2 causes the electron to deviate in a new direction, as continuous energy loss expression: ΔE = (dE/ds)*s  6 Chapter 1 · Electron Beam—Specimen Interactions: Interaction Volume Given a specific set of these parameters, the Monte Carlo elec- dimension y projected onto the x-z plane. (An example of the 1 tron trajectory simulation utilizes geometrical expressions to true three-dimensional trajectories, simulated with the Joy calculate the successive series of locations P1, P2, P3, etc., suc- Monte Carlo, is shown in. Fig. 1.6, in which a small number cessively determining the coordinate locations (x, y, z) that the of trajectories (to minimize overlap) have been rendered as energetic electron follows within the solid. At each location P, an anaglyph stereo representation with the convention left the newly depreciated energy of the electron is known, and after eye = red filter. Inspection of this simulation shows the y the next elastic scattering angle is calculated, the new velocity motion of the electrons in and out of the x-z plane.) The sto- vector components vx, vy, vz are determined to transport the chastic nature of the interaction imposed by the nature of electron to the next location. A trajectory ends when either the elastic scattering is readily apparent in the great variation electron energy falls below a threshold of interest (e.g., 1 keV), among the individual trajectories seen in. Fig. 1.5a, b. It or else the path takes it outside the geometric bounds of the quickly becomes clear that individual beam electrons follow a specimen, which is determined by comparing the current loca- huge range of paths and simulating a small number of trajec- tion (x, y, z) with the specimen boundaries. The capability of tories does not provide an adequate view of the electron beam simulating electron beam interactions in specimens with com- specimen interaction. plex geometrical shapes is one of the major strengths of the Monte Carlo electron trajectory simulation method. Monte Carlo electron trajectory simulation can pro- 1.4.2  onte Carlo Simulation To Visualize M vide visual depictions as well as numerical results of the the Electron Interaction Volume beam–specimen interaction, creating a powerful instructional tool for studying this complex phenomenon. Several power- To capture a reasonable picture representation of the electron ful Monte Carlo simulations appropriate for SEM and X-ray interaction volume, which is the region of the specimen in microanalysis applications are available as free resources: which the beam electrons travel and deposit energy, it is nec- CASINO [7 http://www.gel.usherbrooke.ca/casino/What.html]      essary to calculate many more trajectories.. Figure 1.5c Joy Monte Carlo [7 http://web.utk.edu/~srcutk/htm/      shows the simulation for copper, E0 = 20 keV at 0° tilt simulati.htm] extended to 500 trajectories, which reveals the full extent of NIST DTSA-II [7 http://www.cstl.nist.gov/div837/837.02/      the electron interaction volume. Beyond a few hundred tra- epq/dtsa2/index.html] jectories, superimposing the three-dimensional trajectories to create a two-dimensional representation reaches dimin- While the static images of Monte Carlo simulations pre- ishing returns due to overlap of the plotted lines. While sim- sented below are useful instructional aids, readers are ulating 500 trajectories provides a reasonable qualitative encouraged to perform their own simulations to become view of the electron interaction volume, Monte Carlo calcu- familiar with this powerful tool, which in more elaborate lations of numerical properties of the interaction volume and implementations is an important aid in understanding criti- related processes, such as electron backscattering (discussed cal aspects of SEM imaging. in the backscattered electron module), are subject to statisti- cally predictable variations because of the use of random numbers to select the elastic scattering parameters. Variance 1.4.1  hat Do Individual Monte Carlo W in repeated simulations of the same starting conditions is Trajectories Look Like? related to the number of trajectories and can be described with the properties of the Gaussian (normal) distribution. Perform a Monte Carlo simulation (CASINO simulation) for Thus the precision, p, of the calculation of a parameter of the copper with a beam energy of 20 keV and a tilt of 0° (beam interaction is related to the total number of simulated trajec- perpendicular to the surface) for a small number of trajecto- tories, n, and the fraction, f, of those trajectories that produce ries, for example, 25.. Figure 1.5a, b show two simulations of the effect of interest (e.g., backscattering): 25 trajectories each. The trajectories are actually determined 1/ 2 −1/ 2 in three dimensions (x-y-z, where x-y defines the surface p = ( f n) / ( f n) = ( f n) (1.4) plane and z is perpendicular to the surface) but for plotting are rendered in two dimensions (x-z), with the third 1.4 · Simulating the Effects of Elastic Scattering: Monte Carlo Calculations 7 1..      Fig. 1.5 a Copper, E0 = 20 keV; 0 tilt; 25 trajec- a tories (CASINO Monte Carlo simultion). b Copper, Cu E0 = 20 keV 0.0 nm E0 = 20 keV; 0 tilt; another 25 trajectories. c Copper, E0 = 20 keV; 0 tilt; 200 trajectories 200 nm 200.0 nm 400.0 nm 600.0 nm 800.0 nm -582.5 nm -291.3 nm -0.0 nm 291.3 nm 582.5 nm b 0.0 nm Cu E0 = 20 keV 200 nm 180.0 nm 360.0 nm 540.0 nm 720.0 nm -524.3 nm -262.1 nm -0.0 nm 262.1 nm 524.3 nm c 0.0 nm Cu E0 = 20 keV 200 nm 233.5 nm 466.9 nm 700.4 nm 933.8 nm -680.0 nm -340.0 nm -0.0 nm 340.0 nm 680.0 nm  8 Chapter 1 · Electron Beam—Specimen Interactions: Interaction Volume balance of the energy deposited in a strongly non-linear fash- 1 ion in the much larger portion of the interaction volume.  ow Does the Interaction Volume Change H with Composition?. Figure 1.8 shows the interaction volume in various targets, Energy (keV) 20 C, Si, Cu, Ag, and Au, at fixed beam energy, E0 = 20 keV, and 0° tilt. As the atomic number of the target increases, the lin- Tilt/TOA 0 ear dimensions of the interaction volume decrease. The Number 35 form also changes from pear-shaped with a dense conical region below the beam impact for low atomic number tar- Select gets to a more hemispherical shape for high atomic number 500 nm Repeat targets. computed BS yield = 0.31 Exit kNote the dramatic change of scale Approximately 12 gold atoms were encountered within the..      Fig. 1.6 Three-dimensional representation of a Monte Carlo simulation (Cu, 20 keV, 0° tilt) using the anaglyph stereo method (left footprint of a 1-nm diameter at the surface. Without consid- eye = red filter) (Joy Monte Carlo) ering the effects of elastic scattering, the Bethe range for Au at an incident beam energy of 20 keV limited the penetra- tion of the beam to approximately 1200 nm and a cylindri- 1.4.3  sing the Monte Carlo Electron U cal volume of approximately 940 nm3, containing Trajectory Simulation to Study approximately 5.6 × 104 Au atoms. The effect of elastic scat- the Interaction Volume tering is to create a three-dimensional hemispherical inter- action volume with a radius of approximately 600 nm and a  hat Are the Main Features of the Beam W volume of 4.5 × 108 nm3, containing 2.7 × 1010 Au atoms, an Electron Interaction Volume? increase of nine orders-of-magnitude over the number of In. Fig. 1.5c, the beam electron interaction volume is seen to atoms encountered in the initial beam footprint on the be a very complex structure with dimensions extending over surface. hundreds to thousands of nanometers from the beam impact point, depending on target material and the beam energy. At  ow Does the Interaction Volume Change H 0° tilt, the interaction volume is rotationally symmetric with Incident Beam Energy? around the beam. While the electron trajectories provide a. Figure 1.9 shows the interaction volume for copper at 0° tilt strong visual representation of the interaction volume, more over a range of incident beam energy from 5 to 30 keV. The informative numerical information is needed. The Monte shape of the interaction volume is relatively independent of Carlo simulation can provide detailed information on many beam energy, but the size increases rapidly as the incident aspects of the electron beam–specimen interaction. The beam energy increases. color-encoding of the energy deposited along each trajectory, as implemented in the Joy Monte Carlo shown in. Fig. 1.11,  ow Does the Interaction Volume Change H creates a view that reveals the general three-dimensional with Specimen Tilt? complexity of energy deposition within the interaction vol-. Figure 1.10 shows the interaction volume for copper at an ume. The CASINO Monte Carlo provides an even more incident beam energy of 20 keV and a series of tilt angles. As detailed view of energy deposition, as shown in. Fig. 1.7. the tilt angle increases so that the beam approaches the surface The energy deposition per unit volume is greatest just under at a progressively more shallow angle, the shape of the interac- the beam impact location and rapidly falls off as the periph- tion volume changes significantly. At 0° tilt, the interaction ery of the interaction volume is approached. This calculation volume is rotationally symmetric around the beam, but as the reveals that a small cylindrical volume under the beam tilt angle increases the interaction volume becomes asymmet- impact point, shown in more detail in. Fig. 1.7b, receives ric, with the dense portion of the distribution shifting progres- half of the total energy deposited by the beam in the speci- sively away from the beam impact point. The maximum men (that is, the volume within the 50% contour), with the penetration of the beam is reduced as the tilt angle increases. 1.4 · Simulating the Effects of Elastic Scattering: Monte Carlo Calculations 9 1..      Fig. 1.7 a Isocontours of a energy loss showing fraction 0.0 nm remaining; Cu, 20 keV, 0° tilt; Cu E0 = 20 keV 90% 50,000 trajectories (CASINO Monte Carlo simulation). b Expanded 75% view of high density region of 1.7a 50% 206.3 nm 25% 412.6 nm 10% 5% 618.9 nm 5.0% 10.0% 825.2 nm 25.0% 50.0% 75.0% 90.0% -600.9 nm -300.4 nm 0.0 nm 300.4 nm 600.9 nm b 10% 90% 25% 75% 50% 200 nm  10 Chapter 1 · Electron Beam—Specimen Interactions: Interaction Volume 0.0 nm 1 C